A comparative study of the behaviour of a direct solver and a preconditioned iterative solver for the equations arising from the discretization by Chebyshev collocation of a second-order partial differential equation on a square

The behaviour is compared of two solvers for the discrete equations arising from the discretization using Chebyshev collocation of a second-order linear partial differential equation on a square. The alternative solvers considered are a direct solver and an iterative solver based on preconditioning with the matrix arising from finite-difference discretization of the governing equation. The total error of the collocation derivatives and the separate contributions from round-off and discretization error are examined. The efficiency of the two solvers is compared. The iterative solver is more efficient than the direct solver on fine grids for equations similar to the Poisson equation, provided that there are Dirichlet boundary conditions on at least three of the sides of the square.     

A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid

This work investigates matrix-free algorithms for problems in quasi-static finite-strain hyperelasticity. Iterative solvers with matrix-free operator evaluation have emerged as an attractive alternative to sparse matrices in the fluid dynamics and wave propagation communities because they significantly reduce the memory traffic, the limiting factor in classical finite element solvers. Specifically, we study different matrix-free realizations of the finite element tangent operator and determine whether generalized methods of incorporating complex constitutive behavior might be feasible. In order to improve the convergence behavior of iterative solvers, we also propose a method by which to construct level tangent operators and employ them to define a geometric multigrid preconditioner. The performance of the matrix-free operator and the geometric multigrid preconditioner is compared to the matrix-based implementation with an algebraic multigrid (AMG) preconditioner on a single node for a representative numerical example of a heterogeneous hyperelastic material in two and three dimensions. We find that matrix-free methods for finite-strain solid mechanics are very promising, outperforming linear matrix-based schemes by two to five times, and that it is possible to develop numerically efficient implementations that are independent of the hyperelastic constitutive law.     

A fast and robust iterative solver for nonlinear contact problems using a primal‐dual active set strategy and algebraic multigrid

For extending the usability of implicit FE codes for large‐scale forming simulations, the computation time has to be decreased dramatically. In principle this can be achieved by using iterative solvers. In order to facilitate the use of this kind of solvers, one needs a contact algorithm which does not deteriorate the condition number of the system matrix and therefore does not slow down the convergence of iterative solvers like penalty formulations do. Additionally, an algorithm is desirable which does not blow up the size of the system matrix like methods using standard Lagrange multipliers. The work detailed in this paper shows that a contact algorithm based on a primal‐dual active set strategy provides these advantages and therefore is highly efficient with respect to computation time in combination with fast iterative solvers, especially algebraic multigrid methods. Copyright © 2006 John Wiley & Sons, Ltd.     

Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximations of linear elasticity with and without discontinuous coefficients

Adaptive finite element methods (FEM) generate linear equation systems that require dynamic and irregular patterns of storage, access, and computation, making their parallelization difficult. Additional difficulties are generated for problems in which the coefficients of the governing partial differential equations have large discontinuities. We describe in this paper the development of a set of iterative substructuring based solvers and domain decomposition preconditioners with an algebraic coarse‐grid component that address these difficulties for adaptive hp approximations of linear elasticity with both homogeneous and inhomogeneous material properties. Our solvers are robust and efficient and place no restrictions on the mesh or partitioning. Copyright © 2003 John Wiley & Sons, Ltd.     

Solution of Helmholtz equation by Trefftz method

The application of the Trefftz method for calculating wave forces on offshore structures is presented. Indirect and direct formulations using complete and non-singular systems of Trefftz functions for the Helmholtz equation are posed in this paper. An effective technique using different interpolation functions for the velocity potential and wave force are suggested to improve the computational accuracy of the wave force. The numerical examples show that the present method is highly efficient and accurate.     

Hybrid finite element/volume method for shallow water equations

A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.     

Efficient implicit finite element analysis of sheet forming processes

The computation time for implicit finite element analyses tends to increase disproportionally with increasing problem size. This is due to the repeated solution of linear sets of equations, if direct solvers are used. By using iterative linear equation solvers the total analysis time can be reduced for large systems. For plate or shell element models, however, the condition of the matrix is so ill that iterative solvers do not reach the huge time‐savings that are realized with solid elements. By introducing inertial effects into the implicit finite element code the condition number can be improved and iterative solvers perform much better. An additional advantage is that the inertial effects stabilize the Newton–Raphson iterations. This also applies to quasi‐static processes, for which the inertial effects finally do not affect the results. The presented method can readily be implemented in existing implicit finite element codes. Industrial size deep drawing simulations are executed to investigate the performance of the recommended strategy. It is concluded that the computation time is decreased by a factor of 5 to 10. Copyright © 2003 John Wiley & Sons, Ltd.     

Wide-angle, finite-difference beam propagation in oblique coordinate system

The one-way wave equation in the oblique coordinate system in terms of the square root operator is derived. This equation forms the basis for the development of efficient algorithms using the beam propagation method for the design and optimization of integrated optical devices. In an illustrative example, using the derived one-way wave equation, Anada's very-wide-angle algorithm is generalized to the oblique coordinate system. Since in the oblique coordinate system the direction of propagation can be selected freely to follow the path of the optical beam and to minimize the stair-casing errors, the algorithm is expected to show superior performance, which is confirmed by the results obtained.     

Parallel Solution of Linear Systems

Computational scientists generally seek more accurate results in shorter times, and to achieve this a knowledge of evolving programming paradigms and hardware is important. In particular, optimising solvers for linear systems is a major challenge in scientific computation, and numerical algorithms must be modified or new ones created to fully use the parallel architecture of new computers. Parallel space discretisation solvers for Partial Differential Equations (PDE) such as Domain Decomposition Methods (DDM) are efficient and well documented. At first glance, parallelisation seems to be inconsistent with inherently sequential time evolution, but parallelisation is not limited to space directions. In this article, we present a new and simple method for time parallelisation, based on partial fraction decomposition of the inverse of some special matrices. We discuss its application to the heat equation and some limitations, in associated numerical experiments.     

A block solver for large,unsymmetric, sparse,banded matrices with symmetric profiles

A block equation solver for the solution of large, sparse, banded unsymmetric system of linear equations is presented in this paper. The method employs Crout variation of Gauss elimination technique for the solution. The solver ensures the efficient use of the available memory by doing block factorization and storage. It uses a skyline storage scheme which will avoid unnecessary operations on zero elements above the skyline which has found widespread use in banded symmetric solvers. A FORTRAN code with ample comments is provided. © 1997 John Wiley & Sons, Ltd.     

A stabilization technique for the regularization of nearly singular extended finite elements

In this contribution a simple, robust and efficient stabilization technique for extended finite element (XFEM) simulations is presented. It is useful for arbitrary crack geometries in two or three dimensions that may lead to very bad condition numbers of the global stiffness matrix or even ill-conditioning of the equation system. The method is based on an eigenvalue decomposition of the element stiffness matrix of elements that only possess enriched nodes. Physically meaningful zero eigenmodes as well as enrichment scheme dependent numerically reasonable zero eigenmodes are filtered out. The remaining subspace is stabilized depending on the magnitude of the respective eigenvalues. One of the main advantages is the fact that neither the equation solvers need to be changed nor the solution method is restricted. The efficiency and robustness of the method is demonstrated in numerous examples for 2D and 3D fracture mechanics.     

Solution of the Fourth-moment Equation by an Adaptive Grid Method

Abstract

The parabolic fourth moment equation for a plane wave is integrated numerically using an adaptive grid algorithm. Results are compared with those from an accurate, but far less efficient, operator splitting method, and agree very closely. The adaptive approach has several advantages over current fixed grid methods. The primary one is that significant reductions in computer memory usage can be obtained without a concomitant increase in run time. It is also ideally suited for accurate integration of the fourth-moment equation for an extremely large range of values of both the scattering parameter, Γ, and the wave propagation distance.     

An efficient algorithm for modelling progressive damage accumulation in disordered materials

This paper presents an efficient algorithm for the simulation of progressive fracture in disordered quasi‐brittle materials using discrete lattice networks. The main computational bottleneck involved in modelling the fracture simulations using large discrete lattice networks stems from the fact that a new large set of linear equations needs to be solved every time a lattice bond is broken. Using the present algorithm, the computational complexity of solving the new set of linear equations after breaking a bond reduces to a simple triangular solves (forward elimination and backward substitution) using the already Cholesky factored matrix. This algorithm using the direct sparse solver is faster than the Fourier accelerated iterative solvers such as the preconditioned conjugate gradient (PCG) solvers, and eliminates the critical slowing down associated with the iterative solvers that is especially severe close to the percolation critical points. Numerical results using random resistor networks for modelling the fracture and damage evolution in disordered materials substantiate the efficiency of the present algorithm. In particular, the proposed algorithm is especially advantageous for fracture simulations wherein ensemble averaging of numerical results is necessary to obtain a realistic lattice system response. Copyright © 2005 John Wiley & Sons, Ltd.     

Computationally efficient model for refrigeration compressor gas dynamics

In this paper a computationally efficient steady state model for a typical refrigeration reciprocating compressor is proposed. The plenum cavity is modelled using the acoustic plane wave theory, while the compression process is modelled as a one-dimensional gas dynamics equation. Valve dynamic models, based on a single vibration mode approximation, are coupled with the gas dynamics equation and acoustic plenum models. The steady-state solution of the resultant coupled non-linear equations are posed as a boundary value problem and solved using Warner's algorithm. The Warner's algorithm applied to compressor simulation is shown to be computationally more efficient as compared to conventional techniques such as shooting methods. Comparisons are based on the number of iterations and time taken for convergence. Effect of operating conditions on the overall compressor performance is also investigated.     

A finite element Laplace transform solution technique for the wave equation

A technique is described for the solution of the wave equation with time dependent boundary conditions. The finite element solution accompanied by the numerical Laplace inversion process seems to be an efficient procedure to treat such problems. The programming involved is straightforward in the sense that numerical Laplace inversion routines can be directly used as a time integration procedure after obtaining standard finite element differential equation solutions in the transformed domain. Some results are presented for one- and two- dimensional applications, such as wave propagation in longitudinal bars and wave propagation in harbours.     

几类典型网格下三维弹性问题的代数多层网格法

有限元方法是数值求解三维弹性问题的一类重要的离散化方法.在有限元分析中,网格的几何形状及网格质量会对有限元离散代数系统的求解产生很大影响.该文系统研究了几类典型网格对几种常用AMG法计算效率的影响,并进行了详细的性能测试与比较.利用容易获知的部分几何与分析信息(如方程类型,节点自由度信息),再结合经典AMG法中的网格粗...     

接近爆炸作用下板状介质破坏数值模拟

利用有限元计算软件LS-DYNA建立板状介质破坏的三维模型,并用LS-DYNA970软件的求解器和后处理器对板状介质在爆炸作用下的破坏过程进行了数值模拟。通过数值模拟,再现了炸药爆炸后,冲击波的传播过程以及冲击波作用下板状介质的变形过程;并对爆破动载作用下板状介质破坏情况进行了相应的理论分析。     

球形装药浅层水中水底爆炸冲击波压力测试及分析

介绍了四个球形装药浅层水中水底裸爆时冲击波荷载的测试技术 ,包括实验方案、测试系统、波形分析及数据处理 ,并总结出部分相关结论和经验公式     

Accuracy Investigations of Boundary Element Methods for the Solution of Laplace Equations

Boundary element methods (BEMs) are approved methods for an efficient numerical solution of problems, which are based on a Laplace equation. Here, the solution of electrostatic field problems, steady current flow field problems, and magnetostatic field problems is considered. Focus of this paper is on investigations of accuracy of direct formulations, which are based on Green's theorem. Different types of coupling of computational domains are examined with respect to accuracy and convergence behavior of iterative solvers of the linear system of equations. Furthermore, the influence of singular and nearly singular integrals and the influence of matrix compression techniques to the accuracy of the solution are observed