Application of Stochastic and Deterministic Approaches to Modeling Interstellar Chemistry

NASA Astrophysics Data System (ADS)

Pei, Yezhe

This work is about simulations of interstellar chemistry using the deterministic rate equation (RE) method and the stochastic moment equation (ME) method. Primordial metal-poor interstellar medium (ISM) is of our interest and the socalled “Population-II” stars could have been formed in this environment during the “Epoch of Reionization” in the baby universe. We build a gas phase model using the RE scheme to describe the ionization-powered interstellar chemistry. We demonstrate that OH replaces CO as the most abundant metal-bearing molecule in such interstellar clouds of the early universe. Grain surface reactions play an important role in the studies of astrochemistry. But the lack of an accurate yet effective simulation method still presents a challenge, especially for large, practical gas-grain system. We develop a hybrid scheme of moment equations and rate equations (HMR) for large gas-grain network to model astrochemical reactions in the interstellar clouds. Specifically, we have used a large chemical gas-grain model, with stochastic moment equations to treat the surface chemistry and deterministic rate equations to treat the gas phase chemistry, to simulate astrochemical systems as of the ISM in the Milky Way, the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC). We compare the results to those of pure rate equations and modified rate equations and present a discussion about how moment equations improve our theoretical modeling and how the abundances of the assorted species are changed by varied metallicity. We also model the observed composition of H2O, CO and CO2 ices toward Young Stellar Objects in the LMC and show that the HMR method gives a better match to the observation than the pure RE method.

Data-driven discovery of partial differential equations

PubMed Central

Rudy, Samuel H.; Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan

2017-01-01

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg–de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable. PMID:28508044

An Equation-Free Reduced-Order Modeling Approach to Tropical Pacific Simulation

NASA Astrophysics Data System (ADS)

Wang, Ruiwen; Zhu, Jiang; Luo, Zhendong; Navon, I. M.

2009-03-01

The “equation-free” (EF) method is often used in complex, multi-scale problems. In such cases it is necessary to know the closed form of the required evolution equations about oscopic variables within some applied fields. Conceptually such equations exist, however, they are not available in closed form. The EF method can bypass this difficulty. This method can obtain oscopic information by implementing models at a microscopic level. Given an initial oscopic variable, through lifting we can obtain the associated microscopic variable, which may be evolved using Direct Numerical Simulations (DNS) and by restriction, we can obtain the necessary oscopic information and the projective integration to obtain the desired quantities. In this paper we apply the EF POD-assisted method to the reduced modeling of a large-scale upper ocean circulation in the tropical Pacific domain. The computation cost is reduced dramatically. Compared with the POD method, the method provided more accurate results and it did not require the availability of any explicit equations or the right-hand side (RHS) of the evolution equation.

Solitons of the Kadomtsev-Petviashvili equation based on lattice Boltzmann model

NASA Astrophysics Data System (ADS)

Wang, Huimin

2017-01-01

In this paper, a lattice Boltzmann model for the Kadomtsev-Petviashvili equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales are obtained. Due to the asymmetry in x direction and y direction of the equation, the moments of the equilibrium distribution function are selected are asymmetric. The numerical results demonstrate the lattice Boltzmann method is an effective method to simulate the solitons of the Kadomtsev-Petviashvili equation.

3-D Forward modeling of Induced Polarization Effects of Transient Electromagnetic Method

NASA Astrophysics Data System (ADS)

Wu, Y.; Ji, Y.; Guan, S.; Li, D.; Wang, A.

2017-12-01

In transient electromagnetic (TEM) detection, Induced polarization (IP) effects are so important that they cannot be ignored. The authors simulate the three-dimensional (3-D) induced polarization effects in time-domain directly by applying the finite-difference time-domain method (FDTD) based on Cole-Cole model. Due to the frequency dispersion characteristics of the electrical conductivity, the computations of convolution in the generalized Ohm's law of fractional order system makes the forward modeling particularly complicated. Firstly, we propose a method to approximate the fractional order function of Cole-Cole model using a lower order rational transfer function based on error minimum theory in the frequency domain. In this section, two auxiliary variables are introduced to transform nonlinear least square fitting problem of the fractional order system into a linear programming problem, thus avoiding having to solve a system of equations and nonlinear problems. Secondly, the time-domain expression of Cole-Cole model is obtained by using Inverse Laplace transform. Then, for the calculation of Ohm's law, we propose an e-index auxiliary equation of conductivity to transform the convolution to non-convolution integral; in this section, the trapezoid rule is applied to compute the integral. We then substitute the recursion equation into Maxwell's equations to derive the iterative equations of electromagnetic field using the FDTD method. Finally, we finish the stimulation of 3-D model and evaluate polarization parameters. The results are compared with those obtained from the digital filtering solution of the analytical equation in the homogeneous half space, as well as with the 3-D model results from the auxiliary ordinary differential equation method (ADE). Good agreements are obtained across the three methods. In terms of the 3-D model, the proposed method has higher efficiency and lower memory requirements as execution times and memory usage were reduced by 20% compared with ADE method.

Protein electrostatics: a review of the equations and methods used to model electrostatic equations in biomolecules--applications in biotechnology.

PubMed

Neves-Petersen, Maria Teresa; Petersen, Steffen B

2003-01-01

The molecular understanding of the initial interaction between a protein and, e.g., its substrate, a surface or an inhibitor is essentially an understanding of the role of electrostatics in intermolecular interactions. When studying biomolecules it is becoming increasingly evident that electrostatic interactions play a role in folding, conformational stability, enzyme activity and binding energies as well as in protein-protein interactions. In this chapter we present the key basic equations of electrostatics necessary to derive the equations used to model electrostatic interactions in biomolecules. We will also address how to solve such equations. This chapter is divided into two major sections. In the first part we will review the basic Maxwell equations of electrostatics equations called the Laws of Electrostatics that combined will result in the Poisson equation. This equation is the starting point of the Poisson-Boltzmann (PB) equation used to model electrostatic interactions in biomolecules. Concepts as electric field lines, equipotential surfaces, electrostatic energy and when can electrostatics be applied to study interactions between charges will be addressed. In the second part we will arrive at the electrostatic equations for dielectric media such as a protein. We will address the theory of dielectrics and arrive at the Poisson equation for dielectric media and at the PB equation, the main equation used to model electrostatic interactions in biomolecules (e.g., proteins, DNA). It will be shown how to compute forces and potentials in a dielectric medium. In order to solve the PB equation we will present the continuum electrostatic models, namely the Tanford-Kirkwood and the modified Tandord-Kirkwood methods. Priority will be given to finding the protonation state of proteins prior to solving the PB equation. We also present some methods that can be used to map and study the electrostatic potential distribution on the molecular surface of proteins. The combination of graphical visualisation of the electrostatic fields combined with knowledge about the location of key residues on the protein surface allows us to envision atomic models for enzyme function. Finally, we exemplify the use of some of these methods on the enzymes of the lipase family.

Maximum Likelihood and Restricted Likelihood Solutions in Multiple-Method Studies

PubMed Central

Rukhin, Andrew L.

2011-01-01

A formulation of the problem of combining data from several sources is discussed in terms of random effects models. The unknown measurement precision is assumed not to be the same for all methods. We investigate maximum likelihood solutions in this model. By representing the likelihood equations as simultaneous polynomial equations, the exact form of the Groebner basis for their stationary points is derived when there are two methods. A parametrization of these solutions which allows their comparison is suggested. A numerical method for solving likelihood equations is outlined, and an alternative to the maximum likelihood method, the restricted maximum likelihood, is studied. In the situation when methods variances are considered to be known an upper bound on the between-method variance is obtained. The relationship between likelihood equations and moment-type equations is also discussed. PMID:26989583

Maximum Likelihood and Restricted Likelihood Solutions in Multiple-Method Studies.

PubMed

Rukhin, Andrew L

2011-01-01

A formulation of the problem of combining data from several sources is discussed in terms of random effects models. The unknown measurement precision is assumed not to be the same for all methods. We investigate maximum likelihood solutions in this model. By representing the likelihood equations as simultaneous polynomial equations, the exact form of the Groebner basis for their stationary points is derived when there are two methods. A parametrization of these solutions which allows their comparison is suggested. A numerical method for solving likelihood equations is outlined, and an alternative to the maximum likelihood method, the restricted maximum likelihood, is studied. In the situation when methods variances are considered to be known an upper bound on the between-method variance is obtained. The relationship between likelihood equations and moment-type equations is also discussed.

Local Analysis of Shock Capturing Using Discontinuous Galerkin Methodology

NASA Technical Reports Server (NTRS)

Atkins, H. L.

1997-01-01

The compact form of the discontinuous Galerkin method allows for a detailed local analysis of the method in the neighborhood of the shock for a non-linear model problem. Insight gained from the analysis leads to new flux formulas that are stable and that preserve the compactness of the method. Although developed for a model equation, the flux formulas are applicable to systems such as the Euler equations. This article presents the analysis for methods with a degree up to 5. The analysis is accompanied by supporting numerical experiments using Burgers' equation and the Euler equations.

Identification of unmeasured variables in the set of model constraints of the data reconciliation in a power unit

NASA Astrophysics Data System (ADS)

Szega, Marcin; Nowak, Grzegorz Tadeusz

2013-12-01

In generalized method of data reconciliation as equations of conditions beside substance and energy balances can be used equations which don't have precisely the status of conservation lows. Empirical coefficients in these equations are traded as unknowns' values. To this kind of equations, in application of the generalized method of data reconciliation in supercritical power unit, can be classified: steam flow capacity of a turbine for a group of stages, adiabatic internal efficiency of group of stages, equations for pressure drop in pipelines and equations for heat transfer in regeneration heat exchangers. Mathematical model of a power unit was developed in the code Thermoflex. Using this model the off-design calculation has been made in several points of loads for the power unit. Using these calculations identification of unknown values and empirical coefficients for generalized method of data reconciliation used in power unit has been made. Additional equations of conditions will be used in the generalized method of data reconciliation which will be used in optimization of measurement placement in redundant measurement system in power unit for new control systems

A multivariate model and statistical method for validating tree grade lumber yield equations

Treesearch

Donald W. Seegrist

1975-01-01

Lumber yields within lumber grades can be described by a multivariate linear model. A method for validating lumber yield prediction equations when there are several tree grades is presented. The method is based on multivariate simultaneous test procedures.

A multivariate variational objective analysis-assimilation method. Part 1: Development of the basic model

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.; Ochs, Harry T., III

1988-01-01

The variational method of undetermined multipliers is used to derive a multivariate model for objective analysis. The model is intended for the assimilation of 3-D fields of rawinsonde height, temperature and wind, and mean level temperature observed by satellite into a dynamically consistent data set. Relative measurement errors are taken into account. The dynamic equations are the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation. The model Euler-Lagrange equations are eleven linear and/or nonlinear partial differential and/or algebraic equations. A cyclical solution sequence is described. Other model features include a nonlinear terrain-following vertical coordinate that eliminates truncation error in the pressure gradient terms of the horizontal momentum equations and easily accommodates satellite observed mean layer temperatures in the middle and upper troposphere. A projection of the pressure gradient onto equivalent pressure surfaces removes most of the adverse impacts of the lower coordinate surface on the variational adjustment.

Review of literature on the finite-element solution of the equations of two-dimensional surface-water flow in the horizontal plane

USGS Publications Warehouse

Lee, Jonathan K.; Froehlich, David C.

1987-01-01

Published literature on the application of the finite-element method to solving the equations of two-dimensional surface-water flow in the horizontal plane is reviewed in this report. The finite-element method is ideally suited to modeling two-dimensional flow over complex topography with spatially variable resistance. A two-dimensional finite-element surface-water flow model with depth and vertically averaged velocity components as dependent variables allows the user great flexibility in defining geometric features such as the boundaries of a water body, channels, islands, dikes, and embankments. The following topics are reviewed in this report: alternative formulations of the equations of two-dimensional surface-water flow in the horizontal plane; basic concepts of the finite-element method; discretization of the flow domain and representation of the dependent flow variables; treatment of boundary conditions; discretization of the time domain; methods for modeling bottom, surface, and lateral stresses; approaches to solving systems of nonlinear equations; techniques for solving systems of linear equations; finite-element alternatives to Galerkin's method of weighted residuals; techniques of model validation; and preparation of model input data. References are listed in the final chapter.

Fitting ARMA Time Series by Structural Equation Models.

ERIC Educational Resources Information Center

van Buuren, Stef

1997-01-01

This paper outlines how the stationary ARMA (p,q) model (G. Box and G. Jenkins, 1976) can be specified as a structural equation model. Maximum likelihood estimates for the parameters in the ARMA model can be obtained by software for fitting structural equation models. The method is applied to three problem types. (SLD)

Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

NASA Astrophysics Data System (ADS)

Vitanov, Nikolay K.; Dimitrova, Zlatinka I.

2018-03-01

We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

A new Newton-like method for solving nonlinear equations.

PubMed

Saheya, B; Chen, Guo-Qing; Sui, Yun-Kang; Wu, Cai-Ying

2016-01-01

This paper presents an iterative scheme for solving nonline ar equations. We establish a new rational approximation model with linear numerator and denominator which has generalizes the local linear model. We then employ the new approximation for nonlinear equations and propose an improved Newton's method to solve it. The new method revises the Jacobian matrix by a rank one matrix each iteration and obtains the quadratic convergence property. The numerical performance and comparison show that the proposed method is efficient.

A solution to neural field equations by a recurrent neural network method

NASA Astrophysics Data System (ADS)

Alharbi, Abir

2012-09-01

Neural field equations (NFE) are used to model the activity of neurons in the brain, it is introduced from a single neuron 'integrate-and-fire model' starting point. The neural continuum is spatially discretized for numerical studies, and the governing equations are modeled as a system of ordinary differential equations. In this article the recurrent neural network approach is used to solve this system of ODEs. This consists of a technique developed by combining the standard numerical method of finite-differences with the Hopfield neural network. The architecture of the net, energy function, updating equations, and algorithms are developed for the NFE model. A Hopfield Neural Network is then designed to minimize the energy function modeling the NFE. Results obtained from the Hopfield-finite-differences net show excellent performance in terms of accuracy and speed. The parallelism nature of the Hopfield approaches may make them easier to implement on fast parallel computers and give them the speed advantage over the traditional methods.

Adjoint Method and Predictive Control for 1-D Flow in NASA Ames 11-Foot Transonic Wind Tunnel

NASA Technical Reports Server (NTRS)

Nguyen, Nhan; Ardema, Mark

2006-01-01

This paper describes a modeling method and a new optimal control approach to investigate a Mach number control problem for the NASA Ames 11-Foot Transonic Wind Tunnel. The flow in the wind tunnel is modeled by the 1-D unsteady Euler equations whose boundary conditions prescribe a controlling action by a compressor. The boundary control inputs to the compressor are in turn controlled by a drive motor system and an inlet guide vane system whose dynamics are modeled by ordinary differential equations. The resulting Euler equations are thus coupled to the ordinary differential equations via the boundary conditions. Optimality conditions are established by an adjoint method and are used to develop a model predictive linear-quadratic optimal control for regulating the Mach number due to a test model disturbance during a continuous pitch

An Examination of Higher-Order Treatments of Boundary Conditions in Split-Step Fourier Parabolic Equation Models

DTIC Science & Technology

2015-06-01

method provides improved agreement with a benchmark solution at longer ranges. 14. SUBJECT TERMS parabolic equation , Monterey Miami...elliptic Helmholtz wave equation dates back to mid-1940s, when Leontovich and Fock introduced the PE method to the problem of radio-wave propagation in...improvements in the solutions . B. PROBLEM STATEMENT The Monterey-Miami Parabolic Equation (MMPE) model was developed in the mid-1990s and since then has

Material point method modeling in oil and gas reservoirs

DOEpatents

Vanderheyden, William Brian; Zhang, Duan

2016-06-28

A computer system and method of simulating the behavior of an oil and gas reservoir including changes in the margins of frangible solids. A system of equations including state equations such as momentum, and conservation laws such as mass conservation and volume fraction continuity, are defined and discretized for at least two phases in a modeled volume, one of which corresponds to frangible material. A material point model technique for numerically solving the system of discretized equations, to derive fluid flow at each of a plurality of mesh nodes in the modeled volume, and the velocity of at each of a plurality of particles representing the frangible material in the modeled volume. A time-splitting technique improves the computational efficiency of the simulation while maintaining accuracy on the deformation scale. The method can be applied to derive accurate upscaled model equations for larger volume scale simulations.

A gradual update method for simulating the steady-state solution of stiff differential equations in metabolic circuits.

PubMed

Shiraishi, Emi; Maeda, Kazuhiro; Kurata, Hiroyuki

2009-02-01

Numerical simulation of differential equation systems plays a major role in the understanding of how metabolic network models generate particular cellular functions. On the other hand, the classical and technical problems for stiff differential equations still remain to be solved, while many elegant algorithms have been presented. To relax the stiffness problem, we propose new practical methods: the gradual update of differential-algebraic equations based on gradual application of the steady-state approximation to stiff differential equations, and the gradual update of the initial values in differential-algebraic equations. These empirical methods show a high efficiency for simulating the steady-state solutions for the stiff differential equations that existing solvers alone cannot solve. They are effective in extending the applicability of dynamic simulation to biochemical network models.

Parameter estimation of Monod model by the Least-Squares method for microalgae Botryococcus Braunii sp

NASA Astrophysics Data System (ADS)

See, J. J.; Jamaian, S. S.; Salleh, R. M.; Nor, M. E.; Aman, F.

2018-04-01

This research aims to estimate the parameters of Monod model of microalgae Botryococcus Braunii sp growth by the Least-Squares method. Monod equation is a non-linear equation which can be transformed into a linear equation form and it is solved by implementing the Least-Squares linear regression method. Meanwhile, Gauss-Newton method is an alternative method to solve the non-linear Least-Squares problem with the aim to obtain the parameters value of Monod model by minimizing the sum of square error ( SSE). As the result, the parameters of the Monod model for microalgae Botryococcus Braunii sp can be estimated by the Least-Squares method. However, the estimated parameters value obtained by the non-linear Least-Squares method are more accurate compared to the linear Least-Squares method since the SSE of the non-linear Least-Squares method is less than the linear Least-Squares method.

A high-order multiscale finite-element method for time-domain acoustic-wave modeling

NASA Astrophysics Data System (ADS)

Gao, Kai; Fu, Shubin; Chung, Eric T.

2018-05-01

Accurate and efficient wave equation modeling is vital for many applications in such as acoustics, electromagnetics, and seismology. However, solving the wave equation in large-scale and highly heterogeneous models is usually computationally expensive because the computational cost is directly proportional to the number of grids in the model. We develop a novel high-order multiscale finite-element method to reduce the computational cost of time-domain acoustic-wave equation numerical modeling by solving the wave equation on a coarse mesh based on the multiscale finite-element theory. In contrast to existing multiscale finite-element methods that use only first-order multiscale basis functions, our new method constructs high-order multiscale basis functions from local elliptic problems which are closely related to the Gauss-Lobatto-Legendre quadrature points in a coarse element. Essentially, these basis functions are not only determined by the order of Legendre polynomials, but also by local medium properties, and therefore can effectively convey the fine-scale information to the coarse-scale solution with high-order accuracy. Numerical tests show that our method can significantly reduce the computation time while maintain high accuracy for wave equation modeling in highly heterogeneous media by solving the corresponding discrete system only on the coarse mesh with the new high-order multiscale basis functions.

A high-order multiscale finite-element method for time-domain acoustic-wave modeling

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gao, Kai; Fu, Shubin; Chung, Eric T.

Accurate and efficient wave equation modeling is vital for many applications in such as acoustics, electromagnetics, and seismology. However, solving the wave equation in large-scale and highly heterogeneous models is usually computationally expensive because the computational cost is directly proportional to the number of grids in the model. We develop a novel high-order multiscale finite-element method to reduce the computational cost of time-domain acoustic-wave equation numerical modeling by solving the wave equation on a coarse mesh based on the multiscale finite-element theory. In contrast to existing multiscale finite-element methods that use only first-order multiscale basis functions, our new method constructsmore » high-order multiscale basis functions from local elliptic problems which are closely related to the Gauss–Lobatto–Legendre quadrature points in a coarse element. Essentially, these basis functions are not only determined by the order of Legendre polynomials, but also by local medium properties, and therefore can effectively convey the fine-scale information to the coarse-scale solution with high-order accuracy. Numerical tests show that our method can significantly reduce the computation time while maintain high accuracy for wave equation modeling in highly heterogeneous media by solving the corresponding discrete system only on the coarse mesh with the new high-order multiscale basis functions.« less

A high-order multiscale finite-element method for time-domain acoustic-wave modeling

DOE PAGES

Gao, Kai; Fu, Shubin; Chung, Eric T.

2018-02-04

Accurate and efficient wave equation modeling is vital for many applications in such as acoustics, electromagnetics, and seismology. However, solving the wave equation in large-scale and highly heterogeneous models is usually computationally expensive because the computational cost is directly proportional to the number of grids in the model. We develop a novel high-order multiscale finite-element method to reduce the computational cost of time-domain acoustic-wave equation numerical modeling by solving the wave equation on a coarse mesh based on the multiscale finite-element theory. In contrast to existing multiscale finite-element methods that use only first-order multiscale basis functions, our new method constructsmore » high-order multiscale basis functions from local elliptic problems which are closely related to the Gauss–Lobatto–Legendre quadrature points in a coarse element. Essentially, these basis functions are not only determined by the order of Legendre polynomials, but also by local medium properties, and therefore can effectively convey the fine-scale information to the coarse-scale solution with high-order accuracy. Numerical tests show that our method can significantly reduce the computation time while maintain high accuracy for wave equation modeling in highly heterogeneous media by solving the corresponding discrete system only on the coarse mesh with the new high-order multiscale basis functions.« less

Integral equation methods for computing likelihoods and their derivatives in the stochastic integrate-and-fire model.

PubMed

Paninski, Liam; Haith, Adrian; Szirtes, Gabor

2008-02-01

We recently introduced likelihood-based methods for fitting stochastic integrate-and-fire models to spike train data. The key component of this method involves the likelihood that the model will emit a spike at a given time t. Computing this likelihood is equivalent to computing a Markov first passage time density (the probability that the model voltage crosses threshold for the first time at time t). Here we detail an improved method for computing this likelihood, based on solving a certain integral equation. This integral equation method has several advantages over the techniques discussed in our previous work: in particular, the new method has fewer free parameters and is easily differentiable (for gradient computations). The new method is also easily adaptable for the case in which the model conductance, not just the input current, is time-varying. Finally, we describe how to incorporate large deviations approximations to very small likelihoods.

An entropy correction method for unsteady full potential flows with strong shocks

NASA Technical Reports Server (NTRS)

Whitlow, W., Jr.; Hafez, M. M.; Osher, S. J.

1986-01-01

An entropy correction method for the unsteady full potential equation is presented. The unsteady potential equation is modified to account for entropy jumps across shock waves. The conservative form of the modified equation is solved in generalized coordinates using an implicit, approximate factorization method. A flux-biasing differencing method, which generates the proper amounts of artificial viscosity in supersonic regions, is used to discretize the flow equations in space. Comparisons between the present method and solutions of the Euler equations and between the present method and experimental data are presented. The comparisons show that the present method more accurately models solutions of the Euler equations and experiment than does the isentropic potential formulation.

Navier-Stokes Dynamics by a Discrete Boltzmann Model

NASA Technical Reports Server (NTRS)

Rubinstein, Robet

2010-01-01

This work investigates the possibility of particle-based algorithms for the Navier-Stokes equations and higher order continuum approximations of the Boltzmann equation; such algorithms would generalize the well-known Pullin scheme for the Euler equations. One such method is proposed in the context of a discrete velocity model of the Boltzmann equation. Preliminary results on shock structure are consistent with the expectation that the shock should be much broader than the near discontinuity predicted by the Pullin scheme, yet narrower than the prediction of the Boltzmann equation. We discuss the extension of this essentially deterministic method to a stochastic particle method that, like DSMC, samples the distribution function rather than resolving it completely.

Comparison between results of solution of Burgers' equation and Laplace's equation by Galerkin and least-square finite element methods

NASA Astrophysics Data System (ADS)

Adib, Arash; Poorveis, Davood; Mehraban, Farid

2018-03-01

In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers' equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace's equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace's equation in an earth dam is considered. By solution of Laplace's equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers' equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers' equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace's equation, Galerkin and least square methods can show water table correctly in earth dam.

Derivation and computation of discrete-delay and continuous-delay SDEs in mathematical biology.

PubMed

Allen, Edward J

2014-06-01

Stochastic versions of several discrete-delay and continuous-delay differential equations, useful in mathematical biology, are derived from basic principles carefully taking into account the demographic, environmental, or physiological randomness in the dynamic processes. In particular, stochastic delay differential equation (SDDE) models are derived and studied for Nicholson's blowflies equation, Hutchinson's equation, an SIS epidemic model with delay, bacteria/phage dynamics, and glucose/insulin levels. Computational methods for approximating the SDDE models are described. Comparisons between computational solutions of the SDDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations and of the computational methods.

The study of the plasmon modes of square atomic clusters based on the eigen-oscillation equation of charge under the free-electron gas model

NASA Astrophysics Data System (ADS)

Xue, Hong-Jie; Wu, Reng-Lai; Hu, Cheng-Xi; Zhang, Ming

2018-04-01

In atomic clusters, plasmon modes are generally gained by the resonant responses for external fields. However, these resonant methods still carry some defects: some plasmon modes may not have been found as that may not have been excited by the external fields. Recently, by employing the extended Hubbard model to describe electron systems of atomic clusters, we have presented the eigen-oscillation equation of charge to study plasmon modes. In this work, based on the free-electron gas model, we further explore the eigen-equation method. Under different external electric fields, some of the plasmon mode spectrums with obvious differences are found, which display the defects of the resonant methods. All the plasmon modes obtained by the resonant methods are predicted by the eigen-equation method. This effectively shows that the eigen-equation method is feasible and reliable in the process of finding plasmon. In addition, various kinds of plasmons are displayed by charge distributions, and the evolution features of plasmon with system parameters are gained by the energy absorption spectrum.

The fifth-order partial differential equation for the description of the α + β Fermi-Pasta-Ulam model

NASA Astrophysics Data System (ADS)

Kudryashov, Nikolay A.; Volkov, Alexandr K.

2017-01-01

We study a new nonlinear partial differential equation of the fifth order for the description of perturbations in the Fermi-Pasta-Ulam mass chain. This fifth-order equation is an expansion of the Gardner equation for the description of the Fermi-Pasta-Ulam model. We use the potential of interaction between neighbouring masses with both quadratic and cubic terms. The equation is derived using the continuous limit. Unlike the previous works, we take into account higher order terms in the Taylor series expansions. We investigate the equation using the Painlevé approach. We show that the equation does not pass the Painlevé test and can not be integrated by the inverse scattering transform. We use the logistic function method and the Laurent expansion method to find travelling wave solutions of the fifth-order equation. We use the pseudospectral method for the numerical simulation of wave processes, described by the equation.

Testing the new BPS method in some models of nonabelian magnetic monopole

NASA Astrophysics Data System (ADS)

Prasetyo, I.; Atmaja, A. N.; Ramadhan, H. S.

2018-03-01

The proposed method in [Phys. Lett. B768 351-358 (2017)], which can obtain BPS equations of some models of vortices, is used here to see whether it is still usable for some models of magnetic monopole. Other than the standard Yang-Mills-Higgs, here we report that the method is able to give us the BPS equations from two different magnetic monopole models.

A mixed finite-element method for solving the poroelastic Biot equations with electrokinetic coupling

NASA Astrophysics Data System (ADS)

Pain, C. C.; Saunders, J. H.; Worthington, M. H.; Singer, J. M.; Stuart-Bruges, W.; Mason, G.; Goddard, A.

2005-02-01

In this paper, a numerical method for solving the Biot poroelastic equations is developed. These equations comprise acoustic (typically water) and elastic (porous medium frame) equations, which are coupled mainly through fluid/solid drag terms. This wave solution is coupled to a simplified form of Maxwell's equations, which is solved for the streaming potential resulting from electrokinesis. The ultimate aim is to use the generated electrical signals to provide porosity, permeability and other information about the formation surrounding a borehole. The electrical signals are generated through electrokinesis by seismic waves causing movement of the fluid through pores or fractures of a porous medium. The focus of this paper is the numerical solution of the Biot equations in displacement form, which is achieved using a mixed finite-element formulation with a different finite-element representation for displacements and stresses. The mixed formulation is used in order to reduce spurious displacement modes and fluid shear waves in the numerical solutions. These equations are solved in the time domain using an implicit unconditionally stable time-stepping method using iterative solution methods amenable to solving large systems of equations. The resulting model is embodied in the MODELLING OF ACOUSTICS, POROELASTICS AND ELECTROKINETICS (MAPEK) computer model for electroseismic analysis.

Simulating Chemical Kinetics Without Differential Equations: A Quantitative Theory Based on Chemical Pathways.

PubMed

Bai, Shirong; Skodje, Rex T

2017-08-17

A new approach is presented for simulating the time-evolution of chemically reactive systems. This method provides an alternative to conventional modeling of mass-action kinetics that involves solving differential equations for the species concentrations. The method presented here avoids the need to solve the rate equations by switching to a representation based on chemical pathways. In the Sum Over Histories Representation (or SOHR) method, any time-dependent kinetic observable, such as concentration, is written as a linear combination of probabilities for chemical pathways leading to a desired outcome. In this work, an iterative method is introduced that allows the time-dependent pathway probabilities to be generated from a knowledge of the elementary rate coefficients, thus avoiding the pitfalls involved in solving the differential equations of kinetics. The method is successfully applied to the model Lotka-Volterra system and to a realistic H 2 combustion model.

Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model

NASA Astrophysics Data System (ADS)

Dehghan, Mehdi; Mohammadi, Vahid

2017-03-01

As is said in [27], the tumor-growth model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models [27]. Simulations of this practical model using numerical methods can be applied for evaluating it. The present paper investigates the solution of the tumor growth model with meshless techniques. Meshless methods are applied based on the collocation technique which employ multiquadrics (MQ) radial basis function (RBFs) and generalized moving least squares (GMLS) procedures. The main advantages of these choices come back to the natural behavior of meshless approaches. As well as, a method based on meshless approach can be applied easily for finding the solution of partial differential equations in high-dimension using any distributions of points on regular and irregular domains. The present paper involves a time-dependent system of partial differential equations that describes four-species tumor growth model. To overcome the time variable, two procedures will be used. One of them is a semi-implicit finite difference method based on Crank-Nicolson scheme and another one is based on explicit Runge-Kutta time integration. The first case gives a linear system of algebraic equations which will be solved at each time-step. The second case will be efficient but conditionally stable. The obtained numerical results are reported to confirm the ability of these techniques for solving the two and three-dimensional tumor-growth equations.

A Numerical Model for Trickle Bed Reactors

NASA Astrophysics Data System (ADS)

Propp, Richard M.; Colella, Phillip; Crutchfield, William Y.; Day, Marcus S.

2000-12-01

Trickle bed reactors are governed by equations of flow in porous media such as Darcy's law and the conservation of mass. Our numerical method for solving these equations is based on a total-velocity splitting, sequential formulation which leads to an implicit pressure equation and a semi-implicit mass conservation equation. We use high-resolution finite-difference methods to discretize these equations. Our solution scheme extends previous work in modeling porous media flows in two ways. First, we incorporate physical effects due to capillary pressure, a nonlinear inlet boundary condition, spatial porosity variations, and inertial effects on phase mobilities. In particular, capillary forces introduce a parabolic component into the recast evolution equation, and the inertial effects give rise to hyperbolic nonconvexity. Second, we introduce a modification of the slope-limiting algorithm to prevent our numerical method from producing spurious shocks. We present a numerical algorithm for accommodating these difficulties, show the algorithm is second-order accurate, and demonstrate its performance on a number of simplified problems relevant to trickle bed reactor modeling.

Finite difference methods for transient signal propagation in stratified dispersive media

NASA Technical Reports Server (NTRS)

Lam, D. H.

1975-01-01

Explicit difference equations are presented for the solution of a signal of arbitrary waveform propagating in an ohmic dielectric, a cold plasma, a Debye model dielectric, and a Lorentz model dielectric. These difference equations are derived from the governing time-dependent integro-differential equations for the electric fields by a finite difference method. A special difference equation is derived for the grid point at the boundary of two different media. Employing this difference equation, transient signal propagation in an inhomogeneous media can be solved provided that the medium is approximated in a step-wise fashion. The solutions are generated simply by marching on in time. It is concluded that while the classical transform methods will remain useful in certain cases, with the development of the finite difference methods described, an extensive class of problems of transient signal propagating in stratified dispersive media can be effectively solved by numerical methods.

Modeling Individual Damped Linear Oscillator Processes with Differential Equations: Using Surrogate Data Analysis to Estimate the Smoothing Parameter

ERIC Educational Resources Information Center

Deboeck, Pascal R.; Boker, Steven M.; Bergeman, C. S.

2008-01-01

Among the many methods available for modeling intraindividual time series, differential equation modeling has several advantages that make it promising for applications to psychological data. One interesting differential equation model is that of the damped linear oscillator (DLO), which can be used to model variables that have a tendency to…

Numerical solution of boundary-integral equations for molecular electrostatics.

PubMed

Bardhan, Jaydeep P

2009-03-07

Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.

Peridynamic Multiscale Finite Element Methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Costa, Timothy; Bond, Stephen D.; Littlewood, David John

The problem of computing quantum-accurate design-scale solutions to mechanics problems is rich with applications and serves as the background to modern multiscale science research. The prob- lem can be broken into component problems comprised of communicating across adjacent scales, which when strung together create a pipeline for information to travel from quantum scales to design scales. Traditionally, this involves connections between a) quantum electronic structure calculations and molecular dynamics and between b) molecular dynamics and local partial differ- ential equation models at the design scale. The second step, b), is particularly challenging since the appropriate scales of molecular dynamic andmore » local partial differential equation models do not overlap. The peridynamic model for continuum mechanics provides an advantage in this endeavor, as the basic equations of peridynamics are valid at a wide range of scales limiting from the classical partial differential equation models valid at the design scale to the scale of molecular dynamics. In this work we focus on the development of multiscale finite element methods for the peridynamic model, in an effort to create a mathematically consistent channel for microscale information to travel from the upper limits of the molecular dynamics scale to the design scale. In particular, we first develop a Nonlocal Multiscale Finite Element Method which solves the peridynamic model at multiple scales to include microscale information at the coarse-scale. We then consider a method that solves a fine-scale peridynamic model to build element-support basis functions for a coarse- scale local partial differential equation model, called the Mixed Locality Multiscale Finite Element Method. Given decades of research and development into finite element codes for the local partial differential equation models of continuum mechanics there is a strong desire to couple local and nonlocal models to leverage the speed and state of the art of local models with the flexibility and accuracy of the nonlocal peridynamic model. In the mixed locality method this coupling occurs across scales, so that the nonlocal model can be used to communicate material heterogeneity at scales inappropriate to local partial differential equation models. Additionally, the computational burden of the weak form of the peridynamic model is reduced dramatically by only requiring that the model be solved on local patches of the simulation domain which may be computed in parallel, taking advantage of the heterogeneous nature of next generation computing platforms. Addition- ally, we present a novel Galerkin framework, the 'Ambulant Galerkin Method', which represents a first step towards a unified mathematical analysis of local and nonlocal multiscale finite element methods, and whose future extension will allow the analysis of multiscale finite element methods that mix models across scales under certain assumptions of the consistency of those models.« less

An Efficient Multiscale Finite-Element Method for Frequency-Domain Seismic Wave Propagation

DOE PAGES

Gao, Kai; Fu, Shubin; Chung, Eric T.

2018-02-13

The frequency-domain seismic-wave equation, that is, the Helmholtz equation, has many important applications in seismological studies, yet is very challenging to solve, particularly for large geological models. Iterative solvers, domain decomposition, or parallel strategies can partially alleviate the computational burden, but these approaches may still encounter nontrivial difficulties in complex geological models where a sufficiently fine mesh is required to represent the fine-scale heterogeneities. We develop a novel numerical method to solve the frequency-domain acoustic wave equation on the basis of the multiscale finite-element theory. We discretize a heterogeneous model with a coarse mesh and employ carefully constructed high-order multiscalemore » basis functions to form the basis space for the coarse mesh. Solved from medium- and frequency-dependent local problems, these multiscale basis functions can effectively capture themedium’s fine-scale heterogeneity and the source’s frequency information, leading to a discrete system matrix with a much smaller dimension compared with those from conventional methods.We then obtain an accurate solution to the acoustic Helmholtz equation by solving only a small linear system instead of a large linear system constructed on the fine mesh in conventional methods.We verify our new method using several models of complicated heterogeneities, and the results show that our new multiscale method can solve the Helmholtz equation in complex models with high accuracy and extremely low computational costs.« less

An Efficient Multiscale Finite-Element Method for Frequency-Domain Seismic Wave Propagation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gao, Kai; Fu, Shubin; Chung, Eric T.

The frequency-domain seismic-wave equation, that is, the Helmholtz equation, has many important applications in seismological studies, yet is very challenging to solve, particularly for large geological models. Iterative solvers, domain decomposition, or parallel strategies can partially alleviate the computational burden, but these approaches may still encounter nontrivial difficulties in complex geological models where a sufficiently fine mesh is required to represent the fine-scale heterogeneities. We develop a novel numerical method to solve the frequency-domain acoustic wave equation on the basis of the multiscale finite-element theory. We discretize a heterogeneous model with a coarse mesh and employ carefully constructed high-order multiscalemore » basis functions to form the basis space for the coarse mesh. Solved from medium- and frequency-dependent local problems, these multiscale basis functions can effectively capture themedium’s fine-scale heterogeneity and the source’s frequency information, leading to a discrete system matrix with a much smaller dimension compared with those from conventional methods.We then obtain an accurate solution to the acoustic Helmholtz equation by solving only a small linear system instead of a large linear system constructed on the fine mesh in conventional methods.We verify our new method using several models of complicated heterogeneities, and the results show that our new multiscale method can solve the Helmholtz equation in complex models with high accuracy and extremely low computational costs.« less

Numerical discretization-based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research.

PubMed

Wu, Hulin; Xue, Hongqi; Kumar, Arun

2012-06-01

Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this article, we propose a novel class of methods for estimating parameters in ordinary differential equation (ODE) models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations. First, a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. We consider three different order of discretization methods, Euler's method, trapezoidal rule, and Runge-Kutta method. A higher-order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher. To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators are established. Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods t an HIV study to further illustrate the usefulness of the proposed approaches. © 2012, The International Biometric Society.

Simulation of 2D rarefied gas flows based on the numerical solution of the Boltzmann equation

NASA Astrophysics Data System (ADS)

Poleshkin, Sergey O.; Malkov, Ewgenij A.; Kudryavtsev, Alexey N.; Shershnev, Anton A.; Bondar, Yevgeniy A.; Kohanchik, A. A.

2017-10-01

There are various methods for calculating rarefied gas flows, in particular, statistical methods and deterministic methods based on the finite-difference solutions of the Boltzmann nonlinear kinetic equation and on the solutions of model kinetic equations. There is no universal method; each has its disadvantages in terms of efficiency or accuracy. The choice of the method depends on the problem to be solved and on parameters of calculated flows. Qualitative theoretical arguments help to determine the range of parameters of effectively solved problems for each method; however, it is advisable to perform comparative tests of calculations of the classical problems performed by different methods and with different parameters to have quantitative confirmation of this reasoning. The paper provides the results of the calculations performed by the authors with the help of the Direct Simulation Monte Carlo method and finite-difference methods of solving the Boltzmann equation and model kinetic equations. Based on this comparison, conclusions are made on selecting a particular method for flow simulations in various ranges of flow parameters.

A mesoscopic bridging scale method for fluids and coupling dissipative particle dynamics with continuum finite element method

PubMed Central

Kojic, Milos; Filipovic, Nenad; Tsuda, Akira

2012-01-01

A multiscale procedure to couple a mesoscale discrete particle model and a macroscale continuum model of incompressible fluid flow is proposed in this study. We call this procedure the mesoscopic bridging scale (MBS) method since it is developed on the basis of the bridging scale method for coupling molecular dynamics and finite element models [G.J. Wagner, W.K. Liu, Coupling of atomistic and continuum simulations using a bridging scale decomposition, J. Comput. Phys. 190 (2003) 249–274]. We derive the governing equations of the MBS method and show that the differential equations of motion of the mesoscale discrete particle model and finite element (FE) model are only coupled through the force terms. Based on this coupling, we express the finite element equations which rely on the Navier–Stokes and continuity equations, in a way that the internal nodal FE forces are evaluated using viscous stresses from the mesoscale model. The dissipative particle dynamics (DPD) method for the discrete particle mesoscale model is employed. The entire fluid domain is divided into a local domain and a global domain. Fluid flow in the local domain is modeled with both DPD and FE method, while fluid flow in the global domain is modeled by the FE method only. The MBS method is suitable for modeling complex (colloidal) fluid flows, where continuum methods are sufficiently accurate only in the large fluid domain, while small, local regions of particular interest require detailed modeling by mesoscopic discrete particles. Solved examples – simple Poiseuille and driven cavity flows illustrate the applicability of the proposed MBS method. PMID:23814322

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (3).

PubMed

Murase, Kenya

2016-01-01

In this issue, simultaneous differential equations were introduced. These differential equations are often used in the field of medical physics. The methods for solving them were also introduced, which include Laplace transform and matrix methods. Some examples were also introduced, in which Laplace transform and matrix methods were applied to solving simultaneous differential equations derived from a three-compartment kinetic model for analyzing the glucose metabolism in tissues and Bloch equations for describing the behavior of the macroscopic magnetization in magnetic resonance imaging.In the next (final) issue, partial differential equations and various methods for solving them will be introduced together with some examples in medical physics.

Bridging the Knowledge Gaps between Richards' Equation and Budyko Equation

NASA Astrophysics Data System (ADS)

Wang, D.

2017-12-01

The empirical Budyko equation represents the partitioning of mean annual precipitation into evaporation and runoff. Richards' equation, based on Darcy's law, represents the movement of water in unsaturated soils. The linkage between Richards' equation and Budyko equation is presented by invoking the empirical Soil Conservation Service curve number (SCS-CN) model for computing surface runoff at the event-scale. The basis of the SCS-CN method is the proportionality relationship, i.e., the ratio of continuing abstraction to its potential is equal to the ratio of surface runoff to its potential value. The proportionality relationship can be derived from the Richards' equation for computing infiltration excess and saturation excess models at the catchment scale. Meanwhile, the generalized proportionality relationship is demonstrated as the common basis of SCS-CN method, monthly "abcd" model, and Budyko equation. Therefore, the linkage between Darcy's law and the emergent pattern of mean annual water balance at the catchment scale is presented through the proportionality relationship.

Bogomolny equations in certain generalized baby BPS Skyrme models

NASA Astrophysics Data System (ADS)

Stępień, Ł. T.

2018-01-01

By using the concept of strong necessary conditions (CSNCs), we derive Bogomolny equations and Bogomol’nyi-Prasad-Sommerfield (BPS) bounds for two certain modifications of the baby BPS Skyrme model: the nonminimal coupling to the gauge field and the k-deformed ungauged model. In particular, we study how the Bogomolny equations and the equation for the potential reflect these two modifications. In both examples, the CSNC method appears to be a very useful tool. We also find certain localized solutions of these Bogomolny equations.

A lattice Boltzmann model for the Burgers-Fisher equation.

PubMed

Zhang, Jianying; Yan, Guangwu

2010-06-01

A lattice Boltzmann model is developed for the one- and two-dimensional Burgers-Fisher equation based on the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. In order to obtain the two-dimensional Burgers-Fisher equation, vector sigma(j) has been used. And in order to overcome the drawbacks of "error rebound," a new assumption of additional distribution is presented, where two additional terms, in first order and second order separately, are used. Comparisons with the results obtained by other methods reveal that the numerical solutions obtained by the proposed method converge to exact solutions. The model under new assumption gives better results than that with second order assumption. (c) 2010 American Institute of Physics.

Efficient modeling of phase jitter in dispersion-managed soliton systems.

PubMed

McKinstrie, C J; Xie, C; Lakoba, T I

2002-11-01

The variational method is used to derive correlation equations that model phase jitter in dispersion-managed soliton systems. The predictions of these correlation equations are consistent with numerical solutions of the nonlinear Schrödinger equation on which they are based.

Stable Algorithm For Estimating Airdata From Flush Surface Pressure Measurements

NASA Technical Reports Server (NTRS)

Whitmore, Stephen, A. (Inventor); Cobleigh, Brent R. (Inventor); Haering, Edward A., Jr. (Inventor)

2001-01-01

An airdata estimation and evaluation system and method, including a stable algorithm for estimating airdata from nonintrusive surface pressure measurements. The airdata estimation and evaluation system is preferably implemented in a flush airdata sensing (FADS) system. The system and method of the present invention take a flow model equation and transform it into a triples formulation equation. The triples formulation equation eliminates the pressure related states from the flow model equation by strategically taking the differences of three surface pressures, known as triples. This triples formulation equation is then used to accurately estimate and compute vital airdata from nonintrusive surface pressure measurements.

Evolution of nonlinear waves in a blood-filled artery with an aneurysm

NASA Astrophysics Data System (ADS)

Nikolova, E. V.; Jordanov, I. P.; Dimitrova, Z. I.; Vitanov, N. K.

2017-10-01

We discuss propagation of traveling waves in a blood-filled hyper-elastic artery with a local dilatation (an aneurysm). The processes in the injured artery are modeled by an equation of the motion of the arterial wall and by equations of the motion of the fluid (the blood). Taking into account the specific arterial geometry and applying the reductive perturbation method in long-wave approximation we reduce the model equations to a version of the perturbed Korteweg-de Vries kind equation with variable coefficients. Exact traveling-wave solutions of this equation are obtained by the modified method of simplest equation where the differential equation of Abel is used as a simplest equation. A particular case of the obtained exact solution is numerically simulated and discussed from the point of view of arterial disease mechanics.

A theory of fine structure image models with an application to detection and classification of dementia.

PubMed

O'Neill, William; Penn, Richard; Werner, Michael; Thomas, Justin

2015-06-01

Estimation of stochastic process models from data is a common application of time series analysis methods. Such system identification processes are often cast as hypothesis testing exercises whose intent is to estimate model parameters and test them for statistical significance. Ordinary least squares (OLS) regression and the Levenberg-Marquardt algorithm (LMA) have proven invaluable computational tools for models being described by non-homogeneous, linear, stationary, ordinary differential equations. In this paper we extend stochastic model identification to linear, stationary, partial differential equations in two independent variables (2D) and show that OLS and LMA apply equally well to these systems. The method employs an original nonparametric statistic as a test for the significance of estimated parameters. We show gray scale and color images are special cases of 2D systems satisfying a particular autoregressive partial difference equation which estimates an analogous partial differential equation. Several applications to medical image modeling and classification illustrate the method by correctly classifying demented and normal OLS models of axial magnetic resonance brain scans according to subject Mini Mental State Exam (MMSE) scores. Comparison with 13 image classifiers from the literature indicates our classifier is at least 14 times faster than any of them and has a classification accuracy better than all but one. Our modeling method applies to any linear, stationary, partial differential equation and the method is readily extended to 3D whole-organ systems. Further, in addition to being a robust image classifier, estimated image models offer insights into which parameters carry the most diagnostic image information and thereby suggest finer divisions could be made within a class. Image models can be estimated in milliseconds which translate to whole-organ models in seconds; such runtimes could make real-time medicine and surgery modeling possible.

Differential equation models for sharp threshold dynamics.

PubMed

Schramm, Harrison C; Dimitrov, Nedialko B

2014-01-01

We develop an extension to differential equation models of dynamical systems to allow us to analyze probabilistic threshold dynamics that fundamentally and globally change system behavior. We apply our novel modeling approach to two cases of interest: a model of infectious disease modified for malware where a detection event drastically changes dynamics by introducing a new class in competition with the original infection; and the Lanchester model of armed conflict, where the loss of a key capability drastically changes the effectiveness of one of the sides. We derive and demonstrate a step-by-step, repeatable method for applying our novel modeling approach to an arbitrary system, and we compare the resulting differential equations to simulations of the system's random progression. Our work leads to a simple and easily implemented method for analyzing probabilistic threshold dynamics using differential equations. Published by Elsevier Inc.

Meta-Analytic Methods of Pooling Correlation Matrices for Structural Equation Modeling under Different Patterns of Missing Data

ERIC Educational Resources Information Center

Furlow, Carolyn F.; Beretvas, S. Natasha

2005-01-01

Three methods of synthesizing correlations for meta-analytic structural equation modeling (SEM) under different degrees and mechanisms of missingness were compared for the estimation of correlation and SEM parameters and goodness-of-fit indices by using Monte Carlo simulation techniques. A revised generalized least squares (GLS) method for…

A parameter study of the two-fluid solar wind

NASA Technical Reports Server (NTRS)

Sandbaek, Ornulf; Leer, Egil; Holzer, Thomas E.

1992-01-01

A two-fluid model of the solar wind was introduced by Sturrock and Hartle (1966) and Hartle and Sturrock (1968). In these studies the proton energy equation was integrated neglecting the heat conductive term. Later several authors solved the equations for the two-fluid solar wind model keeping the proton heat conductive term. Methods where the equations are integrated simultaneously outward and inward from the critical point were used. The equations were also integrated inward from a large heliocentric distance. These methods have been applied to cases with low coronal base electron densities and high base temperatures. In this paper we present a method of integrating the two-fluid solar wind equations using an iteration procedure where the equations are integrated separately and the proton flux is kept constant during the integrations. The technique is applicable for a wide range of coronal base densities and temperatures. The method is used to carry out a parameter study of the two-fluid solar wind.

Density-Dependent Conformable Space-time Fractional Diffusion-Reaction Equation and Its Exact Solutions

NASA Astrophysics Data System (ADS)

Hosseini, Kamyar; Mayeli, Peyman; Bekir, Ahmet; Guner, Ozkan

2018-01-01

In this article, a special type of fractional differential equations (FDEs) named the density-dependent conformable fractional diffusion-reaction (DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the \\exp (-φ (\\varepsilon )) -expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.

Relative Performance of Rescaling and Resampling Approaches to Model Chi Square and Parameter Standard Error Estimation in Structural Equation Modeling.

ERIC Educational Resources Information Center

Nevitt, Johnathan; Hancock, Gregory R.

Though common structural equation modeling (SEM) methods are predicated upon the assumption of multivariate normality, applied researchers often find themselves with data clearly violating this assumption and without sufficient sample size to use distribution-free estimation methods. Fortunately, promising alternatives are being integrated into…

Reliability of Summed Item Scores Using Structural Equation Modeling: An Alternative to Coefficient Alpha

ERIC Educational Resources Information Center

Green, Samuel B.; Yang, Yanyun

2009-01-01

A method is presented for estimating reliability using structural equation modeling (SEM) that allows for nonlinearity between factors and item scores. Assuming the focus is on consistency of summed item scores, this method for estimating reliability is preferred to those based on linear SEM models and to the most commonly reported estimate of…

Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method

NASA Astrophysics Data System (ADS)

Jain, Sonal

2018-01-01

In this paper, we aim to use the alternative numerical scheme given by Gnitchogna and Atangana for solving partial differential equations with integer and non-integer differential operators. We applied this method to fractional diffusion model and fractional Buckmaster models with non-local fading memory. The method yields a powerful numerical algorithm for fractional order derivative to implement. Also we present in detail the stability analysis of the numerical method for solving the diffusion equation. This proof shows that this method is very stable and also converges very quickly to exact solution and finally some numerical simulation is presented.

FFT-based computation of the bioheat transfer equation for the HCC ultrasound surgery therapy modeling.

PubMed

Dillenseger, Jean-Louis; Esneault, Simon; Garnier, Carole

2008-01-01

This paper describes a modeling method of the tissue temperature evolution over time in hyperthermia. More precisely, this approach is used to simulate the hepatocellular carcinoma curative treatment by a percutaneous high intensity ultrasound surgery. The tissue temperature evolution over time is classically described by Pennes' bioheat transfer equation which is generally solved by a finite difference method. In this paper we will present a method where the bioheat transfer equation can be algebraically solved after a Fourier transformation over the space coordinates. The implementation and boundary conditions of this method will be shown and compared with the finite difference method.

The Modelling of Axially Translating Flexible Beams

NASA Astrophysics Data System (ADS)

Theodore, R. J.; Arakeri, J. H.; Ghosal, A.

1996-04-01

The axially translating flexible beam with a prismatic joint can be modelled by using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separation of variables cannot be applied to solve this partial differential equation. In this paper, a non-dimensional form of the Euler Bernoulli beam equation is presented, obtained by using the concept of group velocity, and also the conditions under which separation of variables and assumed modes method can be used. The use of clamped-mass boundary conditions leads to a time-dependent frequency equation for the translating flexible beam. A novel method is presented for solving this time dependent frequency equation by using a differential form of the frequency equation. The assume mode/Lagrangian formulation of dynamics is employed to derive closed form equations of motion. It is shown by using Lyapunov's first method that the dynamic responses of flexural modal variables become unstable during retraction of the flexible beam, which the dynamic response during extension of the beam is stable. Numerical simulation results are presented for the uniform axial motion induced transverse vibration for a typical flexible beam.

Transforming parts of a differential equations system to difference equations as a method for run-time savings in NONMEM.

PubMed

Petersson, K J F; Friberg, L E; Karlsson, M O

2010-10-01

Computer models of biological systems grow more complex as computing power increase. Often these models are defined as differential equations and no analytical solutions exist. Numerical integration is used to approximate the solution; this can be computationally intensive, time consuming and be a large proportion of the total computer runtime. The performance of different integration methods depend on the mathematical properties of the differential equations system at hand. In this paper we investigate the possibility of runtime gains by calculating parts of or the whole differential equations system at given time intervals, outside of the differential equations solver. This approach was tested on nine models defined as differential equations with the goal to reduce runtime while maintaining model fit, based on the objective function value. The software used was NONMEM. In four models the computational runtime was successfully reduced (by 59-96%). The differences in parameter estimates, compared to using only the differential equations solver were less than 12% for all fixed effects parameters. For the variance parameters, estimates were within 10% for the majority of the parameters. Population and individual predictions were similar and the differences in OFV were between 1 and -14 units. When computational runtime seriously affects the usefulness of a model we suggest evaluating this approach for repetitive elements of model building and evaluation such as covariate inclusions or bootstraps.

Estimation of Ordinary Differential Equation Parameters Using Constrained Local Polynomial Regression.

PubMed

Ding, A Adam; Wu, Hulin

2014-10-01

We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method.

Estimation of Ordinary Differential Equation Parameters Using Constrained Local Polynomial Regression

PubMed Central

Ding, A. Adam; Wu, Hulin

2015-01-01

We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method. PMID:26401093

Chronology of DIC technique based on the fundamental mathematical modeling and dehydration impact.

PubMed

Alias, Norma; Saipol, Hafizah Farhah Saipan; Ghani, Asnida Che Abd

2014-12-01

A chronology of mathematical models for heat and mass transfer equation is proposed for the prediction of moisture and temperature behavior during drying using DIC (Détente Instantanée Contrôlée) or instant controlled pressure drop technique. DIC technique has the potential as most commonly used dehydration method for high impact food value including the nutrition maintenance and the best possible quality for food storage. The model is governed by the regression model, followed by 2D Fick's and Fourier's parabolic equation and 2D elliptic-parabolic equation in a rectangular slice. The models neglect the effect of shrinkage and radiation effects. The simulations of heat and mass transfer equations with parabolic and elliptic-parabolic types through some numerical methods based on finite difference method (FDM) have been illustrated. Intel®Core™2Duo processors with Linux operating system and C programming language have been considered as a computational platform for the simulation. Qualitative and quantitative differences between DIC technique and the conventional drying methods have been shown as a comparative.

Optimization of GM(1,1) power model

NASA Astrophysics Data System (ADS)

Luo, Dang; Sun, Yu-ling; Song, Bo

2013-10-01

GM (1,1) power model is the expansion of traditional GM (1,1) model and Grey Verhulst model. Compared with the traditional models, GM (1,1) power model has the following advantage: The power exponent in the model which best matches the actual data values can be found by certain technology. So, GM (1,1) power model can reflect nonlinear features of the data, simulate and forecast with high accuracy. It's very important to determine the best power exponent during the modeling process. In this paper, according to the GM(1,1) power model of albino equation is Bernoulli equation, through variable substitution, turning it into the GM(1,1) model of the linear albino equation form, and then through the grey differential equation properly built, established GM(1,1) power model, and parameters with pattern search method solution. Finally, we illustrate the effectiveness of the new methods with the example of simulating and forecasting the promotion rates from senior secondary schools to higher education in China.

2-Dimensional B-Spline Algorithms with Applications to Ray Tracing in Media of Spatially-Varying Refractive Index

DTIC Science & Technology

2007-08-01

In the approach, photon trajectories are computed using a solution of the Eikonal equation (ray-tracing methods) rather than linear trajectories. The...coupling the radiative transport solution into heat transfer and damage models. 15. SUBJECT TERMS: B-Splines, Ray-Tracing, Eikonal Equation...multi-layer biological tissue model. In the approach, photon trajectories are computed using a solution of the Eikonal equation (ray-tracing methods

Visco-acoustic wave-equation traveltime inversion and its sensitivity to attenuation errors

NASA Astrophysics Data System (ADS)

Yu, Han; Chen, Yuqing; Hanafy, Sherif M.; Huang, Jiangping

2018-04-01

A visco-acoustic wave-equation traveltime inversion method is presented that inverts for the shallow subsurface velocity distribution. Similar to the classical wave equation traveltime inversion, this method finds the velocity model that minimizes the squared sum of the traveltime residuals. Even though, wave-equation traveltime inversion can partly avoid the cycle skipping problem, a good initial velocity model is required for the inversion to converge to a reasonable tomogram with different attenuation profiles. When Q model is far away from the real model, the final tomogram is very sensitive to the starting velocity model. Nevertheless, a minor or moderate perturbation of the Q model from the true one does not strongly affect the inversion if the low wavenumber information of the initial velocity model is mostly correct. These claims are validated with numerical tests on both the synthetic and field data sets.

An Improved Estimation Using Polya-Gamma Augmentation for Bayesian Structural Equation Models with Dichotomous Variables

ERIC Educational Resources Information Center

Kim, Seohyun; Lu, Zhenqiu; Cohen, Allan S.

2018-01-01

Bayesian algorithms have been used successfully in the social and behavioral sciences to analyze dichotomous data particularly with complex structural equation models. In this study, we investigate the use of the Polya-Gamma data augmentation method with Gibbs sampling to improve estimation of structural equation models with dichotomous variables.…

Generalized Path Analysis and Generalized Simultaneous Equations Model for Recursive Systems with Responses of Mixed Types

ERIC Educational Resources Information Center

Tsai, Tien-Lung; Shau, Wen-Yi; Hu, Fu-Chang

2006-01-01

This article generalizes linear path analysis (PA) and simultaneous equations models (SiEM) to deal with mixed responses of different types in a recursive or triangular system. An efficient instrumental variable (IV) method for estimating the structural coefficients of a 2-equation partially recursive generalized path analysis (GPA) model and…

Consistent three-equation model for thin films

NASA Astrophysics Data System (ADS)

Richard, Gael; Gisclon, Marguerite; Ruyer-Quil, Christian; Vila, Jean-Paul

2017-11-01

Numerical simulations of thin films of newtonian fluids down an inclined plane use reduced models for computational cost reasons. These models are usually derived by averaging over the fluid depth the physical equations of fluid mechanics with an asymptotic method in the long-wave limit. Two-equation models are based on the mass conservation equation and either on the momentum balance equation or on the work-energy theorem. We show that there is no two-equation model that is both consistent and theoretically coherent and that a third variable and a three-equation model are required to solve all theoretical contradictions. The linear and nonlinear properties of two and three-equation models are tested on various practical problems. We present a new consistent three-equation model with a simple mathematical structure which allows an easy and reliable numerical resolution. The numerical calculations agree fairly well with experimental measurements or with direct numerical resolutions for neutral stability curves, speed of kinematic waves and of solitary waves and depth profiles of wavy films. The model can also predict the flow reversal at the first capillary trough ahead of the main wave hump.

A class of traveling wave solutions for space-time fractional biological population model in mathematical physics

NASA Astrophysics Data System (ADS)

Akram, Ghazala; Batool, Fiza

2017-10-01

The (G'/G)-expansion method is utilized for a reliable treatment of space-time fractional biological population model. The method has been applied in the sense of the Jumarie's modified Riemann-Liouville derivative. Three classes of exact traveling wave solutions, hyperbolic, trigonometric and rational solutions of the associated equation are characterized with some free parameters. A generalized fractional complex transform is applied to convert the fractional equations to ordinary differential equations which subsequently resulted in number of exact solutions. It should be mentioned that the (G'/G)-expansion method is very effective and convenient for solving nonlinear partial differential equations of fractional order whose balancing number is a negative integer.

Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation.

PubMed

Erban, Radek; Kevrekidis, Ioannis G; Adalsteinsson, David; Elston, Timothy C

2006-02-28

We present computer-assisted methods for analyzing stochastic models of gene regulatory networks. The main idea that underlies this equation-free analysis is the design and execution of appropriately initialized short bursts of stochastic simulations; the results of these are processed to estimate coarse-grained quantities of interest, such as mesoscopic transport coefficients. In particular, using a simple model of a genetic toggle switch, we illustrate the computation of an effective free energy Phi and of a state-dependent effective diffusion coefficient D that characterize an unavailable effective Fokker-Planck equation. Additionally we illustrate the linking of equation-free techniques with continuation methods for performing a form of stochastic "bifurcation analysis"; estimation of mean switching times in the case of a bistable switch is also implemented in this equation-free context. The accuracy of our methods is tested by direct comparison with long-time stochastic simulations. This type of equation-free analysis appears to be a promising approach to computing features of the long-time, coarse-grained behavior of certain classes of complex stochastic models of gene regulatory networks, circumventing the need for long Monte Carlo simulations.

Automated reverse engineering of nonlinear dynamical systems

PubMed Central

Bongard, Josh; Lipson, Hod

2007-01-01

Complex nonlinear dynamics arise in many fields of science and engineering, but uncovering the underlying differential equations directly from observations poses a challenging task. The ability to symbolically model complex networked systems is key to understanding them, an open problem in many disciplines. Here we introduce for the first time a method that can automatically generate symbolic equations for a nonlinear coupled dynamical system directly from time series data. This method is applicable to any system that can be described using sets of ordinary nonlinear differential equations, and assumes that the (possibly noisy) time series of all variables are observable. Previous automated symbolic modeling approaches of coupled physical systems produced linear models or required a nonlinear model to be provided manually. The advance presented here is made possible by allowing the method to model each (possibly coupled) variable separately, intelligently perturbing and destabilizing the system to extract its less observable characteristics, and automatically simplifying the equations during modeling. We demonstrate this method on four simulated and two real systems spanning mechanics, ecology, and systems biology. Unlike numerical models, symbolic models have explanatory value, suggesting that automated “reverse engineering” approaches for model-free symbolic nonlinear system identification may play an increasing role in our ability to understand progressively more complex systems in the future. PMID:17553966

Automated reverse engineering of nonlinear dynamical systems.

PubMed

Bongard, Josh; Lipson, Hod

2007-06-12

Complex nonlinear dynamics arise in many fields of science and engineering, but uncovering the underlying differential equations directly from observations poses a challenging task. The ability to symbolically model complex networked systems is key to understanding them, an open problem in many disciplines. Here we introduce for the first time a method that can automatically generate symbolic equations for a nonlinear coupled dynamical system directly from time series data. This method is applicable to any system that can be described using sets of ordinary nonlinear differential equations, and assumes that the (possibly noisy) time series of all variables are observable. Previous automated symbolic modeling approaches of coupled physical systems produced linear models or required a nonlinear model to be provided manually. The advance presented here is made possible by allowing the method to model each (possibly coupled) variable separately, intelligently perturbing and destabilizing the system to extract its less observable characteristics, and automatically simplifying the equations during modeling. We demonstrate this method on four simulated and two real systems spanning mechanics, ecology, and systems biology. Unlike numerical models, symbolic models have explanatory value, suggesting that automated "reverse engineering" approaches for model-free symbolic nonlinear system identification may play an increasing role in our ability to understand progressively more complex systems in the future.

Three-dimensional forward modeling of DC resistivity using the aggregation-based algebraic multigrid method

NASA Astrophysics Data System (ADS)

Chen, Hui; Deng, Ju-Zhi; Yin, Min; Yin, Chang-Chun; Tang, Wen-Wu

2017-03-01

To speed up three-dimensional (3D) DC resistivity modeling, we present a new multigrid method, the aggregation-based algebraic multigrid method (AGMG). We first discretize the differential equation of the secondary potential field with mixed boundary conditions by using a seven-point finite-difference method to obtain a large sparse system of linear equations. Then, we introduce the theory behind the pairwise aggregation algorithms for AGMG and use the conjugate-gradient method with the V-cycle AGMG preconditioner (AGMG-CG) to solve the linear equations. We use typical geoelectrical models to test the proposed AGMG-CG method and compare the results with analytical solutions and the 3DDCXH algorithm for 3D DC modeling (3DDCXH). In addition, we apply the AGMG-CG method to different grid sizes and geoelectrical models and compare it to different iterative methods, such as ILU-BICGSTAB, ILU-GCR, and SSOR-CG. The AGMG-CG method yields nearly linearly decreasing errors, whereas the number of iterations increases slowly with increasing grid size. The AGMG-CG method is precise and converges fast, and thus can improve the computational efficiency in forward modeling of three-dimensional DC resistivity.

Modification of 2-D Time-Domain Shallow Water Wave Equation using Asymptotic Expansion Method

NASA Astrophysics Data System (ADS)

Khairuman, Teuku; Nasruddin, MN; Tulus; Ramli, Marwan

2018-01-01

Generally, research on the tsunami wave propagation model can be conducted by using a linear model of shallow water theory, where a non-linear side on high order is ignored. In line with research on the investigation of the tsunami waves, the Boussinesq equation model underwent a change aimed to obtain an improved quality of the dispersion relation and non-linearity by increasing the order to be higher. To solve non-linear sides at high order is used a asymptotic expansion method. This method can be used to solve non linear partial differential equations. In the present work, we found that this method needs much computational time and memory with the increase of the number of elements.

Generalized Ordinary Differential Equation Models 1

PubMed Central

Miao, Hongyu; Wu, Hulin; Xue, Hongqi

2014-01-01

Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method. PMID:25544787

Generalized Ordinary Differential Equation Models.

PubMed

Miao, Hongyu; Wu, Hulin; Xue, Hongqi

2014-10-01

Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method.

Biot-Gassmann theory for velocities of gas hydrate-bearing sediments

USGS Publications Warehouse

Lee, M.W.

2002-01-01

Elevated elastic velocities are a distinct physical property of gas hydrate-bearing sediments. A number of velocity models and equations (e.g., pore-filling model, cementation model, effective medium theories, weighted equations, and time-average equations) have been used to describe this effect. In particular, the weighted equation and effective medium theory predict reasonably well the elastic properties of unconsolidated gas hydrate-bearing sediments. A weakness of the weighted equation is its use of the empirical relationship of the time-average equation as one element of the equation. One drawback of the effective medium theory is its prediction of unreasonably higher shear-wave velocity at high porosities, so that the predicted velocity ratio does not agree well with the observed velocity ratio. To overcome these weaknesses, a method is proposed, based on Biot-Gassmann theories and assuming the formation velocity ratio (shear to compressional velocity) of an unconsolidated sediment is related to the velocity ratio of the matrix material of the formation and its porosity. Using the Biot coefficient calculated from either the weighted equation or from the effective medium theory, the proposed method accurately predicts the elastic properties of unconsolidated sediments with or without gas hydrate concentration. This method was applied to the observed velocities at the Mallik 2L-39 well, Mackenzie Delta, Canada.

Numerical method based on the lattice Boltzmann model for the Fisher equation.

PubMed

Yan, Guangwu; Zhang, Jianying; Dong, Yinfeng

2008-06-01

In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis.

PubMed

MacDonald, G; Mackenzie, J A; Nolan, M; Insall, R H

2016-03-15

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.

Development of a Standalone Thermal Wellbore Simulator

NASA Astrophysics Data System (ADS)

Xiong, Wanqiang

With continuous developments of various different sophisticated wells in the petroleum industry, wellbore modeling and simulation have increasingly received more attention. Especially in unconventional oil and gas recovery processes, there is a growing demand for more accurate wellbore modeling. Despite notable advancements made in wellbore modeling, none of the existing wellbore simulators has been as successful as reservoir simulators such as Eclipse and CMG's and further research works on handling issues such as accurate heat loss modeling and multi-tubing wellbore modeling are really necessary. A series of mathematical equations including main governing equations, auxiliary equations, PVT equations, thermodynamic equations, drift-flux model equations, and wellbore heat loss calculation equations are collected and screened from publications. Based on these modeling equations, workflows for wellbore simulation and software development are proposed. Research works are conducted in key steps for developing a wellbore simulator: discretization, a grid system, a solution method, a linear equation solver, and computer language. A standalone thermal wellbore simulator is developed by using standard C++ language. This wellbore simulator can simulate single-phase injection and production, two-phase steam injection and two-phase oil and water production. By implementing a multi-part scheme which divides a wellbore with sophisticated configuration into several relative simple simulation running units, this simulator can handle different complex wellbores: wellbore with multistage casings, horizontal wells, multilateral wells and double tubing. In pursuance of improved accuracy of heat loss calculations to surrounding formations, a semi-numerical method is proposed and a series of FLUENT simulations have been conducted in this study. This semi-numerical method involves extending the 2D formation heat transfer simulation to include a casing wall and cement and adopting new correlations regressed by this study. Meanwhile, a correlation for handling heat transfer in double-tubing annulus is regressed. This work initiates the research on heat transfer in a double-tubing wellbore system. A series of validation and test works are performed in hot water injection, steam injection, real filed data, a horizontal well, a double-tubing well and comparison with the Ramey method. The program in this study also performs well in matching with real measured field data, simulation in horizontal wells and double-tubing wells.

A theory of fine structure image models with an application to detection and classification of dementia

PubMed Central

Penn, Richard; Werner, Michael; Thomas, Justin

2015-01-01

Background Estimation of stochastic process models from data is a common application of time series analysis methods. Such system identification processes are often cast as hypothesis testing exercises whose intent is to estimate model parameters and test them for statistical significance. Ordinary least squares (OLS) regression and the Levenberg-Marquardt algorithm (LMA) have proven invaluable computational tools for models being described by non-homogeneous, linear, stationary, ordinary differential equations. Methods In this paper we extend stochastic model identification to linear, stationary, partial differential equations in two independent variables (2D) and show that OLS and LMA apply equally well to these systems. The method employs an original nonparametric statistic as a test for the significance of estimated parameters. Results We show gray scale and color images are special cases of 2D systems satisfying a particular autoregressive partial difference equation which estimates an analogous partial differential equation. Several applications to medical image modeling and classification illustrate the method by correctly classifying demented and normal OLS models of axial magnetic resonance brain scans according to subject Mini Mental State Exam (MMSE) scores. Comparison with 13 image classifiers from the literature indicates our classifier is at least 14 times faster than any of them and has a classification accuracy better than all but one. Conclusions Our modeling method applies to any linear, stationary, partial differential equation and the method is readily extended to 3D whole-organ systems. Further, in addition to being a robust image classifier, estimated image models offer insights into which parameters carry the most diagnostic image information and thereby suggest finer divisions could be made within a class. Image models can be estimated in milliseconds which translate to whole-organ models in seconds; such runtimes could make real-time medicine and surgery modeling possible. PMID:26029638

A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pelanti, Marica, E-mail: marica.pelanti@ensta-paristech.fr; Shyue, Keh-Ming, E-mail: shyue@ntu.edu.tw

2014-02-15

We model liquid–gas flows with cavitation by a variant of the six-equation single-velocity two-phase model with stiff mechanical relaxation of Saurel–Petitpas–Berry (Saurel et al., 2009) [9]. In our approach we employ phasic total energy equations instead of the phasic internal energy equations of the classical six-equation system. This alternative formulation allows us to easily design a simple numerical method that ensures consistency with mixture total energy conservation at the discrete level and agreement of the relaxed pressure at equilibrium with the correct mixture equation of state. Temperature and Gibbs free energy exchange terms are included in the equations as relaxationmore » terms to model heat and mass transfer and hence liquid–vapor transition. The algorithm uses a high-resolution wave propagation method for the numerical approximation of the homogeneous hyperbolic portion of the model. In two dimensions a fully-discretized scheme based on a hybrid HLLC/Roe Riemann solver is employed. Thermo-chemical terms are handled numerically via a stiff relaxation solver that forces thermodynamic equilibrium at liquid–vapor interfaces under metastable conditions. We present numerical results of sample tests in one and two space dimensions that show the ability of the proposed model to describe cavitation mechanisms and evaporation wave dynamics.« less

Modeling of nonequilibrium space plasma flows

NASA Technical Reports Server (NTRS)

Gombosi, Tamas

1995-01-01

Godunov-type numerical solution of the 20 moment plasma transport equations. One of the centerpieces of our proposal was the development of a higher order Godunov-type numerical scheme to solve the gyration dominated 20 moment transport equations. In the first step we explored some fundamental analytic properties of the 20 moment transport equations for a low b plasma, including the eigenvectors and eigenvalues of propagating disturbances. The eigenvalues correspond to wave speeds, while the eigenvectors characterize the transported physical quantities. In this paper we also explored the physically meaningful parameter range of the normalized heat flow components. In the second step a new Godunov scheme type numerical method was developed to solve the coupled set of 20 moment transport equations for a quasineutral single-ion plasma. The numerical method and the first results were presented at several national and international meetings and a paper describing the method has been published in the Journal of Computational Physics. To our knowledge this is the first numerical method which is capable of producing stable time-dependent solutions to the full 20 (or 16) moment set of transport equations, including the full heat flow equation. Previous attempts resulted in unstable (oscillating) solutions of the heat flow equations. Our group invested over two man-years into the development and implementation of the new method. The present model solves the 20 moment transport equations for an ion species and thermal electrons in 8 domain extending from a collision dominated to a collisionless region (200 km to 12,000 km). This model has been applied to study O+ acceleration due to Joule heating in the lower ionosphere.

Comparison of artificial intelligence methods and empirical equations to estimate daily solar radiation

NASA Astrophysics Data System (ADS)

Mehdizadeh, Saeid; Behmanesh, Javad; Khalili, Keivan

2016-08-01

In the present research, three artificial intelligence methods including Gene Expression Programming (GEP), Artificial Neural Networks (ANN) and Adaptive Neuro-Fuzzy Inference System (ANFIS) as well as, 48 empirical equations (10, 12 and 26 equations were temperature-based, sunshine-based and meteorological parameters-based, respectively) were used to estimate daily solar radiation in Kerman, Iran in the period of 1992-2009. To develop the GEP, ANN and ANFIS models, depending on the used empirical equations, various combinations of minimum air temperature, maximum air temperature, mean air temperature, extraterrestrial radiation, actual sunshine duration, maximum possible sunshine duration, sunshine duration ratio, relative humidity and precipitation were considered as inputs in the mentioned intelligent methods. To compare the accuracy of empirical equations and intelligent models, root mean square error (RMSE), mean absolute error (MAE), mean absolute relative error (MARE) and determination coefficient (R2) indices were used. The results showed that in general, sunshine-based and meteorological parameters-based scenarios in ANN and ANFIS models presented high accuracy than mentioned empirical equations. Moreover, the most accurate method in the studied region was ANN11 scenario with five inputs. The values of RMSE, MAE, MARE and R2 indices for the mentioned model were 1.850 MJ m-2 day-1, 1.184 MJ m-2 day-1, 9.58% and 0.935, respectively.

Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

ERIC Educational Resources Information Center

Deboeck, Pascal R.; Boker, Steven M.

2010-01-01

Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for…

Bayesian parameter estimation for nonlinear modelling of biological pathways.

PubMed

Ghasemi, Omid; Lindsey, Merry L; Yang, Tianyi; Nguyen, Nguyen; Huang, Yufei; Jin, Yu-Fang

2011-01-01

The availability of temporal measurements on biological experiments has significantly promoted research areas in systems biology. To gain insight into the interaction and regulation of biological systems, mathematical frameworks such as ordinary differential equations have been widely applied to model biological pathways and interpret the temporal data. Hill equations are the preferred formats to represent the reaction rate in differential equation frameworks, due to their simple structures and their capabilities for easy fitting to saturated experimental measurements. However, Hill equations are highly nonlinearly parameterized functions, and parameters in these functions cannot be measured easily. Additionally, because of its high nonlinearity, adaptive parameter estimation algorithms developed for linear parameterized differential equations cannot be applied. Therefore, parameter estimation in nonlinearly parameterized differential equation models for biological pathways is both challenging and rewarding. In this study, we propose a Bayesian parameter estimation algorithm to estimate parameters in nonlinear mathematical models for biological pathways using time series data. We used the Runge-Kutta method to transform differential equations to difference equations assuming a known structure of the differential equations. This transformation allowed us to generate predictions dependent on previous states and to apply a Bayesian approach, namely, the Markov chain Monte Carlo (MCMC) method. We applied this approach to the biological pathways involved in the left ventricle (LV) response to myocardial infarction (MI) and verified our algorithm by estimating two parameters in a Hill equation embedded in the nonlinear model. We further evaluated our estimation performance with different parameter settings and signal to noise ratios. Our results demonstrated the effectiveness of the algorithm for both linearly and nonlinearly parameterized dynamic systems. Our proposed Bayesian algorithm successfully estimated parameters in nonlinear mathematical models for biological pathways. This method can be further extended to high order systems and thus provides a useful tool to analyze biological dynamics and extract information using temporal data.

Taguchi method for partial differential equations with application in tumor growth.

PubMed

Ilea, M; Turnea, M; Rotariu, M; Arotăriţei, D; Popescu, Marilena

2014-01-01

The growth of tumors is a highly complex process. To describe this process, mathematical models are needed. A variety of partial differential mathematical models for tumor growth have been developed and studied. Most of those models are based on the reaction-diffusion equations and mass conservation law. A variety of modeling strategies have been developed, each focusing on tumor growth. Systems of time-dependent partial differential equations occur in many branches of applied mathematics. The vast majority of mathematical models in tumor growth are formulated in terms of partial differential equations. We propose a mathematical model for the interactions between these three cancer cell populations. The Taguchi methods are widely used by quality engineering scientists to compare the effects of multiple variables, together with their interactions, with a simple and manageable experimental design. In Taguchi's design of experiments, variation is more interesting to study than the average. First, Taguchi methods are utilized to search for the significant factors and the optimal level combination of parameters. Except the three parameters levels, other factors levels other factors levels would not be considered. Second, cutting parameters namely, cutting speed, depth of cut, and feed rate are designed using the Taguchi method. Finally, the adequacy of the developed mathematical model is proved by ANOVA. According to the results of ANOVA, since the percentage contribution of the combined error is as small. Many mathematical models can be quantitatively characterized by partial differential equations. The use of MATLAB and Taguchi method in this article illustrates the important role of informatics in research in mathematical modeling. The study of tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

Pseudo-time algorithms for the Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Swanson, R. C.; Turkel, E.

1986-01-01

A pseudo-time method is introduced to integrate the compressible Navier-Stokes equations to a steady state. This method is a generalization of a method used by Crocco and also by Allen and Cheng. We show that for a simple heat equation that this is just a renormalization of the time. For a convection-diffusion equation the renormalization is dependent only on the viscous terms. We implement the method for the Navier-Stokes equations using a Runge-Kutta type algorithm. This permits the time step to be chosen based on the inviscid model only. We also discuss the use of residual smoothing when viscous terms are present.

Review of Railgun Modeling Techniques: The Computation of Railgun Force and Other Key Factors

NASA Astrophysics Data System (ADS)

Eckert, Nathan James

Currently, railgun force modeling either uses the simple "railgun force equation" or finite element methods. It is proposed here that a middle ground exists that does not require the solution of partial differential equations, is more readily implemented than finite element methods, and is more accurate than the traditional force equation. To develop this method, it is necessary to examine the core railgun factors: power supply mechanisms, the distribution of current in the rails and in the projectile which slides between them (called the armature), the magnetic field created by the current flowing through these rails, the inductance gradient (a key factor in simplifying railgun analysis, referred to as L'), the resultant Lorentz force, and the heating which accompanies this action. Common power supply technologies are investigated, and the shape of their current pulses are modeled. The main causes of current concentration are described, and a rudimentary method for computing current distribution in solid rails and a rectangular armature is shown to have promising accuracy with respect to outside finite element results. The magnetic field is modeled with two methods using the Biot-Savart law, and generally good agreement is obtained with respect to finite element methods (5.8% error on average). To get this agreement, a factor of 2 is added to the original formulation after seeing a reliable offset with FEM results. Three inductance gradient calculations are assessed, and though all agree with FEM results, the Kerrisk method and a regression analysis method developed by Murugan et al. (referred to as the LRM here) perform the best. Six railgun force computation methods are investigated, including the traditional railgun force equation, an equation produced by Waindok and Piekielny, and four methods inspired by the work of Xu et al. Overall, good agreement between the models and outside data is found, but each model's accuracy varies significantly between comparisons. Lastly, an approximation of the temperature profile in railgun rails originally presented by McCorkle and Bahder is replicated. In total, this work describes railgun technology and moderately complex railgun modeling methods, but is inconclusive about the presence of a middle-ground modeling method.

Reduced order modeling of fluid/structure interaction.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Barone, Matthew Franklin; Kalashnikova, Irina; Segalman, Daniel Joseph

2009-11-01

This report describes work performed from October 2007 through September 2009 under the Sandia Laboratory Directed Research and Development project titled 'Reduced Order Modeling of Fluid/Structure Interaction.' This project addresses fundamental aspects of techniques for construction of predictive Reduced Order Models (ROMs). A ROM is defined as a model, derived from a sequence of high-fidelity simulations, that preserves the essential physics and predictive capability of the original simulations but at a much lower computational cost. Techniques are developed for construction of provably stable linear Galerkin projection ROMs for compressible fluid flow, including a method for enforcing boundary conditions that preservesmore » numerical stability. A convergence proof and error estimates are given for this class of ROM, and the method is demonstrated on a series of model problems. A reduced order method, based on the method of quadratic components, for solving the von Karman nonlinear plate equations is developed and tested. This method is applied to the problem of nonlinear limit cycle oscillations encountered when the plate interacts with an adjacent supersonic flow. A stability-preserving method for coupling the linear fluid ROM with the structural dynamics model for the elastic plate is constructed and tested. Methods for constructing efficient ROMs for nonlinear fluid equations are developed and tested on a one-dimensional convection-diffusion-reaction equation. These methods are combined with a symmetrization approach to construct a ROM technique for application to the compressible Navier-Stokes equations.« less

Modified cable equation incorporating transverse polarization of neuronal membranes for accurate coupling of electric fields.

PubMed

Wang, Boshuo; Aberra, Aman S; Grill, Warren M; Peterchev, Angel V

2018-04-01

We present a theory and computational methods to incorporate transverse polarization of neuronal membranes into the cable equation to account for the secondary electric field generated by the membrane in response to transverse electric fields. The effect of transverse polarization on nonlinear neuronal activation thresholds is quantified and discussed in the context of previous studies using linear membrane models. The response of neuronal membranes to applied electric fields is derived under two time scales and a unified solution of transverse polarization is given for spherical and cylindrical cell geometries. The solution is incorporated into the cable equation re-derived using an asymptotic model that separates the longitudinal and transverse dimensions. Two numerical methods are proposed to implement the modified cable equation. Several common neural stimulation scenarios are tested using two nonlinear membrane models to compare thresholds of the conventional and modified cable equations. The implementations of the modified cable equation incorporating transverse polarization are validated against previous results in the literature. The test cases show that transverse polarization has limited effect on activation thresholds. The transverse field only affects thresholds of unmyelinated axons for short pulses and in low-gradient field distributions, whereas myelinated axons are mostly unaffected. The modified cable equation captures the membrane's behavior on different time scales and models more accurately the coupling between electric fields and neurons. It addresses the limitations of the conventional cable equation and allows sound theoretical interpretations. The implementation provides simple methods that are compatible with current simulation approaches to study the effect of transverse polarization on nonlinear membranes. The minimal influence by transverse polarization on axonal activation thresholds for the nonlinear membrane models indicates that predictions of stronger effects in linear membrane models with a fixed activation threshold are inaccurate. Thus, the conventional cable equation works well for most neuroengineering applications, and the presented modeling approach is well suited to address the exceptions.

An extension of the Derrida-Lebowitz-Speer-Spohn equation

NASA Astrophysics Data System (ADS)

Bordenave, Charles; Germain, Pierre; Trogdon, Thomas

2015-12-01

We show how the derivation of the Derrida-Lebowitz-Speer-Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy-Widom GOE distribution.

Approximating a nonlinear advanced-delayed equation from acoustics

NASA Astrophysics Data System (ADS)

Teodoro, M. Filomena

2016-10-01

We approximate the solution of a particular non-linear mixed type functional differential equation from physiology, the mucosal wave model of the vocal oscillation during phonation. The mathematical equation models a superficial wave propagating through the tissues. The numerical scheme is adapted from the work presented in [1, 2, 3], using homotopy analysis method (HAM) to solve the non linear mixed type equation under study.

Cavity master equation for the continuous time dynamics of discrete-spin models.

PubMed

Aurell, E; Del Ferraro, G; Domínguez, E; Mulet, R

2017-05-01

We present an alternate method to close the master equation representing the continuous time dynamics of interacting Ising spins. The method makes use of the theory of random point processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean-field ferromagnet and the dynamics of the one-dimensional Ising system. We present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the random field Ising model, and the Viana-Bray spin-glass model.

Cavity master equation for the continuous time dynamics of discrete-spin models

NASA Astrophysics Data System (ADS)

Aurell, E.; Del Ferraro, G.; Domínguez, E.; Mulet, R.

2017-05-01

We present an alternate method to close the master equation representing the continuous time dynamics of interacting Ising spins. The method makes use of the theory of random point processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean-field ferromagnet and the dynamics of the one-dimensional Ising system. We present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the random field Ising model, and the Viana-Bray spin-glass model.

Quasi-Maximum Likelihood Estimation of Structural Equation Models with Multiple Interaction and Quadratic Effects

ERIC Educational Resources Information Center

Klein, Andreas G.; Muthen, Bengt O.

2007-01-01

In this article, a nonlinear structural equation model is introduced and a quasi-maximum likelihood method for simultaneous estimation and testing of multiple nonlinear effects is developed. The focus of the new methodology lies on efficiency, robustness, and computational practicability. Monte-Carlo studies indicate that the method is highly…

Bias and Efficiency in Structural Equation Modeling: Maximum Likelihood versus Robust Methods

ERIC Educational Resources Information Center

Zhong, Xiaoling; Yuan, Ke-Hai

2011-01-01

In the structural equation modeling literature, the normal-distribution-based maximum likelihood (ML) method is most widely used, partly because the resulting estimator is claimed to be asymptotically unbiased and most efficient. However, this may not hold when data deviate from normal distribution. Outlying cases or nonnormally distributed data,…

Development of a numerical model for vehicle-bridge interaction analysis of railway bridges

NASA Astrophysics Data System (ADS)

Kim, Hee Ju; Cho, Eun Sang; Ham, Jun Su; Park, Ki Tae; Kim, Tae Heon

2016-04-01

In the field of civil engineering, analyzing dynamic response was main concern for a long time. These analysis methods can be divided into moving load analysis method and moving mass analysis method, and formulating each an equation of motion has recently been studied after dividing vehicles and bridges. In this study, the numerical method is presented, which can consider the various train types and can solve the equations of motion for a vehicle-bridge interaction analysis by non-iteration procedure through formulating the coupled equations for motion. Also, 3 dimensional accurate numerical models was developed by KTX-vehicle in order to analyze dynamic response characteristics. The equations of motion for the conventional trains are derived, and the numerical models of the conventional trains are idealized by a set of linear springs and dashpots with 18 degrees of freedom. The bridge models are simplified by the 3 dimensional space frame element which is based on the Euler-Bernoulli theory. The rail irregularities of vertical and lateral directions are generated by PSD functions of the Federal Railroad Administration (FRA).

Immersed boundary method for Boltzmann model kinetic equations

NASA Astrophysics Data System (ADS)

Pekardan, Cem; Chigullapalli, Sruti; Sun, Lin; Alexeenko, Alina

2012-11-01

Three different immersed boundary method formulations are presented for Boltzmann model kinetic equations such as Bhatnagar-Gross-Krook (BGK) and Ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) model equations. 1D unsteady IBM solution for a moving piston is compared with the DSMC results and 2D quasi-steady microscale gas damping solutions are verified by a conformal finite volume method solver. Transient analysis for a sinusoidally moving beam is also carried out for the different pressure conditions (1 atm, 0.1 atm and 0.01 atm) corresponding to Kn=0.05,0.5 and 5. Interrelaxation method (Method 2) is shown to provide a faster convergence as compared to the traditional interpolation scheme used in continuum IBM formulations. Unsteady damping in rarefied regime is characterized by a significant phase-lag which is not captured by quasi-steady approximations.

A Variational Formalism for the Radiative Transfer Equation and a Geostrophic, Hydrostatic Atmosphere: Prelude to Model 3

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.

1991-01-01

The second step in development of MODEL III is summarized. It combines the four radiative transfer equations of the first step with the equations for a geostrophic and hydrostatic atmosphere. This step is intended to bring radiance into a three dimensional balance with wind, height, and temperature. The use of the geostrophic approximation in place of the full set of primitive equations allows for an easier evaluation of how the inclusion of the radiative transfer equation increases the complexity of the variational equations. Seven different variational formulations were developed for geostrophic, hydrostatic, and radiative transfer equations. The first derivation was too complex to yield solutions that were physically meaningful. For the remaining six derivations, the variational method gave the same physical interpretation (the observed brightness temperatures could provide no meaningful input to a geostrophic, hydrostatic balance) at least through the problem solving methodology used in these studies. The variational method is presented and the Euler-Lagrange equations rederived for the geostrophic, hydrostatic, and radiative transfer equations.

Computational methods for vortex dominated compressible flows

NASA Technical Reports Server (NTRS)

Murman, Earll M.

1987-01-01

The principal objectives were to: understand the mechanisms by which Euler equation computations model leading edge vortex flows; understand the vortical and shock wave structures that may exist for different wing shapes, angles of incidence, and Mach numbers; and compare calculations with experiments in order to ascertain the limitations and advantages of Euler equation models. The initial approach utilized the cell centered finite volume Jameson scheme. The final calculation utilized a cell vertex finite volume method on an unstructured grid. Both methods used Runge-Kutta four stage schemes for integrating the equations. The principal findings are briefly summarized.

CFD-ACE+: a CAD system for simulation and modeling of MEMS

NASA Astrophysics Data System (ADS)

Stout, Phillip J.; Yang, H. Q.; Dionne, Paul; Leonard, Andy; Tan, Zhiqiang; Przekwas, Andrzej J.; Krishnan, Anantha

1999-03-01

Computer aided design (CAD) systems are a key to designing and manufacturing MEMS with higher performance/reliability, reduced costs, shorter prototyping cycles and improved time- to-market. One such system is CFD-ACE+MEMS, a modeling and simulation environment for MEMS which includes grid generation, data visualization, graphical problem setup, and coupled fluidic, thermal, mechanical, electrostatic, and magnetic physical models. The fluid model is a 3D multi- block, structured/unstructured/hybrid, pressure-based, implicit Navier-Stokes code with capabilities for multi- component diffusion, multi-species transport, multi-step gas phase chemical reactions, surface reactions, and multi-media conjugate heat transfer. The thermal model solves the total enthalpy from of the energy equation. The energy equation includes unsteady, convective, conductive, species energy, viscous dissipation, work, and radiation terms. The electrostatic model solves Poisson's equation. Both the finite volume method and the boundary element method (BEM) are available for solving Poisson's equation. The BEM method is useful for unbounded problems. The magnetic model solves for the vector magnetic potential from Maxwell's equations including eddy currents but neglecting displacement currents. The mechanical model is a finite element stress/deformation solver which has been coupled to the flow, heat, electrostatic, and magnetic calculations to study flow, thermal electrostatically, and magnetically included deformations of structures. The mechanical or structural model can accommodate elastic and plastic materials, can handle large non-linear displacements, and can model isotropic and anisotropic materials. The thermal- mechanical coupling involves the solution of the steady state Navier equation with thermoelastic deformation. The electrostatic-mechanical coupling is a calculation of the pressure force due to surface charge on the mechanical structure. Results of CFD-ACE+MEMS modeling of MEMS such as cantilever beams, accelerometers, and comb drives are discussed.

Soliton and periodic solutions for time-dependent coefficient non-linear equation

NASA Astrophysics Data System (ADS)

Guner, Ozkan

2016-01-01

In this article, we establish exact solutions for the generalized (3+1)-dimensional variable coefficient Kadomtsev-Petviashvili (GVCKP) equation. Using solitary wave ansatz in terms of ? functions and the modified sine-cosine method, we find exact analytical bright soliton solutions and exact periodic solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The effectiveness and reliability of the method are shown by its application to the GVCKP equation.

Analytical approximate solutions for a general class of nonlinear delay differential equations.

PubMed

Căruntu, Bogdan; Bota, Constantin

2014-01-01

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

Numerical solution of the Saint-Venant equations by an efficient hybrid finite-volume/finite-difference method

NASA Astrophysics Data System (ADS)

Lai, Wencong; Khan, Abdul A.

2018-04-01

A computationally efficient hybrid finite-volume/finite-difference method is proposed for the numerical solution of Saint-Venant equations in one-dimensional open channel flows. The method adopts a mass-conservative finite volume discretization for the continuity equation and a semi-implicit finite difference discretization for the dynamic-wave momentum equation. The spatial discretization of the convective flux term in the momentum equation employs an upwind scheme and the water-surface gradient term is discretized using three different schemes. The performance of the numerical method is investigated in terms of efficiency and accuracy using various examples, including steady flow over a bump, dam-break flow over wet and dry downstream channels, wetting and drying in a parabolic bowl, and dam-break floods in laboratory physical models. Numerical solutions from the hybrid method are compared with solutions from a finite volume method along with analytic solutions or experimental measurements. Comparisons demonstrates that the hybrid method is efficient, accurate, and robust in modeling various flow scenarios, including subcritical, supercritical, and transcritical flows. In this method, the QUICK scheme for the surface slope discretization is more accurate and less diffusive than the center difference and the weighted average schemes.

Estimating and Interpreting Latent Variable Interactions: A Tutorial for Applying the Latent Moderated Structural Equations Method

ERIC Educational Resources Information Center

Maslowsky, Julie; Jager, Justin; Hemken, Douglas

2015-01-01

Latent variables are common in psychological research. Research questions involving the interaction of two variables are likewise quite common. Methods for estimating and interpreting interactions between latent variables within a structural equation modeling framework have recently become available. The latent moderated structural equations (LMS)…

A Model for the Oxidation of Carbon Silicon Carbide Composite Structures

NASA Technical Reports Server (NTRS)

Sullivan, Roy M.

2004-01-01

A mathematical theory and an accompanying numerical scheme have been developed for predicting the oxidation behavior of carbon silicon carbide (C/SiC) composite structures. The theory is derived from the mechanics of the flow of ideal gases through a porous solid. The result of the theoretical formulation is a set of two coupled nonlinear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations are solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of the spatial location and time. The local rate of carbon oxidation is determined using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The nonlinear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual method, allowing for the solution of the differential equations numerically. The numerical method is demonstrated by utilizing the method to model the carbon oxidation and weight loss behavior of C/SiC specimens during thermogravimetric experiments. The numerical method is used to study the physics of carbon oxidation in carbon silicon carbide composites.

APPLE - An aeroelastic analysis system for turbomachines and propfans

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Bakhle, Milind A.; Srivastava, R.; Mehmed, Oral

1992-01-01

This paper reviews aeroelastic analysis methods for propulsion elements (advanced propellers, compressors and turbines) being developed and used at NASA Lewis Research Center. These aeroelastic models include both structural and aerodynamic components. The structural models include the typical section model, the beam model with and without disk flexibility, and the finite element blade model with plate bending elements. The aerodynamic models are based on the solution of equations ranging from the two-dimensional linear potential equation for a cascade to the three-dimensional Euler equations for multi-blade configurations. Typical results are presented for each aeroelastic model. Suggestions for further research are indicated. All the available aeroelastic models and analysis methods are being incorporated into a unified computer program named APPLE (Aeroelasticity Program for Propulsion at LEwis).

A Review of Recent Aeroelastic Analysis Methods for Propulsion at NASA Lewis Research Center

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Bakhle, Milind A.; Srivastava, R.; Mehmed, Oral; Stefko, George L.

1993-01-01

This report reviews aeroelastic analyses for propulsion components (propfans, compressors and turbines) being developed and used at NASA LeRC. These aeroelastic analyses include both structural and aerodynamic models. The structural models include a typical section, a beam (with and without disk flexibility), and a finite-element blade model (with plate bending elements). The aerodynamic models are based on the solution of equations ranging from the two-dimensional linear potential equation to the three-dimensional Euler equations for multibladed configurations. Typical calculated results are presented for each aeroelastic model. Suggestions for further research are made. Many of the currently available aeroelastic models and analysis methods are being incorporated in a unified computer program, APPLE (Aeroelasticity Program for Propulsion at LEwis).

A Variational Assimilation Method for Satellite and Conventional Data: Development of Basic Model for Diagnosis of Cyclone Systems

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.; Scott, Robert W.; Chen, J.

1991-01-01

A summary is presented of the progress toward the completion of a comprehensive diagnostic objective analysis system based upon the calculus of variations. The approach was to first develop the objective analysis subject to the constraints that the final product satisfies the five basic primitive equations for a dry inviscid atmosphere: the two nonlinear horizontal momentum equations, the continuity equation, the hydrostatic equation, and the thermodynamic equation. Then, having derived the basic model, there would be added to it the equations for moist atmospheric processes and the radiative transfer equation.

BADGER v1.0: A Fortran equation of state library

NASA Astrophysics Data System (ADS)

Heltemes, T. A.; Moses, G. A.

2012-12-01

The BADGER equation of state library was developed to enable inertial confinement fusion plasma codes to more accurately model plasmas in the high-density, low-temperature regime. The code had the capability to calculate 1- and 2-T plasmas using the Thomas-Fermi model and an individual electron accounting model. Ion equation of state data can be calculated using an ideal gas model or via a quotidian equation of state with scaled binding energies. Electron equation of state data can be calculated via the ideal gas model or with an adaptation of the screened hydrogenic model with ℓ-splitting. The ionization and equation of state calculations can be done in local thermodynamic equilibrium or in a non-LTE mode using a variant of the Busquet equivalent temperature method. The code was written as a stand-alone Fortran library for ease of implementation by external codes. EOS results for aluminum are presented that show good agreement with the SESAME library and ionization calculations show good agreement with the FLYCHK code. Program summaryProgram title: BADGERLIB v1.0 Catalogue identifier: AEND_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEND_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 41 480 No. of bytes in distributed program, including test data, etc.: 2 904 451 Distribution format: tar.gz Programming language: Fortran 90. Computer: 32- or 64-bit PC, or Mac. Operating system: Windows, Linux, MacOS X. RAM: 249.496 kB plus 195.630 kB per isotope record in memory Classification: 19.1, 19.7. Nature of problem: Equation of State (EOS) calculations are necessary for the accurate simulation of high energy density plasmas. Historically, most EOS codes used in these simulations have relied on an ideal gas model. This model is inadequate for low-temperature, high-density plasma conditions; the gaseous and liquid phases; and the solid phase. The BADGER code was developed to give more realistic EOS data in these regimes. Solution method: BADGER has multiple, user-selectable models to treat the ions, average-atom ionization state and electrons. Ion models are ideal gas and quotidian equation of state (QEOS), ionization models are Thomas-Fermi and individual accounting method (IEM) formulation of the screened hydrogenic model (SHM) with l-splitting, electron ionization models are ideal gas and a Helmholtz free energy minimization method derived from the SHM. The default equation of state and ionization models are appropriate for plasmas in local thermodynamic equilibrium (LTE). The code can calculate non-LTE equation of state (EOS) and ionization data using a simplified form of the Busquet equivalent-temperature method. Restrictions: Physical data are only provided for elements Z=1 to Z=86. Multiple solid phases are not currently supported. Liquid, gas and plasma phases are combined into a generalized "fluid" phase. Unusual features: BADGER divorces the calculation of average-atom ionization from the electron equation of state model, allowing the user to select ionization and electron EOS models that are most appropriate to the simulation. The included ion ideal gas model uses ground-state nuclear spin data to differentiate between isotopes of a given element. Running time: Example provided only takes a few seconds to run.

Reconstruction de defauts a partir de donnees issues de capteurs a courants de foucault avec modele direct differentiel

NASA Astrophysics Data System (ADS)

Trillon, Adrien

Eddy current tomography can be employed to caracterize flaws in metal plates in steam generators of nuclear power plants. Our goal is to evaluate a map of the relative conductivity that represents the flaw. This nonlinear ill-posed problem is difficult to solve and a forward model is needed. First, we studied existing forward models to chose the one that is the most adapted to our case. Finite difference and finite element methods matched very good to our application. We adapted contrast source inversion (CSI) type methods to the chosen model and a new criterion was proposed. These methods are based on the minimization of the weighted errors of the model equations, coupling and observation. They allow an error on the equations. It appeared that reconstruction quality grows with the decay of the error on the coupling equation. We resorted to augmented Lagrangian techniques to constrain coupling equation and to avoid conditioning problems. In order to overcome the ill-posed character of the problem, prior information was introduced about the shape of the flaw and the values of the relative conductivity. Efficiency of the methods are illustrated with simulated flaws in 2D case.

Diffusion Coefficients from Molecular Dynamics Simulations in Binary and Ternary Mixtures

NASA Astrophysics Data System (ADS)

Liu, Xin; Schnell, Sondre K.; Simon, Jean-Marc; Krüger, Peter; Bedeaux, Dick; Kjelstrup, Signe; Bardow, André; Vlugt, Thijs J. H.

2013-07-01

Multicomponent diffusion in liquids is ubiquitous in (bio)chemical processes. It has gained considerable and increasing interest as it is often the rate limiting step in a process. In this paper, we review methods for calculating diffusion coefficients from molecular simulation and predictive engineering models. The main achievements of our research during the past years can be summarized as follows: (1) we introduced a consistent method for computing Fick diffusion coefficients using equilibrium molecular dynamics simulations; (2) we developed a multicomponent Darken equation for the description of the concentration dependence of Maxwell-Stefan diffusivities. In the case of infinite dilution, the multicomponent Darken equation provides an expression for [InlineEquation not available: see fulltext.] which can be used to parametrize the generalized Vignes equation; and (3) a predictive model for self-diffusivities was proposed for the parametrization of the multicomponent Darken equation. This equation accurately describes the concentration dependence of self-diffusivities in weakly associating systems. With these methods, a sound framework for the prediction of mutual diffusion in liquids is achieved.

MATLAB algorithm to implement soil water data assimilation with the Ensemble Kalman Filter using HYDRUS.

PubMed

Valdes-Abellan, Javier; Pachepsky, Yakov; Martinez, Gonzalo

2018-01-01

Data assimilation is becoming a promising technique in hydrologic modelling to update not only model states but also to infer model parameters, specifically to infer soil hydraulic properties in Richard-equation-based soil water models. The Ensemble Kalman Filter method is one of the most widely employed method among the different data assimilation alternatives. In this study the complete Matlab© code used to study soil data assimilation efficiency under different soil and climatic conditions is shown. The code shows the method how data assimilation through EnKF was implemented. Richards equation was solved by the used of Hydrus-1D software which was run from Matlab. •MATLAB routines are released to be used/modified without restrictions for other researchers•Data assimilation Ensemble Kalman Filter method code.•Soil water Richard equation flow solved by Hydrus-1D.

Prediction of fat-free body mass from bioelectrical impedance and anthropometry among 3-year-old children using DXA

PubMed Central

Ejlerskov, Katrine T.; Jensen, Signe M.; Christensen, Line B.; Ritz, Christian; Michaelsen, Kim F.; Mølgaard, Christian

2014-01-01

For 3-year-old children suitable methods to estimate body composition are sparse. We aimed to develop predictive equations for estimating fat-free mass (FFM) from bioelectrical impedance (BIA) and anthropometry using dual-energy X-ray absorptiometry (DXA) as reference method using data from 99 healthy 3-year-old Danish children. Predictive equations were derived from two multiple linear regression models, a comprehensive model (height2/resistance (RI), six anthropometric measurements) and a simple model (RI, height, weight). Their uncertainty was quantified by means of 10-fold cross-validation approach. Prediction error of FFM was 3.0% for both equations (root mean square error: 360 and 356 g, respectively). The derived equations produced BIA-based prediction of FFM and FM near DXA scan results. We suggest that the predictive equations can be applied in similar population samples aged 2–4 years. The derived equations may prove useful for studies linking body composition to early risk factors and early onset of obesity. PMID:24463487

Prediction of fat-free body mass from bioelectrical impedance and anthropometry among 3-year-old children using DXA.

PubMed

Ejlerskov, Katrine T; Jensen, Signe M; Christensen, Line B; Ritz, Christian; Michaelsen, Kim F; Mølgaard, Christian

2014-01-27

For 3-year-old children suitable methods to estimate body composition are sparse. We aimed to develop predictive equations for estimating fat-free mass (FFM) from bioelectrical impedance (BIA) and anthropometry using dual-energy X-ray absorptiometry (DXA) as reference method using data from 99 healthy 3-year-old Danish children. Predictive equations were derived from two multiple linear regression models, a comprehensive model (height(2)/resistance (RI), six anthropometric measurements) and a simple model (RI, height, weight). Their uncertainty was quantified by means of 10-fold cross-validation approach. Prediction error of FFM was 3.0% for both equations (root mean square error: 360 and 356 g, respectively). The derived equations produced BIA-based prediction of FFM and FM near DXA scan results. We suggest that the predictive equations can be applied in similar population samples aged 2-4 years. The derived equations may prove useful for studies linking body composition to early risk factors and early onset of obesity.

Hybrid finite element method for describing the electrical response of biological cells to applied fields.

PubMed

Ying, Wenjun; Henriquez, Craig S

2007-04-01

A novel hybrid finite element method (FEM) for modeling the response of passive and active biological membranes to external stimuli is presented. The method is based on the differential equations that describe the conservation of electric flux and membrane currents. By introducing the electric flux through the cell membrane as an additional variable, the algorithm decouples the linear partial differential equation part from the nonlinear ordinary differential equation part that defines the membrane dynamics of interest. This conveniently results in two subproblems: a linear interface problem and a nonlinear initial value problem. The linear interface problem is solved with a hybrid FEM. The initial value problem is integrated by a standard ordinary differential equation solver such as the Euler and Runge-Kutta methods. During time integration, these two subproblems are solved alternatively. The algorithm can be used to model the interaction of stimuli with multiple cells of almost arbitrary geometries and complex ion-channel gating at the plasma membrane. Numerical experiments are presented demonstrating the uses of the method for modeling field stimulation and action potential propagation.

Regularized Moment Equations and Shock Waves for Rarefied Granular Gas

NASA Astrophysics Data System (ADS)

Reddy, Lakshminarayana; Alam, Meheboob

2016-11-01

It is well-known that the shock structures predicted by extended hydrodynamic models are more accurate than the standard Navier-Stokes model in the rarefied regime, but they fail to predict continuous shock structures when the Mach number exceeds a critical value. Regularization or parabolization is one method to obtain smooth shock profiles at all Mach numbers. Following a Chapman-Enskog-like method, we have derived the "regularized" version 10-moment equations ("R10" moment equations) for inelastic hard-spheres. In order to show the advantage of R10 moment equations over standard 10-moment equations, the R10 moment equations have been employed to solve the Riemann problem of plane shock waves for both molecular and granular gases. The numerical results are compared between the 10-moment and R10-moment models and it is found that the 10-moment model fails to produce continuous shock structures beyond an upstream Mach number of 1 . 34 , while the R10-moment model predicts smooth shock profiles beyond the upstream Mach number of 1 . 34 . The density and granular temperature profiles are found to be asymmetric, with their maxima occurring within the shock-layer.

Oscillations and Rolling for Duffing's Equation

NASA Astrophysics Data System (ADS)

Aref'eva, I. Ya.; Piskovskiy, E. V.; Volovich, I. V.

2013-01-01

The Duffing equation has been used to model nonlinear dynamics not only in mechanics and electronics but also in biology and in neurology for the brain process modeling. Van der Pol's method is often used in nonlinear dynamics to improve perturbation theory results when describing small oscillations. However, in some other problems of nonlinear dynamics particularly in case of Duffing-Higgs equation in field theory, for the Einsten-Friedmann equations in cosmology and for relaxation processes in neurology not only small oscillations regime is of interest but also the regime of slow rolling. In the present work a method for approximate solution to nonlinear dynamics equations in the rolling regime is developed. It is shown that in order to improve perturbation theory in the rolling regime it turns out to be effective to use an expansion in hyperbolic functions instead of trigonometric functions as it is done in van der Pol's method in case of small oscillations. In particular the Duffing equation in the rolling regime is investigated using solution expressed in terms of elliptic functions. Accuracy of obtained approximation is estimated. The Duffing equation with dissipation is also considered.

Multi-scale diffuse interface modeling of multi-component two-phase flow with partial miscibility

NASA Astrophysics Data System (ADS)

Kou, Jisheng; Sun, Shuyu

2016-08-01

In this paper, we introduce a diffuse interface model to simulate multi-component two-phase flow with partial miscibility based on a realistic equation of state (e.g. Peng-Robinson equation of state). Because of partial miscibility, thermodynamic relations are used to model not only interfacial properties but also bulk properties, including density, composition, pressure, and realistic viscosity. As far as we know, this effort is the first time to use diffuse interface modeling based on equation of state for modeling of multi-component two-phase flow with partial miscibility. In numerical simulation, the key issue is to resolve the high contrast of scales from the microscopic interface composition to macroscale bulk fluid motion since the interface has a nanoscale thickness only. To efficiently solve this challenging problem, we develop a multi-scale simulation method. At the microscopic scale, we deduce a reduced interfacial equation under reasonable assumptions, and then we propose a formulation of capillary pressure, which is consistent with macroscale flow equations. Moreover, we show that Young-Laplace equation is an approximation of this capillarity formulation, and this formulation is also consistent with the concept of Tolman length, which is a correction of Young-Laplace equation. At the macroscopical scale, the interfaces are treated as discontinuous surfaces separating two phases of fluids. Our approach differs from conventional sharp-interface two-phase flow model in that we use the capillary pressure directly instead of a combination of surface tension and Young-Laplace equation because capillarity can be calculated from our proposed capillarity formulation. A compatible condition is also derived for the pressure in flow equations. Furthermore, based on the proposed capillarity formulation, we design an efficient numerical method for directly computing the capillary pressure between two fluids composed of multiple components. Finally, numerical tests are carried out to verify the effectiveness of the proposed multi-scale method.

Multi-scale diffuse interface modeling of multi-component two-phase flow with partial miscibility

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kou, Jisheng; Sun, Shuyu, E-mail: shuyu.sun@kaust.edu.sa; School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049

2016-08-01

In this paper, we introduce a diffuse interface model to simulate multi-component two-phase flow with partial miscibility based on a realistic equation of state (e.g. Peng–Robinson equation of state). Because of partial miscibility, thermodynamic relations are used to model not only interfacial properties but also bulk properties, including density, composition, pressure, and realistic viscosity. As far as we know, this effort is the first time to use diffuse interface modeling based on equation of state for modeling of multi-component two-phase flow with partial miscibility. In numerical simulation, the key issue is to resolve the high contrast of scales from themore » microscopic interface composition to macroscale bulk fluid motion since the interface has a nanoscale thickness only. To efficiently solve this challenging problem, we develop a multi-scale simulation method. At the microscopic scale, we deduce a reduced interfacial equation under reasonable assumptions, and then we propose a formulation of capillary pressure, which is consistent with macroscale flow equations. Moreover, we show that Young–Laplace equation is an approximation of this capillarity formulation, and this formulation is also consistent with the concept of Tolman length, which is a correction of Young–Laplace equation. At the macroscopical scale, the interfaces are treated as discontinuous surfaces separating two phases of fluids. Our approach differs from conventional sharp-interface two-phase flow model in that we use the capillary pressure directly instead of a combination of surface tension and Young–Laplace equation because capillarity can be calculated from our proposed capillarity formulation. A compatible condition is also derived for the pressure in flow equations. Furthermore, based on the proposed capillarity formulation, we design an efficient numerical method for directly computing the capillary pressure between two fluids composed of multiple components. Finally, numerical tests are carried out to verify the effectiveness of the proposed multi-scale method.« less

Dispersive traveling wave solutions of the Equal-Width and Modified Equal-Width equations via mathematical methods and its applications

NASA Astrophysics Data System (ADS)

Lu, Dianchen; Seadawy, Aly R.; Ali, Asghar

2018-06-01

The Equal-Width and Modified Equal-Width equations are used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. In this article we have employed extend simple equation method and the exp(-varphi(ξ)) expansion method to construct the exact traveling wave solutions of equal width and modified equal width equations. The obtained results are novel and have numerous applications in current areas of research in mathematical physics. It is exposed that our method, with the help of symbolic computation, provides a effective and powerful mathematical tool for solving different kind nonlinear wave problems.

Constitutive Modelling of Resins in the Stiffness Domain

NASA Astrophysics Data System (ADS)

Klasztorny, M.

2004-09-01

An analytic method for inverting the constitutive compliance equations of viscoelasticity for resins is developed. These equations describe the HWKK/H rheological model, which makes it possible to simulate, with a good accuracy, short-, medium- and long-term viscoelastic processes in epoxy and polyester resins. These processes are of first-rank reversible isothermal type. The time histories of deviatoric stresses are simulated with three independent strain history functions of fractional and normal exponential types. The stiffness equations are described by two elastic and six viscoelastic constants having a clear physic meaning (three long-term relaxation coefficients and three relaxation times). The time histories of axiatoric stresses are simulated as perfectly elastic. The inversion method utilizes approximate constitutive stiffness equations of viscoelasticity for the HWKK/H model. The constitutive compliance equations for the model are a basis for determining the exact complex shear stiffness, whereas the approximate constitutive stiffness equations are used for determining the approximate complex shear stiffness. The viscoelastic constants in the stiffness domain are derived by equating the exact and approximate complex shear stiffnesses. The viscoelastic constants are obtained for Epidian 53 epoxy and Polimal 109 polyester resins. The accuracy of the approximate constitutive stiffness equations are assessed by comparing the approximate and exact complex shear stiffnesses. The constitutive stiffness equations for the HWKK/H model are presented in uncoupled (shear/bulk) and coupled forms. Formulae for converting the constants of shear viscoelasticity into the constants of coupled viscoelasticity are given as well.

Water-budget methods

USGS Publications Warehouse

Healy, Richard W.; Scanlon, Bridget R.

2010-01-01

A water budget is an accounting of water movement into and out of, and storage change within, some control volume. Universal and adaptable are adjectives that reflect key features of water-budget methods for estimating recharge. The universal concept of mass conservation of water implies that water-budget methods are applicable over any space and time scales (Healy et al., 2007). The water budget of a soil column in a laboratory can be studied at scales of millimeters and seconds. A water-budget equation is also an integral component of atmospheric general circulation models used to predict global climates over periods of decades or more. Water-budget equations can be easily customized by adding or removing terms to accurately portray the peculiarities of any hydrologic system. The equations are generally not bound by assumptions on mechanisms by which water moves into, through, and out of the control volume of interest. So water-budget methods can be used to estimate both diffuse and focused recharge, and recharge estimates are unaffected by phenomena such as preferential flow paths within the unsaturated zone.Water-budget methods represent the largest class of techniques for estimating recharge. Most hydrologic models are derived from a water-budget equation and can therefore be classified as water-budget models. It is not feasible to address all water-budget methods in a single chapter. This chapter is limited to discussion of the “residual” water-budget approach, whereby all variables in a water-budget equation, except for recharge, are independently measured or estimated and recharge is set equal to the residual. This chapter is closely linked with Chapter 3, on modeling methods, because the equations presented here form the basis of many models and because models are often used to estimate individual components in water-budget studies. Water budgets for streams and other surface-water bodies are addressed in Chapter 4. The use of soil-water budgets and lysimeters for determining potential recharge and evapotranspiration from changes in water storage is discussed in Chapter 5. Aquifer water-budget methods based on the measurement of groundwater levels are described in Chapter 6.

FDDO and DSMC analyses of rarefied gas flow through 2D nozzles

NASA Technical Reports Server (NTRS)

Chung, Chan-Hong; De Witt, Kenneth J.; Jeng, Duen-Ren; Penko, Paul F.

1992-01-01

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas expanding through a two-dimensional nozzle and into a surrounding low-density environment. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, the variable hard sphere model is used as a molecular model and the no time counter method is employed as a collision sampling technique. The results of both the FDDO and the DSMC methods show good agreement. The FDDO method requires less computational effort than the DSMC method by factors of 10 to 40 in CPU time, depending on the degree of rarefaction.

Lattice Boltzmann model for simulation of magnetohydrodynamics

NASA Technical Reports Server (NTRS)

Chen, Shiyi; Chen, Hudong; Martinez, Daniel; Matthaeus, William

1991-01-01

A numerical method, based on a discrete Boltzmann equation, is presented for solving the equations of magnetohydrodynamics (MHD). The algorithm provides advantages similar to the cellular automaton method in that it is local and easily adapted to parallel computing environments. Because of much lower noise levels and less stringent requirements on lattice size, the method appears to be more competitive with traditional solution methods. Examples show that the model accurately reproduces both linear and nonlinear MHD phenomena.

Meshless collocation methods for the numerical solution of elliptic boundary valued problems the rotational shallow water equations on the sphere

NASA Astrophysics Data System (ADS)

Blakely, Christopher D.

This dissertation thesis has three main goals: (1) To explore the anatomy of meshless collocation approximation methods that have recently gained attention in the numerical analysis community; (2) Numerically demonstrate why the meshless collocation method should clearly become an attractive alternative to standard finite-element methods due to the simplicity of its implementation and its high-order convergence properties; (3) Propose a meshless collocation method for large scale computational geophysical fluid dynamics models. We provide numerical verification and validation of the meshless collocation scheme applied to the rotational shallow-water equations on the sphere and demonstrate computationally that the proposed model can compete with existing high performance methods for approximating the shallow-water equations such as the SEAM (spectral-element atmospheric model) developed at NCAR. A detailed analysis of the parallel implementation of the model, along with the introduction of parallel algorithmic routines for the high-performance simulation of the model will be given. We analyze the programming and computational aspects of the model using Fortran 90 and the message passing interface (mpi) library along with software and hardware specifications and performance tests. Details from many aspects of the implementation in regards to performance, optimization, and stabilization will be given. In order to verify the mathematical correctness of the algorithms presented and to validate the performance of the meshless collocation shallow-water model, we conclude the thesis with numerical experiments on some standardized test cases for the shallow-water equations on the sphere using the proposed method.

Turbulence modeling for hypersonic flows

NASA Technical Reports Server (NTRS)

Marvin, J. G.; Coakley, T. J.

1989-01-01

Turbulence modeling for high speed compressible flows is described and discussed. Starting with the compressible Navier-Stokes equations, methods of statistical averaging are described by means of which the Reynolds-averaged Navier-Stokes equations are developed. Unknown averages in these equations are approximated using various closure concepts. Zero-, one-, and two-equation eddy viscosity models, algebraic stress models and Reynolds stress transport models are discussed. Computations of supersonic and hypersonic flows obtained using several of the models are discussed and compared with experimental results. Specific examples include attached boundary layer flows, shock wave boundary layer interactions and compressible shear layers. From these examples, conclusions regarding the status of modeling and recommendations for future studies are discussed.

Constructive methods of invariant manifolds for kinetic problems

NASA Astrophysics Data System (ADS)

Gorban, Alexander N.; Karlin, Iliya V.; Zinovyev, Andrei Yu.

2004-06-01

The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space ( the invariance equation). The equation of motion for immersed manifolds is obtained ( the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability. A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions. The following examples of applications are presented: nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn∼1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, etc.

Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry.

PubMed

Tang, Chen; Lu, Wenjing; Chen, Song; Zhang, Zhen; Li, Botao; Wang, Wenping; Han, Lin

2007-10-20

We extend and refine previous work [Appl. Opt. 46, 2907 (2007)]. Combining the coupled nonlinear partial differential equations (PDEs) denoising model with the ordinary differential equations enhancement method, we propose the new denoising and enhancing model for electronic speckle pattern interferometry (ESPI) fringe patterns. Meanwhile, we propose the backpropagation neural networks (BPNN) method to obtain unwrapped phase values based on a skeleton map instead of traditional interpolations. We test the introduced methods on the computer-simulated speckle ESPI fringe patterns and experimentally obtained fringe pattern, respectively. The experimental results show that the coupled nonlinear PDEs denoising model is capable of effectively removing noise, and the unwrapped phase values obtained by the BPNN method are much more accurate than those obtained by the well-known traditional interpolation. In addition, the accuracy of the BPNN method is adjustable by changing the parameters of networks such as the number of neurons.

Numerical Analysis of a Class of THM Coupled Model for Porous Materials

NASA Astrophysics Data System (ADS)

Liu, Tangwei; Zhou, Jingying; Lu, Hongzhi

2018-01-01

We consider the coupled models of the Thermo-hydro-mechanical (THM) problem for porous materials which arises in many engineering applications. Firstly, mathematical models of the THM coupled problem for porous materials were discussed. Secondly, for different cases, some numerical difference schemes of coupled model were constructed, respectively. Finally, aassuming that the original water vapour effect is neglectable and that the volume fraction of liquid phase and the solid phase are constants, the nonlinear equations can be reduced to linear equations. The discrete equations corresponding to the linear equations were solved by the Arnodli method.

Tidal, Residual, Intertidal Mudflat (TRIM) Model and its Applications to San Francisco Bay, California

USGS Publications Warehouse

Cheng, R.T.; Casulli, V.; Gartner, J.W.

1993-01-01

A numerical model using a semi-implicit finite-difference method for solving the two-dimensional shallow-water equations is presented. The gradient of the water surface elevation in the momentum equations and the velocity divergence in the continuity equation are finite-differenced implicitly, the remaining terms are finite-differenced explicitly. The convective terms are treated using an Eulerian-Lagrangian method. The combination of the semi-implicit finite-difference solution for the gravity wave propagation, and the Eulerian-Lagrangian treatment of the convective terms renders the numerical model unconditionally stable. When the baroclinic forcing is included, a salt transport equation is coupled to the momentum equations, and the numerical method is subject to a weak stability condition. The method of solution and the properties of the numerical model are given. This numerical model is particularly suitable for applications to coastal plain estuaries and tidal embayments in which tidal currents are dominant, and tidally generated residual currents are important. The model is applied to San Francisco Bay, California where extensive historical tides and current-meter data are available. The model calibration is considered by comparing time-series of the field data and of the model results. Alternatively, and perhaps more meaningfully, the model is calibrated by comparing the harmonic constants of tides and tidal currents derived from field data with those derived from the model. The model is further verified by comparing the model results with an independent data set representing the wet season. The strengths and the weaknesses of the model are assessed based on the results of model calibration and verification. Using the model results, the properties of tides and tidal currents in San Francisco Bay are characterized and discussed. Furthermore, using the numerical model, estimates of San Francisco Bay's volume, surface area, mean water depth, tidal prisms, and tidal excursions at spring and neap tides are computed. Additional applications of the model reveal, qualitatively the spatial distribution of residual variables. ?? 1993 Academic Press. All rights reserved.

Development of Semi-Empirical Damping Equation for Baffled Tank with Oblate Spheroidal Dome

NASA Technical Reports Server (NTRS)

Yang, H. Q.; West, Jeff; Brodnick, Jacob; Eberhart, Chad

2016-01-01

Propellant slosh is a potential source of disturbance that can significantly impact the stability of space vehicles. The slosh dynamics are typically represented by a mechanical model of a spring-mass-damper. This mechanical model is then included in the equation of motion of the entire vehicle for Guidance, Navigation and Control analysis. The typical parameters required by the mechanical model include natural frequency of the slosh, slosh mass, slosh mass center location, and the critical damping ratio. A fundamental study has been undertaken at NASA MSFC to understand the fluid damping physics from a ring baffle in the barrel section of a propellant tank. An asymptotic damping equation and CFD blended equation have been derived by NASA MSFC team to complement the popularly used Miles equation at different flow regimes. The new development has found success in providing a nonlinear damping model for the Space Launch System. The purpose of this study is to further extend the semi-empirical damping equations into the oblate spheroidal dome section of the propellant tanks. First, previous experimental data from the spherical baffled tank are collected and analyzed. Several methods of taking the dome curvature effect, including a generalized Miles equation, area projection method, and equalized fill height method, are assessed. CFD simulation is used to shed light on the interaction of vorticity around the baffle with the locally curved wall and liquid-gas interface. The final damping equation will be validated by a recent subscale test with an oblate spheroidal dome conducted at NASA MSFC.

Thermal constitutive matrix applied to asynchronous electrical machine using the cell method

NASA Astrophysics Data System (ADS)

Domínguez, Pablo Ignacio González; Monzón-Verona, José Miguel; Rodríguez, Leopoldo Simón; Sánchez, Adrián de Pablo

2018-03-01

This work demonstrates the equivalence of two constitutive equations. One is used in Fourier's law of the heat conduction equation, the other in electric conduction equation; both are based on the numerical Cell Method, using the Finite Formulation (FF-CM). A 3-D pure heat conduction model is proposed. The temperatures are in steady state and there are no internal heat sources. The obtained results are compared with an equivalent model developed using the Finite Elements Method (FEM). The particular case of 2-D was also studied. The errors produced are not significant at less than 0.2%. The number of nodes is the number of the unknowns and equations to resolve. There is no significant gain in precision with increasing density of the mesh.

An efficient numerical method for solving the Boltzmann equation in multidimensions

NASA Astrophysics Data System (ADS)

Dimarco, Giacomo; Loubère, Raphaël; Narski, Jacek; Rey, Thomas

2018-01-01

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26]) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3 D × 3 D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.

New Equating Methods and Their Relationships with Levine Observed Score Linear Equating under the Kernel Equating Framework

ERIC Educational Resources Information Center

Chen, Haiwen; Holland, Paul

2010-01-01

In this paper, we develop a new curvilinear equating for the nonequivalent groups with anchor test (NEAT) design under the assumption of the classical test theory model, that we name curvilinear Levine observed score equating. In fact, by applying both the kernel equating framework and the mean preserving linear transformation of…

Estimating, Testing, and Comparing Specific Effects in Structural Equation Models: The Phantom Model Approach

ERIC Educational Resources Information Center

Macho, Siegfried; Ledermann, Thomas

2011-01-01

The phantom model approach for estimating, testing, and comparing specific effects within structural equation models (SEMs) is presented. The rationale underlying this novel method consists in representing the specific effect to be assessed as a total effect within a separate latent variable model, the phantom model that is added to the main…

Reduced modeling of signal transduction – a modular approach

PubMed Central

Koschorreck, Markus; Conzelmann, Holger; Ebert, Sybille; Ederer, Michael; Gilles, Ernst Dieter

2007-01-01

Background Combinatorial complexity is a challenging problem in detailed and mechanistic mathematical modeling of signal transduction. This subject has been discussed intensively and a lot of progress has been made within the last few years. A software tool (BioNetGen) was developed which allows an automatic rule-based set-up of mechanistic model equations. In many cases these models can be reduced by an exact domain-oriented lumping technique. However, the resulting models can still consist of a very large number of differential equations. Results We introduce a new reduction technique, which allows building modularized and highly reduced models. Compared to existing approaches further reduction of signal transduction networks is possible. The method also provides a new modularization criterion, which allows to dissect the model into smaller modules that are called layers and can be modeled independently. Hallmarks of the approach are conservation relations within each layer and connection of layers by signal flows instead of mass flows. The reduced model can be formulated directly without previous generation of detailed model equations. It can be understood and interpreted intuitively, as model variables are macroscopic quantities that are converted by rates following simple kinetics. The proposed technique is applicable without using complex mathematical tools and even without detailed knowledge of the mathematical background. However, we provide a detailed mathematical analysis to show performance and limitations of the method. For physiologically relevant parameter domains the transient as well as the stationary errors caused by the reduction are negligible. Conclusion The new layer based reduced modeling method allows building modularized and strongly reduced models of signal transduction networks. Reduced model equations can be directly formulated and are intuitively interpretable. Additionally, the method provides very good approximations especially for macroscopic variables. It can be combined with existing reduction methods without any difficulties. PMID:17854494

Equation-free modeling unravels the behavior of complex ecological systems

USGS Publications Warehouse

DeAngelis, Donald L.; Yurek, Simeon

2015-01-01

Ye et al. (1) address a critical problem confronting the management of natural ecosystems: How can we make forecasts of possible future changes in populations to help guide management actions? This problem is especially acute for marine and anadromous fisheries, where the large interannual fluctuations of populations, arising from complex nonlinear interactions among species and with varying environmental factors, have defied prediction over even short time scales. The empirical dynamic modeling (EDM) described in Ye et al.’s report, the latest in a series of papers by Sugihara and his colleagues, offers a promising quantitative approach to building models using time series to successfully project dynamics into the future. With the term “equation-free” in the article title, Ye et al. (1) are suggesting broader implications of their approach, considering the centrality of equations in modern science. From the 1700s on, nature has been increasingly described by mathematical equations, with differential or difference equations forming the basic framework for describing dynamics. The use of mathematical equations for ecological systems came much later, pioneered by Lotka and Volterra, who showed that population cycles might be described in terms of simple coupled nonlinear differential equations. It took decades for Lotka–Volterra-type models to become established, but the development of appropriate differential equations is now routine in modeling ecological dynamics. There is no question that the injection of mathematical equations, by forcing “clarity and precision into conjecture” (2), has led to increased understanding of population and community dynamics. As in science in general, in ecology equations are a key method of communication and of framing hypotheses. These equations serve as compact representations of an enormous amount of empirical data and can be analyzed by the powerful methods of mathematics.

A near-wall turbulence model and its application to fully developed turbulent channel and pipe flows

NASA Technical Reports Server (NTRS)

Kim, S.-W.

1988-01-01

A near wall turbulence model and its incorporation into a multiple-time-scale turbulence model are presented. In the method, the conservation of mass, momentum, and the turbulent kinetic energy equations are integrated up to the wall; and the energy transfer rate and the dissipation rate inside the near wall layer are obtained from algebraic equations. The algebraic equations for the energy transfer rate and the dissipation rate inside the near wall layer were obtained from a k-equation turbulence model and the near wall analysis. A fully developed turbulent channel flow and fully developed turbulent pipe flows were solved using a finite element method to test the predictive capability of the turbulence model. The computational results compared favorably with experimental data. It is also shown that the present turbulence model could resolve the over shoot phenomena of the turbulent kinetic energy and the dissipation rate in the region very close to the wall.

Differential Equations Models to Study Quorum Sensing.

PubMed

Pérez-Velázquez, Judith; Hense, Burkhard A

2018-01-01

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Near-wall turbulence model and its application to fully developed turbulent channel and pipe flows

NASA Technical Reports Server (NTRS)

Kim, S.-W.

1990-01-01

A near-wall turbulence model and its incorporation into a multiple-timescale turbulence model are presented. The near-wall turbulence model is obtained from a k-equation turbulence model and a near-wall analysis. In the method, the equations for the conservation of mass, momentum, and turbulent kinetic energy are integrated up to the wall, and the energy transfer and the dissipation rates inside the near-wall layer are obtained from algebraic equations. Fully developed turbulent channel and pipe flows are solved using a finite element method. The computational results compare favorably with experimental data. It is also shown that the turbulence model can resolve the overshoot phenomena of the turbulent kinetic energy and the dissipation rate in the region very close to the wall.

Stability analysis of a liquid fuel annular combustion chamber. M.S. Thesis

NASA Technical Reports Server (NTRS)

Mcdonald, G. H.

1978-01-01

High frequency combustion instability problems in a liquid fuel annular combustion chamber are examined. A modified Galerkin method was used to produce a set of modal amplitude equations from the general nonlinear partial differential acoustic wave equation in order to analyze the problem of instability. From these modal amplitude equations, the two variable perturbation method was used to develop a set of approximate equations of a given order of magnitude. These equations were modeled to show the effects of velocity sensitive combustion instabilities by evaluating the effects of certain parameters in the given set of equations.

Using Bond Graphs for Articulated, Flexible Multi-bodies, Sensors, Actuators, and Controllers with Application to the International Space Station

NASA Technical Reports Server (NTRS)

Montgomery, Raymond C.; Granda, Jose J.

2003-01-01

Conceptually, modeling of flexible, multi-body systems involves a formulation as a set of time-dependent partial differential equations. However, for practical, engineering purposes, this modeling is usually done using the method of Finite Elements, which approximates the set of partial differential equations, thus generalizing the approach to all continuous media. This research investigates the links between the Bond Graph method and the classical methods used to develop system models and advocates the Bond Graph Methodology and current bond graph tools as alternate approaches that will lead to a quick and precise understanding of a flexible multi-body system under automatic control. For long endurance, complex spacecraft, because of articulation and mission evolution the model of the physical system may change frequently. So a method of automatic generation and regeneration of system models that does not lead to implicit equations, as does the Lagrange equation approach, is desirable. The bond graph method has been shown to be amenable to automatic generation of equations with appropriate consideration of causality. Indeed human-interactive software now exists that automatically generates both symbolic and numeric system models and evaluates causality as the user develops the model, e.g. the CAMP-G software package. In this paper the CAMP-G package is used to generate a bond graph model of the International Space Station (ISS) at an early stage in its assembly, Zvezda. The ISS is an ideal example because it is a collection of bodies that are articulated, many of which are highly flexible. Also many reaction jets are used to control translation and attitude, and many electric motors are used to articulate appendages, which consist of photovoltaic arrays and composite assemblies. The Zvezda bond graph model is compared to an existing model, which was generated by the NASA Johnson Space Center during the Verification and Analysis Cycle of Zvezda.

Modelling biochemical reaction systems by stochastic differential equations with reflection.

PubMed

Niu, Yuanling; Burrage, Kevin; Chen, Luonan

2016-05-07

In this paper, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain D, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain D is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain D, and if not perform an orthogonal projection. It is found that the projection onto the closure D¯ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach. Copyright © 2016 Elsevier Ltd. All rights reserved.

A linearized Euler analysis of unsteady flows in turbomachinery

NASA Technical Reports Server (NTRS)

Hall, Kenneth C.; Crawley, Edward F.

1987-01-01

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

The Kadomtsev{endash}Petviashvili equation as a source of integrable model equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maccari, A.

1996-12-01

A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev{endash}Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev{endash}Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs. {copyright} {ital 1996 American Institute of Physics.}

Model-based control strategies for systems with constraints of the program type

NASA Astrophysics Data System (ADS)

Jarzębowska, Elżbieta

2006-08-01

The paper presents a model-based tracking control strategy for constrained mechanical systems. Constraints we consider can be material and non-material ones referred to as program constraints. The program constraint equations represent tasks put upon system motions and they can be differential equations of orders higher than one or two, and be non-integrable. The tracking control strategy relies upon two dynamic models: a reference model, which is a dynamic model of a system with arbitrary order differential constraints and a dynamic control model. The reference model serves as a motion planner, which generates inputs to the dynamic control model. It is based upon a generalized program motion equations (GPME) method. The method enables to combine material and program constraints and merge them both into the motion equations. Lagrange's equations with multipliers are the peculiar case of the GPME, since they can be applied to systems with constraints of first orders. Our tracking strategy referred to as a model reference program motion tracking control strategy enables tracking of any program motion predefined by the program constraints. It extends the "trajectory tracking" to the "program motion tracking". We also demonstrate that our tracking strategy can be extended to a hybrid program motion/force tracking.

Approximating a retarded-advanced differential equation that models human phonation

NASA Astrophysics Data System (ADS)

Teodoro, M. Filomena

2017-11-01

In [1, 2, 3] we have got the numerical solution of a linear mixed type functional differential equation (MTFDE) introduced initially in [4], considering the autonomous and non-autonomous case by collocation, least squares and finite element methods considering B-splines basis set. The present work introduces a numerical scheme using least squares method (LSM) and Gaussian basis functions to solve numerically a nonlinear mixed type equation with symmetric delay and advance which models human phonation. The preliminary results are promising. We obtain an accuracy comparable with the previous results.

New optical solitons of space-time conformable fractional perturbed Gerdjikov-Ivanov equation by sine-Gordon equation method

NASA Astrophysics Data System (ADS)

Yaşar, Elif; Yıldırım, Yakup; Yaşar, Emrullah

2018-06-01

This paper devotes to conformable fractional space-time perturbed Gerdjikov-Ivanov (GI) equation which appears in nonlinear fiber optics and photonic crystal fibers (PCF). We consider the model with full nonlinearity in order to give a generalized flavor. The sine-Gordon equation approach is carried out to model equation for retrieving the dark, bright, dark-bright, singular and combined singular optical solitons. The constraint conditions are also reported for guaranteeing the existence of these solitons. We also present some graphical simulations of the solutions for better understanding the physical phenomena of the behind the considered model.

Dimension reduction method for SPH equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tartakovsky, Alexandre M.; Scheibe, Timothy D.

2011-08-26

Smoothed Particle Hydrodynamics model of a complex multiscale processe often results in a system of ODEs with an enormous number of unknowns. Furthermore, a time integration of the SPH equations usually requires time steps that are smaller than the observation time by many orders of magnitude. A direct solution of these ODEs can be extremely expensive. Here we propose a novel dimension reduction method that gives an approximate solution of the SPH ODEs and provides an accurate prediction of the average behavior of the modeled system. The method consists of two main elements. First, effective equationss for evolution of averagemore » variables (e.g. average velocity, concentration and mass of a mineral precipitate) are obtained by averaging the SPH ODEs over the entire computational domain. These effective ODEs contain non-local terms in the form of volume integrals of functions of the SPH variables. Second, a computational closure is used to close the system of the effective equations. The computational closure is achieved via short bursts of the SPH model. The dimension reduction model is used to simulate flow and transport with mixing controlled reactions and mineral precipitation. An SPH model is used model transport at the porescale. Good agreement between direct solutions of the SPH equations and solutions obtained with the dimension reduction method for different boundary conditions confirms the accuracy and computational efficiency of the dimension reduction model. The method significantly accelerates SPH simulations, while providing accurate approximation of the solution and accurate prediction of the average behavior of the system.« less

Dispersive optical soliton solutions for higher order nonlinear Sasa-Satsuma equation in mono mode fibers via new auxiliary equation method

NASA Astrophysics Data System (ADS)

Khater, Mostafa M. A.; Seadawy, Aly R.; Lu, Dianchen

2018-01-01

In this research, we apply new technique for higher order nonlinear Schrödinger equation which is representing the propagation of short light pulses in the monomode optical fibers and the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Nonlinear Schrödinger equation is one of the basic model in fiber optics. We apply new auxiliary equation method for nonlinear Sasa-Satsuma equation to obtain a new optical forms of solitary traveling wave solutions. Exact and solitary traveling wave solutions are obtained in different kinds like trigonometric, hyperbolic, exponential, rational functions, …, etc. These forms of solutions that we represent in this research prove the superiority of our new technique on almost thirteen powerful methods. The main merits of this method over the other methods are that it gives more general solutions with some free parameters.

Topological soliton solutions for three shallow water waves models

NASA Astrophysics Data System (ADS)

Liu, Jiangen; Zhang, Yufeng; Wang, Yan

2018-07-01

In this article, we investigate three distinct physical structures for shallow water waves models by the improved ansatz method. The method was improved and can be used to obtain more generalized form topological soliton solutions than the original method. As a result, some new exact solutions of the shallow water equations are successfully established and the obtained results are exhibited graphically. The results showed that the improved ansatz method can be applied to solve other nonlinear differential equations arising from mathematical physics.

OpenMP performance for benchmark 2D shallow water equations using LBM

NASA Astrophysics Data System (ADS)

Sabri, Khairul; Rabbani, Hasbi; Gunawan, Putu Harry

2018-03-01

Shallow water equations or commonly referred as Saint-Venant equations are used to model fluid phenomena. These equations can be solved numerically using several methods, like Lattice Boltzmann method (LBM), SIMPLE-like Method, Finite Difference Method, Godunov-type Method, and Finite Volume Method. In this paper, the shallow water equation will be approximated using LBM or known as LABSWE and will be simulated in performance of parallel programming using OpenMP. To evaluate the performance between 2 and 4 threads parallel algorithm, ten various number of grids Lx and Ly are elaborated. The results show that using OpenMP platform, the computational time for solving LABSWE can be decreased. For instance using grid sizes 1000 × 500, the speedup of 2 and 4 threads is observed 93.54 s and 333.243 s respectively.

A highly accurate finite-difference method with minimum dispersion error for solving the Helmholtz equation

NASA Astrophysics Data System (ADS)

Wu, Zedong; Alkhalifah, Tariq

2018-07-01

Numerical simulation of the acoustic wave equation in either isotropic or anisotropic media is crucial to seismic modeling, imaging and inversion. Actually, it represents the core computation cost of these highly advanced seismic processing methods. However, the conventional finite-difference method suffers from severe numerical dispersion errors and S-wave artifacts when solving the acoustic wave equation for anisotropic media. We propose a method to obtain the finite-difference coefficients by comparing its numerical dispersion with the exact form. We find the optimal finite difference coefficients that share the dispersion characteristics of the exact equation with minimal dispersion error. The method is extended to solve the acoustic wave equation in transversely isotropic (TI) media without S-wave artifacts. Numerical examples show that the method is highly accurate and efficient.

Maximum Likelihood Methods in Treating Outliers and Symmetrically Heavy-Tailed Distributions for Nonlinear Structural Equation Models with Missing Data

ERIC Educational Resources Information Center

Lee, Sik-Yum; Xia, Ye-Mao

2006-01-01

By means of more than a dozen user friendly packages, structural equation models (SEMs) are widely used in behavioral, education, social, and psychological research. As the underlying theory and methods in these packages are vulnerable to outliers and distributions with longer-than-normal tails, a fundamental problem in the field is the…

Iterative Methods to Solve Linear RF Fields in Hot Plasma

NASA Astrophysics Data System (ADS)

Spencer, Joseph; Svidzinski, Vladimir; Evstatiev, Evstati; Galkin, Sergei; Kim, Jin-Soo

2014-10-01

Most magnetic plasma confinement devices use radio frequency (RF) waves for current drive and/or heating. Numerical modeling of RF fields is an important part of performance analysis of such devices and a predictive tool aiding design and development of future devices. Prior attempts at this modeling have mostly used direct solvers to solve the formulated linear equations. Full wave modeling of RF fields in hot plasma with 3D nonuniformities is mostly prohibited, with memory demands of a direct solver placing a significant limitation on spatial resolution. Iterative methods can significantly increase spatial resolution. We explore the feasibility of using iterative methods in 3D full wave modeling. The linear wave equation is formulated using two approaches: for cold plasmas the local cold plasma dielectric tensor is used (resolving resonances by particle collisions), while for hot plasmas the conductivity kernel (which includes a nonlocal dielectric response) is calculated by integrating along test particle orbits. The wave equation is discretized using a finite difference approach. The initial guess is important in iterative methods, and we examine different initial guesses including the solution to the cold plasma wave equation. Work is supported by the U.S. DOE SBIR program.

Off-policy reinforcement learning for H∞ control design.

PubMed

Luo, Biao; Wu, Huai-Ning; Huang, Tingwen

2015-01-01

The H∞ control design problem is considered for nonlinear systems with unknown internal system model. It is known that the nonlinear H∞ control problem can be transformed into solving the so-called Hamilton-Jacobi-Isaacs (HJI) equation, which is a nonlinear partial differential equation that is generally impossible to be solved analytically. Even worse, model-based approaches cannot be used for approximately solving HJI equation, when the accurate system model is unavailable or costly to obtain in practice. To overcome these difficulties, an off-policy reinforcement leaning (RL) method is introduced to learn the solution of HJI equation from real system data instead of mathematical system model, and its convergence is proved. In the off-policy RL method, the system data can be generated with arbitrary policies rather than the evaluating policy, which is extremely important and promising for practical systems. For implementation purpose, a neural network (NN)-based actor-critic structure is employed and a least-square NN weight update algorithm is derived based on the method of weighted residuals. Finally, the developed NN-based off-policy RL method is tested on a linear F16 aircraft plant, and further applied to a rotational/translational actuator system.

Hermite Functional Link Neural Network for Solving the Van der Pol-Duffing Oscillator Equation.

PubMed

Mall, Susmita; Chakraverty, S

2016-08-01

Hermite polynomial-based functional link artificial neural network (FLANN) is proposed here to solve the Van der Pol-Duffing oscillator equation. A single-layer hermite neural network (HeNN) model is used, where a hidden layer is replaced by expansion block of input pattern using Hermite orthogonal polynomials. A feedforward neural network model with the unsupervised error backpropagation principle is used for modifying the network parameters and minimizing the computed error function. The Van der Pol-Duffing and Duffing oscillator equations may not be solved exactly. Here, approximate solutions of these types of equations have been obtained by applying the HeNN model for the first time. Three mathematical example problems and two real-life application problems of Van der Pol-Duffing oscillator equation, extracting the features of early mechanical failure signal and weak signal detection problems, are solved using the proposed HeNN method. HeNN approximate solutions have been compared with results obtained by the well known Runge-Kutta method. Computed results are depicted in term of graphs. After training the HeNN model, we may use it as a black box to get numerical results at any arbitrary point in the domain. Thus, the proposed HeNN method is efficient. The results reveal that this method is reliable and can be applied to other nonlinear problems too.

Analytical Dynamics and Nonrigid Spacecraft Simulation

NASA Technical Reports Server (NTRS)

Likins, P. W.

1974-01-01

Application to the simulation of idealized spacecraft are considered both for multiple-rigid-body models and for models consisting of combination of rigid bodies and elastic bodies, with the elastic bodies being defined either as continua, as finite-element systems, or as a collection of given modal data. Several specific examples are developed in detail by alternative methods of analytical mechanics, and results are compared to a Newton-Euler formulation. The following methods are developed from d'Alembert's principle in vector form: (1) Lagrange's form of d'Alembert's principle for independent generalized coordinates; (2) Lagrange's form of d'Alembert's principle for simply constrained systems; (3) Kane's quasi-coordinate formulation of D'Alembert's principle; (4) Lagrange's equations for independent generalized coordinates; (5) Lagrange's equations for simply constrained systems; (6) Lagrangian quasi-coordinate equations (or the Boltzmann-Hamel equations); (7) Hamilton's equations for simply constrained systems; and (8) Hamilton's equations for independent generalized coordinates.

Modeling and simulation of ocean wave propagation using lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Nuraiman, Dian

2017-10-01

In this paper, we present on modeling and simulation of ocean wave propagation from the deep sea to the shoreline. This requires high computational cost for simulation with large domain. We propose to couple a 1D shallow water equations (SWE) model with a 2D incompressible Navier-Stokes equations (NSE) model in order to reduce the computational cost. The coupled model is solved using the lattice Boltzmann method (LBM) with the lattice Bhatnagar-Gross-Krook (BGK) scheme. Additionally, a special method is implemented to treat the complex behavior of free surface close to the shoreline. The result shows the coupled model can reduce computational cost significantly compared to the full NSE model.

Mean-field message-passing equations in the Hopfield model and its generalizations

NASA Astrophysics Data System (ADS)

Mézard, Marc

2017-02-01

Motivated by recent progress in using restricted Boltzmann machines as preprocessing algorithms for deep neural network, we revisit the mean-field equations [belief-propagation and Thouless-Anderson Palmer (TAP) equations] in the best understood of such machines, namely the Hopfield model of neural networks, and we explicit how they can be used as iterative message-passing algorithms, providing a fast method to compute the local polarizations of neurons. In the "retrieval phase", where neurons polarize in the direction of one memorized pattern, we point out a major difference between the belief propagation and TAP equations: The set of belief propagation equations depends on the pattern which is retrieved, while one can use a unique set of TAP equations. This makes the latter method much better suited for applications in the learning process of restricted Boltzmann machines. In the case where the patterns memorized in the Hopfield model are not independent, but are correlated through a combinatorial structure, we show that the TAP equations have to be modified. This modification can be seen either as an alteration of the reaction term in TAP equations or, more interestingly, as the consequence of message passing on a graphical model with several hidden layers, where the number of hidden layers depends on the depth of the correlations in the memorized patterns. This layered structure is actually necessary when one deals with more general restricted Boltzmann machines.

An efficient method for model refinement in diffuse optical tomography

NASA Astrophysics Data System (ADS)

Zirak, A. R.; Khademi, M.

2007-11-01

Diffuse optical tomography (DOT) is a non-linear, ill-posed, boundary value and optimization problem which necessitates regularization. Also, Bayesian methods are suitable owing to measurements data are sparse and correlated. In such problems which are solved with iterative methods, for stabilization and better convergence, the solution space must be small. These constraints subject to extensive and overdetermined system of equations which model retrieving criteria specially total least squares (TLS) must to refine model error. Using TLS is limited to linear systems which is not achievable when applying traditional Bayesian methods. This paper presents an efficient method for model refinement using regularized total least squares (RTLS) for treating on linearized DOT problem, having maximum a posteriori (MAP) estimator and Tikhonov regulator. This is done with combination Bayesian and regularization tools as preconditioner matrices, applying them to equations and then using RTLS to the resulting linear equations. The preconditioning matrixes are guided by patient specific information as well as a priori knowledge gained from the training set. Simulation results illustrate that proposed method improves the image reconstruction performance and localize the abnormally well.

An adaptive finite element method for the inequality-constrained Reynolds equation

NASA Astrophysics Data System (ADS)

Gustafsson, Tom; Rajagopal, Kumbakonam R.; Stenberg, Rolf; Videman, Juha

2018-07-01

We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem and approximated by a residual-based stabilized method. Based on our recent results on the classical obstacle problem, we present optimal a priori estimates and derive novel a posteriori error estimators. The method is implemented as a Nitsche-type finite element technique and shown in numerical computations to be superior to the usually applied penalty methods.

Optimal harvesting for a predator-prey agent-based model using difference equations.

PubMed

Oremland, Matthew; Laubenbacher, Reinhard

2015-03-01

In this paper, a method known as Pareto optimization is applied in the solution of a multi-objective optimization problem. The system in question is an agent-based model (ABM) wherein global dynamics emerge from local interactions. A system of discrete mathematical equations is formulated in order to capture the dynamics of the ABM; while the original model is built up analytically from the rules of the model, the paper shows how minor changes to the ABM rule set can have a substantial effect on model dynamics. To address this issue, we introduce parameters into the equation model that track such changes. The equation model is amenable to mathematical theory—we show how stability analysis can be performed and validated using ABM data. We then reduce the equation model to a simpler version and implement changes to allow controls from the ABM to be tested using the equations. Cohen's weighted κ is proposed as a measure of similarity between the equation model and the ABM, particularly with respect to the optimization problem. The reduced equation model is used to solve a multi-objective optimization problem via a technique known as Pareto optimization, a heuristic evolutionary algorithm. Results show that the equation model is a good fit for ABM data; Pareto optimization provides a suite of solutions to the multi-objective optimization problem that can be implemented directly in the ABM.

Some problems of the calculation of three-dimensional boundary layer flows on general configurations

NASA Technical Reports Server (NTRS)

Cebeci, T.; Kaups, K.; Mosinskis, G. J.; Rehn, J. A.

1973-01-01

An accurate solution of the three-dimensional boundary layer equations over general configurations such as those encountered in aircraft and space shuttle design requires a very efficient, fast, and accurate numerical method with suitable turbulence models for the Reynolds stresses. The efficiency, speed, and accuracy of a three-dimensional numerical method together with the turbulence models for the Reynolds stresses are examined. The numerical method is the implicit two-point finite difference approach (Box Method) developed by Keller and applied to the boundary layer equations by Keller and Cebeci. In addition, a study of some of the problems that may arise in the solution of these equations for three-dimensional boundary layer flows over general configurations.

Estimating Dynamical Systems: Derivative Estimation Hints From Sir Ronald A. Fisher.

PubMed

Deboeck, Pascal R

2010-08-06

The fitting of dynamical systems to psychological data offers the promise of addressing new and innovative questions about how people change over time. One method of fitting dynamical systems is to estimate the derivatives of a time series and then examine the relationships between derivatives using a differential equation model. One common approach for estimating derivatives, Local Linear Approximation (LLA), produces estimates with correlated errors. Depending on the specific differential equation model used, such correlated errors can lead to severely biased estimates of differential equation model parameters. This article shows that the fitting of dynamical systems can be improved by estimating derivatives in a manner similar to that used to fit orthogonal polynomials. Two applications using simulated data compare the proposed method and a generalized form of LLA when used to estimate derivatives and when used to estimate differential equation model parameters. A third application estimates the frequency of oscillation in observations of the monthly deaths from bronchitis, emphysema, and asthma in the United Kingdom. These data are publicly available in the statistical program R, and functions in R for the method presented are provided.

Stochastic differential equations as a tool to regularize the parameter estimation problem for continuous time dynamical systems given discrete time measurements.

PubMed

Leander, Jacob; Lundh, Torbjörn; Jirstrand, Mats

2014-05-01

In this paper we consider the problem of estimating parameters in ordinary differential equations given discrete time experimental data. The impact of going from an ordinary to a stochastic differential equation setting is investigated as a tool to overcome the problem of local minima in the objective function. Using two different models, it is demonstrated that by allowing noise in the underlying model itself, the objective functions to be minimized in the parameter estimation procedures are regularized in the sense that the number of local minima is reduced and better convergence is achieved. The advantage of using stochastic differential equations is that the actual states in the model are predicted from data and this will allow the prediction to stay close to data even when the parameters in the model is incorrect. The extended Kalman filter is used as a state estimator and sensitivity equations are provided to give an accurate calculation of the gradient of the objective function. The method is illustrated using in silico data from the FitzHugh-Nagumo model for excitable media and the Lotka-Volterra predator-prey system. The proposed method performs well on the models considered, and is able to regularize the objective function in both models. This leads to parameter estimation problems with fewer local minima which can be solved by efficient gradient-based methods. Copyright © 2014 The Authors. Published by Elsevier Inc. All rights reserved.

Receptor binding kinetics equations: Derivation using the Laplace transform method.

PubMed

Hoare, Sam R J

Measuring unlabeled ligand receptor binding kinetics is valuable in optimizing and understanding drug action. Unfortunately, deriving equations for estimating kinetic parameters is challenging because it involves calculus; integration can be a frustrating barrier to the pharmacologist seeking to measure simple rate parameters. Here, a well-known tool for simplifying the derivation, the Laplace transform, is applied to models of receptor-ligand interaction. The method transforms differential equations to a form in which simple algebra can be applied to solve for the variable of interest, for example the concentration of ligand-bound receptor. The goal is to provide instruction using familiar examples, to enable investigators familiar with handling equilibrium binding equations to derive kinetic equations for receptor-ligand interaction. First, the Laplace transform is used to derive the equations for association and dissociation of labeled ligand binding. Next, its use for unlabeled ligand kinetic equations is exemplified by a full derivation of the kinetics of competitive binding equation. Finally, new unlabeled ligand equations are derived using the Laplace transform. These equations incorporate a pre-incubation step with unlabeled or labeled ligand. Four equations for measuring unlabeled ligand kinetics were compared and the two new equations verified by comparison with numerical solution. Importantly, the equations have not been verified with experimental data because no such experiments are evident in the literature. Equations were formatted for use in the curve-fitting program GraphPad Prism 6.0 and fitted to simulated data. This description of the Laplace transform method will enable pharmacologists to derive kinetic equations for their model or experimental paradigm under study. Application of the transform will expand the set of equations available for the pharmacologist to measure unlabeled ligand binding kinetics, and for other time-dependent pharmacological activities. Copyright © 2017 Elsevier Inc. All rights reserved.

Video Extrapolation Method Based on Time-Varying Energy Optimization and CIP.

PubMed

Sakaino, Hidetomo

2016-09-01

Video extrapolation/prediction methods are often used to synthesize new videos from images. For fluid-like images and dynamic textures as well as moving rigid objects, most state-of-the-art video extrapolation methods use non-physics-based models that learn orthogonal bases from a number of images but at high computation cost. Unfortunately, data truncation can cause image degradation, i.e., blur, artifact, and insufficient motion changes. To extrapolate videos that more strictly follow physical rules, this paper proposes a physics-based method that needs only a few images and is truncation-free. We utilize physics-based equations with image intensity and velocity: optical flow, Navier-Stokes, continuity, and advection equations. These allow us to use partial difference equations to deal with the local image feature changes. Image degradation during extrapolation is minimized by updating model parameters, where a novel time-varying energy balancer model that uses energy based image features, i.e., texture, velocity, and edge. Moreover, the advection equation is discretized by high-order constrained interpolation profile for lower quantization error than can be achieved by the previous finite difference method in long-term videos. Experiments show that the proposed energy based video extrapolation method outperforms the state-of-the-art video extrapolation methods in terms of image quality and computation cost.

An ill-posed problem for the Black-Scholes equation for a profitable forecast of prices of stock options on real market data

NASA Astrophysics Data System (ADS)

Klibanov, Michael V.; Kuzhuget, Andrey V.; Golubnichiy, Kirill V.

2016-01-01

A new empirical mathematical model for the Black-Scholes equation is proposed to forecast option prices. This model includes new interval for the price of the underlying stock, new initial and new boundary conditions. Conventional notions of maturity time and strike prices are not used. The Black-Scholes equation is solved as a parabolic equation with the reversed time, which is an ill-posed problem. Thus, a regularization method is used to solve it. To verify the validity of our model, real market data for 368 randomly selected liquid options are used. A new trading strategy is proposed. Our results indicates that our method is profitable on those options. Furthermore, it is shown that the performance of two simple extrapolation-based techniques is much worse. We conjecture that our method might lead to significant profits of those financial insitutions which trade large amounts of options. We caution, however, that further studies are necessary to verify this conjecture.

Equations of motion for a spectrum-generating algebra: Lipkin Meshkov Glick model

NASA Astrophysics Data System (ADS)

Rosensteel, G.; Rowe, D. J.; Ho, S. Y.

2008-01-01

For a spectrum-generating Lie algebra, a generalized equations-of-motion scheme determines numerical values of excitation energies and algebra matrix elements. In the approach to the infinite particle number limit or, more generally, whenever the dimension of the quantum state space is very large, the equations-of-motion method may achieve results that are impractical to obtain by diagonalization of the Hamiltonian matrix. To test the method's effectiveness, we apply it to the well-known Lipkin-Meshkov-Glick (LMG) model to find its low-energy spectrum and associated generator matrix elements in the eigenenergy basis. When the dimension of the LMG representation space is 106, computation time on a notebook computer is a few minutes. For a large particle number in the LMG model, the low-energy spectrum makes a quantum phase transition from a nondegenerate harmonic vibrator to a twofold degenerate harmonic oscillator. The equations-of-motion method computes critical exponents at the transition point.

Integral-equation based methods for parameter estimation in output pulses of radiation detectors: Application in nuclear medicine and spectroscopy

NASA Astrophysics Data System (ADS)

Mohammadian-Behbahani, Mohammad-Reza; Saramad, Shahyar

2018-04-01

Model based analysis methods are relatively new approaches for processing the output data of radiation detectors in nuclear medicine imaging and spectroscopy. A class of such methods requires fast algorithms for fitting pulse models to experimental data. In order to apply integral-equation based methods for processing the preamplifier output pulses, this article proposes a fast and simple method for estimating the parameters of the well-known bi-exponential pulse model by solving an integral equation. The proposed method needs samples from only three points of the recorded pulse as well as its first and second order integrals. After optimizing the sampling points, the estimation results were calculated and compared with two traditional integration-based methods. Different noise levels (signal-to-noise ratios from 10 to 3000) were simulated for testing the functionality of the proposed method, then it was applied to a set of experimental pulses. Finally, the effect of quantization noise was assessed by studying different sampling rates. Promising results by the proposed method endorse it for future real-time applications.

Dynamics of a prey-predator system under Poisson white noise excitation

NASA Astrophysics Data System (ADS)

Pan, Shan-Shan; Zhu, Wei-Qiu

2014-10-01

The classical Lotka-Volterra (LV) model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural environment has been modeled as Poisson white noise, is investigated by using the stochastic averaging method. The averaged generalized Itô stochastic differential equation and Fokker-Planck-Kolmogorov (FPK) equation are derived for prey-predator ecosystem driven by Poisson white noise. Approximate stationary solution for the averaged generalized FPK equation is obtained by using the perturbation method. The effect of prey self-competition parameter ɛ2 s on ecosystem behavior is evaluated. The analytical result is confirmed by corresponding Monte Carlo (MC) simulation.

From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Angstmann, C.N.; Donnelly, I.C.; Henry, B.I., E-mail: B.Henry@unsw.edu.au

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also showmore » that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.« less

Structure-preserving spectral element method in attenuating seismic wave modeling

NASA Astrophysics Data System (ADS)

Cai, Wenjun; Zhang, Huai

2016-04-01

This work describes the extension of the conformal symplectic method to solve the damped acoustic wave equation and the elastic wave equations in the framework of the spectral element method. The conformal symplectic method is a variation of conventional symplectic methods to treat non-conservative time evolution problems which has superior behaviors in long-time stability and dissipation preservation. To construct the conformal symplectic method, we first reformulate the damped acoustic wave equation and the elastic wave equations in their equivalent conformal multi-symplectic structures, which naturally reveal the intrinsic properties of the original systems, especially, the dissipation laws. We thereafter separate each structures into a conservative Hamiltonian system and a purely dissipative ordinary differential equation system. Based on the splitting methodology, we solve the two subsystems respectively. The dissipative one is cheaply solved by its analytic solution. While for the conservative system, we combine a fourth-order symplectic Nyström method in time and the spectral element method in space to cover the circumstances in realistic geological structures involving complex free-surface topography. The Strang composition method is adopted thereby to concatenate the corresponding two parts of solutions and generate the completed numerical scheme, which is conformal symplectic and can therefore guarantee the numerical stability and dissipation preservation after a large time modeling. Additionally, a relative larger Courant number than that of the traditional Newmark scheme is found in the numerical experiments in conjunction with a spatial sampling of approximately 5 points per wavelength. A benchmark test for the damped acoustic wave equation validates the effectiveness of our proposed method in precisely capturing dissipation rate. The classical Lamb problem is used to demonstrate the ability of modeling Rayleigh-wave propagation. More comprehensive numerical experiments are presented to investigate the long-time simulation, low dispersion and energy conservation properties of the conformal symplectic method in both the attenuating homogeneous and heterogeneous mediums.

Integral equation methods for vesicle electrohydrodynamics in three dimensions

NASA Astrophysics Data System (ADS)

Veerapaneni, Shravan

2016-12-01

In this paper, we develop a new boundary integral equation formulation that describes the coupled electro- and hydro-dynamics of a vesicle suspended in a viscous fluid and subjected to external flow and electric fields. The dynamics of the vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor-Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response. The coupled governing equations for the vesicle position and its transmembrane electric potential are solved using a numerical method that is spectrally accurate in space and first-order in time. The method uses a semi-implicit time-stepping scheme to overcome the numerical stiffness associated with the governing equations.

The construction of a two-dimensional reproducing kernel function and its application in a biomedical model.

PubMed

Guo, Qi; Shen, Shu-Ting

2016-04-29

There are two major classes of cardiac tissue models: the ionic model and the FitzHugh-Nagumo model. During computer simulation, each model entails solving a system of complex ordinary differential equations and a partial differential equation with non-flux boundary conditions. The reproducing kernel method possesses significant applications in solving partial differential equations. The derivative of the reproducing kernel function is a wavelet function, which has local properties and sensitivities to singularity. Therefore, study on the application of reproducing kernel would be advantageous. Applying new mathematical theory to the numerical solution of the ventricular muscle model so as to improve its precision in comparison with other methods at present. A two-dimensional reproducing kernel function inspace is constructed and applied in computing the solution of two-dimensional cardiac tissue model by means of the difference method through time and the reproducing kernel method through space. Compared with other methods, this method holds several advantages such as high accuracy in computing solutions, insensitivity to different time steps and a slow propagation speed of error. It is suitable for disorderly scattered node systems without meshing, and can arbitrarily change the location and density of the solution on different time layers. The reproducing kernel method has higher solution accuracy and stability in the solutions of the two-dimensional cardiac tissue model.

The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics

NASA Technical Reports Server (NTRS)

Bayliss, A.; Goldstein, C. I.; Turkel, E.

1984-01-01

The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner.

Second-order kinetic model for the sorption of cadmium onto tree fern: a comparison of linear and non-linear methods.

PubMed

Ho, Yuh-Shan

2006-01-01

A comparison was made of the linear least-squares method and a trial-and-error non-linear method of the widely used pseudo-second-order kinetic model for the sorption of cadmium onto ground-up tree fern. Four pseudo-second-order kinetic linear equations are discussed. Kinetic parameters obtained from the four kinetic linear equations using the linear method differed but they were the same when using the non-linear method. A type 1 pseudo-second-order linear kinetic model has the highest coefficient of determination. Results show that the non-linear method may be a better way to obtain the desired parameters.

Numerical methods for the weakly compressible Generalized Langevin Model in Eulerian reference frame

DOE Office of Scientific and Technical Information (OSTI.GOV)

Azarnykh, Dmitrii, E-mail: d.azarnykh@tum.de; Litvinov, Sergey; Adams, Nikolaus A.

2016-06-01

A well established approach for the computation of turbulent flow without resolving all turbulent flow scales is to solve a filtered or averaged set of equations, and to model non-resolved scales by closures derived from transported probability density functions (PDF) for velocity fluctuations. Effective numerical methods for PDF transport employ the equivalence between the Fokker–Planck equation for the PDF and a Generalized Langevin Model (GLM), and compute the PDF by transporting a set of sampling particles by GLM (Pope (1985) [1]). The natural representation of GLM is a system of stochastic differential equations in a Lagrangian reference frame, typically solvedmore » by particle methods. A representation in a Eulerian reference frame, however, has the potential to significantly reduce computational effort and to allow for the seamless integration into a Eulerian-frame numerical flow solver. GLM in a Eulerian frame (GLMEF) formally corresponds to the nonlinear fluctuating hydrodynamic equations derived by Nakamura and Yoshimori (2009) [12]. Unlike the more common Landau–Lifshitz Navier–Stokes (LLNS) equations these equations are derived from the underdamped Langevin equation and are not based on a local equilibrium assumption. Similarly to LLNS equations the numerical solution of GLMEF requires special considerations. In this paper we investigate different numerical approaches to solving GLMEF with respect to the correct representation of stochastic properties of the solution. We find that a discretely conservative staggered finite-difference scheme, adapted from a scheme originally proposed for turbulent incompressible flow, in conjunction with a strongly stable (for non-stochastic PDE) Runge–Kutta method performs better for GLMEF than schemes adopted from those proposed previously for the LLNS. We show that equilibrium stochastic fluctuations are correctly reproduced.« less

A (2+1)-dimensional Korteweg-de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws

NASA Astrophysics Data System (ADS)

Adem, Abdullahi Rashid

2016-05-01

We consider a (2+1)-dimensional Korteweg-de Vries type equation which models the shallow-water waves, surface and internal waves. In the analysis, we use the Lie symmetry method and the multiple exp-function method. Furthermore, conservation laws are computed using the multiplier method.

Feynman path integral application on deriving black-scholes diffusion equation for european option pricing

NASA Astrophysics Data System (ADS)

Utama, Briandhika; Purqon, Acep

2016-08-01

Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods.

Numerical techniques for the solution of the compressible Navier-Stokes equations and implementation of turbulence models. [separated turbulent boundary layer flow problems

NASA Technical Reports Server (NTRS)

Baldwin, B. S.; Maccormack, R. W.; Deiwert, G. S.

1975-01-01

The time-splitting explicit numerical method of MacCormack is applied to separated turbulent boundary layer flow problems. Modifications of this basic method are developed to counter difficulties associated with complicated geometry and severe numerical resolution requirements of turbulence model equations. The accuracy of solutions is investigated by comparison with exact solutions for several simple cases. Procedures are developed for modifying the basic method to improve the accuracy. Numerical solutions of high-Reynolds-number separated flows over an airfoil and shock-separated flows over a flat plate are obtained. A simple mixing length model of turbulence is used for the transonic flow past an airfoil. A nonorthogonal mesh of arbitrary configuration facilitates the description of the flow field. For the simpler geometry associated with the flat plate, a rectangular mesh is used, and solutions are obtained based on a two-equation differential model of turbulence.

Hierarchical Boltzmann simulations and model error estimation

NASA Astrophysics Data System (ADS)

Torrilhon, Manuel; Sarna, Neeraj

2017-08-01

A hierarchical simulation approach for Boltzmann's equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics, but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. We use Hermite discretization, or moment equations, for the steady linearized Boltzmann equation for a proof-of-concept of such a framework. All representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous Galerkin formulation. We demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains. The hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations. We present various model errors for a flow through a curved channel with obstacles.

Modeling of Graphene Planar Grating in the THz Range by the Method of Singular Integral Equations

NASA Astrophysics Data System (ADS)

Kaliberda, Mstislav E.; Lytvynenko, Leonid M.; Pogarsky, Sergey A.

2018-04-01

Diffraction of the H-polarized electromagnetic wave by the planar graphene grating in the THz range is considered. The scattering and absorption characteristics are studied. The scattered field is represented in the spectral domain via unknown spectral function. The mathematical model is based on the graphene surface impedance and the method of singular integral equations. The numerical solution is obtained by the Nystrom-type method of discrete singularities.

An improved method to determine neuromuscular properties using force laws - From single muscle to applications in human movements.

PubMed

Siebert, T; Sust, M; Thaller, S; Tilp, M; Wagner, H

2007-04-01

We evaluate an improved method for individually determining neuromuscular properties in vivo. The method is based on Hill's equation used as a force law combined with Newton's equation of motion. To ensure the range of validity of Hill's equation, we first perform detailed investigations on in vitro single muscles. The force-velocity relation determined with the model coincides well with results obtained by standard methods (r=.99) above 20% of the isometric force. In addition, the model-predicted force curves during work loop contractions very well agree with measurements (mean difference: 2-3%). Subsequently, we deduce theoretically under which conditions it is possible to combine several muscles of the human body to model muscles. This leads to a model equation for human leg extension movements containing parameters for the muscle properties and for the activation. To numerically determine these invariant neuromuscular properties we devise an experimental method based on concentric and isometric leg extensions. With this method we determine individual muscle parameters from experiments such that the simulated curves agree well with experiments (r=.99). A reliability test with 12 participants revealed correlations r=.72-.91 for the neuromuscular parameters (p<.01). Predictions of similar movements under different conditions show mean errors of about 5%. In addition, we present applications in sports practise and theory.

Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution.

PubMed

Lu, Benzhuo; Zhou, Y C; Huber, Gary A; Bond, Stephen D; Holst, Michael J; McCammon, J Andrew

2007-10-07

A computational framework is presented for the continuum modeling of cellular biomolecular diffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equations solutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.

Numerical solution of special ultra-relativistic Euler equations using central upwind scheme

NASA Astrophysics Data System (ADS)

Ghaffar, Tayabia; Yousaf, Muhammad; Qamar, Shamsul

2018-06-01

This article is concerned with the numerical approximation of one and two-dimensional special ultra-relativistic Euler equations. The governing equations are coupled first-order nonlinear hyperbolic partial differential equations. These equations describe perfect fluid flow in terms of the particle density, the four-velocity and the pressure. A high-resolution shock-capturing central upwind scheme is employed to solve the model equations. To avoid excessive numerical diffusion, the considered scheme avails the specific information of local propagation speeds. By using Runge-Kutta time stepping method and MUSCL-type initial reconstruction, we have obtained 2nd order accuracy of the proposed scheme. After discussing the model equations and the numerical technique, several 1D and 2D test problems are investigated. For all the numerical test cases, our proposed scheme demonstrates very good agreement with the results obtained by well-established algorithms, even in the case of highly relativistic 2D test problems. For validation and comparison, the staggered central scheme and the kinetic flux-vector splitting (KFVS) method are also implemented to the same model. The robustness and efficiency of central upwind scheme is demonstrated by the numerical results.

Numerical and experimental investigation on static electric charge model at stable cone-jet region

NASA Astrophysics Data System (ADS)

Hashemi, Ali Reza; Pishevar, Ahmad Reza; Valipouri, Afsaneh; Pǎrǎu, Emilian I.

2018-03-01

In a typical electro-spinning process, the steady stretching process of the jet beyond the Taylor cone has a significant effect on the dimensions of resulting nanofibers. Also, it sets up the conditions for the onset of the bending instability. The focus of this work is the modeling and simulation of the initial stable jet phase seen during the electro-spinning process. The perturbation method was applied to solve hydrodynamic equations, and the electrostatic equation was solved by a boundary integral method. These equations were coupled with the stress boundary conditions derived appropriate at the fluid-fluid interface. Perturbation equations were discretized by the second-order finite difference method, and the Newton method was implemented to solve the discretized nonlinear system. Also, the boundary element method was utilized to solve the electrostatic equation. In the theoretical study, the fluid is described as a leaky dielectric with charges only on the jet surface in dielectric air. In this study, electric charges were modeled as static. Comparison of numerical and experimental results shows that at low flow rates and high electric field, good agreement was achieved because of the superior importance of the charge transport by conduction rather than convection and charge concentration. In addition, the effect of unevenness of the electric field around the nozzle tip was experimentally studied through plate-plate geometry as well as point-plate geometry.

Percent body fat estimations in college men using field and laboratory methods: A three-compartment model approach

PubMed Central

Moon, Jordan R; Tobkin, Sarah E; Smith, Abbie E; Roberts, Michael D; Ryan, Eric D; Dalbo, Vincent J; Lockwood, Chris M; Walter, Ashley A; Cramer, Joel T; Beck, Travis W; Stout, Jeffrey R

2008-01-01

Background Methods used to estimate percent body fat can be classified as a laboratory or field technique. However, the validity of these methods compared to multiple-compartment models has not been fully established. The purpose of this study was to determine the validity of field and laboratory methods for estimating percent fat (%fat) in healthy college-age men compared to the Siri three-compartment model (3C). Methods Thirty-one Caucasian men (22.5 ± 2.7 yrs; 175.6 ± 6.3 cm; 76.4 ± 10.3 kg) had their %fat estimated by bioelectrical impedance analysis (BIA) using the BodyGram™ computer program (BIA-AK) and population-specific equation (BIA-Lohman), near-infrared interactance (NIR) (Futrex® 6100/XL), four circumference-based military equations [Marine Corps (MC), Navy and Air Force (NAF), Army (A), and Friedl], air-displacement plethysmography (BP), and hydrostatic weighing (HW). Results All circumference-based military equations (MC = 4.7% fat, NAF = 5.2% fat, A = 4.7% fat, Friedl = 4.7% fat) along with NIR (NIR = 5.1% fat) produced an unacceptable total error (TE). Both laboratory methods produced acceptable TE values (HW = 2.5% fat; BP = 2.7% fat). The BIA-AK, and BIA-Lohman field methods produced acceptable TE values (2.1% fat). A significant difference was observed for the MC and NAF equations compared to both the 3C model and HW (p < 0.006). Conclusion Results indicate that the BP and HW are valid laboratory methods when compared to the 3C model to estimate %fat in college-age Caucasian men. When the use of a laboratory method is not feasible, BIA-AK, and BIA-Lohman are acceptable field methods to estimate %fat in this population. PMID:18426582

A quadrature based method of moments for nonlinear Fokker-Planck equations

NASA Astrophysics Data System (ADS)

Otten, Dustin L.; Vedula, Prakash

2011-09-01

Fokker-Planck equations which are nonlinear with respect to their probability densities and occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, fermions and bosons can be challenging to solve numerically. To address some underlying challenges, we propose the application of the direct quadrature based method of moments (DQMOM) for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations (NLFPEs). In DQMOM, probability density (or other distribution) functions are represented using a finite collection of Dirac delta functions, characterized by quadrature weights and locations (or abscissas) that are determined based on constraints due to evolution of generalized moments. Three particular examples of nonlinear Fokker-Planck equations considered in this paper include descriptions of: (i) the Shimizu-Yamada model, (ii) the Desai-Zwanzig model (both of which have been developed as models of muscular contraction) and (iii) fermions and bosons. Results based on DQMOM, for the transient and stationary solutions of the nonlinear Fokker-Planck equations, have been found to be in good agreement with other available analytical and numerical approaches. It is also shown that approximate reconstruction of the underlying probability density function from moments obtained from DQMOM can be satisfactorily achieved using a maximum entropy method.

Seismic modeling with radial basis function-generated finite differences (RBF-FD) – a simplified treatment of interfaces

DOE Office of Scientific and Technical Information (OSTI.GOV)

Martin, Bradley, E-mail: brma7253@colorado.edu; Fornberg, Bengt, E-mail: Fornberg@colorado.edu

In a previous study of seismic modeling with radial basis function-generated finite differences (RBF-FD), we outlined a numerical method for solving 2-D wave equations in domains with material interfaces between different regions. The method was applicable on a mesh-free set of data nodes. It included all information about interfaces within the weights of the stencils (allowing the use of traditional time integrators), and was shown to solve problems of the 2-D elastic wave equation to 3rd-order accuracy. In the present paper, we discuss a refinement of that method that makes it simpler to implement. It can also improve accuracy formore » the case of smoothly-variable model parameter values near interfaces. We give several test cases that demonstrate the method solving 2-D elastic wave equation problems to 4th-order accuracy, even in the presence of smoothly-curved interfaces with jump discontinuities in the model parameters.« less

Seismic modeling with radial basis function-generated finite differences (RBF-FD) - a simplified treatment of interfaces

NASA Astrophysics Data System (ADS)

Martin, Bradley; Fornberg, Bengt

2017-04-01

In a previous study of seismic modeling with radial basis function-generated finite differences (RBF-FD), we outlined a numerical method for solving 2-D wave equations in domains with material interfaces between different regions. The method was applicable on a mesh-free set of data nodes. It included all information about interfaces within the weights of the stencils (allowing the use of traditional time integrators), and was shown to solve problems of the 2-D elastic wave equation to 3rd-order accuracy. In the present paper, we discuss a refinement of that method that makes it simpler to implement. It can also improve accuracy for the case of smoothly-variable model parameter values near interfaces. We give several test cases that demonstrate the method solving 2-D elastic wave equation problems to 4th-order accuracy, even in the presence of smoothly-curved interfaces with jump discontinuities in the model parameters.

MODFLOW-2000, The U.S. Geological Survey Modular Ground-Water Model - User Guide to Modularization Concepts and the Ground-Water Flow Process

USGS Publications Warehouse

Harbaugh, Arlen W.; Banta, Edward R.; Hill, Mary C.; McDonald, Michael G.

2000-01-01

MODFLOW is a computer program that numerically solves the three-dimensional ground-water flow equation for a porous medium by using a finite-difference method. Although MODFLOW was designed to be easily enhanced, the design was oriented toward additions to the ground-water flow equation. Frequently there is a need to solve additional equations; for example, transport equations and equations for estimating parameter values that produce the closest match between model-calculated heads and flows and measured values. This report documents a new version of MODFLOW, called MODFLOW-2000, which is designed to accommodate the solution of equations in addition to the ground-water flow equation. This report is a user's manual. It contains an overview of the old and added design concepts, documents one new package, and contains input instructions for using the model to solve the ground-water flow equation.

Measurement-based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry

NASA Astrophysics Data System (ADS)

Xu, Peiliang

2018-06-01

The numerical integration method has been routinely used by major institutions worldwide, for example, NASA Goddard Space Flight Center and German Research Center for Geosciences (GFZ), to produce global gravitational models from satellite tracking measurements of CHAMP and/or GRACE types. Such Earth's gravitational products have found widest possible multidisciplinary applications in Earth Sciences. The method is essentially implemented by solving the differential equations of the partial derivatives of the orbit of a satellite with respect to the unknown harmonic coefficients under the conditions of zero initial values. From the mathematical and statistical point of view, satellite gravimetry from satellite tracking is essentially the problem of estimating unknown parameters in the Newton's nonlinear differential equations from satellite tracking measurements. We prove that zero initial values for the partial derivatives are incorrect mathematically and not permitted physically. The numerical integration method, as currently implemented and used in mathematics and statistics, chemistry and physics, and satellite gravimetry, is groundless, mathematically and physically. Given the Newton's nonlinear governing differential equations of satellite motion with unknown equation parameters and unknown initial conditions, we develop three methods to derive new local solutions around a nominal reference orbit, which are linked to measurements to estimate the unknown corrections to approximate values of the unknown parameters and the unknown initial conditions. Bearing in mind that satellite orbits can now be tracked almost continuously at unprecedented accuracy, we propose the measurement-based perturbation theory and derive global uniformly convergent solutions to the Newton's nonlinear governing differential equations of satellite motion for the next generation of global gravitational models. Since the solutions are global uniformly convergent, theoretically speaking, they are able to extract smallest possible gravitational signals from modern and future satellite tracking measurements, leading to the production of global high-precision, high-resolution gravitational models. By directly turning the nonlinear differential equations of satellite motion into the nonlinear integral equations, and recognizing the fact that satellite orbits are measured with random errors, we further reformulate the links between satellite tracking measurements and the global uniformly convergent solutions to the Newton's governing differential equations as a condition adjustment model with unknown parameters, or equivalently, the weighted least squares estimation of unknown differential equation parameters with equality constraints, for the reconstruction of global high-precision, high-resolution gravitational models from modern (and future) satellite tracking measurements.

a Numerical Comparison of Langrange and Kane's Methods of AN Arm Segment

NASA Astrophysics Data System (ADS)

Rambely, Azmin Sham; Halim, Norhafiza Ab.; Ahmad, Rokiah Rozita

A 2-D model of a two-link kinematic chain is developed using two dynamics equations of motion, namely Kane's and Lagrange Methods. The dynamics equations are reduced to first order differential equation and solved using modified Euler and fourth order Runge Kutta to approximate the shoulder and elbow joint angles during a smash performance in badminton. Results showed that Runge-Kutta produced a better and exact approximation than that of modified Euler and both dynamic equations produced better absolute errors.

Exact soliton of (2 + 1)-dimensional fractional Schrödinger equation

NASA Astrophysics Data System (ADS)

Rizvi, S. T. R.; Ali, K.; Bashir, S.; Younis, M.; Ashraf, R.; Ahmad, M. O.

2017-07-01

The nonlinear fractional Schrödinger equation is the basic equation of fractional quantum mechanics introduced by Nick Laskin in 2002. We apply three tools to solve this mathematical-physical model. First, we find the solitary wave solutions including the trigonometric traveling wave solutions, bell and kink shape solitons using the F-expansion and Improve F-expansion method. We also obtain the soliton solution, singular soliton solutions, rational function solution and elliptic integral function solutions, with the help of the extended trial equation method.

Multiple and exact soliton solutions of the perturbed Korteweg-de Vries equation of long surface waves in a convective fluid via Painlevé analysis, factorization, and simplest equation methods.

PubMed

Selima, Ehab S; Yao, Xiaohua; Wazwaz, Abdul-Majid

2017-06-01

In this research, the surface waves of a horizontal fluid layer open to air under gravity field and vertical temperature gradient effects are studied. The governing equations of this model are reformulated and converted to a nonlinear evolution equation, the perturbed Korteweg-de Vries (pKdV) equation. We investigate the latter equation, which includes dispersion, diffusion, and instability effects, in order to examine the evolution of long surface waves in a convective fluid. Dispersion relation of the pKdV equation and its properties are discussed. The Painlevé analysis is applied not only to check the integrability of the pKdV equation but also to establish the Bäcklund transformation form. In addition, traveling wave solutions and a general form of the multiple-soliton solutions of the pKdV equation are obtained via Bäcklund transformation, the simplest equation method using Bernoulli, Riccati, and Burgers' equations as simplest equations, and the factorization method.

Using an EM Covariance Matrix to Estimate Structural Equation Models with Missing Data: Choosing an Adjusted Sample Size to Improve the Accuracy of Inferences

ERIC Educational Resources Information Center

Enders, Craig K.; Peugh, James L.

2004-01-01

Two methods, direct maximum likelihood (ML) and the expectation maximization (EM) algorithm, can be used to obtain ML parameter estimates for structural equation models with missing data (MD). Although the 2 methods frequently produce identical parameter estimates, it may be easier to satisfy missing at random assumptions using EM. However, no…

Estimating parameters for tree basal area growth with a system of equations and seemingly unrelated regressions

Treesearch

Charles E. Rose; Thomas B. Lynch

2001-01-01

A method was developed for estimating parameters in an individual tree basal area growth model using a system of equations based on dbh rank classes. The estimation method developed is a compromise between an individual tree and a stand level basal area growth model that accounts for the correlation between trees within a plot by using seemingly unrelated regression (...

The study of the Boltzmann equation of solid-gas two-phase flow with three-dimensional BGK model

NASA Astrophysics Data System (ADS)

Liu, Chang-jiang; Pang, Song; Xu, Qiang; He, Ling; Yang, Shao-peng; Qing, Yun-jie

2018-06-01

The motion of many solid-gas two-phase flows can be described by the Boltzmann equation. In order to simplify the Boltzmann equation, the convective-diffusion term is reserved and the collision term is replaced by the three-dimensional Bharnagar-Gross-Krook (BGK) model. Then the simplified Boltzmann equation is solved by homotopy perturbation method (HPM), and its approximate analytical solution is obtained. Through the analyzing, it is proved that the analytical solution satisfies all the constraint conditions, and its formation is in accord with the formation of the solution that is obtained by traditional Chapman-Enskog method, and the solving process of HPM is much more simple and convenient. This preliminarily shows the effectiveness and rapidness of HPM to solve the Boltzmann equation. The results obtained herein provide some theoretical basis for the further study of dynamic model of solid-gas two-phase flows, such as the sturzstrom of high-speed distant landslide caused by microseism and the sand storm caused by strong breeze.

Analytical Theory of the Destruction Terms in Dissipation Rate Transport Equations

NASA Technical Reports Server (NTRS)

Rubinstein, Robert; Zhou, Ye

1996-01-01

Modeled dissipation rate transport equations are often derived by invoking various hypotheses to close correlations in the corresponding exact equations. D. C. Leslie suggested that these models might be derived instead from Kraichnan's wavenumber space integrals for inertial range transport power. This suggestion is applied to the destruction terms in the dissipation rate equations for incompressible turbulence, buoyant turbulence, rotating incompressible turbulence, and rotating buoyant turbulence. Model constants like C(epsilon 2) are expressed as integrals; convergence of these integrals implies the absence of Reynolds number dependence in the corresponding destruction term. The dependence of C(epsilon 2) on rotation rate emerges naturally; sensitization of the modeled dissipation rate equation to rotation is not required. A buoyancy related effect which is absent in the exact transport equation for temperature variance dissipation, but which sometimes improves computational predictions, also arises naturally. Both the presence of this effect and the appropriate time scale in the modeled transport equation depend on whether Bolgiano or Kolmogorov inertial range scaling applies. A simple application of these methods leads to a preliminary, dissipation rate equation for rotating buoyant turbulence.

Point model equations for neutron correlation counting: Extension of Böhnel's equations to any order

DOE PAGES

Favalli, Andrea; Croft, Stephen; Santi, Peter

2015-06-15

Various methods of autocorrelation neutron analysis may be used to extract information about a measurement item containing spontaneously fissioning material. The two predominant approaches being the time correlation analysis (that make use of a coincidence gate) methods of multiplicity shift register logic and Feynman sampling. The common feature is that the correlated nature of the pulse train can be described by a vector of reduced factorial multiplet rates. We call these singlets, doublets, triplets etc. Within the point reactor model the multiplet rates may be related to the properties of the item, the parameters of the detector, and basic nuclearmore » data constants by a series of coupled algebraic equations – the so called point model equations. Solving, or inverting, the point model equations using experimental calibration model parameters is how assays of unknown items is performed. Currently only the first three multiplets are routinely used. In this work we develop the point model equations to higher order multiplets using the probability generating functions approach combined with the general derivative chain rule, the so called Faà di Bruno Formula. Explicit expression up to 5th order are provided, as well the general iterative formula to calculate any order. This study represents the first necessary step towards determining if higher order multiplets can add value to nondestructive measurement practice for nuclear materials control and accountancy.« less

Least Squares Method for Equating Logistic Ability Scales: A General Approach and Evaluation. Iowa Testing Programs Occasional Papers, Number 30.

ERIC Educational Resources Information Center

Haebara, Tomokazu

When several ability scales in item response models are separately derived from different test forms administered to different samples of examinees, these scales must be equated to a common scale because their units and origins are arbitrarily determined and generally different from scale to scale. A general method for equating logistic ability…

Combining existing numerical models with data assimilation using weighted least-squares finite element methods.

PubMed

Rajaraman, Prathish K; Manteuffel, T A; Belohlavek, M; Heys, Jeffrey J

2017-01-01

A new approach has been developed for combining and enhancing the results from an existing computational fluid dynamics model with experimental data using the weighted least-squares finite element method (WLSFEM). Development of the approach was motivated by the existence of both limited experimental blood velocity in the left ventricle and inexact numerical models of the same flow. Limitations of the experimental data include measurement noise and having data only along a two-dimensional plane. Most numerical modeling approaches do not provide the flexibility to assimilate noisy experimental data. We previously developed an approach that could assimilate experimental data into the process of numerically solving the Navier-Stokes equations, but the approach was limited because it required the use of specific finite element methods for solving all model equations and did not support alternative numerical approximation methods. The new approach presented here allows virtually any numerical method to be used for approximately solving the Navier-Stokes equations, and then the WLSFEM is used to combine the experimental data with the numerical solution of the model equations in a final step. The approach dynamically adjusts the influence of the experimental data on the numerical solution so that more accurate data are more closely matched by the final solution and less accurate data are not closely matched. The new approach is demonstrated on different test problems and provides significantly reduced computational costs compared with many previous methods for data assimilation. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

Mathematical model of one-man air revitalization system

NASA Technical Reports Server (NTRS)

1976-01-01

A mathematical model was developed for simulating the steady state performance in electrochemical CO2 concentrators which utilize (NMe4)2 CO3 (aq.) electrolyte. This electrolyte, which accommodates a wide range of air relative humidity, is most suitable for one-man air revitalization systems. The model is based on the solution of coupled nonlinear ordinary differential equations derived from mass transport and rate equations for the processes which take place in the cell. The boundary conditions are obtained by solving the mass and energy transport equations. A shooting method is used to solve the differential equations.

Bifurcations of solitary wave solutions for (two and three)-dimensional nonlinear partial differential equation in quantum and magnetized plasma by using two different methods

NASA Astrophysics Data System (ADS)

Khater, Mostafa M. A.; Seadawy, Aly R.; Lu, Dianchen

2018-06-01

In this research, we study new two techniques that called the extended simple equation method and the novel (G‧/G) -expansion method. The extended simple equation method depend on the auxiliary equation (dϕ/dξ = α + λϕ + μϕ2) which has three ways for solving depends on the specific condition on the parameters as follow: When (λ = 0) this auxiliary equation reduces to Riccati equation, when (α = 0) this auxiliary equation reduces to Bernoulli equation and when (α ≠ 0, λ ≠ 0, μ ≠ 0) we the general solutions of this auxiliary equation while the novel (G‧/G) -expansion method depends also on similar auxiliary equation (G‧/G)‧ = μ + λ(G‧/G) + (v - 1)(G‧/G) 2 which depend also on the value of (λ2 - 4 (v - 1) μ) and the specific condition on the parameters as follow: When (λ = 0) this auxiliary equation reduces to Riccati equation, when (μ = 0) this auxiliary equation reduces to Bernoulli equation and when (λ2 ≠ 4 (v - 1) μ) we the general solutions of this auxiliary equation. This show how both of these auxiliary equation are special cases of Riccati equation. We apply these methods on two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. We obtain the exact traveling wave solutions of these important models and under special condition on the parameters, we get solitary traveling wave solutions. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions.

Control of Supercavitation Flow and Stability of Supercavitating Motion of Bodies

DTIC Science & Technology

2001-02-01

sign opposite to a sign of angle Vf - accidental deflection of the model Sgn M = -Sgn i. 4.3. EQUATIONS OF THE SCM DYNAMICS The most effective method of...the motion stability in interactive regime "researcher - computer" [ 16]. The complete mathematical model of the SCM motion includes a set of equations ...of solid body dynamics, equations to calculate the unsteady cavity shape and relations to calculate the acting forces. A set of dynamic equations of

Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications

NASA Astrophysics Data System (ADS)

Ali, Asghar; Seadawy, Aly R.; Lu, Dianchen

2018-05-01

The aim of this article is to construct some new traveling wave solutions and investigate localized structures for fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) water wave dynamical equation. The simple equation method (SEM) and the modified simple equation method (MSEM) are applied in this paper to construct the analytical traveling wave solutions of AKNS equation. The different waves solutions are derived by assigning special values to the parameters. The obtained results have their importance in the field of physics and other areas of applied sciences. All the solutions are also graphically represented. The constructed results are often helpful for studying several new localized structures and the waves interaction in the high-dimensional models.

A new theoretical basis for numerical simulations of nonlinear acoustic fields

NASA Astrophysics Data System (ADS)

Wójcik, Janusz

2000-07-01

Nonlinear acoustic equations can be considerably simplified. The presented model retains the accuracy of a more complex description of nonlinearity and a uniform description of near and far fields (in contrast to the KZK equation). A method has been presented for obtaining solutions of Kuznetsov's equation from the solutions of the model under consideration. Results of numerical calculations, including comparative ones, are presented.

Non-Linear Acoustic Concealed Weapons Detector

DTIC Science & Technology

2006-05-01

signature analysis 8 the interactions of the beams with concealed objects. The Khokhlov- Zabolotskaya-Kuznetsov ( KZK ) equation is the most widely used...Hamilton developed a finite difference method based on the KZK equation to model pulsed acoustic emissions from axial symmetric sources. Using a...College of William & Mary, we have developed a simulation code using the KZK equation to model non-linear acoustic beams and visualize beam patterns

Quantum cluster variational method and message passing algorithms revisited

NASA Astrophysics Data System (ADS)

Domínguez, E.; Mulet, Roberto

2018-02-01

We present a general framework to study quantum disordered systems in the context of the Kikuchi's cluster variational method (CVM). The method relies in the solution of message passing-like equations for single instances or in the iterative solution of complex population dynamic algorithms for an average case scenario. We first show how a standard application of the Kikuchi's CVM can be easily translated to message passing equations for specific instances of the disordered system. We then present an "ad hoc" extension of these equations to a population dynamic algorithm representing an average case scenario. At the Bethe level, these equations are equivalent to the dynamic population equations that can be derived from a proper cavity ansatz. However, at the plaquette approximation, the interpretation is more subtle and we discuss it taking also into account previous results in classical disordered models. Moreover, we develop a formalism to properly deal with the average case scenario using a replica-symmetric ansatz within this CVM for quantum disordered systems. Finally, we present and discuss numerical solutions of the different approximations for the quantum transverse Ising model and the quantum random field Ising model in two-dimensional lattices.

An efficient mode-splitting method for a curvilinear nearshore circulation model

USGS Publications Warehouse

Shi, Fengyan; Kirby, James T.; Hanes, Daniel M.

2007-01-01

A mode-splitting method is applied to the quasi-3D nearshore circulation equations in generalized curvilinear coordinates. The gravity wave mode and the vorticity wave mode of the equations are derived using the two-step projection method. Using an implicit algorithm for the gravity mode and an explicit algorithm for the vorticity mode, we combine the two modes to derive a mixed difference–differential equation with respect to surface elevation. McKee et al.'s [McKee, S., Wall, D.P., and Wilson, S.K., 1996. An alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms. J. Comput. Phys., 126, 64–76.] ADI scheme is then used to solve the parabolic-type equation in dealing with the mixed derivative and convective terms from the curvilinear coordinate transformation. Good convergence rates are found in two typical cases which represent respectively the motions dominated by the gravity mode and the vorticity mode. Time step limitations imposed by the vorticity convective Courant number in vorticity-mode-dominant cases are discussed. Model efficiency and accuracy are verified in model application to tidal current simulations in San Francisco Bight.

Percent body fat estimations in college men using field and laboratory methods: a three-compartment model approach.

PubMed

Moon, Jordan R; Tobkin, Sarah E; Smith, Abbie E; Roberts, Michael D; Ryan, Eric D; Dalbo, Vincent J; Lockwood, Chris M; Walter, Ashley A; Cramer, Joel T; Beck, Travis W; Stout, Jeffrey R

2008-04-21

Methods used to estimate percent body fat can be classified as a laboratory or field technique. However, the validity of these methods compared to multiple-compartment models has not been fully established. The purpose of this study was to determine the validity of field and laboratory methods for estimating percent fat (%fat) in healthy college-age men compared to the Siri three-compartment model (3C). Thirty-one Caucasian men (22.5 +/- 2.7 yrs; 175.6 +/- 6.3 cm; 76.4 +/- 10.3 kg) had their %fat estimated by bioelectrical impedance analysis (BIA) using the BodyGram computer program (BIA-AK) and population-specific equation (BIA-Lohman), near-infrared interactance (NIR) (Futrex(R) 6100/XL), four circumference-based military equations [Marine Corps (MC), Navy and Air Force (NAF), Army (A), and Friedl], air-displacement plethysmography (BP), and hydrostatic weighing (HW). All circumference-based military equations (MC = 4.7% fat, NAF = 5.2% fat, A = 4.7% fat, Friedl = 4.7% fat) along with NIR (NIR = 5.1% fat) produced an unacceptable total error (TE). Both laboratory methods produced acceptable TE values (HW = 2.5% fat; BP = 2.7% fat). The BIA-AK, and BIA-Lohman field methods produced acceptable TE values (2.1% fat). A significant difference was observed for the MC and NAF equations compared to both the 3C model and HW (p < 0.006). Results indicate that the BP and HW are valid laboratory methods when compared to the 3C model to estimate %fat in college-age Caucasian men. When the use of a laboratory method is not feasible, BIA-AK, and BIA-Lohman are acceptable field methods to estimate %fat in this population.

Fast and accurate calculation of dilute quantum gas using Uehling–Uhlenbeck model equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yano, Ryosuke, E-mail: ryosuke.yano@tokiorisk.co.jp

The Uehling–Uhlenbeck (U–U) model equation is studied for the fast and accurate calculation of a dilute quantum gas. In particular, the direct simulation Monte Carlo (DSMC) method is used to solve the U–U model equation. DSMC analysis based on the U–U model equation is expected to enable the thermalization to be accurately obtained using a small number of sample particles and the dilute quantum gas dynamics to be calculated in a practical time. Finally, the applicability of DSMC analysis based on the U–U model equation to the fast and accurate calculation of a dilute quantum gas is confirmed by calculatingmore » the viscosity coefficient of a Bose gas on the basis of the Green–Kubo expression and the shock layer of a dilute Bose gas around a cylinder.« less

A moist Boussinesq shallow water equations set for testing atmospheric models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zerroukat, M., E-mail: mohamed.zerroukat@metoffice.gov.uk; Allen, T.

The shallow water equations have long been used as an initial test for numerical methods applied to atmospheric models with the test suite of Williamson et al. being used extensively for validating new schemes and assessing their accuracy. However the lack of physics forcing within this simplified framework often requires numerical techniques to be reworked when applied to fully three dimensional models. In this paper a novel two-dimensional shallow water equations system that retains moist processes is derived. This system is derived from three-dimensional Boussinesq approximation of the hydrostatic Euler equations where, unlike the classical shallow water set, we allowmore » the density to vary slightly with temperature. This results in extra (or buoyancy) terms for the momentum equations, through which a two-way moist-physics dynamics feedback is achieved. The temperature and moisture variables are advected as separate tracers with sources that interact with the mean-flow through a simplified yet realistic bulk moist-thermodynamic phase-change model. This moist shallow water system provides a unique tool to assess the usually complex and highly non-linear dynamics–physics interactions in atmospheric models in a simple yet realistic way. The full non-linear shallow water equations are solved numerically on several case studies and the results suggest quite realistic interaction between the dynamics and physics and in particular the generation of cloud and rain. - Highlights: • Novel shallow water equations which retains moist processes are derived from the three-dimensional hydrostatic Boussinesq equations. • The new shallow water set can be seen as a more general one, where the classical equations are a special case of these equations. • This moist shallow water system naturally allows a feedback mechanism from the moist physics increments to the momentum via buoyancy. • Like full models, temperature and moistures are advected as tracers that interact through a simplified yet realistic phase-change model. • This model is a unique tool to test numerical methods for atmospheric models, and physics–dynamics coupling, in a very realistic and simple way.« less

Solving the Rational Polynomial Coefficients Based on L Curve

NASA Astrophysics Data System (ADS)

Zhou, G.; Li, X.; Yue, T.; Huang, W.; He, C.; Huang, Y.

2018-05-01

The rational polynomial coefficients (RPC) model is a generalized sensor model, which can achieve high approximation accuracy. And it is widely used in the field of photogrammetry and remote sensing. Least square method is usually used to determine the optimal parameter solution of the rational function model. However the distribution of control points is not uniform or the model is over-parameterized, which leads to the singularity of the coefficient matrix of the normal equation. So the normal equation becomes ill conditioned equation. The obtained solutions are extremely unstable and even wrong. The Tikhonov regularization can effectively improve and solve the ill conditioned equation. In this paper, we calculate pathological equations by regularization method, and determine the regularization parameters by L curve. The results of the experiments on aerial format photos show that the accuracy of the first-order RPC with the equal denominators has the highest accuracy. The high order RPC model is not necessary in the processing of dealing with frame images, as the RPC model and the projective model are almost the same. The result shows that the first-order RPC model is basically consistent with the strict sensor model of photogrammetry. Orthorectification results both the firstorder RPC model and Camera Model (ERDAS9.2 platform) are similar to each other, and the maximum residuals of X and Y are 0.8174 feet and 0.9272 feet respectively. This result shows that RPC model can be used in the aerial photographic compensation replacement sensor model.

Estimating vapor pressures of pure liquids

DOE Office of Scientific and Technical Information (OSTI.GOV)

Haraburda, S.S.

1996-03-01

Calculating the vapor pressures for pure liquid chemicals is a key step in designing equipment for separation of liquid mixtures. Here is a useful way to develop an equation for predicting vapor pressures over a range of temperatures. The technique uses known vapor pressure points for different temperatures. Although a vapor-pressure equation is being showcased in this article, the basic method has much broader applicability -- in fact, users can apply it to develop equations for any temperature-dependent model. The method can be easily adapted for use in software programs for mathematics evaluation, minimizing the need for any programming. Themore » model used is the Antoine equation, which typically provides a good correlation with experimental or measured data.« less

Solvers for the Cardiac Bidomain Equations

PubMed Central

Vigmond, E.J.; Weber dos Santos, R.; Prassl, A.J.; Deo, M.; Plank, G.

2010-01-01

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modelling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations, and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient. PMID:17900668

One-dimensional transport equation models for sound energy propagation in long spaces: theory.

PubMed

Jing, Yun; Larsen, Edward W; Xiang, Ning

2010-04-01

In this paper, a three-dimensional transport equation model is developed to describe the sound energy propagation in a long space. Then this model is reduced to a one-dimensional model by approximating the solution using the method of weighted residuals. The one-dimensional transport equation model directly describes the sound energy propagation in the "long" dimension and deals with the sound energy in the "short" dimensions by prescribed functions. Also, the one-dimensional model consists of a coupled set of N transport equations. Only N=1 and N=2 are discussed in this paper. For larger N, although the accuracy could be improved, the calculation time is expected to significantly increase, which diminishes the advantage of the model in terms of its computational efficiency.

Solution of the Bagley Torvik equation by fractional DTM

NASA Astrophysics Data System (ADS)

Arora, Geeta; Pratiksha

2017-07-01

In this paper, fractional differential transform method(DTM) is implemented on the Bagley Torvik equation. This equation models the viscoelastic behavior of geological strata, metals, glasses etc. It explains the motion of a rigid plate immersed in a Newtonian fluid. DTM is a simple, reliable and efficient method that gives a series solution. Caputo fractional derivative is considered throughout this work. Two examples are given to demonstrate the validity and applicability of the method and comparison is made with the existing results.

Evaluation of an Approximate Method for Synthesizing Covariance Matrices for Use in Meta-Analytic SEM

ERIC Educational Resources Information Center

Beretvas, S. Natasha; Furlow, Carolyn F.

2006-01-01

Meta-analytic structural equation modeling (MA-SEM) is increasingly being used to assess model-fit for variables' interrelations synthesized across studies. MA-SEM researchers have analyzed synthesized correlation matrices using structural equation modeling (SEM) estimation that is designed for covariance matrices. This can produce incorrect…

Maximum Likelihood Estimation in Meta-Analytic Structural Equation Modeling

ERIC Educational Resources Information Center

Oort, Frans J.; Jak, Suzanne

2016-01-01

Meta-analytic structural equation modeling (MASEM) involves fitting models to a common population correlation matrix that is estimated on the basis of correlation coefficients that are reported by a number of independent studies. MASEM typically consist of two stages. The method that has been found to perform best in terms of statistical…

Application of different variants of the BEM in numerical modeling of bioheat transfer problems.

PubMed

Majchrzak, Ewa

2013-09-01

Heat transfer processes proceeding in the living organisms are described by the different mathematical models. In particular, the typical continuous model of bioheat transfer bases on the most popular Pennes equation, but the Cattaneo-Vernotte equation and the dual phase lag equation are also used. It should be pointed out that in parallel are also examined the vascular models, and then for the large blood vessels and tissue domain the energy equations are formulated separately. In the paper the different variants of the boundary element method as a tool of numerical solution of bioheat transfer problems are discussed. For the steady state problems and the vascular models the classical BEM algorithm and also the multiple reciprocity BEM are presented. For the transient problems connected with the heating of tissue, the various tissue models are considered for which the 1st scheme of the BEM, the BEM using discretization in time and the general BEM are applied. Examples of computations illustrate the possibilities of practical applications of boundary element method in the scope of bioheat transfer problems.

Two- and three-dimensional natural and mixed convection simulation using modular zonal models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wurtz, E.; Nataf, J.M.; Winkelmann, F.

We demonstrate the use of the zonal model approach, which is a simplified method for calculating natural and mixed convection in rooms. Zonal models use a coarse grid and use balance equations, state equations, hydrostatic pressure drop equations and power law equations of the form {ital m} = {ital C}{Delta}{sup {ital n}}. The advantage of the zonal approach and its modular implementation are discussed. The zonal model resolution of nonlinear equation systems is demonstrated for three cases: a 2-D room, a 3-D room and a pair of 3-D rooms separated by a partition with an opening. A sensitivity analysis withmore » respect to physical parameters and grid coarseness is presented. Results are compared to computational fluid dynamics (CFD) calculations and experimental data.« less

A Variational Assimilation Method for Satellite and Conventional Data: Model 2 (version 1)

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.

1991-01-01

The Model II variational data assimilation model is the second of the four variational models designed to blend diverse meteorological data into a dynamically constrained data set. Model II differs from Model I in that it includes the thermodynamic equation as the fifth dynamical constraint. Thus, Model II includes all five of the primative equations that govern atmospheric flow for a dry atmosphere.

Modal Substructuring of Geometrically Nonlinear Finite Element Models with Interface Reduction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kuether, Robert J.; Allen, Matthew S.; Hollkamp, Joseph J.

Substructuring methods have been widely used in structural dynamics to divide large, complicated finite element models into smaller substructures. For linear systems, many methods have been developed to reduce the subcomponents down to a low order set of equations using a special set of component modes, and these are then assembled to approximate the dynamics of a large scale model. In this paper, a substructuring approach is developed for coupling geometrically nonlinear structures, where each subcomponent is drastically reduced to a low order set of nonlinear equations using a truncated set of fixedinterface and characteristic constraint modes. The method usedmore » to extract the coefficients of the nonlinear reduced order model (NLROM) is non-intrusive in that it does not require any modification to the commercial FEA code, but computes the NLROM from the results of several nonlinear static analyses. The NLROMs are then assembled to approximate the nonlinear differential equations of the global assembly. The method is demonstrated on the coupling of two geometrically nonlinear plates with simple supports at all edges. The plates are joined at a continuous interface through the rotational degrees-of-freedom (DOF), and the nonlinear normal modes (NNMs) of the assembled equations are computed to validate the models. The proposed substructuring approach reduces a 12,861 DOF nonlinear finite element model down to only 23 DOF, while still accurately reproducing the first three NNMs of the full order model.« less

Modal Substructuring of Geometrically Nonlinear Finite Element Models with Interface Reduction

DOE PAGES

Kuether, Robert J.; Allen, Matthew S.; Hollkamp, Joseph J.

2017-03-29

Substructuring methods have been widely used in structural dynamics to divide large, complicated finite element models into smaller substructures. For linear systems, many methods have been developed to reduce the subcomponents down to a low order set of equations using a special set of component modes, and these are then assembled to approximate the dynamics of a large scale model. In this paper, a substructuring approach is developed for coupling geometrically nonlinear structures, where each subcomponent is drastically reduced to a low order set of nonlinear equations using a truncated set of fixedinterface and characteristic constraint modes. The method usedmore » to extract the coefficients of the nonlinear reduced order model (NLROM) is non-intrusive in that it does not require any modification to the commercial FEA code, but computes the NLROM from the results of several nonlinear static analyses. The NLROMs are then assembled to approximate the nonlinear differential equations of the global assembly. The method is demonstrated on the coupling of two geometrically nonlinear plates with simple supports at all edges. The plates are joined at a continuous interface through the rotational degrees-of-freedom (DOF), and the nonlinear normal modes (NNMs) of the assembled equations are computed to validate the models. The proposed substructuring approach reduces a 12,861 DOF nonlinear finite element model down to only 23 DOF, while still accurately reproducing the first three NNMs of the full order model.« less

Large-scale computations in fluid mechanics; Proceedings of the Fifteenth Summer Seminar on Applied Mathematics, University of California, La Jolla, CA, June 27-July 8, 1983. Parts 1 & 2

NASA Technical Reports Server (NTRS)

Engquist, B. E. (Editor); Osher, S. (Editor); Somerville, R. C. J. (Editor)

1985-01-01

Papers are presented on such topics as the use of semi-Lagrangian advective schemes in meteorological modeling; computation with high-resolution upwind schemes for hyperbolic equations; dynamics of flame propagation in a turbulent field; a modified finite element method for solving the incompressible Navier-Stokes equations; computational fusion magnetohydrodynamics; and a nonoscillatory shock capturing scheme using flux-limited dissipation. Consideration is also given to the use of spectral techniques in numerical weather prediction; numerical methods for the incorporation of mountains in atmospheric models; techniques for the numerical simulation of large-scale eddies in geophysical fluid dynamics; high-resolution TVD schemes using flux limiters; upwind-difference methods for aerodynamic problems governed by the Euler equations; and an MHD model of the earth's magnetosphere.

An explicit mixed numerical method for mesoscale model

NASA Technical Reports Server (NTRS)

Hsu, H.-M.

1981-01-01

A mixed numerical method has been developed for mesoscale models. The technique consists of a forward difference scheme for time tendency terms, an upstream scheme for advective terms, and a central scheme for the other terms in a physical system. It is shown that the mixed method is conditionally stable and highly accurate for approximating the system of either shallow-water equations in one dimension or primitive equations in three dimensions. Since the technique is explicit and two time level, it conserves computer and programming resources.

Beyond single-stream with the Schrödinger method

NASA Astrophysics Data System (ADS)

Uhlemann, Cora; Kopp, Michael

2016-10-01

We investigate large scale structure formation of collisionless dark matter in the phase space description based on the Vlasov-Poisson equation. We present the Schrödinger method, originally proposed by \\cite{WK93} as numerical technique based on the Schrödinger Poisson equation, as an analytical tool which is superior to the common standard pressureless fluid model. Whereas the dust model fails and develops singularities at shell crossing the Schrödinger method encompasses multi-streaming and even virialization.

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

NASA Astrophysics Data System (ADS)

Läuter, Matthias; Giraldo, Francis X.; Handorf, Dörthe; Dethloff, Klaus

2008-12-01

A global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R3, are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge-Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step. The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992) 211-224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of O(Δx) was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step Δt is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k.

A coupling method for a cardiovascular simulation model which includes the Kalman filter.

PubMed

Hasegawa, Yuki; Shimayoshi, Takao; Amano, Akira; Matsuda, Tetsuya

2012-01-01

Multi-scale models of the cardiovascular system provide new insight that was unavailable with in vivo and in vitro experiments. For the cardiovascular system, multi-scale simulations provide a valuable perspective in analyzing the interaction of three phenomenons occurring at different spatial scales: circulatory hemodynamics, ventricular structural dynamics, and myocardial excitation-contraction. In order to simulate these interactions, multiscale cardiovascular simulation systems couple models that simulate different phenomena. However, coupling methods require a significant amount of calculation, since a system of non-linear equations must be solved for each timestep. Therefore, we proposed a coupling method which decreases the amount of calculation by using the Kalman filter. In our method, the Kalman filter calculates approximations for the solution to the system of non-linear equations at each timestep. The approximations are then used as initial values for solving the system of non-linear equations. The proposed method decreases the number of iterations required by 94.0% compared to the conventional strong coupling method. When compared with a smoothing spline predictor, the proposed method required 49.4% fewer iterations.

The Equation, the Whole Equation, and Nothing but the Equation! One Approach to the Teaching of Linear Equations.

ERIC Educational Resources Information Center

Pirie, Susan E. B.; Martin, Lyndon

1997-01-01

Presents the results of a case study which looked at the mathematics classroom of one teacher trying to teach mathematics with meaning to pupils or lower ability at the secondary level. Contrasts methods of teaching linear equations to a variety of ability levels and uses the Pirie-Kieren model to account for the successful growth in understanding…

A Comparison of Four Linear Equating Methods for the Common-Item Nonequivalent Groups Design Using Simulation Methods. ACT Research Report Series, 2013 (2)

ERIC Educational Resources Information Center

Topczewski, Anna; Cui, Zhongmin; Woodruff, David; Chen, Hanwei; Fang, Yu

2013-01-01

This paper investigates four methods of linear equating under the common item nonequivalent groups design. Three of the methods are well known: Tucker, Angoff-Levine, and Congeneric-Levine. A fourth method is presented as a variant of the Congeneric-Levine method. Using simulation data generated from the three-parameter logistic IRT model we…

A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives

NASA Astrophysics Data System (ADS)

Macías-Díaz, J. E.; Hendy, A. S.; De Staelen, R. H.

2018-03-01

In this work, we investigate a general nonlinear wave equation with Riesz space-fractional derivatives that generalizes various classical hyperbolic models, including the sine-Gordon and the Klein-Gordon equations from relativistic quantum mechanics. A finite-difference discretization of the model is provided using fractional centered differences. The method is a technique that is capable of preserving an energy-like quantity at each iteration. Some computational comparisons against solutions available in the literature are performed in order to assess the capability of the method to preserve the invariant. Our experiments confirm that the technique yields good approximations to the solutions considered. As an application of our scheme, we provide simulations that confirm, for the first time in the literature, the presence of the phenomenon of nonlinear supratransmission in Riesz space-fractional Klein-Gordon equations driven by a harmonic perturbation at the boundary.

A new computational method for reacting hypersonic flows

NASA Astrophysics Data System (ADS)

Niculescu, M. L.; Cojocaru, M. G.; Pricop, M. V.; Fadgyas, M. C.; Pepelea, D.; Stoican, M. G.

2017-07-01

Hypersonic gas dynamics computations are challenging due to the difficulties to have reliable and robust chemistry models that are usually added to Navier-Stokes equations. From the numerical point of view, it is very difficult to integrate together Navier-Stokes equations and chemistry model equations because these partial differential equations have different specific time scales. For these reasons, almost all known finite volume methods fail shortly to solve this second order partial differential system. Unfortunately, the heating of Earth reentry vehicles such as space shuttles and capsules is very close linked to endothermic chemical reactions. A better prediction of wall heat flux leads to smaller safety coefficient for thermal shield of space reentry vehicle; therefore, the size of thermal shield decreases and the payload increases. For these reasons, the present paper proposes a new computational method based on chemical equilibrium, which gives accurate prediction of hypersonic heating in order to support the Earth reentry capsule design.

Incompressible Navier-Stokes Computations with Heat Transfer

NASA Technical Reports Server (NTRS)

Kiris, Cetin; Kwak, Dochan; Rogers, Stuart; Kutler, Paul (Technical Monitor)

1994-01-01

The existing pseudocompressibility method for the system of incompressible Navier-Stokes equations is extended to heat transfer problems by including the energy equation. The solution method is based on the pseudo compressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. Current computations use one-equation Baldwin-Barth turbulence model which is derived from a simplified form of the standard k-epsilon model equations. Both forced and natural convection problems are examined. Numerical results from turbulent reattaching flow behind a backward-facing step will be compared against experimental measurements for the forced convection case. The validity of Boussinesq approximation to simplify the buoyancy force term will be investigated. The natural convective flow structure generated by heat transfer in a vertical rectangular cavity will be studied. The numerical results will be compared by experimental measurements by Morrison and Tran.

Method of moving frames to solve time-dependent Maxwell's equations on anisotropic curved surfaces: Applications to invisible cloak and ELF propagation

NASA Astrophysics Data System (ADS)

Chun, Sehun

2017-07-01

Applying the method of moving frames to Maxwell's equations yields two important advancements for scientific computing. The first is the use of upwind flux for anisotropic materials in Maxwell's equations, especially in the context of discontinuous Galerkin (DG) methods. Upwind flux has been available only to isotropic material, because of the difficulty of satisfying the Rankine-Hugoniot conditions in anisotropic media. The second is to solve numerically Maxwell's equations on curved surfaces without the metric tensor and composite meshes. For numerical validation, spectral convergences are displayed for both two-dimensional anisotropic media and isotropic spheres. In the first application, invisible two-dimensional metamaterial cloaks are simulated with a relatively coarse mesh by both the lossless Drude model and the piecewisely-parametered layered model. In the second application, extremely low frequency propagation on various surfaces such as spheres, irregular surfaces, and non-convex surfaces is demonstrated.

Genetic network inference as a series of discrimination tasks.

PubMed

Kimura, Shuhei; Nakayama, Satoshi; Hatakeyama, Mariko

2009-04-01

Genetic network inference methods based on sets of differential equations generally require a great deal of time, as the equations must be solved many times. To reduce the computational cost, researchers have proposed other methods for inferring genetic networks by solving sets of differential equations only a few times, or even without solving them at all. When we try to obtain reasonable network models using these methods, however, we must estimate the time derivatives of the gene expression levels with great precision. In this study, we propose a new method to overcome the drawbacks of inference methods based on sets of differential equations. Our method infers genetic networks by obtaining classifiers capable of predicting the signs of the derivatives of the gene expression levels. For this purpose, we defined a genetic network inference problem as a series of discrimination tasks, then solved the defined series of discrimination tasks with a linear programming machine. Our experimental results demonstrated that the proposed method is capable of correctly inferring genetic networks, and doing so more than 500 times faster than the other inference methods based on sets of differential equations. Next, we applied our method to actual expression data of the bacterial SOS DNA repair system. And finally, we demonstrated that our approach relates to the inference method based on the S-system model. Though our method provides no estimation of the kinetic parameters, it should be useful for researchers interested only in the network structure of a target system. Supplementary data are available at Bioinformatics online.

Advantages of multigrid methods for certifying the accuracy of PDE modeling

NASA Technical Reports Server (NTRS)

Forester, C. K.

1981-01-01

Numerical techniques for assessing and certifying the accuracy of the modeling of partial differential equations (PDE) to the user's specifications are analyzed. Examples of the certification process with conventional techniques are summarized for the three dimensional steady state full potential and the two dimensional steady Navier-Stokes equations using fixed grid methods (FG). The advantages of the Full Approximation Storage (FAS) scheme of the multigrid technique of A. Brandt compared with the conventional certification process of modeling PDE are illustrated in one dimension with the transformed potential equation. Inferences are drawn for how MG will improve the certification process of the numerical modeling of two and three dimensional PDE systems. Elements of the error assessment process that are common to FG and MG are analyzed.

On the dynamics of chain systems. [applications in manipulator and human body models

NASA Technical Reports Server (NTRS)

Huston, R. L.; Passerello, C. E.

1974-01-01

A computer-oriented method for obtaining dynamical equations of motion for chain systems is presented. A chain system is defined as an arbitrarily assembled set of rigid bodies such that adjoining bodies have at least one common point and such that closed loops are not formed. The equations of motion are developed through the use of Lagrange's form of d'Alembert's principle. The method and procedure is illustrated with an elementary study of a tripod space manipulator. The method is designed for application with systems such as human body models, chains and cables, and dynamic finite-segment models.

The Complex-Step-Finite-Difference method

NASA Astrophysics Data System (ADS)

Abreu, Rafael; Stich, Daniel; Morales, Jose

2015-07-01

We introduce the Complex-Step-Finite-Difference method (CSFDM) as a generalization of the well-known Finite-Difference method (FDM) for solving the acoustic and elastic wave equations. We have found a direct relationship between modelling the second-order wave equation by the FDM and the first-order wave equation by the CSFDM in 1-D, 2-D and 3-D acoustic media. We present the numerical methodology in order to apply the introduced CSFDM and show an example for wave propagation in simple homogeneous and heterogeneous models. The CSFDM may be implemented as an extension into pre-existing numerical techniques in order to obtain fourth- or sixth-order accurate results with compact three time-level stencils. We compare advantages of imposing various types of initial motion conditions of the CSFDM and demonstrate its higher-order accuracy under the same computational cost and dispersion-dissipation properties. The introduced method can be naturally extended to solve different partial differential equations arising in other fields of science and engineering.

Two-dimensional electronic spectra from the hierarchical equations of motion method: Application to model dimers

NASA Astrophysics Data System (ADS)

Chen, Liping; Zheng, Renhui; Shi, Qiang; Yan, YiJing

2010-01-01

We extend our previous study of absorption line shapes of molecular aggregates using the Liouville space hierarchical equations of motion (HEOM) method [L. P. Chen, R. H. Zheng, Q. Shi, and Y. J. Yan, J. Chem. Phys. 131, 094502 (2009)] to calculate third order optical response functions and two-dimensional electronic spectra of model dimers. As in our previous work, we have focused on the applicability of several approximate methods related to the HEOM method. We show that while the second order perturbative quantum master equations are generally inaccurate in describing the peak shapes and solvation dynamics, they can give reasonable peak amplitude evolution even in the intermediate coupling regime. The stochastic Liouville equation results in good peak shapes, but does not properly describe the excited state dynamics due to the lack of detailed balance. A modified version of the high temperature approximation to the HEOM gives the best agreement with the exact result.

Artificial neural network methods in quantum mechanics

NASA Astrophysics Data System (ADS)

Lagaris, I. E.; Likas, A.; Fotiadis, D. I.

1997-08-01

In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for differential and integrodifferential operators, using ANNs. We start by considering the Schrödinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schrödinger and the Dirac equations for a muonic atom, as well as with a nonlocal Schrödinger integrodifferential equation that models the n + α system in the framework of the resonating group method. In two dimensions we consider the well-studied Henon-Heiles Hamiltonian and in three dimensions the model problem of three coupled anharmonic oscillators. The method in all of the treated cases proved to be highly accurate, robust and efficient. Hence it is a promising tool for tackling problems of higher complexity and dimensionality.

Nonlinear analysis of 0-3 polarized PLZT microplate based on the new modified couple stress theory

NASA Astrophysics Data System (ADS)

Wang, Liming; Zheng, Shijie

2018-02-01

In this study, based on the new modified couple stress theory, the size- dependent model for nonlinear bending analysis of a pure 0-3 polarized PLZT plate is developed for the first time. The equilibrium equations are derived from a variational formulation based on the potential energy principle and the new modified couple stress theory. The Galerkin method is adopted to derive the nonlinear algebraic equations from governing differential equations. And then the nonlinear algebraic equations are solved by using Newton-Raphson method. After simplification, the new model includes only a material length scale parameter. In addition, numerical examples are carried out to study the effect of material length scale parameter on the nonlinear bending of a simply supported pure 0-3 polarized PLZT plate subjected to light illumination and uniform distributed load. The results indicate the new model is able to capture the size effect and geometric nonlinearity.

Photonic band gap structure simulator

DOEpatents

Chen, Chiping; Shapiro, Michael A.; Smirnova, Evgenya I.; Temkin, Richard J.; Sirigiri, Jagadishwar R.

2006-10-03

A system and method for designing photonic band gap structures. The system and method provide a user with the capability to produce a model of a two-dimensional array of conductors corresponding to a unit cell. The model involves a linear equation. Boundary conditions representative of conditions at the boundary of the unit cell are applied to a solution of the Helmholtz equation defined for the unit cell. The linear equation can be approximated by a Hermitian matrix. An eigenvalue of the Helmholtz equation is calculated. One computation approach involves calculating finite differences. The model can include a symmetry element, such as a center of inversion, a rotation axis, and a mirror plane. A graphical user interface is provided for the user's convenience. A display is provided to display to a user the calculated eigenvalue, corresponding to a photonic energy level in the Brilloin zone of the unit cell.

Dissipation-preserving spectral element method for damped seismic wave equations

NASA Astrophysics Data System (ADS)

Cai, Wenjun; Zhang, Huai; Wang, Yushun

2017-12-01

This article describes the extension of the conformal symplectic method to solve the damped acoustic wave equation and the elastic wave equations in the framework of the spectral element method. The conformal symplectic method is a variation of conventional symplectic methods to treat non-conservative time evolution problems, which has superior behaviors in long-time stability and dissipation preservation. To reveal the intrinsic dissipative properties of the model equations, we first reformulate the original systems in their equivalent conformal multi-symplectic structures and derive the corresponding conformal symplectic conservation laws. We thereafter separate each system into a conservative Hamiltonian system and a purely dissipative ordinary differential equation system. Based on the splitting methodology, we solve the two subsystems respectively. The dissipative one is cheaply solved by its analytic solution. While for the conservative system, we combine a fourth-order symplectic Nyström method in time and the spectral element method in space to cover the circumstances in realistic geological structures involving complex free-surface topography. The Strang composition method is adopted thereby to concatenate the corresponding two parts of solutions and generate the completed conformal symplectic method. A relative larger Courant number than that of the traditional Newmark scheme is found in the numerical experiments in conjunction with a spatial sampling of approximately 5 points per wavelength. A benchmark test for the damped acoustic wave equation validates the effectiveness of our proposed method in precisely capturing dissipation rate. The classical Lamb problem is used to demonstrate the ability of modeling Rayleigh wave in elastic wave propagation. More comprehensive numerical experiments are presented to investigate the long-time simulation, low dispersion and energy conservation properties of the conformal symplectic methods in both the attenuating homogeneous and heterogeneous media.

Gaussian closure technique applied to the hysteretic Bouc model with non-zero mean white noise excitation

NASA Astrophysics Data System (ADS)

Waubke, Holger; Kasess, Christian H.

2016-11-01

Devices that emit structure-borne sound are commonly decoupled by elastic components to shield the environment from acoustical noise and vibrations. The elastic elements often have a hysteretic behavior that is typically neglected. In order to take hysteretic behavior into account, Bouc developed a differential equation for such materials, especially joints made of rubber or equipped with dampers. In this work, the Bouc model is solved by means of the Gaussian closure technique based on the Kolmogorov equation. Kolmogorov developed a method to derive probability density functions for arbitrary explicit first-order vector differential equations under white noise excitation using a partial differential equation of a multivariate conditional probability distribution. Up to now no analytical solution of the Kolmogorov equation in conjunction with the Bouc model exists. Therefore a wide range of approximate solutions, especially the statistical linearization, were developed. Using the Gaussian closure technique that is an approximation to the Kolmogorov equation assuming a multivariate Gaussian distribution an analytic solution is derived in this paper for the Bouc model. For the stationary case the two methods yield equivalent results, however, in contrast to statistical linearization the presented solution allows to calculate the transient behavior explicitly. Further, stationary case leads to an implicit set of equations that can be solved iteratively with a small number of iterations and without instabilities for specific parameter sets.

Evolutionary optimization with data collocation for reverse engineering of biological networks.

PubMed

Tsai, Kuan-Yao; Wang, Feng-Sheng

2005-04-01

Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.

A FEniCS-based programming framework for modeling turbulent flow by the Reynolds-averaged Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Mortensen, Mikael; Langtangen, Hans Petter; Wells, Garth N.

2011-09-01

Finding an appropriate turbulence model for a given flow case usually calls for extensive experimentation with both models and numerical solution methods. This work presents the design and implementation of a flexible, programmable software framework for assisting with numerical experiments in computational turbulence. The framework targets Reynolds-averaged Navier-Stokes models, discretized by finite element methods. The novel implementation makes use of Python and the FEniCS package, the combination of which leads to compact and reusable code, where model- and solver-specific code resemble closely the mathematical formulation of equations and algorithms. The presented ideas and programming techniques are also applicable to other fields that involve systems of nonlinear partial differential equations. We demonstrate the framework in two applications and investigate the impact of various linearizations on the convergence properties of nonlinear solvers for a Reynolds-averaged Navier-Stokes model.

Simulation on Thermocapillary-Driven Drop Coalescence by Hybrid Lattice Boltzmann Method

NASA Astrophysics Data System (ADS)

Xie, Haiqiong; Zeng, Zhong; Zhang, Liangqi; Yokota, Yuui; Kawazoe, Yoshiyuki; Yoshikawa, Akira

2016-04-01

A hybrid two-phase model, incorporating lattice Boltzmann method (LBM) and finite difference method (FDM), was developed to investigate the coalescence of two drops during their thermocapillary migration. The lattice Boltzmann method with a multi-relaxation-time (MRT) collision model was applied to solve the flow field for incompressible binary fluids, and the method was implemented in an axisymmetric form. The deformation of the drop interface was captured with the phase-field theory, and the continuum surface force model (CSF) was adopted to introduce the surface tension, which depends on the temperature. Both phase-field equation and the energy equation were solved with the finite difference method. The effects of Marangoni number and Capillary numbers on the drop's motion and coalescence were investigated.

Generating a Multiphase Equation of State with Swarm Intelligence

NASA Astrophysics Data System (ADS)

Cox, Geoffrey

2017-06-01

Hydrocode calculations require knowledge of the variation of pressure of a material with density and temperature, which is given by the equation of state. An accurate model needs to account for discontinuities in energy, density and properties of a material across a phase boundary. When generating a multiphase equation of state the modeller attempts to balance the agreement between the available data for compression, expansion and phase boundary location. However, this can prove difficult because minor adjustments in the equation of state for a single phase can have a large impact on the overall phase diagram. Recently, Cox and Christie described a method for combining statistical-mechanics-based condensed matter physics models with a stochastic analysis technique called particle swarm optimisation. The models produced show good agreement with experiment over a wide range of pressure-temperature space. This talk details the general implementation of this technique, shows example results, and describes the types of analysis that can be performed with this method.

Update to core reporting practices in structural equation modeling.

PubMed

Schreiber, James B

This paper is a technical update to "Core Reporting Practices in Structural Equation Modeling." 1 As such, the content covered in this paper includes, sample size, missing data, specification and identification of models, estimation method choices, fit and residual concerns, nested, alternative, and equivalent models, and unique issues within the SEM family of techniques. Copyright © 2016 Elsevier Inc. All rights reserved.

A Correlation-Based Transition Model using Local Variables. Part 2; Test Cases and Industrial Applications

NASA Technical Reports Server (NTRS)

Langtry, R. B.; Menter, F. R.; Likki, S. R.; Suzen, Y. B.; Huang, P. G.; Volker, S.

2006-01-01

A new correlation-based transition model has been developed, which is built strictly on local variables. As a result, the transition model is compatible with modern computational fluid dynamics (CFD) methods using unstructured grids and massive parallel execution. The model is based on two transport equations, one for the intermittency and one for the transition onset criteria in terms of momentum thickness Reynolds number. The proposed transport equations do not attempt to model the physics of the transition process (unlike, e.g., turbulence models), but form a framework for the implementation of correlation-based models into general-purpose CFD methods.

A Dynamic Model for Modern Military Conflict.

DTIC Science & Technology

1982-10-01

within the generalized form of the Lotka - Volterra equations, that can account for the important interactions of modern military conflicts described in...Mx + These equations are seen to be of the generalized Lotka - Volterra form; however, only 20 of the 32 parameters that might possibly be included...determine one equilibrium as the solution of a system of linear equations. The method is derived for the general nth order Lotka - Volterra model in

A collocation--Galerkin finite element model of cardiac action potential propagation.

PubMed

Rogers, J M; McCulloch, A D

1994-08-01

A new computational method was developed for modeling the effects of the geometric complexity, nonuniform muscle fiber orientation, and material inhomogeneity of the ventricular wall on cardiac impulse propagation. The method was used to solve a modification to the FitzHugh-Nagumo system of equations. The geometry, local muscle fiber orientation, and material parameters of the domain were defined using linear Lagrange or cubic Hermite finite element interpolation. Spatial variations of time-dependent excitation and recovery variables were approximated using cubic Hermite finite element interpolation, and the governing finite element equations were assembled using the collocation method. To overcome the deficiencies of conventional collocation methods on irregular domains, Galerkin equations for the no-flux boundary conditions were used instead of collocation equations for the boundary degrees-of-freedom. The resulting system was evolved using an adaptive Runge-Kutta method. Converged two-dimensional simulations of normal propagation showed that this method requires less CPU time than a traditional finite difference discretization. The model also reproduced several other physiologic phenomena known to be important in arrhythmogenesis including: Wenckebach periodicity, slowed propagation and unidirectional block due to wavefront curvature, reentry around a fixed obstacle, and spiral wave reentry. In a new result, we observed wavespeed variations and block due to nonuniform muscle fiber orientation. The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques.

Computation of incompressible viscous flows through turbopump components

NASA Technical Reports Server (NTRS)

Kiris, Cetin; Chang, Leon

1993-01-01

Flow through pump components, such as an inducer and an impeller, is efficiently simulated by solving the incompressible Navier-Stokes equations. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. the equations are solved in steadily rotating reference frames and the centrifugal force and the Coriolis force are added to the equation of motion. Current computations use a one-equation Baldwin-Barth turbulence model which is derived from a simplified form of the standard k-epsilon model equations. The resulting computer code is applied to the flow analysis inside a generic rocket engine pump inducer, a fuel pump impeller, and SSME high pressure fuel turbopump impeller. Numerical results of inducer flow are compared with experimental measurements. In the fuel pump impeller, the effect of downstream boundary conditions is investigated. Flow analyses at 80 percent, 100 percent, and 120 percent of design conditions are presented.

Simultaneous Heat and Mass Transfer Model for Convective Drying of Building Material

NASA Astrophysics Data System (ADS)

Upadhyay, Ashwani; Chandramohan, V. P.

2018-04-01

A mathematical model of simultaneous heat and moisture transfer is developed for convective drying of building material. A rectangular brick is considered for sample object. Finite-difference method with semi-implicit scheme is used for solving the transient governing heat and mass transfer equation. Convective boundary condition is used, as the product is exposed in hot air. The heat and mass transfer equations are coupled through diffusion coefficient which is assumed as the function of temperature of the product. Set of algebraic equations are generated through space and time discretization. The discretized algebraic equations are solved by Gauss-Siedel method via iteration. Grid and time independent studies are performed for finding the optimum number of nodal points and time steps respectively. A MATLAB computer code is developed to solve the heat and mass transfer equations simultaneously. Transient heat and mass transfer simulations are performed to find the temperature and moisture distribution inside the brick.

Numerical Modeling of Saturated Boiling in a Heated Tube

NASA Technical Reports Server (NTRS)

Majumdar, Alok; LeClair, Andre; Hartwig, Jason

2017-01-01

This paper describes a mathematical formulation and numerical solution of boiling in a heated tube. The mathematical formulation involves a discretization of the tube into a flow network consisting of fluid nodes and branches and a thermal network consisting of solid nodes and conductors. In the fluid network, the mass, momentum and energy conservation equations are solved and in the thermal network, the energy conservation equation of solids is solved. A pressure-based, finite-volume formulation has been used to solve the equations in the fluid network. The system of equations is solved by a hybrid numerical scheme which solves the mass and momentum conservation equations by a simultaneous Newton-Raphson method and the energy conservation equation by a successive substitution method. The fluid network and thermal network are coupled through heat transfer between the solid and fluid nodes which is computed by Chen's correlation of saturated boiling heat transfer. The computer model is developed using the Generalized Fluid System Simulation Program and the numerical predictions are compared with test data.

Stoichiometric network analysis and associated dimensionless kinetic equations. Application to a model of the Bray-Liebhafsky reaction.

PubMed

Schmitz, Guy; Kolar-Anić, Ljiljana Z; Anić, Slobodan R; Cupić, Zeljko D

2008-12-25

The stoichiometric network analysis (SNA) introduced by B. L. Clarke is applied to a simplified model of the complex oscillating Bray-Liebhafsky reaction under batch conditions, which was not examined by this method earlier. This powerful method for the analysis of steady-states stability is also used to transform the classical differential equations into dimensionless equations. This transformation is easy and leads to a form of the equations combining the advantages of classical dimensionless equations with the advantages of the SNA. The used dimensionless parameters have orders of magnitude given by the experimental information about concentrations and currents. This simplifies greatly the study of the slow manifold and shows which parameters are essential for controlling its shape and consequently have an important influence on the trajectories. The effectiveness of these equations is illustrated on two examples: the study of the bifurcations points and a simple sensitivity analysis, different from the classical one, more based on the chemistry of the studied system.

Score Equating and Item Response Theory: Some Practical Considerations.

ERIC Educational Resources Information Center

Cook, Linda L.; Eignor, Daniel R.

The purposes of this paper are five-fold to discuss: (1) when item response theory (IRT) equating methods should provide better results than traditional methods; (2) which IRT model, the three-parameter logistic or the one-parameter logistic (Rasch), is the most reasonable to use; (3) what unique contributions IRT methods can offer the equating…

Acoustic 3D modeling by the method of integral equations

NASA Astrophysics Data System (ADS)

Malovichko, M.; Khokhlov, N.; Yavich, N.; Zhdanov, M.

2018-02-01

This paper presents a parallel algorithm for frequency-domain acoustic modeling by the method of integral equations (IE). The algorithm is applied to seismic simulation. The IE method reduces the size of the problem but leads to a dense system matrix. A tolerable memory consumption and numerical complexity were achieved by applying an iterative solver, accompanied by an effective matrix-vector multiplication operation, based on the fast Fourier transform (FFT). We demonstrate that, the IE system matrix is better conditioned than that of the finite-difference (FD) method, and discuss its relation to a specially preconditioned FD matrix. We considered several methods of matrix-vector multiplication for the free-space and layered host models. The developed algorithm and computer code were benchmarked against the FD time-domain solution. It was demonstrated that, the method could accurately calculate the seismic field for the models with sharp material boundaries and a point source and receiver located close to the free surface. We used OpenMP to speed up the matrix-vector multiplication, while MPI was used to speed up the solution of the system equations, and also for parallelizing across multiple sources. The practical examples and efficiency tests are presented as well.

Final Report for''Numerical Methods and Studies of High-Speed Reactive and Non-Reactive Flows''

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schwendeman, D W

2002-11-20

The work carried out under this subcontract involved the development and use of an adaptive numerical method for the accurate calculation of high-speed reactive flows on overlapping grids. The flow is modeled by the reactive Euler equations with an assumed equation of state and with various reaction rate models. A numerical method has been developed to solve the nonlinear hyperbolic partial differential equations in the model. The method uses an unsplit, shock-capturing scheme, and uses a Godunov-type scheme to compute fluxes and a Runge-Kutta error control scheme to compute the source term modeling the chemical reactions. An adaptive mesh refinementmore » (AMR) scheme has been implemented in order to locally increase grid resolution. The numerical method uses composite overlapping grids to handle complex flow geometries. The code is part of the ''Overture-OverBlown'' framework of object-oriented codes [1, 2], and the development has occurred in close collaboration with Bill Henshaw and David Brown, and other members of the Overture team within CASC. During the period of this subcontract, a number of tasks were accomplished, including: (1) an extension of the numerical method to handle ''ignition and grow'' reaction models and a JWL equations of state; (2) an improvement in the efficiency of the AMR scheme and the error estimator; (3) an addition of a scheme of numerical dissipation designed to suppress numerical oscillations/instabilities near expanding detonations and along grid overlaps; and (4) an exploration of the evolution to detonation in an annulus and of detonation failure in an expanding channel.« less

Well-posedness and decay for the dissipative system modeling electro-hydrodynamics in negative Besov spaces

NASA Astrophysics Data System (ADS)

Zhao, Jihong; Liu, Qiao

2017-07-01

In Guo and Wang (2012) [10], Y. Guo and Y. Wang developed a general new energy method for proving the optimal time decay rates of the solutions to dissipative equations. In this paper, we generalize this method in the framework of homogeneous Besov spaces. Moreover, we apply this method to a model arising from electro-hydrodynamics, which is a strongly coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms. We show that some weighted negative Besov norms of solutions are preserved along time evolution, and obtain the optimal time decay rates of the higher-order spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques.

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

NASA Technical Reports Server (NTRS)

Molenaar, J.

1996-01-01

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Exact solutions and conservation laws of the system of two-dimensional viscous Burgers equations

NASA Astrophysics Data System (ADS)

Abdulwahhab, Muhammad Alim

2016-10-01

Fluid turbulence is one of the phenomena that has been studied extensively for many decades. Due to its huge practical importance in fluid dynamics, various models have been developed to capture both the indispensable physical quality and the mathematical structure of turbulent fluid flow. Among the prominent equations used for gaining in-depth insight of fluid turbulence is the two-dimensional Burgers equations. Its solutions have been studied by researchers through various methods, most of which are numerical. Being a simplified form of the two-dimensional Navier-Stokes equations and its wide range of applicability in various fields of science and engineering, development of computationally efficient methods for the solution of the two-dimensional Burgers equations is still an active field of research. In this study, Lie symmetry method is used to perform detailed analysis on the system of two-dimensional Burgers equations. Optimal system of one-dimensional subalgebras up to conjugacy is derived and used to obtain distinct exact solutions. These solutions not only help in understanding the physical effects of the model problem but also, can serve as benchmarks for constructing algorithms and validation of numerical solutions of the system of Burgers equations under consideration at finite Reynolds numbers. Independent and nontrivial conserved vectors are also constructed.

Non-linear analysis of wave progagation using transform methods and plates and shells using integral equations

NASA Astrophysics Data System (ADS)

Pipkins, Daniel Scott

Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and non-linear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the non-linear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed non-linear terms. The non-linear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed non-linear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and non-linear systems. The linear systems modeled are longitudinal and torsional rods and Bernoulli-Euler and Timoshenko beams. For non-linear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells under-going finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially.

Modeling Latent Interactions at Level 2 in Multilevel Structural Equation Models: An Evaluation of Mean-Centered and Residual-Centered Unconstrained Approaches

ERIC Educational Resources Information Center

Leite, Walter L.; Zuo, Youzhen

2011-01-01

Among the many methods currently available for estimating latent variable interactions, the unconstrained approach is attractive to applied researchers because of its relatively easy implementation with any structural equation modeling (SEM) software. Using a Monte Carlo simulation study, we extended and evaluated the unconstrained approach to…

Stochastic differential equation (SDE) model of opening gold share price of bursa saham malaysia

NASA Astrophysics Data System (ADS)

Hussin, F. N.; Rahman, H. A.; Bahar, A.

2017-09-01

Black and Scholes option pricing model is one of the most recognized stochastic differential equation model in mathematical finance. Two parameter estimation methods have been utilized for the Geometric Brownian model (GBM); historical and discrete method. The historical method is a statistical method which uses the property of independence and normality logarithmic return, giving out the simplest parameter estimation. Meanwhile, discrete method considers the function of density of transition from the process of diffusion normal log which has been derived from maximum likelihood method. These two methods are used to find the parameter estimates samples of Malaysians Gold Share Price data such as: Financial Times and Stock Exchange (FTSE) Bursa Malaysia Emas, and Financial Times and Stock Exchange (FTSE) Bursa Malaysia Emas Shariah. Modelling of gold share price is essential since fluctuation of gold affects worldwide economy nowadays, including Malaysia. It is found that discrete method gives the best parameter estimates than historical method due to the smallest Root Mean Square Error (RMSE) value.

Two-Layer Viscous Shallow-Water Equations and Conservation Laws

NASA Astrophysics Data System (ADS)

Kanayama, Hiroshi; Dan, Hiroshi

In our previous papers, the two-layer viscous shallow-water equations were derived from the three-dimensional Navier-Stokes equations under the hydrostatic assumption. Also, it was noted that the combination of upper and lower equations in the two-layer model produces the classical one-layer equations if the density of each layer is the same. Then, the two-layer equations were approximated by a finite element method which followed our numerical scheme established for the one-layer model in 1978. Also, it was numerically demonstrated that the interfacial instability generated when the densities are the same can be eliminated by providing a sufficient density difference. In this paper, we newly show that conservation laws are still valid in the two-layer model. Also, we show results of a new physical experiment for the interfacial instability.

Application of a soft computing technique in predicting the percentage of shear force carried by walls in a rectangular channel with non-homogeneous roughness.

PubMed

Khozani, Zohreh Sheikh; Bonakdari, Hossein; Zaji, Amir Hossein

2016-01-01

Two new soft computing models, namely genetic programming (GP) and genetic artificial algorithm (GAA) neural network (a combination of modified genetic algorithm and artificial neural network methods) were developed in order to predict the percentage of shear force in a rectangular channel with non-homogeneous roughness. The ability of these methods to estimate the percentage of shear force was investigated. Moreover, the independent parameters' effectiveness in predicting the percentage of shear force was determined using sensitivity analysis. According to the results, the GP model demonstrated superior performance to the GAA model. A comparison was also made between the GP program determined as the best model and five equations obtained in prior research. The GP model with the lowest error values (root mean square error ((RMSE) of 0.0515) had the best function compared with the other equations presented for rough and smooth channels as well as smooth ducts. The equation proposed for rectangular channels with rough boundaries (RMSE of 0.0642) outperformed the prior equations for smooth boundaries.

External intermittency prediction using AMR solutions of RANS turbulence and transported PDF models

NASA Astrophysics Data System (ADS)

Olivieri, D. A.; Fairweather, M.; Falle, S. A. E. G.

2011-12-01

External intermittency in turbulent round jets is predicted using a Reynolds-averaged Navier-Stokes modelling approach coupled to solutions of the transported probability density function (pdf) equation for scalar variables. Solutions to the descriptive equations are obtained using a finite-volume method, combined with an adaptive mesh refinement algorithm, applied in both physical and compositional space. This method contrasts with conventional approaches to solving the transported pdf equation which generally employ Monte Carlo techniques. Intermittency-modified eddy viscosity and second-moment turbulence closures are used to accommodate the effects of intermittency on the flow field, with the influence of intermittency also included, through modifications to the mixing model, in the transported pdf equation. Predictions of the overall model are compared with experimental data on the velocity and scalar fields in a round jet, as well as against measurements of intermittency profiles and scalar pdfs in a number of flows, with good agreement obtained. For the cases considered, predictions based on the second-moment turbulence closure are clearly superior, although both turbulence models give realistic predictions of the bimodal scalar pdfs observed experimentally.

Sensitivity Analysis of Hydraulic Head to Locations of Model Boundaries

DOE PAGES

Lu, Zhiming

2018-01-30

Sensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g. hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, wemore » derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. Finally, the approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.« less

Sensitivity Analysis of Hydraulic Head to Locations of Model Boundaries

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lu, Zhiming

Sensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g. hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, wemore » derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. Finally, the approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.« less

Seismic wavefield modeling based on time-domain symplectic and Fourier finite-difference method

NASA Astrophysics Data System (ADS)

Fang, Gang; Ba, Jing; Liu, Xin-xin; Zhu, Kun; Liu, Guo-Chang

2017-06-01

Seismic wavefield modeling is important for improving seismic data processing and interpretation. Calculations of wavefield propagation are sometimes not stable when forward modeling of seismic wave uses large time steps for long times. Based on the Hamiltonian expression of the acoustic wave equation, we propose a structure-preserving method for seismic wavefield modeling by applying the symplectic finite-difference method on time grids and the Fourier finite-difference method on space grids to solve the acoustic wave equation. The proposed method is called the symplectic Fourier finite-difference (symplectic FFD) method, and offers high computational accuracy and improves the computational stability. Using acoustic approximation, we extend the method to anisotropic media. We discuss the calculations in the symplectic FFD method for seismic wavefield modeling of isotropic and anisotropic media, and use the BP salt model and BP TTI model to test the proposed method. The numerical examples suggest that the proposed method can be used in seismic modeling of strongly variable velocities, offering high computational accuracy and low numerical dispersion. The symplectic FFD method overcomes the residual qSV wave of seismic modeling in anisotropic media and maintains the stability of the wavefield propagation for large time steps.

Multiphysics modeling of non-linear laser-matter interactions for optically active semiconductors

NASA Astrophysics Data System (ADS)

Kraczek, Brent; Kanp, Jaroslaw

Development of photonic devices for sensors and communications devices has been significantly enhanced by computational modeling. We present a new computational method for modelling laser propagation in optically-active semiconductors within the paraxial wave approximation (PWA). Light propagation is modeled using the Streamline-upwind/Petrov-Galerkin finite element method (FEM). Material response enters through the non-linear polarization, which serves as the right-hand side of the FEM calculation. Maxwell's equations for classical light propagation within the PWA can be written solely in terms of the electric field, producing a wave equation that is a form of the advection-diffusion-reaction equations (ADREs). This allows adaptation of the computational machinery developed for solving ADREs in fluid dynamics to light-propagation modeling. The non-linear polarization is incorporated using a flexible framework to enable the use of multiple methods for carrier-carrier interactions (e.g. relaxation-time-based or Monte Carlo) to enter through the non-linear polarization, as appropriate to the material type. We demonstrate using a simple carrier-carrier model approximating the response of GaN. Supported by ARL Materials Enterprise.

Flow discharge prediction in compound channels using linear genetic programming

NASA Astrophysics Data System (ADS)

Azamathulla, H. Md.; Zahiri, A.

2012-08-01

SummaryFlow discharge determination in rivers is one of the key elements in mathematical modelling in the design of river engineering projects. Because of the inundation of floodplains and sudden changes in river geometry, flow resistance equations are not applicable for compound channels. Therefore, many approaches have been developed for modification of flow discharge computations. Most of these methods have satisfactory results only in laboratory flumes. Due to the ability to model complex phenomena, the artificial intelligence methods have recently been employed for wide applications in various fields of water engineering. Linear genetic programming (LGP), a branch of artificial intelligence methods, is able to optimise the model structure and its components and to derive an explicit equation based on the variables of the phenomena. In this paper, a precise dimensionless equation has been derived for prediction of flood discharge using LGP. The proposed model was developed using published data compiled for stage-discharge data sets for 394 laboratories, and field of 30 compound channels. The results indicate that the LGP model has a better performance than the existing models.

Scaling Laws Applied to a Modal Formulation of the Aeroservoelastic Equations

NASA Technical Reports Server (NTRS)

Pototzky, Anthony S.

2002-01-01

A method of scaling is described that easily converts the aeroelastic equations of motion of a full-sized aircraft into ones of a wind-tunnel model. To implement the method, a set of rules is provided for the conversion process involving matrix operations with scale factors. In addition, a technique for analytically incorporating a spring mounting system into the aeroelastic equations is also presented. As an example problem, a finite element model of a full-sized aircraft is introduced from the High Speed Research (HSR) program to exercise the scaling method. With a set of scale factor values, a brief outline is given of a procedure to generate the first-order aeroservoelastic analytical model representing the wind-tunnel model. To verify the scaling process as applied to the example problem, the root-locus patterns from the full-sized vehicle and the wind-tunnel model are compared to see if the root magnitudes scale with the frequency scale factor value. Selected time-history results are given from a numerical simulation of an active-controlled wind-tunnel model to demonstrate the utility of the scaling process.

High Order Accurate Finite Difference Modeling of Seismo-Acoustic Wave Propagation in a Moving Atmosphere and a Heterogeneous Earth Model Coupled Across a Realistic Topography

DOE Office of Scientific and Technical Information (OSTI.GOV)

Petersson, N. Anders; Sjogreen, Bjorn

Here, we develop a numerical method for simultaneously simulating acoustic waves in a realistic moving atmosphere and seismic waves in a heterogeneous earth model, where the motions are coupled across a realistic topography. We model acoustic wave propagation by solving the linearized Euler equations of compressible fluid mechanics. The seismic waves are modeled by the elastic wave equation in a heterogeneous anisotropic material. The motion is coupled by imposing continuity of normal velocity and normal stresses across the topographic interface. Realistic topography is resolved on a curvilinear grid that follows the interface. The governing equations are discretized using high ordermore » accurate finite difference methods that satisfy the principle of summation by parts. We apply the energy method to derive the discrete interface conditions and to show that the coupled discretization is stable. The implementation is verified by numerical experiments, and we demonstrate a simulation of coupled wave propagation in a windy atmosphere and a realistic earth model with non-planar topography.« less

High Order Accurate Finite Difference Modeling of Seismo-Acoustic Wave Propagation in a Moving Atmosphere and a Heterogeneous Earth Model Coupled Across a Realistic Topography

DOE PAGES

Petersson, N. Anders; Sjogreen, Bjorn

2017-04-18

Here, we develop a numerical method for simultaneously simulating acoustic waves in a realistic moving atmosphere and seismic waves in a heterogeneous earth model, where the motions are coupled across a realistic topography. We model acoustic wave propagation by solving the linearized Euler equations of compressible fluid mechanics. The seismic waves are modeled by the elastic wave equation in a heterogeneous anisotropic material. The motion is coupled by imposing continuity of normal velocity and normal stresses across the topographic interface. Realistic topography is resolved on a curvilinear grid that follows the interface. The governing equations are discretized using high ordermore » accurate finite difference methods that satisfy the principle of summation by parts. We apply the energy method to derive the discrete interface conditions and to show that the coupled discretization is stable. The implementation is verified by numerical experiments, and we demonstrate a simulation of coupled wave propagation in a windy atmosphere and a realistic earth model with non-planar topography.« less

Sparse Additive Ordinary Differential Equations for Dynamic Gene Regulatory Network Modeling.

PubMed

Wu, Hulin; Lu, Tao; Xue, Hongqi; Liang, Hua

2014-04-02

The gene regulation network (GRN) is a high-dimensional complex system, which can be represented by various mathematical or statistical models. The ordinary differential equation (ODE) model is one of the popular dynamic GRN models. High-dimensional linear ODE models have been proposed to identify GRNs, but with a limitation of the linear regulation effect assumption. In this article, we propose a sparse additive ODE (SA-ODE) model, coupled with ODE estimation methods and adaptive group LASSO techniques, to model dynamic GRNs that could flexibly deal with nonlinear regulation effects. The asymptotic properties of the proposed method are established and simulation studies are performed to validate the proposed approach. An application example for identifying the nonlinear dynamic GRN of T-cell activation is used to illustrate the usefulness of the proposed method.

An adaptive grid algorithm for one-dimensional nonlinear equations

NASA Technical Reports Server (NTRS)

Gutierrez, William E.; Hills, Richard G.

1990-01-01

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

Mathematical Modeling of Chemical Stoichiometry

ERIC Educational Resources Information Center

Croteau, Joshua; Fox, William P.; Varazo, Kristofoland

2007-01-01

In beginning chemistry classes, students are taught a variety of techniques for balancing chemical equations. The most common method is inspection. This paper addresses using a system of linear mathematical equations to solve for the stoichiometric coefficients. Many linear algebra books carry the standard balancing of chemical equations as an…

Frequentist Model Averaging in Structural Equation Modelling.

PubMed

Jin, Shaobo; Ankargren, Sebastian

2018-06-04

Model selection from a set of candidate models plays an important role in many structural equation modelling applications. However, traditional model selection methods introduce extra randomness that is not accounted for by post-model selection inference. In the current study, we propose a model averaging technique within the frequentist statistical framework. Instead of selecting an optimal model, the contributions of all candidate models are acknowledged. Valid confidence intervals and a [Formula: see text] test statistic are proposed. A simulation study shows that the proposed method is able to produce a robust mean-squared error, a better coverage probability, and a better goodness-of-fit test compared to model selection. It is an interesting compromise between model selection and the full model.

Percent body fat estimations in college women using field and laboratory methods: a three-compartment model approach

PubMed Central

Moon, Jordan R; Hull, Holly R; Tobkin, Sarah E; Teramoto, Masaru; Karabulut, Murat; Roberts, Michael D; Ryan, Eric D; Kim, So Jung; Dalbo, Vincent J; Walter, Ashley A; Smith, Abbie T; Cramer, Joel T; Stout, Jeffrey R

2007-01-01

Background Methods used to estimate percent body fat can be classified as a laboratory or field technique. However, the validity of these methods compared to multiple-compartment models has not been fully established. This investigation sought to determine the validity of field and laboratory methods for estimating percent fat (%fat) in healthy college-age women compared to the Siri three-compartment model (3C). Methods Thirty Caucasian women (21.1 ± 1.5 yrs; 164.8 ± 4.7 cm; 61.2 ± 6.8 kg) had their %fat estimated by BIA using the BodyGram™ computer program (BIA-AK) and population-specific equation (BIA-Lohman), NIR (Futrex® 6100/XL), a quadratic (SF3JPW) and linear (SF3WB) skinfold equation, air-displacement plethysmography (BP), and hydrostatic weighing (HW). Results All methods produced acceptable total error (TE) values compared to the 3C model. Both laboratory methods produced similar TE values (HW, TE = 2.4%fat; BP, TE = 2.3%fat) when compared to the 3C model, though a significant constant error (CE) was detected for HW (1.5%fat, p ≤ 0.006). The field methods produced acceptable TE values ranging from 1.8 – 3.8 %fat. BIA-AK (TE = 1.8%fat) yielded the lowest TE among the field methods, while BIA-Lohman (TE = 2.1%fat) and NIR (TE = 2.7%fat) produced lower TE values than both skinfold equations (TE > 2.7%fat) compared to the 3C model. Additionally, the SF3JPW %fat estimation equation resulted in a significant CE (2.6%fat, p ≤ 0.007). Conclusion Data suggest that the BP and HW are valid laboratory methods when compared to the 3C model to estimate %fat in college-age Caucasian women. When the use of a laboratory method is not feasible, NIR, BIA-AK, BIA-Lohman, SF3JPW, and SF3WB are acceptable field methods to estimate %fat in this population. PMID:17988393

Modeling Pulse Transmission in the Monterey Bay Using Parabolic Equation Methods

DTIC Science & Technology

1991-12-01

Collins 9-13 was chosen for this purpose due its energy conservation scheme , and its ability to efficiently incorporate higher order terms in its...pressure field generated by the PE model into normal modes. Additionally, this process provides increased physical understanding of mode coupling and...separation of variables (i.e. normal modes or fast field), as well as pure numerical schemes such as the parabolic equation methods, can be used. However, as

Methods of Investigation of Equations that Describe Waves in Tubes with Elastic Walls and Application of the Theory of Reversible and Weak Dissipative Shocks

NASA Astrophysics Data System (ADS)

Bakholdin, Igor

2018-02-01

Various models of a tube with elastic walls are investigated: with controlled pressure, filled with incompressible fluid, filled with compressible gas. The non-linear theory of hyperelasticity is applied. The walls of a tube are described with complete membrane model. It is proposed to use linear model of plate in order to take the bending resistance of walls into account. The walls of the tube were treated previously as inviscid and incompressible. Compressibility of material of walls and viscosity of material, either gas or liquid are considered. Equations are solved numerically. Three-layer time and space centered reversible numerical scheme and similar two-layer space reversible numerical scheme with approximation of time derivatives by Runge-Kutta method are used. A method of correction of numerical schemes by inclusion of terms with highorder derivatives is developed. Simplified hyperbolic equations are derived.

A numerical investigation of the boundary layer flow of an Eyring-Powell fluid over a stretching sheet via rational Chebyshev functions

NASA Astrophysics Data System (ADS)

Parand, Kourosh; Mahdi Moayeri, Mohammad; Latifi, Sobhan; Delkhosh, Mehdi

2017-07-01

In this paper, a spectral method based on the four kinds of rational Chebyshev functions is proposed to approximate the solution of the boundary layer flow of an Eyring-Powell fluid over a stretching sheet. First, by using the quasilinearization method (QLM), the model which is a nonlinear ordinary differential equation is converted to a sequence of linear ordinary differential equations (ODEs). By applying the proposed method on the ODEs in each iteration, the equations are converted to a system of linear algebraic equations. The results indicate the high accuracy and convergence of our method. Moreover, the effects of the Eyring-Powell fluid material parameters are discussed.

Numerical optimization using flow equations.

PubMed

Punk, Matthias

2014-12-01

We develop a method for multidimensional optimization using flow equations. This method is based on homotopy continuation in combination with a maximum entropy approach. Extrema of the optimizing functional correspond to fixed points of the flow equation. While ideas based on Bayesian inference such as the maximum entropy method always depend on a prior probability, the additional step in our approach is to perform a continuous update of the prior during the homotopy flow. The prior probability thus enters the flow equation only as an initial condition. We demonstrate the applicability of this optimization method for two paradigmatic problems in theoretical condensed matter physics: numerical analytic continuation from imaginary to real frequencies and finding (variational) ground states of frustrated (quantum) Ising models with random or long-range antiferromagnetic interactions.

Numerical optimization using flow equations

NASA Astrophysics Data System (ADS)

Punk, Matthias

2014-12-01

We develop a method for multidimensional optimization using flow equations. This method is based on homotopy continuation in combination with a maximum entropy approach. Extrema of the optimizing functional correspond to fixed points of the flow equation. While ideas based on Bayesian inference such as the maximum entropy method always depend on a prior probability, the additional step in our approach is to perform a continuous update of the prior during the homotopy flow. The prior probability thus enters the flow equation only as an initial condition. We demonstrate the applicability of this optimization method for two paradigmatic problems in theoretical condensed matter physics: numerical analytic continuation from imaginary to real frequencies and finding (variational) ground states of frustrated (quantum) Ising models with random or long-range antiferromagnetic interactions.

Equivalent model of a dually-fed machine for electric drive control systems

NASA Astrophysics Data System (ADS)

Ostrovlyanchik, I. Yu; Popolzin, I. Yu

2018-05-01

The article shows that the mathematical model of a dually-fed machine is complicated because of the presence of a controlled voltage source in the rotor circuit. As a method of obtaining a mathematical model, the method of a generalized two-phase electric machine is applied and a rotating orthogonal coordinate system is chosen that is associated with the representing vector of a stator current. In the chosen coordinate system in the operator form the differential equations of electric equilibrium for the windings of the generalized machine (the Kirchhoff equation) are written together with the expression for the moment, which determines the electromechanical energy transformation in the machine. Equations are transformed so that they connect the currents of the windings, that determine the moment of the machine, and the voltages on these windings. The structural diagram of the machine is assigned to the written equations. Based on the written equations and accepted assumptions, expressions were obtained for the balancing the EMF of windings, and on the basis of these expressions an equivalent mathematical model of a dually-fed machine is proposed, convenient for use in electric drive control systems.

Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in unmagnetized dust plasma

NASA Astrophysics Data System (ADS)

Seadawy, A. R.; El-Rashidy, K.

2018-03-01

The Kadomtsev-Petviashvili (KP) and modified KP equations are two of the most universal models in nonlinear wave theory, which arises as a reduction of system with quadratic nonlinearity which admit weakly dispersive waves. The generalized extended tanh method and the F-expansion method are used to derive exact solitary waves solutions of KP and modified KP equations. The region of solutions are displayed graphically.

Smoothed Particle Hydrodynamics and its applications for multiphase flow and reactive transport in porous media

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tartakovsky, Alexandre M.; Trask, Nathaniel; Pan, K.

2016-03-11

Smoothed Particle Hydrodynamics (SPH) is a Lagrangian method based on a meshless discretization of partial differential equations. In this review, we present SPH discretization of the Navier-Stokes and Advection-Diffusion-Reaction equations, implementation of various boundary conditions, and time integration of the SPH equations, and we discuss applications of the SPH method for modeling pore-scale multiphase flows and reactive transport in porous and fractured media.

Numerical solutions to the time-dependent Bloch equations revisited.

PubMed

Murase, Kenya; Tanki, Nobuyoshi

2011-01-01

The purpose of this study was to demonstrate a simple and fast method for solving the time-dependent Bloch equations. First, the time-dependent Bloch equations were reduced to a homogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. The validity of this method was investigated by comparing with the analytical solutions in the case of constant radiofrequency irradiation. There was a good agreement between them, indicating the validity of this method. As a further example, this method was applied to the time-dependent Bloch equations in the two-pool exchange model for chemical exchange saturation transfer (CEST) or amide proton transfer (APT) magnetic resonance imaging (MRI), and the Z-spectra and asymmetry spectra were calculated from their solutions. They were also calculated using the fourth/fifth-order Runge-Kutta-Fehlberg (RKF) method for comparison. There was also a good agreement between them, and this method was much faster than the RKF method. In conclusion, this method will be useful for analyzing the complex CEST or APT contrast mechanism and/or investigating the optimal conditions for CEST or APT MRI. Copyright © 2011 Elsevier Inc. All rights reserved.

A direct method for the solution of unsteady two-dimensional incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Ghia, K. N.; Osswald, G. A.; Ghia, U.

1983-01-01

The unsteady incompressible Navier-Stokes equations are formulated in terms of vorticity and stream function in generalized curvilinear orthogonal coordinates to facilitiate analysis of flow configurations with general geometries. The numerical method developed solves the conservative form of the transport equation using the alternating-direction implicit method, whereas the stream-function equation is solved by direct block Gaussian elimination. The method is applied to a model problem of flow over a back-step in a doubly infinite channel, using clustered conformal coordinates. One-dimensional stretching functions, dependent on the Reynolds number and the asymptotic behavior of the flow, are used to provide suitable grid distribution in the separation and reattachment regions, as well as in the inflow and outflow regions. The optimum grid distribution selected attempts to honor the multiple length scales of the separated-flow model problem. The asymptotic behavior of the finite-differenced transport equation near infinity is examined and the numerical method is carefully developed so as to lead to spatially second-order accurate wiggle-free solutions, i.e., with minimum dispersive error. Results have been obtained in the entire laminar range for the backstep channel and are in good agreement with the available experimental data for this flow problem.

Simulation of the pulse propagation by the interacting mode parabolic equation method

NASA Astrophysics Data System (ADS)

Trofimov, M. Yu.; Kozitskiy, S. B.; Zakharenko, A. D.

2018-07-01

A broadband modeling of pulses has been performed by using the previously derived interacting mode parabolic equation through the Fourier synthesis. Test examples on the wedge with the angle 2.86∘ (known as the ASA benchmark) show excellent agreement with the source images method.

Forcing scheme analysis for the axisymmetric lattice Boltzmann method under incompressible limit.

PubMed

Zhang, Liangqi; Yang, Shiliang; Zeng, Zhong; Chen, Jie; Yin, Linmao; Chew, Jia Wei

2017-04-01

Because the standard lattice Boltzmann (LB) method is proposed for Cartesian Navier-Stokes (NS) equations, additional source terms are necessary in the axisymmetric LB method for representing the axisymmetric effects. Therefore, the accuracy and applicability of the axisymmetric LB models depend on the forcing schemes adopted for discretization of the source terms. In this study, three forcing schemes, namely, the trapezium rule based scheme, the direct forcing scheme, and the semi-implicit centered scheme, are analyzed theoretically by investigating their derived macroscopic equations in the diffusive scale. Particularly, the finite difference interpretation of the standard LB method is extended to the LB equations with source terms, and then the accuracy of different forcing schemes is evaluated for the axisymmetric LB method. Theoretical analysis indicates that the discrete lattice effects arising from the direct forcing scheme are part of the truncation error terms and thus would not affect the overall accuracy of the standard LB method with general force term (i.e., only the source terms in the momentum equation are considered), but lead to incorrect macroscopic equations for the axisymmetric LB models. On the other hand, the trapezium rule based scheme and the semi-implicit centered scheme both have the advantage of avoiding the discrete lattice effects and recovering the correct macroscopic equations. Numerical tests applied for validating the theoretical analysis show that both the numerical stability and the accuracy of the axisymmetric LB simulations are affected by the direct forcing scheme, which indicate that forcing schemes free of the discrete lattice effects are necessary for the axisymmetric LB method.

Second-order oriented partial-differential equations for denoising in electronic-speckle-pattern interferometry fringes.

PubMed

Tang, Chen; Han, Lin; Ren, Hongwei; Zhou, Dongjian; Chang, Yiming; Wang, Xiaohang; Cui, Xiaolong

2008-10-01

We derive the second-order oriented partial-differential equations (PDEs) for denoising in electronic-speckle-pattern interferometry fringe patterns from two points of view. The first is based on variational methods, and the second is based on controlling diffusion direction. Our oriented PDE models make the diffusion along only the fringe orientation. The main advantage of our filtering method, based on oriented PDE models, is that it is very easy to implement compared with the published filtering methods along the fringe orientation. We demonstrate the performance of our oriented PDE models via application to two computer-simulated and experimentally obtained speckle fringes and compare with related PDE models.

Direct numerical solution of the Ornstein-Zernike integral equation and spatial distribution of water around hydrophobic molecules

NASA Astrophysics Data System (ADS)

Ikeguchi, Mitsunori; Doi, Junta

1995-09-01

The Ornstein-Zernike integral equation (OZ equation) has been used to evaluate the distribution function of solvents around solutes, but its numerical solution is difficult for molecules with a complicated shape. This paper proposes a numerical method to directly solve the OZ equation by introducing the 3D lattice. The method employs no approximation the reference interaction site model (RISM) equation employed. The method enables one to obtain the spatial distribution of spherical solvents around solutes with an arbitrary shape. Numerical accuracy is sufficient when the grid-spacing is less than 0.5 Å for solvent water. The spatial water distribution around a propane molecule is demonstrated as an example of a nonspherical hydrophobic molecule using iso-value surfaces. The water model proposed by Pratt and Chandler is used. The distribution agrees with the molecular dynamics simulation. The distribution increases offshore molecular concavities. The spatial distribution of water around 5α-cholest-2-ene (C27H46) is visualized using computer graphics techniques and a similar trend is observed.

A mathematical model of the heat and fluid flows in direct-chill casting of aluminum sheet ingots and billets

NASA Astrophysics Data System (ADS)

Mortensen, Dag

1999-02-01

A finite-element method model for the time-dependent heat and fluid flows that develop during direct-chill (DC) semicontinuous casting of aluminium ingots is presented. Thermal convection and turbulence are included in the model formulation and, in the mushy zone, the momentum equations are modified with a Darcy-type source term dependent on the liquid fraction. The boundary conditions involve calculations of the air gap along the mold wall as well as the heat transfer to the falling water film with forced convection, nucleate boiling, and film boiling. The mold wall and the starting block are included in the computational domain. In the start-up period of the casting, the ingot domain expands over the starting-block level. The numerical method applies a fractional-step method for the dynamic Navier-Stokes equations and the “streamline upwind Petrov-Galerkin” (SUPG) method for mixed diffusion and convection in the momentum and energy equations. The modeling of the start-up period of the casting is demonstrated and compared to temperature measurements in an AA1050 200×600 mm sheet ingot.

A simple orbit-attitude coupled modelling method for large solar power satellites

NASA Astrophysics Data System (ADS)

Li, Qingjun; Wang, Bo; Deng, Zichen; Ouyang, Huajiang; Wei, Yi

2018-04-01

A simple modelling method is proposed to study the orbit-attitude coupled dynamics of large solar power satellites based on natural coordinate formulation. The generalized coordinates are composed of Cartesian coordinates of two points and Cartesian components of two unitary vectors instead of Euler angles and angular velocities, which is the reason for its simplicity. Firstly, in order to develop natural coordinate formulation to take gravitational force and gravity gradient torque of a rigid body into account, Taylor series expansion is adopted to approximate the gravitational potential energy. The equations of motion are constructed through constrained Hamilton's equations. Then, an energy- and constraint-conserving algorithm is presented to solve the differential-algebraic equations. Finally, the proposed method is applied to simulate the orbit-attitude coupled dynamics and control of a large solar power satellite considering gravity gradient torque and solar radiation pressure. This method is also applicable to dynamic modelling of other rigid multibody aerospace systems.

A numerical method for computing unsteady 2-D boundary layer flows

NASA Technical Reports Server (NTRS)

Krainer, Andreas

1988-01-01

A numerical method for computing unsteady two-dimensional boundary layers in incompressible laminar and turbulent flows is described and applied to a single airfoil changing its incidence angle in time. The solution procedure adopts a first order panel method with a simple wake model to solve for the inviscid part of the flow, and an implicit finite difference method for the viscous part of the flow. Both procedures integrate in time in a step-by-step fashion, in the course of which each step involves the solution of the elliptic Laplace equation and the solution of the parabolic boundary layer equations. The Reynolds shear stress term of the boundary layer equations is modeled by an algebraic eddy viscosity closure. The location of transition is predicted by an empirical data correlation originating from Michel. Since transition and turbulence modeling are key factors in the prediction of viscous flows, their accuracy will be of dominant influence to the overall results.

The Philosophical Aspects of IRT Equating: Modeling Drift to Evaluate Cohort Growth in Large-Scale Assessments

ERIC Educational Resources Information Center

Taherbhai, Husein; Seo, Daeryong

2013-01-01

Calibration and equating is the quintessential necessity for most large-scale educational assessments. However, there are instances when no consideration is given to the equating process in terms of context and substantive realization, and the methods used in its execution. In the view of the authors, equating is not merely an exhibit of the…

A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients Based on the State Transition Matrix

NASA Technical Reports Server (NTRS)

Bartels, Robert E.

2002-01-01

A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.

Reduced basis ANOVA methods for partial differential equations with high-dimensional random inputs

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liao, Qifeng, E-mail: liaoqf@shanghaitech.edu.cn; Lin, Guang, E-mail: guanglin@purdue.edu

2016-07-15

In this paper we present a reduced basis ANOVA approach for partial deferential equations (PDEs) with random inputs. The ANOVA method combined with stochastic collocation methods provides model reduction in high-dimensional parameter space through decomposing high-dimensional inputs into unions of low-dimensional inputs. In this work, to further reduce the computational cost, we investigate spatial low-rank structures in the ANOVA-collocation method, and develop efficient spatial model reduction techniques using hierarchically generated reduced bases. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.

A Time Integration Algorithm Based on the State Transition Matrix for Structures with Time Varying and Nonlinear Properties

NASA Technical Reports Server (NTRS)

Bartels, Robert E.

2003-01-01

A variable order method of integrating the structural dynamics equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. When the time variation of the system can be modeled exactly by a polynomial it produces nearly exact solutions for a wide range of time step sizes. Solutions of a model nonlinear dynamic response exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with solutions obtained by established methods.

A shallow water model for the propagation of tsunami via Lattice Boltzmann method

NASA Astrophysics Data System (ADS)

Zergani, Sara; Aziz, Z. A.; Viswanathan, K. K.

2015-01-01

An efficient implementation of the lattice Boltzmann method (LBM) for the numerical simulation of the propagation of long ocean waves (e.g. tsunami), based on the nonlinear shallow water (NSW) wave equation is presented. The LBM is an alternative numerical procedure for the description of incompressible hydrodynamics and has the potential to serve as an efficient solver for incompressible flows in complex geometries. This work proposes the NSW equations for the irrotational surface waves in the case of complex bottom elevation. In recent time, equation involving shallow water is the current norm in modelling tsunami operations which include the propagation zone estimation. Several test-cases are presented to verify our model. Some implications to tsunami wave modelling are also discussed. Numerical results are found to be in excellent agreement with theory.

An Estimating Equations Approach for the LISCOMP Model.

ERIC Educational Resources Information Center

Reboussin, Beth A.; Liang, Kung-Lee

1998-01-01

A quadratic estimating equations approach for the LISCOMP model is proposed that only requires specification of the first two moments. This method is compared with a three-stage generalized least squares approach through a numerical study and application to a study of life events and neurotic illness. (SLD)

Using Mixed-Effects Structural Equation Models to Study Student Academic Development.

ERIC Educational Resources Information Center

Pike, Gary R.

1992-01-01

A study at the University of Tennessee Knoxville used mixed-effect structural equation models incorporating latent variables as an alternative to conventional methods of analyzing college students' (n=722) first-year-to-senior academic gains. Results indicate, contrary to previous analysis, that coursework and student characteristics interact to…

Angular motion equations for a satellite with hinged flexible solar panel

NASA Astrophysics Data System (ADS)

Ovchinnikov, M. Yu.; Tkachev, S. S.; Roldugin, D. S.; Nuralieva, A. B.; Mashtakov, Y. V.

2016-11-01

Non-linear mathematical model for the satellite with hinged flexible solar panel is presented. Normal modes of flexible elements are used for motion description. Motion equations are derived using virtual work principle. A comparison of normal modes calculation between finite element method and developed model is presented.

Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations.

PubMed

Islam, S M Rayhanul; Khan, Kamruzzaman; Akbar, M Ali

2015-01-01

In this paper, we implement the exp(-Φ(ξ))-expansion method to construct the exact traveling wave solutions for nonlinear evolution equations (NLEEs). Here we consider two model equations, namely the Korteweg-de Vries (KdV) equation and the time regularized long wave (TRLW) equation. These equations play significant role in nonlinear sciences. We obtained four types of explicit function solutions, namely hyperbolic, trigonometric, exponential and rational function solutions of the variables in the considered equations. It has shown that the applied method is quite efficient and is practically well suited for the aforementioned problems and so for the other NLEEs those arise in mathematical physics and engineering fields. PACS numbers: 02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fq.

A three-dimensional parabolic equation model of sound propagation using higher-order operator splitting and Padé approximants.

PubMed

Lin, Ying-Tsong; Collis, Jon M; Duda, Timothy F

2012-11-01

An alternating direction implicit (ADI) three-dimensional fluid parabolic equation solution method with enhanced accuracy is presented. The method uses a square-root Helmholtz operator splitting algorithm that retains cross-multiplied operator terms that have been previously neglected. With these higher-order cross terms, the valid angular range of the parabolic equation solution is improved. The method is tested for accuracy against an image solution in an idealized wedge problem. Computational efficiency improvements resulting from the ADI discretization are also discussed.

Inattentive Drivers: Making the Solution Method the Model

ERIC Educational Resources Information Center

McCartney, Mark

2003-01-01

A simple car following model based on the solution of coupled ordinary differential equations is considered. The model is solved using Euler's method and this method of solution is itself interpreted as a mathematical model for car following. Examples of possible classroom use are given. (Contains 6 figures.)

A New LES/PDF Method for Computational Modeling of Turbulent Reacting Flows

NASA Astrophysics Data System (ADS)

Turkeri, Hasret; Muradoglu, Metin; Pope, Stephen B.

2013-11-01

A new LES/PDF method is developed for computational modeling of turbulent reacting flows. The open source package, OpenFOAM, is adopted as the LES solver and combined with the particle-based Monte Carlo method to solve the LES/PDF model equations. The dynamic Smagorinsky model is employed to account for the subgrid-scale motions. The LES solver is first validated for the Sandia Flame D using a steady flamelet method in which the chemical compositions, density and temperature fields are parameterized by the mean mixture fraction and its variance. In this approach, the modeled transport equations for the mean mixture fraction and the square of the mixture fraction are solved and the variance is then computed from its definition. The results are found to be in a good agreement with the experimental data. Then the LES solver is combined with the particle-based Monte Carlo algorithm to form a complete solver for the LES/PDF model equations. The in situ adaptive tabulation (ISAT) algorithm is incorporated into the LES/PDF method for efficient implementation of detailed chemical kinetics. The LES/PDF method is also applied to the Sandia Flame D using the GRI-Mech 3.0 chemical mechanism and the results are compared with the experimental data and the earlier PDF simulations. The Scientific and Technical Research Council of Turkey (TUBITAK), Grant No. 111M067.

Study on the variable cycle engine modeling techniques based on the component method

NASA Astrophysics Data System (ADS)

Zhang, Lihua; Xue, Hui; Bao, Yuhai; Li, Jijun; Yan, Lan

2016-01-01

Based on the structure platform of the gas turbine engine, the components of variable cycle engine were simulated by using the component method. The mathematical model of nonlinear equations correspondeing to each component of the gas turbine engine was established. Based on Matlab programming, the nonlinear equations were solved by using Newton-Raphson steady-state algorithm, and the performance of the components for engine was calculated. The numerical simulation results showed that the model bulit can describe the basic performance of the gas turbine engine, which verified the validity of the model.

Joint modelling rationale for chained equations

PubMed Central

2014-01-01

Background Chained equations imputation is widely used in medical research. It uses a set of conditional models, so is more flexible than joint modelling imputation for the imputation of different types of variables (e.g. binary, ordinal or unordered categorical). However, chained equations imputation does not correspond to drawing from a joint distribution when the conditional models are incompatible. Concurrently with our work, other authors have shown the equivalence of the two imputation methods in finite samples. Methods Taking a different approach, we prove, in finite samples, sufficient conditions for chained equations and joint modelling to yield imputations from the same predictive distribution. Further, we apply this proof in four specific cases and conduct a simulation study which explores the consequences when the conditional models are compatible but the conditions otherwise are not satisfied. Results We provide an additional “non-informative margins” condition which, together with compatibility, is sufficient. We show that the non-informative margins condition is not satisfied, despite compatible conditional models, in a situation as simple as two continuous variables and one binary variable. Our simulation study demonstrates that as a consequence of this violation order effects can occur; that is, systematic differences depending upon the ordering of the variables in the chained equations algorithm. However, the order effects appear to be small, especially when associations between variables are weak. Conclusions Since chained equations is typically used in medical research for datasets with different types of variables, researchers must be aware that order effects are likely to be ubiquitous, but our results suggest they may be small enough to be negligible. PMID:24559129

THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN.

PubMed

Jiang, H; Liu, F; Meerschaert, M M; McGough, R J

2013-01-01

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n ) ( n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko's Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.

A fast marching algorithm for the factored eikonal equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Treister, Eran, E-mail: erantreister@gmail.com; Haber, Eldad, E-mail: haber@math.ubc.ca; Department of Mathematics, The University of British Columbia, Vancouver, BC

The eikonal equation is instrumental in many applications in several fields ranging from computer vision to geoscience. This equation can be efficiently solved using the iterative Fast Sweeping (FS) methods and the direct Fast Marching (FM) methods. However, when used for a point source, the original eikonal equation is known to yield inaccurate numerical solutions, because of a singularity at the source. In this case, the factored eikonal equation is often preferred, and is known to yield a more accurate numerical solution. One application that requires the solution of the eikonal equation for point sources is travel time tomography. Thismore » inverse problem may be formulated using the eikonal equation as a forward problem. While this problem has been solved using FS in the past, the more recent choice for applying it involves FM methods because of the efficiency in which sensitivities can be obtained using them. However, while several FS methods are available for solving the factored equation, the FM method is available only for the original eikonal equation. In this paper we develop a Fast Marching algorithm for the factored eikonal equation, using both first and second order finite-difference schemes. Our algorithm follows the same lines as the original FM algorithm and requires the same computational effort. In addition, we show how to obtain sensitivities using this FM method and apply travel time tomography, formulated as an inverse factored eikonal equation. Numerical results in two and three dimensions show that our algorithm solves the factored eikonal equation efficiently, and demonstrate the achieved accuracy for computing the travel time. We also demonstrate a recovery of a 2D and 3D heterogeneous medium by travel time tomography using the eikonal equation for forward modeling and inversion by Gauss–Newton.« less

Reflected stochastic differential equation models for constrained animal movement

USGS Publications Warehouse

Hanks, Ephraim M.; Johnson, Devin S.; Hooten, Mevin B.

2017-01-01

Movement for many animal species is constrained in space by barriers such as rivers, shorelines, or impassable cliffs. We develop an approach for modeling animal movement constrained in space by considering a class of constrained stochastic processes, reflected stochastic differential equations. Our approach generalizes existing methods for modeling unconstrained animal movement. We present methods for simulation and inference based on augmenting the constrained movement path with a latent unconstrained path and illustrate this augmentation with a simulation example and an analysis of telemetry data from a Steller sea lion (Eumatopias jubatus) in southeast Alaska.

Fourth order difference methods for hyperbolic IBVP's

NASA Technical Reports Server (NTRS)

Gustafsson, Bertil; Olsson, Pelle

1994-01-01

Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory

NASA Astrophysics Data System (ADS)

Zhou, L.-Q.; Meleshko, S. V.

2017-07-01

The group analysis method is applied to a system of integro-differential equations corresponding to a linear thermoviscoelastic model. A recently developed approach for calculating the symmetry groups of such equations is used. The general solution of the determining equations for the system is obtained. Using subalgebras of the admitted Lie algebra, two classes of partially invariant solutions of the considered system of integro-differential equations are studied.

Partial differential equation techniques for analysing animal movement: A comparison of different methods.

PubMed

Wang, Yi-Shan; Potts, Jonathan R

2017-03-07

Recent advances in animal tracking have allowed us to uncover the drivers of movement in unprecedented detail. This has enabled modellers to construct ever more realistic models of animal movement, which aid in uncovering detailed patterns of space use in animal populations. Partial differential equations (PDEs) provide a popular tool for mathematically analysing such models. However, their construction often relies on simplifying assumptions which may greatly affect the model outcomes. Here, we analyse the effect of various PDE approximations on the analysis of some simple movement models, including a biased random walk, central-place foraging processes and movement in heterogeneous landscapes. Perhaps the most commonly-used PDE method dates back to a seminal paper of Patlak from 1953. However, our results show that this can be a very poor approximation in even quite simple models. On the other hand, more recent methods, based on transport equation formalisms, can provide more accurate results, as long as the kernel describing the animal's movement is sufficiently smooth. When the movement kernel is not smooth, we show that both the older and newer methods can lead to quantitatively misleading results. Our detailed analysis will aid future researchers in the appropriate choice of PDE approximation for analysing models of animal movement. Copyright © 2017 Elsevier Ltd. All rights reserved.

Indriect Measurement Of Nitrogen In A Mult-Component Natural Gas By Heating The Gas

DOEpatents

Morrow, Thomas B.; Behring, II, Kendricks A.

2004-06-22

Methods of indirectly measuring the nitrogen concentration in a natural gas by heating the gas. In two embodiments, the heating energy is correlated to the speed of sound in the gas, the diluent concentrations in the gas, and constant values, resulting in a model equation. Regression analysis is used to calculate the constant values, which can then be substituted into the model equation. If the diluent concentrations other than nitrogen (typically carbon dioxide) are known, the model equation can be solved for the nitrogen concentration.

Two-Level Hierarchical FEM Method for Modeling Passive Microwave Devices

NASA Astrophysics Data System (ADS)

Polstyanko, Sergey V.; Lee, Jin-Fa

1998-03-01

In recent years multigrid methods have been proven to be very efficient for solving large systems of linear equations resulting from the discretization of positive definite differential equations by either the finite difference method or theh-version of the finite element method. In this paper an iterative method of the multiple level type is proposed for solving systems of algebraic equations which arise from thep-version of the finite element analysis applied to indefinite problems. A two-levelV-cycle algorithm has been implemented and studied with a Gauss-Seidel iterative scheme used as a smoother. The convergence of the method has been investigated, and numerical results for a number of numerical examples are presented.

Hpm of Estrogen Model on the Dynamics of Breast Cancer

NASA Astrophysics Data System (ADS)

Govindarajan, A.; Balamuralitharan, S.; Sundaresan, T.

2018-04-01

We enhance a deterministic mathematical model involving universal dynamics on breast cancer with immune response. This is population model so includes Normal cells class, Tumor cells, Immune cells and Estrogen. The eects regarding Estrogen are below incorporated in the model. The effects show to that amount the arrival of greater Estrogen increases the danger over growing breast cancer. Furthermore, approximate solution regarding nonlinear differential equations is arrived by Homotopy Perturbation Method (HPM). Hes HPM is good and correct technique after solve nonlinear differential equation directly. Approximate solution learnt with the support of that method is suitable same as like the actual results in accordance with this models.

Structural Equation Modelling of Multiple Facet Data: Extending Models for Multitrait-Multimethod Data

ERIC Educational Resources Information Center

Bechger, Timo M.; Maris, Gunter

2004-01-01

This paper is about the structural equation modelling of quantitative measures that are obtained from a multiple facet design. A facet is simply a set consisting of a finite number of elements. It is assumed that measures are obtained by combining each element of each facet. Methods and traits are two such facets, and a multitrait-multimethod…

Numerical solution of a logistic growth model for a population with Allee effect considering fuzzy initial values and fuzzy parameters

NASA Astrophysics Data System (ADS)

Amarti, Z.; Nurkholipah, N. S.; Anggriani, N.; Supriatna, A. K.

2018-03-01

Predicting the future of population number is among the important factors that affect the consideration in preparing a good management for the population. This has been done by various known method, one among them is by developing a mathematical model describing the growth of the population. The model usually takes form in a differential equation or a system of differential equations, depending on the complexity of the underlying properties of the population. The most widely used growth models currently are those having a sigmoid solution of time series, including the Verhulst logistic equation and the Gompertz equation. In this paper we consider the Allee effect of the Verhulst’s logistic population model. The Allee effect is a phenomenon in biology showing a high correlation between population size or density and the mean individual fitness of the population. The method used to derive the solution is the Runge-Kutta numerical scheme, since it is in general regarded as one among the good numerical scheme which is relatively easy to implement. Further exploration is done via the fuzzy theoretical approach to accommodate the impreciseness of the initial values and parameters in the model.

A consistent hierarchy of generalized kinetic equation approximations to the master equation applied to surface catalysis.

PubMed

Herschlag, Gregory J; Mitran, Sorin; Lin, Guang

2015-06-21

We develop a hierarchy of approximations to the master equation for systems that exhibit translational invariance and finite-range spatial correlation. Each approximation within the hierarchy is a set of ordinary differential equations that considers spatial correlations of varying lattice distance; the assumption is that the full system will have finite spatial correlations and thus the behavior of the models within the hierarchy will approach that of the full system. We provide evidence of this convergence in the context of one- and two-dimensional numerical examples. Lower levels within the hierarchy that consider shorter spatial correlations are shown to be up to three orders of magnitude faster than traditional kinetic Monte Carlo methods (KMC) for one-dimensional systems, while predicting similar system dynamics and steady states as KMC methods. We then test the hierarchy on a two-dimensional model for the oxidation of CO on RuO2(110), showing that low-order truncations of the hierarchy efficiently capture the essential system dynamics. By considering sequences of models in the hierarchy that account for longer spatial correlations, successive model predictions may be used to establish empirical approximation of error estimates. The hierarchy may be thought of as a class of generalized phenomenological kinetic models since each element of the hierarchy approximates the master equation and the lowest level in the hierarchy is identical to a simple existing phenomenological kinetic models.

A comparison of numerical solutions of partial differential equations with probabilistic and possibilistic parameters for the quantification of uncertainty in subsurface solute transport.

PubMed

Zhang, Kejiang; Achari, Gopal; Li, Hua

2009-11-03

Traditionally, uncertainty in parameters are represented as probabilistic distributions and incorporated into groundwater flow and contaminant transport models. With the advent of newer uncertainty theories, it is now understood that stochastic methods cannot properly represent non random uncertainties. In the groundwater flow and contaminant transport equations, uncertainty in some parameters may be random, whereas those of others may be non random. The objective of this paper is to develop a fuzzy-stochastic partial differential equation (FSPDE) model to simulate conditions where both random and non random uncertainties are involved in groundwater flow and solute transport. Three potential solution techniques namely, (a) transforming a probability distribution to a possibility distribution (Method I) then a FSPDE becomes a fuzzy partial differential equation (FPDE), (b) transforming a possibility distribution to a probability distribution (Method II) and then a FSPDE becomes a stochastic partial differential equation (SPDE), and (c) the combination of Monte Carlo methods and FPDE solution techniques (Method III) are proposed and compared. The effects of these three methods on the predictive results are investigated by using two case studies. The results show that the predictions obtained from Method II is a specific case of that got from Method I. When an exact probabilistic result is needed, Method II is suggested. As the loss or gain of information during a probability-possibility (or vice versa) transformation cannot be quantified, their influences on the predictive results is not known. Thus, Method III should probably be preferred for risk assessments.

A characteristics-based method for solving the transport equation and its application to the process of mantle differentiation and continental root growth

NASA Astrophysics Data System (ADS)

de Smet, Jeroen H.; van den Berg, Arie P.; Vlaar, Nico J.; Yuen, David A.

2000-03-01

Purely advective transport of composition is of major importance in the Geosciences, and efficient and accurate solution methods are needed. A characteristics-based method is used to solve the transport equation. We employ a new hybrid interpolation scheme, which allows for the tuning of stability and accuracy through a threshold parameter ɛth. Stability is established by bilinear interpolations, and bicubic splines are used to maintain accuracy. With this scheme, numerical instabilities can be suppressed by allowing numerical diffusion to work in time and locally in space. The scheme can be applied efficiently for preliminary modelling purposes. This can be followed by detailed high-resolution experiments. First, the principal effects of this hybrid interpolation method are illustrated and some tests are presented for numerical solutions of the transport equation. Second, we illustrate that this approach works successfully for a previously developed continental evolution model for the convecting upper mantle. In this model the transport equation contains a source term, which describes the melt production in pressure-released partial melting. In this model, a characteristic phenomenon of small-scale melting diapirs is observed (De Smet et al.1998; De Smet et al. 1999). High-resolution experiments with grid cells down to 700m horizontally and 515m vertically result in highly detailed observations of the diapiric melting phenomenon.

Three-dimensional time-dependent computer modeling of the electrothermal atomizers for analytical spectrometry

NASA Astrophysics Data System (ADS)

Tsivilskiy, I. V.; Nagulin, K. Yu.; Gilmutdinov, A. Kh.

2016-02-01

A full three-dimensional nonstationary numerical model of graphite electrothermal atomizers of various types is developed. The model is based on solution of a heat equation within solid walls of the atomizer with a radiative heat transfer and numerical solution of a full set of Navier-Stokes equations with an energy equation for a gas. Governing equations for the behavior of a discrete phase, i.e., atomic particles suspended in a gas (including gas-phase processes of evaporation and condensation), are derived from the formal equations molecular kinetics by numerical solution of the Hertz-Langmuir equation. The following atomizers test the model: a Varian standard heated electrothermal vaporizer (ETV), a Perkin Elmer standard THGA transversely heated graphite tube with integrated platform (THGA), and the original double-stage tube-helix atomizer (DSTHA). The experimental verification of computer calculations is carried out by a method of shadow spectral visualization of the spatial distributions of atomic and molecular vapors in an analytical space of an atomizer.

Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations.

PubMed

Li, Q; He, Y L; Wang, Y; Tao, W Q

2007-11-01

A coupled double-distribution-function lattice Boltzmann method is developed for the compressible Navier-Stokes equations. Different from existing thermal lattice Boltzmann methods, this method can recover the compressible Navier-Stokes equations with a flexible specific-heat ratio and Prandtl number. In the method, a density distribution function based on a multispeed lattice is used to recover the compressible continuity and momentum equations, while the compressible energy equation is recovered by an energy distribution function. The energy distribution function is then coupled to the density distribution function via the thermal equation of state. In order to obtain an adjustable specific-heat ratio, a constant related to the specific-heat ratio is introduced into the equilibrium energy distribution function. Two different coupled double-distribution-function lattice Boltzmann models are also proposed in the paper. Numerical simulations are performed for the Riemann problem, the double-Mach-reflection problem, and the Couette flow with a range of specific-heat ratios and Prandtl numbers. The numerical results are found to be in excellent agreement with analytical and/or other solutions.

On the selection of ordinary differential equation models with application to predator-prey dynamical models.

PubMed

Zhang, Xinyu; Cao, Jiguo; Carroll, Raymond J

2015-03-01

We consider model selection and estimation in a context where there are competing ordinary differential equation (ODE) models, and all the models are special cases of a "full" model. We propose a computationally inexpensive approach that employs statistical estimation of the full model, followed by a combination of a least squares approximation (LSA) and the adaptive Lasso. We show the resulting method, here called the LSA method, to be an (asymptotically) oracle model selection method. The finite sample performance of the proposed LSA method is investigated with Monte Carlo simulations, in which we examine the percentage of selecting true ODE models, the efficiency of the parameter estimation compared to simply using the full and true models, and coverage probabilities of the estimated confidence intervals for ODE parameters, all of which have satisfactory performances. Our method is also demonstrated by selecting the best predator-prey ODE to model a lynx and hare population dynamical system among some well-known and biologically interpretable ODE models. © 2014, The International Biometric Society.

A Unique Finite Element Modeling of the Periodic Wave Transformation over Sloping and Barred Beaches by Beji and Nadaoka's Extended Boussinesq Equations

PubMed Central

Jabbari, Mohammad Hadi; Sayehbani, Mesbah; Reisinezhad, Arsham

2013-01-01

This paper presents a numerical model based on one-dimensional Beji and Nadaoka's Extended Boussinesq equations for simulation of periodic wave shoaling and its decomposition over morphological beaches. A unique Galerkin finite element and Adams-Bashforth-Moulton predictor-corrector methods are employed for spatial and temporal discretization, respectively. For direct application of linear finite element method in spatial discretization, an auxiliary variable is hereby introduced, and a particular numerical scheme is offered to rewrite the equations in lower-order form. Stability of the suggested numerical method is also analyzed. Subsequently, in order to display the ability of the presented model, four different test cases are considered. In these test cases, dispersive and nonlinearity effects of the periodic waves over sloping beaches and barred beaches, which are the common coastal profiles, are investigated. Outputs are compared with other existing numerical and experimental data. Finally, it is concluded that the current model can be further developed to model any morphological development of coastal profiles. PMID:23853534

Features in simulation of crystal growth using the hyperbolic PFC equation and the dependence of the numerical solution on the parameters of the computational grid

DOE Office of Scientific and Technical Information (OSTI.GOV)

Starodumov, Ilya; Kropotin, Nikolai

2016-08-10

We investigate the three-dimensional mathematical model of crystal growth called PFC (Phase Field Crystal) in a hyperbolic modification. This model is also called the modified model PFC (originally PFC model is formulated in parabolic form) and allows to describe both slow and rapid crystallization processes on atomic length scales and on diffusive time scales. Modified PFC model is described by the differential equation in partial derivatives of the sixth order in space and second order in time. The solution of this equation is possible only by numerical methods. Previously, authors created the software package for the solution of the Phasemore » Field Crystal problem, based on the method of isogeometric analysis (IGA) and PetIGA program library. During further investigation it was found that the quality of the solution can strongly depends on the discretization parameters of a numerical method. In this report, we show the features that should be taken into account during constructing the computational grid for the numerical simulation.« less

A Two-Stage Estimation Method for Random Coefficient Differential Equation Models with Application to Longitudinal HIV Dynamic Data.

PubMed

Fang, Yun; Wu, Hulin; Zhu, Li-Xing

2011-07-01

We propose a two-stage estimation method for random coefficient ordinary differential equation (ODE) models. A maximum pseudo-likelihood estimator (MPLE) is derived based on a mixed-effects modeling approach and its asymptotic properties for population parameters are established. The proposed method does not require repeatedly solving ODEs, and is computationally efficient although it does pay a price with the loss of some estimation efficiency. However, the method does offer an alternative approach when the exact likelihood approach fails due to model complexity and high-dimensional parameter space, and it can also serve as a method to obtain the starting estimates for more accurate estimation methods. In addition, the proposed method does not need to specify the initial values of state variables and preserves all the advantages of the mixed-effects modeling approach. The finite sample properties of the proposed estimator are studied via Monte Carlo simulations and the methodology is also illustrated with application to an AIDS clinical data set.

Parametric optimal control of uncertain systems under an optimistic value criterion

NASA Astrophysics Data System (ADS)

Li, Bo; Zhu, Yuanguo

2018-01-01

It is well known that the optimal control of a linear quadratic model is characterized by the solution of a Riccati differential equation. In many cases, the corresponding Riccati differential equation cannot be solved exactly such that the optimal feedback control may be a complex time-oriented function. In this article, a parametric optimal control problem of an uncertain linear quadratic model under an optimistic value criterion is considered for simplifying the expression of optimal control. Based on the equation of optimality for the uncertain optimal control problem, an approximation method is presented to solve it. As an application, a two-spool turbofan engine optimal control problem is given to show the utility of the proposed model and the efficiency of the presented approximation method.

Indirect Measurement Of Nitrogen In A Multi-Component Gas By Measuring The Speed Of Sound At Two States Of The Gas.

DOEpatents

Morrow, Thomas B.; Behring, II, Kendricks A.

2004-10-12

A methods of indirectly measuring the nitrogen concentration in a gas mixture. The molecular weight of the gas is modeled as a function of the speed of sound in the gas, the diluent concentrations in the gas, and constant values, resulting in a model equation. Regression analysis is used to calculate the constant values, which can then be substituted into the model equation. If the speed of sound in the gas is measured at two states and diluent concentrations other than nitrogen (typically carbon dioxide) are known, two equations for molecular weight can be equated and solved for the nitrogen concentration in the gas mixture.

Technical report series on global modeling and data assimilation. Volume 2: Direct solution of the implicit formulation of fourth order horizontal diffusion for gridpoint models on the sphere

NASA Technical Reports Server (NTRS)

Li, Yong; Moorthi, S.; Bates, J. Ray; Suarez, Max J.

1994-01-01

High order horizontal diffusion of the form K Delta(exp 2m) is widely used in spectral models as a means of preventing energy accumulation at the shortest resolved scales. In the spectral context, an implicit formation of such diffusion is trivial to implement. The present note describes an efficient method of implementing implicit high order diffusion in global finite difference models. The method expresses the high order diffusion equation as a sequence of equations involving Delta(exp 2). The solution is obtained by combining fast Fourier transforms in longitude with a finite difference solver for the second order ordinary differential equation in latitude. The implicit diffusion routine is suitable for use in any finite difference global model that uses a regular latitude/longitude grid. The absence of a restriction on the timestep makes it particularly suitable for use in semi-Lagrangian models. The scale selectivity of the high order diffusion gives it an advantage over the uncentering method that has been used to control computational noise in two-time-level semi-Lagrangian models.

A two-layer multiple-time-scale turbulence model and grid independence study

NASA Technical Reports Server (NTRS)

Kim, S.-W.; Chen, C.-P.

1989-01-01

A two-layer multiple-time-scale turbulence model is presented. The near-wall model is based on the classical Kolmogorov-Prandtl turbulence hypothesis and the semi-empirical logarithmic law of the wall. In the two-layer model presented, the computational domain of the conservation of mass equation and the mean momentum equation penetrated up to the wall, where no slip boundary condition has been prescribed; and the near wall boundary of the turbulence equations has been located at the fully turbulent region, yet very close to the wall, where the standard wall function method has been applied. Thus, the conservation of mass constraint can be satisfied more rigorously in the two-layer model than in the standard wall function method. In most of the two-layer turbulence models, the number of grid points to be used inside the near-wall layer posed the issue of computational efficiency. The present finite element computational results showed that the grid independent solutions were obtained with as small as two grid points, i.e., one quadratic element, inside the near wall layer. Comparison of the computational results obtained by using the two-layer model and those obtained by using the wall function method is also presented.

A Multi-Fidelity Surrogate Model for Handling Real Gas Equations of State

NASA Astrophysics Data System (ADS)

Ouellet, Frederick; Park, Chanyoung; Rollin, Bertrand; Balachandar, S."bala"

2016-11-01

The explosive dispersal of particles is an example of a complex multiphase and multi-species fluid flow problem. This problem has many engineering applications including particle-laden explosives. In these flows, the detonation products of the explosive cannot be treated as a perfect gas so a real gas equation of state is used to close the governing equations (unlike air, which uses the ideal gas equation for closure). As the products expand outward from the detonation point, they mix with ambient air and create a mixing region where both of the state equations must be satisfied. One of the more accurate, yet computationally expensive, methods to deal with this is a scheme that iterates between the two equations of state until pressure and thermal equilibrium are achieved inside of each computational cell. This work strives to create a multi-fidelity surrogate model of this process. We then study the performance of the model with respect to the iterative method by performing both gas-only and particle laden flow simulations using an Eulerian-Lagrangian approach with a finite volume code. Specifically, the model's (i) computational speed, (ii) memory requirements and (iii) computational accuracy are analyzed to show the benefits of this novel modeling approach. This work was supported by the U.S. Department of Energy, National Nuclear Security Administration, Advanced Simulation and Computing Program, as a Cooperative Agreement under the Predictive Science Academic Alliance Program, under Contract No. DE-NA00023.

2.5-D poroelastic wave modelling in double porosity media

NASA Astrophysics Data System (ADS)

Liu, Xu; Greenhalgh, Stewart; Wang, Yanghua

2011-09-01

To approximate seismic wave propagation in double porosity media, the 2.5-D governing equations of poroelastic waves are developed and numerically solved. The equations are obtained by taking a Fourier transform in the strike or medium-invariant direction over all of the field quantities in the 3-D governing equations. The new memory variables from the Zener model are suggested as a way to represent the sum of the convolution integrals for both the solid particle velocity and the macroscopic fluid flux in the governing equations. By application of the memory equations, the field quantities at every time step need not be stored. However, this approximation allows just two Zener relaxation times to represent the very complex double porosity and dual permeability attenuation mechanism, and thus reduce the difficulty. The 2.5-D governing equations are numerically solved by a time-splitting method for the non-stiff parts and an explicit fourth-order Runge-Kutta method for the time integration and a Fourier pseudospectral staggered-grid for handling the spatial derivative terms. The 2.5-D solution has the advantage of producing a 3-D wavefield (point source) for a 2-D model but is much more computationally efficient than the full 3-D solution. As an illustrative example, we firstly show the computed 2.5-D wavefields in a homogeneous single porosity model for which we reformulated an analytic solution. Results for a two-layer, water-saturated double porosity model and a laterally heterogeneous double porosity structure are also presented.

Scalar field dark energy with a minimal coupling in a spherically symmetric background

NASA Astrophysics Data System (ADS)

Matsumoto, Jiro

Dark energy models and modified gravity theories have been actively studied and the behaviors in the solar system have been also carefully investigated in a part of the models. However, the isotropic solutions of the field equations in the simple models of dark energy, e.g. quintessence model without matter coupling, have not been well investigated. One of the reason would be the nonlinearity of the field equations. In this paper, a method to evaluate the solution of the field equations is constructed, and it is shown that there is a model that can easily pass the solar system tests, whereas, there is also a model that is constrained from the solar system tests.

Applications of IBSOM and ETEM for solving the nonlinear chains of atoms with long-range interactions

NASA Astrophysics Data System (ADS)

Foroutan, Mohammadreza; Zamanpour, Isa; Manafian, Jalil

2017-10-01

This paper presents a number of new solutions obtained for solving a complex nonlinear equation describing dynamics of nonlinear chains of atoms via the improved Bernoulli sub-ODE method (IBSOM) and the extended trial equation method (ETEM). The proposed solutions are kink solitons, anti-kink solitons, soliton solutions, hyperbolic solutions, trigonometric solutions, and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and succinct and can be also applied to other nonlinear models. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to establish many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in Hamilton's equations and generalized momentum where solitons are used for long-range interactions.

A regularized vortex-particle mesh method for large eddy simulation

NASA Astrophysics Data System (ADS)

Spietz, H. J.; Walther, J. H.; Hejlesen, M. M.

2017-11-01

We present recent developments of the remeshed vortex particle-mesh method for simulating incompressible fluid flow. The presented method relies on a parallel higher-order FFT based solver for the Poisson equation. Arbitrary high order is achieved through regularization of singular Green's function solutions to the Poisson equation and recently we have derived novel high order solutions for a mixture of open and periodic domains. With this approach the simulated variables may formally be viewed as the approximate solution to the filtered Navier Stokes equations, hence we use the method for Large Eddy Simulation by including a dynamic subfilter-scale model based on test-filters compatible with the aforementioned regularization functions. Further the subfilter-scale model uses Lagrangian averaging, which is a natural candidate in light of the Lagrangian nature of vortex particle methods. A multiresolution variation of the method is applied to simulate the benchmark problem of the flow past a square cylinder at Re = 22000 and the obtained results are compared to results from the literature.

Applying transfer matrix method to the estimation of the modal characteristics of the NASA Mini-Mass Truss

NASA Technical Reports Server (NTRS)

Shen, Ji-Yao; Taylor, Lawrence W., Jr.

1994-01-01

It is beneficial to use a distributed parameter model for large space structures because the approach minimizes the number of model parameters. Holzer's transfer matrix method provides a useful means to simplify and standardize the procedure for solving the system of partial differential equations. Any large space structures can be broken down into sub-structures with simple elastic and dynamical properties. For each single element, such as beam, tether, or rigid body, we can derive the corresponding transfer matrix. Combining these elements' matrices enables the solution of the global system equations. The characteristics equation can then be formed by satisfying the appropriate boundary conditions. Then natural frequencies and mode shapes can be determined by searching the roots of the characteristic equation at frequencies within the range of interest. This paper applies this methodology, and the maximum likelihood estimation method, to refine the modal characteristics of the NASA Mini-Mast Truss by successively matching the theoretical response to the test data of the truss. The method is being applied to more complex configurations.

A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion equation

NASA Astrophysics Data System (ADS)

Sun, HongGuang; Liu, Xiaoting; Zhang, Yong; Pang, Guofei; Garrard, Rhiannon

2017-09-01

Fractional-order diffusion equations (FDEs) extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media. Real-world diffusion can be multi-dimensional, requiring efficient numerical solvers that can handle long-term memory embedded in mass transport. To address this challenge, a semi-discrete Kansa method is developed to approximate the two-dimensional spatiotemporal FDE, where the Kansa approach first discretizes the FDE, then the Gauss-Jacobi quadrature rule solves the corresponding matrix, and finally the Mittag-Leffler function provides an analytical solution for the resultant time-fractional ordinary differential equation. Numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes. Applications further show that the numerical method can efficiently solve two-dimensional spatiotemporal FDE models with either a continuous or discrete mixing measure. Hence this study provides an efficient and fast computational method for modeling super-diffusive, sub-diffusive, and mixed diffusive processes in large, two-dimensional domains with irregular shapes.

Solution of a few nonlinear problems in aerodynamics by the finite elements and functional least squares methods. Ph.D. Thesis - Paris Univ.; [mathematical models of transonic flow using nonlinear equations

NASA Technical Reports Server (NTRS)

Periaux, J.

1979-01-01

The numerical simulation of the transonic flows of idealized fluids and of incompressible viscous fluids, by the nonlinear least squares methods is presented. The nonlinear equations, the boundary conditions, and the various constraints controlling the two types of flow are described. The standard iterative methods for solving a quasi elliptical nonlinear equation with partial derivatives are reviewed with emphasis placed on two examples: the fixed point method applied to the Gelder functional in the case of compressible subsonic flows and the Newton method used in the technique of decomposition of the lifting potential. The new abstract least squares method is discussed. It consists of substituting the nonlinear equation by a problem of minimization in a H to the minus 1 type Sobolev functional space.

Benchmarking FEniCS for mantle convection simulations

NASA Astrophysics Data System (ADS)

Vynnytska, L.; Rognes, M. E.; Clark, S. R.

2013-01-01

This paper evaluates the usability of the FEniCS Project for mantle convection simulations by numerical comparison to three established benchmarks. The benchmark problems all concern convection processes in an incompressible fluid induced by temperature or composition variations, and cover three cases: (i) steady-state convection with depth- and temperature-dependent viscosity, (ii) time-dependent convection with constant viscosity and internal heating, and (iii) a Rayleigh-Taylor instability. These problems are modeled by the Stokes equations for the fluid and advection-diffusion equations for the temperature and composition. The FEniCS Project provides a novel platform for the automated solution of differential equations by finite element methods. In particular, it offers a significant flexibility with regard to modeling and numerical discretization choices; we have here used a discontinuous Galerkin method for the numerical solution of the advection-diffusion equations. Our numerical results are in agreement with the benchmarks, and demonstrate the applicability of both the discontinuous Galerkin method and FEniCS for such applications.

A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Cooke, C. H.

1976-01-01

An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C deg-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1,000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and asymptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.

On the comparison of perturbation-iteration algorithm and residual power series method to solve fractional Zakharov-Kuznetsov equation

NASA Astrophysics Data System (ADS)

Şenol, Mehmet; Alquran, Marwan; Kasmaei, Hamed Daei

2018-06-01

In this paper, we present analytic-approximate solution of time-fractional Zakharov-Kuznetsov equation. This model demonstrates the behavior of weakly nonlinear ion acoustic waves in a plasma bearing cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Basic definitions of fractional derivatives are described in the Caputo sense. Perturbation-iteration algorithm (PIA) and residual power series method (RPSM) are applied to solve this equation with success. The convergence analysis is also presented for both methods. Numerical results are given and then they are compared with the exact solutions. Comparison of the results reveal that both methods are competitive, powerful, reliable, simple to use and ready to apply to wide range of fractional partial differential equations.

Bukhvostov-Lipatov model and quantum-classical duality

NASA Astrophysics Data System (ADS)

Bazhanov, Vladimir V.; Lukyanov, Sergei L.; Runov, Boris A.

2018-02-01

The Bukhvostov-Lipatov model is an exactly soluble model of two interacting Dirac fermions in 1 + 1 dimensions. The model describes weakly interacting instantons and anti-instantons in the O (3) non-linear sigma model. In our previous work [arxiv:arXiv:1607.04839] we have proposed an exact formula for the vacuum energy of the Bukhvostov-Lipatov model in terms of special solutions of the classical sinh-Gordon equation, which can be viewed as an example of a remarkable duality between integrable quantum field theories and integrable classical field theories in two dimensions. Here we present a complete derivation of this duality based on the classical inverse scattering transform method, traditional Bethe ansatz techniques and analytic theory of ordinary differential equations. In particular, we show that the Bethe ansatz equations defining the vacuum state of the quantum theory also define connection coefficients of an auxiliary linear problem for the classical sinh-Gordon equation. Moreover, we also present details of the derivation of the non-linear integral equations determining the vacuum energy and other spectral characteristics of the model in the case when the vacuum state is filled by 2-string solutions of the Bethe ansatz equations.

Applications of the ETEM for obtaining optical soliton solutions for the Lakshmanan-Porsezian-Daniel model

NASA Astrophysics Data System (ADS)

Manafian, Jalil; Foroutan, Mohammadreza; Guzali, Aref

2017-11-01

This paper examines the effectiveness of an integration scheme which is called the extended trial equation method (ETEM) for solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the Lakshmanan-Porsezian-Daniel (LPD) equation with Kerr and power laws of nonlinearity which describes higher-order dispersion, full nonlinearity and spatiotemporal dispersion is considered, and as an achievement, a series of exact travelling-wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of LPD equation. The movement of obtained solutions is shown graphically, which helps to understand the physical phenomena of this optical soliton equation. Many other such types of nonlinear equations arising in basic fabric of communications network technology and nonlinear optics can also be solved by this method.

A spectral mimetic least-squares method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bochev, Pavel; Gerritsma, Marc

We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are alsomore » satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.« less

A spectral mimetic least-squares method

DOE PAGES

Bochev, Pavel; Gerritsma, Marc

2014-09-01

We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are alsomore » satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.« less

Testing Mediation Using Multiple Regression and Structural Equation Modeling Analyses in Secondary Data

ERIC Educational Resources Information Center

Li, Spencer D.

2011-01-01

Mediation analysis in child and adolescent development research is possible using large secondary data sets. This article provides an overview of two statistical methods commonly used to test mediated effects in secondary analysis: multiple regression and structural equation modeling (SEM). Two empirical studies are presented to illustrate the…

An Extension of IRT-Based Equating to the Dichotomous Testlet Response Theory Model

ERIC Educational Resources Information Center

Tao, Wei; Cao, Yi

2016-01-01

Current procedures for equating number-correct scores using traditional item response theory (IRT) methods assume local independence. However, when tests are constructed using testlets, one concern is the violation of the local item independence assumption. The testlet response theory (TRT) model is one way to accommodate local item dependence.…

Classical Yang-Baxter equations and quantum integrable systems

NASA Astrophysics Data System (ADS)

Jurčo, Branislav

1989-06-01

Quantum integrable models associated with nondegenerate solutions of classical Yang-Baxter equations related to the simple Lie algebras are investigated. These models are diagonalized for rational and trigonometric solutions in the cases of sl(N)/gl(N)/, o(N) and sp(N) algebras. The analogy with the quantum inverse scattering method is demonstrated.

Construction of reduced order models for the non-linear Navier-Stokes equations using the proper orthogonal fecomposition (POD)/Galerkin method.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fike, Jeffrey A.

2013-08-01

The construction of stable reduced order models using Galerkin projection for the Euler or Navier-Stokes equations requires a suitable choice for the inner product. The standard L2 inner product is expected to produce unstable ROMs. For the non-linear Navier-Stokes equations this means the use of an energy inner product. In this report, Galerkin projection for the non-linear Navier-Stokes equations using the L2 inner product is implemented as a first step toward constructing stable ROMs for this set of physics.

From the Boltzmann to the Lattice-Boltzmann Equation:. Beyond BGK Collision Models

NASA Astrophysics Data System (ADS)

Philippi, Paulo Cesar; Hegele, Luiz Adolfo; Surmas, Rodrigo; Siebert, Diogo Nardelli; Dos Santos, Luís Orlando Emerich

In this work, we present a derivation for the lattice-Boltzmann equation directly from the linearized Boltzmann equation, combining the following main features: multiple relaxation times and thermodynamic consistency in the description of non isothermal compressible flows. The method presented here is based on the discretization of increasingly order kinetic models of the Boltzmann equation. Following a Gross-Jackson procedure, the linearized collision term is developed in Hermite polynomial tensors and the resulting infinite series is diagonalized after a chosen integer N, establishing the order of approximation of the collision term. The velocity space is discretized, in accordance with a quadrature method based on prescribed abscissas (Philippi et al., Phys. Rev E 73, 056702, 2006). The problem of describing the energy transfer is discussed, in relation with the order of approximation of a two relaxation-times lattice Boltzmann model. The velocity-step, temperature-step and the shock tube problems are investigated, adopting lattices with 37, 53 and 81 velocities.

Global boundary flattening transforms for acoustic propagation under rough sea surfaces.

PubMed

Oba, Roger M

2010-07-01

This paper introduces a conformal transform of an acoustic domain under a one-dimensional, rough sea surface onto a domain with a flat top. This non-perturbative transform can include many hundreds of wavelengths of the surface variation. The resulting two-dimensional, flat-topped domain allows direct application of any existing, acoustic propagation model of the Helmholtz or wave equation using transformed sound speeds. Such a transform-model combination applies where the surface particle velocity is much slower than sound speed, such that the boundary motion can be neglected. Once the acoustic field is computed, the bijective (one-to-one and onto) mapping permits the field interpolation in terms of the original coordinates. The Bergstrom method for inverse Riemann maps determines the transform by iterated solution of an integral equation for a surface matching term. Rough sea surface forward scatter test cases provide verification of the method using a particular parabolic equation model of the Helmholtz equation.

Multipath analysis diffraction calculations

NASA Technical Reports Server (NTRS)

Statham, Richard B.

1996-01-01

This report describes extensions of the Kirchhoff diffraction equation to higher edge terms and discusses their suitability to model diffraction multipath effects of a small satellite structure. When receiving signals, at a satellite, from the Global Positioning System (GPS), reflected signals from the satellite structure result in multipath errors in the determination of the satellite position. Multipath error can be caused by diffraction of the reflected signals and a method of calculating this diffraction is required when using a facet model of the satellite. Several aspects of the Kirchhoff equation are discussed and numerical examples, in the near and far fields, are shown. The vector form of the extended Kirchhoff equation, by adding the Larmor-Tedone and Kottler edge terms, is given as a mathematical model in an appendix. The Kirchhoff equation was investigated as being easily implemented and of good accuracy in the basic form, especially in phase determination. The basic Kirchhoff can be extended for higher accuracy if desired. A brief discussion of the method of moments and the geometric theory of diffraction is included, but seems to offer no clear advantage in implementation over the Kirchhoff for facet models.

Theoretical and computational analyses of LNG evaporator

NASA Astrophysics Data System (ADS)

Chidambaram, Palani Kumar; Jo, Yang Myung; Kim, Heuy Dong

2017-04-01

Theoretical and numerical analysis on the fluid flow and heat transfer inside a LNG evaporator is conducted in this work. Methane is used instead of LNG as the operating fluid. This is because; methane constitutes over 80% of natural gas. The analytical calculations are performed using simple mass and energy balance equations. The analytical calculations are made to assess the pressure and temperature variations in the steam tube. Multiphase numerical simulations are performed by solving the governing equations (basic flow equations of continuity, momentum and energy equations) in a portion of the evaporator domain consisting of a single steam pipe. The flow equations are solved along with equations of species transport. Multiphase modeling is incorporated using VOF method. Liquid methane is the primary phase. It vaporizes into the secondary phase gaseous methane. Steam is another secondary phase which flows through the heating coils. Turbulence is modeled by a two equation turbulence model. Both the theoretical and numerical predictions are seen to match well with each other. Further parametric studies are planned based on the current research.

Modeling techniques for quantum cascade lasers

NASA Astrophysics Data System (ADS)

Jirauschek, Christian; Kubis, Tillmann

2014-03-01

Quantum cascade lasers are unipolar semiconductor lasers covering a wide range of the infrared and terahertz spectrum. Lasing action is achieved by using optical intersubband transitions between quantized states in specifically designed multiple-quantum-well heterostructures. A systematic improvement of quantum cascade lasers with respect to operating temperature, efficiency, and spectral range requires detailed modeling of the underlying physical processes in these structures. Moreover, the quantum cascade laser constitutes a versatile model device for the development and improvement of simulation techniques in nano- and optoelectronics. This review provides a comprehensive survey and discussion of the modeling techniques used for the simulation of quantum cascade lasers. The main focus is on the modeling of carrier transport in the nanostructured gain medium, while the simulation of the optical cavity is covered at a more basic level. Specifically, the transfer matrix and finite difference methods for solving the one-dimensional Schrödinger equation and Schrödinger-Poisson system are discussed, providing the quantized states in the multiple-quantum-well active region. The modeling of the optical cavity is covered with a focus on basic waveguide resonator structures. Furthermore, various carrier transport simulation methods are discussed, ranging from basic empirical approaches to advanced self-consistent techniques. The methods include empirical rate equation and related Maxwell-Bloch equation approaches, self-consistent rate equation and ensemble Monte Carlo methods, as well as quantum transport approaches, in particular the density matrix and non-equilibrium Green's function formalism. The derived scattering rates and self-energies are generally valid for n-type devices based on one-dimensional quantum confinement, such as quantum well structures.

Modeling techniques for quantum cascade lasers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jirauschek, Christian; Kubis, Tillmann

2014-03-15

Quantum cascade lasers are unipolar semiconductor lasers covering a wide range of the infrared and terahertz spectrum. Lasing action is achieved by using optical intersubband transitions between quantized states in specifically designed multiple-quantum-well heterostructures. A systematic improvement of quantum cascade lasers with respect to operating temperature, efficiency, and spectral range requires detailed modeling of the underlying physical processes in these structures. Moreover, the quantum cascade laser constitutes a versatile model device for the development and improvement of simulation techniques in nano- and optoelectronics. This review provides a comprehensive survey and discussion of the modeling techniques used for the simulation ofmore » quantum cascade lasers. The main focus is on the modeling of carrier transport in the nanostructured gain medium, while the simulation of the optical cavity is covered at a more basic level. Specifically, the transfer matrix and finite difference methods for solving the one-dimensional Schrödinger equation and Schrödinger-Poisson system are discussed, providing the quantized states in the multiple-quantum-well active region. The modeling of the optical cavity is covered with a focus on basic waveguide resonator structures. Furthermore, various carrier transport simulation methods are discussed, ranging from basic empirical approaches to advanced self-consistent techniques. The methods include empirical rate equation and related Maxwell-Bloch equation approaches, self-consistent rate equation and ensemble Monte Carlo methods, as well as quantum transport approaches, in particular the density matrix and non-equilibrium Green's function formalism. The derived scattering rates and self-energies are generally valid for n-type devices based on one-dimensional quantum confinement, such as quantum well structures.« less

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models

PubMed Central

Rosenblatt, Marcus; Timmer, Jens; Kaschek, Daniel

2016-01-01

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization. PMID:27243005

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models.

PubMed

Rosenblatt, Marcus; Timmer, Jens; Kaschek, Daniel

2016-01-01

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization.

A low noise discrete velocity method for the Boltzmann equation with quantized rotational and vibrational energy

NASA Astrophysics Data System (ADS)

Clarke, Peter; Varghese, Philip; Goldstein, David

2018-01-01

A discrete velocity method is developed for gas mixtures of diatomic molecules with both rotational and vibrational energy states. A full quantized model is described, and rotation-translation and vibration-translation energy exchanges are simulated using a Larsen-Borgnakke exchange model. Elastic and inelastic molecular interactions are modeled during every simulated collision to help produce smooth internal energy distributions. The method is verified by comparing simulations of homogeneous relaxation by our discrete velocity method to numerical solutions of the Jeans and Landau-Teller equations, and to direct simulation Monte Carlo. We compute the structure of a 1D shock using this method, and determine how the rotational energy distribution varies with spatial location in the shock and with position in velocity space.

A Sub-filter Scale Noise Equation far Hybrid LES Simulations

NASA Technical Reports Server (NTRS)

Goldstein, Marvin E.

2006-01-01

Hybrid LES/subscale modeling approaches have an important advantage over the current noise prediction methods in that they only involve modeling of the relatively universal subscale motion and not the configuration dependent larger scale turbulence . Previous hybrid approaches use approximate statistical techniques or extrapolation methods to obtain the requisite information about the sub-filter scale motion. An alternative approach would be to adopt the modeling techniques used in the current noise prediction methods and determine the unknown stresses from experimental data. The present paper derives an equation for predicting the sub scale sound from information that can be obtained with currently available experimental procedures. The resulting prediction method would then be intermediate between the current noise prediction codes and previously proposed hybrid techniques.

Parallel iterative methods for sparse linear and nonlinear equations

NASA Technical Reports Server (NTRS)

Saad, Youcef

1989-01-01

As three-dimensional models are gaining importance, iterative methods will become almost mandatory. Among these, preconditioned Krylov subspace methods have been viewed as the most efficient and reliable, when solving linear as well as nonlinear systems of equations. There has been several different approaches taken to adapt iterative methods for supercomputers. Some of these approaches are discussed and the methods that deal more specifically with general unstructured sparse matrices, such as those arising from finite element methods, are emphasized.

Numerical modeling method on the movement of water flow and suspended solids in two-dimensional sedimentation tanks in the wastewater treatment plant.

PubMed

Zeng, Guang-Ming; Jiang, Yi-Min; Qin, Xiao-Sheng; Huang, Guo-He; Li, Jian-Bing

2003-01-01

Taking the distributing calculation of velocity and concentration as an example, the paper established a series of governing equations by the vorticity-stream function method, and dispersed the equations by the finite differencing method. After figuring out the distribution field of velocity, the paper also calculated the concentration distribution in sedimentation tank by using the two-dimensional concentration transport equation. The validity and feasibility of the numerical method was verified through comparing with experimental data. Furthermore, the paper carried out a tentative exploration into the application of numerical simulation of sedimentation tanks.

Terrain and refractivity effects on non-optical paths

NASA Astrophysics Data System (ADS)

Barrios, Amalia E.

1994-07-01

The split-step parabolic equation (SSPE) has been used extensively to model tropospheric propagation over the sea, but recent efforts have extended this method to propagation over arbitrary terrain. At the Naval Command, Control and Ocean Surveillance Center (NCCOSC), Research, Development, Test and Evaluation Division, a split-step Terrain Parabolic Equation Model (TPEM) has been developed that takes into account variable terrain and range-dependent refractivity profiles. While TPEM has been previously shown to compare favorably with measured data and other existing terrain models, two alternative methods to model radiowave propagation over terrain, implemented within TPEM, will be presented that give a two to ten-fold decrease in execution time. These two methods are also shown to agree well with measured data.

Evaluation of a Consistent LES/PDF Method Using a Series of Experimental Spray Flames

NASA Astrophysics Data System (ADS)

Heye, Colin; Raman, Venkat

2012-11-01

A consistent method for the evolution of the joint-scalar probability density function (PDF) transport equation is proposed for application to large eddy simulation (LES) of turbulent reacting flows containing evaporating spray droplets. PDF transport equations provide the benefit of including the chemical source term in closed form, however, additional terms describing LES subfilter mixing must be modeled. The recent availability of detailed experimental measurements provide model validation data for a wide range of evaporation rates and combustion regimes, as is well-known to occur in spray flames. In this work, the experimental data will used to investigate the impact of droplet mass loading and evaporation rates on the subfilter scalar PDF shape in comparison with conventional flamelet models. In addition, existing model term closures in the PDF transport equations are evaluated with a focus on their validity in the presence of regime changes.

Modeling of outgassing and matrix decomposition in carbon-phenolic composites

NASA Technical Reports Server (NTRS)

Mcmanus, Hugh L.

1994-01-01

Work done in the period Jan. - June 1994 is summarized. Two threads of research have been followed. First, the thermodynamics approach was used to model the chemical and mechanical responses of composites exposed to high temperatures. The thermodynamics approach lends itself easily to the usage of variational principles. This thermodynamic-variational approach has been applied to the transpiration cooling problem. The second thread is the development of a better algorithm to solve the governing equations resulting from the modeling. Explicit finite difference method is explored for solving the governing nonlinear, partial differential equations. The method allows detailed material models to be included and solution on massively parallel supercomputers. To demonstrate the feasibility of the explicit scheme in solving nonlinear partial differential equations, a transpiration cooling problem was solved. Some interesting transient behaviors were captured such as stress waves and small spatial oscillations of transient pressure distribution.

Multiple-reflection model of human skin and estimation of pigment concentrations

NASA Astrophysics Data System (ADS)

Ohtsuki, Rie; Tominaga, Shoji; Tanno, Osamu

2012-07-01

We describe a new method for estimating the concentrations of pigments in the human skin using surface spectral reflectance. We derive an equation that expresses the surface spectral reflectance of the human skin. First, we propose an optical model of the human skin that accounts for the stratum corneum. We also consider the difference between the scattering coefficient of the epidermis and that of the dermis. We then derive an equation by applying the Kubelka-Munk theory to an optical model of the human skin. Unlike a model developed in a recent study, the present equation considers pigments as well as multiple reflections and the thicknesses of the skin layers as factors that affect the color of the human skin. In two experiments, we estimate the pigment concentrations using the measured surface spectral reflectances. Finally, we confirm the feasibility of the concentrations estimated by the proposed method by evaluating the estimated pigment concentrations in the skin.

Personal computer study of finite-difference methods for the transonic small disturbance equation

NASA Technical Reports Server (NTRS)

Bland, Samuel R.

1989-01-01

Calculation of unsteady flow phenomena requires careful attention to the numerical treatment of the governing partial differential equations. The personal computer provides a convenient and useful tool for the development of meshes, algorithms, and boundary conditions needed to provide time accurate solution of these equations. The one-dimensional equation considered provides a suitable model for the study of wave propagation in the equations of transonic small disturbance potential flow. Numerical results for effects of mesh size, extent, and stretching, time step size, and choice of far-field boundary conditions are presented. Analysis of the discretized model problem supports these numerical results. Guidelines for suitable mesh and time step choices are given.

A Study of Multigrid Preconditioners Using Eigensystem Analysis

NASA Technical Reports Server (NTRS)

Roberts, Thomas W.; Swanson, R. C.

2005-01-01

The convergence properties of numerical schemes for partial differential equations are studied by examining the eigensystem of the discrete operator. This method of analysis is very general, and allows the effects of boundary conditions and grid nonuniformities to be examined directly. Algorithms for the Laplace equation and a two equation model hyperbolic system are examined.

Empirical Sampling Distributions of Equating Coefficients for Graded and Nominal Response Instruments.

ERIC Educational Resources Information Center

Baker, Frank B.

1997-01-01

Examined the sampling distributions of equating coefficients produced by the characteristic curve method for tests using graded and nominal response scoring using simulated data. For both models and across all three equating situations, the sampling distributions were generally bell-shaped and peaked, and occasionally had a small degree of…

Dissolution process analysis using model-free Noyes-Whitney integral equation.

PubMed

Hattori, Yusuke; Haruna, Yoshimasa; Otsuka, Makoto

2013-02-01

Drug dissolution process of solid dosages is theoretically described by Noyes-Whitney-Nernst equation. However, the analysis of the process is demonstrated assuming some models. Normally, the model-dependent methods are idealized and require some limitations. In this study, Noyes-Whitney integral equation was proposed and applied to represent the drug dissolution profiles of a solid formulation via the non-linear least squares (NLLS) method. The integral equation is a model-free formula involving the dissolution rate constant as a parameter. In the present study, several solid formulations were prepared via changing the blending time of magnesium stearate (MgSt) with theophylline monohydrate, α-lactose monohydrate, and crystalline cellulose. The formula could excellently represent the dissolution profile, and thereby the rate constant and specific surface area could be obtained by NLLS method. Since the long time blending coated the particle surface with MgSt, it was found that the water permeation was disturbed by its layer dissociating into disintegrant particles. In the end, the solid formulations were not disintegrated; however, the specific surface area gradually increased during the process of dissolution. The X-ray CT observation supported this result and demonstrated that the rough surface was dominant as compared to dissolution, and thus, specific surface area of the solid formulation gradually increased. Copyright © 2012 Elsevier B.V. All rights reserved.

Physical lumping methods for developing linear reduced models for high speed propulsion systems

NASA Technical Reports Server (NTRS)

Immel, S. M.; Hartley, Tom T.; Deabreu-Garcia, J. Alex

1991-01-01

In gasdynamic systems, information travels in one direction for supersonic flow and in both directions for subsonic flow. A shock occurs at the transition from supersonic to subsonic flow. Thus, to simulate these systems, any simulation method implemented for the quasi-one-dimensional Euler equations must have the ability to capture the shock. In this paper, a technique combining both backward and central differencing is presented. The equations are subsequently linearized about an operating point and formulated into a linear state space model. After proper implementation of the boundary conditions, the model order is reduced from 123 to less than 10 using the Schur method of balancing. Simulations comparing frequency and step response of the reduced order model and the original system models are presented.

Finite element approximation of the radiative transport equation in a medium with piece-wise constant refractive index

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lehtikangas, O., E-mail: Ossi.Lehtikangas@uef.fi; Tarvainen, T.; Department of Computer Science, University College London, Gower Street, London WC1E 6BT

2015-02-01

The radiative transport equation can be used as a light transport model in a medium with scattering particles, such as biological tissues. In the radiative transport equation, the refractive index is assumed to be constant within the medium. However, in biomedical media, changes in the refractive index can occur between different tissue types. In this work, light propagation in a medium with piece-wise constant refractive index is considered. Light propagation in each sub-domain with a constant refractive index is modeled using the radiative transport equation and the equations are coupled using boundary conditions describing Fresnel reflection and refraction phenomena onmore » the interfaces between the sub-domains. The resulting coupled system of radiative transport equations is numerically solved using a finite element method. The approach is tested with simulations. The results show that this coupled system describes light propagation accurately through comparison with the Monte Carlo method. It is also shown that neglecting the internal changes of the refractive index can lead to erroneous boundary measurements of scattered light.« less

A three-dimensional method-of-characteristics solute-transport model (MOC3D)

USGS Publications Warehouse

Konikow, Leonard F.; Goode, D.J.; Hornberger, G.Z.

1996-01-01

This report presents a model, MOC3D, that simulates three-dimensional solute transport in flowing ground water. The model computes changes in concentration of a single dissolved chemical constituent over time that are caused by advective transport, hydrodynamic dispersion (including both mechanical dispersion and diffusion), mixing (or dilution) from fluid sources, and mathematically simple chemical reactions (including linear sorption, which is represented by a retardation factor, and decay). The transport model is integrated with MODFLOW, a three-dimensional ground-water flow model that uses implicit finite-difference methods to solve the transient flow equation. MOC3D uses the method of characteristics to solve the transport equation on the basis of the hydraulic gradients computed with MODFLOW for a given time step. This implementation of the method of characteristics uses particle tracking to represent advective transport and explicit finite-difference methods to calculate the effects of other processes. However, the explicit procedure has several stability criteria that may limit the size of time increments for solving the transport equation; these are automatically determined by the program. For improved efficiency, the user can apply MOC3D to a subgrid of the primary MODFLOW grid that is used to solve the flow equation. However, the transport subgrid must have uniform grid spacing along rows and columns. The report includes a description of the theoretical basis of the model, a detailed description of input requirements and output options, and the results of model testing and evaluation. The model was evaluated for several problems for which exact analytical solutions are available and by benchmarking against other numerical codes for selected complex problems for which no exact solutions are available. These test results indicate that the model is very accurate for a wide range of conditions and yields minimal numerical dispersion for advection-dominated problems. Mass-balance errors are generally less than 10 percent, and tend to decrease and stabilize with time.

Techniques for estimating flood-peak discharges of rural, unregulated streams in Ohio

USGS Publications Warehouse

Koltun, G.F.

2003-01-01

Regional equations for estimating 2-, 5-, 10-, 25-, 50-, 100-, and 500-year flood-peak discharges at ungaged sites on rural, unregulated streams in Ohio were developed by means of ordinary and generalized least-squares (GLS) regression techniques. One-variable, simple equations and three-variable, full-model equations were developed on the basis of selected basin characteristics and flood-frequency estimates determined for 305 streamflow-gaging stations in Ohio and adjacent states. The average standard errors of prediction ranged from about 39 to 49 percent for the simple equations, and from about 34 to 41 percent for the full-model equations. Flood-frequency estimates determined by means of log-Pearson Type III analyses are reported along with weighted flood-frequency estimates, computed as a function of the log-Pearson Type III estimates and the regression estimates. Values of explanatory variables used in the regression models were determined from digital spatial data sets by means of a geographic information system (GIS), with the exception of drainage area, which was determined by digitizing the area within basin boundaries manually delineated on topographic maps. Use of GIS-based explanatory variables represents a major departure in methodology from that described in previous reports on estimating flood-frequency characteristics of Ohio streams. Examples are presented illustrating application of the regression equations to ungaged sites on ungaged and gaged streams. A method is provided to adjust regression estimates for ungaged sites by use of weighted and regression estimates for a gaged site on the same stream. A region-of-influence method, which employs a computer program to estimate flood-frequency characteristics for ungaged sites based on data from gaged sites with similar characteristics, was also tested and compared to the GLS full-model equations. For all recurrence intervals, the GLS full-model equations had superior prediction accuracy relative to the simple equations and therefore are recommended for use.

A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries.

PubMed

Ge, Liang; Sotiropoulos, Fotis

2007-08-01

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries

PubMed Central

Ge, Liang; Sotiropoulos, Fotis

2008-01-01

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [1]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus. PMID:19194533

Hybrid diffusion-P3 equation in N-layered turbid media: steady-state domain.

PubMed

Shi, Zhenzhi; Zhao, Huijuan; Xu, Kexin

2011-10-01

This paper discusses light propagation in N-layered turbid media. The hybrid diffusion-P3 equation is solved for an N-layered finite or infinite turbid medium in the steady-state domain for one point source using the extrapolated boundary condition. The Fourier transform formalism is applied to derive the analytical solutions of the fluence rate in Fourier space. Two inverse Fourier transform methods are developed to calculate the fluence rate in real space. In addition, the solutions of the hybrid diffusion-P3 equation are compared to the solutions of the diffusion equation and the Monte Carlo simulation. For the case of small absorption coefficients, the solutions of the N-layered diffusion equation and hybrid diffusion-P3 equation are almost equivalent and are in agreement with the Monte Carlo simulation. For the case of large absorption coefficients, the model of the hybrid diffusion-P3 equation is more precise than that of the diffusion equation. In conclusion, the model of the hybrid diffusion-P3 equation can replace the diffusion equation for modeling light propagation in the N-layered turbid media for a wide range of absorption coefficients.

Iterative Methods for Solving Nonlinear Parabolic Problem in Pension Saving Management

NASA Astrophysics Data System (ADS)

Koleva, M. N.

2011-11-01

In this work we consider a nonlinear parabolic equation, obtained from Riccati like transformation of the Hamilton-Jacobi-Bellman equation, arising in pension saving management. We discuss two numerical iterative methods for solving the model problem—fully implicit Picard method and mixed Picard-Newton method, which preserves the parabolic characteristics of the differential problem. Numerical experiments for comparison the accuracy and effectiveness of the algorithms are discussed. Finally, observations are given.

A boundary integral approach to the scattering of nonplanar acoustic waves by rigid bodies

NASA Technical Reports Server (NTRS)

Gallman, Judith M.; Myers, M. K.; Farassat, F.

1990-01-01

The acoustic scattering of an incident wave by a rigid body can be described by a singular Fredholm integral equation of the second kind. This equation is derived by solving the wave equation using generalized function theory, Green's function for the wave equation in unbounded space, and the acoustic boundary condition for a perfectly rigid body. This paper will discuss the derivation of the wave equation, its reformulation as a boundary integral equation, and the solution of the integral equation by the Galerkin method. The accuracy of the Galerkin method can be assessed by applying the technique outlined in the paper to reproduce the known pressure fields that are due to various point sources. From the analysis of these simpler cases, the accuracy of the Galerkin solution can be inferred for the scattered pressure field caused by the incidence of a dipole field on a rigid sphere. The solution by the Galerkin technique can then be applied to such problems as a dipole model of a propeller whose pressure field is incident on a rigid cylinder. This is the groundwork for modeling the scattering of rotating blade noise by airplane fuselages.

Field data-based mathematical modeling by Bode equations and vector fitting algorithm for renewable energy applications.

PubMed

Sabry, A H; W Hasan, W Z; Ab Kadir, M Z A; Radzi, M A M; Shafie, S

2018-01-01

The power system always has several variations in its profile due to random load changes or environmental effects such as device switching effects when generating further transients. Thus, an accurate mathematical model is important because most system parameters vary with time. Curve modeling of power generation is a significant tool for evaluating system performance, monitoring and forecasting. Several numerical techniques compete to fit the curves of empirical data such as wind, solar, and demand power rates. This paper proposes a new modified methodology presented as a parametric technique to determine the system's modeling equations based on the Bode plot equations and the vector fitting (VF) algorithm by fitting the experimental data points. The modification is derived from the familiar VF algorithm as a robust numerical method. This development increases the application range of the VF algorithm for modeling not only in the frequency domain but also for all power curves. Four case studies are addressed and compared with several common methods. From the minimal RMSE, the results show clear improvements in data fitting over other methods. The most powerful features of this method is the ability to model irregular or randomly shaped data and to be applied to any algorithms that estimating models using frequency-domain data to provide state-space or transfer function for the model.

Field data-based mathematical modeling by Bode equations and vector fitting algorithm for renewable energy applications

PubMed Central

W. Hasan, W. Z.

2018-01-01

The power system always has several variations in its profile due to random load changes or environmental effects such as device switching effects when generating further transients. Thus, an accurate mathematical model is important because most system parameters vary with time. Curve modeling of power generation is a significant tool for evaluating system performance, monitoring and forecasting. Several numerical techniques compete to fit the curves of empirical data such as wind, solar, and demand power rates. This paper proposes a new modified methodology presented as a parametric technique to determine the system’s modeling equations based on the Bode plot equations and the vector fitting (VF) algorithm by fitting the experimental data points. The modification is derived from the familiar VF algorithm as a robust numerical method. This development increases the application range of the VF algorithm for modeling not only in the frequency domain but also for all power curves. Four case studies are addressed and compared with several common methods. From the minimal RMSE, the results show clear improvements in data fitting over other methods. The most powerful features of this method is the ability to model irregular or randomly shaped data and to be applied to any algorithms that estimating models using frequency-domain data to provide state-space or transfer function for the model. PMID:29351554

Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

NASA Astrophysics Data System (ADS)

Schnoerr, David; Sanguinetti, Guido; Grima, Ramon

2017-03-01

Stochastic fluctuations of molecule numbers are ubiquitous in biological systems. Important examples include gene expression and enzymatic processes in living cells. Such systems are typically modelled as chemical reaction networks whose dynamics are governed by the chemical master equation. Despite its simple structure, no analytic solutions to the chemical master equation are known for most systems. Moreover, stochastic simulations are computationally expensive, making systematic analysis and statistical inference a challenging task. Consequently, significant effort has been spent in recent decades on the development of efficient approximation and inference methods. This article gives an introduction to basic modelling concepts as well as an overview of state of the art methods. First, we motivate and introduce deterministic and stochastic methods for modelling chemical networks, and give an overview of simulation and exact solution methods. Next, we discuss several approximation methods, including the chemical Langevin equation, the system size expansion, moment closure approximations, time-scale separation approximations and hybrid methods. We discuss their various properties and review recent advances and remaining challenges for these methods. We present a comparison of several of these methods by means of a numerical case study and highlight some of their respective advantages and disadvantages. Finally, we discuss the problem of inference from experimental data in the Bayesian framework and review recent methods developed the literature. In summary, this review gives a self-contained introduction to modelling, approximations and inference methods for stochastic chemical kinetics.

A one-step method for modelling longitudinal data with differential equations.

PubMed

Hu, Yueqin; Treinen, Raymond

2018-04-06

Differential equation models are frequently used to describe non-linear trajectories of longitudinal data. This study proposes a new approach to estimate the parameters in differential equation models. Instead of estimating derivatives from the observed data first and then fitting a differential equation to the derivatives, our new approach directly fits the analytic solution of a differential equation to the observed data, and therefore simplifies the procedure and avoids bias from derivative estimations. A simulation study indicates that the analytic solutions of differential equations (ASDE) approach obtains unbiased estimates of parameters and their standard errors. Compared with other approaches that estimate derivatives first, ASDE has smaller standard error, larger statistical power and accurate Type I error. Although ASDE obtains biased estimation when the system has sudden phase change, the bias is not serious and a solution is also provided to solve the phase problem. The ASDE method is illustrated and applied to a two-week study on consumers' shopping behaviour after a sale promotion, and to a set of public data tracking participants' grammatical facial expression in sign language. R codes for ASDE, recommendations for sample size and starting values are provided. Limitations and several possible expansions of ASDE are also discussed. © 2018 The British Psychological Society.

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

PubMed Central

Chaudhry, Jehanzeb Hameed; Comer, Jeffrey; Aksimentiev, Aleksei; Olson, Luke N.

2013-01-01

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes. To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current. PMID:24363784

Sensitivity analysis of dynamic biological systems with time-delays.

PubMed

Wu, Wu Hsiung; Wang, Feng Sheng; Chang, Maw Shang

2010-10-15

Mathematical modeling has been applied to the study and analysis of complex biological systems for a long time. Some processes in biological systems, such as the gene expression and feedback control in signal transduction networks, involve a time delay. These systems are represented as delay differential equation (DDE) models. Numerical sensitivity analysis of a DDE model by the direct method requires the solutions of model and sensitivity equations with time-delays. The major effort is the computation of Jacobian matrix when computing the solution of sensitivity equations. The computation of partial derivatives of complex equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors. To address this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary. We have proposed an efficient algorithm with an adaptive step size control to compute the solution and dynamic sensitivities of biological systems described by ordinal differential equations (ODEs). The adaptive direct-decoupled algorithm is extended to solve the solution and dynamic sensitivities of time-delay systems describing by DDEs. To save the human effort and avoid the human errors in the computation of partial derivatives, an automatic differentiation technique is embedded in the extended algorithm to evaluate the Jacobian matrix. The extended algorithm is implemented and applied to two realistic models with time-delays: the cardiovascular control system and the TNF-α signal transduction network. The results show that the extended algorithm is a good tool for dynamic sensitivity analysis on DDE models with less user intervention. By comparing with direct-coupled methods in theory, the extended algorithm is efficient, accurate, and easy to use for end users without programming background to do dynamic sensitivity analysis on complex biological systems with time-delays.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Thompson, S.

This report describes the use of several subroutines from the CORLIB core mathematical subroutine library for the solution of a model fluid flow problem. The model consists of the Euler partial differential equations. The equations are spatially discretized using the method of pseudo-characteristics. The resulting system of ordinary differential equations is then integrated using the method of lines. The stiff ordinary differential equation solver LSODE (2) from CORLIB is used to perform the time integration. The non-stiff solver ODE (4) is used to perform a related integration. The linear equation solver subroutines DECOMP and SOLVE are used to solve linearmore » systems whose solutions are required in the calculation of the time derivatives. The monotone cubic spline interpolation subroutines PCHIM and PCHFE are used to approximate water properties. The report describes the use of each of these subroutines in detail. It illustrates the manner in which modules from a standard mathematical software library such as CORLIB can be used as building blocks in the solution of complex problems of practical interest. 9 refs., 2 figs., 4 tabs.« less

An Efficient Numerical Approach for Nonlinear Fokker-Planck equations

NASA Astrophysics Data System (ADS)

Otten, Dustin; Vedula, Prakash

2009-03-01

Fokker-Planck equations which are nonlinear with respect to their probability densities that occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, classical fermions and bosons can be challenging to solve numerically. To address some underlying challenges in obtaining numerical solutions, we propose a quadrature based moment method for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations. In this approach the distribution function is represented as a collection of Dirac delta functions with corresponding quadrature weights and locations, that are in turn determined from constraints based on evolution of generalized moments. Properties of the distribution function can be obtained by solution of transport equations for quadrature weights and locations. We will apply this computational approach to study a wide range of problems, including the Desai-Zwanzig Model (for nonlinear muscular contraction) and multivariate nonlinear Fokker-Planck equations describing classical fermions and bosons, and will also demonstrate good agreement with results obtained from Monte Carlo and other standard numerical methods.

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

PubMed Central

Koehl, Patrice; Delarue, Marc

2010-01-01

The Poisson–Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson–Boltzmann–Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available. PMID:20151727

AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.

PubMed

Koehl, Patrice; Delarue, Marc

2010-02-14

The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

Introduction to Numerical Methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schoonover, Joseph A.

2016-06-14

These are slides for a lecture for the Parallel Computing Summer Research Internship at the National Security Education Center. This gives an introduction to numerical methods. Repetitive algorithms are used to obtain approximate solutions to mathematical problems, using sorting, searching, root finding, optimization, interpolation, extrapolation, least squares regresion, Eigenvalue problems, ordinary differential equations, and partial differential equations. Many equations are shown. Discretizations allow us to approximate solutions to mathematical models of physical systems using a repetitive algorithm and introduce errors that can lead to numerical instabilities if we are not careful.

A general theory of kinetics and thermodynamics of steady-state copolymerization.

PubMed

Shu, Yao-Gen; Song, Yong-Shun; Ou-Yang, Zhong-Can; Li, Ming

2015-06-17

Kinetics of steady-state copolymerization has been investigated since the 1940s. Irreversible terminal and penultimate models were successfully applied to a number of comonomer systems, but failed for systems where depropagation is significant. Although a general mathematical treatment of the terminal model with depropagation was established in the 1980s, a penultimate model and higher-order terminal models with depropagation have not been systematically studied, since depropagation leads to hierarchically-coupled and unclosed kinetic equations which are hard to solve analytically. In this work, we propose a truncation method to solve the steady-state kinetic equations of any-order terminal models with depropagation in a unified way, by reducing them into closed steady-state equations which give the exact solution of the original kinetic equations. Based on the steady-state equations, we also derive a general thermodynamic equality in which the Shannon entropy of the copolymer sequence is explicitly introduced as part of the free energy dissipation of the whole copolymerization system.

The continuous adjoint approach to the k-ε turbulence model for shape optimization and optimal active control of turbulent flows

NASA Astrophysics Data System (ADS)

Papoutsis-Kiachagias, E. M.; Zymaris, A. S.; Kavvadias, I. S.; Papadimitriou, D. I.; Giannakoglou, K. C.

2015-03-01

The continuous adjoint to the incompressible Reynolds-averaged Navier-Stokes equations coupled with the low Reynolds number Launder-Sharma k-ε turbulence model is presented. Both shape and active flow control optimization problems in fluid mechanics are considered, aiming at minimum viscous losses. In contrast to the frequently used assumption of frozen turbulence, the adjoint to the turbulence model equations together with appropriate boundary conditions are derived, discretized and solved. This is the first time that the adjoint equations to the Launder-Sharma k-ε model have been derived. Compared to the formulation that neglects turbulence variations, the impact of additional terms and equations is evaluated. Sensitivities computed using direct differentiation and/or finite differences are used for comparative purposes. To demonstrate the need for formulating and solving the adjoint to the turbulence model equations, instead of merely relying upon the 'frozen turbulence assumption', the gain in the optimization turnaround time offered by the proposed method is quantified.

PDF modeling of near-wall turbulent flows

NASA Astrophysics Data System (ADS)

Dreeben, Thomas David

1997-06-01

Pdf methods are extended to include modeling of wall- bounded turbulent flows. For flows in which resolution of the viscous sublayer is desired, a Pdf near-wall model is developed in which the Generalized Langevin model is combined with an exact model for viscous transport. Durbin's method of elliptic relaxation is used to incorporate the wall effects into the governing equations without the use of wall functions or damping functions. Close to the wall, the Generalized Langevin model provides an analogy to the effect of the fluctuating continuity equation. This enables accurate modeling of the near-wall turbulent statistics. Demonstrated accuracy for fully-developed channel flow is achieved with a Pdf/Monte Carlo simulation, and with its related Reynolds-stress closure. For flows in which the details of the viscous sublayer are not important, a Pdf wall- function method is developed with the Simplified Langevin model.

The use of copulas to practical estimation of multivariate stochastic differential equation mixed effects models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rupšys, P.

A system of stochastic differential equations (SDE) with mixed-effects parameters and multivariate normal copula density function were used to develop tree height model for Scots pine trees in Lithuania. A two-step maximum likelihood parameter estimation method is used and computational guidelines are given. After fitting the conditional probability density functions to outside bark diameter at breast height, and total tree height, a bivariate normal copula distribution model was constructed. Predictions from the mixed-effects parameters SDE tree height model calculated during this research were compared to the regression tree height equations. The results are implemented in the symbolic computational language MAPLE.

NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION.

PubMed

Liu, F; Meerschaert, M M; McGough, R J; Zhuang, P; Liu, Q

2013-03-01

In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION

PubMed Central

Liu, F.; Meerschaert, M.M.; McGough, R.J.; Zhuang, P.; Liu, Q.

2013-01-01

In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian. PMID:23772179

Pyroclastic flow transport dynamics for a Montserrat volcano eruption

NASA Astrophysics Data System (ADS)

Cordoba, G.; Sparks, S.; del Risco, E.

2003-04-01

A two phase model of pyroclastic flows dynamics which account for the bed load and suspended load is shown. The model uses the compressible Navier-Stokes equations coupled with the convection-diffusion equation in order to take into account for the sedimentation. The skin friction is taken into account by using the wall functions. In despite of the complex mathematical formulation of the model, it has been implemented in a Personal Computer due to an assumption of two phase one velocity model which reduce the number of equations in the system. This non-linear equation system is solved numerically by using the Finite Element Method. This numerical method let us move the mesh in the direction of the deposition and then accounting for the shape of the bed and the thickness of the deposit The model is applied to the Montserrat's White River basin which extend from the dome to the sea, located about 4 Km away and then compared with the field data from the Boxing Day (26 December, 1997) eruption. Additionally some features as the temporary evolution of the dynamical pressure, particle concentration and temperature along the path at each time step is shown.

Three-phase Power Flow Calculation of Low Voltage Distribution Network Considering Characteristics of Residents Load

NASA Astrophysics Data System (ADS)

Wang, Yaping; Lin, Shunjiang; Yang, Zhibin

2017-05-01

In the traditional three-phase power flow calculation of the low voltage distribution network, the load model is described as constant power. Since this model cannot reflect the characteristics of actual loads, the result of the traditional calculation is always different from the actual situation. In this paper, the load model in which dynamic load represented by air conditioners parallel with static load represented by lighting loads is used to describe characteristics of residents load, and the three-phase power flow calculation model is proposed. The power flow calculation model includes the power balance equations of three-phase (A,B,C), the current balance equations of phase 0, and the torque balancing equations of induction motors in air conditioners. And then an alternating iterative algorithm of induction motor torque balance equations with each node balance equations is proposed to solve the three-phase power flow model. This method is applied to an actual low voltage distribution network of residents load, and by the calculation of three different operating states of air conditioners, the result demonstrates the effectiveness of the proposed model and the algorithm.

New non-linear model of groundwater recharge: Inclusion of memory, heterogeneity and visco-elasticity

NASA Astrophysics Data System (ADS)

Spannenberg, Jescica; Atangana, Abdon; Vermeulen, P. D.

2017-09-01

Fractional differentiation has adequate use for investigating real world scenarios related to geological formations associated with elasticity, heterogeneity, viscoelasticity, and the memory effect. Since groundwater systems exist in these geological formations, modelling groundwater recharge as a real world scenario is a challenging task to do because existing recharge estimation methods are governed by linear equations which make use of constant field parameters. This is inadequate because in reality these parameters are a function of both space and time. This study therefore concentrates on modifying the recharge equation governing the EARTH model, by application of the Eton approach. Accordingly, this paper presents a modified equation which is non-linear, and accounts for parameters in a way that it is a function of both space and time. To be more specific, herein, recharge and drainage resistance which are parameters within the equation, became a function of both space and time. Additionally, the study entailed solving the non-linear equation using an iterative method as well as numerical solutions by means of the Crank-Nicolson scheme. The numerical solutions were used alongside the Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu derivatives, so that account was taken for elasticity, heterogeneity, viscoelasticity, and the memory effect. In essence, this paper presents a more adequate model for recharge estimation.

A reaction-based paradigm to model reactive chemical transport in groundwater with general kinetic and equilibrium reactions.

PubMed

Zhang, Fan; Yeh, Gour-Tsyh; Parker, Jack C; Brooks, Scott C; Pace, Molly N; Kim, Young-Jin; Jardine, Philip M; Watson, David B

2007-06-16

This paper presents a reaction-based water quality transport model in subsurface flow systems. Transport of chemical species with a variety of chemical and physical processes is mathematically described by M partial differential equations (PDEs). Decomposition via Gauss-Jordan column reduction of the reaction network transforms M species reactive transport equations into two sets of equations: a set of thermodynamic equilibrium equations representing N(E) equilibrium reactions and a set of reactive transport equations of M-N(E) kinetic-variables involving no equilibrium reactions (a kinetic-variable is a linear combination of species). The elimination of equilibrium reactions from reactive transport equations allows robust and efficient numerical integration. The model solves the PDEs of kinetic-variables rather than individual chemical species, which reduces the number of reactive transport equations and simplifies the reaction terms in the equations. A variety of numerical methods are investigated for solving the coupled transport and reaction equations. Simulation comparisons with exact solutions were performed to verify numerical accuracy and assess the effectiveness of various numerical strategies to deal with different application circumstances. Two validation examples involving simulations of uranium transport in soil columns are presented to evaluate the ability of the model to simulate reactive transport with complex reaction networks involving both kinetic and equilibrium reactions.

Estimating varying coefficients for partial differential equation models.

PubMed

Zhang, Xinyu; Cao, Jiguo; Carroll, Raymond J

2017-09-01

Partial differential equations (PDEs) are used to model complex dynamical systems in multiple dimensions, and their parameters often have important scientific interpretations. In some applications, PDE parameters are not constant but can change depending on the values of covariates, a feature that we call varying coefficients. We propose a parameter cascading method to estimate varying coefficients in PDE models from noisy data. Our estimates of the varying coefficients are shown to be consistent and asymptotically normally distributed. The performance of our method is evaluated by a simulation study and by an empirical study estimating three varying coefficients in a PDE model arising from LIDAR data. © 2017, The International Biometric Society.

An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir; The Laboratory of Quantum Information Processing, Yazd University, Yazd; Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir

Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Errormore » analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.« less

Non-Linear Steady State Vibrations of Beams Excited by Vortex Shedding

NASA Astrophysics Data System (ADS)

LEWANDOWSKI, R.

2002-05-01

In this paper the non-linear vibrations of beams excited by vortex-shedding are considered. In particular, the steady state responses of beams near the synchronization region are taken into account. The main aerodynamic properties of wind are described by using the semi-empirical model proposed by Hartlen and Currie. The finite element method and the strip method are used to formulate the equation of motion of the system treated. The harmonic balance method is adopted to derive the amplitude equations. These equations are solved with the help of the continuation method which is very convenient to perform the parametric studies of the problem and to determine the response curve in the synchronization region. Moreover, the equations of motion are also integrated using the Newmark method. The results of calculations of several example problems are also shown to confirm the efficiency and accuracy of the presented method. The results obtained by the harmonic balance method and by the Newmark methods are in good agreement with each other.

Simplified method for numerical modeling of fiber lasers.

PubMed

Shtyrina, O V; Yarutkina, I A; Fedoruk, M P

2014-12-29

A simplified numerical approach to modeling of dissipative dispersion-managed fiber lasers is examined. We present a new numerical iteration algorithm for finding the periodic solutions of the system of nonlinear ordinary differential equations describing the intra-cavity dynamics of the dissipative soliton characteristics in dispersion-managed fiber lasers. We demonstrate that results obtained using simplified model are in good agreement with full numerical modeling based on the corresponding partial differential equations.

Transfer matrix method for dynamics modeling and independent modal space vibration control design of linear hybrid multibody system

NASA Astrophysics Data System (ADS)

Rong, Bao; Rui, Xiaoting; Lu, Kun; Tao, Ling; Wang, Guoping; Ni, Xiaojun

2018-05-01

In this paper, an efficient method of dynamics modeling and vibration control design of a linear hybrid multibody system (MS) is studied based on the transfer matrix method. The natural vibration characteristics of a linear hybrid MS are solved by using low-order transfer equations. Then, by constructing the brand-new body dynamics equation, augmented operator and augmented eigenvector, the orthogonality of augmented eigenvector of a linear hybrid MS is satisfied, and its state space model expressed in each independent model space is obtained easily. According to this dynamics model, a robust independent modal space-fuzzy controller is designed for vibration control of a general MS, and the genetic optimization of some critical control parameters of fuzzy tuners is also presented. Two illustrative examples are performed, which results show that this method is computationally efficient and with perfect control performance.

Modeling Electrokinetic Flows by the Smoothed Profile Method

PubMed Central

Luo, Xian; Beskok, Ali; Karniadakis, George Em

2010-01-01

We propose an efficient modeling method for electrokinetic flows based on the Smoothed Profile Method (SPM) [1–4] and spectral element discretizations. The new method allows for arbitrary differences in the electrical conductivities between the charged surfaces and the the surrounding electrolyte solution. The electrokinetic forces are included into the flow equations so that the Poisson-Boltzmann and electric charge continuity equations are cast into forms suitable for SPM. The method is validated by benchmark problems of electroosmotic flow in straight channels and electrophoresis of charged cylinders. We also present simulation results of electrophoresis of charged microtubules, and show that the simulated electrophoretic mobility and anisotropy agree with the experimental values. PMID:20352076

Multigrid and Krylov Subspace Methods for the Discrete Stokes Equations

NASA Technical Reports Server (NTRS)

Elman, Howard C.

1996-01-01

Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations, a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper, we compare the performance of four such methods: variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual, and multigrid methods, for solving several two-dimensional model problems. The results indicate that where it is applicable, multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantage of being both independent of iteration parameters and widely applicable.

Simulation of Two-Fluid Flows by the Least-Squares Finite Element Method Using a Continuum Surface Tension Model

NASA Technical Reports Server (NTRS)

Wu, Jie; Yu, Sheng-Tao; Jiang, Bo-nan

1996-01-01

In this paper a numerical procedure for simulating two-fluid flows is presented. This procedure is based on the Volume of Fluid (VOF) method proposed by Hirt and Nichols and the continuum surface force (CSF) model developed by Brackbill, et al. In the VOF method fluids of different properties are identified through the use of a continuous field variable (color function). The color function assigns a unique constant (color) to each fluid. The interfaces between different fluids are distinct due to sharp gradients of the color function. The evolution of the interfaces is captured by solving the convective equation of the color function. The CSF model is used as a means to treat surface tension effect at the interfaces. Here a modified version of the CSF model, proposed by Jacqmin, is used to calculate the tension force. In the modified version, the force term is obtained by calculating the divergence of a stress tensor defined by the gradient of the color function. In its analytical form, this stress formulation is equivalent to the original CSF model. Numerically, however, the use of the stress formulation has some advantages over the original CSF model, as it bypasses the difficulty in approximating the curvatures of the interfaces. The least-squares finite element method (LSFEM) is used to discretize the governing equation systems. The LSFEM has proven to be effective in solving incompressible Navier-Stokes equations and pure convection equations, making it an ideal candidate for the present applications. The LSFEM handles all the equations in a unified manner without any additional special treatment such as upwinding or artificial dissipation. Various bench mark tests have been carried out for both two dimensional planar and axisymmetric flows, including a dam breaking, oscillating and stationary bubbles and a conical liquid sheet in a pressure swirl atomizer.

Ray Modeling Methods for Range Dependent Ocean Environments

DTIC Science & Technology

1983-12-01

the eikonal equation, gives rise to equations for ray paths which are perpendicular to the wave fronts. Equation II.4, the transport equation, leads... databases for use by MEDUSA. The author has assisted in the installation of MEDUSA at computer facilities which possess databases containing archives of...sound velocity profiles, bathymetry, and bottom loss data. At each computer site, programs convert the archival data retrieved by the database system

Numerical Solution of the Extended Nernst-Planck Model.

PubMed

Samson; Marchand

1999-07-01

The main features of a numerical model aiming at predicting the drift of ions in an electrolytic solution upon a chemical potential gradient are presented. The mechanisms of ionic diffusion are described by solving the extended Nernst-Planck system of equations. The electrical coupling between the various ionic fluxes is accounted for by the Poisson equation. Furthermore, chemical activity effects are considered in the model. The whole system of nonlinear equations is solved using the finite-element method. Results yielded by the model for simple test cases are compared to those obtained using an analytical solution. Applications of the model to more complex problems are also presented and discussed. Copyright 1999 Academic Press.

Solitary traveling wave solutions of pressure equation of bubbly liquids with examination for viscosity and heat transfer

NASA Astrophysics Data System (ADS)

Khater, Mostafa M. A.; Seadawy, Aly R.; Lu, Dianchen

2018-03-01

In this research, we investigate one of the most popular model in nature and also industrial which is the pressure equation of bubbly liquids with examination for viscosity and heat transfer which has many application in nature and engineering. Understanding the physical meaning of exact and solitary traveling wave solutions for this equation gives the researchers in this field a great clear vision of the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer and also dynamics of contrast agents in the blood flow at ultrasonic researches. To achieve our goal, we apply three different methods which are extended tanh-function method, extended simple equation method and a new auxiliary equation method on this equation. We obtained exact and solitary traveling wave solutions and we also discuss the similarity and difference between these three method and make a comparison between results that we obtained with another results that obtained with the different researchers using different methods. All of these results and discussion explained the fact that our new auxiliary equation method is considered to be the most general, powerful and the most result-oriented. These kinds of solutions and discussion allow for the understanding of the phenomenon and its intrinsic properties as well as the ease of way of application and its applicability to other phenomena.

Thermochemical nonequilibrium in atomic hydrogen at elevated temperatures

NASA Technical Reports Server (NTRS)

Scott, R. K.

1972-01-01

A numerical study of the nonequilibrium flow of atomic hydrogen in a cascade arc was performed to obtain insight into the physics of the hydrogen cascade arc. A rigorous mathematical model of the flow problem was formulated, incorporating the important nonequilibrium transport phenomena and atomic processes which occur in atomic hydrogen. Realistic boundary conditions, including consideration of the wall electrostatic sheath phenomenon, were included in the model. The governing equations of the asymptotic region of the cascade arc were obtained by writing conservation of mass and energy equations for the electron subgas, an energy conservation equation for heavy particles and an equation of state. Finite-difference operators for variable grid spacing were applied to the governing equations and the resulting system of strongly coupled, stiff equations were solved numerically by the Newton-Raphson method.

Guidelines for a graph-theoretic implementation of structural equation modeling

USGS Publications Warehouse

Grace, James B.; Schoolmaster, Donald R.; Guntenspergen, Glenn R.; Little, Amanda M.; Mitchell, Brian R.; Miller, Kathryn M.; Schweiger, E. William

2012-01-01

Structural equation modeling (SEM) is increasingly being chosen by researchers as a framework for gaining scientific insights from the quantitative analyses of data. New ideas and methods emerging from the study of causality, influences from the field of graphical modeling, and advances in statistics are expanding the rigor, capability, and even purpose of SEM. Guidelines for implementing the expanded capabilities of SEM are currently lacking. In this paper we describe new developments in SEM that we believe constitute a third-generation of the methodology. Most characteristic of this new approach is the generalization of the structural equation model as a causal graph. In this generalization, analyses are based on graph theoretic principles rather than analyses of matrices. Also, new devices such as metamodels and causal diagrams, as well as an increased emphasis on queries and probabilistic reasoning, are now included. Estimation under a graph theory framework permits the use of Bayesian or likelihood methods. The guidelines presented start from a declaration of the goals of the analysis. We then discuss how theory frames the modeling process, requirements for causal interpretation, model specification choices, selection of estimation method, model evaluation options, and use of queries, both to summarize retrospective results and for prospective analyses. The illustrative example presented involves monitoring data from wetlands on Mount Desert Island, home of Acadia National Park. Our presentation walks through the decision process involved in developing and evaluating models, as well as drawing inferences from the resulting prediction equations. In addition to evaluating hypotheses about the connections between human activities and biotic responses, we illustrate how the structural equation (SE) model can be queried to understand how interventions might take advantage of an environmental threshold to limit Typha invasions. The guidelines presented provide for an updated definition of the SEM process that subsumes the historical matrix approach under a graph-theory implementation. The implementation is also designed to permit complex specifications and to be compatible with various estimation methods. Finally, they are meant to foster the use of probabilistic reasoning in both retrospective and prospective considerations of the quantitative implications of the results.

Modeling of high pressure arc-discharge with a fully-implicit Navier-Stokes stabilized finite element flow solver

NASA Astrophysics Data System (ADS)

Sahai, A.; Mansour, N. N.; Lopez, B.; Panesi, M.

2017-05-01

This work addresses the modeling of high pressure electric discharge in an arc-heated wind tunnel. The combined numerical solution of Poisson’s equation, radiative transfer equations, and the set of Favre-averaged thermochemical nonequilibrium Navier-Stokes equations allows for the determination of the electric, radiation, and flow fields, accounting for their mutual interaction. Semi-classical statistical thermodynamics is used to determine the plasma thermodynamic properties, while transport properties are obtained from kinetic principles with the Chapman-Enskog method. A multi-temperature formulation is used to account for thermal non-equilibrium. Finally, the turbulence closure of the flow equations is obtained by means of the Spalart-Allmaras model, which requires the solution of an additional scalar transport equation. A Streamline upwind Petrov-Galerkin stabilized finite element formulation is employed to solve the Navier-Stokes equation. The electric field equation is solved using the standard Galerkin formulation. A stable formulation for the radiative transfer equations is obtained using the least-squares finite element method. The developed simulation framework has been applied to investigate turbulent plasma flows in the 20 MW Aerodynamic Heating Facility at NASA Ames Research Center. The current model is able to predict the process of energy addition and re-distribution due to Joule heating and thermal radiation, resulting in a hot central core surrounded by colder flow. The use of an unsteady three-dimensional treatment also allows the asymmetry due to a dynamic electric arc attachment point in the cathode chamber to be captured accurately. The current work paves the way for detailed estimation of operating characteristics for arc-heated wind tunnels which are critical in testing thermal protection systems.

Comparison of methods for the prediction of human clearance from hepatocyte intrinsic clearance for a set of reference compounds and an external evaluation set.

PubMed

Yamagata, Tetsuo; Zanelli, Ugo; Gallemann, Dieter; Perrin, Dominique; Dolgos, Hugues; Petersson, Carl

2017-09-01

1. We compared direct scaling, regression model equation and the so-called "Poulin et al." methods to scale clearance (CL) from in vitro intrinsic clearance (CL int ) measured in human hepatocytes using two sets of compounds. One reference set comprised of 20 compounds with known elimination pathways and one external evaluation set based on 17 compounds development in Merck (MS). 2. A 90% prospective confidence interval was calculated using the reference set. This interval was found relevant for the regression equation method. The three outliers identified were justified on the basis of their elimination mechanism. 3. The direct scaling method showed a systematic underestimation of clearance in both the reference and evaluation sets. The "Poulin et al." and the regression equation methods showed no obvious bias in either the reference or evaluation sets. 4. The regression model equation was slightly superior to the "Poulin et al." method in the reference set and showed a better absolute average fold error (AAFE) of value 1.3 compared to 1.6. A larger difference was observed in the evaluation set were the regression method and "Poulin et al." resulted in an AAFE of 1.7 and 2.6, respectively (removing the three compounds with known issues mentioned above). A similar pattern was observed for the correlation coefficient. Based on these data we suggest the regression equation method combined with a prospective confidence interval as the first choice for the extrapolation of human in vivo hepatic metabolic clearance from in vitro systems.

Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique

NASA Technical Reports Server (NTRS)

Nordmann, R.; Weiser, P.

1989-01-01

The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively.

A constitutive material model for nonlinear finite element structural analysis using an iterative matrix approach

NASA Technical Reports Server (NTRS)

Koenig, Herbert A.; Chan, Kwai S.; Cassenti, Brice N.; Weber, Richard

1988-01-01

A unified numerical method for the integration of stiff time dependent constitutive equations is presented. The solution process is directly applied to a constitutive model proposed by Bodner. The theory confronts time dependent inelastic behavior coupled with both isotropic hardening and directional hardening behaviors. Predicted stress-strain responses from this model are compared to experimental data from cyclic tests on uniaxial specimens. An algorithm is developed for the efficient integration of the Bodner flow equation. A comparison is made with the Euler integration method. An analysis of computational time is presented for the three algorithms.

Full nonlinear treatment of the global thermospheric wind system. Part 1: Mathematical method and analysis of forces

NASA Technical Reports Server (NTRS)

Blum, P. W.; Harris, I.

1973-01-01

The equations of horizontal motion of the neutral atmosphere between 120 and 500 km are integrated with the inclusion of all the nonlinear terms of the convective derivative and the viscous forces due to vertical and horizontal velocity gradients. Empirical models of the distribution of neutral and charged particles are assumed to be known. The model of velocities developed is a steady state model. In part 1 the mathematical method used in the integration of the Navier-Stokes equations is described and the various forces are analysed.

Semi-analytical solutions of the Schnakenberg model of a reaction-diffusion cell with feedback

NASA Astrophysics Data System (ADS)

Al Noufaey, K. S.

2018-06-01

This paper considers the application of a semi-analytical method to the Schnakenberg model of a reaction-diffusion cell. The semi-analytical method is based on the Galerkin method which approximates the original governing partial differential equations as a system of ordinary differential equations. Steady-state curves, bifurcation diagrams and the region of parameter space in which Hopf bifurcations occur are presented for semi-analytical solutions and the numerical solution. The effect of feedback control, via altering various concentrations in the boundary reservoirs in response to concentrations in the cell centre, is examined. It is shown that increasing the magnitude of feedback leads to destabilization of the system, whereas decreasing this parameter to negative values of large magnitude stabilizes the system. The semi-analytical solutions agree well with numerical solutions of the governing equations.

The Principle of Energetic Consistency: Application to the Shallow-Water Equations

NASA Technical Reports Server (NTRS)

Cohn, Stephen E.

2009-01-01

If the complete state of the earth's atmosphere (e.g., pressure, temperature, winds and humidity, everywhere throughout the atmosphere) were known at any particular initial time, then solving the equations that govern the dynamical behavior of the atmosphere would give the complete state at all subsequent times. Part of the difficulty of weather prediction is that the governing equations can only be solved approximately, which is what weather prediction models do. But weather forecasts would still be far from perfect even if the equations could be solved exactly, because the atmospheric state is not and cannot be known completely at any initial forecast time. Rather, the initial state for a weather forecast can only be estimated from incomplete observations taken near the initial time, through a process known as data assimilation. Weather prediction models carry out their computations on a grid of points covering the earth's atmosphere. The formulation of these models is guided by a mathematical convergence theory which guarantees that, given the exact initial state, the model solution approaches the exact solution of the governing equations as the computational grid is made more fine. For the data assimilation process, however, there does not yet exist a convergence theory. This book chapter represents an effort to begin establishing a convergence theory for data assimilation methods. The main result, which is called the principle of energetic consistency, provides a necessary condition that a convergent method must satisfy. Current methods violate this principle, as shown in earlier work of the author, and therefore are not convergent. The principle is illustrated by showing how to apply it as a simple test of convergence for proposed methods.

Smoothed Particle Hydrodynamics Continuous Boundary Force method for Navier-Stokes equations subject to Robin boundary condition

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pan, Wenxiao; Bao, Jie; Tartakovsky, Alexandre M.

2014-02-15

Robin boundary condition for the Navier-Stokes equations is used to model slip conditions at the fluid-solid boundaries. A novel Continuous Boundary Force (CBF) method is proposed for solving the Navier-Stokes equations subject to Robin boundary condition. In the CBF method, the Robin boundary condition at boundary is replaced by the homogeneous Neumann boundary condition at the boundary and a volumetric force term added to the momentum conservation equation. Smoothed Particle Hydrodynamics (SPH) method is used to solve the resulting Navier-Stokes equations. We present solutions for two-dimensional and three-dimensional flows in domains bounded by flat and curved boundaries subject to variousmore » forms of the Robin boundary condition. The numerical accuracy and convergence are examined through comparison of the SPH-CBF results with the solutions of finite difference or finite element method. Taken the no-slip boundary condition as a special case of slip boundary condition, we demonstrate that the SPH-CBF method describes accurately both no-slip and slip conditions.« less

Power Series Solution to the Pendulum Equation

ERIC Educational Resources Information Center

Benacka, Jan

2009-01-01

This note gives a power series solution to the pendulum equation that enables to investigate the system in an analytical way only, i.e. to avoid numeric methods. A method of determining the number of the terms for getting a required relative error is presented that uses bigger and lesser geometric series. The solution is suitable for modelling the…

Direct computation of stochastic flow in reservoirs with uncertain parameters

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dainton, M.P.; Nichols, N.K.; Goldwater, M.H.

1997-01-15

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point andmore » to the field convariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data. 14 refs., 14 figs., 3 tabs.« less

Modeling the Gross-Pitaevskii Equation Using the Quantum Lattice Gas Method

NASA Astrophysics Data System (ADS)

Oganesov, Armen

We present an improved Quantum Lattice Gas (QLG) algorithm as a mesoscopic unitary perturbative representation of the mean field Gross Pitaevskii (GP) equation for Bose-Einstein Condensates (BECs). The method employs an interleaved sequence of unitary collide and stream operators. QLG is applicable to many different scalar potentials in the weak interaction regime and has been used to model the Korteweg-de Vries (KdV), Burgers and GP equations. It can be implemented on both quantum and classical computers and is extremely scalable. We present results for 1D soliton solutions with positive and negative internal interactions, as well as vector solitons with inelastic scattering. In higher dimensions we look at the behavior of vortex ring reconnection. A further improvement is considered with a proper operator splitting technique via a Fourier transformation. This is great for quantum computers since the quantum FFT is exponentially faster than its classical counterpart which involves non-local data on the entire lattice (Quantum FFT is the backbone of the Shor algorithm for quantum factorization). We also present an imaginary time method in which we transform the Schrodinger equation into a diffusion equation for recovering ground state initial conditions of a quantum system suitable for the QLG algorithm.

Parallel But Not Equivalent: Challenges and Solutions for Repeated Assessment of Cognition over Time

PubMed Central

Gross, Alden L.; Inouye, Sharon K.; Rebok, George W.; Brandt, Jason; Crane, Paul K.; Parisi, Jeanine M.; Tommet, Doug; Bandeen-Roche, Karen; Carlson, Michelle C.; Jones, Richard N.

2013-01-01

Objective Analyses of individual differences in change may be unintentionally biased when versions of a neuropsychological test used at different follow-ups are not of equivalent difficulty. This study’s objective was to compare mean, linear, and equipercentile equating methods and demonstrate their utility in longitudinal research. Study Design and Setting The Advanced Cognitive Training for Independent and Vital Elderly (ACTIVE, N=1,401) study is a longitudinal randomized trial of cognitive training. The Alzheimer’s Disease Neuroimaging Initiative (ADNI, n=819) is an observational cohort study. Nonequivalent alternate versions of the Auditory Verbal Learning Test (AVLT) were administered in both studies. Results Using visual displays, raw and mean-equated AVLT scores in both studies showed obvious nonlinear trajectories in reference groups that should show minimal change, poor equivalence over time (ps≤0.001), and raw scores demonstrated poor fits in models of within-person change (RMSEAs>0.12). Linear and equipercentile equating produced more similar means in reference groups (ps≥0.09) and performed better in growth models (RMSEAs<0.05). Conclusion Equipercentile equating is the preferred equating method because it accommodates tests more difficult than a reference test at different percentiles of performance and performs well in models of within-person trajectory. The method has broad applications in both clinical and research settings to enhance the ability to use nonequivalent test forms. PMID:22540849

On one solution of Volterra integral equations of second kind

NASA Astrophysics Data System (ADS)

Myrhorod, V.; Hvozdeva, I.

2016-10-01

A solution of Volterra integral equations of the second kind with separable and difference kernels based on solutions of corresponding equations linking the kernel and resolvent is suggested. On the basis of a discrete functions class, the equations linking the kernel and resolvent are obtained and the methods of their analytical solutions are proposed. A mathematical model of the gas-turbine engine state modification processes in the form of Volterra integral equation of the second kind with separable kernel is offered.

Statistically qualified neuro-analytic failure detection method and system

DOEpatents

Vilim, Richard B.; Garcia, Humberto E.; Chen, Frederick W.

2002-03-02

An apparatus and method for monitoring a process involve development and application of a statistically qualified neuro-analytic (SQNA) model to accurately and reliably identify process change. The development of the SQNA model is accomplished in two stages: deterministic model adaption and stochastic model modification of the deterministic model adaptation. Deterministic model adaption involves formulating an analytic model of the process representing known process characteristics, augmenting the analytic model with a neural network that captures unknown process characteristics, and training the resulting neuro-analytic model by adjusting the neural network weights according to a unique scaled equation error minimization technique. Stochastic model modification involves qualifying any remaining uncertainty in the trained neuro-analytic model by formulating a likelihood function, given an error propagation equation, for computing the probability that the neuro-analytic model generates measured process output. Preferably, the developed SQNA model is validated using known sequential probability ratio tests and applied to the process as an on-line monitoring system. Illustrative of the method and apparatus, the method is applied to a peristaltic pump system.

Finite element analysis of flowfield in the single hole film cooling technique.

PubMed

Bazdidi-Tehrani, F; Mahmoodi, A A

2001-05-01

Film cooling is currently used in gas turbine hot sections, such as the combustor wall and the turbine blades, to prevent those sections from failing at elevated temperatures. In the single hole film cooling method, coolant air is injected from a hole into the mainstream and thus the flow is naturally three dimensional. In this paper, the Navier-Stokes and the energy equations are solved on a flat plate by the Finite Element Method (FEM) using brick elements. Algebraic equations are obtained by use of the Petrov-Galerkin method. The pressure term is removed from the momentum equations, by employing the Penalty method. The governing equations are transient and the flow is incompressible and turbulent. The model of turbulence in the near wall region is the wall function method, and in the fully turbulent region is the k-epsilon model. The system of the algebraic equations are solved by the Frontal method. The coolant injection angle and the blowing rate are among the parameters which are studied. In order to examine the present computer code, the results are compared with the Blasius (exact) solution and also with the empirical 1/7th power-law and good agreement is shown. Also, the optimum cooling performance is shown to be at 35 degree angle of coolant injection and the optimum blowing rate is 0.5. The film cooling effectiveness data, at the optimum conditions, is directly compared with the experimental results of Goldstein et al. and good agreement is demonstrated.

An efficient nonlinear finite-difference approach in the computational modeling of the dynamics of a nonlinear diffusion-reaction equation in microbial ecology.

PubMed

Macías-Díaz, J E; Macías, Siegfried; Medina-Ramírez, I E

2013-12-01

In this manuscript, we present a computational model to approximate the solutions of a partial differential equation which describes the growth dynamics of microbial films. The numerical technique reported in this work is an explicit, nonlinear finite-difference methodology which is computationally implemented using Newton's method. Our scheme is compared numerically against an implicit, linear finite-difference discretization of the same partial differential equation, whose computer coding requires an implementation of the stabilized bi-conjugate gradient method. Our numerical results evince that the nonlinear approach results in a more efficient approximation to the solutions of the biofilm model considered, and demands less computer memory. Moreover, the positivity of initial profiles is preserved in the practice by the nonlinear scheme proposed. Copyright © 2013 Elsevier Ltd. All rights reserved.

Model-free simulations of turbulent reactive flows

NASA Technical Reports Server (NTRS)

Givi, Peyman

1989-01-01

The current computational methods for solving transport equations of turbulent reacting single-phase flows are critically reviewed, with primary attention given to those methods that lead to model-free simulations. In particular, consideration is given to direct numerical simulations using spectral (Galerkin) and pseudospectral (collocation) methods, spectral element methods, and Lagrangian methods. The discussion also covers large eddy simulations and turbulence modeling.

Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches.

PubMed

Xu, Zhenli; Ma, Manman; Liu, Pei

2014-07-01

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the system composed of the PNP and DH equations. Particularly, toward the numerical challenge of solving the high-dimensional DH equation, we developed an analytical WKB approximation and a numerical approach based on the selective inversion of sparse matrices. The model and numerical methods are validated by simulating the charge diffusion in electrolytes between two electrodes, for which effects of dielectrics and correlation are investigated by comparing the results with the prediction by the classical PNP theory. We find that, at the length scale of the interface separation comparable to the Bjerrum length, the results of the modified equations are significantly different from the classical PNP predictions mostly due to the dielectric effect. It is also shown that when the ion self energy is in weak or mediate strength, the WKB approximation presents a high accuracy, compared to precise finite-difference results.

Modelling emission turbulence-radiation interaction by using a hybrid flamelet/stochastic Eulerian field method

NASA Astrophysics Data System (ADS)

Consalvi, Jean-Louis

2017-01-01

The time-averaged Radiative Transfer Equation (RTE) introduces two unclosed terms, known as `absorption Turbulence Radiation Interaction (TRI)' and `emission TRI'. Emission TRI is related to the non-linear coupling between fluctuations of the absorption coefficient and fluctuations of the Planck function and can be described without introduction any approximation by using a transported PDF method. In this study, a hybrid flamelet/ Stochastic Eulerian Field Model is used to solve the transport equation of the one-point one-time PDF. In this formulation, the steady laminar flamelet model (SLF) is coupled to a joint Probability Density Function (PDF) of mixture fraction, enthalpy defect, scalar dissipation rate, and soot quantities and the PDF transport equation is solved by using a Stochastic Eulerian Field (SEF) method. Soot production is modeled by a semi-empirical model and the spectral dependence of the radiatively participating species, namely combustion products and soot, are computed by using a Narrow Band Correlated-k (NBCK) model. The model is applied to simulate an ethylene/methane turbulent jet flame burning in an oxygen-enriched environment. Model results are compared with the experiments and the effects of taken into account Emission TRI on flame structure, soot production and radiative loss are discussed.

An efficient and guaranteed stable numerical method for continuous modeling of infiltration and redistribution with a shallow dynamic water table

NASA Astrophysics Data System (ADS)

Lai, Wencong; Ogden, Fred L.; Steinke, Robert C.; Talbot, Cary A.

2015-03-01

We have developed a one-dimensional numerical method to simulate infiltration and redistribution in the presence of a shallow dynamic water table. This method builds upon the Green-Ampt infiltration with Redistribution (GAR) model and incorporates features from the Talbot-Ogden (T-O) infiltration and redistribution method in a discretized moisture content domain. The redistribution scheme is more physically meaningful than the capillary weighted redistribution scheme in the T-O method. Groundwater dynamics are considered in this new method instead of hydrostatic groundwater front. It is also computationally more efficient than the T-O method. Motion of water in the vadose zone due to infiltration, redistribution, and interactions with capillary groundwater are described by ordinary differential equations. Numerical solutions to these equations are computationally less expensive than solutions of the highly nonlinear Richards' (1931) partial differential equation. We present results from numerical tests on 11 soil types using multiple rain pulses with different boundary conditions, with and without a shallow water table and compare against the numerical solution of Richards' equation (RE). Results from the new method are in satisfactory agreement with RE solutions in term of ponding time, deponding time, infiltration rate, and cumulative infiltrated depth. The new method, which we call "GARTO" can be used as an alternative to the RE for 1-D coupled surface and groundwater models in general situations with homogeneous soils with dynamic water table. The GARTO method represents a significant advance in simulating groundwater surface water interactions because it very closely matches the RE solution while being computationally efficient, with guaranteed mass conservation, and no stability limitations that can affect RE solvers in the case of a near-surface water table.

Additivity and maximum likelihood estimation of nonlinear component biomass models

Treesearch

David L.R. Affleck

2015-01-01

Since Parresol's (2001) seminal paper on the subject, it has become common practice to develop nonlinear tree biomass equations so as to ensure compatibility among total and component predictions and to fit equations jointly using multi-step least squares (MSLS) methods. In particular, many researchers have specified total tree biomass models by aggregating the...

Reporting Multiple-Group Mean and Covariance Structure across Occasions with Structural Equation Modeling

ERIC Educational Resources Information Center

Okech, David

2012-01-01

Objectives: Using baseline and second wave data, the study evaluated the measurement and structural properties of parenting stress, personal mastery, and economic strain with N = 381 lower income parents who decided to join and those who did not join in a child development savings account program. Methods: Structural equation modeling mean and…

Load influence on gear noise. [mathematical model for determining acoustic pressure level as function of load

NASA Technical Reports Server (NTRS)

Merticaru, V.

1974-01-01

An original mathematical model is proposed to derive equations for calculation of gear noise. These equations permit the acoustic pressure level to be determined as a function of load. Application of this method to three parallel gears is reported. The logical calculation scheme is given, as well as the results obtained.

Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gamba, Irene M.; Tharkabhushanam, Sri Harsha

We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation (BTE) in d-dimensions for variable hard sphere (VHS) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computation is reduced to a separate integral over the unit sphere S{sup d-1}. The conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibilitymore » (inelastic interactions) or elastic models of slowing down process. Our simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation, both for elastic and inelastic VHS interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in Bobylev et al. [A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, Journal of Statistical Physics 111 (2003) 403-417] and generalized to a wide range of related models in Bobylev et al. [A.V. Bobylev, C. Cercignani, I.M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Communication in Mathematical Physics, in press. URL: ()]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods [T.V. Noije, M. Ernst, Velocity distributions in homogeneously cooling and heated granular fluids, Granular Matter 1(57) (1998); M.H. Ernst, R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, Journal of Statistical Physics 109 (2002) 407-432; S.J. Moon, M.D. Shattuck, J. Swift, Velocity distributions and correlations in homogeneously heated granular media, Physical Review E 64 (2001) 031303; I.M. Gamba, S. Rjasanow, W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Mathematical and Computer Modelling 42 (2005) 683-700] and rigorously proven in Gamba et al. [I.M. Gamba, V. Panferov, C. Villani, On the Boltzmann equation for diffusively excited granular media, Communications in Mathematical Physics 246 (2004) 503-541(39)] and [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, Journal of Statistical Physics 116 (2004) 1651-1682].« less

Moving mesh finite element simulation for phase-field modeling of brittle fracture and convergence of Newton's iteration

NASA Astrophysics Data System (ADS)

Zhang, Fei; Huang, Weizhang; Li, Xianping; Zhang, Shicheng

2018-03-01

A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the decomposition of the strain tensor into tensile and compressive components is essential for the success of the phase-field modeling of brittle fracture but results in a non-smooth elastic energy and stronger nonlinearity in the governing equation. This makes the governing equation much more difficult to solve and, in particular, Newton's iteration often fails to converge. Three regularization methods are proposed to smooth out the decomposition of the strain tensor. Numerical examples of fracture propagation under quasi-static load demonstrate that all of the methods can effectively improve the convergence of Newton's iteration for relatively small values of the regularization parameter but without compromising the accuracy of the numerical solution. They also show that the moving mesh finite element method is able to adaptively concentrate the mesh elements around propagating cracks and handle multiple and complex crack systems.

Four-Dimensional Data Assimilation Using the Adjoint Method

NASA Astrophysics Data System (ADS)

Bao, Jian-Wen

The calculus of variations is used to confirm that variational four-dimensional data assimilation (FDDA) using the adjoint method can be implemented when the numerical model equations have a finite number of first-order discontinuous points. These points represent the on/off switches associated with physical processes, for which the Jacobian matrix of the model equation does not exist. Numerical evidence suggests that, in some situations when the adjoint method is used for FDDA, the temperature field retrieved using horizontal wind data is numerically not unique. A physical interpretation of this type of non-uniqueness of the retrieval is proposed in terms of energetics. The adjoint equations of a numerical model can also be used for model-parameter estimation. A general computational procedure is developed to determine the size and distribution of any internal model parameter. The procedure is then applied to a one-dimensional shallow -fluid model in the context of analysis-nudging FDDA: the weighting coefficients used by the Newtonian nudging technique are determined. The sensitivity of these nudging coefficients to the optimal objectives and constraints is investigated. Experiments of FDDA using the adjoint method are conducted using the dry version of the hydrostatic Penn State/NCAR mesoscale model (MM4) and its adjoint. The minimization procedure converges and the initialization experiment is successful. Temperature-retrieval experiments involving an assimilation of the horizontal wind are also carried out using the adjoint of MM4.

Model predictive control of attitude maneuver of a geostationary flexible satellite based on genetic algorithm

NASA Astrophysics Data System (ADS)

TayyebTaher, M.; Esmaeilzadeh, S. Majid

2017-07-01

This article presents an application of Model Predictive Controller (MPC) to the attitude control of a geostationary flexible satellite. SIMO model has been used for the geostationary satellite, using the Lagrange equations. Flexibility is also included in the modelling equations. The state space equations are expressed in order to simplify the controller. Naturally there is no specific tuning rule to find the best parameters of an MPC controller which fits the desired controller. Being an intelligence method for optimizing problem, Genetic Algorithm has been used for optimizing the performance of MPC controller by tuning the controller parameter due to minimum rise time, settling time, overshoot of the target point of the flexible structure and its mode shape amplitudes to make large attitude maneuvers possible. The model included geosynchronous orbit environment and geostationary satellite parameters. The simulation results of the flexible satellite with attitude maneuver shows the efficiency of proposed optimization method in comparison with LQR optimal controller.

Linear network representation of multistate models of transport.

PubMed Central

Sandblom, J; Ring, A; Eisenman, G

1982-01-01

By introducing external driving forces in rate-theory models of transport we show how the Eyring rate equations can be transformed into Ohm's law with potentials that obey Kirchhoff's second law. From such a formalism the state diagram of a multioccupancy multicomponent system can be directly converted into linear network with resistors connecting nodal (branch) points and with capacitances connecting each nodal point with a reference point. The external forces appear as emf or current generators in the network. This theory allows the algebraic methods of linear network theory to be used in solving the flux equations for multistate models and is particularly useful for making proper simplifying approximation in models of complex membrane structure. Some general properties of linear network representation are also deduced. It is shown, for instance, that Maxwell's reciprocity relationships of linear networks lead directly to Onsager's relationships in the near equilibrium region. Finally, as an example of the procedure, the equivalent circuit method is used to solve the equations for a few transport models. PMID:7093425

Adaptive moving mesh methods for simulating one-dimensional groundwater problems with sharp moving fronts

USGS Publications Warehouse

Huang, W.; Zheng, Lingyun; Zhan, X.

2002-01-01

Accurate modelling of groundwater flow and transport with sharp moving fronts often involves high computational cost, when a fixed/uniform mesh is used. In this paper, we investigate the modelling of groundwater problems using a particular adaptive mesh method called the moving mesh partial differential equation approach. With this approach, the mesh is dynamically relocated through a partial differential equation to capture the evolving sharp fronts with a relatively small number of grid points. The mesh movement and physical system modelling are realized by solving the mesh movement and physical partial differential equations alternately. The method is applied to the modelling of a range of groundwater problems, including advection dominated chemical transport and reaction, non-linear infiltration in soil, and the coupling of density dependent flow and transport. Numerical results demonstrate that sharp moving fronts can be accurately and efficiently captured by the moving mesh approach. Also addressed are important implementation strategies, e.g. the construction of the monitor function based on the interpolation error, control of mesh concentration, and two-layer mesh movement. Copyright ?? 2002 John Wiley and Sons, Ltd.

Numerical modelling of ultrasonic waves in a bubbly Newtonian liquid using a high-order acoustic cavitation model.

PubMed

Lebon, G S Bruno; Tzanakis, I; Djambazov, G; Pericleous, K; Eskin, D G

2017-07-01

To address difficulties in treating large volumes of liquid metal with ultrasound, a fundamental study of acoustic cavitation in liquid aluminium, expressed in an experimentally validated numerical model, is presented in this paper. To improve the understanding of the cavitation process, a non-linear acoustic model is validated against reference water pressure measurements from acoustic waves produced by an immersed horn. A high-order method is used to discretize the wave equation in both space and time. These discretized equations are coupled to the Rayleigh-Plesset equation using two different time scales to couple the bubble and flow scales, resulting in a stable, fast, and reasonably accurate method for the prediction of acoustic pressures in cavitating liquids. This method is then applied to the context of treatment of liquid aluminium, where it predicts that the most intense cavitation activity is localised below the vibrating horn and estimates the acoustic decay below the sonotrode with reasonable qualitative agreement with experimental data. Copyright © 2017 The Author(s). Published by Elsevier B.V. All rights reserved.

Radiative transfer equation accounting for rotational Raman scattering and its solution by the discrete-ordinates method

NASA Astrophysics Data System (ADS)

Rozanov, Vladimir V.; Vountas, Marco

2014-01-01

Rotational Raman scattering of solar light in Earth's atmosphere leads to the filling-in of Fraunhofer and telluric lines observed in the reflected spectrum. The phenomenological derivation of the inelastic radiative transfer equation including rotational Raman scattering is presented. The different forms of the approximate radiative transfer equation with first-order rotational Raman scattering terms are obtained employing the Cabannes, Rayleigh, and Cabannes-Rayleigh scattering models. The solution of these equations is considered in the framework of the discrete-ordinates method using rigorous and approximate approaches to derive particular integrals. An alternative forward-adjoint technique is suggested as well. A detailed description of the model including the exact spectral matching and a binning scheme that significantly speeds up the calculations is given. The considered solution techniques are implemented in the radiative transfer software package SCIATRAN and a specified benchmark setup is presented to enable readers to compare with own results transparently.

A probabilistic approach to radiative energy loss calculations for optically thick atmospheres - Hydrogen lines and continua

NASA Technical Reports Server (NTRS)

Canfield, R. C.; Ricchiazzi, P. J.

1980-01-01

An approximate probabilistic radiative transfer equation and the statistical equilibrium equations are simultaneously solved for a model hydrogen atom consisting of three bound levels and ionization continuum. The transfer equation for L-alpha, L-beta, H-alpha, and the Lyman continuum is explicitly solved assuming complete redistribution. The accuracy of this approach is tested by comparing source functions and radiative loss rates to values obtained with a method that solves the exact transfer equation. Two recent model solar-flare chromospheres are used for this test. It is shown that for the test atmospheres the probabilistic method gives values of the radiative loss rate that are characteristically good to a factor of 2. The advantage of this probabilistic approach is that it retains a description of the dominant physical processes of radiative transfer in the complete redistribution case, yet it achieves a major reduction in computational requirements.

Solution of the equations for one-dimensional, two-phase, immiscible flow by geometric methods

NASA Astrophysics Data System (ADS)

Boronin, Ivan; Shevlyakov, Andrey

2018-03-01

Buckley-Leverett equations describe non viscous, immiscible, two-phase filtration, which is often of interest in modelling of oil production. For many parameters and initial conditions, the solutions of these equations exhibit non-smooth behaviour, namely discontinuities in form of shock waves. In this paper we obtain a novel method for the solution of Buckley-Leverett equations, which is based on geometry of differential equations. This method is fast, accurate, stable, and describes non-smooth phenomena. The main idea of the method is that classic discontinuous solutions correspond to the continuous surfaces in the space of jets - the so-called multi-valued solutions (Bocharov et al., Symmetries and conservation laws for differential equations of mathematical physics. American Mathematical Society, Providence, 1998). A mapping of multi-valued solutions from the jet space onto the plane of the independent variables is constructed. This mapping is not one-to-one, and its singular points form a curve on the plane of the independent variables, which is called the caustic. The real shock occurs at the points close to the caustic and is determined by the Rankine-Hugoniot conditions.

Application of satellite data in variational analysis for global cyclonic systems

NASA Technical Reports Server (NTRS)

Achtemeier, G. L.

1988-01-01

The goal of the research is a variational data assimilation method that incorporates as dynamical constraints, the primitive equations for a moist, convectively unstable atmosphere and the radiative transfer equation. Variables to be adjusted include the three-dimensional vector wind, height, temperature, and moisture from rawinsonde data, and cloud-wind vectors, moisture, and radiance from satellite data. In order to facilitate thorough analysis of each of the model components, four variational models that divide the problem naturally according to increasing complexity were defined. The research performed during the second year fall into four areas: sensitivity studies involving Model 1; evaluation of Model 2; reformation of Model 1 for greater compatibility with Model 2; development of Model 3 (radiative transfer equation); and making the model more responsive to the observations.

Free-Surface Flow and Fluid-Object Interaction Modeling With Emphasis on Ship Hydrodynamics

DTIC Science & Technology

2012-01-01

0 on Cawt (21) in a weak sense. Equation (20) is the Eikonal partial differential equation subject to the interior constraint given by Eq. (21). To...tion, respectively. The formulation given by Eq. (22) is the SUPG method [30] applied to the Eikonal equation. At the steady state, the above problem

Matrix Pseudospectral Method for (Visco)Elastic Tides Modeling of Planetary Bodies

NASA Astrophysics Data System (ADS)

Zabranova, Eliska; Hanyk, Ladidslav; Matyska, Ctirad

2010-05-01

We deal with the equations and boundary conditions describing deformation and gravitational potential of prestressed spherically symmetric elastic bodies by decomposing governing equations into a series of boundary value problems (BVP) for ordinary differential equations (ODE) of the second order. In contrast to traditional Runge-Kutta integration techniques, highly accurate pseudospectral schemes are employed to directly discretize the BVP on Chebyshev grids and a set of linear algebraic equations with an almost block diagonal matrix is derived. As a consequence of keeping the governing ODEs of the second order instead of the usual first-order equations, the resulting algebraic system is half-sized but derivatives of the model parameters are required. Moreover, they can be easily evaluated for models, where structural parametres are piecewise polynomially dependent. Both accuracy and efficiency of the method are tested by evaluating the tidal Love numbers for the Earth's model PREM. Finally, we also derive complex Love numbers for models with the Maxwell viscoelastic rheology, where viscosity is a depth-dependent function. The method is applied to evaluation of the tidal Love numbers for models of Mars and Venus. The Love numbers of the two Martian models - the former optimized to cosmochemical data and the latter to the moment of inertia (Sohl and Spohn, 1997) - are h2=0.172 (0.212) and k2=0.093 (0.113). For Venus, the value of k2=0.295 (Konopliv and Yoder, 1996), obtained from the gravity-field analysis, is consistent with the results for our model with the liquid-core radius of 3110 km (Zábranová et al., 2009). Together with rapid evaluation of free oscillation periods by an analogous method, this combined matrix approach could by employed as an efficient numerical tool in structural studies of planetary bodies. REFERENCES Konopliv, A. S. and Yoder, C. F., 1996. Venusian k2 tidal Love number from Magellan and PVO tracking data, Geophys. Res. Lett., 23, 1857-1860. Sohl, F., and Spohn, T., 1997. The interior structure of Mars: Implications from SNC meteorites, J. Geophys. Res., 102, 1613-1635. Zabranova, E., Hanyk L. and Matyska, C.: Matrix Pseudospectral Method for Elastic Tides Modeling. In: Holota P. (Ed.): Mission and Passion: Science. A volume dedicated to Milan Bursa on the occasion of his 80th birthday. Published by the Czech National Committee of Geodesy and Geophysics. Prague, 2009, pp. 243-260.

A zonally symmetric model for the monsoon-Hadley circulation with stochastic convective forcing

NASA Astrophysics Data System (ADS)

De La Chevrotière, Michèle; Khouider, Boualem

2017-02-01

Idealized models of reduced complexity are important tools to understand key processes underlying a complex system. In climate science in particular, they are important for helping the community improve our ability to predict the effect of climate change on the earth system. Climate models are large computer codes based on the discretization of the fluid dynamics equations on grids of horizontal resolution in the order of 100 km, whereas unresolved processes are handled by subgrid models. For instance, simple models are routinely used to help understand the interactions between small-scale processes due to atmospheric moist convection and large-scale circulation patterns. Here, a zonally symmetric model for the monsoon circulation is presented and solved numerically. The model is based on the Galerkin projection of the primitive equations of atmospheric synoptic dynamics onto the first modes of vertical structure to represent free tropospheric circulation and is coupled to a bulk atmospheric boundary layer (ABL) model. The model carries bulk equations for water vapor in both the free troposphere and the ABL, while the processes of convection and precipitation are represented through a stochastic model for clouds. The model equations are coupled through advective nonlinearities, and the resulting system is not conservative and not necessarily hyperbolic. This makes the design of a numerical method for the solution of this system particularly difficult. Here, we develop a numerical scheme based on the operator time-splitting strategy, which decomposes the system into three pieces: a conservative part and two purely advective parts, each of which is solved iteratively using an appropriate method. The conservative system is solved via a central scheme, which does not require hyperbolicity since it avoids the Riemann problem by design. One of the advective parts is a hyperbolic diagonal matrix, which is easily handled by classical methods for hyperbolic equations, while the other advective part is a nilpotent matrix, which is solved via the method of lines. Validation tests using a synthetic exact solution are presented, and formal second-order convergence under grid refinement is demonstrated. Moreover, the model is tested under realistic monsoon conditions, and the ability of the model to simulate key features of the monsoon circulation is illustrated in two distinct parameter regimes.

Dynamics in a Maximally Symmetric Universe

NASA Astrophysics Data System (ADS)

Bewketu, Asnakew

2016-03-01

Our present understanding of the evolution of the universe relies upon the Friedmann- Robertson- Walker cosmological models. This model is so successful that it is now being considered as the Standard Model of Cosmology. So in this work we derive the Fried- mann equations using the Friedmann-Robertson-Walker metric together with Einstein field equation and then we give a simple method to reduce Friedmann equations to a second order linear differential equation when it is supplemented with a time dependent equation of state. Furthermore, as illustrative examples, we solve this equation for some specific time dependent equation of states. And also by using the Friedmann equations with some time dependent equation of state we try to determine the cosmic scale factor(the rate at which the universe expands) and age of the Friedmann universe, for the matter dominated era, radiation dominated era and for both matter and radiation dominated era by considering different cases. We have finally discussed the observable quantities that can be evidences for the accelerated expansion of the Friedmann universe. I would like to acknowledge Addis Ababa University for its financial and material support to my work on the title mentioned above.

GVE-Based Dynamics and Control for Formation Flying Spacecraft

NASA Technical Reports Server (NTRS)

Breger, Louis; How, Jonathan P.

2004-01-01

Formation flying is an enabling technology for many future space missions. This paper presents extensions to the equations of relative motion expressed in Keplerian orbital elements, including new initialization techniques for general formation configurations. A new linear time-varying form of the equations of relative motion is developed from Gauss Variational Equations and used in a model predictive controller. The linearizing assumptions for these equations are shown to be consistent with typical formation flying scenarios. Several linear, convex initialization techniques are presented, as well as a general, decentralized method for coordinating a tetrahedral formation using differential orbital elements. Control methods are validated using a commercial numerical propagator.

Efficient numerical method for solving Cauchy problem for the Gamma equation

NASA Astrophysics Data System (ADS)

Koleva, Miglena N.

2011-12-01

In this work we consider Cauchy problem for the so called Gamma equation, derived by transforming the fully nonlinear Black-Scholes equation for option price into a quasilinear parabolic equation for the second derivative (Greek) Γ = VSS of the option price V. We develop an efficient numerical method for solving the model problem concerning different volatility terms. Using suitable change of variables the problem is transformed on finite interval, keeping original behavior of the solution at the infinity. Then we construct Picard-Newton algorithm with adaptive mesh step in time, which can be applied also in the case of non-differentiable functions. Results of numerical simulations are given.

Population density equations for stochastic processes with memory kernels

NASA Astrophysics Data System (ADS)

Lai, Yi Ming; de Kamps, Marc

2017-06-01

We present a method for solving population density equations (PDEs)-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation—a recent result from random network theory—describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.

Identifying effective connectivity parameters in simulated fMRI: a direct comparison of switching linear dynamic system, stochastic dynamic causal, and multivariate autoregressive models

PubMed Central

Smith, Jason F.; Chen, Kewei; Pillai, Ajay S.; Horwitz, Barry

2013-01-01

The number and variety of connectivity estimation methods is likely to continue to grow over the coming decade. Comparisons between methods are necessary to prune this growth to only the most accurate and robust methods. However, the nature of connectivity is elusive with different methods potentially attempting to identify different aspects of connectivity. Commonalities of connectivity definitions across methods upon which base direct comparisons can be difficult to derive. Here, we explicitly define “effective connectivity” using a common set of observation and state equations that are appropriate for three connectivity methods: dynamic causal modeling (DCM), multivariate autoregressive modeling (MAR), and switching linear dynamic systems for fMRI (sLDSf). In addition while deriving this set, we show how many other popular functional and effective connectivity methods are actually simplifications of these equations. We discuss implications of these connections for the practice of using one method to simulate data for another method. After mathematically connecting the three effective connectivity methods, simulated fMRI data with varying numbers of regions and task conditions is generated from the common equation. This simulated data explicitly contains the type of the connectivity that the three models were intended to identify. Each method is applied to the simulated data sets and the accuracy of parameter identification is analyzed. All methods perform above chance levels at identifying correct connectivity parameters. The sLDSf method was superior in parameter estimation accuracy to both DCM and MAR for all types of comparisons. PMID:23717258

Development of a Pressure-Dependent Constitutive Model with Combined Multilinear Kinematic and Isotropic Hardening

NASA Technical Reports Server (NTRS)

Allen Phillip A.; Wilson, Christopher D.

2003-01-01

The development of a pressure-dependent constitutive model with combined multilinear kinematic and isotropic hardening is presented. The constitutive model is developed using the ABAQUS user material subroutine (UMAT). First the pressure-dependent plasticity model is derived. Following this, the combined bilinear and combined multilinear hardening equations are developed for von Mises plasticity theory. The hardening rule equations are then modified to include pressure dependency. The method for implementing the new constitutive model into ABAQUS is given.

Field verification of reconstructed dam-break flood, Laurel Run, Pennsylvania

USGS Publications Warehouse

Chen, Cheng-lung; Armbruster, Jeffrey T.

1979-01-01

A one-dimensional dam-break flood routing model is verified by using observed data on the flash flood resulting from the failure of Laurel Run Reservoir Dam near Johnstown, Pennsylvania. The model has been developed on the basis of an explicit scheme of the characteristics method with specified time intervals. The model combines one of the characteristic equations with the Rankine-Hugoniot shock equations to trace the corresponding characteristic backward to the known state for solving the depth and velocity of flow at the wave front. The previous version of the model has called for a modification of the method of solution to overcome the computational difficulty at the narrow breach and at any geomorphological constraints where channel geometry changes rapidly. The large reduction in the computational inaccuracies and oscillations was achieved by introducing the actual "storage width" in the equation of continuity and the imaginary "conveyance width" in the equation of motion. Close agreement between observed and computed peak stages at several stations downstream of the dam strongly suggests the validity and applicability of the model. However, small numerical noise appearing in the computed stage and discharge hydrographs at the dam site as well as discrepancy of attenuated peaks in the discharge hydrographs indicate the need for further model improvement.

Two-dimensional coupled mathematical modeling of fluvial processes with intense sediment transport and rapid bed evolution

NASA Astrophysics Data System (ADS)

Yue, Zhiyuan; Cao, Zhixian; Li, Xin; Che, Tao

2008-09-01

Alluvial rivers may experience intense sediment transport and rapid bed evolution under a high flow regime, for which traditional decoupled mathematical river models based on simplified conservation equations are not applicable. A two-dimensional coupled mathematical model is presented, which is generally applicable to the fluvial processes with either intense or weak sediment transport. The governing equations of the model comprise the complete shallow water hydrodynamic equations closed with Manning roughness for boundary resistance and empirical relationships for sediment exchange with the erodible bed. The second-order Total-Variation-Diminishing version of the Weighted-Average-Flux method, along with the HLLC approximate Riemann Solver, is adapted to solve the governing equations, which can properly resolve shock waves and contact discontinuities. The model is applied to the pilot study of the flooding due to a sudden outburst of a real glacial-lake.

A mathematical model for simulating noise suppression of lined ejectors

NASA Technical Reports Server (NTRS)

Watson, Willie R.

1994-01-01

A mathematical model containing the essential features embodied in the noise suppression of lined ejectors is presented. Although some simplification of the physics is necessary to render the model mathematically tractable, the current model is the most versatile and technologically advanced at the current time. A system of linearized equations and the boundary conditions governing the sound field are derived starting from the equations of fluid dynamics. A nonreflecting boundary condition is developed. In view of the complex nature of the equations, a parametric study requires the use of numerical techniques and modern computers. A finite element algorithm that solves the differential equations coupled with the boundary condition is then introduced. The numerical method results in a matrix equation with several hundred thousand degrees of freedom that is solved efficiently on a supercomputer. The model is validated by comparing results either with exact solutions or with approximate solutions from other works. In each case, excellent correlations are obtained. The usefulness of the model as an optimization tool and the importance of variable impedance liners as a mechanism for achieving broadband suppression within a lined ejector are demonstrated.

Optimal sensor placement for control of a supersonic mixed-compression inlet with variable geometry

NASA Astrophysics Data System (ADS)

Moore, Kenneth Thomas

A method of using fluid dynamics models for the generation of models that are useable for control design and analysis is investigated. The problem considered is the control of the normal shock location in the VDC inlet, which is a mixed-compression, supersonic, variable-geometry inlet of a jet engine. A quasi-one-dimensional set of fluid equations incorporating bleed and moving walls is developed. An object-oriented environment is developed for simulation of flow systems under closed-loop control. A public interface between the controller and fluid classes is defined. A linear model representing the dynamics of the VDC inlet is developed from the finite difference equations, and its eigenstructure is analyzed. The order of this model is reduced using the square root balanced model reduction method to produce a reduced-order linear model that is suitable for control design and analysis tasks. A modification to this method that improves the accuracy of the reduced-order linear model for the purpose of sensor placement is presented and analyzed. The reduced-order linear model is used to develop a sensor placement method that quantifies as a function of the sensor location the ability of a sensor to provide information on the variable of interest for control. This method is used to develop a sensor placement metric for the VDC inlet. The reduced-order linear model is also used to design a closed loop control system to control the shock position in the VDC inlet. The object-oriented simulation code is used to simulate the nonlinear fluid equations under closed-loop control.

Fitted Fourier-pseudospectral methods for solving a delayed reaction-diffusion partial differential equation in biology

NASA Astrophysics Data System (ADS)

Adam, A. M. A.; Bashier, E. B. M.; Hashim, M. H. A.; Patidar, K. C.

2017-07-01

In this work, we design and analyze a fitted numerical method to solve a reaction-diffusion model with time delay, namely, a delayed version of a population model which is an extension of the logistic growth (LG) equation for a food-limited population proposed by Smith [F.E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology 44 (1963) 651-663]. Seeing that the analytical solution (in closed form) is hard to obtain, we seek for a robust numerical method. The method consists of a Fourier-pseudospectral semi-discretization in space and a fitted operator implicit-explicit scheme in temporal direction. The proposed method is analyzed for convergence and we found that it is unconditionally stable. Illustrative numerical results will be presented at the conference.

Simulation electromagnetic scattering on bodies through integral equation and neural networks methods

NASA Astrophysics Data System (ADS)

Lvovich, I. Ya; Preobrazhenskiy, A. P.; Choporov, O. N.

2018-05-01

The paper deals with the issue of electromagnetic scattering on a perfectly conducting diffractive body of a complex shape. Performance calculation of the body scattering is carried out through the integral equation method. Fredholm equation of the second time was used for calculating electric current density. While solving the integral equation through the moments method, the authors have properly described the core singularity. The authors determined piecewise constant functions as basic functions. The chosen equation was solved through the moments method. Within the Kirchhoff integral approach it is possible to define the scattered electromagnetic field, in some way related to obtained electrical currents. The observation angles sector belongs to the area of the front hemisphere of the diffractive body. To improve characteristics of the diffractive body, the authors used a neural network. All the neurons contained a logsigmoid activation function and weighted sums as discriminant functions. The paper presents the matrix of weighting factors of the connectionist model, as well as the results of the optimized dimensions of the diffractive body. The paper also presents some basic steps in calculation technique of the diffractive bodies, based on the combination of integral equation and neural networks methods.

Quantitative Estimation of Seismic Velocity Changes Using Time-Lapse Seismic Data and Elastic-Wave Sensitivity Approach

NASA Astrophysics Data System (ADS)

Denli, H.; Huang, L.

2008-12-01

Quantitative monitoring of reservoir property changes is essential for safe geologic carbon sequestration. Time-lapse seismic surveys have the potential to effectively monitor fluid migration in the reservoir that causes geophysical property changes such as density, and P- and S-wave velocities. We introduce a novel method for quantitative estimation of seismic velocity changes using time-lapse seismic data. The method employs elastic sensitivity wavefields, which are the derivatives of elastic wavefield with respect to density, P- and S-wave velocities of a target region. We derive the elastic sensitivity equations from analytical differentiations of the elastic-wave equations with respect to seismic-wave velocities. The sensitivity equations are coupled with the wave equations in a way that elastic waves arriving in a target reservoir behave as a secondary source to sensitivity fields. We use a staggered-grid finite-difference scheme with perfectly-matched layers absorbing boundary conditions to simultaneously solve the elastic-wave equations and the elastic sensitivity equations. By elastic-wave sensitivities, a linear relationship between relative seismic velocity changes in the reservoir and time-lapse seismic data at receiver locations can be derived, which leads to an over-determined system of equations. We solve this system of equations using a least- square method for each receiver to obtain P- and S-wave velocity changes. We validate the method using both surface and VSP synthetic time-lapse seismic data for a multi-layered model and the elastic Marmousi model. Then we apply it to the time-lapse field VSP data acquired at the Aneth oil field in Utah. A total of 10.5K tons of CO2 was injected into the oil reservoir between the two VSP surveys for enhanced oil recovery. The synthetic and field data studies show that our new method can quantitatively estimate changes in seismic velocities within a reservoir due to CO2 injection/migration.

High-frequency sound waves to eliminate a horizon in the mixmaster universe.

NASA Technical Reports Server (NTRS)

Chitre, D. M.

1972-01-01

From the linear wave equation for small-amplitude sound waves in a curved space-time, there is derived a geodesiclike differential equation for sound rays to describe the motion of wave packets. These equations are applied in the generic, nonrotating, homogeneous closed-model universe (the 'mixmaster universe,' Bianchi type IX). As for light rays described by Doroshkevich and Novikov (DN), these sound rays can circumnavigate the universe near the singularity to remove particle horizons only for a small class of these models and in special directions. Although these results parallel those of DN, different Hamiltonian methods are used for treating the Einstein equations.