Parallel implementation of finite-element/newton method for solution of steady-state and transient nonlinear partial differential equations

Domain decomposition by nested dissection for concurrent factorization and storage (CFS) of asymmetric matrices is coupled with finite element and spectral element discretizations and with Newton's method to yield an algorithm for parallel solution of nonlinear initial-and boundary-value problem. The efficiency of the CFS algorithm implemented on a MIMD computer is demonstrated by analysis of the solution of the two-dimensional, Poisson equation discretized using both finite and spectral elements. Computation rates and speedups for the LU-decomposition algorithm, which is the most time consuming portion of the solution algorithm, scale with the number of processors. The spectral element discretization with high-order interpolating polynomials yields especially high speedups because the ratio of communication to computation is lower than for low-order finite element discretizations. The robustness of the parallel implementation of the finite-element/Newton algorithm is demonstrated by solution of steady and transient natural convection in a two-dimensional cavity, a standard test problem for low Prandtl number convection. Time integration is performed using a fully implicit algorithm with a modified Newton's method for solution of nonlinear equations at each time step. The efficiency of the CFS version of the finite-element/Newton algorithm compares well with a spectral element algorithm implemented on a MIMD computer using iterative matrix methods.Submitted toJ. Scientific Computing, August 25, 1994.     

基于有限元方法的极小曲面造型

讨论极小曲面方程的求解。极小曲面方程是一个高度非线性的二阶椭圆偏微分方程，求解十分困难。该文基于有限元方法，使用一个简单而有效的线性化策略，将问题转化为一系列线性问题，从而大大简化了求解过程。数值结果表明该方法简单有效，能产生合理的结果。     

A sequential coupling of optimal topology and multilevel shape design applied to two-dimensional nonlinear magnetostatics

In this paper, a sequential coupling of two-dimensional (2D) optimal topology and shape design is proposed so that a coarsely discretized and optimized topology is the initial guess for the following shape optimization. In between, we approximate the optimized topology by piecewise Bézier shapes via least square fitting. For the topology optimization, we use the steepest descent method. The state problem is a nonlinear Poisson equation discretized by the finite element method and eliminated within Newton iterations, while the particular linear systems are solved using a multigrid preconditioned conjugate gradients method. The shape optimization is also solved in a multilevel fashion, where at each level the sequential quadratic programming is employed. We further propose an adjoint sensitivity analysis method for the nested nonlinear state system. At the end, the machinery is applied to optimal design of a direct electric current electromagnet. The results correspond to physical experiments. This research has been supported by the Austrian Science Fund FWF within the SFB “Numerical and Symbolic Scientific Computing” under the grant SFB F013, subprojects F1309 and F1315, by the Czech Ministry of Education under the grant AVČR 1ET400300415, by the Czech Grant Agency under the grant GAČR 201/05/P008 and by the Slovak Grant Agency under the project VEGA 1/0262/03.     

Application of a modified semismooth Newton method to some elasto-plastic problems

Some elasto-plasticity models with hardening are discussed and some incremental finite element methods with different time discretisation schemes are considered. The corresponding one-time-step problems lead to variational equations with various non-linear operators. Common properties of the non-linear operators are derived and consequently a general problem is formulated. The problem can be solved by Newton-like methods. First, the semismooth Newton method is analysed. The local superlinear convergence is proved in dependence on the finite element discretisation parameter. Then it is introduced a modified semismooth Newton method which contain suitable “damping” in each Newton iteration in addition. The determination of the damping coefficients uses the fact that the investigated problem can be formulated as a minimisation one. The method is globally convergent, independently on the discretisation parameter. Moreover the local superlinear convergence also holds. The influence of inexact inner solvers is also discussed. The method is illustrated on a numerical example.     

Parallel-vector computations for geometrically nonlinear finite element analysis

Existing procedures for nonlinear finite element analysis are reviewed. Common computational steps among existing methods are identified. Parallel-vector solution strategies for the generation and assembly of element matrices, solution of the resulting system of linear equations, calculations of the unbalanced loads, displacements and stresses are all incorporated into the Newton-Raphson (NR), modified Newton-Raphson (mNR), and BFGS methods. Furthermore, a mixed parallel-vector Choleski-Preconditioned Conjugate Gradient (C-PCG) equation solver is also developed and incorporated into the piecewise linear procedure for nonlinear finite element analysis. Numerical results have indicated that the Newton-Raphson method is the most effective nonlinear procedure and the mixed C-PCG equation solver offers substantial computational advantages in a parallel-vector computer environment.     

A two-stage algorithm integrating genetic algorithm and modified Newton method for neural network training in engineering systems

A two-stage algorithm combining the advantages of adaptive genetic algorithm and modified Newton method is developed for effective training in feedforward neural networks. The genetic algorithm with adaptive reproduction, crossover, and mutation operators is to search for initial weight and bias of the neural network, while the modified Newton method, similar to BFGS algorithm, is to increase network training performance. The benchmark tests show that the two-stage algorithm is superior to many conventional ones: steepest descent, steepest descent with adaptive learning rate, conjugate gradient, and Newton-based methods and is suitable to small network in engineering applications. In addition to numerical simulation, the effectiveness of the two-stage algorithm is validated by experiments of system identification and vibration suppression.     

Geometrical nonlinearities on the static analysis of highly flexible steel cable-stayed bridges

An evaluation of the importance of geometrically nonlinear effects on the structural static analysis of steel cable-stayed bridges is presented. A finite element model is analyzed using linear, pseudo-linear and nonlinear methods. The pseudo-linear approach is based on the modified elastic modulus. The nonlinear analysis involves cable sag, large displacement and beam-column effects. The results confirm that both cable sag and large displacement originate the most important nonlinear effects in those structures. Beam-column effects are irrelevant for service loads. Both the pseudo-linear approach and the modified modulus element prove to be very limited or even inappropriate.     

Two-Level Newton’s Method for Nonlinear Elliptic PDEs

A combination method of Newton’s method and two-level piecewise linear finite element algorithm is applied for solving second-order nonlinear elliptic partial differential equations numerically. Newton’s method is to find a finite element solution by solving $m$ Newton equations on a fine mesh. The two-level Newton’s method solves $m-1$ Newton equations on a coarse mesh and processes one Newton iteration on a fine mesh. Moreover, the optimal error estimates of Newton’s method and the two-level Newton’s method are provided to justify the efficiency of the two-level Newton’s method. If we choose $H$ such that $h=O(|\log h|^{1-2/{p}}H^2)$ for the $W^{1,p}(\Omega )$ -error estimates, the two-level Newton’s method is asymptotically as accurate as Newton’s method on the fine mesh. Meanwhile, the numerical investigations provided a sufficient support for the theoretical analysis. Finally, these investigations also proved that the proposed method is efficient for solving the nonlinear elliptic problems.     

A Solution Method for a Certain Nonlinear Interface Problem in Unbounded Domains

In this paper, we consider a kind of nonlinear interface problem in unbounded domains. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive the optimal error estimate of finite element approximation to the coupled FEM-BEM problem. Then we introduce a preconditioning steepest descent method for solving the discrete system by constructing a cheap domain decomposition preconditioner. Moreover, we give a complete analysis to the convergence speed of this iterative method. Received March 30, 2000; revised November 29, 2000     

Interactive-adaptive geometrically nonlinear analysis of shell structures

This paper presents an investigation of interactive-adaptive techniques for nonlinear finite element structural analysis. In particular, effective methods leading to reliable automated, finite element solutions of nonlinear shell problems are of primary interest here. This includes automated adaptive nonlinear solution procedures based on error estimation and adaptive step length control, reliable finite elements that account for finite deformations and finite rotations, three-dimensional finite element modeling, and an easy-to-use, easy-to-learn graphical user interface with three-dimensional graphics. A computational environment, which interactively couples a comprehensive geometric modeler, an automatic three-dimensional mesh generator and an advanced nonlinear finite element analysis program with real-time computer graphics and animation tools, is presented. Three examples illustrate the merit and potential of the approaches adopted here and confirm the feasibility of developing fully automated computer aided engineering environments.     

Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

Crippling analysis of composite stringers based on complete unloading method

This paper addresses a nonlinear finite element method for the crippling analysis of composite laminated stringers. For the finite element modeling, a nine-node laminated shell element based on the first order shear deformation theory is used. Failure-induced stiffness degradation is simulated by the complete unloading method. A modified arc-length algorithm is incorporated in the nonlinear finite element method to trace the post-failure equilibrium path after a local buckling. Finite element results show excellent agreement with those of previous experiment. A parametric study is performed to assess the effect of the flange-width, web-height, and stacking sequence on the buckling, local buckling, and crippling stresses of stringers.     

Numerical nonlinear observers using pseudo‐Newton‐type solvers

In constructing a globally convergent numerical nonlinear observer of Newton‐type for a continuous‐time nonlinear system, a globally convergent nonlinear equation solver with a guaranteed rate of convergence is necessary. In particular, the solver should be Jacobian free, because an analytic form of the state transition map of the nonlinear system is generally unavailable. In this paper, two Jacobian‐free nonlinear equation solvers of pseudo‐Newton type that fulfill these requirements are proposed. One of them is based on the finite difference approximation of the Jacobian with variable step size together with the line search. The other uses a similar idea, but the estimate of the Jacobian is mostly updated through a BFGS‐type law. Then, by using these solvers, globally stable numerical nonlinear observers are constructed. Numerical results are included to illustrate the effectiveness of the proposed methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Multipoint flux mixed finite element methods for slightly compressible flow in porous media

In this paper, we consider multipoint flux mixed finite element discretizations for slightly compressible Darcy flow in porous media. The methods are formulated on general meshes composed of triangles, quadrilaterals, tetrahedra or hexahedra. An inexact Newton method that allows for local velocity elimination is proposed for the solution of the nonlinear fully discrete scheme. We derive optimal error estimates for both the scalar and vector unknowns in the semidiscrete formulation. Numerical examples illustrate the convergence behavior of the methods, and their performance on test problems including permeability coefficients with increasing heterogeneity.     

Least square finite element solution of stagnation point flow

A detailed analysis of the least square finite element solution of nonlinear boundary value problems is presented with reference to a particular example of nonlinear coupled differential equations governing the flow of an incompressible viscous fluid in the vicinity of a forward stagnation point at a blunt body. The numerical solutions are presented for different cases. The results obtained by the least square finite element method are in very good agreement with the results available in the literature confirming the versatility and usefulness of the application of the method to nonlinear boundary value problems governing the fluid flow problems.     

On the use of parallel processors for implicit Runge-Kutta methods

An iteration scheme, for solving the non-linear equations arising in the implementation of implicit Runge-Kutta methods, is proposed. This scheme is particularly suitable for parallel computation and can be applied to any method which has a coefficient matrixA with all eigenvalues real (and positive). For such methods, the efficiency of a modified Newton scheme may often be improved by the use of a similarity transformation ofA but, even when this is the case, the proposed scheme can have advantages for parallel computation. Numerical results illustrate this. The new scheme converges in a finite number of iterations when applied to linear systems of differential equations, achieving this by using the nilpotency of a strictly lower triangular matrixS ^?1 AS — Λ, with Λ a diagonal matrix. The scheme reduces to the modified Newton scheme whenS ^?1 AS is diagonal.A convergence result is obtained which is applicable to nonlinear stiff systems.     

折半搜索法预测二元混合物的共沸点

在化工分离过程设计中,共沸点预测的作用十分重要,目前常用的方法有牛顿迭代和牛顿同伦等,都需要求解大型非线性方程组,且牛顿迭代法易发散,本文提出修正UNIFAC模型的逐次代入法同时与折半搜索联合的算法,既克服牛顿迭代计算时因取初值不合适时而容易发散的缺点,又不需要求解大型非线性方程组,且计算速度快,计算中对逐次代入法进行改进,使温度初值的取法更简捷,且无发散现象,通过验证乙醇-苯等10多种二元混合物,计算过程均可在1 MS以内完成,计算所得共沸点与文献所载实验值比较,平均误差＜1%,共沸点组成与文献所载实验值比较,平均误差＜2%,证明该法不但可用于二元混合物共沸点预测,又可在相应大型数据库中查找可产生共沸效果的混合物.     

改进的牛顿预测–校正格式

在数值分析领域中,牛顿算法由于其形式的简单性及快速的收敛性而被广泛地应用于求解非线性方程问题.受一类求解方程的预测–校正技术的启示,本文针对求解非线性方程单根的问题提出了一种牛顿预测–校正格式,并将其推广到多维向量值函数情况.为此,首先用图描述了这种新的预测–校正格式并导出了其收敛阶.这种新格式每步迭代仅需计算一次函数值和一次导函数值.然后,经过测试函数的检验,并与牛顿算法及其他高阶算法(1+√2阶、3阶、4阶、5阶、6阶)比较,表明新算法具有较快的收敛性.最后,将这种新格式推广到多维向量值函数,采用泰勒公式证明了其收敛性,并给出了一个二维算例来验证其收敛的有效性.     

基于空间组合式单元模型的表面求交算法

在空间埋置组合式单元模型中，钢筋单元可埋置于混凝土单元任何位置，混凝土单 元网格剖分不受钢筋位置的限制，方便实用，但需确定钢筋单元两端在混凝土单元表面的位置坐 标。因此，求解钢筋线与混凝土单元表面的交点坐标是应用该单元模型的前提，现有的求解方法 只适用于混凝土单元表面是平面的情况。为此，提出了牛顿迭代法和分块解析法两种处理方法， 能求解钢筋线与混凝土单元表面为任何形状时的交点坐标，增强了该模型的适用性。通过算例验 证了这两种方法的正确性。从适用性而言，分块解析法要优于牛顿迭代法。