The solution of nonlinear finite element equations

An algorithm is described which appears to give an efficient solution of nonlinear finite element equations. It is a quisi-Nowton method, and we compare it with some of the alternatives. Initial tests of its application to both material and geometric nonlinearities are discussed.     

A finite element solution program for large structures

A straightforward and general computer program for assembling and solving (using Gauss elimination technique) widely sparsed finite element matrix equations with very large bandwidth and capable of handling different degrees-of-freedom and variable bandwidth at different nodes, is described herein. The program assembles any type of finite elements having arbitrary number of nodes and each node may have differnt degrees-of-freedom. It requires only a small core memory in the computer, although a fast random access device is also needed. The two very important features of this program are (i) it does not store any zero submatrices within the band and (ii) during the solution of equations all operations dealing with zero submatrices within the band are automatically skipped and thus the savings of a considerable amount of disc storage space and computer time can be effected in many cases. Another feature is that many right hand sides can be handled simultaneously. Hence the program is very economical for structures having widely sparsed matrix equations. A listing of the computer program written in FORTRAN IV for CDC 6400 computer is readily available from the authors, but unfortunately could not be given here because of lack of space. The program is so general that it can be used to solve a wide class of finite element problems without actually having to understand fully the techniques behind it.     

Formulation and solution of hierarchical finite element equations

The paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems. Static and eigenvalue linear elastic problems as well as convection–diffusion problems are studied. The hierarchical formulation is well suited for adaptive procedures. For the convection-diffusion problem the hierarchical approximation is made in time only. Different hierarchical functions are proposed for different types of problems. Both weighted residual and least-squares formulations are applied. A combination of these two gives a penalty method with a constraint equation corresponding to the least-squares method. A whole class of time integration formulae is obtained. These are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain. If a linear discontinuous hierarchical base function is used in the Galerkin weak formulation, the method so obtained corresponds to the discontinuous Galerkin method in time and is especially suited for convection dominated problems. The streamline-diffusion method is found to be the aforementioned penalty method. This paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation. Properties of these systems arising from static problems are investigated. The paper presents some new possibilities for iterative solution of hierarchic element equations, and different procedures are compared in a numerical example. Finally, a simple ID convection-diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow.     

An iterative solution scheme for systems of boundary element equations

In this paper an iterative scheme of first order is developed for the purpose of solving linear systems of equations. In particular, systems that are derived from boundary integral equations are investigated. The iterative schemes to be considered are of the form Ex^(k+1) = Dx^(k) + d, where E and D are square matrices. It will be assumed that E is a lower matrix, i.e. the coefficients above the central diagonal are zero. It will be shown that by considering matrix D embedded in a vector space and reducing its size with respect to a chosen metric, that convergence rates can be substantially improved. Equation ordering and parameter matrices are used to reduce the magnitude of D. A number of examples are tested to illustrate the importance of the choice of metric, equation ordering and the parameter matrix. Computation times are determined for both the iterative procedure and Gauss elimination indicating the usefulness of iteration which can be orders of magnitude faster.     

Bordering method for solving systems of linear equations generated by the finite element method in the plate bending problem

A combined iteration algorithm based on the bordering and conjugate gradient methods is proposed to solve systems of linear equations generated by the finite element method in the plate bending problem. The numerical results for the analysis of the convergence rate of the iterative process are presented in the solution of model problems using a classical and modified algorithm of the method of conjugate gradients. The possibility of acceleration of the iterative algorithm is shown. __________ Translated from Problemy Prochnosti, No. 4, pp. 137–145, July–August, 2007.     

An efficient finite element solution using a large pre-solved regular element

In this paper a finite element algorithm is presented using a large pre-solved hyper element. Utilizing the largest rectangle/cuboid inside an arbitrary domain, a large hyper element is developed that is solved using graph product rules. This pre-solved hyper element is efficiently inserted into the finite element formulation of partial differential equations (PDE) and engineering problems to reduce the computational complexity and execution time of the solution. A general solution of the large pre-solved element for a uniform mesh of triangular and rectangular elements is formulated for second-order PDEs. The efficiency of the algorithm depends on the relative size of the large element and the domain; however, the method remains as efficient as a classic method for even relatively small sizes. The application of the method is demonstrated using different examples.     

An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations

Finite element simulations of very large‐scale soil–structure interaction problems (e.g. excavations, tunnelling, pile‐rafts, etc.) typically involve the solution of a very large, ill‐conditioned, and indefinite Biot system of equations. The traditional preconditioned conjugate gradient solver coupled with the standard Jacobi (SJ) preconditioner can be very inefficient for this class of problems. This paper presents a robust generalized Jacobi (GJ) preconditioner that is extremely effective for solving very large‐scale Biot's finite element equations using the symmetric quasi‐minimal residual method. The GJ preconditioner can be formed, inverted, and implemented within an ‘element‐by‐element’ framework as readily as the SJ preconditioner. It was derived as a diagonal approximation to a theoretical form, which can be proven mathematically to possess an attractive eigenvalue clustering property. The effectiveness of the GJ preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three‐dimensional footing problem. This paper casts a new perspective on the potentialities of the simple diagonal preconditioner, which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix. Copyright © 2002 John Wiley & Sons, Ltd.     

A finite element method for the general solution of ordinary differential equations

A finite element method is developed by which it is possible to obtain the general solution of an ordinary differential equation directly. The procedure consists of approximating the differential equation with a rectangular matrix equation and of solving the latter equation by using generalized matrix inversion. It is shown in the paper that the homogeneous and inhomogeneous solutions of the two systems correspond and that the approximate solutions produced form the complete general solution of the original differential equation.     

A split Band-Cholesky equation solving strategy for finite element analysis of transisent field problems

An efficient strategy is outlined for out-of-core solution of the large systems of equations which specify nodal point time derivatives in finite element models of transient flow problems. The positive definiteness, symmetry, and band structure of the finite element mass matrices, as well as the nature of the equation assemblage process, is exploited by the method. Computational results are indicated for systems on the order of several thousand unknowns in size.     

Isoparametric,finite element,variational solution of integral equations for three-dimensional fields

An improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields. The technique provides higher-order approximation of the unknown function over a bounding surface described by two-parameter, non-planar elements. The integral equation is discretized through the Rayleigh–Ritz procedure. Convergence to the solution for operators having a positive-definite component is guaranteed. Kernel singularities are treated by removing them from the relevant integrals and dealing with them analytically. A successive element iterative process, which produces the solution of the large dense matrix of the complete structure, is described. The discretization and equation solution take place one element at a time resulting in storage and computational savings. Results obtained for classical test models, involving scalar electrostatic potential and vector elastostatic displacement fields, demonstrate the technique for the solution of the Fredholm integral equation of the first kind. Solution of the Fredholm equation of the second kind is to be reported subsequently.     

Adaptive finite element analysis with quadrilateral elements using a newh-refinement strategy

The theory and mathematical bases ofa-posteriori error estimates are explained. It is shown that theMedial Axis of a body can be used to decompose it into a set of mutually non-overlapping quadrilateral and triangular primitives. A mesh generation scheme used to generate quadrilaterals inside these primitives is also presented together with its relevant implementation aspects. A newh-refinement strategy based on weighted average energy norm and enhanced by strain energy density ratios is proposed and two typical problems are solved to demonstrate its efficiency over the conventional refinement strategy in the relative improvement of global asymptotic convergence.     

The problem of selecting the shape functions for a p-type finite element

The paper addresses the question of the optimal selection of the shape functions for p-type finite elements and discusses the effectivity of the conjugate gradient and multilevel iteration method for solving the corresponding linear system.     

Transformation of dependent variables and the finite element solution of nonlinear evolution equations

Transformation of dependent variables as, for example, the Kirchhoff transformation, is a classical tool for solving nonlinear partial differential equations. This approach is used here in connection with the finite element method and explained first in case of nonlinear heat conduction problems without phase change. The main applications of the method given in the paper concern a nonlinear degenerate parabolic equation for fluid flow through a porous medium and Stefan (moving boundary) problems.     

Adaptive finite element remeshing in a large deformation analysis of metal powder forming

In this paper, a general framework for the finite element simulation of powder forming processes is presented. A large displacement formulation, based on a total and updated Lagrangian formulation and an adaptive finite element strategy based on error estimates and automatic remeshing techniques are utilized. To describe the constitutive model of the highly non‐linear behaviour of powder materials, an elliptical cap model based on a hardening rule to define the dependence of the yield surface on the degree of plastic straining is applied. The interfacial behaviour between the die and powder is modelled by using a plasticity theory of friction in the context of an interface element formulation. Finally, the powder behaviour during the compaction of a set of complex shapes are analysed numerically. The simulation of the deformation is shown as well as the distribution of relative density contours at different time stages. The results clearly indicate that the algorithm makes it possible to simulate the powder forming problems efficiently and automatically. Copyright © 1999 John Wiley & Sons, Ltd.     

Adaptive finite element methodology for tumour angiogenesis modelling

In this paper, we consider an adaptive finite element approach for reliable, efficient solution of a class of continuum models for tumour‐induced angiogenesis. The ideas are demonstrated using an established three equation reaction/transport model that simulates aspects of tumour‐induced angiogenesis in a deterministic manner. The weak variational formulation and finite element approximation scheme for the model are developed, and a statistical approach for concurrent adaptive mesh refinement and coarsening is described. The appropriate form of the model and solution dependence on choice of parameters are explored. Computational results are presented for 1D, 2D and 3D geometry models. The effectiveness of the open‐source, parallel adaptive software library (LibMesh) that is being developed in the CFDLab at the University of Texas is also demonstrated. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive mesh refinement processes for finite element solutions

The nature of adaptive processes is reviewed using as an example a specific finite element problem. Several possible formulations for the objective of mesh refinement processes are given. For the most natural of these objectives the A*-algorithms of artificial intelligence turn out to provide a solution process which is known to be optimal in a certain sense. But practically, there is insufficient information and these processes tend to be inefficient. On the other hand, if the objective is replaced by a strictly local objective based on Pareto-optimality, the error estimates of the resulting mesh sequence are shown to exhibit the order of convergence which is known to be best possible for the type of elements used.     

Computer program for solution of large,sparse, unsymmetric systems of linear equations

A computer program is presented for in-core solution of a large, sparse, unsymmetric, unbanded system of linear equations. The program employs two partially packed arrays (one for storing non-zero elements, and one for column identifications). Used as the pivotal row is the row with the minimum number of non-zero elements. To avoid instability, the pivot is the largest absolute element in the pivotal row. The method was tested on a system of equations encountered in field application of the three-dimensional Galerkin finite element solution of flow and mass transport through porous media. Performance is compared with that of available alternatives.     

The finite element solution of elliptical systems on a data parallel computer

A study is conducted of the finite element solution of elliptic partial differential equations on a data parallel computer. A nodal assembly technique is introduced which maps a single node to a single processor. The system of equations is first assembled and then solved in parallel using a conjugate gradient algorithm for unsymmetric, non-positive definite systems. Using this technique and a massively parallel machine, problems in excess of 100k nodes are solved. Results of electromagnetic scattering, governed by the 2-d scalar Helmholtz equation, are presented for both an infinite cylinder and an airfoil cross-section. Solutions are demonstrated for a wide range of object sizes. A summary of performance data is given for a set of test problems.