A solution algorithm for linear constraint equations in finite element analysis

A general solution algorithm is presented for the incorporation of a general set of linear constraint equations into a linear algebraic system; such situations arise in the application of the finite element method to a variety of physical problems. Implementation of the algorithm, without need for pre-arranging the equations, into an equation solver using Gauss elimination is developed. The method is most attractive as compared to other approaches for constrained systems.     

Treatment of skew boundary conditions in finite element analysis

Finite element analyses in both structural and geotechnical applications sometimes require the freedom directions at certain nodes to differ from the global Cartesian directions. Examples of this include modelling of oblique interface behaviour and implementation of ‘skew’ boundary conditions. A simple method is described for performing the transformations at the element level, and a FORTRAN 77 subroutine is included to perform the operations.     

Implementation of nonlinear boundary conditions in finite element analysis

In finite element analysis sometimes special boundary conditions such as no-tension, friction contact or other nonlinear types occur. A family of boundary condition elements is presented. Based on the Winkler assumption for elastic foundations the stiffness matrix for two and three dimensional isoparametric elements is developed which account for linear or nonlinear bedding of structures.     

Accurate evaluation of support reactions in finite element plate-bending analysis

Two simple approaches are presented which allow the distribution of support reactions to be predicted with as high degree of accuracy as the displacements. In the first approach the plate element assembly is completed with special one-dimensional elastic support elements. If their Winkler coefficient is suitably tuned, an accurate prediction of reactions is obtained as a part of the finite element analysis without unduly affecting the displacements and moments of the plate. In the second approach, a standard finite element calculation (without elastic support elements) is performed first and the distribution of reactions is then evaluated based on the known nodal forces at boundary nodes of the plate.

The two approaches are indiscriminately applicable with Kirchhoff and Reissner-Mindlin plate bending elements. Their practical efficiency is illustrated by numerical examples.     

Characteristic-based operator-splitting finite element method for Navier-Stokes equations

A new finite element method, which is the characteristic-based operator-splitting (CBOS) algorithm, is developed to solve Navier-Stokes (N-S) equations. In each time step, the equations are split into the diffusive part and the convective part by adopting the operator-splitting algorithm. For the diffusive part, the temporal discretization is performed by the backward difference method which yields an implicit scheme and the spatial discretization is performed by the standard Galerkin method. The convective...     

Multigrid for finite element discretizations of aerodynamics equations

A multigrid algorithm for the solution of a finite element stabilized discretization of compressible fluid dynamics equations on unstructured grids is described. The solution of the stationary problems is sought by time-stepping and a linearization of the nonlinear discrete systems leads to a very large system of linear equations. These systems are ill-conditioned and require efficient computational procedures. The numerical experiments for Navier-Stokes and Euler systems are presented. The method can be easily included in a parallel library as a preconditioner.     

A stable finite element for the stokes equations

We present in this paper a new velocity-pressure finite element for the computation of Stokes flow. We discretize the velocity field with continuous piecewise linear functions enriched by bubble functions, and the pressure by piecewise linear functions. We show that this element satisfies the usual inf-sup condition and converges with first order for both velocities and pressure. Finally we relate this element to families of higer order elements and to the popular Taylor-Hood element.     

Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations

Formulations which complement the streamline-upwind/Petrov-Galerkin procedure are presented. These formulations minimize the oscillations about sharp internal and boundary layers in convection-dominated and reaction-dominated flows. The proposed methods are tested on various single- and multi-component transport problems.     

Penalty finite element method for the Navier-Stokes equations

We present an analysis of a penalty formulation of the stationary Navier-Stokes equations for an incompressible fluid. Subject to restrictions on the viscosity and prescribed body force, it is shown that there exists a unique solution to this penalty problem. The solution to the penalty problem is shown to converge to the solution of the Navier-Stokes problem as O(ε) where ε → 0 is the penalty parameter.Existence, uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized Navier-Stokes problem presented here. Numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases.     

Combined finite volume–finite element method for shallow water equations

This paper is concerned with solving the viscous and inviscid shallow water equations. The numerical method is based on second-order finite volume–finite element (FV–FE) discretization: the convective inviscid terms of the shallow water equations are computed by a finite volume method, while the diffusive viscous terms are computed with a finite element method. The method is implemented on unstructured meshes. The inviscid fluxes are evaluated with the approximate Riemann solver coupled with a second-order upwind reconstruction. Herein, the Roe and the Osher approximate Riemann solvers are used respectively and a comparison between them is made. Appropriate limiters are used to suppress spurious oscillations and the performance of three different limiters is assessed. Moreover, the second-order conforming piecewise linear finite elements are used. The second-order TVD Runge–Kutta method is applied to the time integration. Verification of the method for the full viscous system and the inviscid equations is carried out. By solving an advection–diffusion problem, the performance assessment for the FV–FE method, the full finite volume method, and the discontinuous Galerkin method is presented.     

A class of iterative methods for finite element equations

A general method in the form of an accelerated preconditioned iterative refinement method (including some wellknown iterative methods and direct factorization methods) is presented for the solution of symmetric, sparse matrix problems. An analysis of one such approximate factorization, the SSOR method, is given, and some inherently advantageous properties of the conjugate gradient acceleration method are pointed out. A comparison is made of the computational complexity and storage in the SSOR preconditioned method with some direct methods applied to second order discretized boundary value problems. For plane problems of average size the direct methods are somewhat faster if enough right hand sides are present. For large enough problems (large number of nodes) the iterative method is faster. For three-dimensional problems no Cholesky factorization method can compete with the SSOR preconditioned method, not even for average sized problems.     

Hybrid finite element methods for the von Karman equations

We analyse the «assumed stresses» hybrid approximation of the Von Karman equations; we provide convergence results and optimal error bounds for a large class of finite element discretizations.     

A space-time least-square finite element scheme for advection-diffusion equations

A space-time least-square finite element scheme is presented for the advection-diffusion problems at moderate to high Peclet numbers. This scheme is designed to eliminate spurious oscillations and can be used to define the steady-state solution as the asymptotic transient solution for large time. Numerical results, using linear elements in a 1D space and bilinear elements in a 2D space, demonstrate the accuracy and the stability of the new scheme.     

Support system for finite element analysis

This paper describes work aimed at developing an intelligent support system for finite element modeling and a methodology for managing input data model. Analyzing various statement structures of input data, three structural interface models — the hierarchical browser, the spread sheet and the model generator — are proposed for advanced representation and editing. Two knowledge models composed of macro visual data representation (user oriented model) and micro regularized data representation (processor oriented model) are revealed in conformity with the approach of object-orientation. Moreover, an extended relational schema composed of a composite object (assembly of functional elements) and several abstracted scalar indexes has been implemented for case retrieval.     

The finite element approximation of Hamilton-Jacobi-Bellman equations

This paper deals with the finite element approximation of Hamilton-Jacobi-Bellman equations. We establish a convergence and a quasi-optimal L^∞-error estimate, involving a weakly coupled systems of quasi-variational inequalities for the solution of which an interative scheme of monotone kind is introduced and analyzed.     

Knowledge-based control for finite element analysis

This paper shows that control logic may be separated from analysis software and that a knowledge-based expert system can use this logic to perform interactive computation. Heuristics that control a simple interactive finite element analysis program are represented using a rule-based format and are used by a goal-driven logic processor to invoke analysis activity.Traditional algorithm-oriented control and the proposed knowledge-based control are compared in a simple displacement computation scenario to identify the advantages/disadvantages of the two approaches. General activities and constraints, practical methods of reasoning and representation, and knowledge-based expert systems are discussed with emphasis on applications to interactive finite element analysis.An analysis control expert system has been developed for use in the numerical analysis of two-dimensional linear problems in solid and structural mechanics. An example problem is used to clarify the methods used to direct activity and to identify the problems associated with conditional task processing for interactive analysis.The main difference between the analysis program described in this paper and conventional analysis programs is related to the control architecture. The general conclusion of this paper is that knowledge-based control is more effective and flexible than algorithm-oriented control.     

三维Euler方程的隐式间断有限元算法

为了求解三维欧拉方程,对隐式时间离散格式间断有限元方法进行了研究。根据间断Galerkin有限元方法思想,构造内迭代SOR-LU-SGS隐式时间离散格式,结合当地时间步长技术、多重网格方法,实现了三维流场的计算。数值计算了ONERAM6机翼、大攻角尖前缘三角翼以及DLR-F4翼身组合体的亚声速绕流问题。结果表明,加入SOR内迭代步的LU-SGS隐式算法具有较大的优势,相较于GMRES算法所占用的内存少且收敛速度相当,是LU-SGS算法的三倍以上。针对三维算例,具有较好的稳定性和较高的收敛速度,能够给出准确的流场信息。与原方法相比,SOR-LU-SGS方法无论是在迭代步数上还是在CPU时间上,效率均有明显提高,适合于三维复杂流场计算。