Error analysis of finite element method for Poisson–Nernst–Planck equations

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

Sun, Yuzhou; Sun, Pengtao; Zheng, Bin

A priori error estimates of finite element method for time-dependent Poisson-Nernst-Planck equations are studied in this work. We obtain the optimal error estimates in L∞(H1) and L2(H1) norms, and suboptimal error estimates in L∞(L2) norm, with linear element, and optimal error estimates in L∞(L2) norm with quadratic or higher-order element, for both semi- and fully discrete finite element approximations. Numerical experiments are also given to validate the theoretical results.

A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

NASA Astrophysics Data System (ADS)

Lin, Zeng; Wang, Dongdong

2017-10-01

Due to the nonlocal property of the fractional derivative, the finite element analysis of fractional diffusion equation often leads to a dense and non-symmetric stiffness matrix, in contrast to the conventional finite element formulation with a particularly desirable symmetric and banded stiffness matrix structure for the typical diffusion equation. This work first proposes a finite element formulation that preserves the symmetry and banded stiffness matrix characteristics for the fractional diffusion equation. The key point of the proposed formulation is the symmetric weak form construction through introducing a fractional weight function. It turns out that the stiffness part of the present formulation is identical to its counterpart of the finite element method for the conventional diffusion equation and thus the stiffness matrix formulation becomes trivial. Meanwhile, the fractional derivative effect in the discrete formulation is completely transferred to the force vector, which is obviously much easier and efficient to compute than the dense fractional derivative stiffness matrix. Subsequently, it is further shown that for the general fractional advection-diffusion-reaction equation, the symmetric and banded structure can also be maintained for the diffusion stiffness matrix, although the total stiffness matrix is not symmetric in this case. More importantly, it is demonstrated that under certain conditions this symmetric diffusion stiffness matrix formulation is capable of producing very favorable numerical solutions in comparison with the conventional non-symmetric diffusion stiffness matrix finite element formulation. The effectiveness of the proposed methodology is illustrated through a series of numerical examples.

A modular finite-element model (MODFE) for areal and axisymmetric ground-water-flow problems, Part 2: Derivation of finite-element equations and comparisons with analytical solutions

USGS Publications Warehouse

Cooley, Richard L.

1992-01-01

MODFE, a modular finite-element model for simulating steady- or unsteady-state, area1 or axisymmetric flow of ground water in a heterogeneous anisotropic aquifer is documented in a three-part series of reports. In this report, part 2, the finite-element equations are derived by minimizing a functional of the difference between the true and approximate hydraulic head, which produces equations that are equivalent to those obtained by either classical variational or Galerkin techniques. Spatial finite elements are triangular with linear basis functions, and temporal finite elements are one dimensional with linear basis functions. Physical processes that can be represented by the model include (1) confined flow, unconfined flow (using the Dupuit approximation), or a combination of both; (2) leakage through either rigid or elastic confining units; (3) specified recharge or discharge at points, along lines, or areally; (4) flow across specified-flow, specified-head, or head-dependent boundaries; (5) decrease of aquifer thickness to zero under extreme water-table decline and increase of aquifer thickness from zero as the water table rises; and (6) head-dependent fluxes from springs, drainage wells, leakage across riverbeds or confining units combined with aquifer dewatering, and evapotranspiration. The matrix equations produced by the finite-element method are solved by the direct symmetric-Doolittle method or the iterative modified incomplete-Cholesky conjugate-gradient method. The direct method can be efficient for small- to medium-sized problems (less than about 500 nodes), and the iterative method is generally more efficient for larger-sized problems. Comparison of finite-element solutions with analytical solutions for five example problems demonstrates that the finite-element model can yield accurate solutions to ground-water flow problems.

Flow Applications of the Least Squares Finite Element Method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan

1998-01-01

The main thrust of the effort has been towards the development, analysis and implementation of the least-squares finite element method (LSFEM) for fluid dynamics and electromagnetics applications. In the past year, there were four major accomplishments: 1) special treatments in computational fluid dynamics and computational electromagnetics, such as upwinding, numerical dissipation, staggered grid, non-equal order elements, operator splitting and preconditioning, edge elements, and vector potential are unnecessary; 2) the analysis of the LSFEM for most partial differential equations can be based on the bounded inverse theorem; 3) the finite difference and finite volume algorithms solve only two Maxwell equations and ignore the divergence equations; and 4) the first numerical simulation of three-dimensional Marangoni-Benard convection was performed using the LSFEM.

Analysis of Large Quasistatic Deformations of Inelastic Solids by a New Stress Based Finite Element Method. Ph.D. Thesis Final Report

NASA Technical Reports Server (NTRS)

Reed, Kenneth W.

1992-01-01

A new hybrid stress finite element algorithm suitable for analyses of large quasistatic deformation of inelastic solids is presented. Principal variables in the formulation are the nominal stress rate and spin. The finite element equations which result are discrete versions of the equations of compatibility and angular momentum balance. Consistent reformulation of the constitutive equation and accurate and stable time integration of the stress are discussed at length. Examples which bring out the feasibility and performance of the algorithm conclude the work.

Efficiency trade-offs of steady-state methods using FEM and FDM. [iterative solutions for nonlinear flow equations

NASA Technical Reports Server (NTRS)

Gartling, D. K.; Roache, P. J.

1978-01-01

The efficiency characteristics of finite element and finite difference approximations for the steady-state solution of the Navier-Stokes equations are examined. The finite element method discussed is a standard Galerkin formulation of the incompressible, steady-state Navier-Stokes equations. The finite difference formulation uses simple centered differences that are O(delta x-squared). Operation counts indicate that a rapidly converging Newton-Raphson-Kantorovitch iteration scheme is generally preferable over a Picard method. A split NOS Picard iterative algorithm for the finite difference method was most efficient.

Least-squares finite element methods for compressible Euler equations

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Carey, G. F.

1990-01-01

A method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L-sq-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L-sq method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H1-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

Finite elements and fluid dynamics. [instability effects on solution of nonlinear equations

NASA Technical Reports Server (NTRS)

Fix, G.

1975-01-01

Difficulties concerning a use of the finite element method in the solution of the nonlinear equations of fluid dynamics are partly related to various 'hidden' instabilities which often arise in fluid calculations. The instabilities are typically due to boundary effects or nonlinearities. It is shown that in certain cases these instabilities can be avoided if certain conservation laws are satisfied, and that the latter are often intimately related to finite elements.

Time-independent hybrid enrichment for finite element solution of transient conduction–radiation in diffusive grey media

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

Mohamed, M. Shadi, E-mail: m.s.mohamed@durham.ac.uk; Seaid, Mohammed; Trevelyan, Jon

2013-10-15

We investigate the effectiveness of the partition-of-unity finite element method for transient conduction–radiation problems in diffusive grey media. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary diffusion approximation to the radiation in grey media. The coupled equations are integrated in time using a semi-implicit method in the finite element framework. We show that for the considered problems, a combination of hyperbolic and exponential enrichment functions based on an approximation of the boundary layer leads to improved accuracy compared to the conventional finite element method. It is illustrated that this approach canmore » be more efficient than using h adaptivity to increase the accuracy of the finite element method near the boundary walls. The performance of the proposed partition-of-unity method is analyzed on several test examples for transient conduction–radiation problems in two space dimensions.« less

Three-dimensional finite element analysis for high velocity impact. [of projectiles from space debris

NASA Technical Reports Server (NTRS)

Chan, S. T. K.; Lee, C. H.; Brashears, M. R.

1975-01-01

A finite element algorithm for solving unsteady, three-dimensional high velocity impact problems is presented. A computer program was developed based on the Eulerian hydroelasto-viscoplastic formulation and the utilization of the theorem of weak solutions. The equations solved consist of conservation of mass, momentum, and energy, equation of state, and appropriate constitutive equations. The solution technique is a time-dependent finite element analysis utilizing three-dimensional isoparametric elements, in conjunction with a generalized two-step time integration scheme. The developed code was demonstrated by solving one-dimensional as well as three-dimensional impact problems for both the inviscid hydrodynamic model and the hydroelasto-viscoplastic model.

Finite elements: Theory and application

NASA Technical Reports Server (NTRS)

Dwoyer, D. L. (Editor); Hussaini, M. Y. (Editor); Voigt, R. G. (Editor)

1988-01-01

Recent advances in FEM techniques and applications are discussed in reviews and reports presented at the ICASE/LaRC workshop held in Hampton, VA in July 1986. Topics addressed include FEM approaches for partial differential equations, mixed FEMs, singular FEMs, FEMs for hyperbolic systems, iterative methods for elliptic finite-element equations on general meshes, mathematical aspects of FEMS for incompressible viscous flows, and gradient weighted moving finite elements in two dimensions. Consideration is given to adaptive flux-corrected FEM transport techniques for CFD, mixed and singular finite elements and the field BEM, p and h-p versions of the FEM, transient analysis methods in computational dynamics, and FEMs for integrated flow/thermal/structural analysis.

Nonlinear initial-boundary value solutions by the finite element method. [for Navier-Stokes equations of two dimensional flow

NASA Technical Reports Server (NTRS)

Baker, A. J.

1974-01-01

The finite-element method is used to establish a numerical solution algorithm for the Navier-Stokes equations for two-dimensional flows of a viscous compressible fluid. Numerical experiments confirm the advection property for the finite-element equivalent of the nonlinear convection term for both unidirectional and recirculating flowfields. For linear functionals, the algorithm demonstrates good accuracy using coarse discretizations and h squared convergence with discretization refinement.

A Posteriori Finite Element Bounds for Sensitivity Derivatives of Partial-Differential-Equation Outputs. Revised

NASA Technical Reports Server (NTRS)

Lewis, Robert Michael; Patera, Anthony T.; Peraire, Jaume

1998-01-01

We present a Neumann-subproblem a posteriori finite element procedure for the efficient and accurate calculation of rigorous, 'constant-free' upper and lower bounds for sensitivity derivatives of functionals of the solutions of partial differential equations. The design motivation for sensitivity derivative error control is discussed; the a posteriori finite element procedure is described; the asymptotic bounding properties and computational complexity of the method are summarized; and illustrative numerical results are presented.

Finite elements of nonlinear continua.

NASA Technical Reports Server (NTRS)

Oden, J. T.

1972-01-01

The finite element method is extended to a broad class of practical nonlinear problems, treating both theory and applications from a general and unifying point of view. The thermomechanical principles of continuous media and the properties of the finite element method are outlined, and are brought together to produce discrete physical models of nonlinear continua. The mathematical properties of the models are analyzed, and the numerical solution of the equations governing the discrete models is examined. The application of the models to nonlinear problems in finite elasticity, viscoelasticity, heat conduction, and thermoviscoelasticity is discussed. Other specific topics include the topological properties of finite element models, applications to linear and nonlinear boundary value problems, convergence, continuum thermodynamics, finite elasticity, solutions to nonlinear partial differential equations, and discrete models of the nonlinear thermomechanical behavior of dissipative media.

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

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

Mu, Lin; Wang, Junping; Ye, Xiu

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for bothmore » the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.« less

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2017-08-17

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for bothmore » the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.« less

A simple finite element method for non-divergence form elliptic equation

DOE PAGES

Mu, Lin; Ye, Xiu

2017-03-01

Here, we develop a simple finite element method for solving second order elliptic equations in non-divergence form by combining least squares concept with discontinuous approximations. This simple method has a symmetric and positive definite system and can be easily analyzed and implemented. We could have also used general meshes with polytopal element and hanging node in the method. We prove that our finite element solution approaches to the true solution when the mesh size approaches to zero. Numerical examples are tested that demonstrate the robustness and flexibility of the method.

A simple finite element method for non-divergence form elliptic equation

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

Mu, Lin; Ye, Xiu

Here, we develop a simple finite element method for solving second order elliptic equations in non-divergence form by combining least squares concept with discontinuous approximations. This simple method has a symmetric and positive definite system and can be easily analyzed and implemented. We could have also used general meshes with polytopal element and hanging node in the method. We prove that our finite element solution approaches to the true solution when the mesh size approaches to zero. Numerical examples are tested that demonstrate the robustness and flexibility of the method.

Dynamic characterization, monitoring and control of rotating flexible beam-mass structures via piezo-embedded techniques

NASA Technical Reports Server (NTRS)

Lai, Steven H.-Y.

1992-01-01

A variational principle and a finite element discretization technique were used to derive the dynamic equations for a high speed rotating flexible beam-mass system embedded with piezo-electric materials. The dynamic equation thus obtained allows the development of finite element models which accommodate both the original structural element and the piezoelectric element. The solutions of finite element models provide system dynamics needed to design a sensing system. The characterization of gyroscopic effect and damping capacity of smart rotating devices are addressed. Several simulation examples are presented to validate the analytical solution.

Finite element analysis of low speed viscous and inviscid aerodynamic flows

NASA Technical Reports Server (NTRS)

Baker, A. J.; Manhardt, P. D.

1977-01-01

A weak interaction solution algorithm was established for aerodynamic flow about an isolated airfoil. Finite element numerical methodology was applied to solution of each of differential equations governing potential flow, and viscous and turbulent boundary layer and wake flow downstream of the sharp trailing edge. The algorithm accounts for computed viscous displacement effects on the potential flow. Closure for turbulence was accomplished using both first and second order models. The COMOC finite element fluid mechanics computer program was modified to solve the identified equation systems for two dimensional flows. A numerical program was completed to determine factors affecting solution accuracy, convergence and stability for the combined potential, boundary layer, and parabolic Navier-Stokes equation systems. Good accuracy and convergence are demonstrated. Each solution is obtained within the identical finite element framework of COMOC.

FINITE-ELEMENT ANALYSIS OF MULTIPHASE IMMISCIBLE FLOW THROUGH SOILS

EPA Science Inventory

A finite-element model is developed for multiphase flow through soil involving three immiscible fluids: namely, air, water, and a nonaqueous phase liquid (NAPL). A variational method is employed for the finite-element formulation corresponding to the coupled differential equation...

Solution of a tridiagonal system of equations on the finite element machine

NASA Technical Reports Server (NTRS)

Bostic, S. W.

1984-01-01

Two parallel algorithms for the solution of tridiagonal systems of equations were implemented on the Finite Element Machine. The Accelerated Parallel Gauss method, an iterative method, and the Buneman algorithm, a direct method, are discussed and execution statistics are presented.

On some theoretical and practical aspects of multigrid methods. [to solve finite element systems from elliptic equations

NASA Technical Reports Server (NTRS)

Nicolaides, R. A.

1979-01-01

A description and explanation of a simple multigrid algorithm for solving finite element systems is given. Numerical results for an implementation are reported for a number of elliptic equations, including cases with singular coefficients and indefinite equations. The method shows the high efficiency, essentially independent of the grid spacing, predicted by the theory.

Analysis of the transient behavior of rubbing components

NASA Technical Reports Server (NTRS)

Quezdou, M. B.; Mullen, R. L.

1986-01-01

Finite element equations are developed for studying deformations and temperatures resulting from frictional heating in sliding system. The formulation is done for linear steady state motion in two dimensions. The equations include the effect of the velocity on the moving components. This gives spurious oscillations in their solutions by Galerkin finite element methods. A method called streamline upwind scheme is used to try to deal with this deficiency. The finite element program is then used to investigate the friction of heating in gas path seal.

A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations

NASA Technical Reports Server (NTRS)

Hu, Changqing; Shu, Chi-Wang

1998-01-01

In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method.

Solving the incompressible surface Navier-Stokes equation by surface finite elements

NASA Astrophysics Data System (ADS)

Reuther, Sebastian; Voigt, Axel

2018-01-01

We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus g (S ) . The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding R3, penalization of the normal component, a Chorin projection method, and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincaré-Hopf theorem on n-tori.

Adaptive macro finite elements for the numerical solution of monodomain equations in cardiac electrophysiology.

PubMed

Heidenreich, Elvio A; Ferrero, José M; Doblaré, Manuel; Rodríguez, José F

2010-07-01

Many problems in biology and engineering are governed by anisotropic reaction-diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction-diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two- and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.

A comparative study of finite element and finite difference methods for Cauchy-Riemann type equations

NASA Technical Reports Server (NTRS)

Fix, G. J.; Rose, M. E.

1983-01-01

A least squares formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods. Closely related arguments are shown to establish convergence estimates.

The application of the least squares finite element method to Abel's integral equation. [with application to glow discharge problem

NASA Technical Reports Server (NTRS)

Balasubramanian, R.; Norrie, D. H.; De Vries, G.

1979-01-01

Abel's integral equation is the governing equation for certain problems in physics and engineering, such as radiation from distributed sources. The finite element method for the solution of this non-linear equation is presented for problems with cylindrical symmetry and the extension to more general integral equations is indicated. The technique was applied to an axisymmetric glow discharge problem and the results show excellent agreement with previously obtained solutions

A simple, stable, and accurate linear tetrahedral finite element for transient, nearly, and fully incompressible solid dynamics: A dynamic variational multiscale approach [A simple, stable, and accurate tetrahedral finite element for transient, nearly incompressible, linear and nonlinear elasticity: A dynamic variational multiscale approach

DOE PAGES

Scovazzi, Guglielmo; Carnes, Brian; Zeng, Xianyi; ...

2015-11-12

Here, we propose a new approach for the stabilization of linear tetrahedral finite elements in the case of nearly incompressible transient solid dynamics computations. Our method is based on a mixed formulation, in which the momentum equation is complemented by a rate equation for the evolution of the pressure field, approximated with piece-wise linear, continuous finite element functions. The pressure equation is stabilized to prevent spurious pressure oscillations in computations. Incidentally, it is also shown that many stabilized methods previously developed for the static case do not generalize easily to transient dynamics. Extensive tests in the context of linear andmore » nonlinear elasticity are used to corroborate the claim that the proposed method is robust, stable, and accurate.« less

A simple, stable, and accurate linear tetrahedral finite element for transient, nearly, and fully incompressible solid dynamics: A dynamic variational multiscale approach [A simple, stable, and accurate tetrahedral finite element for transient, nearly incompressible, linear and nonlinear elasticity: A dynamic variational multiscale approach

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

Scovazzi, Guglielmo; Carnes, Brian; Zeng, Xianyi

Here, we propose a new approach for the stabilization of linear tetrahedral finite elements in the case of nearly incompressible transient solid dynamics computations. Our method is based on a mixed formulation, in which the momentum equation is complemented by a rate equation for the evolution of the pressure field, approximated with piece-wise linear, continuous finite element functions. The pressure equation is stabilized to prevent spurious pressure oscillations in computations. Incidentally, it is also shown that many stabilized methods previously developed for the static case do not generalize easily to transient dynamics. Extensive tests in the context of linear andmore » nonlinear elasticity are used to corroborate the claim that the proposed method is robust, stable, and accurate.« less

POSTPROCESSING MIXED FINITE ELEMENT METHODS FOR SOLVING CAHN-HILLIARD EQUATION: METHODS AND ERROR ANALYSIS

PubMed Central

Wang, Wansheng; Chen, Long; Zhou, Jie

2015-01-01

A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures. PMID:27110063

The Relation of Finite Element and Finite Difference Methods

NASA Technical Reports Server (NTRS)

Vinokur, M.

1976-01-01

Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.

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.

UXO Discrimination in Cases with Overlapping Signatures

DTIC Science & Technology

2007-03-07

13. APPENDIX B: HFE -BIEM ..........................................................................................................290 - 7...First principals numerical solutions developed were a Hybrid Finite Element – Boundary Integral Equation Method ( HFE -BIEM) body of revolution (BOR...attacks, namely the Method of Auxiliary Sources (MAS) and the Hybrid Finite Element – Boundary Integral Equation Method ( HFE -BIEM). These work

High-Order Thermal Radiative Transfer

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

Woods, Douglas Nelson; Cleveland, Mathew Allen; Wollaeger, Ryan Thomas

2017-09-18

The objective of this research is to asses the sensitivity of the linearized thermal radiation transport equations to finite element order on unstructured meshes and to investigate the sensitivity of the nonlinear TRT equations due to evaluating the opacities and heat capacity at nodal temperatures in 2-D using high-order finite elements.

Comparison of Gap Elements and Contact Algorithm for 3D Contact Analysis of Spiral Bevel Gears

NASA Technical Reports Server (NTRS)

Bibel, G. D.; Tiku, K.; Kumar, A.; Handschuh, R.

1994-01-01

Three dimensional stress analysis of spiral bevel gears in mesh using the finite element method is presented. A finite element model is generated by solving equations that identify tooth surface coordinates. Contact is simulated by the automatic generation of nonpenetration constraints. This method is compared to a finite element contact analysis conducted with gap elements.

Stable finite element approximations of two-phase flow with soluble surfactant

NASA Astrophysics Data System (ADS)

Barrett, John W.; Garcke, Harald; Nürnberg, Robert

2015-09-01

A parametric finite element approximation of incompressible two-phase flow with soluble surfactants is presented. The Navier-Stokes equations are coupled to bulk and surfaces PDEs for the surfactant concentrations. At the interface adsorption, desorption and stress balances involving curvature effects and Marangoni forces have to be considered. A parametric finite element approximation for the advection of the interface, which maintains good mesh properties, is coupled to the evolving surface finite element method, which is used to discretize the surface PDE for the interface surfactant concentration. The resulting system is solved together with standard finite element approximations of the Navier-Stokes equations and of the bulk parabolic PDE for the surfactant concentration. Semidiscrete and fully discrete approximations are analyzed with respect to stability, conservation and existence/uniqueness issues. The approach is validated for simple test cases and for complex scenarios, including colliding drops in a shear flow, which are computed in two and three space dimensions.

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

Finite element solution of transient fluid-structure interaction problems

NASA Technical Reports Server (NTRS)

Everstine, Gordon C.; Cheng, Raymond S.; Hambric, Stephen A.

1991-01-01

A finite element approach using NASTRAN is developed for solving time-dependent fluid-structure interaction problems, with emphasis on the transient scattering of acoustic waves from submerged elastic structures. Finite elements are used for modeling both structure and fluid domains to facilitate the graphical display of the wave motion through both media. For the liquid, the use of velocity potential as the fundamental unknown results in a symmetric matrix equation. The approach is illustrated for the problem of transient scattering from a submerged elastic spherical shell subjected to an incident tone burst. The use of an analogy between the equations of elasticity and the wave equation of acoustics, a necessary ingredient to the procedure, is summarized.

Finite element concepts in computational aerodynamics

NASA Technical Reports Server (NTRS)

Baker, A. J.

1978-01-01

Finite element theory was employed to establish an implicit numerical solution algorithm for the time averaged unsteady Navier-Stokes equations. Both the multidimensional and a time-split form of the algorithm were considered, the latter of particular interest for problem specification on a regular mesh. A Newton matrix iteration procedure is outlined for solving the resultant nonlinear algebraic equation systems. Multidimensional discretization procedures are discussed with emphasis on automated generation of specific nonuniform solution grids and accounting of curved surfaces. The time-split algorithm was evaluated with regards to accuracy and convergence properties for hyperbolic equations on rectangular coordinates. An overall assessment of the viability of the finite element concept for computational aerodynamics is made.

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.

Fourier analysis of finite element preconditioned collocation schemes

NASA Technical Reports Server (NTRS)

Deville, Michel O.; Mund, Ernest H.

1990-01-01

The spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes is investigated. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advection-diffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential equations resulting from Fourier analysis (in the tranverse direction) of the 2-D Stokes problem. All results agree with previous conclusions on the numerical efficiency of finite element preconditioning schemes.

A 2-D Interface Element for Coupled Analysis of Independently Modeled 3-D Finite Element Subdomains

NASA Technical Reports Server (NTRS)

Kandil, Osama A.

1998-01-01

Over the past few years, the development of the interface technology has provided an analysis framework for embedding detailed finite element models within finite element models which are less refined. This development has enabled the use of cascading substructure domains without the constraint of coincident nodes along substructure boundaries. The approach used for the interface element is based on an alternate variational principle often used in deriving hybrid finite elements. The resulting system of equations exhibits a high degree of sparsity but gives rise to a non-positive definite system which causes difficulties with many of the equation solvers in general-purpose finite element codes. Hence the global system of equations is generally solved using, a decomposition procedure with pivoting. The research reported to-date for the interface element includes the one-dimensional line interface element and two-dimensional surface interface element. Several large-scale simulations, including geometrically nonlinear problems, have been reported using the one-dimensional interface element technology; however, only limited applications are available for the surface interface element. In the applications reported to-date, the geometry of the interfaced domains exactly match each other even though the spatial discretization within each domain may be different. As such, the spatial modeling of each domain, the interface elements and the assembled system is still laborious. The present research is focused on developing a rapid modeling procedure based on a parametric interface representation of independently defined subdomains which are also independently discretized.

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.

High-Accuracy Finite Element Method: Benchmark Calculations

NASA Astrophysics Data System (ADS)

Gusev, Alexander; Vinitsky, Sergue; Chuluunbaatar, Ochbadrakh; Chuluunbaatar, Galmandakh; Gerdt, Vladimir; Derbov, Vladimir; Góźdź, Andrzej; Krassovitskiy, Pavel

2018-02-01

We describe a new high-accuracy finite element scheme with simplex elements for solving the elliptic boundary-value problems and show its efficiency on benchmark solutions of the Helmholtz equation for the triangle membrane and hypercube.

A finite element method for solving the shallow water equations on the sphere

NASA Astrophysics Data System (ADS)

Comblen, Richard; Legrand, Sébastien; Deleersnijder, Eric; Legat, Vincent

Within the framework of ocean general circulation modeling, the present paper describes an efficient way to discretize partial differential equations on curved surfaces by means of the finite element method on triangular meshes. Our approach benefits from the inherent flexibility of the finite element method. The key idea consists in a dialog between a local coordinate system defined for each element in which integration takes place, and a nodal coordinate system in which all local contributions related to a vectorial degree of freedom are assembled. Since each element of the mesh and each degree of freedom are treated in the same way, the so-called pole singularity issue is fully circumvented. Applied to the shallow water equations expressed in primitive variables, this new approach has been validated against the standard test set defined by [Williamson, D.L., Drake, J.B., Hack, J.J., Jakob, R., Swarztrauber, P.N., 1992. A standard test set for numerical approximations to the shallow water equations in spherical geometry. Journal of Computational Physics 102, 211-224]. Optimal rates of convergence for the P1NC-P1 finite element pair are obtained, for both global and local quantities of interest. Finally, the approach can be extended to three-dimensional thin-layer flows in a straightforward manner.

Element-topology-independent preconditioners for parallel finite element computations

NASA Technical Reports Server (NTRS)

Park, K. C.; Alexander, Scott

1992-01-01

A family of preconditioners for the solution of finite element equations are presented, which are element-topology independent and thus can be applicable to element order-free parallel computations. A key feature of the present preconditioners is the repeated use of element connectivity matrices and their left and right inverses. The properties and performance of the present preconditioners are demonstrated via beam and two-dimensional finite element matrices for implicit time integration computations.

Finite element method for optimal guidance of an advanced launch vehicle

NASA Technical Reports Server (NTRS)

Hodges, Dewey H.; Bless, Robert R.; Calise, Anthony J.; Leung, Martin

1992-01-01

A temporal finite element based on a mixed form of Hamilton's weak principle is summarized for optimal control problems. The resulting weak Hamiltonian finite element method is extended to allow for discontinuities in the states and/or discontinuities in the system equations. An extension of the formulation to allow for control inequality constraints is also presented. The formulation does not require element quadrature, and it produces a sparse system of nonlinear algebraic equations. To evaluate its feasibility for real-time guidance applications, this approach is applied to the trajectory optimization of a four-state, two-stage model with inequality constraints for an advanced launch vehicle. Numerical results for this model are presented and compared to results from a multiple-shooting code. The results show the accuracy and computational efficiency of the finite element method.

On finite element implementation and computational techniques for constitutive modeling of high temperature composites

NASA Technical Reports Server (NTRS)

Saleeb, A. F.; Chang, T. Y. P.; Wilt, T.; Iskovitz, I.

1989-01-01

The research work performed during the past year on finite element implementation and computational techniques pertaining to high temperature composites is outlined. In the present research, two main issues are addressed: efficient geometric modeling of composite structures and expedient numerical integration techniques dealing with constitutive rate equations. In the first issue, mixed finite elements for modeling laminated plates and shells were examined in terms of numerical accuracy, locking property and computational efficiency. Element applications include (currently available) linearly elastic analysis and future extension to material nonlinearity for damage predictions and large deformations. On the material level, various integration methods to integrate nonlinear constitutive rate equations for finite element implementation were studied. These include explicit, implicit and automatic subincrementing schemes. In all cases, examples are included to illustrate the numerical characteristics of various methods that were considered.

A multidimensional finite element method for CFD

NASA Technical Reports Server (NTRS)

Pepper, Darrell W.; Humphrey, Joseph W.

1991-01-01

A finite element method is used to solve the equations of motion for 2- and 3-D fluid flow. The time-dependent equations are solved explicitly using quadrilateral (2-D) and hexahedral (3-D) elements, mass lumping, and reduced integration. A Petrov-Galerkin technique is applied to the advection terms. The method requires a minimum of computational storage, executes quickly, and is scalable for execution on computer systems ranging from PCs to supercomputers.

Discrete maximum principle for the P1 - P0 weak Galerkin finite element approximations

NASA Astrophysics Data System (ADS)

Wang, Junping; Ye, Xiu; Zhai, Qilong; Zhang, Ran

2018-06-01

This paper presents two discrete maximum principles (DMP) for the numerical solution of second order elliptic equations arising from the weak Galerkin finite element method. The results are established by assuming an h-acute angle condition for the underlying finite element triangulations. The mathematical theory is based on the well-known De Giorgi technique adapted in the finite element context. Some numerical results are reported to validate the theory of DMP.

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Ho, Lee-Wing; Maday, Yvon; Patera, Anthony T.; Ronquist, Einar M.

1989-01-01

A high-order Lagrangian-decoupling method is presented for the unsteady convection-diffusion and incompressible Navier-Stokes equations. The method is based upon: (1) Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem; (2) implicit high-order backward-differentiation finite-difference schemes for integration along characteristics; (3) finite element or spectral element spatial discretizations; and (4) mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high order accuracy, and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.

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

Models and finite element approximations for interacting nanosized piezoelectric bodies and acoustic medium

NASA Astrophysics Data System (ADS)

Nasedkin, A. V.

2017-01-01

This research presents the new size-dependent models of piezoelectric materials oriented to finite element applications. The proposed models include the facilities of taking into account different mechanisms of damping for mechanical and electric fields. The coupled models also incorporate the equations of the theory of acoustics for viscous fluids. In particular cases, these models permit to use the mode superposition method with full separation of the finite element systems into independent equations for the independent modes for transient and harmonic problems. The main boundary conditions were supplemented with the facilities of taking into account the coupled surface effects, allowing to explore the nanoscale piezoelectric materials in the framework of theories of continuous media with surface stresses and their generalizations. For the considered problems we have implemented the finite element technologies and various numerical algorithms to maintain a symmetrical structure of the finite element quasi-definite matrices (matrix structure for the problems with a saddle point).

On finite element methods for the Helmholtz equation

NASA Technical Reports Server (NTRS)

Aziz, A. K.; Werschulz, A. G.

1979-01-01

The numerical solution of the Helmholtz equation is considered via finite element methods. A two-stage method which gives the same accuracy in the computed gradient as in the computed solution is discussed. Error estimates for the method using a newly developed proof are given, and the computational considerations which show this method to be computationally superior to previous methods are presented.

User's Guide for ENSAERO_FE Parallel Finite Element Solver

NASA Technical Reports Server (NTRS)

Eldred, Lloyd B.; Guruswamy, Guru P.

1999-01-01

A high fidelity parallel static structural analysis capability is created and interfaced to the multidisciplinary analysis package ENSAERO-MPI of Ames Research Center. This new module replaces ENSAERO's lower fidelity simple finite element and modal modules. Full aircraft structures may be more accurately modeled using the new finite element capability. Parallel computation is performed by breaking the full structure into multiple substructures. This approach is conceptually similar to ENSAERO's multizonal fluid analysis capability. The new substructure code is used to solve the structural finite element equations for each substructure in parallel. NASTRANKOSMIC is utilized as a front end for this code. Its full library of elements can be used to create an accurate and realistic aircraft model. It is used to create the stiffness matrices for each substructure. The new parallel code then uses an iterative preconditioned conjugate gradient method to solve the global structural equations for the substructure boundary nodes.

Stabilization and discontinuity-capturing parameters for space-time flow computations with finite element and isogeometric discretizations

NASA Astrophysics Data System (ADS)

Takizawa, Kenji; Tezduyar, Tayfun E.; Otoguro, Yuto

2018-04-01

Stabilized methods, which have been very common in flow computations for many years, typically involve stabilization parameters, and discontinuity-capturing (DC) parameters if the method is supplemented with a DC term. Various well-performing stabilization and DC parameters have been introduced for stabilized space-time (ST) computational methods in the context of the advection-diffusion equation and the Navier-Stokes equations of incompressible and compressible flows. These parameters were all originally intended for finite element discretization but quite often used also for isogeometric discretization. The stabilization and DC parameters we present here for ST computations are in the context of the advection-diffusion equation and the Navier-Stokes equations of incompressible flows, target isogeometric discretization, and are also applicable to finite element discretization. The parameters are based on a direction-dependent element length expression. The expression is outcome of an easy to understand derivation. The key components of the derivation are mapping the direction vector from the physical ST element to the parent ST element, accounting for the discretization spacing along each of the parametric coordinates, and mapping what we have in the parent element back to the physical element. The test computations we present for pure-advection cases show that the parameters proposed result in good solution profiles.

Application of finite element substructuring to composite micromechanics. M.S. Thesis - Akron Univ., May 1984

NASA Technical Reports Server (NTRS)

Caruso, J. J.

1984-01-01

Finite element substructuring is used to predict unidirectional fiber composite hygral (moisture), thermal, and mechanical properties. COSMIC NASTRAN and MSC/NASTRAN are used to perform the finite element analysis. The results obtained from the finite element model are compared with those obtained from the simplified composite micromechanics equations. A unidirectional composite structure made of boron/HM-epoxy, S-glass/IMHS-epoxy and AS/IMHS-epoxy are studied. The finite element analysis is performed using three dimensional isoparametric brick elements and two distinct models. The first model consists of a single cell (one fiber surrounded by matrix) to form a square. The second model uses the single cell and substructuring to form a nine cell square array. To compare computer time and results with the nine cell superelement model, another nine cell model is constructed using conventional mesh generation techniques. An independent computer program consisting of the simplified micromechanics equation is developed to predict the hygral, thermal, and mechanical properties for this comparison. The results indicate that advanced techniques can be used advantageously for fiber composite micromechanics.

The method of space-time and conservation element and solution element: A new approach for solving the Navier-Stokes and Euler equations

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung

1995-01-01

A new numerical framework for solving conservation laws is being developed. This new framework differs substantially in both concept and methodology from the well-established methods, i.e., finite difference, finite volume, finite element, and spectral methods. It is conceptually simple and designed to overcome several key limitations of the above traditional methods. A two-level scheme for solving the convection-diffusion equation is constructed and used to illuminate the major differences between the present method and those previously mentioned. This explicit scheme, referred to as the a-mu scheme, has two independent marching variables.

Iterative methods for elliptic finite element equations on general meshes

NASA Technical Reports Server (NTRS)

Nicolaides, R. A.; Choudhury, Shenaz

1986-01-01

Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included.

Finite element techniques for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation

NASA Technical Reports Server (NTRS)

Glaisner, F.; Tezduyar, T. E.

1987-01-01

Finite element procedures for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation have been implemented. For both formulations, streamline-upwind/Petrov-Galerkin techniques are used for the discretization of the transport equations. The main problem associated with the vorticity stream-function formulation is the lack of boundary conditions for vorticity at solid surfaces. Here an implicit treatment of the vorticity at no-slip boundaries is incorporated in a predictor-multicorrector time integration scheme. For the primitive variable formulation, mixed finite-element approximations are used. A nine-node element and a four-node + bubble element have been implemented. The latter is shown to exhibit a checkerboard pressure mode and a numerical treatment for this spurious pressure mode is proposed. The two methods are compared from the points of view of simulating internal and external flows and the possibilities of extensions to three dimensions.

Contact Stress Analysis of Spiral Bevel Gears Using Finite Element Analysis

NASA Technical Reports Server (NTRS)

Bibel, G. D.; Kumar, A; Reddy, S.; Handschuh, R.

1995-01-01

A procedure is presented for performing three-dimensional stress analysis of spiral bevel gears in mesh using the finite element method. The procedure involves generating a finite element model by solving equations that identify tooth surface coordinates. Coordinate transformations are used to orientate the gear and pinion for gear meshing. Contact boundary conditions are simulated with gap elements. A solution technique for correct orientation of the gap elements is given. Example models and results are presented.

Study of solution procedures for nonlinear structural equations

NASA Technical Reports Server (NTRS)

Young, C. T., II; Jones, R. F., Jr.

1980-01-01

A method for the redution of the cost of solution of large nonlinear structural equations was developed. Verification was made using the MARC-STRUC structure finite element program with test cases involving single and multiple degrees of freedom for static geometric nonlinearities. The method developed was designed to exist within the envelope of accuracy and convergence characteristic of the particular finite element methodology used.

Phase-space finite elements in a least-squares solution of the transport equation

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

Drumm, C.; Fan, W.; Pautz, S.

2013-07-01

The linear Boltzmann transport equation is solved using a least-squares finite element approximation in the space, angular and energy phase-space variables. The method is applied to both neutral particle transport and also to charged particle transport in the presence of an electric field, where the angular and energy derivative terms are handled with the energy/angular finite elements approximation, in a manner analogous to the way the spatial streaming term is handled. For multi-dimensional problems, a novel approach is used for the angular finite elements: mapping the surface of a unit sphere to a two-dimensional planar region and using a meshingmore » tool to generate a mesh. In this manner, much of the spatial finite-elements machinery can be easily adapted to handle the angular variable. The energy variable and the angular variable for one-dimensional problems make use of edge/beam elements, also building upon the spatial finite elements capabilities. The methods described here can make use of either continuous or discontinuous finite elements in space, angle and/or energy, with the use of continuous finite elements resulting in a smaller problem size and the use of discontinuous finite elements resulting in more accurate solutions for certain types of problems. The work described in this paper makes use of continuous finite elements, so that the resulting linear system is symmetric positive definite and can be solved with a highly efficient parallel preconditioned conjugate gradients algorithm. The phase-space finite elements capability has been built into the Sceptre code and applied to several test problems, including a simple one-dimensional problem with an analytic solution available, a two-dimensional problem with an isolated source term, showing how the method essentially eliminates ray effects encountered with discrete ordinates, and a simple one-dimensional charged-particle transport problem in the presence of an electric field. (authors)« less

A least-squares finite element method for 3D incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Lin, T. L.; Hou, Lin-Jun; Povinelli, Louis A.

1993-01-01

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

A finite element based method for solution of optimal control problems

NASA Technical Reports Server (NTRS)

Bless, Robert R.; Hodges, Dewey H.; Calise, Anthony J.

1989-01-01

A temporal finite element based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables that are expanded in terms of elemental values and simple shape functions. Unlike other variational approaches to optimal control problems, however, time derivatives of the states and costates do not appear in the governing variational equation. Instead, the only quantities whose time derivatives appear therein are virtual states and virtual costates. Also noteworthy among characteristics of the finite element formulation is the fact that in the algebraic equations which contain costates, they appear linearly. Thus, the remaining equations can be solved iteratively without initial guesses for the costates; this reduces the size of the problem by about a factor of two. Numerical results are presented herein for an elementary trajectory optimization problem which show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The goal is to evaluate the feasibility of this approach for real-time guidance applications. To this end, a simplified two-stage, four-state model for an advanced launch vehicle application is presented which is suitable for finite element solution.

A Posteriori Bounds for Linear-Functional Outputs of Crouzeix-Raviart Finite Element Discretizations of the Incompressible Stokes Problem

NASA Technical Reports Server (NTRS)

Patera, Anthony T.; Paraschivoiu, Marius

1998-01-01

We present a finite element technique for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions; the finite element discretization is effected by Crouzeix-Raviart elements, the discontinuous pressure approximation of which is central to our approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic "energy" reformulation of the desired output; the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the intersubdomain continuity conditions on velocity. Appeal to the dual max-min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization; the Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires solution only of local subdomain Stokes problems on the fine-mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flowrate past, and the lift force on, a body immersed in a channel.

Adaptive mesh refinement for time-domain electromagnetics using vector finite elements :a feasibility study.

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

Turner, C. David; Kotulski, Joseph Daniel; Pasik, Michael Francis

This report investigates the feasibility of applying Adaptive Mesh Refinement (AMR) techniques to a vector finite element formulation for the wave equation in three dimensions. Possible error estimators are considered first. Next, approaches for refining tetrahedral elements are reviewed. AMR capabilities within the Nevada framework are then evaluated. We summarize our conclusions on the feasibility of AMR for time-domain vector finite elements and identify a path forward.

A simple finite element method for linear hyperbolic problems

DOE PAGES

Mu, Lin; Ye, Xiu

2017-09-14

Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.

A simple finite element method for linear hyperbolic problems

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

Mu, Lin; Ye, Xiu

Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.

A quasi-Lagrangian finite element method for the Navier-Stokes equations in a time-dependent domain

NASA Astrophysics Data System (ADS)

Lozovskiy, Alexander; Olshanskii, Maxim A.; Vassilevski, Yuri V.

2018-05-01

The paper develops a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method builds on a quasi-Lagrangian formulation of the problem. The paper provides stability and convergence analysis of the fully discrete (finite-difference in time and finite-element in space) method. The analysis does not assume any CFL time-step restriction, it rather needs mild conditions of the form $\\Delta t\\le C$, where $C$ depends only on problem data, and $h^{2m_u+2}\\le c\\,\\Delta t$, $m_u$ is polynomial degree of velocity finite element space. Both conditions result from a numerical treatment of practically important non-homogeneous boundary conditions. The theoretically predicted convergence rate is confirmed by a set of numerical experiments. Further we apply the method to simulate a flow in a simplified model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.

High performance computation of radiative transfer equation using the finite element method

NASA Astrophysics Data System (ADS)

Badri, M. A.; Jolivet, P.; Rousseau, B.; Favennec, Y.

2018-05-01

This article deals with an efficient strategy for numerically simulating radiative transfer phenomena using distributed computing. The finite element method alongside the discrete ordinate method is used for spatio-angular discretization of the monochromatic steady-state radiative transfer equation in an anisotropically scattering media. Two very different methods of parallelization, angular and spatial decomposition methods, are presented. To do so, the finite element method is used in a vectorial way. A detailed comparison of scalability, performance, and efficiency on thousands of processors is established for two- and three-dimensional heterogeneous test cases. Timings show that both algorithms scale well when using proper preconditioners. It is also observed that our angular decomposition scheme outperforms our domain decomposition method. Overall, we perform numerical simulations at scales that were previously unattainable by standard radiative transfer equation solvers.

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.

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

Wavelet-based spectral finite element dynamic analysis for an axially moving Timoshenko beam

NASA Astrophysics Data System (ADS)

Mokhtari, Ali; Mirdamadi, Hamid Reza; Ghayour, Mostafa

2017-08-01

In this article, wavelet-based spectral finite element (WSFE) model is formulated for time domain and wave domain dynamic analysis of an axially moving Timoshenko beam subjected to axial pretension. The formulation is similar to conventional FFT-based spectral finite element (SFE) model except that Daubechies wavelet basis functions are used for temporal discretization of the governing partial differential equations into a set of ordinary differential equations. The localized nature of Daubechies wavelet basis functions helps to rule out problems of SFE model due to periodicity assumption, especially during inverse Fourier transformation and back to time domain. The high accuracy of WSFE model is then evaluated by comparing its results with those of conventional finite element and SFE results. The effects of moving beam speed and axial tensile force on vibration and wave characteristics, and static and dynamic stabilities of moving beam are investigated.

Contact stress analysis of spiral bevel gears using nonlinear finite element static analysis

NASA Technical Reports Server (NTRS)

Bibel, G. D.; Kumar, A.; Reddy, S.; Handschuh, R.

1993-01-01

A procedure is presented for performing three-dimensional stress analysis of spiral bevel gears in mesh using the finite element method. The procedure involves generating a finite element model by solving equations that identify tooth surface coordinates. Coordinate transformations are used to orientate the gear and pinion for gear meshing. Contact boundary conditions are simulated with gap elements. A solution technique for correct orientation of the gap elements is given. Example models and results are presented.

The least-squares finite element method for low-mach-number compressible viscous flows

NASA Technical Reports Server (NTRS)

Yu, Sheng-Tao

1994-01-01

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. Conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique; and, finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: first, driven cavity flows at various Reynolds numbers; and, buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.

Finite element procedures for coupled linear analysis of heat transfer, fluid and solid mechanics

NASA Technical Reports Server (NTRS)

Sutjahjo, Edhi; Chamis, Christos C.

1993-01-01

Coupled finite element formulations for fluid mechanics, heat transfer, and solid mechanics are derived from the conservation laws for energy, mass, and momentum. To model the physics of interactions among the participating disciplines, the linearized equations are coupled by combining domain and boundary coupling procedures. Iterative numerical solution strategy is presented to solve the equations, with the partitioning of temporal discretization implemented.

Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies

NASA Technical Reports Server (NTRS)

Winget, J. M.; Hughes, T. J. R.

1985-01-01

The particular problems investigated in the present study arise from nonlinear transient heat conduction. One of two types of nonlinearities considered is related to a material temperature dependence which is frequently needed to accurately model behavior over the range of temperature of engineering interest. The second nonlinearity is introduced by radiation boundary conditions. The finite element equations arising from the solution of nonlinear transient heat conduction problems are formulated. The finite element matrix equations are temporally discretized, and a nonlinear iterative solution algorithm is proposed. Algorithms for solving the linear problem are discussed, taking into account the form of the matrix equations, Gaussian elimination, cost, and iterative techniques. Attention is also given to approximate factorization, implementational aspects, and numerical results.

Finite Element Solution to the Helmholtz Equation with High Wave Number. Part 1. The h-Version of the FEM

DTIC Science & Technology

1993-11-01

4) between the exact solution and it’s best approximnation on the one and the FE-solution on the other hand. The determining equation for ti. & ielt ...Acknowledgement: The work of the first atitlhor wvas supported by Grant No 517 402 524 3 of the Gerinan Academic Exchange Service (l)AA[)). The work of thle second...methou, mn: A.K. Aziz (ed.), The mathematical foundations of tile finite element, method with applicai.4ons to partial differential equations, Academic

Solving the transport equation with quadratic finite elements: Theory and applications

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

Ferguson, J.M.

1997-12-31

At the 4th Joint Conference on Computational Mathematics, the author presented a paper introducing a new quadratic finite element scheme (QFEM) for solving the transport equation. In the ensuing year the author has obtained considerable experience in the application of this method, including solution of eigenvalue problems, transmission problems, and solution of the adjoint form of the equation as well as the usual forward solution. He will present detailed results, and will also discuss other refinements of his transport codes, particularly for 3-dimensional problems on rectilinear and non-rectilinear grids.

A high order accurate finite element algorithm for high Reynolds number flow prediction

NASA Technical Reports Server (NTRS)

Baker, A. J.

1978-01-01

A Galerkin-weighted residuals formulation is employed to establish an implicit finite element solution algorithm for generally nonlinear initial-boundary value problems. Solution accuracy, and convergence rate with discretization refinement, are quantized in several error norms, by a systematic study of numerical solutions to several nonlinear parabolic and a hyperbolic partial differential equation characteristic of the equations governing fluid flows. Solutions are generated using selective linear, quadratic and cubic basis functions. Richardson extrapolation is employed to generate a higher-order accurate solution to facilitate isolation of truncation error in all norms. Extension of the mathematical theory underlying accuracy and convergence concepts for linear elliptic equations is predicted for equations characteristic of laminar and turbulent fluid flows at nonmodest Reynolds number. The nondiagonal initial-value matrix structure introduced by the finite element theory is determined intrinsic to improved solution accuracy and convergence. A factored Jacobian iteration algorithm is derived and evaluated to yield a consequential reduction in both computer storage and execution CPU requirements while retaining solution accuracy.

Dynamic Fracture of Concrete. Part 1

DTIC Science & Technology

1990-02-14

unnotched) by Mindess and the Charpy type impact tests by Shah. In both cases, dynamic finite element modeling with the adjusted constitutive equavm for the...Mindess and the Charpy type impact tests by Shah. In both cases, dynamic finite element modeling with the adjusted constitutive equations for the...Modeling Shah’s Charpy Impact Tests ................ 190 Figure 7.20 Specimen Configuration and Finite Element Model for Concrete and Mortar Beam Impact

Stress-intensity factor equations for cracks in three-dimensional finite bodies

NASA Technical Reports Server (NTRS)

Newman, J. C., Jr.; Raju, I. S.

1981-01-01

Empirical stress intensity factor equations are presented for embedded elliptical cracks, semi-elliptical surface cracks, quarter-elliptical corner cracks, semi-elliptical surface cracks at a hole, and quarter-elliptical corner cracks at a hole in finite plates. The plates were subjected to remote tensile loading. Equations give stress intensity factors as a function of parametric angle, crack depth, crack length, plate thickness, and where applicable, hole radius. The stress intensity factors used to develop the equations were obtained from three dimensional finite element analyses of these crack configurations.

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.

Vertical discretization with finite elements for a global hydrostatic model on the cubed sphere

NASA Astrophysics Data System (ADS)

Yi, Tae-Hyeong; Park, Ja-Rin

2017-06-01

A formulation of Galerkin finite element with basis-spline functions on a hybrid sigma-pressure coordinate is presented to discretize the vertical terms of global Eulerian hydrostatic equations employed in a numerical weather prediction system, which is horizontally discretized with high-order spectral elements on a cubed sphere grid. This replaces the vertical discretization of conventional central finite difference that is first-order accurate in non-uniform grids and causes numerical instability in advection-dominant flows. Therefore, a model remains in the framework of Galerkin finite elements for both the horizontal and vertical spatial terms. The basis-spline functions, obtained from the de-Boor algorithm, are employed to derive both the vertical derivative and integral operators, since Eulerian advection terms are involved. These operators are used to discretize the vertical terms of the prognostic and diagnostic equations. To verify the vertical discretization schemes and compare their performance, various two- and three-dimensional idealized cases and a hindcast case with full physics are performed in terms of accuracy and stability. It was shown that the vertical finite element with the cubic basis-spline function is more accurate and stable than that of the vertical finite difference, as indicated by faster residual convergence, fewer statistical errors, and reduction in computational mode. This leads to the general conclusion that the overall performance of a global hydrostatic model might be significantly improved with the vertical finite element.

A locally conservative non-negative finite element formulation for anisotropic advective-diffusive-reactive systems

NASA Astrophysics Data System (ADS)

Mudunuru, M. K.; Shabouei, M.; Nakshatrala, K.

2015-12-01

Advection-diffusion-reaction (ADR) equations appear in various areas of life sciences, hydrogeological systems, and contaminant transport. Obtaining stable and accurate numerical solutions can be challenging as the underlying equations are coupled, nonlinear, and non-self-adjoint. Currently, there is neither a robust computational framework available nor a reliable commercial package known that can handle various complex situations. Herein, the objective of this poster presentation is to present a novel locally conservative non-negative finite element formulation that preserves the underlying physical and mathematical properties of a general linear transient anisotropic ADR equation. In continuous setting, governing equations for ADR systems possess various important properties. In general, all these properties are not inherited during finite difference, finite volume, and finite element discretizations. The objective of this poster presentation is two fold: First, we analyze whether the existing numerical formulations (such as SUPG and GLS) and commercial packages provide physically meaningful values for the concentration of the chemical species for various realistic benchmark problems. Furthermore, we also quantify the errors incurred in satisfying the local and global species balance for two popular chemical kinetics schemes: CDIMA (chlorine dioxide-iodine-malonic acid) and BZ (Belousov--Zhabotinsky). Based on these numerical simulations, we show that SUPG and GLS produce unphysical values for concentration of chemical species due to the violation of the non-negative constraint, contain spurious node-to-node oscillations, and have large errors in local and global species balance. Second, we proposed a novel finite element formulation to overcome the above difficulties. The proposed locally conservative non-negative computational framework based on low-order least-squares finite elements is able to preserve these underlying physical and mathematical properties. Several representative numerical examples are discussed to illustrate the importance of the proposed numerical formulations to accurately describe various aspects of mixing process in chaotic flows and to simulate transport in highly heterogeneous anisotropic media.

Finite element approximation of an optimal control problem for the von Karman equations

NASA Technical Reports Server (NTRS)

Hou, L. Steven; Turner, James C.

1994-01-01

This paper is concerned with optimal control problems for the von Karman equations with distributed controls. We first show that optimal solutions exist. We then show that Lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced. Finally we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations.

A finite element solution algorithm for the Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Baker, A. J.

1974-01-01

A finite element solution algorithm is established for the two-dimensional Navier-Stokes equations governing the steady-state kinematics and thermodynamics of a variable viscosity, compressible multiple-species fluid. For an incompressible fluid, the motion may be transient as well. The primitive dependent variables are replaced by a vorticity-streamfunction description valid in domains spanned by rectangular, cylindrical and spherical coordinate systems. Use of derived variables provides a uniformly elliptic partial differential equation description for the Navier-Stokes system, and for which the finite element algorithm is established. Explicit non-linearity is accepted by the theory, since no psuedo-variational principles are employed, and there is no requirement for either computational mesh or solution domain closure regularity. Boundary condition constraints on the normal flux and tangential distribution of all computational variables, as well as velocity, are routinely piecewise enforceable on domain closure segments arbitrarily oriented with respect to a global reference frame.

The Coupling of Finite Element and Integral Equation Representations for Efficient Three-Dimensional Modeling of Electromagnetic Scattering and Radiation

NASA Technical Reports Server (NTRS)

Cwik, Tom; Zuffada, Cinzia; Jamnejad, Vahraz

1996-01-01

Finite element modeling has proven useful for accurtely simulating scattered or radiated fields from complex three-dimensional objects whose geometry varies on the scale of a fraction of a wavelength.

Application of the Finite Element Method in Atomic and Molecular Physics

NASA Technical Reports Server (NTRS)

Shertzer, Janine

2007-01-01

The finite element method (FEM) is a numerical algorithm for solving second order differential equations. It has been successfully used to solve many problems in atomic and molecular physics, including bound state and scattering calculations. To illustrate the diversity of the method, we present here details of two applications. First, we calculate the non-adiabatic dipole polarizability of Hi by directly solving the first and second order equations of perturbation theory with FEM. In the second application, we calculate the scattering amplitude for e-H scattering (without partial wave analysis) by reducing the Schrodinger equation to set of integro-differential equations, which are then solved with FEM.

A fast solver for the Helmholtz equation based on the generalized multiscale finite-element method

NASA Astrophysics Data System (ADS)

Fu, Shubin; Gao, Kai

2017-11-01

Conventional finite-element methods for solving the acoustic-wave Helmholtz equation in highly heterogeneous media usually require finely discretized mesh to represent the medium property variations with sufficient accuracy. Computational costs for solving the Helmholtz equation can therefore be considerably expensive for complicated and large geological models. Based on the generalized multiscale finite-element theory, we develop a novel continuous Galerkin method to solve the Helmholtz equation in acoustic media with spatially variable velocity and mass density. Instead of using conventional polynomial basis functions, we use multiscale basis functions to form the approximation space on the coarse mesh. The multiscale basis functions are obtained from multiplying the eigenfunctions of a carefully designed local spectral problem with an appropriate multiscale partition of unity. These multiscale basis functions can effectively incorporate the characteristics of heterogeneous media's fine-scale variations, thus enable us to obtain accurate solution to the Helmholtz equation without directly solving the large discrete system formed on the fine mesh. Numerical results show that our new solver can significantly reduce the dimension of the discrete Helmholtz equation system, and can also obviously reduce the computational time.

Analysis of spurious oscillation modes for the shallow water and Navier-Stokes equations

USGS Publications Warehouse

Walters, R.A.; Carey, G.F.

1983-01-01

The origin and nature of spurious oscillation modes that appear in mixed finite element methods are examined. In particular, the shallow water equations are considered and a modal analysis for the one-dimensional problem is developed. From the resulting dispersion relations we find that the spurious modes in elevation are associated with zero frequency and large wave number (wavelengths of the order of the nodal spacing) and consequently are zero-velocity modes. The spurious modal behavior is the result of the finite spatial discretization. By means of an artificial compressibility and limiting argument we are able to resolve the similar problem for the Navier-Stokes equations. The relationship of this simpler analysis to alternative consistency arguments is explained. This modal approach provides an explanation of the phenomenon in question and permits us to deduce the cause of the very complex behavior of spurious modes observed in numerical experiments with the shallow water equations and Navier-Stokes equations. Furthermore, this analysis is not limited to finite element formulations, but is also applicable to finite difference formulations. ?? 1983.

Least-squares solution of incompressible Navier-Stokes equations with the p-version of finite elements

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Sonnad, Vijay

1991-01-01

A p-version of the least squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady state incompressible viscous flow problems. The resulting system of symmetric and positive definite linear equations can be solved satisfactorily with the conjugate gradient method. In conjunction with the use of rapid operator application which avoids the formation of either element of global matrices, it is possible to achieve a highly compact and efficient solution scheme for the incompressible Navier-Stokes equations. Numerical results are presented for two-dimensional flow over a backward facing step. The effectiveness of simple outflow boundary conditions is also demonstrated.

A Viscoelastic Hybrid Shell Finite Element

NASA Technical Reports Server (NTRS)

Johnson, Arthur

1999-01-01

An elastic large displacement thick-shell hybrid finite element is modified to allow for the calculation of viscoelastic stresses. Internal strain variables are introduced at he element's stress nodes and are employed to construct a viscous material model. First order ordinary differential equations relate the internal strain variables to the corresponding elastic strains at the stress nodes. The viscous stresses are computed from the internal strain variables using viscous moduli which are a fraction of the elastic moduli. The energy dissipated by the action of the viscous stresses in included in the mixed variational functional. Nonlinear quasi-static viscous equilibrium equations are then obtained. Previously developed Taylor expansions of the equilibrium equations are modified to include the viscous terms. A predictor-corrector time marching solution algorithm is employed to solve the algebraic-differential equations. The viscous shell element is employed to numerically simulate a stair-step loading and unloading of an aircraft tire in contact with a frictionless surface.

A finite element solver for 3-D compressible viscous flows

NASA Technical Reports Server (NTRS)

Reddy, K. C.; Reddy, J. N.; Nayani, S.

1990-01-01

Computation of the flow field inside a space shuttle main engine (SSME) requires the application of state of the art computational fluid dynamic (CFD) technology. Several computer codes are under development to solve 3-D flow through the hot gas manifold. Some algorithms were designed to solve the unsteady compressible Navier-Stokes equations, either by implicit or explicit factorization methods, using several hundred or thousands of time steps to reach a steady state solution. A new iterative algorithm is being developed for the solution of the implicit finite element equations without assembling global matrices. It is an efficient iteration scheme based on a modified nonlinear Gauss-Seidel iteration with symmetric sweeps. The algorithm is analyzed for a model equation and is shown to be unconditionally stable. Results from a series of test problems are presented. The finite element code was tested for couette flow, which is flow under a pressure gradient between two parallel plates in relative motion. Another problem that was solved is viscous laminar flow over a flat plate. The general 3-D finite element code was used to compute the flow in an axisymmetric turnaround duct at low Mach numbers.

Effect of triangular element orientation on finite element solutions of the Helmholtz equation

NASA Technical Reports Server (NTRS)

Baumeister, K. J.

1986-01-01

The Galerkin finite element solutions for the scalar homogeneous Helmholtz equation are presented for no reflection, hard wall, and potential relief exit terminations with a variety of triangular element orientations. For this group of problems, the correlation between the accuracy of the solution and the orientation of the linear triangle is examined. Nonsymmetric element patterns are found to give generally poor results in the model problems investigated, particularly for cases where standing waves exist. For a fixed number of vertical elements, the results showed that symmetric element patterns give much better agreement with corresponding exact analytical results. In laminated wave guide application, the symmetric pyramid pattern is convenient to use and is shown to give excellent results.

A Mechanical Power Flow Capability for the Finite Element Code NASTRAN

DTIC Science & Technology

1989-07-01

perimental methods. statistical energy analysis , the finite element method, and a finite element analog-,y using heat conduction equations. Experimental...weights and inertias of the transducers attached to an experimental structure may produce accuracy problems. Statistical energy analysis (SEA) is a...405-422 (1987). 8. Lyon, R.L., Statistical Energy Analysis of Dynamical Sistems, The M.I.T. Press, (1975). 9. Mickol, J.D., and R.J. Bernhard, "An

A Streamline-Upwind Petrov-Galerkin Finite Element Scheme for Non-Ionized Hypersonic Flows in Thermochemical Nonequilibrium

NASA Technical Reports Server (NTRS)

Kirk, Benjamin S.; Bova, Stephen W.; Bond, Ryan B.

2011-01-01

Presentation topics include background and motivation; physical modeling including governing equations and thermochemistry; finite element formulation; results of inviscid thermal nonequilibrium chemically reacting flow and viscous thermal equilibrium chemical reacting flow; and near-term effort.

A hybridized formulation for the weak Galerkin mixed finite element method

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

Mu, Lin; Wang, Junping; Ye, Xiu

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising frommore » the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.« less

A hybridized formulation for the weak Galerkin mixed finite element method

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2016-01-14

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising frommore » the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. In conclusion, some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.« less

The MHOST finite element program: 3-D inelastic analysis methods for hot section components. Volume 2: User's manual

NASA Technical Reports Server (NTRS)

Nakazawa, Shohei

1989-01-01

The user options available for running the MHOST finite element analysis package is described. MHOST is a solid and structural analysis program based on the mixed finite element technology, and is specifically designed for 3-D inelastic analysis. A family of 2- and 3-D continuum elements along with beam and shell structural elements can be utilized, many options are available in the constitutive equation library, the solution algorithms and the analysis capabilities. The outline of solution algorithms is discussed along with the data input and output, analysis options including the user subroutines and the definition of the finite elements implemented in the program package.

Examining the validity of Stoney-equation for in-situ stress measurements in thin film electrodes using a large-deformation finite-element procedure

NASA Astrophysics Data System (ADS)

Wen, Jici; Wei, Yujie; Cheng, Yang-Tse

2018-05-01

During the lithiation and delithiation of a thin film electrode, stress in the electrode is deduced from the curvature change of the film using the Stoney equation. The accuracy of such a measurement is conditioned on the assumptions that (a) the mechanical properties of the electrode remain unchanged during lithiation and (b) small deformation holds. Here, we demonstrate that the change in elastic properties can influence the measurement of the stress in thin film electrodes. We consider the coupling between diffusion and deformation during lithiation and delithiation of thin film electrodes and implement the constitutive behavior in a finite-deformation finite element procedure. We demonstrate that both the variation in elastic properties in thin film electrodes and finite-deformation during lithiation and delithiation would challenge the applicability of the Stoney-equation for in-situ stress measurements of thin film electrodes.

On mathematical modelling of aeroelastic problems with finite element method

NASA Astrophysics Data System (ADS)

Sváček, Petr

2018-06-01

This paper is interested in solution of two-dimensional aeroelastic problems. Two mathematical models are compared for a benchmark problem. First, the classical approach of linearized aerodynamical forces is described to determine the aeroelastic instability and the aeroelastic response in terms of frequency and damping coefficient. This approach is compared to the coupled fluid-structure model solved with the aid of finite element method used for approximation of the incompressible Navier-Stokes equations. The finite element approximations are coupled to the non-linear motion equations of a flexibly supported airfoil. Both methods are first compared for the case of small displacement, where the linearized approach can be well adopted. The influence of nonlinearities for the case of post-critical regime is discussed.

Error analysis and correction of discrete solutions from finite element codes

NASA Technical Reports Server (NTRS)

Thurston, G. A.; Stein, P. A.; Knight, N. F., Jr.; Reissner, J. E.

1984-01-01

Many structures are an assembly of individual shell components. Therefore, results for stresses and deflections from finite element solutions for each shell component should agree with the equations of shell theory. This paper examines the problem of applying shell theory to the error analysis and the correction of finite element results. The general approach to error analysis and correction is discussed first. Relaxation methods are suggested as one approach to correcting finite element results for all or parts of shell structures. Next, the problem of error analysis of plate structures is examined in more detail. The method of successive approximations is adapted to take discrete finite element solutions and to generate continuous approximate solutions for postbuckled plates. Preliminary numerical results are included.

Optimal least-squares finite element method for elliptic problems

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Povinelli, Louis A.

1991-01-01

An optimal least squares finite element method is proposed for two dimensional and three dimensional elliptic problems and its advantages are discussed over the mixed Galerkin method and the usual least squares finite element method. In the usual least squares finite element method, the second order equation (-Delta x (Delta u) + u = f) is recast as a first order system (-Delta x p + u = f, Delta u - p = 0). The error analysis and numerical experiment show that, in this usual least squares finite element method, the rate of convergence for flux p is one order lower than optimal. In order to get an optimal least squares method, the irrotationality Delta x p = 0 should be included in the first order system.

Modeling Progressive Failure of Bonded Joints Using a Single Joint Finite Element

NASA Technical Reports Server (NTRS)

Stapleton, Scott E.; Waas, Anthony M.; Bednarcyk, Brett A.

2010-01-01

Enhanced finite elements are elements with an embedded analytical solution which can capture detailed local fields, enabling more efficient, mesh-independent finite element analysis. In the present study, an enhanced finite element is applied to generate a general framework capable of modeling an array of joint types. The joint field equations are derived using the principle of minimum potential energy, and the resulting solutions for the displacement fields are used to generate shape functions and a stiffness matrix for a single joint finite element. This single finite element thus captures the detailed stress and strain fields within the bonded joint, but it can function within a broader structural finite element model. The costs associated with a fine mesh of the joint can thus be avoided while still obtaining a detailed solution for the joint. Additionally, the capability to model non-linear adhesive constitutive behavior has been included within the method, and progressive failure of the adhesive can be modeled by using a strain-based failure criteria and re-sizing the joint as the adhesive fails. Results of the model compare favorably with experimental and finite element results.

C1 finite elements on non-tensor-product 2d and 3d manifolds

PubMed Central

Nguyen, Thien; Karčiauskas, Kęstutis; Peters, Jörg

2015-01-01

Geometrically continuous (Gk) constructions naturally yield families of finite elements for isogeometric analysis (IGA) that are Ck also for non-tensor-product layout. This paper describes and analyzes one such concrete C1 geometrically generalized IGA element (short: gIGA element) that generalizes bi-quadratic splines to quad meshes with irregularities. The new gIGA element is based on a recently-developed G1 surface construction that recommends itself by its a B-spline-like control net, low (least) polynomial degree, good shape properties and reproduction of quadratics at irregular (extraordinary) points. Remarkably, for Poisson’s equation on the disk using interior vertices of valence 3 and symmetric layout, we observe O(h3) convergence in the L∞ norm for this family of elements. Numerical experiments confirm the elements to be effective for solving the trivariate Poisson equation on the solid cylinder, deformations thereof (a turbine blade), modeling and computing geodesics on smooth free-form surfaces via the heat equation, for solving the biharmonic equation on the disk and for Koiter-type thin-shell analysis. PMID:26594070

C1 finite elements on non-tensor-product 2d and 3d manifolds.

PubMed

Nguyen, Thien; Karčiauskas, Kęstutis; Peters, Jörg

2016-01-01

Geometrically continuous ( G k ) constructions naturally yield families of finite elements for isogeometric analysis (IGA) that are C k also for non-tensor-product layout. This paper describes and analyzes one such concrete C 1 geometrically generalized IGA element (short: gIGA element) that generalizes bi-quadratic splines to quad meshes with irregularities. The new gIGA element is based on a recently-developed G 1 surface construction that recommends itself by its a B-spline-like control net, low (least) polynomial degree, good shape properties and reproduction of quadratics at irregular (extraordinary) points. Remarkably, for Poisson's equation on the disk using interior vertices of valence 3 and symmetric layout, we observe O ( h 3 ) convergence in the L ∞ norm for this family of elements. Numerical experiments confirm the elements to be effective for solving the trivariate Poisson equation on the solid cylinder, deformations thereof (a turbine blade), modeling and computing geodesics on smooth free-form surfaces via the heat equation, for solving the biharmonic equation on the disk and for Koiter-type thin-shell analysis.

Algorithms and software for solving finite element equations on serial and parallel architectures

NASA Technical Reports Server (NTRS)

George, Alan

1989-01-01

Over the past 15 years numerous new techniques have been developed for solving systems of equations and eigenvalue problems arising in finite element computations. A package called SPARSPAK has been developed by the author and his co-workers which exploits these new methods. The broad objective of this research project is to incorporate some of this software in the Computational Structural Mechanics (CSM) testbed, and to extend the techniques for use on multiprocessor architectures.

Coupled NASTRAN/boundary element formulation for acoustic scattering

NASA Technical Reports Server (NTRS)

Everstine, Gordon C.; Henderson, Francis M.; Schuetz, Luise S.

1987-01-01

A coupled finite element/boundary element capability is described for calculating the sound pressure field scattered by an arbitrary submerged 3-D elastic structure. Structural and fluid impedances are calculated with no approximation other than discretization. The surface fluid pressures and normal velocities are first calculated by coupling a NASTRAN finite element model of the structure with a discretized form of the Helmholtz surface integral equation for the exterior field. Far field pressures are then evaluated from the surface solution using the Helmholtz exterior integral equation. The overall approach is illustrated and validated using a known analytic solution for scattering from submerged spherical shells.

A three-dimensional nonlinear Timoshenko beam based on the core-congruential formulation

NASA Technical Reports Server (NTRS)

Crivelli, Luis A.; Felippa, Carlos A.

1992-01-01

A three-dimensional, geometrically nonlinear two-node Timoshenkoo beam element based on the total Larangrian description is derived. The element behavior is assumed to be linear elastic, but no restrictions are placed on magnitude of finite rotations. The resulting element has twelve degrees of freedom: six translational components and six rotational-vector components. The formulation uses the Green-Lagrange strains and second Piola-Kirchhoff stresses as energy-conjugate variables and accounts for the bending-stretching and bending-torsional coupling effects without special provisions. The core-congruential formulation (CCF) is used to derived the discrete equations in a staged manner. Core equations involving the internal force vector and tangent stiffness matrix are developed at the particle level. A sequence of matrix transformations carries these equations to beam cross-sections and finally to the element nodal degrees of freedom. The choice of finite rotation measure is made in the next-to-last transformation stage, and the choice of over-the-element interpolation in the last one. The tangent stiffness matrix is found to retain symmetry if the rotational vector is chosen to measure finite rotations. An extensive set of numerical examples is presented to test and validate the present element.

Finite element, modal co-ordinate analysis of structures subjected to moving loads

NASA Astrophysics Data System (ADS)

Olsson, M.

1985-03-01

Some of the possibilities of the finite element method in the moving load problem are demonstrated. The bridge-vehicle interaction phenomenon is considered by deriving a general bridge-vehicle element which is believed to be novel. This element may be regarded as a finite element with time-dependent and unsymmetric element matrices. The bridge response is formulated in modal co-ordinates thereby reducing the number of equations to be solved within each time step. Illustrative examples are shown for the special case of a beam bridge model and a one-axle vehicle model.

A Poisson equation formulation for pressure calculations in penalty finite element models for viscous incompressible flows

NASA Technical Reports Server (NTRS)

Sohn, J. L.; Heinrich, J. C.

1990-01-01

The calculation of pressures when the penalty-function approximation is used in finite-element solutions of laminar incompressible flows is addressed. A Poisson equation for the pressure is formulated that involves third derivatives of the velocity field. The second derivatives appearing in the weak formulation of the Poisson equation are calculated from the C0 velocity approximation using a least-squares method. The present scheme is shown to be efficient, free of spurious oscillations, and accurate. Examples of applications are given and compared with results obtained using mixed formulations.

Fem Formulation of Heat Transfer in Cylindrical Porous Medium

NASA Astrophysics Data System (ADS)

Azeem; Khaleed, H. M. T.; Soudagar, Manzoor Elahi M.

2017-08-01

Heat transfer in porous medium can be derived from the fundamental laws of flow in porous region ass given by Henry Darcy. The fluid flow and energy transport inside the porous medium can be described with the help of momentum and energy equations. The heat transfer in cylindrical porous medium differs from its counterpart in radial and axial coordinates. The present work is focused to discuss the finite element formulation of heat transfer in cylindrical porous medium. The basic partial differential equations are derived using Darcy law which is the converted into a set of algebraic equations with the help of finite element method. The resulting equations are solved by matrix method for two solution variables involved in the coupled equations.

Bolt clampup relaxation in a graphite/epoxy laminate

NASA Technical Reports Server (NTRS)

Shivakumar, K. N.; Crews, J. H., Jr.

1982-01-01

A simple bolted joint was analyzed to calculate bolt clampup relaxation for a graphite/epoxy (T300/5208) laminate. A viscoelastic finite element analysis of a double-lap joint with a steel bolt was conducted. Clampup forces were calculated for various steady-state temperature-moisture conditions using a 20-year exposure duration. The finite element analysis predicted that clampup forces relax even for the room-temperature-dry condition. The relaxations were 8, 13, 20, and 30 percent for exposure durations of 1 day, 1 month, 1 year, and 20 years, respectively. As expected, higher temperatures and moisture levels each increased the relaxation rate. The combined viscoelastic effects of steady-state temperature and moisture appeared to be additive. From the finite-element analysis, a simple equation was developed for clampup force relaxation. This generalized equation was used to calculate clampup forces for the same temperature-moisture conditions as used in the finite-element analysis. The two sets of calculated results agreed well.

Mathematical aspects of finite element methods for incompressible viscous flows

NASA Technical Reports Server (NTRS)

Gunzburger, M. D.

1986-01-01

Mathematical aspects of finite element methods are surveyed for incompressible viscous flows, concentrating on the steady primitive variable formulation. The discretization of a weak formulation of the Navier-Stokes equations are addressed, then the stability condition is considered, the satisfaction of which insures the stability of the approximation. Specific choices of finite element spaces for the velocity and pressure are then discussed. Finally, the connection between different weak formulations and a variety of boundary conditions is explored.

FIESTA ROC: A new finite element analysis program for solar cell simulation

NASA Technical Reports Server (NTRS)

Clark, Ralph O.

1991-01-01

The Finite Element Semiconductor Three-dimensional Analyzer by Ralph O. Clark (FIESTA ROC) is a computational tool for investigating in detail the performance of arbitrary solar cell structures. As its name indicates, it uses the finite element technique to solve the fundamental semiconductor equations in the cell. It may be used for predicting the performance (thereby dictating the design parameters) of a proposed cell or for investigating the limiting factors in an established design.

Efficient Preconditioning for the p-Version Finite Element Method in Two Dimensions

DTIC Science & Technology

1989-10-01

paper, we study fast parallel preconditioners for systems of equations arising from the p-version finite element method. The p-version finite element...computations and the solution of a relatively small global auxiliary problem. We study two different methods. In the first (Section 3), the global...20], will be studied in the next section. Problem (3.12) is obviously much more easily solved than the original problem ,nd the procedure is highly

A three-dimensional, finite element model for coastal and estuarine circulation

USGS Publications Warehouse

Walters, R.A.

1992-01-01

This paper describes the development and application of a three-dimensional model for coastal and estuarine circulation. The model uses a harmonic expansion in time and a finite element discretization in space. All nonlinear terms are retained, including quadratic bottom stress, advection and wave transport (continuity nonlinearity). The equations are solved as a global and a local problem, where the global problem is the solution of the wave equation formulation of the shallow water equations, and the local problem is the solution of the momentum equation for the vertical velocity profile. These equations are coupled to the advection-diffusion equation for salt so that density gradient forcing is included in the momentum equations. The model is applied to a study of Delaware Bay, U.S.A., where salinity intrusion is the primary focus. ?? 1991.

Finite element methods for the biomechanics of soft hydrated tissues: nonlinear analysis and adaptive control of meshes.

PubMed

Spilker, R L; de Almeida, E S; Donzelli, P S

1992-01-01

This chapter addresses computationally demanding numerical formulations in the biomechanics of soft tissues. The theory of mixtures can be used to represent soft hydrated tissues in the human musculoskeletal system as a two-phase continuum consisting of an incompressible solid phase (collagen and proteoglycan) and an incompressible fluid phase (interstitial water). We first consider the finite deformation of soft hydrated tissues in which the solid phase is represented as hyperelastic. A finite element formulation of the governing nonlinear biphasic equations is presented based on a mixed-penalty approach and derived using the weighted residual method. Fluid and solid phase deformation, velocity, and pressure are interpolated within each element, and the pressure variables within each element are eliminated at the element level. A system of nonlinear, first-order differential equations in the fluid and solid phase deformation and velocity is obtained. In order to solve these equations, the contributions of the hyperelastic solid phase are incrementally linearized, a finite difference rule is introduced for temporal discretization, and an iterative scheme is adopted to achieve equilibrium at the end of each time increment. We demonstrate the accuracy and adequacy of the procedure using a six-node, isoparametric axisymmetric element, and we present an example problem for which independent numerical solution is available. Next, we present an automated, adaptive environment for the simulation of soft tissue continua in which the finite element analysis is coupled with automatic mesh generation, error indicators, and projection methods. Mesh generation and updating, including both refinement and coarsening, for the two-dimensional examples examined in this study are performed using the finite quadtree approach. The adaptive analysis is based on an error indicator which is the L2 norm of the difference between the finite element solution and a projected finite element solution. Total stress, calculated as the sum of the solid and fluid phase stresses, is used in the error indicator. To allow the finite difference algorithm to proceed in time using an updated mesh, solution values must be transferred to the new nodal locations. This rezoning is accomplished using a projected field for the primary variables. The accuracy and effectiveness of this adaptive finite element analysis is demonstrated using a linear, two-dimensional, axisymmetric problem corresponding to the indentation of a thin sheet of soft tissue. The method is shown to effectively capture the steep gradients and to produce solutions in good agreement with independent, converged, numerical solutions.

Anisotropic constitutive model for nickel base single crystal alloys: Development and finite element implementation

NASA Technical Reports Server (NTRS)

Dame, L. T.; Stouffer, D. C.

1986-01-01

A tool for the mechanical analysis of nickel base single crystal superalloys, specifically Rene N4, used in gas turbine engine components is developed. This is achieved by a rate dependent anisotropic constitutive model implemented in a nonlinear three dimensional finite element code. The constitutive model is developed from metallurigical concepts utilizing a crystallographic approach. A non Schmid's law formulation is used to model the tension/compression asymmetry and orientation dependence in octahedral slip. Schmid's law is a good approximation to the inelastic response of the material in cube slip. The constitutive equations model the tensile behavior, creep response, and strain rate sensitivity of these alloys. Methods for deriving the material constants from standard tests are presented. The finite element implementation utilizes an initial strain method and twenty noded isoparametric solid elements. The ability to model piecewise linear load histories is included in the finite element code. The constitutive equations are accurately and economically integrated using a second order Adams-Moulton predictor-corrector method with a dynamic time incrementing procedure. Computed results from the finite element code are compared with experimental data for tensile, creep and cyclic tests at 760 deg C. The strain rate sensitivity and stress relaxation capabilities of the model are evaluated.

A robust, finite element model for hydrostatic surface water flows

USGS Publications Warehouse

Walters, R.A.; Casulli, V.

1998-01-01

A finite element scheme is introduced for the 2-dimensional shallow water equations using semi-implicit methods in time. A semi-Lagrangian method is used to approximate the effects of advection. A wave equation is formed at the discrete level such that the equations decouple into an equation for surface elevation and a momentum equation for the horizontal velocity. The convergence rates and relative computational efficiency are examined with the use of three test cases representing various degrees of difficulty. A test with a polar-quadrant grid investigates the response to local grid-scale forcing and the presence of spurious modes, a channel test case establishes convergence rates, and a field-scale test case examines problems with highly irregular grids.A finite element scheme is introduced for the 2-dimensional shallow water equations using semi-implicit methods in time. A semi-Lagrangian method is used to approximate the effects of advection. A wave equation is formed at the discrete level such that the equations decouple into an equation for surface elevation and a momentum equation for the horizontal velocity. The convergence rates and relative computational efficiency are examined with the use of three test cases representing various degrees of difficulty. A test with a polar-quadrant grid investigates the response to local grid-scale forcing and the presence of spurious modes, a channel test case establishes convergence rates, and a field-scale test case examines problems with highly irregular grids.

Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Meade, Andrew James, Jr.

1989-01-01

A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, Navier-Stokes schemes. The primary goal is to develop a finite element method for the calculation of 3-D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3-D boundary layer of pointed bodies of arbitrary cross section. The 3-D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3-D partial differential equations are reduced to 2-D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasius-type of similarity variable. This is equivalent to a Dorodnitsyn-type formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2-D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a Petrov-Galerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of 5 and 10 deg. The numerical solutions assume a laminar boundary layer with free stream Mach number of 7. Results include circumferential distribution of skin friction and surface heat transfer, and cross flow velocity distributions across the layer.

Finite element solution of optimal control problems with state-control inequality constraints

NASA Technical Reports Server (NTRS)

Bless, Robert R.; Hodges, Dewey H.

1992-01-01

It is demonstrated that the weak Hamiltonian finite-element formulation is amenable to the solution of optimal control problems with inequality constraints which are functions of both state and control variables. Difficult problems can be treated on account of the ease with which algebraic equations can be generated before having to specify the problem. These equations yield very accurate solutions. Owing to the sparse structure of the resulting Jacobian, computer solutions can be obtained quickly when the sparsity is exploited.

Moving finite elements in 2-D

NASA Technical Reports Server (NTRS)

Gelinas, R. J.; Doss, S. K.; Vajk, J. P.; Djomehri, J.; Miller, K.

1983-01-01

The mathematical background regarding the moving finite element (MFE) method of Miller and Miller (1981) is discussed, taking into account a general system of partial differential equations (PDE) and the amenability of the MFE method in two dimensions to code modularization and to semiautomatic user-construction of numerous PDE systems for both Dirichlet and zero-Neumann boundary conditions. A description of test problem results is presented, giving attention to aspects of single square wave propagation, and a solution of the heat equation.

Application of the control volume mixed finite element method to a triangular discretization

USGS Publications Warehouse

Naff, R.L.

2012-01-01

A two-dimensional control volume mixed finite element method is applied to the elliptic equation. Discretization of the computational domain is based in triangular elements. Shape functions and test functions are formulated on the basis of an equilateral reference triangle with unit edges. A pressure support based on the linear interpolation of elemental edge pressures is used in this formulation. Comparisons are made between results from the standard mixed finite element method and this control volume mixed finite element method. Published 2011. This article is a US Government work and is in the public domain in the USA. ?? 2012 John Wiley & Sons, Ltd. This article is a US Government work and is in the public domain in the USA.

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.

An Integrated Magnetic Circuit Model and Finite Element Model Approach to Magnetic Bearing Design

NASA Technical Reports Server (NTRS)

Provenza, Andrew J.; Kenny, Andrew; Palazzolo, Alan B.

2003-01-01

A code for designing magnetic bearings is described. The code generates curves from magnetic circuit equations relating important bearing performance parameters. Bearing parameters selected from the curves by a designer to meet the requirements of a particular application are input directly by the code into a three-dimensional finite element analysis preprocessor. This means that a three-dimensional computer model of the bearing being developed is immediately available for viewing. The finite element model solution can be used to show areas of magnetic saturation and make more accurate predictions of the bearing load capacity, current stiffness, position stiffness, and inductance than the magnetic circuit equations did at the start of the design process. In summary, the code combines one-dimensional and three-dimensional modeling methods for designing magnetic bearings.

Energy and technology review: Engineering modeling

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

Cabayan, H.S.; Goudreau, G.L.; Ziolkowski, R.W.

1986-10-01

This report presents information concerning: Modeling Canonical Problems in Electromagnetic Coupling Through Apertures; Finite-Element Codes for Computing Electrostatic Fields; Finite-Element Modeling of Electromagnetic Phenomena; Modeling Microwave-Pulse Compression in a Resonant Cavity; Lagrangian Finite-Element Analysis of Penetration Mechanics; Crashworthiness Engineering; Computer Modeling of Metal-Forming Processes; Thermal-Mechanical Modeling of Tungsten Arc Welding; Modeling Air Breakdown Induced by Electromagnetic Fields; Iterative Techniques for Solving Boltzmann's Equations for p-Type Semiconductors; Semiconductor Modeling; and Improved Numerical-Solution Techniques in Large-Scale Stress Analysis.

Least-squares finite element solution of 3D incompressible Navier-Stokes problems

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Lin, Tsung-Liang; Povinelli, Louis A.

1992-01-01

Although significant progress has been made in the finite element solution of incompressible viscous flow problems. Development of more efficient methods is still needed before large-scale computation of 3D problems becomes feasible. This paper presents such a development. The most popular finite element method for the solution of incompressible Navier-Stokes equations is the classic Galerkin mixed method based on the velocity-pressure formulation. The mixed method requires the use of different elements to interpolate the velocity and the pressure in order to satisfy the Ladyzhenskaya-Babuska-Brezzi (LBB) condition for the existence of the solution. On the other hand, due to the lack of symmetry and positive definiteness of the linear equations arising from the mixed method, iterative methods for the solution of linear systems have been hard to come by. Therefore, direct Gaussian elimination has been considered the only viable method for solving the systems. But, for three-dimensional problems, the computer resources required by a direct method become prohibitively large. In order to overcome these difficulties, a least-squares finite element method (LSFEM) has been developed. This method is based on the first-order velocity-pressure-vorticity formulation. In this paper the LSFEM is extended for the solution of three-dimensional incompressible Navier-Stokes equations written in the following first-order quasi-linear velocity-pressure-vorticity formulation.

Exact finite elements for conduction and convection

NASA Technical Reports Server (NTRS)

Thornton, E. A.; Dechaumphai, P.; Tamma, K. K.

1981-01-01

An approach for developing exact one dimensional conduction-convection finite elements is presented. Exact interpolation functions are derived based on solutions to the governing differential equations by employing a nodeless parameter. Exact interpolation functions are presented for combined heat transfer in several solids of different shapes, and for combined heat transfer in a flow passage. Numerical results demonstrate that exact one dimensional elements offer advantages over elements based on approximate interpolation functions.

Higher and lowest order mixed finite element approximation of subsurface flow problems with solutions of low regularity

NASA Astrophysics Data System (ADS)

Bause, Markus

2008-02-01

In this work we study mixed finite element approximations of Richards' equation for simulating variably saturated subsurface flow and simultaneous reactive solute transport. Whereas higher order schemes have proved their ability to approximate reliably reactive solute transport (cf., e.g. [Bause M, Knabner P. Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping. Comput Visual Sci 7;2004:61-78]), the Raviart- Thomas mixed finite element method ( RT0) with a first order accurate flux approximation is popular for computing the underlying water flow field (cf. [Bause M, Knabner P. Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Adv Water Resour 27;2004:565-581, Farthing MW, Kees CE, Miller CT. Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv Water Resour 26;2003:373-394, Starke G. Least-squares mixed finite element solution of variably saturated subsurface flow problems. SIAM J Sci Comput 21;2000:1869-1885, Younes A, Mosé R, Ackerer P, Chavent G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. J Comp Phys 149;1999:148-167, Woodward CS, Dawson CN. Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J Numer Anal 37;2000:701-724]). This combination might be non-optimal. Higher order techniques could increase the accuracy of the flow field calculation and thereby improve the prediction of the solute transport. Here, we analyse the application of the Brezzi- Douglas- Marini element ( BDM1) with a second order accurate flux approximation to elliptic, parabolic and degenerate problems whose solutions lack the regularity that is assumed in optimal order error analyses. For the flow field calculation a superiority of the BDM1 approach to the RT0 one is observed, which however is less significant for the accompanying solute transport.

A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition

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

Gerritsma, Marc; Bochev, Pavel

Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximatedmore » by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.« less

Real-time adaptive finite element solution of time-dependent Kohn-Sham equation

NASA Astrophysics Data System (ADS)

Bao, Gang; Hu, Guanghui; Liu, Di

2015-01-01

In our previous paper (Bao et al., 2012 [1]), a general framework of using adaptive finite element methods to solve the Kohn-Sham equation has been presented. This work is concerned with solving the time-dependent Kohn-Sham equations. The numerical methods are studied in the time domain, which can be employed to explain both the linear and the nonlinear effects. A Crank-Nicolson scheme and linear finite element space are employed for the temporal and spatial discretizations, respectively. To resolve the trouble regions in the time-dependent simulations, a heuristic error indicator is introduced for the mesh adaptive methods. An algebraic multigrid solver is developed to efficiently solve the complex-valued system derived from the semi-implicit scheme. A mask function is employed to remove or reduce the boundary reflection of the wavefunction. The effectiveness of our method is verified by numerical simulations for both linear and nonlinear phenomena, in which the effectiveness of the mesh adaptive methods is clearly demonstrated.

A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition

DOE PAGES

Gerritsma, Marc; Bochev, Pavel

2016-03-22

Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximatedmore » by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.« less

An efficient finite element technique for sound propagation in axisymmetric hard wall ducts carrying high subsonic Mach number flows

NASA Technical Reports Server (NTRS)

Tag, I. A.; Lumsdaine, E.

1978-01-01

The general non-linear three-dimensional equation for acoustic potential is derived by using a perturbation technique. The linearized axisymmetric equation is then solved by using a finite element algorithm based on the Galerkin formulation for a harmonic time dependence. The solution is carried out in complex number notation for the acoustic velocity potential. Linear, isoparametric, quadrilateral elements with non-uniform distribution across the duct section are implemented. The resultant global matrix is stored in banded form and solved by using a modified Gauss elimination technique. Sound pressure levels and acoustic velocities are calculated from post element solutions. Different duct geometries are analyzed and compared with experimental results.

A unique set of micromechanics equations for high temperature metal matrix composites

NASA Technical Reports Server (NTRS)

Hopkins, D. A.; Chamis, C. C.

1985-01-01

A unique set of micromechanic equations is presented for high temperature metal matrix composites. The set includes expressions to predict mechanical properties, thermal properties and constituent microstresses for the unidirectional fiber reinforced ply. The equations are derived based on a mechanics of materials formulation assuming a square array unit cell model of a single fiber, surrounding matrix and an interphase to account for the chemical reaction which commonly occurs between fiber and matrix. A three-dimensional finite element analysis was used to perform a preliminary validation of the equations. Excellent agreement between properties predicted using the micromechanics equations and properties simulated by the finite element analyses are demonstrated. Implementation of the micromechanics equations as part of an integrated computational capability for nonlinear structural analysis of high temperature multilayered fiber composites is illustrated.

Preconditioned conjugate residual methods for the solution of spectral equations

NASA Technical Reports Server (NTRS)

Wong, Y. S.; Zang, T. A.; Hussaini, M. Y.

1986-01-01

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Mixed finite element - discontinuous finite volume element discretization of a general class of multicontinuum models

NASA Astrophysics Data System (ADS)

Ruiz-Baier, Ricardo; Lunati, Ivan

2016-10-01

We present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua. In the framework of mixture theory, we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases. The model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture (barycentric) velocities, phase pressure, and saturation. We construct an accurate, robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume-element discretization of the mass (or transport) equations. The scheme is devised for unstructured meshes and relies on mixed Brezzi-Douglas-Marini approximations of phase and total velocities, on piecewise constant elements for the approximation of phase or total pressures, as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest (such as volume fraction, total density, saturation, etc.). As the discretization scheme is derived for a general formulation of multicontinuum physical systems, it can be readily applied to a large class of simplified multiphase models; on the other, the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate. An extensive set of numerical test cases involving two- and three-dimensional porous media are presented to demonstrate the accuracy of the method (displaying an optimal convergence rate), the physics-preserving properties of the mixed-primal scheme, as well as the robustness of the method (which is successfully used to simulate diverse physical phenomena such as density fingering, Terzaghi's consolidation, deformation of a cantilever bracket, and Boycott effects). The applicability of the method is not limited to flow in porous media, but can also be employed to describe many other physical systems governed by a similar set of equations, including e.g. multi-component materials.

Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions

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

Lu Benzhuo; Holst, Michael J.; Center for Theoretical Biological Physics, University of California San Diego, La Jolla, CA 92093

2010-09-20

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for simulating electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised formore » time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.« less

Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element Solutions

PubMed Central

Lu, Benzhuo; Holst, Michael J.; McCammon, J. Andrew; Zhou, Y. C.

2010-01-01

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems. PMID:21709855

Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element Solutions.

PubMed

Lu, Benzhuo; Holst, Michael J; McCammon, J Andrew; Zhou, Y C

2010-09-20

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.

Survey of the status of finite element methods for partial differential equations

NASA Technical Reports Server (NTRS)

Temam, Roger

1986-01-01

The finite element methods (FEM) have proved to be a powerful technique for the solution of boundary value problems associated with partial differential equations of either elliptic, parabolic, or hyperbolic type. They also have a good potential for utilization on parallel computers particularly in relation to the concept of domain decomposition. This report is intended as an introduction to the FEM for the nonspecialist. It contains a survey which is totally nonexhaustive, and it also contains as an illustration, a report on some new results concerning two specific applications, namely a free boundary fluid-structure interaction problem and the Euler equations for inviscid flows.

A Semi-Analytical Method for Determining the Energy Release Rate of Cracks in Adhesively-Bonded Single-Lap Composite Joints

NASA Technical Reports Server (NTRS)

Yang, Charles; Sun, Wenjun; Tomblin, John S.; Smeltzer, Stanley S., III

2007-01-01

A semi-analytical method for determining the strain energy release rate due to a prescribed interface crack in an adhesively-bonded, single-lap composite joint subjected to axial tension is presented. The field equations in terms of displacements within the joint are formulated by using first-order shear deformable, laminated plate theory together with kinematic relations and force equilibrium conditions. The stress distributions for the adherends and adhesive are determined after the appropriate boundary and loading conditions are applied and the equations for the field displacements are solved. Based on the adhesive stress distributions, the forces at the crack tip are obtained and the strain energy release rate of the crack is determined by using the virtual crack closure technique (VCCT). Additionally, the test specimen geometry from both the ASTM D3165 and D1002 test standards are utilized during the derivation of the field equations in order to correlate analytical models with future test results. The system of second-order differential field equations is solved to provide the adherend and adhesive stress response using the symbolic computation tool, Maple 9. Finite element analyses using J-integral as well as VCCT were performed to verify the developed analytical model. The finite element analyses were conducted using the commercial finite element analysis software ABAQUS. The results determined using the analytical method correlated well with the results from the finite element analyses.

Quasi-Static Viscoelastic Finite Element Model of an Aircraft Tire

NASA Technical Reports Server (NTRS)

Johnson, Arthur R.; Tanner, John A.; Mason, Angela J.

1999-01-01

An elastic large displacement thick-shell mixed finite element is modified to allow for the calculation of viscoelastic stresses. Internal strain variables are introduced at the element's stress nodes and are employed to construct a viscous material model. First order ordinary differential equations relate the internal strain variables to the corresponding elastic strains at the stress nodes. The viscous stresses are computed from the internal strain variables using viscous moduli which are a fraction of the elastic moduli. The energy dissipated by the action of the viscous stresses is included in the mixed variational functional. The nonlinear quasi-static viscous equilibrium equations are then obtained. Previously developed Taylor expansions of the nonlinear elastic equilibrium equations are modified to include the viscous terms. A predictor-corrector time marching solution algorithm is employed to solve the algebraic-differential equations. The viscous shell element is employed to computationally simulate a stair-step loading and unloading of an aircraft tire in contact with a frictionless surface.

Experiences on p-Version Time-Discontinuous Galerkin's Method for Nonlinear Heat Transfer Analysis and Sensitivity Analysis

NASA Technical Reports Server (NTRS)

Hou, Gene

2004-01-01

The focus of this research is on the development of analysis and sensitivity analysis equations for nonlinear, transient heat transfer problems modeled by p-version, time discontinuous finite element approximation. The resulting matrix equation of the state equation is simply in the form ofA(x)x = c, representing a single step, time marching scheme. The Newton-Raphson's method is used to solve the nonlinear equation. Examples are first provided to demonstrate the accuracy characteristics of the resultant finite element approximation. A direct differentiation approach is then used to compute the thermal sensitivities of a nonlinear heat transfer problem. The report shows that only minimal coding effort is required to enhance the analysis code with the sensitivity analysis capability.

MPSalsa Version 1.5: A Finite Element Computer Program for Reacting Flow Problems: Part 1 - Theoretical Development

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

Devine, K.D.; Hennigan, G.L.; Hutchinson, S.A.

1999-01-01

The theoretical background for the finite element computer program, MPSalsa Version 1.5, is presented in detail. MPSalsa is designed to solve laminar or turbulent low Mach number, two- or three-dimensional incompressible and variable density reacting fluid flows on massively parallel computers, using a Petrov-Galerkin finite element formulation. The code has the capability to solve coupled fluid flow (with auxiliary turbulence equations), heat transport, multicomponent species transport, and finite-rate chemical reactions, and to solve coupled multiple Poisson or advection-diffusion-reaction equations. The program employs the CHEMKIN library to provide a rigorous treatment of multicomponent ideal gas kinetics and transport. Chemical reactions occurringmore » in the gas phase and on surfaces are treated by calls to CHEMKIN and SURFACE CHEMK3N, respectively. The code employs unstructured meshes, using the EXODUS II finite element database suite of programs for its input and output files. MPSalsa solves both transient and steady flows by using fully implicit time integration, an inexact Newton method and iterative solvers based on preconditioned Krylov methods as implemented in the Aztec. solver library.« less

Discontinuous Galerkin finite element method for solving population density functions of cortical pyramidal and thalamic neuronal populations.

PubMed

Huang, Chih-Hsu; Lin, Chou-Ching K; Ju, Ming-Shaung

2015-02-01

Compared with the Monte Carlo method, the population density method is efficient for modeling collective dynamics of neuronal populations in human brain. In this method, a population density function describes the probabilistic distribution of states of all neurons in the population and it is governed by a hyperbolic partial differential equation. In the past, the problem was mainly solved by using the finite difference method. In a previous study, a continuous Galerkin finite element method was found better than the finite difference method for solving the hyperbolic partial differential equation; however, the population density function often has discontinuity and both methods suffer from a numerical stability problem. The goal of this study is to improve the numerical stability of the solution using discontinuous Galerkin finite element method. To test the performance of the new approach, interaction of a population of cortical pyramidal neurons and a population of thalamic neurons was simulated. The numerical results showed good agreement between results of discontinuous Galerkin finite element and Monte Carlo methods. The convergence and accuracy of the solutions are excellent. The numerical stability problem could be resolved using the discontinuous Galerkin finite element method which has total-variation-diminishing property. The efficient approach will be employed to simulate the electroencephalogram or dynamics of thalamocortical network which involves three populations, namely, thalamic reticular neurons, thalamocortical neurons and cortical pyramidal neurons. Copyright © 2014 Elsevier Ltd. All rights reserved.

Discontinuous finite element method for vector radiative transfer

NASA Astrophysics Data System (ADS)

Wang, Cun-Hai; Yi, Hong-Liang; Tan, He-Ping

2017-03-01

The discontinuous finite element method (DFEM) is applied to solve the vector radiative transfer in participating media. The derivation in a discrete form of the vector radiation governing equations is presented, in which the angular space is discretized by the discrete-ordinates approach with a local refined modification, and the spatial domain is discretized into finite non-overlapped discontinuous elements. The elements in the whole solution domain are connected by modelling the boundary numerical flux between adjacent elements, which makes the DFEM numerically stable for solving radiative transfer equations. Several various problems of vector radiative transfer are tested to verify the performance of the developed DFEM, including vector radiative transfer in a one-dimensional parallel slab containing a Mie/Rayleigh/strong forward scattering medium and a two-dimensional square medium. The fact that DFEM results agree very well with the benchmark solutions in published references shows that the developed DFEM in this paper is accurate and effective for solving vector radiative transfer problems.

FINITE ELEMENT MODEL FOR TIDAL AND RESIDUAL CIRCULATION.

USGS Publications Warehouse

Walters, Roy A.

1986-01-01

Harmonic decomposition is applied to the shallow water equations, thereby creating a system of equations for the amplitude of the various tidal constituents and for the residual motions. The resulting equations are elliptic in nature, are well posed and in practice are shown to be numerically well-behaved. There are a number of strategies for choosing elements: the two extremes are to use a few high-order elements with continuous derivatives, or to use a large number of simpler linear elements. In this paper simple linear elements are used and prove effective.

Parallel processing in finite element structural analysis

NASA Technical Reports Server (NTRS)

Noor, Ahmed K.

1987-01-01

A brief review is made of the fundamental concepts and basic issues of parallel processing. Discussion focuses on parallel numerical algorithms, performance evaluation of machines and algorithms, and parallelism in finite element computations. A computational strategy is proposed for maximizing the degree of parallelism at different levels of the finite element analysis process including: 1) formulation level (through the use of mixed finite element models); 2) analysis level (through additive decomposition of the different arrays in the governing equations into the contributions to a symmetrized response plus correction terms); 3) numerical algorithm level (through the use of operator splitting techniques and application of iterative processes); and 4) implementation level (through the effective combination of vectorization, multitasking and microtasking, whenever available).

A Runge-Kutta discontinuous finite element method for high speed flows

NASA Technical Reports Server (NTRS)

Bey, Kim S.; Oden, J. T.

1991-01-01

A Runge-Kutta discontinuous finite element method is developed for hyperbolic systems of conservation laws in two space variables. The discontinuous Galerkin spatial approximation to the conservation laws results in a system of ordinary differential equations which are marched in time using Runge-Kutta methods. Numerical results for the two-dimensional Burger's equation show that the method is (p+1)-order accurate in time and space, where p is the degree of the polynomial approximation of the solution within an element and is capable of capturing shocks over a single element without oscillations. Results for this problem also show that the accuracy of the solution in smooth regions is unaffected by the local projection and that the accuracy in smooth regions increases as p increases. Numerical results for the Euler equations show that the method captures shocks without oscillations and with higher resolution than a first-order scheme.

Finite element solution for energy conservation using a highly stable explicit integration algorithm

NASA Technical Reports Server (NTRS)

Baker, A. J.; Manhardt, P. D.

1972-01-01

Theoretical derivation of a finite element solution algorithm for the transient energy conservation equation in multidimensional, stationary multi-media continua with irregular solution domain closure is considered. The complete finite element matrix forms for arbitrarily irregular discretizations are established, using natural coordinate function representations. The algorithm is embodied into a user-oriented computer program (COMOC) which obtains transient temperature distributions at the node points of the finite element discretization using a highly stable explicit integration procedure with automatic error control features. The finite element algorithm is shown to posses convergence with discretization for a transient sample problem. The condensed form for the specific heat element matrix is shown to be preferable to the consistent form. Computed results for diverse problems illustrate the versatility of COMOC, and easily prepared output subroutines are shown to allow quick engineering assessment of solution behavior.

Radiation Heat Transfer Between Diffuse-Gray Surfaces Using Higher Order Finite Elements

NASA Technical Reports Server (NTRS)

Gould, Dana C.

2000-01-01

This paper presents recent work on developing methods for analyzing radiation heat transfer between diffuse-gray surfaces using p-version finite elements. The work was motivated by a thermal analysis of a High Speed Civil Transport (HSCT) wing structure which showed the importance of radiation heat transfer throughout the structure. The analysis also showed that refining the finite element mesh to accurately capture the temperature distribution on the internal structure led to very large meshes with unacceptably long execution times. Traditional methods for calculating surface-to-surface radiation are based on assumptions that are not appropriate for p-version finite elements. Two methods for determining internal radiation heat transfer are developed for one and two-dimensional p-version finite elements. In the first method, higher-order elements are divided into a number of sub-elements. Traditional methods are used to determine radiation heat flux along each sub-element and then mapped back to the parent element. In the second method, the radiation heat transfer equations are numerically integrated over the higher-order element. Comparisons with analytical solutions show that the integration scheme is generally more accurate than the sub-element method. Comparison to results from traditional finite elements shows that significant reduction in the number of elements in the mesh is possible using higher-order (p-version) finite elements.

SUPG Finite Element Simulations of Compressible Flows for Aerothermodynamic Applications

NASA Technical Reports Server (NTRS)

Kirk, Benjamin S.

2007-01-01

This viewgraph presentation reviews the Streamline-Upwind Petrov-Galerkin (SUPG) Finite Element Simulation. It covers the background, governing equations, weak formulation, shock capturing, inviscid flux discretization, time discretization, linearization, and implicit solution strategies. It also reviews some applications such as Type IV Shock Interaction, Forward-Facing Cavity and AEDC Sharp Double Cone.

Large Angle Transient Dynamics (LATDYN) user's manual

NASA Technical Reports Server (NTRS)

Abrahamson, A. Louis; Chang, Che-Wei; Powell, Michael G.; Wu, Shih-Chin; Bingel, Bradford D.; Theophilos, Paula M.

1991-01-01

A computer code for modeling the large angle transient dynamics (LATDYN) of structures was developed to investigate techniques for analyzing flexible deformation and control/structure interaction problems associated with large angular motions of spacecraft. This type of analysis is beyond the routine capability of conventional analytical tools without simplifying assumptions. In some instances, the motion may be sufficiently slow and the spacecraft (or component) sufficiently rigid to simplify analyses of dynamics and controls by making pseudo-static and/or rigid body assumptions. The LATDYN introduces a new approach to the problem by combining finite element structural analysis, multi-body dynamics, and control system analysis in a single tool. It includes a type of finite element that can deform and rotate through large angles at the same time, and which can be connected to other finite elements either rigidly or through mechanical joints. The LATDYN also provides symbolic capabilities for modeling control systems which are interfaced directly with the finite element structural model. Thus, the nonlinear equations representing the structural model are integrated along with the equations representing sensors, processing, and controls as a coupled system.

Modal Substructuring of Geometrically Nonlinear Finite-Element Models

DOE PAGES

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

2015-12-21

The efficiency of a modal substructuring method depends on the component modes used to reduce each subcomponent model. Methods such as Craig–Bampton have been used extensively to reduce linear finite-element models with thousands or even millions of degrees of freedom down orders of magnitude while maintaining acceptable accuracy. A novel reduction method is proposed here for geometrically nonlinear finite-element models using the fixed-interface and constraint modes of the linearized system to reduce each subcomponent model. The geometric nonlinearity requires an additional cubic and quadratic polynomial function in the modal equations, and the nonlinear stiffness coefficients are determined by applying amore » series of static loads and using the finite-element code to compute the response. The geometrically nonlinear, reduced modal equations for each subcomponent are then coupled by satisfying compatibility and force equilibrium. This modal substructuring approach is an extension of the Craig–Bampton method and is readily applied to geometrically nonlinear models built directly within commercial finite-element packages. The efficiency of this new approach is demonstrated on two example problems: one that couples two geometrically nonlinear beams at a shared rotational degree of freedom, and another that couples an axial spring element to the axial degree of freedom of a geometrically nonlinear beam. The nonlinear normal modes of the assembled models are compared with those of a truth model to assess the accuracy of the novel modal substructuring approach.« less

Modal Substructuring of Geometrically Nonlinear Finite-Element Models

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

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

The efficiency of a modal substructuring method depends on the component modes used to reduce each subcomponent model. Methods such as Craig–Bampton have been used extensively to reduce linear finite-element models with thousands or even millions of degrees of freedom down orders of magnitude while maintaining acceptable accuracy. A novel reduction method is proposed here for geometrically nonlinear finite-element models using the fixed-interface and constraint modes of the linearized system to reduce each subcomponent model. The geometric nonlinearity requires an additional cubic and quadratic polynomial function in the modal equations, and the nonlinear stiffness coefficients are determined by applying amore » series of static loads and using the finite-element code to compute the response. The geometrically nonlinear, reduced modal equations for each subcomponent are then coupled by satisfying compatibility and force equilibrium. This modal substructuring approach is an extension of the Craig–Bampton method and is readily applied to geometrically nonlinear models built directly within commercial finite-element packages. The efficiency of this new approach is demonstrated on two example problems: one that couples two geometrically nonlinear beams at a shared rotational degree of freedom, and another that couples an axial spring element to the axial degree of freedom of a geometrically nonlinear beam. The nonlinear normal modes of the assembled models are compared with those of a truth model to assess the accuracy of the novel modal substructuring approach.« less

Finite element analysis and computer graphics visualization of flow around pitching and plunging airfoils

NASA Technical Reports Server (NTRS)

Bratanow, T.; Ecer, A.

1973-01-01

A general computational method for analyzing unsteady flow around pitching and plunging airfoils was developed. The finite element method was applied in developing an efficient numerical procedure for the solution of equations describing the flow around airfoils. The numerical results were employed in conjunction with computer graphics techniques to produce visualization of the flow. The investigation involved mathematical model studies of flow in two phases: (1) analysis of a potential flow formulation and (2) analysis of an incompressible, unsteady, viscous flow from Navier-Stokes equations.

New discretization and solution techniques for incompressible viscous flow problems

NASA Technical Reports Server (NTRS)

Gunzburger, M. D.; Nicolaides, R. A.; Liu, C. H.

1983-01-01

Several topics arising in the finite element solution of the incompressible Navier-Stokes equations are considered. Specifically, the question of choosing finite element velocity/pressure spaces is addressed, particularly from the viewpoint of achieving stable discretizations leading to convergent pressure approximations. The role of artificial viscosity in viscous flow calculations is studied, emphasizing work by several researchers for the anisotropic case. The last section treats the problem of solving the nonlinear systems of equations which arise from the discretization. Time marching methods and classical iterative techniques, as well as some modifications are mentioned.

A simple finite element method for the Stokes equations

DOE PAGES

Mu, Lin; Ye, Xiu

2017-03-21

The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate.

A simple finite element method for the Stokes equations

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

Mu, Lin; Ye, Xiu

The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate.

The Crank Nicolson Time Integrator for EMPHASIS.

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

McGregor, Duncan Alisdair Odum; Love, Edward; Kramer, Richard Michael Jack

2018-03-01

We investigate the use of implicit time integrators for finite element time domain approxi- mations of Maxwell's equations in vacuum. We discretize Maxwell's equations in time using Crank-Nicolson and in 3D space using compatible finite elements. We solve the system by taking a single step of Newton's method and inverting the Eddy-Current Schur complement allowing for the use of standard preconditioning techniques. This approach also generalizes to more complex material models that can include the Unsplit PML. We present verification results and demonstrate performance at CFL numbers up to 1000.

Finite-Strain Fractional-Order Viscoelastic (FOV) Material Models and Numerical Methods for Solving Them

NASA Technical Reports Server (NTRS)

Freed, Alan D.; Diethelm, Kai; Gray, Hugh R. (Technical Monitor)

2002-01-01

Fraction-order viscoelastic (FOV) material models have been proposed and studied in 1D since the 1930's, and were extended into three dimensions in the 1970's under the assumption of infinitesimal straining. It was not until 1997 that Drozdov introduced the first finite-strain FOV constitutive equations. In our presentation, we shall continue in this tradition by extending the standard, FOV, fluid and solid, material models introduced in 1971 by Caputo and Mainardi into 3D constitutive formula applicable for finite-strain analyses. To achieve this, we generalize both the convected and co-rotational derivatives of tensor fields to fractional order. This is accomplished by defining them first as body tensor fields and then mapping them into space as objective Cartesian tensor fields. Constitutive equations are constructed using both variants for fractional rate, and their responses are contrasted in simple shear. After five years of research and development, we now possess a basic suite of numerical tools necessary to study finite-strain FOV constitutive equations and their iterative refinement into a mature collection of material models. Numerical methods still need to be developed for efficiently solving fraction al-order integrals, derivatives, and differential equations in a finite element setting where such constitutive formulae would need to be solved at each Gauss point in each element of a finite model, which can number into the millions in today's analysis.

Incompressible Navier-Stokes and parabolized Navier-Stokes solution procedures and computational techniques

NASA Technical Reports Server (NTRS)

Rubin, S. G.

1982-01-01

Recent developments with finite-difference techniques are emphasized. The quotation marks reflect the fact that any finite discretization procedure can be included in this category. Many so-called finite element collocation and galerkin methods can be reproduced by appropriate forms of the differential equations and discretization formulas. Many of the difficulties encountered in early Navier-Stokes calculations were inherent not only in the choice of the different equations (accuracy), but also in the method of solution or choice of algorithm (convergence and stability, in the manner in which the dependent variables or discretized equations are related (coupling), in the manner that boundary conditions are applied, in the manner that the coordinate mesh is specified (grid generation), and finally, in recognizing that for many high Reynolds number flows not all contributions to the Navier-Stokes equations are necessarily of equal importance (parabolization, preferred direction, pressure interaction, asymptotic and mathematical character). It is these elements that are reviewed. Several Navier-Stokes and parabolized Navier-Stokes formulations are also presented.

Computation of three-dimensional nozzle-exhaust flow fields with the GIM code

NASA Technical Reports Server (NTRS)

Spradley, L. W.; Anderson, P. G.

1978-01-01

A methodology is introduced for constructing numerical analogs of the partial differential equations of continuum mechanics. A general formulation is provided which permits classical finite element and many of the finite difference methods to be derived directly. The approach, termed the General Interpolants Method (GIM), can combined the best features of finite element and finite difference methods. A quasi-variational procedure is used to formulate the element equations, to introduce boundary conditions into the method and to provide a natural assembly sequence. A derivation is given in terms of general interpolation functions from this procedure. Example computations for transonic and supersonic flows in two and three dimensions are given to illustrate the utility of GIM. A three-dimensional nozzle-exhaust flow field is solved including interaction with the freestream and a coupled treatment of the shear layer. Potential applications of the GIM code to a variety of computational fluid dynamics problems is then discussed in terms of existing capability or by extension of the methodology.

Domain decomposition methods for nonconforming finite element spaces of Lagrange-type

NASA Technical Reports Server (NTRS)

Cowsar, Lawrence C.

1993-01-01

In this article, we consider the application of three popular domain decomposition methods to Lagrange-type nonconforming finite element discretizations of scalar, self-adjoint, second order elliptic equations. The additive Schwarz method of Dryja and Widlund, the vertex space method of Smith, and the balancing method of Mandel applied to nonconforming elements are shown to converge at a rate no worse than their applications to the standard conforming piecewise linear Galerkin discretization. Essentially, the theory for the nonconforming elements is inherited from the existing theory for the conforming elements with only modest modification by constructing an isomorphism between the nonconforming finite element space and a space of continuous piecewise linear functions.

A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule

NASA Astrophysics Data System (ADS)

Ying, Jinyong; Xie, Dexuan

2015-10-01

The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model for calculating electrostatics of ionic solvated biomolecule. In this paper, a new finite element and finite difference hybrid method is presented to solve PBE efficiently based on a special seven-overlapped box partition with one central box containing the solute region and surrounded by six neighboring boxes. In particular, an efficient finite element solver is applied to the central box while a fast preconditioned conjugate gradient method using a multigrid V-cycle preconditioning is constructed for solving a system of finite difference equations defined on a uniform mesh of each neighboring box. Moreover, the PBE domain, the box partition, and an interface fitted tetrahedral mesh of the central box can be generated adaptively for a given PQR file of a biomolecule. This new hybrid PBE solver is programmed in C, Fortran, and Python as a software tool for predicting electrostatics of a biomolecule in a symmetric 1:1 ionic solvent. Numerical results on two test models with analytical solutions and 12 proteins validate this new software tool, and demonstrate its high performance in terms of CPU time and memory usage.

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.

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.

Robust Hybrid Finite Element Methods for Antennas and Microwave Circuits

NASA Technical Reports Server (NTRS)

Gong, J.; Volakis, John L.

1996-01-01

One of the primary goals in this dissertation is concerned with the development of robust hybrid finite element-boundary integral (FE-BI) techniques for modeling and design of conformal antennas of arbitrary shape. Both the finite element and integral equation methods will be first overviewed in this chapter with an emphasis on recently developed hybrid FE-BI methodologies for antennas, microwave and millimeter wave applications. The structure of the dissertation is then outlined. We conclude the chapter with discussions of certain fundamental concepts and methods in electromagnetics, which are important to this study.

Finite element modeling of truss structures with frequency-dependent material damping

NASA Technical Reports Server (NTRS)

Lesieutre, George A.

1991-01-01

A physically motivated modelling technique for structural dynamic analysis that accommodates frequency dependent material damping was developed. Key features of the technique are the introduction of augmenting thermodynamic fields (AFT) to interact with the usual mechanical displacement field, and the treatment of the resulting coupled governing equations using finite element analysis methods. The AFT method is fully compatible with current structural finite element analysis techniques. The method is demonstrated in the dynamic analysis of a 10-bay planar truss structure, a structure representative of those contemplated for use in future space systems.

A survey of the core-congruential formulation for geometrically nonlinear TL finite elements

NASA Technical Reports Server (NTRS)

Felippa, Carlos A.; Crivelli, Luis A.; Haugen, Bjorn

1994-01-01

This article presents a survey of the core-congruential formulation (CCF) for geometrically nonlinear mechanical finite elements based on the total Lagrangian (TL) kinematic description. Although the key ideas behind the CCF can be traced back to Rajasekaran and Murray in 1973, it has not subsequently received serious attention. The CCF is distinguished by a two-phase development of the finite element stiffness equations. The initial phase developed equations for individual particles. These equations are expressed in terms of displacement gradients as degrees of freedom. The second phase involves congruential-type transformations that eventually binds the element particles of an individual element in terms of its node-displacement degrees of freedom. Two versions of the CCF, labeled direct and generalized, are distinguished. The direct CCF (DCCF) is first described in general form and then applied to the derivation of geometrically nonlinear bar, and plane stress elements using the Green-Lagrange strain measure. The more complex generalized CCF (GCCF) is described and applied to the derivation of 2D and 3D Timoshenko beam elements. Several advantages of the CCF, notably the physically clean separation of material and geometric stiffnesses, and its independence with respect to the ultimate choice of shape functions and element degrees of freedom, are noted. Application examples involving very large motions solved with the 3D beam element display the range of applicability of this formulation, which transcends the kinematic limitations commonly attributed to the TL description.

Final Report of the Project "From the finite element method to the virtual element method"

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

Manzini, Gianmarco; Gyrya, Vitaliy

The Finite Element Method (FEM) is a powerful numerical tool that is being used in a large number of engineering applications. The FEM is constructed on triangular/tetrahedral and quadrilateral/hexahedral meshes. Extending the FEM to general polygonal/polyhedral meshes in straightforward way turns out to be extremely difficult and leads to very complex and computationally expensive schemes. The reason for this failure is that the construction of the basis functions on elements with a very general shape is a non-trivial and complex task. In this project we developed a new family of numerical methods, dubbed the Virtual Element Method (VEM) for themore » numerical approximation of partial differential equations (PDE) of elliptic type suitable to polygonal and polyhedral unstructured meshes. We successfully formulated, implemented and tested these methods and studied both theoretically and numerically their stability, robustness and accuracy for diffusion problems, convection-reaction-diffusion problems, the Stokes equations and the biharmonic equations.« less

HFEM3D

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

Weiss, Chester J

Software solves the three-dimensional Poisson equation div(k(grad(u)) = f, by the finite element method for the case when material properties, k, are distributed over hierarchy of edges, facets and tetrahedra in the finite element mesh. Method is described in Weiss, CJ, Finite element analysis for model parameters distributed on a hierarchy of geometric simplices, Geophysics, v82, E155-167, doi:10.1190/GEO2017-0058.1 (2017). A standard finite element method for solving Poisson’s equation is augmented by including in the 3D stiffness matrix additional 2D and 1D stiffness matrices representing the contributions from material properties associated with mesh faces and edges, respectively. The resulting linear systemmore » is solved iteratively using the conjugate gradient method with Jacobi preconditioning. To minimize computer storage for program execution, the linear solver computes matrix-vector contractions element-by-element over the mesh, without explicit storage of the global stiffness matrix. Program output vtk compliant for visualization and rendering by 3rd party software. Program uses dynamic memory allocation and as such there are no hard limits on problem size outside of those imposed by the operating system and configuration on which the software is run. Dimension, N, of the finite element solution vector is constrained by the the addressable space in 32-vs-64 bit operating systems. Total storage requirements for the problem. Total working space required for the program is approximately 13*N double precision words.« less

A split band-Cholesky equation solving strategy for finite element analysis of transient field problems. [in fluid mechanics

NASA Technical Reports Server (NTRS)

Cooke, C. H.

1978-01-01

The paper describes the split-Cholesky strategy for banded matrices arising from the large systems of equations in certain fluid mechanics problems. The basic idea is that for a banded matrix the computation can be carried out in pieces, with only a small portion of the matrix residing in core. Mesh considerations are discussed by demonstrating the manner in which the assembly of finite element equations proceeds for linear trial functions on a triangular mesh. The FORTRAN code which implements the out-of-core decomposition strategy for banded symmetric positive definite matrices (mass matrices) of a coupled initial value problem is given.

Exact finite elements for conduction and convection

NASA Technical Reports Server (NTRS)

Thornton, E. A.; Dechaumphai, P.; Tamma, K. K.

1981-01-01

An appproach for developing exact one dimensional conduction-convection finite elements is presented. Exact interpolation functions are derived based on solutions to the governing differential equations by employing a nodeless parameter. Exact interpolation functions are presented for combined heat transfer in several solids of different shapes, and for combined heat transfer in a flow passage. Numerical results demonstrate that exact one dimensional elements offer advantages over elements based on approximate interpolation functions. Previously announced in STAR as N81-31507

DOUAR: A new three-dimensional creeping flow numerical model for the solution of geological problems

NASA Astrophysics Data System (ADS)

Braun, Jean; Thieulot, Cédric; Fullsack, Philippe; DeKool, Marthijn; Beaumont, Christopher; Huismans, Ritske

2008-12-01

We present a new finite element code for the solution of the Stokes and energy (or heat transport) equations that has been purposely designed to address crustal-scale to mantle-scale flow problems in three dimensions. Although it is based on an Eulerian description of deformation and flow, the code, which we named DOUAR ('Earth' in Breton language), has the ability to track interfaces and, in particular, the free surface, by using a dual representation based on a set of particles placed on the interface and the computation of a level set function on the nodes of the finite element grid, thus ensuring accuracy and efficiency. The code also makes use of a new method to compute the dynamic Delaunay triangulation connecting the particles based on non-Euclidian, curvilinear measure of distance, ensuring that the density of particles remains uniform and/or dynamically adapted to the curvature of the interface. The finite element discretization is based on a non-uniform, yet regular octree division of space within a unit cube that allows efficient adaptation of the finite element discretization, i.e. in regions of strong velocity gradient or high interface curvature. The finite elements are cubes (the leaves of the octree) in which a q1- p0 interpolation scheme is used. Nodal incompatibilities across faces separating elements of differing size are dealt with by introducing linear constraints among nodal degrees of freedom. Discontinuities in material properties across the interfaces are accommodated by the use of a novel method (which we called divFEM) to integrate the finite element equations in which the elemental volume is divided by a local octree to an appropriate depth (resolution). A variety of rheologies have been implemented including linear, non-linear and thermally activated creep and brittle (or plastic) frictional deformation. A simple smoothing operator has been defined to avoid checkerboard oscillations in pressure that tend to develop when using a highly irregular octree discretization and the tri-linear (or q1- p0) finite element. A three-dimensional cloud of particles is used to track material properties that depend on the integrated history of deformation (the integrated strain, for example); its density is variable and dynamically adapted to the computed flow. The large system of algebraic equations that results from the finite element discretization and linearization of the basic partial differential equations is solved using a multi-frontal massively parallel direct solver that can efficiently factorize poorly conditioned systems resulting from the highly non-linear rheology and the presence of the free surface. The code is almost entirely parallelized. We present example results including the onset of a Rayleigh-Taylor instability, the indentation of a rigid-plastic material and the formation of a fold beneath a free eroding surface, that demonstrate the accuracy, efficiency and appropriateness of the new code to solve complex geodynamical problems in three dimensions.

Energy Finite Element Analysis Developments for Vibration Analysis of Composite Aircraft Structures

NASA Technical Reports Server (NTRS)

Vlahopoulos, Nickolas; Schiller, Noah H.

2011-01-01

The Energy Finite Element Analysis (EFEA) has been utilized successfully for modeling complex structural-acoustic systems with isotropic structural material properties. In this paper, a formulation for modeling structures made out of composite materials is presented. An approach based on spectral finite element analysis is utilized first for developing the equivalent material properties for the composite material. These equivalent properties are employed in the EFEA governing differential equations for representing the composite materials and deriving the element level matrices. The power transmission characteristics at connections between members made out of non-isotropic composite material are considered for deriving suitable power transmission coefficients at junctions of interconnected members. These coefficients are utilized for computing the joint matrix that is needed to assemble the global system of EFEA equations. The global system of EFEA equations is solved numerically and the vibration levels within the entire system can be computed. The new EFEA formulation for modeling composite laminate structures is validated through comparison to test data collected from a representative composite aircraft fuselage that is made out of a composite outer shell and composite frames and stiffeners. NASA Langley constructed the composite cylinder and conducted the test measurements utilized in this work.

Analysis and Testing of Plates with Piezoelectric Sensors and Actuators

NASA Technical Reports Server (NTRS)

Bevan, Jeffrey S.

1998-01-01

Piezoelectric material inherently possesses coupling between electrostatics and structural dynamics. Utilizing linear piezoelectric theory results in an intrinsically coupled pair of piezoelectric constitutive equations. One equation describes the direct piezoelectric effect where strains produce an electric field and the other describes the converse effect where an applied electrical field produces strain. The purpose of this study is to compare finite element analysis and experiments of a thin plate with bonded piezoelectric material. Since an isotropic plate in combination with a thin piezoelectric layer constitutes a special case of a laminated composite, the classical laminated plate theory is used in the formulation to accommodated generic laminated composite panels with multiple bonded and embedded piezoelectric layers. Additionally, the von Karman large deflection plate theory is incorporated. The formulation results in laminate constitutive equations that are amiable to the inclusion of the piezoelectric constitutive equations yielding in a fully electro-mechanically coupled composite laminate. Using the finite element formulation, the governing differential equations of motion of a composite laminate with embedded piezoelectric layers are derived. The finite element model not only considers structural degrees of freedom (d.o.f.) but an additional electrical d.o.f. for each piezoelectric layer. Comparison between experiment and numerical prediction is performed by first treating the piezoelectric as a sensor and then again treating it as an actuator. To assess the piezoelectric layer as a sensor, various uniformly distributed pressure loads were simulated in the analysis and the corresponding generated voltages were calculated using both linear and nonlinear finite element analyses. Experiments were carried out by applying the same uniformly distributed loads and measuring the resulting generated voltages and corresponding maximum plate deflections. It is found that a highly nonlinear relationship exists between maximum deflection and voltage versus pressure loading. In order to assess comparisons of predicted and measured piezoelectric actuation, sinusoidal excitation voltages are simulated/applied and maximum deflections are calculated/measured. The maximum deflection as a function of time was determined using the linear finite elements analysis. Good correlation between prediction and measurement was achieved in all cases.

A Finite Element Theory for Predicting the Attenuation of Extended-Reacting Liners

NASA Technical Reports Server (NTRS)

Watson, W. R.; Jones, M. G.

2009-01-01

A non-modal finite element theory for predicting the attenuation of an extended-reacting liner containing a porous facesheet and located in a no-flow duct is presented. The mathematical approach is to solve separate wave equations in the liner and duct airway and to couple these two solutions by invoking kinematic constraints at the facesheet that are consistent with a continuum theory of fluid motion. Given the liner intrinsic properties, a weak Galerkin finite element formulation with cubic polynomial basis functions is used as the basis for generating a discrete system of acoustic equations that are solved to obtain the coupled acoustic field. A state-of-the-art, asymmetric, parallel, sparse equation solver is implemented that allows tens of thousands of grid points to be analyzed. A grid refinement study is presented to show that the predicted attenuation converges. Excellent comparison of the numerically predicted attenuation to that of a mode theory (using a Haynes 25 metal foam liner) is used to validate the computational approach. Simulations are also presented for fifteen porous plate, extended-reacting liners. The construction of some of the porous plate liners suggest that they should behave as resonant liners while the construction of others suggest that they should behave as broadband attenuators. In each case the finite element theory is observed to predict the proper attenuation trend.

Optimal control penalty finite elements - Applications to integrodifferential equations

NASA Astrophysics Data System (ADS)

Chung, T. J.

The application of the optimal-control/penalty finite-element method to the solution of integrodifferential equations in radiative-heat-transfer problems (Chung et al.; Chung and Kim, 1982) is discussed and illustrated. The nonself-adjointness of the convective terms in the governing equations is treated by utilizing optimal-control cost functions and employing penalty functions to constrain auxiliary equations which permit the reduction of second-order derivatives to first order. The OCPFE method is applied to combined-mode heat transfer by conduction, convection, and radiation, both without and with scattering and viscous dissipation; the results are presented graphically and compared to those obtained by other methods. The OCPFE method is shown to give good results in cases where standard Galerkin FE fail, and to facilitate the investigation of scattering and dissipation effects.

High-order solution methods for grey discrete ordinates thermal radiative transfer

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

Maginot, Peter G., E-mail: maginot1@llnl.gov; Ragusa, Jean C., E-mail: jean.ragusa@tamu.edu; Morel, Jim E., E-mail: morel@tamu.edu

This work presents a solution methodology for solving the grey radiative transfer equations that is both spatially and temporally more accurate than the canonical radiative transfer solution technique of linear discontinuous finite element discretization in space with implicit Euler integration in time. We solve the grey radiative transfer equations by fully converging the nonlinear temperature dependence of the material specific heat, material opacities, and Planck function. The grey radiative transfer equations are discretized in space using arbitrary-order self-lumping discontinuous finite elements and integrated in time with arbitrary-order diagonally implicit Runge–Kutta time integration techniques. Iterative convergence of the radiation equation ismore » accelerated using a modified interior penalty diffusion operator to precondition the full discrete ordinates transport operator.« less

High-order solution methods for grey discrete ordinates thermal radiative transfer

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

Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E.

This paper presents a solution methodology for solving the grey radiative transfer equations that is both spatially and temporally more accurate than the canonical radiative transfer solution technique of linear discontinuous finite element discretization in space with implicit Euler integration in time. We solve the grey radiative transfer equations by fully converging the nonlinear temperature dependence of the material specific heat, material opacities, and Planck function. The grey radiative transfer equations are discretized in space using arbitrary-order self-lumping discontinuous finite elements and integrated in time with arbitrary-order diagonally implicit Runge–Kutta time integration techniques. Iterative convergence of the radiation equation ismore » accelerated using a modified interior penalty diffusion operator to precondition the full discrete ordinates transport operator.« less

High-order solution methods for grey discrete ordinates thermal radiative transfer

DOE PAGES

Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E.

2016-09-29

This paper presents a solution methodology for solving the grey radiative transfer equations that is both spatially and temporally more accurate than the canonical radiative transfer solution technique of linear discontinuous finite element discretization in space with implicit Euler integration in time. We solve the grey radiative transfer equations by fully converging the nonlinear temperature dependence of the material specific heat, material opacities, and Planck function. The grey radiative transfer equations are discretized in space using arbitrary-order self-lumping discontinuous finite elements and integrated in time with arbitrary-order diagonally implicit Runge–Kutta time integration techniques. Iterative convergence of the radiation equation ismore » accelerated using a modified interior penalty diffusion operator to precondition the full discrete ordinates transport operator.« less

A new parallel-vector finite element analysis software on distributed-memory computers

NASA Technical Reports Server (NTRS)

Qin, Jiangning; Nguyen, Duc T.

1993-01-01

A new parallel-vector finite element analysis software package MPFEA (Massively Parallel-vector Finite Element Analysis) is developed for large-scale structural analysis on massively parallel computers with distributed-memory. MPFEA is designed for parallel generation and assembly of the global finite element stiffness matrices as well as parallel solution of the simultaneous linear equations, since these are often the major time-consuming parts of a finite element analysis. Block-skyline storage scheme along with vector-unrolling techniques are used to enhance the vector performance. Communications among processors are carried out concurrently with arithmetic operations to reduce the total execution time. Numerical results on the Intel iPSC/860 computers (such as the Intel Gamma with 128 processors and the Intel Touchstone Delta with 512 processors) are presented, including an aircraft structure and some very large truss structures, to demonstrate the efficiency and accuracy of MPFEA.

Heat transfer model and finite element formulation for simulation of selective laser melting

NASA Astrophysics Data System (ADS)

Roy, Souvik; Juha, Mario; Shephard, Mark S.; Maniatty, Antoinette M.

2017-10-01

A novel approach and finite element formulation for modeling the melting, consolidation, and re-solidification process that occurs in selective laser melting additive manufacturing is presented. Two state variables are introduced to track the phase (melt/solid) and the degree of consolidation (powder/fully dense). The effect of the consolidation on the absorption of the laser energy into the material as it transforms from a porous powder to a dense melt is considered. A Lagrangian finite element formulation, which solves the governing equations on the unconsolidated reference configuration is derived, which naturally considers the effect of the changing geometry as the powder melts without needing to update the simulation domain. The finite element model is implemented into a general-purpose parallel finite element solver. Results are presented comparing to experimental results in the literature for a single laser track with good agreement. Predictions for a spiral laser pattern are also shown.

The application of super wavelet finite element on temperature-pressure coupled field simulation of LPG tank under jet fire

NASA Astrophysics Data System (ADS)

Zhao, Bin

2015-02-01

Temperature-pressure coupled field analysis of liquefied petroleum gas (LPG) tank under jet fire can offer theoretical guidance for preventing the fire accidents of LPG tank, the application of super wavelet finite element on it is studied in depth. First, review of related researches on heat transfer analysis of LPG tank under fire and super wavelet are carried out. Second, basic theory of super wavelet transform is studied. Third, the temperature-pressure coupled model of gas phase and liquid LPG under jet fire is established based on the equation of state, the VOF model and the RNG k-ɛ model. Then the super wavelet finite element formulation is constructed using the super wavelet scale function as interpolating function. Finally, the simulation is carried out, and results show that the super wavelet finite element method has higher computing precision than wavelet finite element method.

Coupling finite element and spectral methods: First results

NASA Technical Reports Server (NTRS)

Bernardi, Christine; Debit, Naima; Maday, Yvon

1987-01-01

A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares, a finite element approximation is used on the first square and a spectral discretization is used on the second one. Two kinds of matching conditions on the interface are presented and compared. In both cases, error estimates are proved.

Finite element solution of lubrication problems

NASA Technical Reports Server (NTRS)

Reddi, M. M.

1971-01-01

A variational formulation of the transient lubrication problem is presented and the corresponding finite element equations derived for three and six point triangles, and, four and eight point quadrilaterals. Test solutions for a one dimensional slider bearing used in validating the computer program are given. Utility of the method is demonstrated by a solution of the shrouded step bearing.

The Overshoot Phenomenon in Geodynamics Codes

NASA Astrophysics Data System (ADS)

Kommu, R. K.; Heien, E. M.; Kellogg, L. H.; Bangerth, W.; Heister, T.; Studley, E. H.

2013-12-01

The overshoot phenomenon is a common occurrence in numerical software when a continuous function on a finite dimensional discretized space is used to approximate a discontinuous jump, in temperature and material concentration, for example. The resulting solution overshoots, and undershoots, the discontinuous jump. Numerical simulations play an extremely important role in mantle convection research. This is both due to the strong temperature and stress dependence of viscosity and also due to the inaccessibility of deep earth. Under these circumstances, it is essential that mantle convection simulations be extremely accurate and reliable. CitcomS and ASPECT are two finite element based mantle convection simulations developed and maintained by the Computational Infrastructure for Geodynamics. CitcomS is a finite element based mantle convection code that is designed to run on multiple high-performance computing platforms. ASPECT, an adaptive mesh refinement (AMR) code built on the Deal.II library, is also a finite element based mantle convection code that scales well on various HPC platforms. CitcomS and ASPECT both exhibit the overshoot phenomenon. One attempt at controlling the overshoot uses the Entropy Viscosity method, which introduces an artificial diffusion term in the energy equation of mantle convection. This artificial diffusion term is small where the temperature field is smooth. We present results from CitcomS and ASPECT that quantify the effect of the Entropy Viscosity method in reducing the overshoot phenomenon. In the discontinuous Galerkin (DG) finite element method, the test functions used in the method are continuous within each element but are discontinuous across inter-element boundaries. The solution space in the DG method is discontinuous. FEniCS is a collection of free software tools that automate the solution of differential equations using finite element methods. In this work we also present results from a finite element mantle convection simulation implemented in FEniCS that investigates the effect of using DG elements in reducing the overshoot problem.

Energy Finite Element Analysis for Computing the High Frequency Vibration of the Aluminum Testbed Cylinder and Correlating the Results to Test Data

NASA Technical Reports Server (NTRS)

Vlahopoulos, Nickolas

2005-01-01

The Energy Finite Element Analysis (EFEA) is a finite element based computational method for high frequency vibration and acoustic analysis. The EFEA solves with finite elements governing differential equations for energy variables. These equations are developed from wave equations. Recently, an EFEA method for computing high frequency vibration of structures either in vacuum or in contact with a dense fluid has been presented. The presence of fluid loading has been considered through added mass and radiation damping. The EFEA developments were validated by comparing EFEA results to solutions obtained by very dense conventional finite element models and solutions from classical techniques such as statistical energy analysis (SEA) and the modal decomposition method for bodies of revolution. EFEA results have also been compared favorably with test data for the vibration and the radiated noise generated by a large scale submersible vehicle. The primary variable in EFEA is defined as the time averaged over a period and space averaged over a wavelength energy density. A joint matrix computed from the power transmission coefficients is utilized for coupling the energy density variables across any discontinuities, such as change of plate thickness, plate/stiffener junctions etc. When considering the high frequency vibration of a periodically stiffened plate or cylinder, the flexural wavelength is smaller than the interval length between two periodic stiffeners, therefore the stiffener stiffness can not be smeared by computing an equivalent rigidity for the plate or cylinder. The periodic stiffeners must be regarded as coupling components between periodic units. In this paper, Periodic Structure (PS) theory is utilized for computing the coupling joint matrix and for accounting for the periodicity characteristics.

Modeling of Structural-Acoustic Interaction Using Coupled FE/BE Method and Control of Interior Acoustic Pressure Using Piezoelectric Actuators

NASA Technical Reports Server (NTRS)

Mei, Chuh; Shi, Yacheng

1997-01-01

A coupled finite element (FE) and boundary element (BE) approach is presented to model full coupled structural/acoustic/piezoelectric systems. The dual reciprocity boundary element method is used so that the natural frequencies and mode shapes of the coupled system can be obtained, and to extend this approach to time dependent problems. The boundary element method is applied to interior acoustic domains, and the results are very accurate when compared with limited exact solutions. Structural-acoustic problems are then analyzed with the coupled finite element/boundary element method, where the finite element method models the structural domain and the boundary element method models the acoustic domain. Results for a system consisting of an isotropic panel and a cubic cavity are in good agreement with exact solutions and experiment data. The response of a composite panel backed cavity is then obtained. The results show that the mass and stiffness of piezoelectric layers have to be considered. The coupled finite element and boundary element equations are transformed into modal coordinates, which is more convenient for transient excitation. Several transient problems are solved based on this formulation. Two control designs, a linear quadratic regulator (LQR) and a feedforward controller, are applied to reduce the acoustic pressure inside the cavity based on the equations in modal coordinates. The results indicate that both controllers can reduce the interior acoustic pressure and the plate deflection.

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

Aeroelastic Stability of Rotor Blades Using Finite Element Analysis

NASA Technical Reports Server (NTRS)

Chopra, I.; Sivaneri, N.

1982-01-01

The flutter stability of flap bending, lead-lag bending, and torsion of helicopter rotor blades in hover is investigated using a finite element formulation based on Hamilton's principle. The blade is divided into a number of finite elements. Quasi-steady strip theory is used to evaluate the aerodynamic loads. The nonlinear equations of motion are solved for steady-state blade deflections through an iterative procedure. The equations of motion are linearized assuming blade motion to be a small perturbation about the steady deflected shape. The normal mode method based on the coupled rotating natural modes is used to reduce the number of equations in the flutter analysis. First the formulation is applied to single-load-path blades (articulated and hingeless blades). Numerical results show very good agreement with existing results obtained using the modal approach. The second part of the application concerns multiple-load-path blades, i.e. bearingless blades. Numerical results are presented for several analytical models of the bearingless blade. Results are also obtained using an equivalent beam approach wherein a bearingless blade is modelled as a single beam with equivalent properties. Results show the equivalent beam model.

Numerical modeling of guided ultrasonic waves generated and received by piezoelectric wafer in a Delaminated composite beam

NASA Astrophysics Data System (ADS)

Xu, G. D.; Xu, B. Q.; Xu, C. G.; Luo, Y.

2017-05-01

A spectral finite element method (SFEM) is developed to analyze guided ultrasonic waves in a delaminated composite beam excited and received by a pair of surface-bonded piezoelectric wafers. The displacements of the composite beam and the piezoelectric wafer are represented by Timoshenko beam and Euler Bernoulli theory respectively. The linear piezoelectricity is used to model the electrical-mechanical coupling between the piezoelectric wafer and the beam. The coupled governing equations and the boundary conditions in time domain are obtained by using the Hamilton's principle, and then the SFEM are formulated by transforming the coupled governing equations into frequency domain via the discrete Fourier transform. The guided waves are analyzed while the interaction of waves with delamination is also discussed. The elements needed in SFEM is far fewer than those for finite element method (FEM), which result in a much faster solution speed in this study. The high accuracy of the present SFEM is verified by comparing with the finite element results.

Fiber-reinforced materials: finite elements for the treatment of the inextensibility constraint

NASA Astrophysics Data System (ADS)

Auricchio, Ferdinando; Scalet, Giulia; Wriggers, Peter

2017-12-01

The present paper proposes a numerical framework for the analysis of problems involving fiber-reinforced anisotropic materials. Specifically, isotropic linear elastic solids, reinforced by a single family of inextensible fibers, are considered. The kinematic constraint equation of inextensibility in the fiber direction leads to the presence of an undetermined fiber stress in the constitutive equations. To avoid locking-phenomena in the numerical solution due to the presence of the constraint, mixed finite elements based on the Lagrange multiplier, perturbed Lagrangian, and penalty method are proposed. Several boundary-value problems under plane strain conditions are solved and numerical results are compared to analytical solutions, whenever the derivation is possible. The performed simulations allow to assess the performance of the proposed finite elements and to discuss several features of the developed formulations concerning the effective approximation for the displacement and fiber stress fields, mesh convergence, and sensitivity to penalty parameters.

An internal variable constitutive model for the large deformation of metals at high temperatures

NASA Technical Reports Server (NTRS)

Brown, Stuart; Anand, Lallit

1988-01-01

The advent of large deformation finite element methodologies is beginning to permit the numerical simulation of hot working processes whose design until recently has been based on prior industrial experience. Proper application of such finite element techniques requires realistic constitutive equations which more accurately model material behavior during hot working. A simple constitutive model for hot working is the single scalar internal variable model for isotropic thermal elastoplasticity proposed by Anand. The model is recalled and the specific scalar functions, for the equivalent plastic strain rate and the evolution equation for the internal variable, presented are slight modifications of those proposed by Anand. The modified functions are better able to represent high temperature material behavior. The monotonic constant true strain rate and strain rate jump compression experiments on a 2 percent silicon iron is briefly described. The model is implemented in the general purpose finite element program ABAQUS.

Novel numerical techniques for magma dynamics

NASA Astrophysics Data System (ADS)

Rhebergen, S.; Katz, R. F.; Wathen, A.; Alisic, L.; Rudge, J. F.; Wells, G.

2013-12-01

We discuss the development of finite element techniques and solvers for magma dynamics computations. These are implemented within the FEniCS framework. This approach allows for user-friendly, expressive, high-level code development, but also provides access to powerful, scalable numerical solvers and a large family of finite element discretisations. With the recent addition of dolfin-adjoint, FeniCS supports automated adjoint and tangent-linear models, enabling the rapid development of Generalised Stability Analysis. The ability to easily scale codes to three dimensions with large meshes, and/or to apply intricate adjoint calculations means that efficiency of the numerical algorithms is vital. We therefore describe our development and analysis of preconditioners designed specifically for finite element discretizations of equations governing magma dynamics. The preconditioners are based on Elman-Silvester-Wathen methods for the Stokes equation, and we extend these to flows with compaction. Our simulations are validated by comparison of results with laboratory experiments on partially molten aggregates.

Finite element modeling of frictionally restrained composite interfaces

NASA Technical Reports Server (NTRS)

Ballarini, Roberto; Ahmed, Shamim

1989-01-01

The use of special interface finite elements to model frictional restraint in composite interfaces is described. These elements simulate Coulomb friction at the interface, and are incorporated into a standard finite element analysis of a two-dimensional isolated fiber pullout test. Various interfacial characteristics, such as the distribution of stresses at the interface, the extent of slip and delamination, load diffusion from fiber to matrix, and the amount of fiber extraction or depression are studied for different friction coefficients. The results are compared to those obtained analytically using a singular integral equation approach, and those obtained by assuming a constant interface shear strength. The usefulness of these elements in micromechanical modeling of fiber-reinforced composite materials is highlighted.

A new family of stable elements for the Stokes problem based on a mixed Galerkin/least-squares finite element formulation

NASA Technical Reports Server (NTRS)

Franca, Leopoldo P.; Loula, Abimael F. D.; Hughes, Thomas J. R.; Miranda, Isidoro

1989-01-01

Adding to the classical Hellinger-Reissner formulation, a residual form of the equilibrium equation, a new Galerkin/least-squares finite element method is derived. It fits within the framework of a mixed finite element method and is stable for rather general combinations of stress and velocity interpolations, including equal-order discontinuous stress and continuous velocity interpolations which are unstable within the Galerkin approach. Error estimates are presented based on a generalization of the Babuska-Brezzi theory. Numerical results (not presented herein) have confirmed these estimates as well as the good accuracy and stability of the method.

A study of the response of nonlinear springs

NASA Technical Reports Server (NTRS)

Hyer, M. W.; Knott, T. W.; Johnson, E. R.

1991-01-01

The various phases to developing a methodology for studying the response of a spring-reinforced arch subjected to a point load are discussed. The arch is simply supported at its ends with both the spring and the point load assumed to be at midspan. The spring is present to off-set the typical snap through behavior normally associated with arches, and to provide a structure that responds with constant resistance over a finite displacement. The various phases discussed consist of the following: (1) development of the closed-form solution for the shallow arch case; (2) development of a finite difference analysis to study (shallow) arches; and (3) development of a finite element analysis for studying more general shallow and nonshallow arches. The two numerical analyses rely on a continuation scheme to move the solution past limit points, and to move onto bifurcated paths, both characteristics being common to the arch problem. An eigenvalue method is used for a continuation scheme. The finite difference analysis is based on a mixed formulation (force and displacement variables) of the governing equations. The governing equations for the mixed formulation are in first order form, making the finite difference implementation convenient. However, the mixed formulation is not well-suited for the eigenvalue continuation scheme. This provided the motivation for the displacement based finite element analysis. Both the finite difference and the finite element analyses are compared with the closed form shallow arch solution. Agreement is excellent, except for the potential problems with the finite difference analysis and the continuation scheme. Agreement between the finite element analysis and another investigator's numerical analysis for deep arches is also good.

Advances and future directions of research on spectral methods

NASA Technical Reports Server (NTRS)

Patera, A. T.

1986-01-01

Recent advances in spectral methods are briefly reviewed and characterized with respect to their convergence and computational complexity. Classical finite element and spectral approaches are then compared, and spectral element (or p-type finite element) approximations are introduced. The method is applied to the full Navier-Stokes equations, and examples are given of the application of the technique to several transitional flows. Future directions of research in the field are outlined.

Thermal-stress analysis for a wood composite blade

NASA Technical Reports Server (NTRS)

Fu, K. C.; Harb, A.

1984-01-01

A thermal-stress analysis of a wind turbine blade made of wood composite material is reported. First, the governing partial differential equation on heat conduction is derived, then, a finite element procedure using variational approach is developed for the solution of the governing equation. Thus, the temperature distribution throughout the blade is determined. Next, based on the temperature distribution, a finite element procedure using potential energy approach is applied to determine the thermal-stress distribution. A set of results is obtained through the use of a computer, which is considered to be satisfactory. All computer programs are contained in the report.

Chemorheology of reactive systems: Finite element analysis

NASA Technical Reports Server (NTRS)

Douglas, C.; Roylance, D.

1982-01-01

The equations which govern the nonisothermal flow of reactive fluids are outlined, and the means by which finite element analysis is used to solve these equations for the sort of arbitrary boundary conditions encountered in industrial practice are described. The performance of the computer code is illustrated by several trial problems, selected more for their value in providing insight to polymer processing flows than as practical production problems. Although a good deal remains to be learned as to the performance and proper use of this numerical technique, it is undeniably useful in providing better understanding of today's complicated polymer processing problems.

New discretization and solution techniques for incompressible viscous flow problems

NASA Technical Reports Server (NTRS)

Gunzburger, M. D.; Nicolaides, R. A.; Liu, C. H.

1983-01-01

This paper considers several topics arising in the finite element solution of the incompressible Navier-Stokes equations. Specifically, the question of choosing finite element velocity/pressure spaces is addressed, particularly from the viewpoint of achieving stable discretizations leading to convergent pressure approximations. Following this, the role of artificial viscosity in viscous flow calculations is studied, emphasizing recent work by several researchers for the anisotropic case. The last section treats the problem of solving the nonlinear systems of equations which arise from the discretization. Time marching methods and classical iterative techniques, as well as some recent modifications are mentioned.

Dynamic analysis of suspension cable based on vector form intrinsic finite element method

NASA Astrophysics Data System (ADS)

Qin, Jian; Qiao, Liang; Wan, Jiancheng; Jiang, Ming; Xia, Yongjun

2017-10-01

A vector finite element method is presented for the dynamic analysis of cable structures based on the vector form intrinsic finite element (VFIFE) and mechanical properties of suspension cable. Firstly, the suspension cable is discretized into different elements by space points, the mass and external forces of suspension cable are transformed into space points. The structural form of cable is described by the space points at different time. The equations of motion for the space points are established according to the Newton’s second law. Then, the element internal forces between the space points are derived from the flexible truss structure. Finally, the motion equations of space points are solved by the central difference method with reasonable time integration step. The tangential tension of the bearing rope in a test ropeway with the moving concentrated loads is calculated and compared with the experimental data. The results show that the tangential tension of suspension cable with moving loads is consistent with the experimental data. This method has high calculated precision and meets the requirements of engineering application.

A stabilized element-based finite volume method for poroelastic problems

NASA Astrophysics Data System (ADS)

Honório, Hermínio T.; Maliska, Clovis R.; Ferronato, Massimiliano; Janna, Carlo

2018-07-01

The coupled equations of Biot's poroelasticity, consisting of stress equilibrium and fluid mass balance in deforming porous media, are numerically solved. The governing partial differential equations are discretized by an Element-based Finite Volume Method (EbFVM), which can be used in three dimensional unstructured grids composed of elements of different types. One of the difficulties for solving these equations is the numerical pressure instability that can arise when undrained conditions take place. In this paper, a stabilization technique is developed to overcome this problem by employing an interpolation function for displacements that considers also the pressure gradient effect. The interpolation function is obtained by the so-called Physical Influence Scheme (PIS), typically employed for solving incompressible fluid flows governed by the Navier-Stokes equations. Classical problems with analytical solutions, as well as three-dimensional realistic cases are addressed. The results reveal that the proposed stabilization technique is able to eliminate the spurious pressure instabilities arising under undrained conditions at a low computational cost.

Methods for analysis of cracks in three-dimensional solids

NASA Technical Reports Server (NTRS)

Raju, I. S.; Newman, J. C., Jr.

1984-01-01

Various analytical and numerical methods used to evaluate the stress intensity factors for cracks in three-dimensional (3-D) solids are reviewed. Classical exact solutions and many of the approximate methods used in 3-D analyses of cracks are reviewed. The exact solutions for embedded elliptic cracks in infinite solids are discussed. The approximate methods reviewed are the finite element methods, the boundary integral equation (BIE) method, the mixed methods (superposition of analytical and finite element method, stress difference method, discretization-error method, alternating method, finite element-alternating method), and the line-spring model. The finite element method with singularity elements is the most widely used method. The BIE method only needs modeling of the surfaces of the solid and so is gaining popularity. The line-spring model appears to be the quickest way to obtain good estimates of the stress intensity factors. The finite element-alternating method appears to yield the most accurate solution at the minimum cost.

Adaptive finite element modelling of three-dimensional magnetotelluric fields in general anisotropic media

NASA Astrophysics Data System (ADS)

Liu, Ying; Xu, Zhenhuan; Li, Yuguo

2018-04-01

We present a goal-oriented adaptive finite element (FE) modelling algorithm for 3-D magnetotelluric fields in generally anisotropic conductivity media. The model consists of a background layered structure, containing anisotropic blocks. Each block and layer might be anisotropic by assigning to them 3 × 3 conductivity tensors. The second-order partial differential equations are solved using the adaptive finite element method (FEM). The computational domain is subdivided into unstructured tetrahedral elements, which allow for complex geometries including bathymetry and dipping interfaces. The grid refinement process is guided by a global posteriori error estimator and is performed iteratively. The system of linear FE equations for electric field E is solved with a direct solver MUMPS. Then the magnetic field H can be found, in which the required derivatives are computed numerically using cubic spline interpolation. The 3-D FE algorithm has been validated by comparisons with both the 3-D finite-difference solution and 2-D FE results. Two model types are used to demonstrate the effects of anisotropy upon 3-D magnetotelluric responses: horizontal and dipping anisotropy. Finally, a 3D sea hill model is modelled to study the effect of oblique interfaces and the dipping anisotropy.

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

NASA Technical Reports Server (NTRS)

Mostrel, M. M.

1988-01-01

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

A diffuse-interface method for two-phase flows with soluble surfactants

PubMed Central

Teigen, Knut Erik; Song, Peng; Lowengrub, John; Voigt, Axel

2010-01-01

A method is presented to solve two-phase problems involving soluble surfactants. The incompressible Navier–Stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations. A non-linear equation of state is used to relate the surface tension to the interfacial surfactant concentration. The method is based on the use of a diffuse interface, which allows a simple implementation using standard finite difference or finite element techniques. Here, finite difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Results are presented for a drop in shear flow in both 2D and 3D, and the effect of solubility is discussed. PMID:21218125

A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements

NASA Astrophysics Data System (ADS)

Qin, Shanlin; Liu, Fawang; Turner, Ian W.

2018-03-01

The consideration of diffusion processes in magnetic resonance imaging (MRI) signal attenuation is classically described by the Bloch-Torrey equation. However, many recent works highlight the distinct deviation in MRI signal decay due to anomalous diffusion, which motivates the fractional order generalization of the Bloch-Torrey equation. In this work, we study the two-dimensional multi-term time and space fractional diffusion equation generalized from the time and space fractional Bloch-Torrey equation. By using the Galerkin finite element method with a structured mesh consisting of rectangular elements to discretize in space and the L1 approximation of the Caputo fractional derivative in time, a fully discrete numerical scheme is derived. A rigorous analysis of stability and error estimation is provided. Numerical experiments in the square and L-shaped domains are performed to give an insight into the efficiency and reliability of our method. Then the scheme is applied to solve the multi-term time and space fractional Bloch-Torrey equation, which shows that the extra time derivative terms impact the relaxation process.

A finite element approach for solution of the 3D Euler equations

NASA Technical Reports Server (NTRS)

Thornton, E. A.; Ramakrishnan, R.; Dechaumphai, P.

1986-01-01

Prediction of thermal deformations and stresses has prime importance in the design of the next generation of high speed flight vehicles. Aerothermal load computations for complex three-dimensional shapes necessitate development of procedures to solve the full Navier-Stokes equations. This paper details the development of a three-dimensional inviscid flow approach which can be extended for three-dimensional viscous flows. A finite element formulation, based on a Taylor series expansion in time, is employed to solve the compressible Euler equations. Model generation and results display are done using a commercially available program, PATRAN, and vectorizing strategies are incorporated to ensure computational efficiency. Sample problems are presented to demonstrate the validity of the approach for analyzing high speed compressible flows.

The P1-RKDG method for two-dimensional Euler equations of gas dynamics

NASA Technical Reports Server (NTRS)

Cockburn, Bernardo; Shu, Chi-Wang

1991-01-01

A class of nonlinearly stable Runge-Kutta local projection discontinuous Galerkin (RKDG) finite element methods for conservation laws is investigated. Two dimensional Euler equations for gas dynamics are solved using P1 elements. The generalization of the local projections, which for scalar nonlinear conservation laws was designed to satisfy a local maximum principle, to systems of conservation laws such as the Euler equations of gas dynamics using local characteristic decompositions is discussed. Numerical examples include the standard regular shock reflection problem, the forward facing step problem, and the double Mach reflection problem. These preliminary numerical examples are chosen to show the capacity of the approach to obtain nonlinearly stable results comparable with the modern nonoscillatory finite difference methods.

Application of the Green's function method for 2- and 3-dimensional steady transonic flows

NASA Technical Reports Server (NTRS)

Tseng, K.

1984-01-01

A Time-Domain Green's function method for the nonlinear time-dependent three-dimensional aerodynamic potential equation is presented. The Green's theorem is being used to transform the partial differential equation into an integro-differential-delay equation. Finite-element and finite-difference methods are employed for the spatial and time discretizations to approximate the integral equation by a system of differential-delay equations. Solution may be obtained by solving for this nonlinear simultaneous system of equations in time. This paper discusses the application of the method to the Transonic Small Disturbance Equation and numerical results for lifting and nonlifting airfoils and wings in steady flows are presented.

Three-dimensional finite elements for the analysis of soil contamination using a multiple-porosity approach

NASA Astrophysics Data System (ADS)

El-Zein, Abbas; Carter, John P.; Airey, David W.

2006-06-01

A three-dimensional finite-element model of contaminant migration in fissured clays or contaminated sand which includes multiple sources of non-equilibrium processes is proposed. The conceptual framework can accommodate a regular network of fissures in 1D, 2D or 3D and immobile solutions in the macro-pores of aggregated topsoils, as well as non-equilibrium sorption. A Galerkin weighted-residual statement for the three-dimensional form of the equations in the Laplace domain is formulated. Equations are discretized using linear and quadratic prism elements. The system of algebraic equations is solved in the Laplace domain and solution is inverted to the time domain numerically. The model is validated and its scope is illustrated through the analysis of three problems: a waste repository deeply buried in fissured clay, a storage tank leaking into sand and a sanitary landfill leaching into fissured clay over a sand aquifer.

The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes

NASA Astrophysics Data System (ADS)

Liu, Jiangguo; Tavener, Simon; Wang, Zhuoran

2018-04-01

This paper investigates the lowest-order weak Galerkin finite element method for solving the Darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles. In this approach, the pressure is approximated by constants in element interiors and on edges. The discrete weak gradients of these constant basis functions are specified in local Raviart-Thomas spaces, specifically RT0 for triangles and unmapped RT[0] for quadrilaterals. These discrete weak gradients are used to approximate the classical gradient when solving the Darcy equation. The method produces continuous normal fluxes and is locally mass-conservative, regardless of mesh quality, and has optimal order convergence in pressure, velocity, and normal flux, when the quadrilaterals are asymptotically parallelograms. Implementation is straightforward and results in symmetric positive-definite discrete linear systems. We present numerical experiments and comparisons with other existing methods.

Drekar v.2.0

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

Seefeldt, Ben; Sondak, David; Hensinger, David M.

Drekar is an application code that solves partial differential equations for fluids that can be optionally coupled to electromagnetics. Drekar solves low-mach compressible and incompressible computational fluid dynamics (CFD), compressible and incompressible resistive magnetohydrodynamics (MHD), and multiple species plasmas interacting with electromagnetic fields. Drekar discretization technology includes continuous and discontinuous finite element formulations, stabilized finite element formulations, mixed integration finite element bases (nodal, edge, face, volume) and an initial arbitrary Lagrangian Eulerian (ALE) capability. Drekar contains the implementation of the discretized physics and leverages the open source Trilinos project for both parallel solver capabilities and general finite element discretization tools.more » The code will be released open source under a BSD license. The code is used for fundamental research for simulation of fluids and plasmas on high performance computing environments.« less

Numerical Analysis of an H 1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

PubMed Central

Wang, Jinfeng; Zhao, Meng; Zhang, Min; Liu, Yang; Li, Hong

2014-01-01

We discuss and analyze an H 1-Galerkin mixed finite element (H 1-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H 1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H 1-GMFE method. Based on the discussion on the theoretical error analysis in L 2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H 1-norm. Moreover, we derive and analyze the stability of H 1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure. PMID:25184148

Element-by-element Solution Procedures for Nonlinear Structural Analysis

NASA Technical Reports Server (NTRS)

Hughes, T. J. R.; Winget, J. M.; Levit, I.

1984-01-01

Element-by-element approximate factorization procedures are proposed for solving the large finite element equation systems which arise in nonlinear structural mechanics. Architectural and data base advantages of the present algorithms over traditional direct elimination schemes are noted. Results of calculations suggest considerable potential for the methods described.

A class of hybrid finite element methods for electromagnetics: A review

NASA Technical Reports Server (NTRS)

Volakis, J. L.; Chatterjee, A.; Gong, J.

1993-01-01

Integral equation methods have generally been the workhorse for antenna and scattering computations. In the case of antennas, they continue to be the prominent computational approach, but for scattering applications the requirement for large-scale computations has turned researchers' attention to near neighbor methods such as the finite element method, which has low O(N) storage requirements and is readily adaptable in modeling complex geometrical features and material inhomogeneities. In this paper, we review three hybrid finite element methods for simulating composite scatterers, conformal microstrip antennas, and finite periodic arrays. Specifically, we discuss the finite element method and its application to electromagnetic problems when combined with the boundary integral, absorbing boundary conditions, and artificial absorbers for terminating the mesh. Particular attention is given to large-scale simulations, methods, and solvers for achieving low memory requirements and code performance on parallel computing architectures.

The CMC:3DPNS computer program for prediction of three-dimensional, subsonic, turbulent aerodynamic juncture region flow. Volume 1: Theoretical

NASA Technical Reports Server (NTRS)

Baker, A. J.

1982-01-01

An order-of-magnitude analysis of the subsonic three dimensional steady time averaged Navier-Stokes equations, for semibounded aerodynamic juncture geometries, yields the parabolic Navier-Stokes simplification. The numerical solution of the resultant pressure Poisson equation is cast into complementary and particular parts, yielding an iterative interaction algorithm with an exterior three dimensional potential flow solution. A parabolic transverse momentum equation set is constructed, wherein robust enforcement of first order continuity effects is accomplished using a penalty differential constraint concept within a finite element solution algorithm. A Reynolds stress constitutive equation, with low turbulence Reynolds number wall functions, is employed for closure, using parabolic forms of the two-equation turbulent kinetic energy-dissipation equation system. Numerical results document accuracy, convergence, and utility of the developed finite element algorithm, and the CMC:3DPNS computer code applied to an idealized wing-body juncture region. Additional results document accuracy aspects of the algorithm turbulence closure model.

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

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

Kuprat, A.P.; Glasser, A.H.

The authors discuss unstructured grids for application to transport in the tokamak edge SOL. They have developed a new metric with which to judge element elongation and resolution requirements. Using this method, the authors apply a standard moving finite element technique to advance the SOL equations while inserting/deleting dynamically nodes that violate an elongation criterion. In a tokamak plasma, this method achieves a more uniform accuracy, and results in highly stretched triangular finite elements, except near separatrix X-point where transport is more isotropic.

Transient finite element analysis of electric double layer using Nernst-Planck-Poisson equations with a modified Stern layer.

PubMed

Lim, Jongil; Whitcomb, John; Boyd, James; Varghese, Julian

2007-01-01

A finite element implementation of the transient nonlinear Nernst-Planck-Poisson (NPP) and Nernst-Planck-Poisson-modified Stern (NPPMS) models is presented. The NPPMS model uses multipoint constraints to account for finite ion size, resulting in realistic ion concentrations even at high surface potential. The Poisson-Boltzmann equation is used to provide a limited check of the transient models for low surface potential and dilute bulk solutions. The effects of the surface potential and bulk molarity on the electric potential and ion concentrations as functions of space and time are studied. The ability of the models to predict realistic energy storage capacity is investigated. The predicted energy is much more sensitive to surface potential than to bulk solution molarity.

A fictitious domain finite element method for simulations of fluid-structure interactions: The Navier-Stokes equations coupled with a moving solid

NASA Astrophysics Data System (ADS)

Court, Sébastien; Fournié, Michel

2015-05-01

The paper extends a stabilized fictitious domain finite element method initially developed for the Stokes problem to the incompressible Navier-Stokes equations coupled with a moving solid. This method presents the advantage to predict an optimal approximation of the normal stress tensor at the interface. The dynamics of the solid is governed by the Newton's laws and the interface between the fluid and the structure is materialized by a level-set which cuts the elements of the mesh. An algorithm is proposed in order to treat the time evolution of the geometry and numerical results are presented on a classical benchmark of the motion of a disk falling in a channel.

Adaptive computational methods for aerothermal heating analysis

NASA Technical Reports Server (NTRS)

Price, John M.; Oden, J. Tinsley

1988-01-01

The development of adaptive gridding techniques for finite-element analysis of fluid dynamics equations is described. The developmental work was done with the Euler equations with concentration on shock and inviscid flow field capturing. Ultimately this methodology is to be applied to a viscous analysis for the purpose of predicting accurate aerothermal loads on complex shapes subjected to high speed flow environments. The development of local error estimate strategies as a basis for refinement strategies is discussed, as well as the refinement strategies themselves. The application of the strategies to triangular elements and a finite-element flux-corrected-transport numerical scheme are presented. The implementation of these strategies in the GIM/PAGE code for 2-D and 3-D applications is documented and demonstrated.

Finite element analysis of two disk rotor system

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

Dixit, Harsh Kumar

A finite element model of simple horizontal rotor system is developed for evaluating its dynamic behaviour. The model is based on Timoshenko beam element and accounts for the effect of gyroscopic couple and other rotational forces. Present rotor system consists of single shaft which is supported by bearings at both ends and two disks are mounted at different locations. The natural frequencies, mode shapes and orbits of rotating system for a specific range of rotation speed are obtained by developing a MATLAB code for solving the finite element equations of rotary system. Consequently, Campbell diagram is plotted for finding amore » relationship between natural whirl frequencies and rotation of the rotor.« less

Semi-automatic sparse preconditioners for high-order finite element methods on non-uniform meshes

NASA Astrophysics Data System (ADS)

Austin, Travis M.; Brezina, Marian; Jamroz, Ben; Jhurani, Chetan; Manteuffel, Thomas A.; Ruge, John

2012-05-01

High-order finite elements often have a higher accuracy per degree of freedom than the classical low-order finite elements. However, in the context of implicit time-stepping methods, high-order finite elements present challenges to the construction of efficient simulations due to the high cost of inverting the denser finite element matrix. There are many cases where simulations are limited by the memory required to store the matrix and/or the algorithmic components of the linear solver. We are particularly interested in preconditioned Krylov methods for linear systems generated by discretization of elliptic partial differential equations with high-order finite elements. Using a preconditioner like Algebraic Multigrid can be costly in terms of memory due to the need to store matrix information at the various levels. We present a novel method for defining a preconditioner for systems generated by high-order finite elements that is based on a much sparser system than the original high-order finite element system. We investigate the performance for non-uniform meshes on a cube and a cubed sphere mesh, showing that the sparser preconditioner is more efficient and uses significantly less memory. Finally, we explore new methods to construct the sparse preconditioner and examine their effectiveness for non-uniform meshes. We compare results to a direct use of Algebraic Multigrid as a preconditioner and to a two-level additive Schwarz method.

A parallel finite element simulator for ion transport through three-dimensional ion channel systems.

PubMed

Tu, Bin; Chen, Minxin; Xie, Yan; Zhang, Linbo; Eisenberg, Bob; Lu, Benzhuo

2013-09-15

A parallel finite element simulator, ichannel, is developed for ion transport through three-dimensional ion channel systems that consist of protein and membrane. The coordinates of heavy atoms of the protein are taken from the Protein Data Bank and the membrane is represented as a slab. The simulator contains two components: a parallel adaptive finite element solver for a set of Poisson-Nernst-Planck (PNP) equations that describe the electrodiffusion process of ion transport, and a mesh generation tool chain for ion channel systems, which is an essential component for the finite element computations. The finite element method has advantages in modeling irregular geometries and complex boundary conditions. We have built a tool chain to get the surface and volume mesh for ion channel systems, which consists of a set of mesh generation tools. The adaptive finite element solver in our simulator is implemented using the parallel adaptive finite element package Parallel Hierarchical Grid (PHG) developed by one of the authors, which provides the capability of doing large scale parallel computations with high parallel efficiency and the flexibility of choosing high order elements to achieve high order accuracy. The simulator is applied to a real transmembrane protein, the gramicidin A (gA) channel protein, to calculate the electrostatic potential, ion concentrations and I - V curve, with which both primitive and transformed PNP equations are studied and their numerical performances are compared. To further validate the method, we also apply the simulator to two other ion channel systems, the voltage dependent anion channel (VDAC) and α-Hemolysin (α-HL). The simulation results agree well with Brownian dynamics (BD) simulation results and experimental results. Moreover, because ionic finite size effects can be included in PNP model now, we also perform simulations using a size-modified PNP (SMPNP) model on VDAC and α-HL. It is shown that the size effects in SMPNP can effectively lead to reduced current in the channel, and the results are closer to BD simulation results. Copyright © 2013 Wiley Periodicals, Inc.

Nanoscale Transport Optimization

DTIC Science & Technology

2008-12-04

could be argued that the advantage of using ABAQUS for this modeling construct has more to do with its ability to impose a user-defined subroutine that...finite element analysis. This is accomplished by employing a user defined subroutine for fluid properties at the interface within the finite element...package ABAQUS . Model Components: As noted above the governing equation for the material system is given as, ( ) ( ) 4484476444 8444 76

Numerical Methods Using B-Splines

NASA Technical Reports Server (NTRS)

Shariff, Karim; Merriam, Marshal (Technical Monitor)

1997-01-01

The seminar will discuss (1) The current range of applications for which B-spline schemes may be appropriate (2) The property of high-resolution and the relationship between B-spline and compact schemes (3) Comparison between finite-element, Hermite finite element and B-spline schemes (4) Mesh embedding using B-splines (5) A method for the incompressible Navier-Stokes equations in curvilinear coordinates using divergence-free expansions.

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.

Extended Finite Element Method with Simplified Spherical Harmonics Approximation for the Forward Model of Optical Molecular Imaging

PubMed Central

Li, Wei; Yi, Huangjian; Zhang, Qitan; Chen, Duofang; Liang, Jimin

2012-01-01

An extended finite element method (XFEM) for the forward model of 3D optical molecular imaging is developed with simplified spherical harmonics approximation (SPN). In XFEM scheme of SPN equations, the signed distance function is employed to accurately represent the internal tissue boundary, and then it is used to construct the enriched basis function of the finite element scheme. Therefore, the finite element calculation can be carried out without the time-consuming internal boundary mesh generation. Moreover, the required overly fine mesh conforming to the complex tissue boundary which leads to excess time cost can be avoided. XFEM conveniences its application to tissues with complex internal structure and improves the computational efficiency. Phantom and digital mouse experiments were carried out to validate the efficiency of the proposed method. Compared with standard finite element method and classical Monte Carlo (MC) method, the validation results show the merits and potential of the XFEM for optical imaging. PMID:23227108

Extended finite element method with simplified spherical harmonics approximation for the forward model of optical molecular imaging.

PubMed

Li, Wei; Yi, Huangjian; Zhang, Qitan; Chen, Duofang; Liang, Jimin

2012-01-01

An extended finite element method (XFEM) for the forward model of 3D optical molecular imaging is developed with simplified spherical harmonics approximation (SP(N)). In XFEM scheme of SP(N) equations, the signed distance function is employed to accurately represent the internal tissue boundary, and then it is used to construct the enriched basis function of the finite element scheme. Therefore, the finite element calculation can be carried out without the time-consuming internal boundary mesh generation. Moreover, the required overly fine mesh conforming to the complex tissue boundary which leads to excess time cost can be avoided. XFEM conveniences its application to tissues with complex internal structure and improves the computational efficiency. Phantom and digital mouse experiments were carried out to validate the efficiency of the proposed method. Compared with standard finite element method and classical Monte Carlo (MC) method, the validation results show the merits and potential of the XFEM for optical imaging.

On the dynamics of approximating schemes for dissipative nonlinear equations

NASA Technical Reports Server (NTRS)

Jones, Donald A.

1993-01-01

Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.

Exponential convergence through linear finite element discretization of stratified subdomains

NASA Astrophysics Data System (ADS)

Guddati, Murthy N.; Druskin, Vladimir; Vaziri Astaneh, Ali

2016-10-01

Motivated by problems where the response is needed at select localized regions in a large computational domain, we devise a novel finite element discretization that results in exponential convergence at pre-selected points. The key features of the discretization are (a) use of midpoint integration to evaluate the contribution matrices, and (b) an unconventional mapping of the mesh into complex space. Named complex-length finite element method (CFEM), the technique is linked to Padé approximants that provide exponential convergence of the Dirichlet-to-Neumann maps and thus the solution at specified points in the domain. Exponential convergence facilitates drastic reduction in the number of elements. This, combined with sparse computation associated with linear finite elements, results in significant reduction in the computational cost. The paper presents the basic ideas of the method as well as illustration of its effectiveness for a variety of problems involving Laplace, Helmholtz and elastodynamics equations.

Finite element analysis of left ventricle during cardiac cycles in viscoelasticity.

PubMed

Shen, Jing Jin; Xu, Feng Yu; Yang, Wen An

2016-08-01

To investigate the effect of myocardial viscoeslasticity on heart function, this paper presents a finite element model based on a hyper-viscoelastic model for the passive myocardium and Hill's three-element model for the active contraction. The hyper-viscoelastic model considers the myocardium microstructure, while the active model is phenomenologically based on the combination of Hill's equation for the steady tetanized contraction and the specific time-length-force property of the myocardial muscle. To validate the finite element model, the end-diastole strains and the end-systole strain predicted by the model are compared with the experimental values in the literature. It is found that the proposed model not only can estimate well the pumping function of the heart, but also predicts the transverse shear strains. The finite element model is also applied to analyze the influence of viscoelasticity on the residual stresses in the myocardium. Copyright © 2016 Elsevier Ltd. All rights reserved.

A survey of parametrized variational principles and applications to computational mechanics

NASA Technical Reports Server (NTRS)

Felippa, Carlos A.

1993-01-01

This survey paper describes recent developments in the area of parametrized variational principles (PVP's) and selected applications to finite-element computational mechanics. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations. The theory of single-field PVP's based on gauge functions (also known as null Lagrangians) is a subset of the inverse problem of variational calculus that has limited value. On the other hand, multifield PVP's are more interesting from theoretical and practical standpoints. Following a tutorial introduction, the paper describes the recent construction of multifield PVP's in several areas of elasticity and electromagnetics. It then discusses three applications to finite-element computational mechanics: the derivation of high-performance finite elements, the development of element-level error indicators, and the constructions of finite element templates. The paper concludes with an overview of open research areas.

Nonnegative methods for bilinear discontinuous differencing of the S N equations on quadrilaterals

DOE PAGES

Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E.

2016-12-22

Historically, matrix lumping and ad hoc flux fixups have been the only methods used to eliminate or suppress negative angular flux solutions associated with the unlumped bilinear discontinuous (UBLD) finite element spatial discretization of the two-dimensional S N equations. Though matrix lumping inhibits negative angular flux solutions of the S N equations, it does not guarantee strictly positive solutions. In this paper, we develop and define a strictly nonnegative, nonlinear, Petrov-Galerkin finite element method that fully preserves the bilinear discontinuous spatial moments of the transport equation. Additionally, we define two ad hoc fixups that maintain particle balance and explicitly setmore » negative nodes of the UBLD finite element solution to zero but use different auxiliary equations to fully define their respective solutions. We assess the ability to inhibit negative angular flux solutions and the accuracy of every spatial discretization that we consider using a glancing void test problem with a discontinuous solution known to stress numerical methods. Though significantly more computationally intense, the nonlinear Petrov-Galerkin scheme results in a strictly nonnegative solution and is a more accurate solution than all the other methods considered. One fixup, based on shape preserving, results in a strictly nonnegative final solution but has increased numerical diffusion relative to the Petrov-Galerkin scheme and is less accurate than the UBLD solution. The second fixup, which preserves as many spatial moments as possible while setting negative values of the unlumped solution to zero, is less accurate than the Petrov-Galerkin scheme but is more accurate than the other fixup. However, it fails to guarantee a strictly nonnegative final solution. As a result, the fully lumped bilinear discontinuous finite element solution is the least accurate method, with significantly more numerical diffusion than the Petrov-Galerkin scheme and both fixups.« less

Nonnegative methods for bilinear discontinuous differencing of the S N equations on quadrilaterals

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

Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E.

Historically, matrix lumping and ad hoc flux fixups have been the only methods used to eliminate or suppress negative angular flux solutions associated with the unlumped bilinear discontinuous (UBLD) finite element spatial discretization of the two-dimensional S N equations. Though matrix lumping inhibits negative angular flux solutions of the S N equations, it does not guarantee strictly positive solutions. In this paper, we develop and define a strictly nonnegative, nonlinear, Petrov-Galerkin finite element method that fully preserves the bilinear discontinuous spatial moments of the transport equation. Additionally, we define two ad hoc fixups that maintain particle balance and explicitly setmore » negative nodes of the UBLD finite element solution to zero but use different auxiliary equations to fully define their respective solutions. We assess the ability to inhibit negative angular flux solutions and the accuracy of every spatial discretization that we consider using a glancing void test problem with a discontinuous solution known to stress numerical methods. Though significantly more computationally intense, the nonlinear Petrov-Galerkin scheme results in a strictly nonnegative solution and is a more accurate solution than all the other methods considered. One fixup, based on shape preserving, results in a strictly nonnegative final solution but has increased numerical diffusion relative to the Petrov-Galerkin scheme and is less accurate than the UBLD solution. The second fixup, which preserves as many spatial moments as possible while setting negative values of the unlumped solution to zero, is less accurate than the Petrov-Galerkin scheme but is more accurate than the other fixup. However, it fails to guarantee a strictly nonnegative final solution. As a result, the fully lumped bilinear discontinuous finite element solution is the least accurate method, with significantly more numerical diffusion than the Petrov-Galerkin scheme and both fixups.« less

Verification of a non-hydrostatic dynamical core using horizontally spectral element vertically finite difference method: 2-D aspects

NASA Astrophysics Data System (ADS)

Choi, S.-J.; Giraldo, F. X.; Kim, J.; Shin, S.

2014-06-01

The non-hydrostatic (NH) compressible Euler equations of dry atmosphere are solved in a simplified two dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points. The FDM employs a third-order upwind biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative terms and quadrature. The Euler equations used here are in a flux form based on the hydrostatic pressure vertical coordinate, which are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate is implemented in this model. We verified the model by conducting widely used standard benchmark tests: the inertia-gravity wave, rising thermal bubble, density current wave, and linear hydrostatic mountain wave. The results from those tests demonstrate that the horizontally spectral element vertically finite difference model is accurate and robust. By using the 2-D slice model, we effectively show that the combined spatial discretization method of the spectral element and finite difference method in the horizontal and vertical directions, respectively, offers a viable method for the development of a NH dynamical core.

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for high Reynolds number laminar flows

NASA Technical Reports Server (NTRS)

Kim, Sang-Wook

1988-01-01

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for the Navier-Stokes equations is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions and the pressure was interpolated using linear shape functions. For the two dimensional case, the pressure is defined on a triangular element which is contained inside the complete biquadratic element for velocity variables; and for the three dimensional case, the pressure is defined on a tetrahedral element which is again contained inside the complete tri-quadratic element. Thus the pressure is discontinuous across the element boundaries. Example problems considered include: a cavity flow for Reynolds number of 400 through 10,000; a laminar backward facing step flow; and a laminar flow in a square duct of strong curvature. The computational results compared favorable with those of the finite difference methods as well as experimental data available. A finite elememt computer program for incompressible, laminar flows is presented.

Computer-Aided Engineering of Semiconductor Integrated Circuits

DTIC Science & Technology

1979-07-01

equation using a five point finite difference approximation. Section 4.3.6 describes the numerical techniques and iterative algorithms which are used...neighbor points. This is generally referred to as a five point finite difference scheme on a rectangular grid, as described below. The finite difference ...problems in steady state have been analyzed by the finite difference method [4. 16 ] [4.17 3 or finite element method [4. 18 3, [4. 19 3 as reported last

Iterative methods for mixed finite element equations

NASA Technical Reports Server (NTRS)

Nakazawa, S.; Nagtegaal, J. C.; Zienkiewicz, O. C.

1985-01-01

Iterative strategies for the solution of indefinite system of equations arising from the mixed finite element method are investigated in this paper with application to linear and nonlinear problems in solid and structural mechanics. The augmented Hu-Washizu form is derived, which is then utilized to construct a family of iterative algorithms using the displacement method as the preconditioner. Two types of iterative algorithms are implemented. Those are: constant metric iterations which does not involve the update of preconditioner; variable metric iterations, in which the inverse of the preconditioning matrix is updated. A series of numerical experiments is conducted to evaluate the numerical performance with application to linear and nonlinear model problems.

Quadratic Finite Element Method for 1D Deterministic Transport

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

Tolar, Jr., D R; Ferguson, J M

2004-01-06

In the discrete ordinates, or SN, numerical solution of the transport equation, both the spatial ({und r}) and angular ({und {Omega}}) dependences on the angular flux {psi}{und r},{und {Omega}}are modeled discretely. While significant effort has been devoted toward improving the spatial discretization of the angular flux, we focus on improving the angular discretization of {psi}{und r},{und {Omega}}. Specifically, we employ a Petrov-Galerkin quadratic finite element approximation for the differencing of the angular variable ({mu}) in developing the one-dimensional (1D) spherical geometry S{sub N} equations. We develop an algorithm that shows faster convergence with angular resolution than conventional S{sub N} algorithms.

SAGUARO: a finite-element computer program for partially saturated porous flow problems

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

Eaton, R.R.; Gartling, D.K.; Larson, D.E.

1983-06-01

SAGUARO is a finite element computer program designed to calculate two-dimensional flow of mass and energy through porous media. The media may be saturated or partially saturated. SAGUARO solves the parabolic time-dependent mass transport equation which accounts for the presence of partially saturated zones through the use of highly non-linear material characteristic curves. The energy equation accounts for the possibility of partially saturated regions by adjusting the thermal capacitances and thermal conductivities according to the volume fraction of water present in the local pores. Program capabilities, user instructions and a sample problem are presented in this manual.

A study of delamination buckling of laminates

NASA Technical Reports Server (NTRS)

Mukherjee, Yu-Xie; Xie, Zhi-Cheng; Ingraffea, Anthony

1990-01-01

The subject of this paper is the buckling of laminated plates, with a preexisting delamination, subjected to in-plane loading. Each laminate is modelled as an orthotropic Mindlin plate. The analysis is carried out by a combination of the finite element and asymptotic expansion methods. By applying the finite element method, plates with general delamination regions can be studied. The asymptotic expansion method reduces the number of unknown variables of the eigenvalue equation to that of the equation for a single Kirchhoff plate. Numerical results are presented for several examples. The effects of the shape, size, and position of the delamination on the buckling load are studied through these examples.

Finite Element Analysis of Poroelastic Composites Undergoing Thermal and Gas Diffusion

NASA Technical Reports Server (NTRS)

Salamon, N. J. (Principal Investigator); Sullivan, Roy M.; Lee, Sunpyo

1995-01-01

A theory for time-dependent thermal and gas diffusion in mechanically time-rate-independent anisotropic poroelastic composites has been developed. This theory advances previous work by the latter two authors by providing for critical transverse shear through a three-dimensional axisymmetric formulation and using it in a new hypothesis for determining the Biot fluid pressure-solid stress coupling factor. The derived governing equations couple material deformation with temperature and internal pore pressure and more strongly couple gas diffusion and heat transfer than the previous theory. Hence the theory accounts for the interactions between conductive heat transfer in the porous body and convective heat carried by the mass flux through the pores. The Bubnov Galerkin finite element method is applied to the governing equations to transform them into a semidiscrete finite element system. A numerical procedure is developed to solve the coupled equations in the space and time domains. The method is used to simulate two high temperature tests involving thermal-chemical decomposition of carbon-phenolic composites. In comparison with measured data, the results are accurate. Moreover unlike previous work, for a single set of poroelastic parameters, they are consistent with two measurements in a restrained thermal growth test.

Finite element formulation of fluctuating hydrodynamics for fluids filled with rigid particles using boundary fitted meshes

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

De Corato, M., E-mail: marco.decorato@unina.it; Slot, J.J.M., E-mail: j.j.m.slot@tue.nl; Hütter, M., E-mail: m.huetter@tue.nl

In this paper, we present a finite element implementation of fluctuating hydrodynamics with a moving boundary fitted mesh for treating the suspended particles. The thermal fluctuations are incorporated into the continuum equations using the Landau and Lifshitz approach [1]. The proposed implementation fulfills the fluctuation–dissipation theorem exactly at the discrete level. Since we restrict the equations to the creeping flow case, this takes the form of a relation between the diffusion coefficient matrix and friction matrix both at the particle and nodal level of the finite elements. Brownian motion of arbitrarily shaped particles in complex confinements can be considered withinmore » the present formulation. A multi-step time integration scheme is developed to correctly capture the drift term required in the stochastic differential equation (SDE) describing the evolution of the positions of the particles. The proposed approach is validated by simulating the Brownian motion of a sphere between two parallel plates and the motion of a spherical particle in a cylindrical cavity. The time integration algorithm and the fluctuating hydrodynamics implementation are then applied to study the diffusion and the equilibrium probability distribution of a confined circle under an external harmonic potential.« less

SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson-Boltzmann equation.

PubMed

Xie, Yang; Ying, Jinyong; Xie, Dexuan

2017-03-30

SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

Fem Formulation for Heat and Mass Transfer in Porous Medium

NASA Astrophysics Data System (ADS)

Azeem; Soudagar, Manzoor Elahi M.; Salman Ahmed, N. J.; Anjum Badruddin, Irfan

2017-08-01

Heat and mass transfer in porous medium can be modelled using three partial differential equations namely, momentum equation, energy equation and mass diffusion. These three equations are coupled to each other by some common terms that turn the whole phenomenon into a complex problem with inter-dependable variables. The current article describes the finite element formulation of heat and mass transfer in porous medium with respect to Cartesian coordinates. The problem under study is formulated into algebraic form of equations by using Galerkin's method with the help of two-node linear triangular element having three nodes. The domain is meshed with smaller sized elements near the wall region and bigger size away from walls.

A finite-volume Eulerian-Lagrangian Localized Adjoint Method for solution of the advection-dispersion equation

USGS Publications Warehouse

Healy, R.W.; Russell, T.F.

1993-01-01

A new mass-conservative method for solution of the one-dimensional advection-dispersion equation is derived and discussed. Test results demonstrate that the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) outperforms standard finite-difference methods, in terms of accuracy and efficiency, for solute transport problems that are dominated by advection. For dispersion-dominated problems, the performance of the method is similar to that of standard methods. Like previous ELLAM formulations, FVELLAM systematically conserves mass globally with all types of boundary conditions. FVELLAM differs from other ELLAM approaches in that integrated finite differences, instead of finite elements, are used to approximate the governing equation. This approach, in conjunction with a forward tracking scheme, greatly facilitates mass conservation. The mass storage integral is numerically evaluated at the current time level, and quadrature points are then tracked forward in time to the next level. Forward tracking permits straightforward treatment of inflow boundaries, thus avoiding the inherent problem in backtracking, as used by most characteristic methods, of characteristic lines intersecting inflow boundaries. FVELLAM extends previous ELLAM results by obtaining mass conservation locally on Lagrangian space-time elements. Details of the integration, tracking, and boundary algorithms are presented. Test results are given for problems in Cartesian and radial coordinates.

Numerical solution of quadratic matrix equations for free vibration analysis of structures

NASA Technical Reports Server (NTRS)

Gupta, K. K.

1975-01-01

This paper is concerned with the efficient and accurate solution of the eigenvalue problem represented by quadratic matrix equations. Such matrix forms are obtained in connection with the free vibration analysis of structures, discretized by finite 'dynamic' elements, resulting in frequency-dependent stiffness and inertia matrices. The paper presents a new numerical solution procedure of the quadratic matrix equations, based on a combined Sturm sequence and inverse iteration technique enabling economical and accurate determination of a few required eigenvalues and associated vectors. An alternative procedure based on a simultaneous iteration procedure is also described when only the first few modes are the usual requirement. The employment of finite dynamic elements in conjunction with the presently developed eigenvalue routines results in a most significant economy in the dynamic analysis of structures.

Numerical calculations of velocity and pressure distribution around oscillating airfoils

NASA Technical Reports Server (NTRS)

Bratanow, T.; Ecer, A.; Kobiske, M.

1974-01-01

An analytical procedure based on the Navier-Stokes equations was developed for analyzing and representing properties of unsteady viscous flow around oscillating obstacles. A variational formulation of the vorticity transport equation was discretized in finite element form and integrated numerically. At each time step of the numerical integration, the velocity field around the obstacle was determined for the instantaneous vorticity distribution from the finite element solution of Poisson's equation. The time-dependent boundary conditions around the oscillating obstacle were introduced as external constraints, using the Lagrangian Multiplier Technique, at each time step of the numerical integration. The procedure was then applied for determining pressures around obstacles oscillating in unsteady flow. The obtained results for a cylinder and an airfoil were illustrated in the form of streamlines and vorticity and pressure distributions.

Analysis of Flexible Bars and Frames with Large Displacements of Nodes By Finite Element Method in the Form of Classical Mixed Method

NASA Astrophysics Data System (ADS)

Ignatyev, A. V.; Ignatyev, V. A.; Onischenko, E. V.

2017-11-01

This article is the continuation of the work made bt the authors on the development of the algorithms that implement the finite element method in the form of a classical mixed method for the analysis of geometrically nonlinear bar systems [1-3]. The paper describes an improved algorithm of the formation of the nonlinear governing equations system for flexible plane frames and bars with large displacements of nodes based on the finite element method in a mixed classical form and the use of the procedure of step-by-step loading. An example of the analysis is given.

A Linear-Elasticity Solver for Higher-Order Space-Time Mesh Deformation

NASA Technical Reports Server (NTRS)

Diosady, Laslo T.; Murman, Scott M.

2018-01-01

A linear-elasticity approach is presented for the generation of meshes appropriate for a higher-order space-time discontinuous finite-element method. The equations of linear-elasticity are discretized using a higher-order, spatially-continuous, finite-element method. Given an initial finite-element mesh, and a specified boundary displacement, we solve for the mesh displacements to obtain a higher-order curvilinear mesh. Alternatively, for moving-domain problems we use the linear-elasticity approach to solve for a temporally discontinuous mesh velocity on each time-slab and recover a continuous mesh deformation by integrating the velocity. The applicability of this methodology is presented for several benchmark test cases.

magnum.fe: A micromagnetic finite-element simulation code based on FEniCS

NASA Astrophysics Data System (ADS)

Abert, Claas; Exl, Lukas; Bruckner, Florian; Drews, André; Suess, Dieter

2013-11-01

We have developed a finite-element micromagnetic simulation code based on the FEniCS package called magnum.fe. Here we describe the numerical methods that are applied as well as their implementation with FEniCS. We apply a transformation method for the solution of the demagnetization-field problem. A semi-implicit weak formulation is used for the integration of the Landau-Lifshitz-Gilbert equation. Numerical experiments show the validity of simulation results. magnum.fe is open source and well documented. The broad feature range of the FEniCS package makes magnum.fe a good choice for the implementation of novel micromagnetic finite-element algorithms.

A finite element-boundary integral method for cavities in a circular cylinder

NASA Technical Reports Server (NTRS)

Kempel, Leo C.; Volakis, John L.

1992-01-01

Conformal antenna arrays offer many cost and weight advantages over conventional antenna systems. However, due to a lack of rigorous mathematical models for conformal antenna arrays, antenna designers resort to measurement and planar antenna concepts for designing non-planar conformal antennas. Recently, we have found the finite element-boundary integral method to be very successful in modeling large planar arrays of arbitrary composition in a metallic plane. We extend this formulation to conformal arrays on large metallic cylinders. In this report, we develop the mathematical formulation. In particular, we discuss the shape functions, the resulting finite elements and the boundary integral equations, and the solution of the conformal finite element-boundary integral system. Some validation results are presented and we further show how this formulation can be applied with minimal computational and memory resources.

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…

The semi-discrete Galerkin finite element modelling of compressible viscous flow past an airfoil

NASA Technical Reports Server (NTRS)

Meade, Andrew J., Jr.

1992-01-01

A method is developed to solve the two-dimensional, steady, compressible, turbulent boundary-layer equations and is coupled to an existing Euler solver for attached transonic airfoil analysis problems. The boundary-layer formulation utilizes the semi-discrete Galerkin (SDG) method to model the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby permitting the use of a uniform finite element grid which provides high resolution near the wall and automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes, through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past NACA 0012 and RAE 2822 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack. All results show good agreement with experiment, and the coupled code proved to be a computationally-efficient and accurate airfoil analysis tool.

Stress analysis of rotating propellers subject to forced excitations

NASA Astrophysics Data System (ADS)

Akgun, Ulas

Turbine blades experience vibrations due to the flow disturbances. These vibrations are the leading cause for fatigue failure in turbine blades. This thesis presents the finite element analysis methods to estimate the maximum vibrational stresses of rotating structures under forced excitation. The presentation included starts with the derived equations of motion for vibration of rotating beams using energy methods under the Euler Bernoulli beam assumptions. The nonlinear large displacement formulation captures the centrifugal stiffening and gyroscopic effects. The weak form of the equations and their finite element discretization are shown. The methods implemented were used for normal modes analyses and forced vibration analyses of rotating beam structures. The prediction of peak stresses under simultaneous multi-mode excitation show that the maximum vibrational stresses estimated using the linear superposition of the stresses can greatly overestimate the stresses if the phase information due to damping (physical and gyroscopic effects) are neglected. The last section of this thesis also presents the results of a practical study that involves finite element analysis and redesign of a composite propeller.

Mass Efficiency Considerations for Thermally Insulated Structural Skin of an Aerospace Vehicle

NASA Technical Reports Server (NTRS)

Blosser, Max L.

2012-01-01

An approximate equation was derived to predict the mass of insulation required to limit the maximum temperature reached by an insulated structure subjected to a transient heating pulse. In the course of the derivation two figures of merit were identified. One figure of merit correlates to the effectiveness of the heat capacity of the underlying structural material in reducing the amount of required insulation. The second figure of merit provides an indicator of the mass efficiency of the insulator material. An iterative, one dimensional finite element analysis was used to size the external insulation required to protect the structure at a single location on the Space Shuttle Orbiter and a reusable launch vehicle. Required insulation masses were calculated for a range of different materials for both structure and insulator. The required insulation masses calculated using the approximate equation were shown to typically agree with finite element results within 10 to 20 percent over the range of parameters studied. Finite element results closely followed the trends indicated by both figures of merit.

Large-scale computation of incompressible viscous flow by least-squares finite element method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Lin, T. L.; Povinelli, Louis A.

1993-01-01

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations and results in symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli function-vorticity formulation for incompressible viscous flows is also tested. For three-dimensional cases, an additional compatibility equation, i.e., the divergence of the vorticity vector should be zero, is included to make the first-order system elliptic. The simple substitution of the Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this scheme for large-scale computation, we give numerical results for 2D driven cavity problem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1,000 with 50 x 50 x 50 trilinear elements. The Taylor-Goertler-like vortices are observed for Re = 1,000.

COMPARISON OF NUMERICAL SCHEMES FOR SOLVING A SPHERICAL PARTICLE DIFFUSION EQUATION

EPA Science Inventory

A new robust iterative numerical scheme was developed for a nonlinear diffusive model that described sorption dynamics in spherical particle suspensions. he numerical scheme had been applied to finite difference and finite element models that showed rapid convergence and stabilit...

Stabilized finite element methods to simulate the conductances of ion channels

NASA Astrophysics Data System (ADS)

Tu, Bin; Xie, Yan; Zhang, Linbo; Lu, Benzhuo

2015-03-01

We have previously developed a finite element simulator, ichannel, to simulate ion transport through three-dimensional ion channel systems via solving the Poisson-Nernst-Planck equations (PNP) and Size-modified Poisson-Nernst-Planck equations (SMPNP), and succeeded in simulating some ion channel systems. However, the iterative solution between the coupled Poisson equation and the Nernst-Planck equations has difficulty converging for some large systems. One reason we found is that the NP equations are advection-dominated diffusion equations, which causes troubles in the usual FE solution. The stabilized schemes have been applied to compute fluids flow in various research fields. However, they have not been studied in the simulation of ion transport through three-dimensional models based on experimentally determined ion channel structures. In this paper, two stabilized techniques, the SUPG and the Pseudo Residual-Free Bubble function (PRFB) are introduced to enhance the numerical robustness and convergence performance of the finite element algorithm in ichannel. The conductances of the voltage dependent anion channel (VDAC) and the anthrax toxin protective antigen pore (PA) are simulated to validate the stabilization techniques. Those two stabilized schemes give reasonable results for the two proteins, with decent agreement with both experimental data and Brownian dynamics (BD) simulations. For a variety of numerical tests, it is found that the simulator effectively avoids previous numerical instability after introducing the stabilization methods. Comparison based on our test data set between the two stabilized schemes indicates both SUPG and PRFB have similar performance (the latter is slightly more accurate and stable), while SUPG is relatively more convenient to implement.

A split finite element algorithm for the compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Baker, A. J.

1979-01-01

An accurate and efficient numerical solution algorithm is established for solution of the high Reynolds number limit of the Navier-Stokes equations governing the multidimensional flow of a compressible essentially inviscid fluid. Finite element interpolation theory is used within a dissipative formulation established using Galerkin criteria within the Method of Weighted Residuals. An implicit iterative solution algorithm is developed, employing tensor product bases within a fractional steps integration procedure, that significantly enhances solution economy concurrent with sharply reduced computer hardware demands. The algorithm is evaluated for resolution of steep field gradients and coarse grid accuracy using both linear and quadratic tensor product interpolation bases. Numerical solutions for linear and nonlinear, one, two and three dimensional examples confirm and extend the linearized theoretical analyses, and results are compared to competitive finite difference derived algorithms.

Transient Finite Element Computations on a Variable Transputer System

NASA Technical Reports Server (NTRS)

Smolinski, Patrick J.; Lapczyk, Ireneusz

1993-01-01

A parallel program to analyze transient finite element problems was written and implemented on a system of transputer processors. The program uses the explicit time integration algorithm which eliminates the need for equation solving, making it more suitable for parallel computations. An interprocessor communication scheme was developed for arbitrary two dimensional grid processor configurations. Several 3-D problems were analyzed on a system with a small number of processors.

Application of Finite Element to Evaluate Material with Small Modulus of Elasticity

DTIC Science & Technology

2013-03-01

14  Figure 8: Cross-sectional diagram of thorax highlighting the various muscle groups in the Hawkmoth and the interaction with Exoskeleton ...44  Figure 26: Partially Dissected Moth highlighting the point of incision of the exoskeleton (wings are removed...applications to the exoskeleton of the hawkmoth are examined. The formulation of these equations is discussed in Chapter 2 and the finite element model is

Edge equilibrium code for tokamaks

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

Li, Xujing; Zakharov, Leonid E.; Drozdov, Vladimir V.

2014-01-15

The edge equilibrium code (EEC) described in this paper is developed for simulations of the near edge plasma using the finite element method. It solves the Grad-Shafranov equation in toroidal coordinate and uses adaptive grids aligned with magnetic field lines. Hermite finite elements are chosen for the numerical scheme. A fast Newton scheme which is the same as implemented in the equilibrium and stability code (ESC) is applied here to adjust the grids.

A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation

NASA Astrophysics Data System (ADS)

Vergez, Guillaume; Danaila, Ionut; Auliac, Sylvain; Hecht, Frédéric

2016-12-01

We present a new numerical system using classical finite elements with mesh adaptivity for computing stationary solutions of the Gross-Pitaevskii equation. The programs are written as a toolbox for FreeFem++ (www.freefem.org), a free finite-element software available for all existing operating systems. This offers the advantage to hide all technical issues related to the implementation of the finite element method, allowing to easily code various numerical algorithms. Two robust and optimized numerical methods were implemented to minimize the Gross-Pitaevskii energy: a steepest descent method based on Sobolev gradients and a minimization algorithm based on the state-of-the-art optimization library Ipopt. For both methods, mesh adaptivity strategies are used to reduce the computational time and increase the local spatial accuracy when vortices are present. Different run cases are made available for 2D and 3D configurations of Bose-Einstein condensates in rotation. An optional graphical user interface is also provided, allowing to easily run predefined cases or with user-defined parameter files. We also provide several post-processing tools (like the identification of quantized vortices) that could help in extracting physical features from the simulations. The toolbox is extremely versatile and can be easily adapted to deal with different physical models.

Guided waves dispersion equations for orthotropic multilayered pipes solved using standard finite elements code.

PubMed

Predoi, Mihai Valentin

2014-09-01

The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0,n), torsional T(0,n) and flexural F(m,n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method. Copyright © 2014 Elsevier B.V. All rights reserved.

Finite Element Method Analysis of An Out Flow With Free Surface In Transition Zones

NASA Astrophysics Data System (ADS)

Saoula, R. Iddir S.; Mokhtar, K. Ait

The object of this work is to present this part of the fluid mechanics that relates to out-flows of the fluid to big speeds in transitions. Results usually gotten by the classic processes can only have a qualitative aspect. The method fluently used for the count of these out-flows to big speeds is the one of characteristics, this approach remains interesting so much that doesn't appear within the out-flow of intersections of shock waves, as well as of reflections of these. In the simple geometry case, the method of finite differences satisfying result, But when the complexity of this geometry imposes itself, it is the method of finite elements that is proposed to solve this type of prob- lem, in particular for problems Trans critic. The goal of our work is to analyse free surface flows in channels no prismatic has oblong transverse section in zone of tran- sition. (Convergent, divergent). The basic mathematical model of this study is Saint Venant derivatives partial equations. To solve these equations we use the finite ele- ment method, the element of reference is the triangular element with 6 nodes which are quadratic in speed and linear in height (pressure). Our results and their obtains by others are very close to experimental results.

Static aeroelastic analysis of wings using Euler/Navier-Stokes equations coupled with improved wing-box finite element structures

NASA Technical Reports Server (NTRS)

Guruswamy, Guru P.; MacMurdy, Dale E.; Kapania, Rakesh K.

1994-01-01

Strong interactions between flow about an aircraft wing and the wing structure can result in aeroelastic phenomena which significantly impact aircraft performance. Time-accurate methods for solving the unsteady Navier-Stokes equations have matured to the point where reliable results can be obtained with reasonable computational costs for complex non-linear flows with shock waves, vortices and separations. The ability to combine such a flow solver with a general finite element structural model is key to an aeroelastic analysis in these flows. Earlier work involved time-accurate integration of modal structural models based on plate elements. A finite element model was developed to handle three-dimensional wing boxes, and incorporated into the flow solver without the need for modal analysis. Static condensation is performed on the structural model to reduce the structural degrees of freedom for the aeroelastic analysis. Direct incorporation of the finite element wing-box structural model with the flow solver requires finding adequate methods for transferring aerodynamic pressures to the structural grid and returning deflections to the aerodynamic grid. Several schemes were explored for handling the grid-to-grid transfer of information. The complex, built-up nature of the wing-box complicated this transfer. Aeroelastic calculations for a sample wing in transonic flow comparing various simple transfer schemes are presented and discussed.

Development of Finite Elements for Two-Dimensional Structural Analysis Using the Integrated Force Method

NASA Technical Reports Server (NTRS)

Kaljevic, Igor; Patnaik, Surya N.; Hopkins, Dale A.

1996-01-01

The Integrated Force Method has been developed in recent years for the analysis of structural mechanics problems. This method treats all independent internal forces as unknown variables that can be calculated by simultaneously imposing equations of equilibrium and compatibility conditions. In this paper a finite element library for analyzing two-dimensional problems by the Integrated Force Method is presented. Triangular- and quadrilateral-shaped elements capable of modeling arbitrary domain configurations are presented. The element equilibrium and flexibility matrices are derived by discretizing the expressions for potential and complementary energies, respectively. The displacement and stress fields within the finite elements are independently approximated. The displacement field is interpolated as it is in the standard displacement method, and the stress field is approximated by using complete polynomials of the correct order. A procedure that uses the definitions of stress components in terms of an Airy stress function is developed to derive the stress interpolation polynomials. Such derived stress fields identically satisfy the equations of equilibrium. Moreover, the resulting element matrices are insensitive to the orientation of local coordinate systems. A method is devised to calculate the number of rigid body modes, and the present elements are shown to be free of spurious zero-energy modes. A number of example problems are solved by using the present library, and the results are compared with corresponding analytical solutions and with results from the standard displacement finite element method. The Integrated Force Method not only gives results that agree well with analytical and displacement method results but also outperforms the displacement method in stress calculations.

High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation

NASA Astrophysics Data System (ADS)

Anderson, R.; Dobrev, V.; Kolev, Tz.; Kuzmin, D.; Quezada de Luna, M.; Rieben, R.; Tomov, V.

2017-04-01

In this work we present a FCT-like Maximum-Principle Preserving (MPP) method to solve the transport equation. We use high-order polynomial spaces; in particular, we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension. The method combines the concepts of positive basis functions for discontinuous Galerkin finite element spatial discretization, locally defined solution bounds, element-based flux correction, and non-linear local mass redistribution. We consider a simple 1D problem with non-smooth initial data to explain and understand the behavior of different parts of the method. Convergence tests in space indicate that high-order accuracy is achieved. Numerical results from several benchmarks in two and three dimensions are also reported.

The L sub 1 finite element method for pure convection problems

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan

1991-01-01

The least squares (L sub 2) finite element method is introduced for 2-D steady state pure convection problems with smooth solutions. It is proven that the L sub 2 method has the same stability estimate as the original equation, i.e., the L sub 2 method has better control of the streamline derivative. Numerical convergence rates are given to show that the L sub 2 method is almost optimal. This L sub 2 method was then used as a framework to develop an iteratively reweighted L sub 2 finite element method to obtain a least absolute residual (L sub 1) solution for problems with discontinuous solutions. This L sub 1 finite element method produces a nonoscillatory, nondiffusive and highly accurate numerical solution that has a sharp discontinuity in one element on both coarse and fine meshes. A robust reweighting strategy was also devised to obtain the L sub 1 solution in a few iterations. A number of examples solved by using triangle and bilinear elements are presented.

Probabilistic boundary element method

NASA Technical Reports Server (NTRS)

Cruse, T. A.; Raveendra, S. T.

1989-01-01

The purpose of the Probabilistic Structural Analysis Method (PSAM) project is to develop structural analysis capabilities for the design analysis of advanced space propulsion system hardware. The boundary element method (BEM) is used as the basis of the Probabilistic Advanced Analysis Methods (PADAM) which is discussed. The probabilistic BEM code (PBEM) is used to obtain the structural response and sensitivity results to a set of random variables. As such, PBEM performs analogous to other structural analysis codes such as finite elements in the PSAM system. For linear problems, unlike the finite element method (FEM), the BEM governing equations are written at the boundary of the body only, thus, the method eliminates the need to model the volume of the body. However, for general body force problems, a direct condensation of the governing equations to the boundary of the body is not possible and therefore volume modeling is generally required.

Parallel Ellipsoidal Perfectly Matched Layers for Acoustic Helmholtz Problems on Exterior Domains

DOE PAGES

Bunting, Gregory; Prakash, Arun; Walsh, Timothy; ...

2018-01-26

Exterior acoustic problems occur in a wide range of applications, making the finite element analysis of such problems a common practice in the engineering community. Various methods for truncating infinite exterior domains have been developed, including absorbing boundary conditions, infinite elements, and more recently, perfectly matched layers (PML). PML are gaining popularity due to their generality, ease of implementation, and effectiveness as an absorbing boundary condition. PML formulations have been developed in Cartesian, cylindrical, and spherical geometries, but not ellipsoidal. In addition, the parallel solution of PML formulations with iterative solvers for the solution of the Helmholtz equation, and howmore » this compares with more traditional strategies such as infinite elements, has not been adequately investigated. In this study, we present a parallel, ellipsoidal PML formulation for acoustic Helmholtz problems. To faciliate the meshing process, the ellipsoidal PML layer is generated with an on-the-fly mesh extrusion. Though the complex stretching is defined along ellipsoidal contours, we modify the Jacobian to include an additional mapping back to Cartesian coordinates in the weak formulation of the finite element equations. This allows the equations to be solved in Cartesian coordinates, which is more compatible with existing finite element software, but without the necessity of dealing with corners in the PML formulation. Herein we also compare the conditioning and performance of the PML Helmholtz problem with infinite element approach that is based on high order basis functions. On a set of representative exterior acoustic examples, we show that high order infinite element basis functions lead to an increasing number of Helmholtz solver iterations, whereas for PML the number of iterations remains constant for the same level of accuracy. Finally, this provides an additional advantage of PML over the infinite element approach.« less

Parallel Ellipsoidal Perfectly Matched Layers for Acoustic Helmholtz Problems on Exterior Domains

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

Bunting, Gregory; Prakash, Arun; Walsh, Timothy

Exterior acoustic problems occur in a wide range of applications, making the finite element analysis of such problems a common practice in the engineering community. Various methods for truncating infinite exterior domains have been developed, including absorbing boundary conditions, infinite elements, and more recently, perfectly matched layers (PML). PML are gaining popularity due to their generality, ease of implementation, and effectiveness as an absorbing boundary condition. PML formulations have been developed in Cartesian, cylindrical, and spherical geometries, but not ellipsoidal. In addition, the parallel solution of PML formulations with iterative solvers for the solution of the Helmholtz equation, and howmore » this compares with more traditional strategies such as infinite elements, has not been adequately investigated. In this study, we present a parallel, ellipsoidal PML formulation for acoustic Helmholtz problems. To faciliate the meshing process, the ellipsoidal PML layer is generated with an on-the-fly mesh extrusion. Though the complex stretching is defined along ellipsoidal contours, we modify the Jacobian to include an additional mapping back to Cartesian coordinates in the weak formulation of the finite element equations. This allows the equations to be solved in Cartesian coordinates, which is more compatible with existing finite element software, but without the necessity of dealing with corners in the PML formulation. Herein we also compare the conditioning and performance of the PML Helmholtz problem with infinite element approach that is based on high order basis functions. On a set of representative exterior acoustic examples, we show that high order infinite element basis functions lead to an increasing number of Helmholtz solver iterations, whereas for PML the number of iterations remains constant for the same level of accuracy. Finally, this provides an additional advantage of PML over the infinite element approach.« less

Finite Element A Posteriori Error Estimation for Heat Conduction. Degree awarded by George Washington Univ.

NASA Technical Reports Server (NTRS)

Lang, Christapher G.; Bey, Kim S. (Technical Monitor)

2002-01-01

This research investigates residual-based a posteriori error estimates for finite element approximations of heat conduction in single-layer and multi-layered materials. The finite element approximation, based upon hierarchical modelling combined with p-version finite elements, is described with specific application to a two-dimensional, steady state, heat-conduction problem. Element error indicators are determined by solving an element equation for the error with the element residual as a source, and a global error estimate in the energy norm is computed by collecting the element contributions. Numerical results of the performance of the error estimate are presented by comparisons to the actual error. Two methods are discussed and compared for approximating the element boundary flux. The equilibrated flux method provides more accurate results for estimating the error than the average flux method. The error estimation is applied to multi-layered materials with a modification to the equilibrated flux method to approximate the discontinuous flux along a boundary at the material interfaces. A directional error indicator is developed which distinguishes between the hierarchical modeling error and the finite element error. Numerical results are presented for single-layered materials which show that the directional indicators accurately determine which contribution to the total error dominates.

Iterative algorithms for large sparse linear systems on parallel computers

NASA Technical Reports Server (NTRS)

Adams, L. M.

1982-01-01

Algorithms for assembling in parallel the sparse system of linear equations that result from finite difference or finite element discretizations of elliptic partial differential equations, such as those that arise in structural engineering are developed. Parallel linear stationary iterative algorithms and parallel preconditioned conjugate gradient algorithms are developed for solving these systems. In addition, a model for comparing parallel algorithms on array architectures is developed and results of this model for the algorithms are given.

Discontinuous Finite Element Quasidiffusion Methods

DOE PAGES

Anistratov, Dmitriy Yurievich; Warsa, James S.

2018-05-21

Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and themore » LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.« less

Discontinuous Finite Element Quasidiffusion Methods

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

Anistratov, Dmitriy Yurievich; Warsa, James S.

Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and themore » LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.« less

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

MHOST version 4.2. Volume 1: Users' manual

NASA Technical Reports Server (NTRS)

Nakazawa, Shohei

1989-01-01

This manual describes the user options available for running the MHOST finite element analysis package. MHOST is a solid and structural analysis program based on mixed finite element technology, and is specifically designed for three-dimensional inelastic analysis. A family of two- and three-dimensional continuum elements along with beam and shell structural elements can be utilized. Many options are available in the constitutive equation library, the solution algorithms and the analysis capabilities. An overview of the algorithms, a general description of the input data formats, and a discussion of input data for selecting solution algorithms are given.

FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

NASA Astrophysics Data System (ADS)

Regnier, D.; Dubray, N.; Verrière, M.; Schunck, N.

2018-04-01

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this paper, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different types of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank-Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of 240Pu and 256Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. We emphasize that in a realistic calculation of fission mass distributions of 240Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents).

Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation

NASA Technical Reports Server (NTRS)

Cai, Xiao-Chuan; Gropp, William D.; Keyes, David E.; Melvin, Robin G.; Young, David P.

1996-01-01

We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact finite-difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and, economical for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributed-memory parallel computer.

Electroelastic fields in a layered piezoelectric cylindrical shell under dynamic load

NASA Astrophysics Data System (ADS)

Saviz, M. R.; Shakeri, M.; Yas, M. H.

2007-10-01

The objective of this paper is to demonstrate layerwise theory for the analysis of thick laminated piezoelectric shell structures. A general finite element formulation using the layerwise theory is developed for a laminated cylindrical shell with piezoelectric layers, subjected to dynamic loads. The quadratic approximation of the displacement and electric potential in the thickness direction is considered. The governing equations are reduced to two-dimensional (2D) differential equations. The three-dimensional (3D) elasticity solution is also presented. The resulting equations are solved by a proper finite element method. The numerical results for static loading are compared with exact solutions of benchmark problems. Numerical examples of the dynamic problem are presented. The convergence is studied, as is the influence of the electromechanical coupling on the axisymmetric free-vibration characteristics of a thick cylinder.

FINITE ELEMENT MODEL FOR TIDES AND CURRENTS WITH FIELD APPLICATIONS.

USGS Publications Warehouse

Walters, Roy A.

1988-01-01

A finite element model, based upon the shallow water equations, is used to calculate tidal amplitudes and currents for two field-scale test problems. Because tides are characterized by line spectra, the governing equations are subjected to harmonic decomposition. Thus the solution variables are the real and imaginary parts of the amplitude of sea level and velocity rather than a time series of these variables. The time series is recovered through synthesis. This scheme, coupled with a modified form of the governing equations, leads to high computational efficiency and freedom from excessive numerical noise. Two test-cases are presented. The first is a solution for eleven tidal constituents in the English Channel and southern North Sea, and three constituents are discussed. The second is an analysis of the frequency response and tidal harmonics for south San Francisco Bay.

Calculation of the Full Scattering Amplitude without Partial Wave Decomposition II

NASA Technical Reports Server (NTRS)

Shertzer, J.; Temkin, A.

2003-01-01

As is well known, the full scattering amplitude can be expressed as an integral involving the complete scattering wave function. We have shown that the integral can be simplified and used in a practical way. Initial application to electron-hydrogen scattering without exchange was highly successful. The Schrodinger equation (SE) can be reduced to a 2d partial differential equation (pde), and was solved using the finite element method. We have now included exchange by solving the resultant SE, in the static exchange approximation. The resultant equation can be reduced to a pair of coupled pde's, to which the finite element method can still be applied. The resultant scattering amplitudes, both singlet and triplet, as a function of angle can be calculated for various energies. The results are in excellent agreement with converged partial wave results.

Lattice Truss Structural Response Using Energy Methods

NASA Technical Reports Server (NTRS)

Kenner, Winfred Scottson

1996-01-01

A deterministic methodology is presented for developing closed-form deflection equations for two-dimensional and three-dimensional lattice structures. Four types of lattice structures are studied: beams, plates, shells and soft lattices. Castigliano's second theorem, which entails the total strain energy of a structure, is utilized to generate highly accurate results. Derived deflection equations provide new insight into the bending and shear behavior of the four types of lattices, in contrast to classic solutions of similar structures. Lattice derivations utilizing kinetic energy are also presented, and used to examine the free vibration response of simple lattice structures. Derivations utilizing finite element theory for unique lattice behavior are also presented and validated using the finite element analysis code EAL.

Finite-element solutions for geothermal systems

NASA Technical Reports Server (NTRS)

Chen, J. C.; Conel, J. E.

1977-01-01

Vector potential and scalar potential are used to formulate the governing equations for a single-component and single-phase geothermal system. By assuming an initial temperature field, the fluid velocity can be determined which, in turn, is used to calculate the convective heat transfer. The energy equation is then solved by considering convected heat as a distributed source. Using the resulting temperature to compute new source terms, the final results are obtained by iterations of the procedure. Finite-element methods are proposed for modeling of realistic geothermal systems; the advantages of such methods are discussed. The developed methodology is then applied to a sample problem. Favorable agreement is obtained by comparisons with a previous study.

Calculation of compressible boundary layer flow about airfoils by a finite element/finite difference method

NASA Technical Reports Server (NTRS)

Strong, Stuart L.; Meade, Andrew J., Jr.

1992-01-01

Preliminary results are presented of a finite element/finite difference method (semidiscrete Galerkin method) used to calculate compressible boundary layer flow about airfoils, in which the group finite element scheme is applied to the Dorodnitsyn formulation of the boundary layer equations. The semidiscrete Galerkin (SDG) method promises to be fast, accurate and computationally efficient. The SDG method can also be applied to any smoothly connected airfoil shape without modification and possesses the potential capability of calculating boundary layer solutions beyond flow separation. Results are presented for low speed laminar flow past a circular cylinder and past a NACA 0012 airfoil at zero angle of attack at a Mach number of 0.5. Also shown are results for compressible flow past a flat plate for a Mach number range of 0 to 10 and results for incompressible turbulent flow past a flat plate. All numerical solutions assume an attached boundary layer.

Development of Composite Materials with High Passive Damping Properties

DTIC Science & Technology

2006-05-15

frequency response function analysis. Sound transmission through sandwich panels was studied using the statistical energy analysis (SEA). Modal density...2.2.3 Finite element models 14 2.2.4 Statistical energy analysis method 15 CHAPTER 3 ANALYSIS OF DAMPING IN SANDWICH MATERIALS. 24 3.1 Equation of...sheets and the core. 2.2.4 Statistical energy analysis method Finite element models are generally only efficient for problems at low and middle frequencies

Edge Equilibrium Code (EEC) For Tokamaks

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

Li, Xujling

2014-02-24

The edge equilibrium code (EEC) described in this paper is developed for simulations of the near edge plasma using the finite element method. It solves the Grad-Shafranov equation in toroidal coordinate and uses adaptive grids aligned with magnetic field lines. Hermite finite elements are chosen for the numerical scheme. A fast Newton scheme which is the same as implemented in the equilibrium and stability code (ESC) is applied here to adjust the grids

Progress on a generalized coordinates tensor product finite element 3DPNS algorithm for subsonic

NASA Technical Reports Server (NTRS)

Baker, A. J.; Orzechowski, J. A.

1983-01-01

A generalized coordinates form of the penalty finite element algorithm for the 3-dimensional parabolic Navier-Stokes equations for turbulent subsonic flows was derived. This algorithm formulation requires only three distinct hypermatrices and is applicable using any boundary fitted coordinate transformation procedure. The tensor matrix product approximation to the Jacobian of the Newton linear algebra matrix statement was also derived. Tne Newton algorithm was restructured to replace large sparse matrix solution procedures with grid sweeping using alpha-block tridiagonal matrices, where alpha equals the number of dependent variables. Numerical experiments were conducted and the resultant data gives guidance on potentially preferred tensor product constructions for the penalty finite element 3DPNS algorithm.

Plasticity - Theory and finite element applications.

NASA Technical Reports Server (NTRS)

Armen, H., Jr.; Levine, H. S.

1972-01-01

A unified presentation is given of the development and distinctions associated with various incremental solution procedures used to solve the equations governing the nonlinear behavior of structures, and this is discussed within the framework of the finite-element method. Although the primary emphasis here is on material nonlinearities, consideration is also given to geometric nonlinearities acting separately or in combination with nonlinear material behavior. The methods discussed here are applicable to a broad spectrum of structures, ranging from simple beams to general three-dimensional bodies. The finite-element analysis methods for material nonlinearity are general in the sense that any of the available plasticity theories can be incorporated to treat strain hardening or ideally plastic behavior.

Analysis of Transformation Plasticity in Steel Using a Finite Element Method Coupled with a Phase Field Model

PubMed Central

Cho, Yi-Gil; Kim, Jin-You; Cho, Hoon-Hwe; Cha, Pil-Ryung; Suh, Dong-Woo; Lee, Jae Kon; Han, Heung Nam

2012-01-01

An implicit finite element model was developed to analyze the deformation behavior of low carbon steel during phase transformation. The finite element model was coupled hierarchically with a phase field model that could simulate the kinetics and micro-structural evolution during the austenite-to-ferrite transformation of low carbon steel. Thermo-elastic-plastic constitutive equations for each phase were adopted to confirm the transformation plasticity due to the weaker phase yielding that was proposed by Greenwood and Johnson. From the simulations under various possible plastic properties of each phase, a more quantitative understanding of the origin of transformation plasticity was attempted by a comparison with the experimental observation. PMID:22558295

TORO II: A finite element computer program for nonlinear quasi-static problems in electromagnetics: Part 1, Theoretical background

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

Gartling, D.K.

The theoretical and numerical background for the finite element computer program, TORO II, is presented in detail. TORO II is designed for the multi-dimensional analysis of nonlinear, electromagnetic field problems described by the quasi-static form of Maxwell`s equations. A general description of the boundary value problems treated by the program is presented. The finite element formulation and the associated numerical methods used in TORO II are also outlined. Instructions for the use of the code are documented in SAND96-0903; examples of problems analyzed with the code are also provided in the user`s manual. 24 refs., 8 figs.

Approximation theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures

NASA Technical Reports Server (NTRS)

Gibson, J. S.; Adamian, A.

1988-01-01

An approximation theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that approximate closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The approximation scheme approximates the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.

General Rotorcraft Aeromechanical Stability Program (GRASP): Theory manual

NASA Technical Reports Server (NTRS)

Hodges, Dewey H.; Hopkins, A. Stewart; Kunz, Donald L.; Hinnant, Howard E.

1990-01-01

The general rotorcraft aeromechanical stability program (GRASP) was developed to calculate aeroelastic stability for rotorcraft in hovering flight, vertical flight, and ground contact conditions. GRASP is described in terms of its capabilities and its philosophy of modeling. The equations of motion that govern the physical system are described, as well as the analytical approximations used to derive them. The equations include the kinematical equation, the element equations, and the constraint equations. In addition, the solution procedures used by GRASP are described. GRASP is capable of treating the nonlinear static and linearized dynamic behavior of structures represented by arbitrary collections of rigid-body and beam elements. These elements may be connected in an arbitrary fashion, and are permitted to have large relative motions. The main limitation of this analysis is that periodic coefficient effects are not treated, restricting rotorcraft flight conditions to hover, axial flight, and ground contact. Instead of following the methods employed in other rotorcraft programs. GRASP is designed to be a hybrid of the finite-element method and the multibody methods used in spacecraft analysis. GRASP differs from traditional finite-element programs by allowing multiple levels of substructure in which the substructures can move and/or rotate relative to others with no small-angle approximations. This capability facilitates the modeling of rotorcraft structures, including the rotating/nonrotating interface and the details of the blade/root kinematics for various types. GRASP differs from traditional multibody programs by considering aeroelastic effects, including inflow dynamics (simple unsteady aerodynamics) and nonlinear aerodynamic coefficients.

Finite elements and the method of conjugate gradients on a concurrent processor

NASA Technical Reports Server (NTRS)

Lyzenga, G. A.; Raefsky, A.; Hager, G. H.

1985-01-01

An algorithm for the iterative solution of finite element problems on a concurrent processor is presented. The method of conjugate gradients is used to solve the system of matrix equations, which is distributed among the processors of a MIMD computer according to an element-based spatial decomposition. This algorithm is implemented in a two-dimensional elastostatics program on the Caltech Hypercube concurrent processor. The results of tests on up to 32 processors show nearly linear concurrent speedup, with efficiencies over 90 percent for sufficiently large problems.

Finite elements and the method of conjugate gradients on a concurrent processor

NASA Technical Reports Server (NTRS)

Lyzenga, G. A.; Raefsky, A.; Hager, B. H.

1984-01-01

An algorithm for the iterative solution of finite element problems on a concurrent processor is presented. The method of conjugate gradients is used to solve the system of matrix equations, which is distributed among the processors of a MIMD computer according to an element-based spatial decomposition. This algorithm is implemented in a two-dimensional elastostatics program on the Caltech Hypercube concurrent processor. The results of tests on up to 32 processors show nearly linear concurrent speedup, with efficiencies over 90% for sufficiently large problems.

Rigorous joining of advanced reduced-dimensional beam models to three-dimensional finite element models

NASA Astrophysics Data System (ADS)

Song, Huimin

In the aerospace and automotive industries, many finite element analyses use lower-dimensional finite elements such as beams, plates and shells, to simplify the modeling. These simplified models can greatly reduce the computation time and cost; however, reduced-dimensional models may introduce inaccuracies, particularly near boundaries and near portions of the structure where reduced-dimensional models may not apply. Another factor in creation of such models is that beam-like structures frequently have complex geometry, boundaries and loading conditions, which may make them unsuitable for modeling with single type of element. The goal of this dissertation is to develop a method that can accurately and efficiently capture the response of a structure by rigorous combination of a reduced-dimensional beam finite element model with a model based on full two-dimensional (2D) or three-dimensional (3D) finite elements. The first chapter of the thesis gives the background of the present work and some related previous work. The second chapter is focused on formulating a system of equations that govern the joining of a 2D model with a beam model for planar deformation. The essential aspect of this formulation is to find the transformation matrices to achieve deflection and load continuity on the interface. Three approaches are provided to obtain the transformation matrices. An example based on joining a beam to a 2D finite element model is examined, and the accuracy of the analysis is studied by comparing joint results with the full 2D analysis. The third chapter is focused on formulating the system of equations for joining a beam to a 3D finite element model for static and free-vibration problems. The transition between the 3D elements and beam elements is achieved by use of the stress recovery technique of the variational-asymptotic method as implemented in VABS (the Variational Asymptotic Beam Section analysis). The formulations for an interface transformation matrix and the generalized Timoshenko beam are discussed in this chapter. VABS is also used to obtain the beam constitutive properties and warping functions for stress recovery. Several 3D-beam joint examples are presented to show the convergence and accuracy of the analysis. Accuracy is accessed by comparing the joint results with the full 3D analysis. The fourth chapter provides conclusions from present studies and recommendations for future work.

Distribution factors for construction loads and girder capacity equations [project summary].

DOT National Transportation Integrated Search

2017-03-01

This project focused on the use of Florida I-beams (FIBs) in bridge construction. University of Florida researchers used analytical models and finite element analysis to update equations used in the design of bridges using FIBs. They were particularl...

A projection hybrid high order finite volume/finite element method for incompressible turbulent flows

NASA Astrophysics Data System (ADS)

Busto, S.; Ferrín, J. L.; Toro, E. F.; Vázquez-Cendón, M. E.

2018-01-01

In this paper the projection hybrid FV/FE method presented in [1] is extended to account for species transport equations. Furthermore, turbulent regimes are also considered thanks to the k-ε model. Regarding the transport diffusion stage new schemes of high order of accuracy are developed. The CVC Kolgan-type scheme and ADER methodology are extended to 3D. The latter is modified in order to profit from the dual mesh employed by the projection algorithm and the derivatives involved in the diffusion term are discretized using a Galerkin approach. The accuracy and stability analysis of the new method are carried out for the advection-diffusion-reaction equation. Within the projection stage the pressure correction is computed by a piecewise linear finite element method. Numerical results are presented, aimed at verifying the formal order of accuracy of the scheme and to assess the performance of the method on several realistic test problems.

The accuracy of the Gaussian-and-finite-element-Coulomb (GFC) method for the calculation of Coulomb integrals.

PubMed

Przybytek, Michal; Helgaker, Trygve

2013-08-07

We analyze the accuracy of the Coulomb energy calculated using the Gaussian-and-finite-element-Coulomb (GFC) method. In this approach, the electrostatic potential associated with the molecular electronic density is obtained by solving the Poisson equation and then used to calculate matrix elements of the Coulomb operator. The molecular electrostatic potential is expanded in a mixed Gaussian-finite-element (GF) basis set consisting of Gaussian functions of s symmetry centered on the nuclei (with exponents obtained from a full optimization of the atomic potentials generated by the atomic densities from symmetry-averaged restricted open-shell Hartree-Fock theory) and shape functions defined on uniform finite elements. The quality of the GF basis is controlled by means of a small set of parameters; for a given width of the finite elements d, the highest accuracy is achieved at smallest computational cost when tricubic (n = 3) elements are used in combination with two (γ(H) = 2) and eight (γ(1st) = 8) Gaussians on hydrogen and first-row atoms, respectively, with exponents greater than a given threshold (αmin (G)=0.5). The error in the calculated Coulomb energy divided by the number of atoms in the system depends on the system type but is independent of the system size or the orbital basis set, vanishing approximately like d(4) with decreasing d. If the boundary conditions for the Poisson equation are calculated in an approximate way, the GFC method may lose its variational character when the finite elements are too small; with larger elements, it is less sensitive to inaccuracies in the boundary values. As it is possible to obtain accurate boundary conditions in linear time, the overall scaling of the GFC method for large systems is governed by another computational step-namely, the generation of the three-center overlap integrals with three Gaussian orbitals. The most unfavorable (nearly quadratic) scaling is observed for compact, truly three-dimensional systems; however, this scaling can be reduced to linear by introducing more effective techniques for recognizing significant three-center overlap distributions.

A Second Order Semi-Discrete Cosserat Rod Model Suitable for Dynamic Simulations in Real Time

NASA Astrophysics Data System (ADS)

Lang, Holger; Linn, Joachim

2009-09-01

We present an alternative approach for a semi-discrete viscoelastic Cosserat rod model that allows both fast dynamic computations within milliseconds and accurate results compared to detailed finite element solutions. The model is able to represent extension, shearing, bending and torsion. For inner dissipation, a consistent damping potential from Antman is chosen. The continuous equations of motion, which consist a system of nonlinear hyperbolic partial differential algebraic equations, are derived from a two dimensional variational principle. The semi-discrete balance equations are obtained by spatial finite difference schemes on a staggered grid and standard index reduction techniques. The right-hand side of the model and its Jacobian can be chosen free of higher algebraic (e.g. root) or transcendent (e.g. trigonometric or exponential) functions and is therefore extremely cheap to evaluate numerically. For the time integration of the system, we use well established stiff solvers. As our model yields computational times within milliseconds, it is suitable for interactive manipulation. It reflects structural mechanics solutions sufficiently correct, as comparison with detailed finite element results shows.

A Thermo-Optic Propagation Modeling Capability.

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

Schrader, Karl; Akau, Ron

2014-10-01

A new theoretical basis is derived for tracing optical rays within a finite-element (FE) volume. The ray-trajectory equations are cast into the local element coordinate frame and the full finite-element interpolation is used to determine instantaneous index gradient for the ray-path integral equation. The FE methodology (FEM) is also used to interpolate local surface deformations and the surface normal vector for computing the refraction angle when launching rays into the volume, and again when rays exit the medium. The method is implemented in the Matlab(TM) environment and compared to closed- form gradient index models. A software architecture is also developedmore » for implementing the algorithms in the Zemax(TM) commercial ray-trace application. A controlled thermal environment was constructed in the laboratory, and measured data was collected to validate the structural, thermal, and optical modeling methods.« less

Simplified equation for Young's modulus of CNT reinforced concrete

NASA Astrophysics Data System (ADS)

Chandran, RameshBabu; Gifty Honeyta A, Maria

2017-12-01

This research investigation focuses on finite element modeling of carbon nanotube (CNT) reinforced concrete matrix for three grades of concrete namely M40, M60 and M120. Representative volume element (RVE) was adopted and one-eighth model depicting the CNT reinforced concrete matrix was simulated using FEA software ANSYS17.2. Adopting random orientation of CNTs, with nine fibre volume fractions from 0.1% to 0.9%, finite element modeling simulations replicated exactly the CNT reinforced concrete matrix. Upon evaluations of the model, the longitudinal and transverse Young's modulus of elasticity of the CNT reinforced concrete was arrived. The graphical plots between various fibre volume fractions and the concrete grade revealed simplified equation for estimating the young's modulus. It also exploited the fact that the concrete grade does not have significant impact in CNT reinforced concrete matrix.

Finite Element Peen Forming Simulation

NASA Astrophysics Data System (ADS)

Gariépy, Alexandre; Larose, Simon; Perron, Claude; Bocher, Philippe; Lévesque, Martin

Shot peening consists of projecting multiple small particles onto a ductile part in order to induce compressive residual stresses near the surface. Peen forming, a derivative of shot peening, is a process that creates an unbalanced stress state which in turn leads to a deformation to shape thin parts. This versatile and cost-effective process is commonly used to manufacture aluminum wing skins and rocket panels. This paper presents the finite element modelling approach that was developed by the authors to simulate the process. The method relies on shell elements and calculated stress profiles and uses an approximation equation to take into account the incremental nature of the process. Finite element predictions were in good agreement with experimental results for small-scale tests. The method was extended to a hypothetical wing skin model to show its potential applications.

Finite element analysis of inviscid subsonic boattail flow

NASA Technical Reports Server (NTRS)

Chima, R. V.; Gerhart, P. M.

1981-01-01

A finite element code for analysis of inviscid subsonic flows over arbitrary nonlifting planar or axisymmetric bodies is described. The code solves a novel primitive variable formulation of the coupled irrotationality and compressible continuity equations. Results for flow over a cylinder, a sphere, and a NACA 0012 airfoil verify the code. Computed subcritical flows over an axisymmetric boattailed afterbody compare well with finite difference results and experimental data. Interative coupling with an integral turbulent boundary layer code shows strong viscous effects on the inviscid flow. Improvements in code efficiency and extensions to transonic flows are discussed.

Well-Balanced Second-Order Approximation of the Shallow Water Equations With Friction via Continuous Galerkin Finite Elements

NASA Astrophysics Data System (ADS)

Quezada de Luna, M.; Farthing, M.; Guermond, J. L.; Kees, C. E.; Popov, B.

2017-12-01

The Shallow Water Equations (SWEs) are popular for modeling non-dispersive incompressible water waves where the horizontal wavelength is much larger than the vertical scales. They can be derived from the incompressible Navier-Stokes equations assuming a constant vertical velocity. The SWEs are important in Geophysical Fluid Dynamics for modeling surface gravity waves in shallow regimes; e.g., in the deep ocean. Some common geophysical applications are the evolution of tsunamis, river flooding and dam breaks, storm surge simulations, atmospheric flows and others. This work is concerned with the approximation of the time-dependent Shallow Water Equations with friction using explicit time stepping and continuous finite elements. The objective is to construct a method that is at least second-order accurate in space and third or higher-order accurate in time, positivity preserving, well-balanced with respect to rest states, well-balanced with respect to steady sliding solutions on inclined planes and robust with respect to dry states. Methods fulfilling the desired goals are common within the finite volume literature. However, to the best of our knowledge, schemes with the above properties are not well developed in the context of continuous finite elements. We start this work based on a finite element method that is second-order accurate in space, positivity preserving and well-balanced with respect to rest states. We extend it by: modifying the artificial viscosity (via the entropy viscosity method) to deal with issues of loss of accuracy around local extrema, considering a singular Manning friction term handled via an explicit discretization under the usual CFL condition, considering a water height regularization that depends on the mesh size and is consistent with the polynomial approximation, reducing dispersive errors introduced by lumping the mass matrix and others. After presenting the details of the method we show numerical tests that demonstrate the well-balanced nature of the scheme and its convergence properties. We conclude with well-known benchmark problems including the Malpasset dam break (see the attached figure). All numerical experiments are performed and available in the Proteus toolkit, which is an open source python package for modeling continuum mechanical processes and fluid flow.

Mixed models and reduction method for dynamic analysis of anisotropic shells

NASA Technical Reports Server (NTRS)

Noor, A. K.; Peters, J. M.

1985-01-01

A time-domain computational procedure is presented for predicting the dynamic response of laminated anisotropic shells. The two key elements of the procedure are: (1) use of mixed finite element models having independent interpolation (shape) functions for stress resultants and generalized displacements for the spatial discretization of the shell, with the stress resultants allowed to be discontinuous at interelement boundaries; and (2) use of a dynamic reduction method, with the global approximation vectors consisting of the static solution and an orthogonal set of Lanczos vectors. The dynamic reduction is accomplished by means of successive application of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is first used to generate the global approximation vectors. Then the Rayleigh-Ritz technique is used to generate a reduced system of ordinary differential equations in the amplitudes of these modes. The temporal integration of the reduced differential equations is performed by using an explicit half-station central difference scheme (Leap-frog method). The effectiveness of the proposed procedure is demonstrated by means of a numerical example and its advantages over reduction methods used with the displacement formulation are discussed.

Stabilised finite-element methods for solving the level set equation with mass conservation

NASA Astrophysics Data System (ADS)

Kabirou Touré, Mamadou; Fahsi, Adil; Soulaïmani, Azzeddine

2016-01-01

Finite-element methods are studied for solving moving interface flow problems using the level set approach and a stabilised variational formulation proposed in Touré and Soulaïmani (2012; Touré and Soulaïmani To appear in 2016), coupled with a level set correction method. The level set correction is intended to enhance the mass conservation satisfaction property. The stabilised variational formulation (Touré and Soulaïmani 2012; Touré and Soulaïmani, To appear in 2016) constrains the level set function to remain close to the signed distance function, while the mass conservation is a correction step which enforces the mass balance. The eXtended finite-element method (XFEM) is used to take into account the discontinuities of the properties within an element. XFEM is applied to solve the Navier-Stokes equations for two-phase flows. The numerical methods are numerically evaluated on several test cases such as time-reversed vortex flow, a rigid-body rotation of Zalesak's disc, sloshing flow in a tank, a dam-break over a bed, and a rising bubble subjected to buoyancy. The numerical results show the importance of satisfying global mass conservation to accurately capture the interface position.

Finite Element Solution of Unsteady Mixed Convection Flow of Micropolar Fluid over a Porous Shrinking Sheet

PubMed Central

Gupta, Diksha; Singh, Bani

2014-01-01

The objective of this investigation is to analyze the effect of unsteadiness on the mixed convection boundary layer flow of micropolar fluid over a permeable shrinking sheet in the presence of viscous dissipation. At the sheet a variable distribution of suction is assumed. The unsteadiness in the flow and temperature fields is caused by the time dependence of the shrinking velocity and surface temperature. With the aid of similarity transformations, the governing partial differential equations are transformed into a set of nonlinear ordinary differential equations, which are solved numerically, using variational finite element method. The influence of important physical parameters, namely, suction parameter, unsteadiness parameter, buoyancy parameter and Eckert number on the velocity, microrotation, and temperature functions is investigated and analyzed with the help of their graphical representations. Additionally skin friction and the rate of heat transfer have also been computed. Under special conditions, an exact solution for the flow velocity is compared with the numerical results obtained by finite element method. An excellent agreement is observed for the two sets of solutions. Furthermore, to verify the convergence of numerical results, calculations are conducted with increasing number of elements. PMID:24672310

A finite element code for modelling tracer transport in a non-isothermal two-phase flow system for CO2 geological storage characterization

NASA Astrophysics Data System (ADS)

Tong, F.; Niemi, A. P.; Yang, Z.; Fagerlund, F.; Licha, T.; Sauter, M.

2011-12-01

This paper presents a new finite element method (FEM) code for modeling tracer transport in a non-isothermal two-phase flow system. The main intended application is simulation of the movement of so-called novel tracers for the purpose of characterization of geologically stored CO2 and its phase partitioning and migration in deep saline formations. The governing equations are based on the conservation of mass and energy. Among the phenomena accounted for are liquid-phase flow, gas flow, heat transport and the movement of the novel tracers. The movement of tracers includes diffusion and the advection associated with the gas and liquid flow. The temperature, gas pressure, suction, concentration of tracer in liquid phase and concentration of tracer in gas phase are chosen as the five primary variables. Parameters such as the density, viscosity, thermal expansion coefficient are expressed in terms of the primary variables. The governing equations are discretized in space using the Galerkin finite element formulation, and are discretized in time by one-dimensional finite difference scheme. This leads to an ill-conditioned FEM equation that has many small entries along the diagonal of the non-symmetric coefficient matrix. In order to deal with the problem of non-symmetric ill-conditioned matrix equation, special techniques are introduced . Firstly, only nonzero elements of the matrix need to be stored. Secondly, it is avoided to directly solve the whole large matrix. Thirdly, a strategy has been used to keep the diversity of solution methods in the calculation process. Additionally, an efficient adaptive mesh technique is included in the code in order to track the wetting front. The code has been validated against several classical analytical solutions, and will be applied for simulating the CO2 injection experiment to be carried out at the Heletz site, Israel, as part of the EU FP7 project MUSTANG.

Numerical simulations of earthquakes and the dynamics of fault systems using the Finite Element method.

NASA Astrophysics Data System (ADS)

Kettle, L. M.; Mora, P.; Weatherley, D.; Gross, L.; Xing, H.

2006-12-01

Simulations using the Finite Element method are widely used in many engineering applications and for the solution of partial differential equations (PDEs). Computational models based on the solution of PDEs play a key role in earth systems simulations. We present numerical modelling of crustal fault systems where the dynamic elastic wave equation is solved using the Finite Element method. This is achieved using a high level computational modelling language, escript, available as open source software from ACcESS (Australian Computational Earth Systems Simulator), the University of Queensland. Escript is an advanced geophysical simulation software package developed at ACcESS which includes parallel equation solvers, data visualisation and data analysis software. The escript library was implemented to develop a flexible Finite Element model which reliably simulates the mechanism of faulting and the physics of earthquakes. Both 2D and 3D elastodynamic models are being developed to study the dynamics of crustal fault systems. Our final goal is to build a flexible model which can be applied to any fault system with user-defined geometry and input parameters. To study the physics of earthquake processes, two different time scales must be modelled, firstly the quasi-static loading phase which gradually increases stress in the system (~100years), and secondly the dynamic rupture process which rapidly redistributes stress in the system (~100secs). We will discuss the solution of the time-dependent elastic wave equation for an arbitrary fault system using escript. This involves prescribing the correct initial stress distribution in the system to simulate the quasi-static loading of faults to failure; determining a suitable frictional constitutive law which accurately reproduces the dynamics of the stick/slip instability at the faults; and using a robust time integration scheme. These dynamic models generate data and information that can be used for earthquake forecasting.

Matrix elements and duality for type 2 unitary representations of the Lie superalgebra gl(m|n)

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

Werry, Jason L.; Gould, Mark D.; Isaac, Phillip S.

The characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible gl(m|n) modules. In particular, we give matrix element formulae for all gl(m|n) generators, including the non-elementary generators, together with their phases on finite dimensional type 2 unitary irreducible representations which include the contravariant tensor representations and an additional class of essentially typical representations. Remarkably, we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non-vanishing matrix elements up to a phase.

Calculation of Thermally-Induced Displacements in Spherically Domed Ion Engine Grids

NASA Technical Reports Server (NTRS)

Soulas, George C.

2006-01-01

An analytical method for predicting the thermally-induced normal and tangential displacements of spherically domed ion optics grids under an axisymmetric thermal loading is presented. A fixed edge support that could be thermally expanded is used for this analysis. Equations for the displacements both normal and tangential to the surface of the spherical shell are derived. A simplified equation for the displacement at the center of the spherical dome is also derived. The effects of plate perforation on displacements and stresses are determined by modeling the perforated plate as an equivalent solid plate with modified, or effective, material properties. Analytical model results are compared to the results from a finite element model. For the solid shell, comparisons showed that the analytical model produces results that closely match the finite element model results. The simplified equation for the normal displacement of the spherical dome center is also found to accurately predict this displacement. For the perforated shells, the analytical solution and simplified equation produce accurate results for materials with low thermal expansion coefficients.

A finite element simulation of sound attenuation in a finite duct with a peripherally variable liner

NASA Technical Reports Server (NTRS)

Watson, W. R.

1977-01-01

Using multimodal analysis, a variational finite element method is presented for analyzing sound attenuation in a three-dimensional finite duct with a peripherally variable liner in the absence of flow. A rectangular element, with cubic shaped functions, is employed. Once a small portion of a peripheral liner is removed, the attenuation rate near the frequency where maximum attenuation occurs drops significantly. The positioning of the liner segments affects the attenuation characteristics of the liner. Effects of the duct termination are important in the low frequency ranges. The main effect of peripheral variation of the liner is a broadening of the attenuation characteristics in the midfrequency range. Because of matrix size limitations of the presently available computer program, the eigenvalue equations should be solved out of core in order to handle realistic sources.

Finite Element in Angle Unit Sphere Meshing for Charged Particle Transport.

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

Ortega, Mario Ivan; Drumm, Clifton R.

Finite element in angle formulations of the charged particle transport equation require the discretization of the unit sphere. In Sceptre, a three-dimensional surface mesh of a sphere is transformed into a two-dimensional mesh. Projection of a sphere onto a two-dimensional surface is well studied with map makers spending the last few centuries attempting to create maps that preserve proportion and area. Using these techniques, various meshing schemes for the unit sphere were investigated.

KANTBP: A program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

NASA Astrophysics Data System (ADS)

Chuluunbaatar, O.; Gusev, A. A.; Abrashkevich, A. G.; Amaya-Tapia, A.; Kaschiev, M. S.; Larsen, S. Y.; Vinitsky, S. I.

2007-10-01

A FORTRAN 77 program is presented which calculates energy values, reaction matrix and corresponding radial wave functions in a coupled-channel approximation of the hyperspherical adiabatic approach. In this approach, a multi-dimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on the finite interval with homogeneous boundary conditions of the third type. The resulting system of radial equations which contains the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite-element method. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials. Program summaryProgram title: KANTBP Catalogue identifier: ADZH_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZH_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.: 4224 No. of bytes in distributed program, including test data, etc.: 31 232 Distribution format: tar.gz Programming language: FORTRAN 77 Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP RAM: depends on (a) the number of differential equations; (b) the number and order of finite-elements; (c) the number of hyperradial points; and (d) the number of eigensolutions required. Test run requires 30 MB Classification: 2.1, 2.4 External routines: GAULEG and GAUSSJ [W.H. Press, B.F. Flanery, S.A. Teukolsky, W.T. Vetterley, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986] Nature of problem: In the hyperspherical adiabatic approach [J. Macek, J. Phys. B 1 (1968) 831-843; U. Fano, Rep. Progr. Phys. 46 (1983) 97-165; C.D. Lin, Adv. Atom. Mol. Phys. 22 (1986) 77-142], a multi-dimensional Schrödinger equation for a two-electron system [A.G. Abrashkevich, D.G. Abrashkevich, M. Shapiro, Comput. Phys. Comm. 90 (1995) 311-339] or a hydrogen atom in magnetic field [M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337-2352] is reduced by separating the radial coordinate ρ from the angular variables to a system of second-order ordinary differential equations which contain potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present the finite-element method procedure based on the use of high-order accuracy approximations for calculating approximate eigensolutions for such systems of coupled differential equations. Solution method: The boundary problems for coupled differential equations are solved by the finite-element method using high-order accuracy approximations [A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 85 (1995) 40-64]. The generalized algebraic eigenvalue problem AF=EBF with respect to pair unknowns ( E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982]. The generalized algebraic eigenvalue problem (A-EB)F=λDF with respect to pair unknowns (λ,F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the LDL factorization of symmetric matrix and back-substitution methods using the DECOMP and REDBAK programs, respectively [K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice-Hall, New York, 1982]. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials described in [Yu. A. Kuperin, P.B. Kurasov, Yu.B. Melnikov, S.P. Merkuriev, Ann. Phys. 205 (1991) 330-361; O. Chuluunbaatar, A.A. Gusev, S.Y. Larsen, S.I. Vinitsky, J. Phys. A 35 (2002) L513-L525; N.P. Mehta, J.R. Shepard, Phys. Rev. A 72 (2005) 032728-1-11; O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen, S.I. Vinitsky, J. Phys. B 39 (2006) 243-269]. For this benchmark model the needed analytical expressions for the potential matrix elements and first-derivative coupling terms, their asymptotics and asymptotics of radial solutions of the boundary problems for coupled differential equations have been produced with help of a MAPLE computer algebra system. Restrictions: The computer memory requirements depend on: (a) the number of differential equations; (b) the number and order of finite-elements; (c) the total number of hyperradial points; and (d) the number of eigensolutions required. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write-Up and listing for details). The user must also supply subroutine POTCAL for evaluating potential matrix elements. The user should supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMSC (when solving the scattering problem) that evaluate the asymptotics of the radial wave functions at the right boundary point in case of a boundary condition of the third type, respectively. Running time: The running time depends critically upon: (a) the number of differential equations; (b) the number and order of finite-elements; (c) the total number of hyperradial points on interval [0,ρ]; and (d) the number of eigensolutions required. The test run which accompanies this paper took 28.48 s without calculation of matrix potentials on the Intel Pentium IV 2.4 GHz.

A finite element: Boundary integral method for electromagnetic scattering. Ph.D. Thesis Technical Report, Feb. - Sep. 1992

NASA Technical Reports Server (NTRS)

Collins, J. D.; Volakis, John L.

1992-01-01

A method that combines the finite element and boundary integral techniques for the numerical solution of electromagnetic scattering problems is presented. The finite element method is well known for requiring a low order storage and for its capability to model inhomogeneous structures. Of particular emphasis in this work is the reduction of the storage requirement by terminating the finite element mesh on a boundary in a fashion which renders the boundary integrals in convolutional form. The fast Fourier transform is then used to evaluate these integrals in a conjugate gradient solver, without a need to generate the actual matrix. This method has a marked advantage over traditional integral equation approaches with respect to the storage requirement of highly inhomogeneous structures. Rectangular, circular, and ogival mesh termination boundaries are examined for two-dimensional scattering. In the case of axially symmetric structures, the boundary integral matrix storage is reduced by exploiting matrix symmetries and solving the resulting system via the conjugate gradient method. In each case several results are presented for various scatterers aimed at validating the method and providing an assessment of its capabilities. Important in methods incorporating boundary integral equations is the issue of internal resonance. A method is implemented for their removal, and is shown to be effective in the two-dimensional and three-dimensional applications.

Finite-element time-domain modeling of electromagnetic data in general dispersive medium using adaptive Padé series

NASA Astrophysics Data System (ADS)

Cai, Hongzhu; Hu, Xiangyun; Xiong, Bin; Zhdanov, Michael S.

2017-12-01

The induced polarization (IP) method has been widely used in geophysical exploration to identify the chargeable targets such as mineral deposits. The inversion of the IP data requires modeling the IP response of 3D dispersive conductive structures. We have developed an edge-based finite-element time-domain (FETD) modeling method to simulate the electromagnetic (EM) fields in 3D dispersive medium. We solve the vector Helmholtz equation for total electric field using the edge-based finite-element method with an unstructured tetrahedral mesh. We adopt the backward propagation Euler method, which is unconditionally stable, with semi-adaptive time stepping for the time domain discretization. We use the direct solver based on a sparse LU decomposition to solve the system of equations. We consider the Cole-Cole model in order to take into account the frequency-dependent conductivity dispersion. The Cole-Cole conductivity model in frequency domain is expanded using a truncated Padé series with adaptive selection of the center frequency of the series for early and late time. This approach can significantly increase the accuracy of FETD modeling.

Finite element formulation of viscoelastic sandwich beams using fractional derivative operators

NASA Astrophysics Data System (ADS)

Galucio, A. C.; Deü, J.-F.; Ohayon, R.

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler-Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects.

The Mixed Finite Element Multigrid Method for Stokes Equations

PubMed Central

Muzhinji, K.; Shateyi, S.; Motsa, S. S.

2015-01-01

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q 2-Q 1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results. PMID:25945361

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

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.

Generalized Fourier analyses of the advection-diffusion equation - Part I: one-dimensional domains

NASA Astrophysics Data System (ADS)

Christon, Mark A.; Martinez, Mario J.; Voth, Thomas E.

2004-07-01

This paper presents a detailed multi-methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi-discretizations of the scalar advection-diffusion equation. The errors are reported in terms of non-dimensional phase and group speed, discrete diffusivity, artificial diffusivity, and grid-induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew-symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. It is demonstrated that streamline upwind Petrov-Galerkin and its control-volume finite element analogue, the streamline upwind control-volume method, produce both an artificial diffusivity and a concomitant phase speed adjustment in addition to the usual semi-discrete artifacts observed in the phase speed, group speed and diffusivity. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super-convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second-order behaviour. In Part II of this paper, we consider two-dimensional semi-discretizations of the advection-diffusion equation and also assess the affects of grid-induced anisotropy observed in the non-dimensional phase speed, and the discrete and artificial diffusivities. Although this work can only be considered a first step in a comprehensive multi-methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.

Finite element modeling of electromagnetic fields and waves using NASTRAN

NASA Technical Reports Server (NTRS)

Moyer, E. Thomas, Jr.; Schroeder, Erwin

1989-01-01

The various formulations of Maxwell's equations are reviewed with emphasis on those formulations which most readily form analogies with Navier's equations. Analogies involving scalar and vector potentials and electric and magnetic field components are presented. Formulations allowing for media with dielectric and conducting properties are emphasized. It is demonstrated that many problems in electromagnetism can be solved using the NASTRAN finite element code. Several fundamental problems involving time harmonic solutions of Maxwell's equations with known analytic solutions are solved using NASTRAN to demonstrate convergence and mesh requirements. Mesh requirements are studied as a function of frequency, conductivity, and dielectric properties. Applications in both low frequency and high frequency are highlighted. The low frequency problems demonstrate the ability to solve problems involving media inhomogeneity and unbounded domains. The high frequency applications demonstrate the ability to handle problems with large boundary to wavelength ratios.

Experimental and numerical analysis of the constitutive equation of rubber composites reinforced with random ceramic particle

NASA Astrophysics Data System (ADS)

Luo, D. M.; Xie, Y.; Su, X. R.; Zhou, Y. L.

2018-01-01

Based on the four classical models of Mooney-Rivlin (M-R), Yeoh, Ogden and Neo-Hookean (N-H) model, a strain energy constitutive equation with large deformation for rubber composites reinforced with random ceramic particles is proposed from the angle of continuum mechanics theory in this paper. By decoupling the interaction between matrix and random particles, the strain energy of each phase is obtained to derive the explicit constitutive equation for rubber composites. The tests results of uni-axial tensile, pure shear and equal bi-axial tensile are simulated by the non-linear finite element method on the ANSYS platform. The results from finite element method are compared with those from experiment, and the material parameters are determined by fitting the results from different test conditions, and the influence of radius of random ceramic particles on the effective mechanical properties are analyzed.

Mixed variational formulations of finite element analysis of elastoacoustic/slosh fluid-structure interaction

NASA Technical Reports Server (NTRS)

Felippa, Carlos A.; Ohayon, Roger

1991-01-01

A general three-field variational principle is obtained for the motion of an acoustic fluid enclosed in a rigid or flexible container by the method of canonical decomposition applied to a modified form of the wave equation in the displacement potential. The general principle is specialized to a mixed two-field principle that contains the fluid displacement potential and pressure as independent fields. This principle contains a free parameter alpha. Semidiscrete finite-element equations of motion based on this principle are displayed and applied to the transient response and free-vibrations of the coupled fluid-structure problem. It is shown that a particular setting of alpha yields a rich set of formulations that can be customized to fit physical and computational requirements. The variational principle is then extended to handle slosh motions in a uniform gravity field, and used to derive semidiscrete equations of motion that account for such effects.

The solution of non-linear hyperbolic equation systems by the finite element method

NASA Technical Reports Server (NTRS)

Loehner, R.; Morgan, K.; Zienkiewicz, O. C.

1984-01-01

A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations

NASA Astrophysics Data System (ADS)

Whiteley, J. P.

2017-10-01

Large, incompressible elastic deformations are governed by a system of nonlinear partial differential equations. The finite element discretisation of these partial differential equations yields a system of nonlinear algebraic equations that are usually solved using Newton's method. On each iteration of Newton's method, a linear system must be solved. We exploit the structure of the Jacobian matrix to propose a preconditioner, comprising two steps. The first step is the solution of a relatively small, symmetric, positive definite linear system using the preconditioned conjugate gradient method. This is followed by a small number of multigrid V-cycles for a larger linear system. Through the use of exemplar elastic deformations, the preconditioner is demonstrated to facilitate the iterative solution of the linear systems arising. The number of GMRES iterations required has only a very weak dependence on the number of degrees of freedom of the linear systems.

EMPHASIS/Nevada UTDEM user guide. Version 2.0.

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

Turner, C. David; Seidel, David Bruce; Pasik, Michael Francis

The Unstructured Time-Domain ElectroMagnetics (UTDEM) portion of the EMPHASIS suite solves Maxwell's equations using finite-element techniques on unstructured meshes. This document provides user-specific information to facilitate the use of the code for applications of interest. UTDEM is a general-purpose code for solving Maxwell's equations on arbitrary, unstructured tetrahedral meshes. The geometries and the meshes thereof are limited only by the patience of the user in meshing and by the available computing resources for the solution. UTDEM solves Maxwell's equations using finite-element method (FEM) techniques on tetrahedral elements using vector, edge-conforming basis functions. EMPHASIS/Nevada Unstructured Time-Domain ElectroMagnetic Particle-In-Cell (UTDEM PIC) ismore » a superset of the capabilities found in UTDEM. It adds the capability to simulate systems in which the effects of free charge are important and need to be treated in a self-consistent manner. This is done by integrating the equations of motion for macroparticles (a macroparticle is an object that represents a large number of real physical particles, all with the same position and momentum) being accelerated by the electromagnetic forces upon the particle (Lorentz force). The motion of these particles results in a current, which is a source for the fields in Maxwell's equations.« less

Coupled bending-torsion steady-state response of pretwisted, nonuniform rotating beams using a transfer-matrix method

NASA Technical Reports Server (NTRS)

Gray, Carl E., Jr.

1988-01-01

Using the Newtonian method, the equations of motion are developed for the coupled bending-torsion steady-state response of beams rotating at constant angular velocity in a fixed plane. The resulting equations are valid to first order strain-displacement relationships for a long beam with all other nonlinear terms retained. In addition, the equations are valid for beams with the mass centroidal axis offset (eccentric) from the elastic axis, nonuniform mass and section properties, and variable twist. The solution of these coupled, nonlinear, nonhomogeneous, differential equations is obtained by modifying a Hunter linear second-order transfer-matrix solution procedure to solve the nonlinear differential equations and programming the solution for a desk-top personal computer. The modified transfer-matrix method was verified by comparing the solution for a rotating beam with a geometric, nonlinear, finite-element computer code solution; and for a simple rotating beam problem, the modified method demonstrated a significant advantage over the finite-element solution in accuracy, ease of solution, and actual computer processing time required to effect a solution.

Multiscale Concrete Modeling of Aging Degradation

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

Hammi, Yousseff; Gullett, Philipp; Horstemeyer, Mark F.

In this work a numerical finite element framework is implemented to enable the integration of coupled multiscale and multiphysics transport processes. A User Element subroutine (UEL) in Abaqus is used to simultaneously solve stress equilibrium, heat conduction, and multiple diffusion equations for 2D and 3D linear and quadratic elements. Transport processes in concrete structures and their degradation mechanisms are presented along with the discretization of the governing equations. The multiphysics modeling framework is theoretically extended to the linear elastic fracture mechanics (LEFM) by introducing the eXtended Finite Element Method (XFEM) and based on the XFEM user element implementation of Ginermore » et al. [2009]. A damage model that takes into account the damage contribution from the different degradation mechanisms is theoretically developed. The total contribution of damage is forwarded to a Multi-Stage Fatigue (MSF) model to enable the assessment of the fatigue life and the deterioration of reinforced concrete structures in a nuclear power plant. Finally, two examples are presented to illustrate the developed multiphysics user element implementation and the XFEM implementation of Giner et al. [2009].« less

A Dynamic Finite Element Method for Simulating the Physics of Faults Systems

NASA Astrophysics Data System (ADS)

Saez, E.; Mora, P.; Gross, L.; Weatherley, D.

2004-12-01

We introduce a dynamic Finite Element method using a novel high level scripting language to describe the physical equations, boundary conditions and time integration scheme. The library we use is the parallel Finley library: a finite element kernel library, designed for solving large-scale problems. It is incorporated as a differential equation solver into a more general library called escript, based on the scripting language Python. This library has been developed to facilitate the rapid development of 3D parallel codes, and is optimised for the Australian Computational Earth Systems Simulator Major National Research Facility (ACcESS MNRF) supercomputer, a 208 processor SGI Altix with a peak performance of 1.1 TFlops. Using the scripting approach we obtain a parallel FE code able to take advantage of the computational efficiency of the Altix 3700. We consider faults as material discontinuities (the displacement, velocity, and acceleration fields are discontinuous at the fault), with elastic behavior. The stress continuity at the fault is achieved naturally through the expression of the fault interactions in the weak formulation. The elasticity problem is solved explicitly in time, using the Saint Verlat scheme. Finally, we specify a suitable frictional constitutive relation and numerical scheme to simulate fault behaviour. Our model is based on previous work on modelling fault friction and multi-fault systems using lattice solid-like models. We adapt the 2D model for simulating the dynamics of parallel fault systems described to the Finite-Element method. The approach uses a frictional relation along faults that is slip and slip-rate dependent, and the numerical integration approach introduced by Mora and Place in the lattice solid model. In order to illustrate the new Finite Element model, single and multi-fault simulation examples are presented.

The finite element method in low speed aerodynamics

NASA Technical Reports Server (NTRS)

Baker, A. J.; Manhardt, P. D.

1975-01-01

The finite element procedure is shown to be of significant impact in design of the 'computational wind tunnel' for low speed aerodynamics. The uniformity of the mathematical differential equation description, for viscous and/or inviscid, multi-dimensional subsonic flows about practical aerodynamic system configurations, is utilized to establish the general form of the finite element algorithm. Numerical results for inviscid flow analysis, as well as viscous boundary layer, parabolic, and full Navier Stokes flow descriptions verify the capabilities and overall versatility of the fundamental algorithm for aerodynamics. The proven mathematical basis, coupled with the distinct user-orientation features of the computer program embodiment, indicate near-term evolution of a highly useful analytical design tool to support computational configuration studies in low speed aerodynamics.

New Multigrid Method Including Elimination Algolithm Based on High-Order Vector Finite Elements in Three Dimensional Magnetostatic Field Analysis

NASA Astrophysics Data System (ADS)

Hano, Mitsuo; Hotta, Masashi

A new multigrid method based on high-order vector finite elements is proposed in this paper. Low level discretizations in this method are obtained by using low-order vector finite elements for the same mesh. Gauss-Seidel method is used as a smoother, and a linear equation of lowest level is solved by ICCG method. But it is often found that multigrid solutions do not converge into ICCG solutions. An elimination algolithm of constant term using a null space of the coefficient matrix is also described. In three dimensional magnetostatic field analysis, convergence time and number of iteration of this multigrid method are discussed with the convectional ICCG method.

Simulation of Aluminum Micro-mirrors for Space Applications at Cryogenic Temperatures

NASA Technical Reports Server (NTRS)

Kuhn, J. L.; Dutta, S. B.; Greenhouse, M. A.; Mott, D. B.

2000-01-01

Closed form and finite element models are developed to predict the device response of aluminum electrostatic torsion micro-mirrors fabricated on silicon substrate for space applications at operating temperatures of 30K. Initially, closed form expressions for electrostatic pressure arid mechanical restoring torque are used to predict the pull-in and release voltages at room temperature. Subsequently, a detailed mechanical finite element model is developed to predict stresses and vertical beam deflection induced by the electrostatic and thermal loads. An incremental and iterative solution method is used in conjunction with the nonlinear finite element model and closed form electrostatic equations to solve. the coupled electro-thermo-mechanical problem. The simulation results are compared with experimental measurements at room temperature of fabricated micro-mirror devices.

A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D

NASA Astrophysics Data System (ADS)

Boscheri, Walter; Dumbser, Michael

2014-10-01

In this paper we present a new family of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) one-step ADER-WENO finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations with stiff source terms on moving tetrahedral meshes in three space dimensions. A WENO reconstruction technique is used to achieve high order of accuracy in space, while an element-local space-time Discontinuous Galerkin finite element predictor on moving curved meshes is used to obtain a high order accurate one-step time discretization. Within the space-time predictor the physical element is mapped onto a reference element using a high order isoparametric approach, where the space-time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space-time nodes. Since our algorithm is cell-centered, the final mesh motion is computed by using a suitable node solver algorithm. A rezoning step as well as a flattener strategy are used in some of the test problems to avoid mesh tangling or excessive element deformations that may occur when the computation involves strong shocks or shear waves. The ALE algorithm presented in this article belongs to the so-called direct ALE methods because the final Lagrangian finite volume scheme is based directly on a space-time conservation formulation of the governing PDE system, with the rezoned geometry taken already into account during the computation of the fluxes. We apply our new high order unstructured ALE schemes to the 3D Euler equations of compressible gas dynamics, for which a set of classical numerical test problems has been solved and for which convergence rates up to sixth order of accuracy in space and time have been obtained. We furthermore consider the equations of classical ideal magnetohydrodynamics (MHD) as well as the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms.

A meta-model analysis of a finite element simulation for defining poroelastic properties of intervertebral discs.

PubMed

Nikkhoo, Mohammad; Hsu, Yu-Chun; Haghpanahi, Mohammad; Parnianpour, Mohamad; Wang, Jaw-Lin

2013-06-01

Finite element analysis is an effective tool to evaluate the material properties of living tissue. For an interactive optimization procedure, the finite element analysis usually needs many simulations to reach a reasonable solution. The meta-model analysis of finite element simulation can be used to reduce the computation of a structure with complex geometry or a material with composite constitutive equations. The intervertebral disc is a complex, heterogeneous, and hydrated porous structure. A poroelastic finite element model can be used to observe the fluid transferring, pressure deviation, and other properties within the disc. Defining reasonable poroelastic material properties of the anulus fibrosus and nucleus pulposus is critical for the quality of the simulation. We developed a material property updating protocol, which is basically a fitting algorithm consisted of finite element simulations and a quadratic response surface regression. This protocol was used to find the material properties, such as the hydraulic permeability, elastic modulus, and Poisson's ratio, of intact and degenerated porcine discs. The results showed that the in vitro disc experimental deformations were well fitted with limited finite element simulations and a quadratic response surface regression. The comparison of material properties of intact and degenerated discs showed that the hydraulic permeability significantly decreased but Poisson's ratio significantly increased for the degenerated discs. This study shows that the developed protocol is efficient and effective in defining material properties of a complex structure such as the intervertebral disc.

Solution of the 2-D steady-state radiative transfer equation in participating media with specular reflections using SUPG and DG finite elements

NASA Astrophysics Data System (ADS)

Le Hardy, D.; Favennec, Y.; Rousseau, B.

2016-08-01

The 2D radiative transfer equation coupled with specular reflection boundary conditions is solved using finite element schemes. Both Discontinuous Galerkin and Streamline-Upwind Petrov-Galerkin variational formulations are fully developed. These two schemes are validated step-by-step for all involved operators (transport, scattering, reflection) using analytical formulations. Numerical comparisons of the two schemes, in terms of convergence rate, reveal that the quadratic SUPG scheme proves efficient for solving such problems. This comparison constitutes the main issue of the paper. Moreover, the solution process is accelerated using block SOR-type iterative methods, for which the determination of the optimal parameter is found in a very cheap way.

Generalized fourier analyses of the advection-diffusion equation - Part II: two-dimensional domains

NASA Astrophysics Data System (ADS)

Voth, Thomas E.; Martinez, Mario J.; Christon, Mark A.

2004-07-01

Part I of this work presents a detailed multi-methods comparison of the spatial errors associated with the one-dimensional finite difference, finite element and finite volume semi-discretizations of the scalar advection-diffusion equation. In Part II we extend the analysis to two-dimensional domains and also consider the effects of wave propagation direction and grid aspect ratio on the phase speed, and the discrete and artificial diffusivities. The observed dependence of dispersive and diffusive behaviour on propagation direction makes comparison of methods more difficult relative to the one-dimensional results. For this reason, integrated (over propagation direction and wave number) error and anisotropy metrics are introduced to facilitate comparison among the various methods. With respect to these metrics, the consistent mass Galerkin and consistent mass control-volume finite element methods, and their streamline upwind derivatives, exhibit comparable accuracy, and generally out-perform their lumped mass counterparts and finite-difference based schemes. While this work can only be considered a first step in a comprehensive multi-methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common mathematical framework. Published in 2004 by John Wiley & Sons, Ltd.

A combined finite element-boundary element formulation for solution of axially symmetric bodies

NASA Technical Reports Server (NTRS)

Collins, Jeffrey D.; Volakis, John L.

1991-01-01

A new method is presented for the computation of electromagnetic scattering from axially symmetric bodies. To allow the simulation of inhomogeneous cross sections, the method combines the finite element and boundary element techniques. Interior to a fictitious surface enclosing the scattering body, the finite element method is used which results in a sparce submatrix, whereas along the enclosure the Stratton-Chu integral equation is enforced. By choosing the fictitious enclosure to be a right circular cylinder, most of the resulting boundary integrals are convolutional and may therefore be evaluated via the FFT with which the system is iteratively solved. In view of the sparce matrix associated with the interior fields, this reduces the storage requirement of the entire system to O(N) making the method attractive for large scale computations. The details of the corresponding formulation and its numerical implementation are described.

FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

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

Regnier, D.; Dubray, N.; Verriere, M.

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this study, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different typesmore » of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank–Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of 240Pu and 256Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. Finally, we emphasize that in a realistic calculation of fission mass distributions of 240Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents).« less

FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

DOE PAGES

Regnier, D.; Dubray, N.; Verriere, M.; ...

2017-12-20

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this study, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different typesmore » of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank–Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of 240Pu and 256Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. Finally, we emphasize that in a realistic calculation of fission mass distributions of 240Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents).« less

Role of geomechanically grown fractures on dispersive transport in heterogeneous geological formations.

PubMed

Nick, H M; Paluszny, A; Blunt, M J; Matthai, S K

2011-11-01

A second order in space accurate implicit scheme for time-dependent advection-dispersion equations and a discrete fracture propagation model are employed to model solute transport in porous media. We study the impact of the fractures on mass transport and dispersion. To model flow and transport, pressure and transport equations are integrated using a finite-element, node-centered finite-volume approach. Fracture geometries are incrementally developed from a random distributions of material flaws using an adoptive geomechanical finite-element model that also produces fracture aperture distributions. This quasistatic propagation assumes a linear elastic rock matrix, and crack propagation is governed by a subcritical crack growth failure criterion. Fracture propagation, intersection, and closure are handled geometrically. The flow and transport simulations are separately conducted for a range of fracture densities that are generated by the geomechanical finite-element model. These computations show that the most influential parameters for solute transport in fractured porous media are as follows: fracture density and fracture-matrix flux ratio that is influenced by matrix permeability. Using an equivalent fracture aperture size, computed on the basis of equivalent permeability of the system, we also obtain an acceptable prediction of the macrodispersion of poorly interconnected fracture networks. The results hold for fractures at relatively low density.

pK(A) in proteins solving the Poisson-Boltzmann equation with finite elements.

PubMed

Sakalli, Ilkay; Knapp, Ernst-Walter

2015-11-05

Knowledge on pK(A) values is an eminent factor to understand the function of proteins in living systems. We present a novel approach demonstrating that the finite element (FE) method of solving the linearized Poisson-Boltzmann equation (lPBE) can successfully be used to compute pK(A) values in proteins with high accuracy as a possible replacement to finite difference (FD) method. For this purpose, we implemented the software molecular Finite Element Solver (mFES) in the framework of the Karlsberg+ program to compute pK(A) values. This work focuses on a comparison between pK(A) computations obtained with the well-established FD method and with the new developed FE method mFES, solving the lPBE using protein crystal structures without conformational changes. Accurate and coarse model systems are set up with mFES using a similar number of unknowns compared with the FD method. Our FE method delivers results for computations of pK(A) values and interaction energies of titratable groups, which are comparable in accuracy. We introduce different thermodynamic cycles to evaluate pK(A) values and we show for the FE method how different parameters influence the accuracy of computed pK(A) values. © 2015 Wiley Periodicals, Inc.

Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method

NASA Astrophysics Data System (ADS)

Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye

2018-04-01

The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.

Sound transmission analysis of partially treated MR fluid-based sandwich panels using finite element method

NASA Astrophysics Data System (ADS)

Hemmatian, M.; Sedaghati, R.

2017-04-01

This study aims at developing a finite element model to predict the sound transmission loss (STL) of a multilayer panel partially treated with a Magnetorheological (MR) fluid core layer. MR fluids are smart materials with promising controllable rheological characteristics in which the application of an external magnetic field instantly changes their rheological properties. Partial treatment of sandwich panels with MR fluid core layer provides an opportunity to change stiffness and damping of the structure without significantly increasing the mass. The STL of a finite sandwich panel partially treated with MR fluid is modeled using the finite element (FE) method. Circular sandwich panels with clamped boundary condition and elastic face sheets in which the core layer is segmented circumferentially is considered. The MR fluid core layer is considered as a viscoelastic material with complex shear modulus with the magnetic field and frequency dependent storage and loss moduli. Neglecting the effect of the panel's vibration on the pressure forcing function, the work done by the acoustic pressure is expressed as a function of the blocked pressure in order to calculate the force vector in the equation of the motion of the panel. The governing finite element equation of motion of the MR sandwich panel is then developed to predict the transverse vibration of the panel which can then be utilized to obtain the radiated sound using Green's function. The developed model is used to conduct a systematic parametric study on the effect of different locations of MR fluid treatment on the natural frequencies and the STL.

State-constrained booster trajectory solutions via finite elements and shooting

NASA Technical Reports Server (NTRS)

Bless, Robert R.; Hodges, Dewey H.; Seywald, Hans

1993-01-01

This paper presents an extension of a FEM formulation based on variational principles. A general formulation for handling internal boundary conditions and discontinuities in the state equations is presented, and the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions which only touch the state constraint and solutions which have a boundary arc of finite length are considered. Suitable shape and test functions are chosen for a FEM discretization. All element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form. The final form of the algebraic equations is then derived. A simple state-constrained problem is solved. Then, for a practical application of the use of the FEM formulation, a launch vehicle subject to a dynamic pressure constraint (a first-order state inequality constraint) is solved. The results presented for the launch-vehicle trajectory have some interesting features, including a touch-point solution.

The Nonlinear Dynamic Response of an Elastic-Plastic Thin Plate under Impulsive Loading,

DTIC Science & Technology

1987-06-11

Among those numerical methods, the finite element method is the most effective one. The method presented in this paper is an " influence function " numerical...computational time is much less than the finite element method. Its precision is higher also. II. Basic Assumption and the Influence Function of a Simple...calculation. Fig. 1 3 2. The Influence function of a Simple Supported Plate The motion differential equation of a thin plate can be written as DV’w+ _.eluq() (1

A p-version finite element method for steady incompressible fluid flow and convective heat transfer

NASA Technical Reports Server (NTRS)

Winterscheidt, Daniel L.

1993-01-01

A new p-version finite element formulation for steady, incompressible fluid flow and convective heat transfer problems is presented. The steady-state residual equations are obtained by considering a limiting case of the least-squares formulation for the transient problem. The method circumvents the Babuska-Brezzi condition, permitting the use of equal-order interpolation for velocity and pressure, without requiring the use of arbitrary parameters. Numerical results are presented to demonstrate the accuracy and generality of the method.

Proceedings of the Scientific Conference on Obscuration and Aerosol Research Held in Aberdeen Maryland on 27-30 June 1989

DTIC Science & Technology

1990-08-01

corneal structure for both normal and swollen corneas. Other problems of future interest are the understanding of the structure of scarred and dystrophied ...METHOD AND RESULTS The system of equations is solved numerically on a Cray X-MP by a finite element method with 9-node Lagrange quadrilaterals ( Becker ...Appl. Math., 42, 430. Becker , E. B., G. F. Carey, and J. T. Oden, 1981. Finite Elements: An Introduction (Vol. 1), Prentice- Hall, Englewood Cliffs, New

Computational mechanics analysis tools for parallel-vector supercomputers

NASA Technical Reports Server (NTRS)

Storaasli, Olaf O.; Nguyen, Duc T.; Baddourah, Majdi; Qin, Jiangning

1993-01-01

Computational algorithms for structural analysis on parallel-vector supercomputers are reviewed. These parallel algorithms, developed by the authors, are for the assembly of structural equations, 'out-of-core' strategies for linear equation solution, massively distributed-memory equation solution, unsymmetric equation solution, general eigensolution, geometrically nonlinear finite element analysis, design sensitivity analysis for structural dynamics, optimization search analysis and domain decomposition. The source code for many of these algorithms is available.

Comparison of radiated noise from shrouded and unshrouded propellers

NASA Technical Reports Server (NTRS)

Eversman, Walter

1992-01-01

The ducted propeller in a free field is modeled using the finite element method. The generation, propagation, and radiation of sound from a ducted fan is described by the convened wave equation with volumetric body forces. Body forces are used to introduce the blade loading for rotating blades and stationary exit guide vanes. For an axisymmetric nacelle or shroud, the problem is formulated in cylindrical coordinates. For a specified angular harmonic, the angular coordinate is eliminated, resulting in a two-dimensional representation. A finite element discretization based on nine-node quadratic isoparametric elements is used.

Finite element model for brittle fracture and fragmentation

DOE PAGES

Li, Wei; Delaney, Tristan J.; Jiao, Xiangmin; ...

2016-06-01

A new computational model for brittle fracture and fragmentation has been developed based on finite element analysis of non-linear elasticity equations. The proposed model propagates the cracks by splitting the mesh nodes alongside the most over-strained edges based on the principal direction of strain tensor. To prevent elements from overlapping and folding under large deformations, robust geometrical constraints using the method of Lagrange multipliers have been incorporated. In conclusion, the model has been applied to 2D simulations of the formation and propagation of cracks in brittle materials, and the fracture and fragmentation of stretched and compressed materials.

Finite element model for brittle fracture and fragmentation

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

Li, Wei; Delaney, Tristan J.; Jiao, Xiangmin

A new computational model for brittle fracture and fragmentation has been developed based on finite element analysis of non-linear elasticity equations. The proposed model propagates the cracks by splitting the mesh nodes alongside the most over-strained edges based on the principal direction of strain tensor. To prevent elements from overlapping and folding under large deformations, robust geometrical constraints using the method of Lagrange multipliers have been incorporated. In conclusion, the model has been applied to 2D simulations of the formation and propagation of cracks in brittle materials, and the fracture and fragmentation of stretched and compressed materials.

Numerical simulation of self-sustained oscillation of a voice-producing element based on Navier-Stokes equations and the finite element method.

PubMed

de Vries, Martinus P; Hamburg, Marc C; Schutte, Harm K; Verkerke, Gijsbertus J; Veldman, Arthur E P

2003-04-01

Surgical removal of the larynx results in radically reduced production of voice and speech. To improve voice quality a voice-producing element (VPE) is developed, based on the lip principle, called after the lips of a musician while playing a brass instrument. To optimize the VPE, a numerical model is developed. In this model, the finite element method is used to describe the mechanical behavior of the VPE. The flow is described by two-dimensional incompressible Navier-Stokes equations. The interaction between VPE and airflow is modeled by placing the grid of the VPE model in the grid of the aerodynamical model, and requiring continuity of forces and velocities. By applying and increasing pressure to the numerical model, pulses comparable to glottal volume velocity waveforms are obtained. By variation of geometric parameters their influence can be determined. To validate this numerical model, an in vitro test with a prototype of the VPE is performed. Experimental and numerical results show an acceptable agreement.

Finite element flow analysis; Proceedings of the Fourth International Symposium on Finite Element Methods in Flow Problems, Chuo University, Tokyo, Japan, July 26-29, 1982

NASA Astrophysics Data System (ADS)

Kawai, T.

Among the topics discussed are the application of FEM to nonlinear free surface flow, Navier-Stokes shallow water wave equations, incompressible viscous flows and weather prediction, the mathematical analysis and characteristics of FEM, penalty function FEM, convective, viscous, and high Reynolds number FEM analyses, the solution of time-dependent, three-dimensional and incompressible Navier-Stokes equations, turbulent boundary layer flow, FEM modeling of environmental problems over complex terrain, and FEM's application to thermal convection problems and to the flow of polymeric materials in injection molding processes. Also covered are FEMs for compressible flows, including boundary layer flows and transonic flows, hybrid element approaches for wave hydrodynamic loadings, FEM acoustic field analyses, and FEM treatment of free surface flow, shallow water flow, seepage flow, and sediment transport. Boundary element methods and FEM computational technique topics are also discussed. For individual items see A84-25834 to A84-25896

Box compression analysis of world-wide data spanning 46 years

Treesearch

Thomas J. Urbanik; Benjamin Frank

2006-01-01

The state of the art among most industry citations of box compression estimation is the equation by McKee developed in 1963. Because of limitations in computing tools at the time the McKee equation was developed, the equation is a simplification, with many constraints, of a more general relationship. By applying the results of sophisticated finite element modeling, in...

Final Report - Subcontract B623760

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

Bank, R.

2017-11-17

During my visit to LLNL during July 17{27, 2017, I worked on linear system solvers. The two level hierarchical solver that initiated our study was developed to solve linear systems arising from hp adaptive finite element calculations, and is implemented in the PLTMG software package, version 12. This preconditioner typically requires 3-20% of the space used by the stiffness matrix for higher order elements. It has multigrid like convergence rates for a wide variety of PDEs (self-adjoint positive de nite elliptic equations, convection dominated convection-diffusion equations, and highly indefinite Helmholtz equations, among others). The convergence rate is not independent ofmore » the polynomial degree p as p ! 1, but but remains strong for p 9, which is the highest polynomial degree allowed in PLTMG, due to limitations of the numerical quadrature rules implemented in the software package. A more complete description of the method and some numerical experiments illustrating its effectiveness appear in. Like traditional geometric multilevel methods, this scheme relies on knowledge of the underlying finite element space in order to construct the smoother and the coarse grid correction.« less

On the Development of an Efficient Parallel Hybrid Solver with Application to Acoustically Treated Aero-Engine Nacelles

NASA Technical Reports Server (NTRS)

Watson, Willie R.; Nark, Douglas M.; Nguyen, Duc T.; Tungkahotara, Siroj

2006-01-01

A finite element solution to the convected Helmholtz equation in a nonuniform flow is used to model the noise field within 3-D acoustically treated aero-engine nacelles. Options to select linear or cubic Hermite polynomial basis functions and isoparametric elements are included. However, the key feature of the method is a domain decomposition procedure that is based upon the inter-mixing of an iterative and a direct solve strategy for solving the discrete finite element equations. This procedure is optimized to take full advantage of sparsity and exploit the increased memory and parallel processing capability of modern computer architectures. Example computations are presented for the Langley Flow Impedance Test facility and a rectangular mapping of a full scale, generic aero-engine nacelle. The accuracy and parallel performance of this new solver are tested on both model problems using a supercomputer that contains hundreds of central processing units. Results show that the method gives extremely accurate attenuation predictions, achieves super-linear speedup over hundreds of CPUs, and solves upward of 25 million complex equations in a quarter of an hour.

A new weak Galerkin finite element method for elliptic interface problems

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu; ...

2016-08-26

We introduce and analyze a new weak Galerkin (WG) finite element method in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H1 and L2 norms are established for the present WG finite element solutions. We conducted extensive numerical experiments inmore » order to examine the accuracy, flexibility, and robustness of the proposed WG interface approach. In solving regular elliptic interface problems, high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees. Moreover, the WG method is shown to be able to accommodate very complicated interfaces, due to its flexibility in choosing finite element partitions. Finally, in dealing with challenging problems with low regularities, the piecewise linear WG method is capable of delivering a second order of accuracy in L∞ norm for both C1 and H2 continuous solutions.« less

A new weak Galerkin finite element method for elliptic interface problems

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

Mu, Lin; Wang, Junping; Ye, Xiu

We introduce and analyze a new weak Galerkin (WG) finite element method in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H1 and L2 norms are established for the present WG finite element solutions. We conducted extensive numerical experiments inmore » order to examine the accuracy, flexibility, and robustness of the proposed WG interface approach. In solving regular elliptic interface problems, high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees. Moreover, the WG method is shown to be able to accommodate very complicated interfaces, due to its flexibility in choosing finite element partitions. Finally, in dealing with challenging problems with low regularities, the piecewise linear WG method is capable of delivering a second order of accuracy in L∞ norm for both C1 and H2 continuous solutions.« less

A quasi two-dimensional model for sound attenuation by the sonic crystals.

PubMed

Gupta, A; Lim, K M; Chew, C H

2012-10-01

Sound propagation in the sonic crystal (SC) along the symmetry direction is modeled by sound propagation through a variable cross-sectional area waveguide. A one-dimensional (1D) model based on the Webster horn equation is used to obtain sound attenuation through the SC. This model is compared with two-dimensional (2D) finite element simulation and experiment. The 1D model prediction of frequency band for sound attenuation is found to be shifted by around 500 Hz with respect to the finite element simulation. The reason for this shift is due to the assumption involved in the 1D model. A quasi 2D model is developed for sound propagation through the waveguide. Sound pressure profiles from the quasi 2D model are compared with the finite element simulation and the 1D model. The result shows significant improvement over the 1D model and is in good agreement with the 2D finite element simulation. Finally, sound attenuation through the SC is computed based on the quasi 2D model and is found to be in good agreement with the finite element simulation. The quasi 2D model provides an improved method to calculate sound attenuation through the SC.

A Mixed Finite Volume Element Method for Flow Calculations in Porous Media

NASA Technical Reports Server (NTRS)

Jones, Jim E.

1996-01-01

A key ingredient in the simulation of flow in porous media is the accurate determination of the velocities that drive the flow. The large scale irregularities of the geology, such as faults, fractures, and layers suggest the use of irregular grids in the simulation. Work has been done in applying the finite volume element (FVE) methodology as developed by McCormick in conjunction with mixed methods which were developed by Raviart and Thomas. The resulting mixed finite volume element discretization scheme has the potential to generate more accurate solutions than standard approaches. The focus of this paper is on a multilevel algorithm for solving the discrete mixed FVE equations. The algorithm uses a standard cell centered finite difference scheme as the 'coarse' level and the more accurate mixed FVE scheme as the 'fine' level. The algorithm appears to have potential as a fast solver for large size simulations of flow in porous media.

The Reverse Time Migration technique coupled with Interior Penalty Discontinuous Galerkin method.

NASA Astrophysics Data System (ADS)

Baldassari, C.; Barucq, H.; Calandra, H.; Denel, B.; Diaz, J.

2009-04-01

Seismic imaging is based on the seismic reflection method which produces an image of the subsurface from reflected waves recordings by using a tomography process and seismic migration is the industrial standard to improve the quality of the images. The migration process consists in replacing the recorded wavefields at their actual place by using various mathematical and numerical methods but each of them follows the same schedule, according to the pioneering idea of Claerbout: numerical propagation of the source function (propagation) and of the recorded wavefields (retropropagation) and next, construction of the image by applying an imaging condition. The retropropagation step can be realized accouting for the time reversibility of the wave equation and the resulting algorithm is currently called Reverse Time Migration (RTM). To be efficient, especially in three dimensional domain, the RTM requires the solution of the full wave equation by fast numerical methods. Finite element methods are considered as the best discretization method for solving the wave equation, even if they lead to the solution of huge systems with several millions of degrees of freedom, since they use meshes adapted to the domain topography and the boundary conditions are naturally taken into account in the variational formulation. Among the different finite element families, the spectral element one (SEM) is very interesting because it leads to a diagonal mass matrix which dramatically reduces the cost of the numerical computation. Moreover this method is very accurate since it allows the use of high order finite elements. However, SEM uses meshes of the domain made of quadrangles in 2D or hexaedra in 3D which are difficult to compute and not always suitable for complex topographies. Recently, Grote et al. applied the IPDG (Interior Penalty Discontinuous Galerkin) method to the wave equation. This approach is very interesting since it relies on meshes with triangles in 2D or tetrahedra in 3D, which allows to handle the topography of the domain very accurately. Moreover, the fact that the resulting mass matrix is block-diagonal and that IPDG is compatible with the use of high-order finite element may let us suppose that its performances are similar to the ones of the SEM. In this presentation, we study the performances of IDPG through numerical comparisons with the SEM in 1D and 2D. We compare in particular the accuracy of the solutions obtained by the two methods with various order of approximation and the computational burden of the algorithms. The conclusion is IPDG and SEM perform similarly when considering low order finite elements while IPDG outperforms SEM in case of high order finite elements. Next we illustrate the impact of IPDG on the RTM, first through a simple configuration test (two-layered medium), then through realistic industrial applications in 2D.

Parallel aeroelastic computations for wing and wing-body configurations

NASA Technical Reports Server (NTRS)

Byun, Chansup

1994-01-01

The objective of this research is to develop computationally efficient methods for solving fluid-structural interaction problems by directly coupling finite difference Euler/Navier-Stokes equations for fluids and finite element dynamics equations for structures on parallel computers. This capability will significantly impact many aerospace projects of national importance such as Advanced Subsonic Civil Transport (ASCT), where the structural stability margin becomes very critical at the transonic region. This research effort will have direct impact on the High Performance Computing and Communication (HPCC) Program of NASA in the area of parallel computing.

Semi-discrete Galerkin solution of the compressible boundary-layer equations with viscous-inviscid interaction

NASA Technical Reports Server (NTRS)

Day, Brad A.; Meade, Andrew J., Jr.

1993-01-01

A semi-discrete Galerkin (SDG) method is under development to model attached, turbulent, and compressible boundary layers for transonic airfoil analysis problems. For the boundary-layer formulation the method models the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby providing high resolution near the wall and permitting the use of a uniform finite element grid which automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past RAE 2822 and NACA 0012 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack.

Finite Element Multi-scale Modeling of Chemical Segregation in Steel Solidification Taking into Account the Transport of Equiaxed Grains

NASA Astrophysics Data System (ADS)

Nguyen, Thi-Thuy-My; Gandin, Charles-André; Combeau, Hervé; Založnik, Miha; Bellet, Michel

2018-02-01

The transport of solid crystals in the liquid pool during solidification of large ingots is known to have a significant effect on their final grain structure and macrosegregation. Numerical modeling of the associated physics is challenging since complex and strong interactions between heat and mass transfer at the microscopic and macroscopic scales must be taken into account. The paper presents a finite element multi-scale solidification model coupling nucleation, growth, and solute diffusion at the microscopic scale, represented by a single unique grain, while also including transport of the liquid and solid phases at the macroscopic scale of the ingots. The numerical resolution is based on a splitting method which sequentially describes the evolution and interaction of quantities into a transport and a growth stage. This splitting method reduces the non-linear complexity of the set of equations and is, for the first time, implemented using the finite element method. This is possible due to the introduction of an artificial diffusion in all conservation equations solved by the finite element method. Simulations with and without grain transport are compared to demonstrate the impact of solid phase transport on the solidification process as well as the formation of macrosegregation in a binary alloy (Sn-5 wt pct Pb). The model is also applied to the solidification of the binary alloy Fe-0.36 wt pct C in a domain representative of a 3.3-ton steel ingot.

Methods for analysis of cracks in three-dimensional solids

NASA Technical Reports Server (NTRS)

Raju, I. S.; Newman, J. C., Jr.

1984-01-01

Analytical and numerical methods evaluating the stress-intensity factors for three-dimensional cracks in solids are presented, with reference to fatigue failure in aerospace structures. The exact solutions for embedded elliptical and circular cracks in infinite solids, and the approximate methods, including the finite-element, the boundary-integral equation, the line-spring models, and the mixed methods are discussed. Among the mixed methods, the superposition of analytical and finite element methods, the stress-difference, the discretization-error, the alternating, and the finite element-alternating methods are reviewed. Comparison of the stress-intensity factor solutions for some three-dimensional crack configurations showed good agreement. Thus, the choice of a particular method in evaluating the stress-intensity factor is limited only to the availability of resources and computer programs.

Heat transfer monitoring by means of the hot wire technique and finite element analysis software.

PubMed

Hernández Wong, J; Suarez, V; Guarachi, J; Calderón, A; Rojas-Trigos, J B; Juárez, A G; Marín, E

2014-01-01

It is reported the study of the radial heat transfer in a homogeneous and isotropic substance with a heat linear source in its axial axis. For this purpose, the hot wire characterization technique has been used, in order to obtain the temperature distribution as a function of radial distance from the axial axis and time exposure. Also, the solution of the transient heat transport equation for this problem was obtained under appropriate boundary conditions, by means of finite element technique. A comparison between experimental, conventional theoretical model and numerical simulated results is done to demonstrate the utility of the finite element analysis simulation methodology in the investigation of the thermal response of substances. Copyright © 2013 Elsevier Ltd. All rights reserved.

Finite element-integral simulation of static and flight fan noise radiation from the JT15D turbofan engine

NASA Technical Reports Server (NTRS)

Baumeister, K. J.; Horowitz, S. J.

1982-01-01

An iterative finite element integral technique is used to predict the sound field radiated from the JT15D turbofan inlet. The sound field is divided into two regions: the sound field within and near the inlet which is computed using the finite element method and the radiation field beyond the inlet which is calculated using an integral solution technique. The velocity potential formulation of the acoustic wave equation was employed in the program. For some single mode JT15D data, the theory and experiment are in good agreement for the far field radiation pattern as well as suppressor attenuation. Also, the computer program is used to simulate flight effects that cannot be performed on a ground static test stand.

Determination of Nonlinear Stiffness Coefficients for Finite Element Models with Application to the Random Vibration Problem

NASA Technical Reports Server (NTRS)

Muravyov, Alexander A.

1999-01-01

In this paper, a method for obtaining nonlinear stiffness coefficients in modal coordinates for geometrically nonlinear finite-element models is developed. The method requires application of a finite-element program with a geometrically non- linear static capability. The MSC/NASTRAN code is employed for this purpose. The equations of motion of a MDOF system are formulated in modal coordinates. A set of linear eigenvectors is used to approximate the solution of the nonlinear problem. The random vibration problem of the MDOF nonlinear system is then considered. The solutions obtained by application of two different versions of a stochastic linearization technique are compared with linear and exact (analytical) solutions in terms of root-mean-square (RMS) displacements and strains for a beam structure.

Three-Dimensional Finite Element Ablative Thermal Response and Thermostructural Design of Thermal Protection Systems

NASA Technical Reports Server (NTRS)

Dec, John A.; Braun, Robert D.

2011-01-01

A finite element ablation and thermal response program is presented for simulation of three-dimensional transient thermostructural analysis. The three-dimensional governing differential equations and finite element formulation are summarized. A novel probabilistic design methodology for thermal protection systems is presented. The design methodology is an eight step process beginning with a parameter sensitivity study and is followed by a deterministic analysis whereby an optimum design can determined. The design process concludes with a Monte Carlo simulation where the probabilities of exceeding design specifications are estimated. The design methodology is demonstrated by applying the methodology to the carbon phenolic compression pads of the Crew Exploration Vehicle. The maximum allowed values of bondline temperature and tensile stress are used as the design specifications in this study.

A domain decomposition approach to implementing fault slip in finite-element models of quasi-static and dynamic crustal deformation

USGS Publications Warehouse

Aagaard, Brad T.; Knepley, M.G.; Williams, C.A.

2013-01-01

We employ a domain decomposition approach with Lagrange multipliers to implement fault slip in a finite-element code, PyLith, for use in both quasi-static and dynamic crustal deformation applications. This integrated approach to solving both quasi-static and dynamic simulations leverages common finite-element data structures and implementations of various boundary conditions, discretization schemes, and bulk and fault rheologies. We have developed a custom preconditioner for the Lagrange multiplier portion of the system of equations that provides excellent scalability with problem size compared to conventional additive Schwarz methods. We demonstrate application of this approach using benchmarks for both quasi-static viscoelastic deformation and dynamic spontaneous rupture propagation that verify the numerical implementation in PyLith.

Scattering and radiation analysis of three-dimensional cavity arrays via a hybrid finite element method

NASA Technical Reports Server (NTRS)

Jin, Jian-Ming; Volakis, John L.

1992-01-01

A hybrid numerical technique is presented for a characterization of the scattering and radiation properties of three-dimensional cavity arrays recessed in a ground plane. The technique combines the finite element and boundary integral methods and invokes Floquet's representation to formulate a system of equations for the fields at the apertures and those inside the cavities. The system is solved via the conjugate gradient method in conjunction with the Fast Fourier Transform (FFT) thus achieving an O(N) storage requirement. By virtue of the finite element method, the proposed technique is applicable to periodic arrays comprised of cavities having arbitrary shape and filled with inhomogeneous dielectrics. Several numerical results are presented, along with new measured data, which demonstrate the validity, efficiency, and capability of the technique.

An emulator for minimizing finite element analysis implementation resources

NASA Technical Reports Server (NTRS)

Melosh, R. J.; Utku, S.; Salama, M.; Islam, M.

1982-01-01

A finite element analysis emulator providing a basis for efficiently establishing an optimum computer implementation strategy when many calculations are involved is described. The SCOPE emulator determines computer resources required as a function of the structural model, structural load-deflection equation characteristics, the storage allocation plan, and computer hardware capabilities. Thereby, it provides data for trading analysis implementation options to arrive at a best strategy. The models contained in SCOPE lead to micro-operation computer counts of each finite element operation as well as overall computer resource cost estimates. Application of SCOPE to the Memphis-Arkansas bridge analysis provides measures of the accuracy of resource assessments. Data indicate that predictions are within 17.3 percent for calculation times and within 3.2 percent for peripheral storage resources for the ELAS code.

Computational mechanics analysis tools for parallel-vector supercomputers

NASA Technical Reports Server (NTRS)

Storaasli, O. O.; Nguyen, D. T.; Baddourah, M. A.; Qin, J.

1993-01-01

Computational algorithms for structural analysis on parallel-vector supercomputers are reviewed. These parallel algorithms, developed by the authors, are for the assembly of structural equations, 'out-of-core' strategies for linear equation solution, massively distributed-memory equation solution, unsymmetric equation solution, general eigen-solution, geometrically nonlinear finite element analysis, design sensitivity analysis for structural dynamics, optimization algorithm and domain decomposition. The source code for many of these algorithms is available from NASA Langley.

Finite element solution of torsion and other 2-D Poisson equations

NASA Technical Reports Server (NTRS)

Everstine, G. C.

1982-01-01

The NASTRAN structural analysis computer program may be used, without modification, to solve two dimensional Poisson equations such as arise in the classical Saint Venant torsion problem. The nonhomogeneous term (the right-hand side) in the Poisson equation can be handled conveniently by specifying a gravitational load in a "structural" analysis. The use of an analogy between the equations of elasticity and those of classical mathematical physics is summarized in detail.

Calculation methods for compressible turbulent boundary layers, 1976

NASA Technical Reports Server (NTRS)

Bushnell, D. M.; Cary, A. M., Jr.; Harris, J. E.

1977-01-01

Equations and closure methods for compressible turbulent boundary layers are discussed. Flow phenomena peculiar to calculation of these boundary layers were considered, along with calculations of three dimensional compressible turbulent boundary layers. Procedures for ascertaining nonsimilar two and three dimensional compressible turbulent boundary layers were appended, including finite difference, finite element, and mass-weighted residual methods.

Mixed-RKDG Finite Element Methods for the 2-D Hydrodynamic Model for Semiconductor Device Simulation

DOE PAGES

Chen, Zhangxin; Cockburn, Bernardo; Jerome, Joseph W.; ...

1995-01-01

In this paper we introduce a new method for numerically solving the equations of the hydrodynamic model for semiconductor devices in two space dimensions. The method combines a standard mixed finite element method, used to obtain directly an approximation to the electric field, with the so-called Runge-Kutta Discontinuous Galerkin (RKDG) method, originally devised for numerically solving multi-dimensional hyperbolic systems of conservation laws, which is applied here to the convective part of the equations. Numerical simulations showing the performance of the new method are displayed, and the results compared with those obtained by using Essentially Nonoscillatory (ENO) finite difference schemes. Frommore » the perspective of device modeling, these methods are robust, since they are capable of encompassing broad parameter ranges, including those for which shock formation is possible. The simulations presented here are for Gallium Arsenide at room temperature, but we have tested them much more generally with considerable success.« less

A comparison of matrix methods for calculating eigenvalues in acoustically lined ducts

NASA Technical Reports Server (NTRS)

Watson, W.; Lansing, D. L.

1976-01-01

Three approximate methods - finite differences, weighted residuals, and finite elements - were used to solve the eigenvalue problem which arises in finding the acoustic modes and propagation constants in an absorptively lined two-dimensional duct without airflow. The matrix equations derived for each of these methods were solved for the eigenvalues corresponding to various values of wall impedance. Two matrix orders, 20 x 20 and 40 x 40, were used. The cases considered included values of wall admittance for which exact eigenvalues were known and for which several nearly equal roots were present. Ten of the lower order eigenvalues obtained from the three approximate methods were compared with solutions calculated from the exact characteristic equation in order to make an assessment of the relative accuracy and reliability of the three methods. The best results were given by the finite element method using a cubic polynomial. Excellent accuracy was consistently obtained, even for nearly equal eigenvalues, by using a 20 x 20 order matrix.

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.

Probabilistic finite elements for fracture and fatigue analysis

NASA Technical Reports Server (NTRS)

Liu, W. K.; Belytschko, T.; Lawrence, M.; Besterfield, G. H.

1989-01-01

The fusion of the probabilistic finite element method (PFEM) and reliability analysis for probabilistic fracture mechanics (PFM) is presented. A comprehensive method for determining the probability of fatigue failure for curved crack growth was developed. The criterion for failure or performance function is stated as: the fatigue life of a component must exceed the service life of the component; otherwise failure will occur. An enriched element that has the near-crack-tip singular strain field embedded in the element is used to formulate the equilibrium equation and solve for the stress intensity factors at the crack-tip. Performance and accuracy of the method is demonstrated on a classical mode 1 fatigue problem.

Edge-based nonlinear diffusion for finite element approximations of convection-diffusion equations and its relation to algebraic flux-correction schemes.

PubMed

Barrenechea, Gabriel R; Burman, Erik; Karakatsani, Fotini

2017-01-01

For the case of approximation of convection-diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters. The performance of the method is illustrated on some numerical test cases in two space dimensions.

Fiber Composite Sandwich Thermostructural Behavior: Computational Simulation

NASA Technical Reports Server (NTRS)

Chamis, C. C.; Aiello, R. A.; Murthy, P. L. N.

1986-01-01

Several computational levels of progressive sophistication/simplification are described to computationally simulate composite sandwich hygral, thermal, and structural behavior. The computational levels of sophistication include: (1) three-dimensional detailed finite element modeling of the honeycomb, the adhesive and the composite faces; (2) three-dimensional finite element modeling of the honeycomb assumed to be an equivalent continuous, homogeneous medium, the adhesive and the composite faces; (3) laminate theory simulation where the honeycomb (metal or composite) is assumed to consist of plies with equivalent properties; and (4) derivations of approximate, simplified equations for thermal and mechanical properties by simulating the honeycomb as an equivalent homogeneous medium. The approximate equations are combined with composite hygrothermomechanical and laminate theories to provide a simple and effective computational procedure for simulating the thermomechanical/thermostructural behavior of fiber composite sandwich structures.

Efficient Computation of Atmospheric Flows with Tempest: Development of Next-Generation Climate and Weather Prediction Algorithms at Non-Hydrostatic Scales

NASA Astrophysics Data System (ADS)

Guerra, J. E.; Ullrich, P. A.

2015-12-01

Tempest is a next-generation global climate and weather simulation platform designed to allow experimentation with numerical methods at very high spatial resolutions. The atmospheric fluid equations are discretized by continuous / discontinuous finite elements in the horizontal and by a staggered nodal finite element method (SNFEM) in the vertical, coupled with implicit/explicit time integration. At global horizontal resolutions below 10km, many important questions remain on optimal techniques for solving the fluid equations. We present results from a suite of meso-scale test cases to validate the performance of the SNFEM applied in the vertical. Internal gravity wave, mountain wave, convective, and Cartesian baroclinic instability tests will be shown at various vertical orders of accuracy and compared with known results.

TECHNICAL NOTE: Direct finite-element analysis of the frequency response of a Y-Z lithium niobate SAW filter

NASA Astrophysics Data System (ADS)

Xu, Guanshui

2000-12-01

A direct finite-element model is developed for the full-scale analysis of the electromechanical phenomena involved in surface acoustic wave (SAW) devices. The equations of wave propagation in piezoelectric materials are discretized using the Galerkin method, in which an implicit algorithm of the Newmark family with unconditional stability is implemented. The Rayleigh damping coefficients are included in the elements near the boundary to reduce the influence of the reflection of waves. The performance of the model is demonstrated by the analysis of the frequency response of a Y-Z lithium niobate filter with two uniform ports, with emphasis on the influence of the number of electrodes. The frequency response of the filter is obtained through the Fourier transform of the impulse response, which is solved directly from the finite-element simulation. It shows that the finite-element results are in good agreement with the characteristic frequency response of the filter predicted by the simple phase-matching argument. The ability of the method to evaluate the influence of the bulk waves at the high-frequency end of the filter passband and the influence of the number of electrodes on insertion loss is noteworthy. We conclude that the direct finite-element analysis of SAW devices can be used as an effective tool for the design of high-performance SAW devices. Some practical computational challenges of finite-element modeling of SAW devices are discussed.

Hypervelocity Impact Behaviour of CFRP-A1/HC Sandwich Panel: Finite-Element Studies

NASA Astrophysics Data System (ADS)

Phadnis, Vaibhav A.; Roy, Anish; Silberschmidt, Vadim V.

2014-06-01

The mechanical response of CFRP-Al/HC (carbon fibre- reinforced/epoxy composite face sheets with Al honeycomb core) sandwich panels to hyper-velocity impact ( 1 km/s) is studied using a finite-element model developed in ABAQUS/Explicit. The intraply damage of CFRP face sheets is analysed by the means of a user-defined material model (VUMAT) employing a combination of Hashin and Puck criteria and delamination is modelled using cohesive-zone elements. The damage of Al/HC core is assessed on the basis of a Johnson-Cook dynamic failure model while its hydrodynamic response is captured using the Mie- Gruneisen equation of state. The results obtained with the developed finite-element model showed a reasonable correlation to experimental damage patterns. The surface peeling of both face sheets was evident, with a significant delamination around the impact location accompanied by crushing of HC core.

Finite element analysis of hypervelocity impact behaviour of CFRP-Al/HC sandwich panel

NASA Astrophysics Data System (ADS)

Phadnis, Vaibhav A.; Silberschmidt, Vadim V.

2015-09-01

The mechanical response of CFRP-Al/HC (carbon fibre-reinforced/epoxy composite face sheets with Al honeycomb core) sandwich panels to hyper-velocity impact (up to 1 km/s) is studied using a finite-element model developed in ABAQUS/Explicit. The intraply damage of CFRP face sheets is analysed by mean of a user-defined material model (VUMAT) employing a combination of Hashin and Puck criteria, delamination modelled using cohesive-zone elements. The damaged Al/HC core is assessed on the basis of a Johnson Cook dynamic failure model while its hydrodynamic response is captured using the Mie-Gruneisen equation of state. The results obtained with the developed finite-element model showed a reasonable correlation to experimental damage patterns. The surface peeling of both face sheets was evident, with a significant delamination around the impact location accompanied by crushing HC core.

A finite element-boundary integral method for conformal antenna arrays on a circular cylinder

NASA Technical Reports Server (NTRS)

Kempel, Leo C.; Volakis, John L.; Woo, Alex C.; Yu, C. Long

1992-01-01

Conformal antenna arrays offer many cost and weight advantages over conventional antenna systems. In the past, antenna designers have had to resort to expensive measurements in order to develop a conformal array design. This is due to the lack of rigorous mathematical models for conformal antenna arrays, and as a result the design of conformal arrays is primarily based on planar antenna design concepts. Recently, we have found the finite element-boundary integral method to be very successful in modeling large planar arrays of arbitrary composition in a metallic plane. Herewith we shall extend this formulation for conformal arrays on large metallic cylinders. In this we develop the mathematical formulation. In particular we discuss the finite element equations, the shape elements, and the boundary integral evaluation, and it is shown how this formulation can be applied with minimal computation and memory requirements. The implementation shall be discussed in a later report.

A finite element-boundary integral method for conformal antenna arrays on a circular cylinder

NASA Technical Reports Server (NTRS)

Kempel, Leo C.; Volakis, John L.

1992-01-01

Conformal antenna arrays offer many cost and weight advantages over conventional antenna systems. In the past, antenna designers have had to resort to expensive measurements in order to develop a conformal array design. This was due to the lack of rigorous mathematical models for conformal antenna arrays. As a result, the design of conformal arrays was primarily based on planar antenna design concepts. Recently, we have found the finite element-boundary integral method to be very successful in modeling large planar arrays of arbitrary composition in a metallic plane. We are extending this formulation to conformal arrays on large metallic cylinders. In doing so, we will develop a mathematical formulation. In particular, we discuss the finite element equations, the shape elements, and the boundary integral evaluation. It is shown how this formulation can be applied with minimal computation and memory requirements.

Parallel iterative solution for h and p approximations of the shallow water equations

USGS Publications Warehouse

Barragy, E.J.; Walters, R.A.

1998-01-01

A p finite element scheme and parallel iterative solver are introduced for a modified form of the shallow water equations. The governing equations are the three-dimensional shallow water equations. After a harmonic decomposition in time and rearrangement, the resulting equations are a complex Helmholz problem for surface elevation, and a complex momentum equation for the horizontal velocity. Both equations are nonlinear and the resulting system is solved using the Picard iteration combined with a preconditioned biconjugate gradient (PBCG) method for the linearized subproblems. A subdomain-based parallel preconditioner is developed which uses incomplete LU factorization with thresholding (ILUT) methods within subdomains, overlapping ILUT factorizations for subdomain boundaries and under-relaxed iteration for the resulting block system. The method builds on techniques successfully applied to linear elements by introducing ordering and condensation techniques to handle uniform p refinement. The combined methods show good performance for a range of p (element order), h (element size), and N (number of processors). Performance and scalability results are presented for a field scale problem where up to 512 processors are used. ?? 1998 Elsevier Science Ltd. All rights reserved.

Program For Finite-Element Analyses Of Phase-Change Fluids

NASA Technical Reports Server (NTRS)

Viterna, L. A.

1995-01-01

PHASTRAN analyzes heat-transfer and flow behaviors of materials undergoing phase changes. Many phase changes operate over range of accelerations or effective gravitational fields. To analyze such thermal systems, it is necessary to obtain simultaneous solutions for equations of conservation of energy, momentum, and mass, and for equation of state. Written in APL2.

One-Dimensional Ablation with Pyrolysis Gas Flow Using a Full Newton's Method and Finite Control Volume Procedure

NASA Technical Reports Server (NTRS)

Amar, Adam J.; Blackwell, Ben F.; Edwards, Jack R.

2007-01-01

The development and verification of a one-dimensional material thermal response code with ablation is presented. The implicit time integrator, control volume finite element spatial discretization, and Newton's method for nonlinear iteration on the entire system of residual equations have been implemented and verified for the thermochemical ablation of internally decomposing materials. This study is a continuation of the work presented in "One-Dimensional Ablation with Pyrolysis Gas Flow Using a Full Newton's Method and Finite Control Volume Procedure" (AIAA-2006-2910), which described the derivation, implementation, and verification of the constant density solid energy equation terms and boundary conditions. The present study extends the model to decomposing materials including decomposition kinetics, pyrolysis gas flow through the porous char layer, and a mixture (solid and gas) energy equation. Verification results are presented for the thermochemical ablation of a carbon-phenolic ablator which involves the solution of the entire system of governing equations.

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.

Extension of Ko Straight-Beam Displacement Theory to Deformed Shape Predictions of Slender Curved Structures

NASA Technical Reports Server (NTRS)

Ko, William L.; Fleischer, Van Tran

2011-01-01

The Ko displacement theory originally developed for shape predictions of straight beams is extended to shape predictions of curved beams. The surface strains needed for shape predictions were analytically generated from finite-element nodal stress outputs. With the aid of finite-element displacement outputs, mathematical functional forms for curvature-effect correction terms are established and incorporated into straight-beam deflection equations for shape predictions of both cantilever and two-point supported curved beams. The newly established deflection equations for cantilever curved beams could provide quite accurate shape predictions for different cantilever curved beams, including the quarter-circle cantilever beam. Furthermore, the newly formulated deflection equations for two-point supported curved beams could provide accurate shape predictions for a range of two-point supported curved beams, including the full-circular ring. Accuracy of the newly developed curved-beam deflection equations is validated through shape prediction analysis of curved beams embedded in the windward shallow spherical shell of a generic crew exploration vehicle. A single-point collocation method for optimization of shape predictions is discussed in detail

A partially penalty immersed Crouzeix-Raviart finite element method for interface problems.

PubMed

An, Na; Yu, Xijun; Chen, Huanzhen; Huang, Chaobao; Liu, Zhongyan

2017-01-01

The elliptic equations with discontinuous coefficients are often used to describe the problems of the multiple materials or fluids with different densities or conductivities or diffusivities. In this paper we develop a partially penalty immersed finite element (PIFE) method on triangular grids for anisotropic flow models, in which the diffusion coefficient is a piecewise definite-positive matrix. The standard linear Crouzeix-Raviart type finite element space is used on non-interface elements and the piecewise linear Crouzeix-Raviart type immersed finite element (IFE) space is constructed on interface elements. The piecewise linear functions satisfying the interface jump conditions are uniquely determined by the integral averages on the edges as degrees of freedom. The PIFE scheme is given based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. The solvability of the method is proved and the optimal error estimates in the energy norm are obtained. Numerical experiments are presented to confirm our theoretical analysis and show that the newly developed PIFE method has optimal-order convergence in the [Formula: see text] norm as well. In addition, numerical examples also indicate that this method is valid for both the isotropic and the anisotropic elliptic interface problems.

Soft tissue deformation estimation by spatio-temporal Kalman filter finite element method.

PubMed

Yarahmadian, Mehran; Zhong, Yongmin; Gu, Chengfan; Shin, Jaehyun

2018-01-01

Soft tissue modeling plays an important role in the development of surgical training simulators as well as in robot-assisted minimally invasive surgeries. It has been known that while the traditional Finite Element Method (FEM) promises the accurate modeling of soft tissue deformation, it still suffers from a slow computational process. This paper presents a Kalman filter finite element method to model soft tissue deformation in real time without sacrificing the traditional FEM accuracy. The proposed method employs the FEM equilibrium equation and formulates it as a filtering process to estimate soft tissue behavior using real-time measurement data. The model is temporally discretized using the Newmark method and further formulated as the system state equation. Simulation results demonstrate that the computational time of KF-FEM is approximately 10 times shorter than the traditional FEM and it is still as accurate as the traditional FEM. The normalized root-mean-square error of the proposed KF-FEM in reference to the traditional FEM is computed as 0.0116. It is concluded that the proposed method significantly improves the computational performance of the traditional FEM without sacrificing FEM accuracy. The proposed method also filters noises involved in system state and measurement data.

The Programming Language Python In Earth System Simulations

NASA Astrophysics Data System (ADS)

Gross, L.; Imranullah, A.; Mora, P.; Saez, E.; Smillie, J.; Wang, C.

2004-12-01

Mathematical models in earth sciences base on the solution of systems of coupled, non-linear, time-dependent partial differential equations (PDEs). The spatial and time-scale vary from a planetary scale and million years for convection problems to 100km and 10 years for fault systems simulations. Various techniques are in use to deal with the time dependency (e.g. Crank-Nicholson), with the non-linearity (e.g. Newton-Raphson) and weakly coupled equations (e.g. non-linear Gauss-Seidel). Besides these high-level solution algorithms discretization methods (e.g. finite element method (FEM), boundary element method (BEM)) are used to deal with spatial derivatives. Typically, large-scale, three dimensional meshes are required to resolve geometrical complexity (e.g. in the case of fault systems) or features in the solution (e.g. in mantel convection simulations). The modelling environment escript allows the rapid implementation of new physics as required for the development of simulation codes in earth sciences. Its main object is to provide a programming language, where the user can define new models and rapidly develop high-level solution algorithms. The current implementation is linked with the finite element package finley as a PDE solver. However, the design is open and other discretization technologies such as finite differences and boundary element methods could be included. escript is implemented as an extension of the interactive programming environment python (see www.python.org). Key concepts introduced are Data objects, which are holding values on nodes or elements of the finite element mesh, and linearPDE objects, which are defining linear partial differential equations to be solved by the underlying discretization technology. In this paper we will show the basic concepts of escript and will show how escript is used to implement a simulation code for interacting fault systems. We will show some results of large-scale, parallel simulations on an SGI Altix system. Acknowledgements: Project work is supported by Australian Commonwealth Government through the Australian Computational Earth Systems Simulator Major National Research Facility, Queensland State Government Smart State Research Facility Fund, The University of Queensland and SGI.

Layerwise Finite Elements for Smart Piezoceramic Composite Plates in Thermal Environments

NASA Technical Reports Server (NTRS)

Saravanos, Dimitris A.; Lee, Ho-Jun

1996-01-01

Analytical formulations are presented which account for the coupled mechanical, electrical, and thermal response of piezoelectric composite laminates and plate structures. A layerwise theory is formulated with the inherent capability to explicitly model the active and sensory response of piezoelectric composite plates having arbitrary laminate configurations in thermal environments. Finite element equations are derived and implemented for a bilinear 4-noded plate element. Application cases demonstrate the capability to manage thermally induced bending and twisting deformations in symmetric and antisymmetric composite plates with piezoelectric actuators, and show the corresponding electrical response of distributed piezoelectric sensors. Finally, the resultant stresses in the thermal piezoelectric composite laminates are investigated.

A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

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

Schunert, Sebastian; Wang, Yaqi; Gleicher, Frederick

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form ismore » based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.« less

A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

DOE PAGES

Schunert, Sebastian; Wang, Yaqi; Gleicher, Frederick; ...

2017-02-21

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form ismore » based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.« less

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.

Analytical analysis and implementation of a low-speed high-torque permanent magnet vernier in-wheel motor for electric vehicle

NASA Astrophysics Data System (ADS)

Li, Jiangui; Wang, Junhua; Zhigang, Zhao; Yan, Weili

2012-04-01

In this paper, analytical analysis of the permanent magnet vernier (PMV) is presented. The key is to analytically solve the governing Laplacian/quasi-Poissonian field equations in the motor regions. By using the time-stepping finite element method, the analytical method is verified. Hence, the performances of the PMV machine are quantitatively compared with that of the analytical results. The analytical results agree well with the finite element method results. Finally, the experimental results are given to further show the validity of the analysis.

Finite element procedures for time-dependent convection-diffusion-reaction systems

NASA Technical Reports Server (NTRS)

Tezduyar, T. E.; Park, Y. J.; Deans, H. A.

1988-01-01

New finite element procedures based on the streamline-upwind/Petrov-Galerkin formulations are developed for time-dependent convection-diffusion-reaction equations. These procedures minimize spurious oscillations for convection-dominated and reaction-dominated problems. The results obtained for representative numerical examples are accurate with minimal oscillations. As a special application problem, the single-well chemical tracer test (a procedure for measuring oil remaining in a depleted field) is simulated numerically. The results show the importance of temperature effects on the interpreted value of residual oil saturation from such tests.

A new least-squares transport equation compatible with voids

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

Hansen, J. B.; Morel, J. E.

2013-07-01

We define a new least-squares transport equation that is applicable in voids, can be solved using source iteration with diffusion-synthetic acceleration, and requires only the solution of an independent set of second-order self-adjoint equations for each direction during each source iteration. We derive the equation, discretize it using the S{sub n} method in conjunction with a linear-continuous finite-element method in space, and computationally demonstrate various of its properties. (authors)

A numerical spectral approach to solve the dislocation density transport equation

NASA Astrophysics Data System (ADS)

Djaka, K. S.; Taupin, V.; Berbenni, S.; Fressengeas, C.

2015-09-01

A numerical spectral approach is developed to solve in a fast, stable and accurate fashion, the quasi-linear hyperbolic transport equation governing the spatio-temporal evolution of the dislocation density tensor in the mechanics of dislocation fields. The approach relies on using the Fast Fourier Transform algorithm. Low-pass spectral filters are employed to control both the high frequency Gibbs oscillations inherent to the Fourier method and the fast-growing numerical instabilities resulting from the hyperbolic nature of the transport equation. The numerical scheme is validated by comparison with an exact solution in the 1D case corresponding to dislocation dipole annihilation. The expansion and annihilation of dislocation loops in 2D and 3D settings are also produced and compared with finite element approximations. The spectral solutions are shown to be stable, more accurate for low Courant numbers and much less computation time-consuming than the finite element technique based on an explicit Galerkin-least squares scheme.

Three-Dimensional Simulations of Marangoni-Benard Convection in Small Containers by the Least-Squares Finite Element Method

NASA Technical Reports Server (NTRS)

Yu, Sheng-Tao; Jiang, Bo-Nan; Wu, Jie; Duh, J. C.

1996-01-01

This paper reports a numerical study of the Marangoni-Benard (MB) convection in a planar fluid layer. The least-squares finite element method (LSFEM) is employed to solve the three-dimensional Stokes equations and the energy equation. First, the governing equations are reduced to be first-order by introducing variables such as vorticity and heat fluxes. The resultant first-order system is then cast into a div-curl-grad formulation, and its ellipticity and permissible boundary conditions are readily proved. This numerical approach provides an equal-order discretization for velocity, pressure, vorticity, temperature, and heat conduction fluxes, and therefore can provide high fidelity solutions for the complex flow physics of the MB convection. Numerical results reported include the critical Marangoni numbers (M(sub ac)) for the onset of the convection in containers with various aspect ratios, and the planforms of supercritical MB flows. The numerical solutions compared favorably with the experimental results reported by Koschmieder et al..

Discontinuous Galerkin Methods for NonLinear Differential Systems

NASA Technical Reports Server (NTRS)

Barth, Timothy; Mansour, Nagi (Technical Monitor)

2001-01-01

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Optimized growth and reorientation of anisotropic material based on evolution equations

NASA Astrophysics Data System (ADS)

Jantos, Dustin R.; Junker, Philipp; Hackl, Klaus

2018-07-01

Modern high-performance materials have inherent anisotropic elastic properties. The local material orientation can thus be considered to be an additional design variable for the topology optimization of structures containing such materials. In our previous work, we introduced a variational growth approach to topology optimization for isotropic, linear-elastic materials. We solved the optimization problem purely by application of Hamilton's principle. In this way, we were able to determine an evolution equation for the spatial distribution of density mass, which can be evaluated in an iterative process within a solitary finite element environment. We now add the local material orientation described by a set of three Euler angles as additional design variables into the three-dimensional model. This leads to three additional evolution equations that can be separately evaluated for each (material) point. Thus, no additional field unknown within the finite element approach is needed, and the evolution of the spatial distribution of density mass and the evolution of the Euler angles can be evaluated simultaneously.

Discontinuous Galerkin finite element methods for radiative transfer in spherical symmetry

NASA Astrophysics Data System (ADS)

Kitzmann, D.; Bolte, J.; Patzer, A. B. C.

2016-11-01

The discontinuous Galerkin finite element method (DG-FEM) is successfully applied to treat a broad variety of transport problems numerically. In this work, we use the full capacity of the DG-FEM to solve the radiative transfer equation in spherical symmetry. We present a discontinuous Galerkin method to directly solve the spherically symmetric radiative transfer equation as a two-dimensional problem. The transport equation in spherical atmospheres is more complicated than in the plane-parallel case owing to the appearance of an additional derivative with respect to the polar angle. The DG-FEM formalism allows for the exact integration of arbitrarily complex scattering phase functions, independent of the angular mesh resolution. We show that the discontinuous Galerkin method is able to describe accurately the radiative transfer in extended atmospheres and to capture discontinuities or complex scattering behaviour which might be present in the solution of certain radiative transfer tasks and can, therefore, cause severe numerical problems for other radiative transfer solution methods.

Development of an hp-version finite element method for computational optimal control

NASA Technical Reports Server (NTRS)

Hodges, Dewey H.; Warner, Michael S.

1993-01-01

The purpose of this research effort was to begin the study of the application of hp-version finite elements to the numerical solution of optimal control problems. Under NAG-939, the hybrid MACSYMA/FORTRAN code GENCODE was developed which utilized h-version finite elements to successfully approximate solutions to a wide class of optimal control problems. In that code the means for improvement of the solution was the refinement of the time-discretization mesh. With the extension to hp-version finite elements, the degrees of freedom include both nodal values and extra interior values associated with the unknown states, co-states, and controls, the number of which depends on the order of the shape functions in each element. One possible drawback is the increased computational effort within each element required in implementing hp-version finite elements. We are trying to determine whether this computational effort is sufficiently offset by the reduction in the number of time elements used and improved Newton-Raphson convergence so as to be useful in solving optimal control problems in real time. Because certain of the element interior unknowns can be eliminated at the element level by solving a small set of nonlinear algebraic equations in which the nodal values are taken as given, the scheme may turn out to be especially powerful in a parallel computing environment. A different processor could be assigned to each element. The number of processors, strictly speaking, is not required to be any larger than the number of sub-regions which are free of discontinuities of any kind.

Fluid/Structure Interaction Studies of Aircraft Using High Fidelity Equations on Parallel Computers

NASA Technical Reports Server (NTRS)

Guruswamy, Guru; VanDalsem, William (Technical Monitor)

1994-01-01

Abstract Aeroelasticity which involves strong coupling of fluids, structures and controls is an important element in designing an aircraft. Computational aeroelasticity using low fidelity methods such as the linear aerodynamic flow equations coupled with the modal structural equations are well advanced. Though these low fidelity approaches are computationally less intensive, they are not adequate for the analysis of modern aircraft such as High Speed Civil Transport (HSCT) and Advanced Subsonic Transport (AST) which can experience complex flow/structure interactions. HSCT can experience vortex induced aeroelastic oscillations whereas AST can experience transonic buffet associated structural oscillations. Both aircraft may experience a dip in the flutter speed at the transonic regime. For accurate aeroelastic computations at these complex fluid/structure interaction situations, high fidelity equations such as the Navier-Stokes for fluids and the finite-elements for structures are needed. Computations using these high fidelity equations require large computational resources both in memory and speed. Current conventional super computers have reached their limitations both in memory and speed. As a result, parallel computers have evolved to overcome the limitations of conventional computers. This paper will address the transition that is taking place in computational aeroelasticity from conventional computers to parallel computers. The paper will address special techniques needed to take advantage of the architecture of new parallel computers. Results will be illustrated from computations made on iPSC/860 and IBM SP2 computer by using ENSAERO code that directly couples the Euler/Navier-Stokes flow equations with high resolution finite-element structural equations.

Evaluation of finite-element models and stress-intensity factors for surface cracks emanating from stress concentrations

NASA Technical Reports Server (NTRS)

Tan, P. W.; Raju, I. S.; Shivakumar, K. N.; Newman, J. C., Jr.

1990-01-01

A re-evaluation of the 3-D finite-element models and methods used to analyze surface crack at stress concentrations is presented. Previous finite-element models used by Raju and Newman for surface and corner cracks at holes were shown to have ill-shaped elements at the intersection of the hole and crack boundaries. Improved models, without these ill-shaped elements, were developed for a surface crack at a circular hole and at a semi-circular edge notch. Stress-intensity factors were calculated by both the nodal-force and virtual-crack-closure methods. Comparisons made between the previously developed stress-intensity factor equations and the results from the improved models agreed well except for configurations with large notch-radii-to-plate-thickness ratios. Stress-intensity factors for a semi-elliptical surface crack located at the center of a semi-circular edge notch in a plate subjected to remote tensile loadings were calculated using the improved models.

Primal-mixed formulations for reaction-diffusion systems on deforming domains

NASA Astrophysics Data System (ADS)

Ruiz-Baier, Ricardo

2015-10-01

We propose a finite element formulation for a coupled elasticity-reaction-diffusion system written in a fully Lagrangian form and governing the spatio-temporal interaction of species inside an elastic, or hyper-elastic body. A primal weak formulation is the baseline model for the reaction-diffusion system written in the deformed domain, and a finite element method with piecewise linear approximations is employed for its spatial discretization. On the other hand, the strain is introduced as mixed variable in the equations of elastodynamics, which in turn acts as coupling field needed to update the diffusion tensor of the modified reaction-diffusion system written in a deformed domain. The discrete mechanical problem yields a mixed finite element scheme based on row-wise Raviart-Thomas elements for stresses, Brezzi-Douglas-Marini elements for displacements, and piecewise constant pressure approximations. The application of the present framework in the study of several coupled biological systems on deforming geometries in two and three spatial dimensions is discussed, and some illustrative examples are provided and extensively analyzed.

Solving the Helmholtz equation in conformal mapped ARROW structures using homotopy perturbation method.

PubMed

Reck, Kasper; Thomsen, Erik V; Hansen, Ole

2011-01-31

The scalar wave equation, or Helmholtz equation, describes within a certain approximation the electromagnetic field distribution in a given system. In this paper we show how to solve the Helmholtz equation in complex geometries using conformal mapping and the homotopy perturbation method. The solution of the mapped Helmholtz equation is found by solving an infinite series of Poisson equations using two dimensional Fourier series. The solution is entirely based on analytical expressions and is not mesh dependent. The analytical results are compared to a numerical (finite element method) solution.

Development of new vibration energy flow analysis software and its applications to vehicle systems

NASA Astrophysics Data System (ADS)

Kim, D.-J.; Hong, S.-Y.; Park, Y.-H.

2005-09-01

The Energy flow analysis (EFA) offers very promising results in predicting the noise and vibration responses of system structures in medium-to-high frequency ranges. We have developed the Energy flow finite element method (EFFEM) based software, EFADSC++ R4, for the vibration analysis. The software can analyze the system structures composed of beam, plate, spring-damper, rigid body elements and many other components developed, and has many useful functions in analysis. For convenient use of the software, the main functions of the whole software are modularized into translator, model-converter, and solver. The translator module makes it possible to use finite element (FE) model for the vibration analysis. The model-converter module changes FE model into energy flow finite element (EFFE) model, and generates joint elements to cover the vibrational attenuation in the complex structures composed of various elements and can solve the joint element equations by using the wave tra! nsmission approach very quickly. The solver module supports the various direct and iterative solvers for multi-DOF structures. The predictions of vibration for real vehicles by using the developed software were performed successfully.

WARP3D-Release 10.8: Dynamic Nonlinear Analysis of Solids using a Preconditioned Conjugate Gradient Software Architecture

NASA Technical Reports Server (NTRS)

Koppenhoefer, Kyle C.; Gullerud, Arne S.; Ruggieri, Claudio; Dodds, Robert H., Jr.; Healy, Brian E.

1998-01-01

This report describes theoretical background material and commands necessary to use the WARP3D finite element code. WARP3D is under continuing development as a research code for the solution of very large-scale, 3-D solid models subjected to static and dynamic loads. Specific features in the code oriented toward the investigation of ductile fracture in metals include a robust finite strain formulation, a general J-integral computation facility (with inertia, face loading), an element extinction facility to model crack growth, nonlinear material models including viscoplastic effects, and the Gurson-Tver-gaard dilatant plasticity model for void growth. The nonlinear, dynamic equilibrium equations are solved using an incremental-iterative, implicit formulation with full Newton iterations to eliminate residual nodal forces. The history integration of the nonlinear equations of motion is accomplished with Newmarks Beta method. A central feature of WARP3D involves the use of a linear-preconditioned conjugate gradient (LPCG) solver implemented in an element-by-element format to replace a conventional direct linear equation solver. This software architecture dramatically reduces both the memory requirements and CPU time for very large, nonlinear solid models since formation of the assembled (dynamic) stiffness matrix is avoided. Analyses thus exhibit the numerical stability for large time (load) steps provided by the implicit formulation coupled with the low memory requirements characteristic of an explicit code. In addition to the much lower memory requirements of the LPCG solver, the CPU time required for solution of the linear equations during each Newton iteration is generally one-half or less of the CPU time required for a traditional direct solver. All other computational aspects of the code (element stiffnesses, element strains, stress updating, element internal forces) are implemented in the element-by- element, blocked architecture. This greatly improves vectorization of the code on uni-processor hardware and enables straightforward parallel-vector processing of element blocks on multi-processor hardware.

Generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

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

Gao, Kai; Fu, Shubin; Gibson, Richard L.

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale mediummore » property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.« less

Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

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

Gao, Kai, E-mail: kaigao87@gmail.com; Fu, Shubin, E-mail: shubinfu89@gmail.com; Gibson, Richard L., E-mail: gibson@tamu.edu

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale mediummore » property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.« less

Generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media

DOE PAGES

Gao, Kai; Fu, Shubin; Gibson, Richard L.; ...

2015-04-14

It is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale mediummore » property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.« less

A mixed-penalty biphasic finite element formulation incorporating viscous fluids and material interfaces.

PubMed

Chan, B; Donzelli, P S; Spilker, R L

2000-06-01

The fluid viscosity term of the fluid phase constitutive equation and the interface boundary conditions between biphasic, solid and fluid domains have been incorporated into a mixed-penalty finite element formulation of the linear biphasic theory for hydrated soft tissue. The finite element code can now model a single-phase viscous incompressible fluid, or a single-phase elastic solid, as limiting cases of a biphasic material. Interface boundary conditions allow the solution of problems involving combinations of biphasic, fluid and solid regions. To incorporate these conditions, the volume-weighted mixture velocity is introduced as a degree of freedom at interface nodes so that the kinematic continuity conditions are satisfied by conventional finite element assembly techniques. Results comparing our numerical method with an independent, analytic solution for the problem of Couette flow over rigid and deformable porous biphasic layers show that the finite element code accurately predicts the viscous fluid flows and deformation in the porous biphasic region. Thus, the analysis can be used to model the interface between synovial fluid and articular cartilage in diarthrodial joints. This is an important step toward modeling and understanding the mechanisms of joint lubrication and another step toward fully modeling the in vivo behavior of a diarthrodial joint.

A Finite-Volume approach for compressible single- and two-phase flows in flexible pipelines with fluid-structure interaction

NASA Astrophysics Data System (ADS)

Daude, F.; Galon, P.

2018-06-01

A Finite-Volume scheme for the numerical computations of compressible single- and two-phase flows in flexible pipelines is proposed based on an approximate Godunov-type approach. The spatial discretization is here obtained using the HLLC scheme. In addition, the numerical treatment of abrupt changes in area and network including several pipelines connected at junctions is also considered. The proposed approach is based on the integral form of the governing equations making it possible to tackle general equations of state. A coupled approach for the resolution of fluid-structure interaction of compressible fluid flowing in flexible pipes is considered. The structural problem is solved using Euler-Bernoulli beam finite elements. The present Finite-Volume method is applied to ideal gas and two-phase steam-water based on the Homogeneous Equilibrium Model (HEM) in conjunction with a tabulated equation of state in order to demonstrate its ability to tackle general equations of state. The extensive application of the scheme for both shock tube and other transient flow problems demonstrates its capability to resolve such problems accurately and robustly. Finally, the proposed 1-D fluid-structure interaction model appears to be computationally efficient.

A modular finite-element model (MODFE) for areal and axisymmetric ground-water-flow problems, Part 1: Model Description and User's Manual

USGS Publications Warehouse

Torak, L.J.

1993-01-01

A MODular, Finite-Element digital-computer program (MODFE) was developed to simulate steady or unsteady-state, two-dimensional or axisymmetric ground-water flow. Geometric- and hydrologic-aquifer characteristics in two spatial dimensions are represented by triangular finite elements and linear basis functions; one-dimensional finite elements and linear basis functions represent time. Finite-element matrix equations are solved by the direct symmetric-Doolittle method or the iterative modified, incomplete-Cholesky, conjugate-gradient method. Physical processes that can be represented by the model include (1) confined flow, unconfined flow (using the Dupuit approximation), or a combination of both; (2) leakage through either rigid or elastic confining beds; (3) specified recharge or discharge at points, along lines, and over areas; (4) flow across specified-flow, specified-head, or bead-dependent boundaries; (5) decrease of aquifer thickness to zero under extreme water-table decline and increase of aquifer thickness from zero as the water table rises; and (6) head-dependent fluxes from springs, drainage wells, leakage across riverbeds or confining beds combined with aquifer dewatering, and evapotranspiration. The report describes procedures for applying MODFE to ground-water-flow problems, simulation capabilities, and data preparation. Guidelines for designing the finite-element mesh and for node numbering and determining band widths are given. Tables are given that reference simulation capabilities to specific versions of MODFE. Examples of data input and model output for different versions of MODFE are provided.

A modular finite-element model (MODFE) for areal and axisymmetric ground-water-flow problems; Part 1, Model description and user's manual

USGS Publications Warehouse

Torak, Lynn J.

1992-01-01

A MODular, Finite-Element digital-computer program (MODFE) was developed to simulate steady or unsteady-state, two-dimensional or axisymmetric ground-water flow. Geometric- and hydrologic-aquifer characteristics in two spatial dimensions are represented by triangular finite elements and linear basis functions; one-dimensional finite elements and linear basis functions represent time. Finite-element matrix equations are solved by the direct symmetric-Doolittle method or the iterative modified, incomplete-Cholesky, conjugate-gradient method. Physical processes that can be represented by the model include (1) confined flow, unconfined flow (using the Dupuit approximation), or a combination of both; (2) leakage through either rigid or elastic confining beds; (3) specified recharge or discharge at points, along lines, and over areas; (4) flow across specified-flow, specified-head, or head-dependent boundaries; (5) decrease of aquifer thickness to zero under extreme water-table decline and increase of aquifer thickness from zero as the water table rises; and (6) head-dependent fluxes from springs, drainage wells, leakage across riverbeds or confining beds combined with aquifer dewatering, and evapotranspiration.The report describes procedures for applying MODFE to ground-water-flow problems, simulation capabilities, and data preparation. Guidelines for designing the finite-element mesh and for node numbering and determining band widths are given. Tables are given that reference simulation capabilities to specific versions of MODFE. Examples of data input and model output for different versions of MODFE are provided.

A weak Hamiltonian finite element method for optimal guidance of an advanced launch vehicle

NASA Technical Reports Server (NTRS)

Hodges, Dewey H.; Calise, Anthony J.; Bless, Robert R.; Leung, Martin

1989-01-01

A temporal finite-element method based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables, which are expanded in terms of nodal values and simple shape functions. Time derivatives of the states and costates do not appear in the governing variational equation; the only quantities whose time derivatives appear therein are virtual states and virtual costates. Numerical results are presented for an elementary trajectory optimization problem; they show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The feasibility of this approach for real-time guidance applications is evaluated. A simplified model for an advanced launch vehicle application that is suitable for finite-element solution is presented.

MHOST: An efficient finite element program for inelastic analysis of solids and structures

NASA Technical Reports Server (NTRS)

Nakazawa, S.

1988-01-01

An efficient finite element program for 3-D inelastic analysis of gas turbine hot section components was constructed and validated. A novel mixed iterative solution strategy is derived from the augmented Hu-Washizu variational principle in order to nodally interpolate coordinates, displacements, deformation, strains, stresses and material properties. A series of increasingly sophisticated material models incorporated in MHOST include elasticity, secant plasticity, infinitesimal and finite deformation plasticity, creep and unified viscoplastic constitutive model proposed by Walker. A library of high performance elements is built into this computer program utilizing the concepts of selective reduced integrations and independent strain interpolations. A family of efficient solution algorithms is implemented in MHOST for linear and nonlinear equation solution including the classical Newton-Raphson, modified, quasi and secant Newton methods with optional line search and the conjugate gradient method.

A finite-element analysis for steady and oscillatory supersonic flows around complex configurations

NASA Technical Reports Server (NTRS)

Morino, L.; Chen, L. T.

1974-01-01

The problem of small perturbation potential supersonic flow around complex configurations is considered. This problem requires the solution of an integral equation relating the values of the potential on the surface of the body to the values of the normal derivative, which is known from the small perturbation boundary conditions. The surface of the body is divided into small (hyperboloidal quadrilateral) surface elements, sigma sub i, which are described in terms of the Cartesian components of the four corner points. The values of the potential (and its normal derivative) within each element is assumed to be constant and equal to its value at the centroid of the element, and this yields a set of linear algebraic equations. The coefficients of the equation are given by source and doublet integrals over the surface elements, sigma sub i. The results obtained using the above formulation are compared with existing analytical and experimental results.

LATDYN - PROGRAM FOR SIMULATION OF LARGE ANGLE TRANSIENT DYNAMICS OF FLEXIBLE AND RIGID STRUCTURES

NASA Technical Reports Server (NTRS)

Housner, J. M.

1994-01-01

LATDYN is a computer code for modeling the Large Angle Transient DYNamics of flexible articulating structures and mechanisms involving joints about which members rotate through large angles. LATDYN extends and brings together some of the aspects of Finite Element Structural Analysis, Multi-Body Dynamics, and Control System Analysis; three disciplines that have been historically separate. It combines significant portions of their distinct capabilities into one single analysis tool. The finite element formulation for flexible bodies in LATDYN extends the conventional finite element formulation by using a convected coordinate system for constructing the equation of motion. LATDYN's formulation allows for large displacements and rotations of finite elements subject to the restriction that deformations within each are small. Also, the finite element approach implemented in LATDYN provides a convergent path for checking solutions simply by increasing mesh density. For rigid bodies and joints LATDYN borrows extensively from methodology used in multi-body dynamics where rigid bodies may be defined and connected together through joints (hinges, ball, universal, sliders, etc.). Joints may be modeled either by constraints or by adding joint degrees of freedom. To eliminate error brought about by the separation of structural analysis and control analysis, LATDYN provides symbolic capabilities for modeling control systems which are integrated with the structural dynamic analysis itself. Its command language contains syntactical structures which perform symbolic operations which are also interfaced directly with the finite element structural model, bypassing the modal approximation. Thus, when the dynamic equations representing the structural model are integrated, the equations representing the control system are integrated along with them as a coupled system. This procedure also has the side benefit of enabling a dramatic simplification of the user interface for modeling control systems. Three FORTRAN computer programs, the LATDYN Program, the Preprocessor, and the Postprocessor, make up the collective LATDYN System. The Preprocessor translates user commands into a form which can be used while the LATDYN program provides the computational core. The Postprocessor allows the user to interactively plot and manage a database of LATDYN transient analysis results. It also includes special facilities for modeling control systems and for programming changes to the model which take place during analysis sequence. The documentation includes a Demonstration Problem Manual for the evaluation and verification of results and a Postprocessor guide. Because the program should be viewed as a byproduct of research on technology development, LATDYN's scope is limited. It does not have a wide library of finite elements, and 3-D Graphics are not available. Nevertheless, it does have a measure of "user friendliness". The LATDYN program was developed over a period of several years and was implemented on a CDC NOS/VE & Convex Unix computer. It is written in FORTRAN 77 and has a virtual memory requirement of 1.46 MB. The program was validated on a DEC MICROVAX operating under VMS 5.2.

Influence of local meshing size on stress intensity factor of orthopedic lag screw

NASA Astrophysics Data System (ADS)

Husain, M. N.; Daud, R.; Basaruddin, K. S.; Mat, F.; Bajuri, M. Y.; Arifin, A. K.

2017-09-01

Linear elastic fracture mechanics (LEFM) concept is generally used to study the influence of crack on the performance of structures. In order to study the LEFM concept on damaged structure, the usage of finite element analysis software is implemented to do the simulation of the structure. Mesh generation is one of the most crucial procedures in finite element method. For the structure that crack or damaged, it is very important to determine the accurate local meshing size at the crack tip of the crack itself in order to get the accurate value of stress intensity factor, KI. Pre crack will be introduced to the lag screw based on the von mises' stress result that had been performed in previous research. This paper shows the influence of local mesh arrangement on numerical value of the stress intensity factor, KI obtained by the displacement method. This study aims to simulate the effect of local meshing which is the singularity region on stress intensity factor, KI to the critical point of failure in screw. Five different set of wedges meshing size are introduced during the simulation of finite element analysis. The number of wedges used to simulate this research is 8, 10, 14, 16 and 20. There are three set of numerical equations used to validate the results which are brown and srawley, gross and brown and Tada equation. The result obtained from the finite element software (ANSYS APDL) has a positive agreement with the numerical analysis which is Brown and Srawley compared to other numerical formula. Radius of first row size of 0.014 and singularity element with 14 numbers of wedges is proved to be the best local meshing for this study.

A Spectral Finite Element Approach to Modeling Soft Solids Excited with High-Frequency Harmonic Loads

PubMed Central

Brigham, John C.; Aquino, Wilkins; Aguilo, Miguel A.; Diamessis, Peter J.

2010-01-01

An approach for efficient and accurate finite element analysis of harmonically excited soft solids using high-order spectral finite elements is presented and evaluated. The Helmholtz-type equations used to model such systems suffer from additional numerical error known as pollution when excitation frequency becomes high relative to stiffness (i.e. high wave number), which is the case, for example, for soft tissues subject to ultrasound excitations. The use of high-order polynomial elements allows for a reduction in this pollution error, but requires additional consideration to counteract Runge's phenomenon and/or poor linear system conditioning, which has led to the use of spectral element approaches. This work examines in detail the computational benefits and practical applicability of high-order spectral elements for such problems. The spectral elements examined are tensor product elements (i.e. quad or brick elements) of high-order Lagrangian polynomials with non-uniformly distributed Gauss-Lobatto-Legendre nodal points. A shear plane wave example is presented to show the dependence of the accuracy and computational expense of high-order elements on wave number. Then, a convergence study for a viscoelastic acoustic-structure interaction finite element model of an actual ultrasound driven vibroacoustic experiment is shown. The number of degrees of freedom required for a given accuracy level was found to consistently decrease with increasing element order. However, the computationally optimal element order was found to strongly depend on the wave number. PMID:21461402

On mixed and displacement finite element models of a refined shear deformation theory for laminated anisotropic plates

NASA Technical Reports Server (NTRS)

Reddy, J. N.

1986-01-01

An improved plate theory that accounts for the transverse shear deformation is presented, and mixed and displacement finite element models of the theory are developed. The theory is based on an assumed displacement field in which the inplane displacements are expanded in terms of the thickness coordinate up to the cubic term and the transverse deflection is assumed to be independent of the thickness coordinate. The governing equations of motion for the theory are derived from the Hamilton's principle. The theory eliminates the need for shear correction factors because the transverse shear stresses are represented parabolically. A mixed finite element model that uses independent approximations of the displacements and moments, and a displacement model that uses only displacements as degrees of freedom are developed. A comparison of the numerical results for bending with the exact solutions of the new theory and the three-dimensional elasticity theory shows that the present theory (and hence the finite element models) is more accurate than other plate-theories of the same order.

Sensitivity calculations for iteratively solved problems

NASA Technical Reports Server (NTRS)

Haftka, R. T.

1985-01-01

The calculation of sensitivity derivatives of solutions of iteratively solved systems of algebraic equations is investigated. A modified finite difference procedure is presented which improves the accuracy of the calculated derivatives. The procedure is demonstrated for a simple algebraic example as well as an element-by-element preconditioned conjugate gradient iterative solution technique applied to truss examples.

Fluid-structure finite-element vibrational analysis

NASA Technical Reports Server (NTRS)

Feng, G. C.; Kiefling, L.

1974-01-01

A fluid finite element has been developed for a quasi-compressible fluid. Both kinetic and potential energy are expressed as functions of nodal displacements. Thus, the formulation is similar to that used for structural elements, with the only differences being that the fluid can possess gravitational potential, and the constitutive equations for fluid contain no shear coefficients. Using this approach, structural and fluid elements can be used interchangeably in existing efficient sparse-matrix structural computer programs such as SPAR. The theoretical development of the element formulations and the relationships of the local and global coordinates are shown. Solutions of fluid slosh, liquid compressibility, and coupled fluid-shell oscillation problems which were completed using a temporary digital computer program are shown. The frequency correlation of the solutions with classical theory is excellent.

Analytical Finite Element Simulation Model for Structural Crashworthiness Prediction

DOT National Transportation Integrated Search

1974-02-01

The analytical development and appropriate derivations are presented for a simulation model of vehicle crashworthiness prediction. Incremental equations governing the nonlinear elasto-plastic dynamic response of three-dimensional frame structures are...

Iterative and multigrid methods in the finite element solution of incompressible and turbulent fluid flow

NASA Astrophysics Data System (ADS)

Lavery, N.; Taylor, C.

1999-07-01

Multigrid and iterative methods are used to reduce the solution time of the matrix equations which arise from the finite element (FE) discretisation of the time-independent equations of motion of the incompressible fluid in turbulent motion. Incompressible flow is solved by using the method of reduce interpolation for the pressure to satisfy the Brezzi-Babuska condition. The k-l model is used to complete the turbulence closure problem. The non-symmetric iterative matrix methods examined are the methods of least squares conjugate gradient (LSCG), biconjugate gradient (BCG), conjugate gradient squared (CGS), and the biconjugate gradient squared stabilised (BCGSTAB). The multigrid algorithm applied is based on the FAS algorithm of Brandt, and uses two and three levels of grids with a V-cycling schedule. These methods are all compared to the non-symmetric frontal solver. Copyright

Relativistic extension of a charge-conservative finite element solver for time-dependent Maxwell-Vlasov equations

NASA Astrophysics Data System (ADS)

Na, D.-Y.; Moon, H.; Omelchenko, Y. A.; Teixeira, F. L.

2018-01-01

Accurate modeling of relativistic particle motion is essential for physical predictions in many problems involving vacuum electronic devices, particle accelerators, and relativistic plasmas. A local, explicit, and charge-conserving finite-element time-domain (FETD) particle-in-cell (PIC) algorithm for time-dependent (non-relativistic) Maxwell-Vlasov equations on irregular (unstructured) meshes was recently developed by Moon et al. [Comput. Phys. Commun. 194, 43 (2015); IEEE Trans. Plasma Sci. 44, 1353 (2016)]. Here, we extend this FETD-PIC algorithm to the relativistic regime by implementing and comparing three relativistic particle-pushers: (relativistic) Boris, Vay, and Higuera-Cary. We illustrate the application of the proposed relativistic FETD-PIC algorithm for the analysis of particle cyclotron motion at relativistic speeds, harmonic particle oscillation in the Lorentz-boosted frame, and relativistic Bernstein modes in magnetized charge-neutral (pair) plasmas.

Efficient Computation of Atmospheric Flows with Tempest: Validation of Next-Generation Climate and Weather Prediction Algorithms at Non-Hydrostatic Scales

NASA Astrophysics Data System (ADS)

Guerra, Jorge; Ullrich, Paul

2016-04-01

Tempest is a next-generation global climate and weather simulation platform designed to allow experimentation with numerical methods for a wide range of spatial resolutions. The atmospheric fluid equations are discretized by continuous / discontinuous finite elements in the horizontal and by a staggered nodal finite element method (SNFEM) in the vertical, coupled with implicit/explicit time integration. At horizontal resolutions below 10km, many important questions remain on optimal techniques for solving the fluid equations. We present results from a suite of idealized test cases to validate the performance of the SNFEM applied in the vertical with an emphasis on flow features and dynamic behavior. Internal gravity wave, mountain wave, convective bubble, and Cartesian baroclinic instability tests will be shown at various vertical orders of accuracy and compared with known results.

A thermal analysis of a spirally wound battery using a simple mathematical model

NASA Technical Reports Server (NTRS)

Evans, T. I.; White, R. E.

1989-01-01

A two-dimensional thermal model for spirally wound batteries has been developed. The governing equation of the model is the energy balance. Convective and insulated boundary conditions are used, and the equations are solved using a finite element code called TOPAZ2D. The finite element mesh is generated using a preprocessor to TOPAZ2D called MAZE. The model is used to estimate temperature profiles within a spirally wound D-size cell. The model is applied to the lithium/thionyl chloride cell because of the thermal management problems that this cell exhibits. Simplified one-dimensional models are presented that can be used to predict best and worst temperature profiles. The two-dimensional model is used to predict the regions of maximum temperature within the spirally wound cell. Normal discharge as well as thermal runaway conditions are investigated.

Coupling 2D Finite Element Models and Circuit Equations Using a Bottom-Up Methodology

DTIC Science & Technology

2002-11-01

EQUATIONS USING A BOTTOM-UP METHODOLOGY E. G6mezl, J. Roger-Folch2 , A. Gabald6nt and A. Molina’ ’Dpto. de Ingenieria Eldctrica. Universidad Polit...de Ingenieria Elictrica. ETSII. Universidad Politdcnica de Valencia. PO Box 22012, 46071. Valencia, Spain. E-mail: iroger adie.upv.es ABSTRACT The

A multigrid solver for the semiconductor equations

NASA Technical Reports Server (NTRS)

Bachmann, Bernhard

1993-01-01

We present a multigrid solver for the exponential fitting method. The solver is applied to the current continuity equations of semiconductor device simulation in two dimensions. The exponential fitting method is based on a mixed finite element discretization using the lowest-order Raviart-Thomas triangular element. This discretization method yields a good approximation of front layers and guarantees current conservation. The corresponding stiffness matrix is an M-matrix. 'Standard' multigrid solvers, however, cannot be applied to the resulting system, as this is dominated by an unsymmetric part, which is due to the presence of strong convection in part of the domain. To overcome this difficulty, we explore the connection between Raviart-Thomas mixed methods and the nonconforming Crouzeix-Raviart finite element discretization. In this way we can construct nonstandard prolongation and restriction operators using easily computable weighted L(exp 2)-projections based on suitable quadrature rules and the upwind effects of the discretization. The resulting multigrid algorithm shows very good results, even for real-world problems and for locally refined grids.

An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

NASA Astrophysics Data System (ADS)

de Almeida, Valmor F.

2017-07-01

A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

Using Plate Finite Elements for Modeling Fillets in Design, Optimization, and Dynamic Analysis

NASA Technical Reports Server (NTRS)

Brown, A. M.; Seugling, R. M.

2003-01-01

A methodology has been developed that allows the use of plate elements instead of numerically inefficient solid elements for modeling structures with 90 degree fillets. The technique uses plate bridges with pseudo Young's modulus (Eb) and thickness (tb) values placed between the tangent points of the fillets. These parameters are obtained by solving two nonlinear simultaneous equations in terms of the independent variables rlt and twallt. These equations are generated by equating the rotation at the tangent point of a bridge system with that of a fillet, where both rotations are derived using beam theory. Accurate surface fits of the solutions are also presented to provide the user with closed-form equations for the parameters. The methodology was verified on the subcomponent level and with a representative filleted structure, where the technique yielded a plate model exhibiting a level of accuracy better than or equal to a high-fidelity solid model and with a 90-percent reduction in the number of DOFs. The application of this method for parametric design studies, optimization, and dynamic analysis should prove extremely beneficial for the finite element practitioner. Although the method does not attempt to produce accurate stresses in the filleted region, it can also be used to obtain stresses elsewhere in the structure for preliminary analysis. A future avenue of study is to extend the theory developed here to other fillet geometries, including fillet angles other than 90 and multifaceted intersections.

Parallel Element Agglomeration Algebraic Multigrid and Upscaling Library

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

Barker, Andrew T.; Benson, Thomas R.; Lee, Chak Shing

ParELAG is a parallel C++ library for numerical upscaling of finite element discretizations and element-based algebraic multigrid solvers. It provides optimal complexity algorithms to build multilevel hierarchies and solvers that can be used for solving a wide class of partial differential equations (elliptic, hyperbolic, saddle point problems) on general unstructured meshes. Additionally, a novel multilevel solver for saddle point problems with divergence constraint is implemented.

Nonlinear finite element simulation of non-local tension softening for high strength steel material

NASA Astrophysics Data System (ADS)

Tong, F. M.

The capability of current finite element softwares in simulating the stress-strain relation beyond the elastic-plastic region has been limited by the inability for non- positivity in the computational finite elements' stiffness matrixes. Although analysis up to the peak stress has been proved adequate for analysis and design, it provides no indication of the possible failure predicament that is to follow. Therefore an attempt was made to develop a modelling technique capable of capturing the complete stress-deformation response in an analysis beyond the limit point. This proposed model characterizes a cyclic loading and unloading procedure, as observed in a typical laboratory uniaxial cyclic test, along with a series of material properties updates. The Voce equation and a polynomial function were proposed to define the monotonic elastoplastic hardening and softening behaviour respectively. A modified form of the Voce equation was used to capture the reloading response in the softening region. To accommodate the reduced load capacity of the material at each subsequent softening point, an optimization macro was written to control this optimum load at which the material could withstand. This preliminary study has ignored geometrical effect and is thus incapable of capturing the localized necking phenomenon that accompanies many ductile materials. The current softening model is sufficient if a global measure is considered. Several validation cases were performed to investigate the feasibility of the modelling technique and the results have been proved satisfactory. The ANSYS finite element software is used as the platform at which the modelling technique operates.

Finite element computation of multi-physical micropolar transport phenomena from an inclined moving plate in porous media

NASA Astrophysics Data System (ADS)

Shamshuddin, MD.; Anwar Bég, O.; Sunder Ram, M.; Kadir, A.

2018-02-01

Non-Newtonian flows arise in numerous industrial transport processes including materials fabrication systems. Micropolar theory offers an excellent mechanism for exploring the fluid dynamics of new non-Newtonian materials which possess internal microstructure. Magnetic fields may also be used for controlling electrically-conducting polymeric flows. To explore numerical simulation of transport in rheological materials processing, in the current paper, a finite element computational solution is presented for magnetohydrodynamic, incompressible, dissipative, radiative and chemically-reacting micropolar fluid flow, heat and mass transfer adjacent to an inclined porous plate embedded in a saturated homogenous porous medium. Heat generation/absorption effects are included. Rosseland's diffusion approximation is used to describe the radiative heat flux in the energy equation. A Darcy model is employed to simulate drag effects in the porous medium. The governing transport equations are rendered into non-dimensional form under the assumption of low Reynolds number and also low magnetic Reynolds number. Using a Galerkin formulation with a weighted residual scheme, finite element solutions are presented to the boundary value problem. The influence of plate inclination, Eringen coupling number, radiation-conduction number, heat absorption/generation parameter, chemical reaction parameter, plate moving velocity parameter, magnetic parameter, thermal Grashof number, species (solutal) Grashof number, permeability parameter, Eckert number on linear velocity, micro-rotation, temperature and concentration profiles. Furthermore, the influence of selected thermo-physical parameters on friction factor, surface heat transfer and mass transfer rate is also tabulated. The finite element solutions are verified with solutions from several limiting cases in the literature. Interesting features in the flow are identified and interpreted.

Analytical and Numerical Results for an Adhesively Bonded Joint Subjected to Pure Bending

NASA Technical Reports Server (NTRS)

Smeltzer, Stanley S., III; Lundgren, Eric

2006-01-01

A one-dimensional, semi-analytical methodology that was previously developed for evaluating adhesively bonded joints composed of anisotropic adherends and adhesives that exhibit inelastic material behavior is further verified in the present paper. A summary of the first-order differential equations and applied joint loading used to determine the adhesive response from the methodology are also presented. The method was previously verified against a variety of single-lap joint configurations from the literature that subjected the joints to cases of axial tension and pure bending. Using the same joint configuration and applied bending load presented in a study by Yang, the finite element analysis software ABAQUS was used to further verify the semi-analytical method. Linear static ABAQUS results are presented for two models, one with a coarse and one with a fine element meshing, that were used to verify convergence of the finite element analyses. Close agreement between the finite element results and the semi-analytical methodology were determined for both the shear and normal stress responses of the adhesive bondline. Thus, the semi-analytical methodology was successfully verified using the ABAQUS finite element software and a single-lap joint configuration subjected to pure bending.

Hybrid DG/FV schemes for magnetohydrodynamics and relativistic hydrodynamics

NASA Astrophysics Data System (ADS)

Núñez-de la Rosa, Jonatan; Munz, Claus-Dieter

2018-01-01

This paper presents a high order hybrid discontinuous Galerkin/finite volume scheme for solving the equations of the magnetohydrodynamics (MHD) and of the relativistic hydrodynamics (SRHD) on quadrilateral meshes. In this approach, for the spatial discretization, an arbitrary high order discontinuous Galerkin spectral element (DG) method is combined with a finite volume (FV) scheme in order to simulate complex flow problems involving strong shocks. Regarding the time discretization, a fourth order strong stability preserving Runge-Kutta method is used. In the proposed hybrid scheme, a shock indicator is computed at the beginning of each Runge-Kutta stage in order to flag those elements containing shock waves or discontinuities. Subsequently, the DG solution in these troubled elements and in the current time step is projected onto a subdomain composed of finite volume subcells. Right after, the DG operator is applied to those unflagged elements, which, in principle, are oscillation-free, meanwhile the troubled elements are evolved with a robust second/third order FV operator. With this approach we are able to numerically simulate very challenging problems in the context of MHD and SRHD in one, and two space dimensions and with very high order polynomials. We make convergence tests and show a comprehensive one- and two dimensional testbench for both equation systems, focusing in problems with strong shocks. The presented hybrid approach shows that numerical schemes of very high order of accuracy are able to simulate these complex flow problems in an efficient and robust manner.

Finite Volume Element (FVE) discretization and multilevel solution of the axisymmetric heat equation

NASA Astrophysics Data System (ADS)

Litaker, Eric T.

1994-12-01

The axisymmetric heat equation, resulting from a point-source of heat applied to a metal block, is solved numerically; both iterative and multilevel solutions are computed in order to compare the two processes. The continuum problem is discretized in two stages: finite differences are used to discretize the time derivatives, resulting is a fully implicit backward time-stepping scheme, and the Finite Volume Element (FVE) method is used to discretize the spatial derivatives. The application of the FVE method to a problem in cylindrical coordinates is new, and results in stencils which are analyzed extensively. Several iteration schemes are considered, including both Jacobi and Gauss-Seidel; a thorough analysis of these schemes is done, using both the spectral radii of the iteration matrices and local mode analysis. Using this discretization, a Gauss-Seidel relaxation scheme is used to solve the heat equation iteratively. A multilevel solution process is then constructed, including the development of intergrid transfer and coarse grid operators. Local mode analysis is performed on the components of the amplification matrix, resulting in the two-level convergence factors for various combinations of the operators. A multilevel solution process is implemented by using multigrid V-cycles; the iterative and multilevel results are compared and discussed in detail. The computational savings resulting from the multilevel process are then discussed.

A model for tides and currents in the English Channel and southern North Sea

USGS Publications Warehouse

Walters, Roy A.

1987-01-01

The amplitude and phase of 11 tidal constituents for the English Channel and southern North Sea are calculated using a frequency domain, finite element model. The governing equations - the shallow water equations - are modifed such that sea level is calculated using an elliptic equation of the Helmholz type followed by a back-calculation of velocity using the primitive momentum equations. Triangular elements with linear basis functions are used. The modified form of the governing equations provides stable solutions with little numerical noise. In this field-scale test problem, the model was able to produce the details of the structure of 11 tidal constituents including O1, K1, M2, S2, N2, K2, M4, MS4, MN4, M6, and 2MS6.

A rigorous solution of the Navier-Stokes equations for unsteady viscous flow at high Reynolds numbers around oscillating airfoils

NASA Technical Reports Server (NTRS)

Bratanow, T.; Aksu, H.; Spehert, T.

1975-01-01

A method based on the Navier-Stokes equations was developed for analyzing the unsteady incompressible viscous flow around oscillating airfoils at high Reynolds numbers. The Navier-Stokes equations have been integrated in their classical Helmholtz vorticity transport equation form, and the instantaneous velocity field at each time step was determined by the solution of Poisson's equation. A refined finite element was utilized to allow for a conformable solution of the stream function and its first space derivatives at the element interfaces. A corresponding set of accurate boundary conditions was applied; thus obtaining a rigorous solution for the velocity field. The details of the computational procedure and examples of computed results describing the unsteady flow characteristics around the airfoil are presented.

A finite difference solution for the propagation of sound in near sonic flows

NASA Technical Reports Server (NTRS)

Hariharan, S. I.; Lester, H. C.

1983-01-01

An explicit time/space finite difference procedure is used to model the propagation of sound in a quasi one-dimensional duct containing high Mach number subsonic flow. Nonlinear acoustic equations are derived by perturbing the time-dependent Euler equations about a steady, compressible mean flow. The governing difference relations are based on a fourth-order, two-step (predictor-corrector) MacCormack scheme. The solution algorithm functions by switching on a time harmonic source and allowing the difference equations to iterate to a steady state. The principal effect of the non-linearities was to shift acoustical energy to higher harmonics. With increased source strengths, wave steepening was observed. This phenomenon suggests that the acoustical response may approach a shock behavior at at higher sound pressure level as the throat Mach number aproaches unity. On a peak level basis, good agreement between the nonlinear finite difference and linear finite element solutions was observed, even through a peak sound pressure level of about 150 dB occurred in the throat region. Nonlinear steady state waveform solutions are shown to be in excellent agreement with a nonlinear asymptotic theory.

Finite element methods and Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Cuvelier, C.; Segal, A.; van Steenhoven, A. A.

This book is devoted to two and three-dimensional FEM analysis of the Navier-Stokes (NS) equations describing one flow of a viscous incompressible fluid. Three different approaches to the NS equations are described: a direct method, a penalty method, and a method that constructs discrete solenoidal vector fields. Subjects of current research which are important from the industrial/technological viewpoint are considered, including capillary-free boundaries, nonisothermal flows, turbulence, and non-Newtonian fluids.

M-step preconditioned conjugate gradient methods

NASA Technical Reports Server (NTRS)

Adams, L.

1983-01-01

Preconditioned conjugate gradient methods for solving sparse symmetric and positive finite systems of linear equations are described. Necessary and sufficient conditions are given for when these preconditioners can be used and an analysis of their effectiveness is given. Efficient computer implementations of these methods are discussed and results on the CYBER 203 and the Finite Element Machine under construction at NASA Langley Research Center are included.

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.

A finite element beam propagation method for simulation of liquid crystal devices.

PubMed

Vanbrabant, Pieter J M; Beeckman, Jeroen; Neyts, Kristiaan; James, Richard; Fernandez, F Anibal

2009-06-22

An efficient full-vectorial finite element beam propagation method is presented that uses higher order vector elements to calculate the wide angle propagation of an optical field through inhomogeneous, anisotropic optical materials such as liquid crystals. The full dielectric permittivity tensor is considered in solving Maxwell's equations. The wide applicability of the method is illustrated with different examples: the propagation of a laser beam in a uniaxial medium, the tunability of a directional coupler based on liquid crystals and the near-field diffraction of a plane wave in a structure containing micrometer scale variations in the transverse refractive index, similar to the pixels of a spatial light modulator.

Finite-element numerical modeling of atmospheric turbulent boundary layer

NASA Technical Reports Server (NTRS)

Lee, H. N.; Kao, S. K.

1979-01-01

A dynamic turbulent boundary-layer model in the neutral atmosphere is constructed, using a dynamic turbulent equation of the eddy viscosity coefficient for momentum derived from the relationship among the turbulent dissipation rate, the turbulent kinetic energy and the eddy viscosity coefficient, with aid of the turbulent second-order closure scheme. A finite-element technique was used for the numerical integration. In preliminary results, the behavior of the neutral planetary boundary layer agrees well with the available data and with the existing elaborate turbulent models, using a finite-difference scheme. The proposed dynamic formulation of the eddy viscosity coefficient for momentum is particularly attractive and can provide a viable alternative approach to study atmospheric turbulence, diffusion and air pollution.

A triangular thin shell finite element: Nonlinear analysis. [structural analysis

NASA Technical Reports Server (NTRS)

Thomas, G. R.; Gallagher, R. H.

1975-01-01

Aspects of the formulation of a triangular thin shell finite element which pertain to geometrically nonlinear (small strain, finite displacement) behavior are described. The procedure for solution of the resulting nonlinear algebraic equations combines a one-step incremental (tangent stiffness) approach with one iteration in the Newton-Raphson mode. A method is presented which permits a rational estimation of step size in this procedure. Limit points are calculated by means of a superposition scheme coupled to the incremental side of the solution procedure while bifurcation points are calculated through a process of interpolation of the determinants of the tangent-stiffness matrix. Numerical results are obtained for a flat plate and two curved shell problems and are compared with alternative solutions.

Numerical simulation of fluid flow around a scramaccelerator projectile

NASA Technical Reports Server (NTRS)

Pepper, Darrell W.; Humphrey, Joseph W.; Sobota, Thomas H.

1991-01-01

Numerical simulations of the fluid motion and temperature distribution around a 'scramaccelerator' projectile are obtained for Mach numbers in the 5-10 range. A finite element method is used to solve the equations of motion for inviscid and viscous two-dimensional or axisymmetric compressible flow. The time-dependent equations are solved explicitly, using bilinear isoparametric quadrilateral elements, mass lumping, and a shock-capturing Petrov-Galerkin formulation. Computed results indicate that maintaining on-design performance for controlling and stabilizing oblique detonation waves is critically dependent on projectile shape and Mach number.

Local-Mesh, Local-Order, Adaptive Finite Element Methods with a Posteriori Error Estimators for Elliptic Partial Differential Equations.

DTIC Science & Technology

1981-12-01

I I I I I o-F--o -- oIl lI I I 0--0------0I Im I I o--G--o ] II I I ...C-0076, the Department of Energy (DOE Grant DE-AC02-77ET53053), The National Science Foundation (Graduate Fellowship), and Yale University. " i o V.IM...element method, the choice of discretization i eft to the user, who must base his decision on experience with similar equations. - In recent years,

Numerical Simulation of Two-Fluid Mingling Using the Particle Finite Element Method with Applications to Magmatic and Volcanic Processes

NASA Astrophysics Data System (ADS)

de Mier, M.; Costa, F.; Idelsohn, S.

2008-12-01

Many magmatic and volcanic processes (e.g., magma differentiation, mingling, transport in the volcanic conduit) are controlled by the physical properties and flow styles of high-temperature silicate melts. Such processes can be experimentally investigated using analog systems and scaling methods, but it is difficult to find the suitable material and it is generally not possible to quantitatively extrapolate the results to the natural system. An alternative means of studying fluid dynamics in volcanic systems is with numerical models. We have chosen the Particle Finite Element Method (PFEM), which is based on a Delaunay mesh that moves with the fluid velocity, the Navier-Stokes equations in Lagrangian formulation, and linear elements for velocity, pressure, and temperature. Remeshing is performed when the grid becomes too distorted [E. Oñate et al., 2004. The Particle Finite Element Method: An Overview. Int. J. Comput. Meth. 1, 267-307]. The method is ideal for tracking material interfaces between different fluids or media. Methods based on Eulerian reference frames need special techniques, such as level-set or volume-of-fluid, to capture the interface position, and these techniques add a significant numerical diffusion at the interface. We have performed a series of two-dimensional simulations of a classical problem of fluid dynamics in magmatic and volcanic systems: intrusion of a basaltic melt in a silica-rich magma reservoir. We have used realistic physical properties and equations of state for the silicate melts (e.g., temperature, viscosity, and density) and tracked the changes in the system for geologically relevant time scales (up to 100 years). The problem is modeled by the low-Mach-number equations derived from an asymptotic analysis of the compressible Navier-Stokes equations that removes shock waves from the flow but allows however large variations of density due to temperature variations. Non-constant viscosity and volume changes are taken into account in the momentum conservation equation through the full shear-stress tensor. The implications of different magma intrusion rates, volumes, and times will be discussed in the context of mafic-silicic magma mixing and eruption triggers.

Finite elements for the calculation of turbulent flows in three-dimensional complex geometries

NASA Astrophysics Data System (ADS)

Ruprecht, A.

A finite element program for the calculation of incompressible turbulent flows is presented. In order to reduce the required storage an iterative algorithm is used which solves the necessary equations sequentially. The state of turbulence is defined by the k-epsilon model. In addition to the standard k-epsilon model, the modification of Bardina et al., taking into account the rotation of the mean flow, is investigated. With this program, the flow in the draft tube of a Kaplan turbine is examined. Calculations are carried out for swirling and nonswirling entrance flow. The results are compared with measurements.

Domain decomposition for a mixed finite element method in three dimensions

USGS Publications Warehouse

Cai, Z.; Parashkevov, R.R.; Russell, T.F.; Wilson, J.D.; Ye, X.

2003-01-01

We consider the solution of the discrete linear system resulting from a mixed finite element discretization applied to a second-order elliptic boundary value problem in three dimensions. Based on a decomposition of the velocity space, these equations can be reduced to a discrete elliptic problem by eliminating the pressure through the use of substructures of the domain. The practicality of the reduction relies on a local basis, presented here, for the divergence-free subspace of the velocity space. We consider additive and multiplicative domain decomposition methods for solving the reduced elliptic problem, and their uniform convergence is established.

In-memory integration of existing software components for parallel adaptive unstructured mesh workflows

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

Smith, Cameron W.; Granzow, Brian; Diamond, Gerrett

Unstructured mesh methods, like finite elements and finite volumes, support the effective analysis of complex physical behaviors modeled by partial differential equations over general threedimensional domains. The most reliable and efficient methods apply adaptive procedures with a-posteriori error estimators that indicate where and how the mesh is to be modified. Although adaptive meshes can have two to three orders of magnitude fewer elements than a more uniform mesh for the same level of accuracy, there are many complex simulations where the meshes required are so large that they can only be solved on massively parallel systems.

In-memory integration of existing software components for parallel adaptive unstructured mesh workflows

DOE PAGES

Smith, Cameron W.; Granzow, Brian; Diamond, Gerrett; ...

2017-01-01

Unstructured mesh methods, like finite elements and finite volumes, support the effective analysis of complex physical behaviors modeled by partial differential equations over general threedimensional domains. The most reliable and efficient methods apply adaptive procedures with a-posteriori error estimators that indicate where and how the mesh is to be modified. Although adaptive meshes can have two to three orders of magnitude fewer elements than a more uniform mesh for the same level of accuracy, there are many complex simulations where the meshes required are so large that they can only be solved on massively parallel systems.

An arbitrary Lagrangian–Eulerian finite element formulation for a poroelasticity problem stemming from mixture theory

DOE PAGES

Costanzo, Francesco; Miller, Scott T.

2017-05-22

In this paper, a finite element formulation is developed for a poroelastic medium consisting of an incompressible hyperelastic skeleton saturated by an incompressible fluid. The governing equations stem from mixture theory and the application is motivated by the study of interstitial fluid flow in brain tissue. The formulation is based on the adoption of an arbitrary Lagrangian–Eulerian (ALE) perspective. We focus on a flow regime in which inertia forces are negligible. Finally, the stability and convergence of the formulation is discussed, and numerical results demonstrate agreement with the theory.

An arbitrary Lagrangian–Eulerian finite element formulation for a poroelasticity problem stemming from mixture theory

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

Costanzo, Francesco; Miller, Scott T.

In this paper, a finite element formulation is developed for a poroelastic medium consisting of an incompressible hyperelastic skeleton saturated by an incompressible fluid. The governing equations stem from mixture theory and the application is motivated by the study of interstitial fluid flow in brain tissue. The formulation is based on the adoption of an arbitrary Lagrangian–Eulerian (ALE) perspective. We focus on a flow regime in which inertia forces are negligible. Finally, the stability and convergence of the formulation is discussed, and numerical results demonstrate agreement with the theory.