MULTIVARIATE HERMITE APPROXIMATION FOR DESIGN OPTIMIZATION

An approximation based on multiple function and gradient information is developed using Hermite interpolation concepts. The goal is to build a high-quality approximation for complex and multidisciplinary design optimization problems employing analysis such as aeroservoelasticity, structural control, probability, etc. The proposed multidimensional approximation utilizes exact analyses data generated during the course of iterative optimization. The approximation possesses the property of reproducing the function and gradient information of known data points. The accuracy of the new approach is compared with linear, reciprocal and other standard approximations. Because the proposed algorithm uses more data points, its efficiency has to be compared in the context of iterative optimization.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

Generic domain decomposition and iterative solvers for 3D BEM problems

In the past two decades, considerable improvements concerning integration algorithms and solvers involved in boundary‐element formulations have been obtained. First, a great deal of efficient techniques for evaluating singular and quasi‐singular boundary‐element integrals have been, definitely, established, and second, iterative Krylov solvers have proven to be advantageous when compared to direct ones also including non‐Hermitian matrices. The former fact has implied in CPU‐time reduction during the assembling of the system of equations and the latter fact in its faster solution. In this paper, a triangle‐polar‐co‐ordinate transformation and the Telles co‐ordinate transformation, applied in previous works independently for evaluating singular and quasi‐singular integrals, are combined to increase the efficiency of the integration algorithms, and so, to improve the performance of the matrix‐assembly routines. In addition, the Jacobi‐preconditioned biconjugate gradient (J‐BiCG) solver is used to develop a generic substructuring boundary‐element algorithm. In this way, it is not only the system solution accelerated but also the computer memory optimized. Discontinuous boundary elements are implemented to simplify the coupling algorithm for a generic number of subregions. Several numerical experiments are carried out to show the performance of the computer code with regard to matrix assembly and the system solving. In the discussion of results, expressed in terms of accuracy and CPU time, advantages and potential applications of the BE code developed are highlighted. Copyright © 2006 John Wiley & Sons, Ltd.     

多变量矩阵方程异类约束解的修正共轭梯度法

基于求解线性代数方程组的共轭梯度法,通过对相关矩阵和系数的修改,建立了一种求多矩阵变量矩阵方程异类约束解的修正共轭梯度法.该算法不要求等价线性代数方程组的系数矩阵具备正定性、可逆性或者列满秩性,因此算法总是可行的.利用该算法不仅可以判断矩阵方程的异类约束解是否存在,而且在有异类约束解,不考虑舍入误差时,可在有限步计算后求得矩阵方程的一组异类约束解;选取特殊初始矩阵时,可求得矩阵方程的极小范数异类约束解.另外,还可求得指定矩阵在异类约束解集合中的最佳逼近.算例验证了该算法的有效性.     

Coarse division transform based preconditioner for boundary element problems

This paper is devoted to the development of efficient preconditioners for an iterative solution of equation sets arising in the Boundary Element Method (BEM). A standard collocation system of equations is transformed to a new basis associated with an auxiliary coarse division model for boundary unknowns and solved in this basis. New systems have rapidly decreasing coefficients and by neglecting a large number of them it is possible to construct readily invertible, sparse preconditioners for iterative procedures. The specific features of the transformed matrix can be attributed to the analytical properties of integral equations. Although the transformation is based on an auxiliary coarse division model, it does not require any additional operations with boundary elements. All manipulations necessary to construct the mapping are performed on the level of algebraic equations. Numerical experiments included in the paper confirm a high rate of convergence of the developed iterative scheme.     

An algorithm for the matrix‐free solution of quasistatic frictional contact problems

A contact enforcement algorithm has been developed for matrix‐free quasistatic finite element techniques. Matrix‐free (iterative) solution algorithms such as non‐linear conjugate gradients (CG) and dynamic relaxation (DR) are desirable for large solid mechanics applications where direct linear equation solving is prohibitively expensive, but in contrast to more traditional Newton–Raphson and quasi‐Newton iteration strategies, the number of iterations required for convergence is typically of the same order as the number of degrees of freedom of the model. It is therefore crucial that each of these iterations be inexpensive to per‐form, which is of course the essence of a matrix free method. In applying such methods to contact problems we emphasize here two requirements: first, that the treatment of the contact should not make an average equilibrium iteration considerably more expensive; and second, that the contact constraints should be imposed in such a way that they do not introduce spurious energy that acts against the iterative solver. These practical concerns are utilized to develop an iterative technique for accurate constraint enforcement that is suitable for non‐linear conjugate gradient and dynamic relaxation iterative schemes. Copyright © 1999 John Wiley & Sons, Ltd.     

多边形比例边界有限元非线性高效分析方法

比例边界有限元法作为一种高精度的半解析数值求解方法,特别适合于求解无限域与应力奇异性等问题,多边形比例边界单元在模拟裂纹扩展过程、处理局部网格重剖分等方面相较于有限单元法具有明显优势。目前,比例边界有限元法更多关注的是线弹性问题的求解,而非线性比例边界单元的研究则处于起步阶段。该文将高效的隔离非线性有限元法用于比例边界单元的非线性分析,提出了一种高效的隔离非线性比例边界有限元法。该方法认为每个边界线单元覆盖的区域为相互独立的扇形子单元,其形函数以及应变-位移矩阵可通过半解析的弹性解获得;每个扇形区的非线性应变场通过设置非线性应变插值点来表达,引入非线性本构关系即可实现多边形比例边界单元高效非线性分析。多边形比例边界单元的刚度通过集成每个扇形子单元的刚度获取,扇形子单元的刚度可采用高斯积分方案进行求解,其精度保持不变。由于引入了较多的非线性应变插值点,舒尔补矩阵维数较大,该文采用Woodbury近似法对隔离非线性比例边界单元的控制方程进行求解。该方法对大规模非线性问题的计算具有较高的计算效率,数值算例验证了算法的正确性以及高效性,将该方法进行推广,对实际工程分析具有重要意义。     

Algebraic multilevel iteration method for lowest order Raviart–Thomas space and applications

An optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems, that have their weak formulation in the space H(div), is presented. The algorithm is developed for the discrete problem obtained by using the lowest‐order Raviart–Thomas space. The method is theoretically analyzed and supporting numerical examples are presented. Furthermore, as a particular application, the algorithm is used for the solution of the discrete minimization problem which arises in the functional‐type a posteriori error estimates for the discontinuous Galerkin approximation of elliptic problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

This paper is concerned with energy–momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy–momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three‐field formulation, in which the deformation field, the velocity field and a strain‐like viscous internal variable field are treated as independent quantities. The new non‐linear viscous evolution equation satisfies a non‐negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy–momentum consistency is based on the collocation property and an enhanced second Piola–Kirchhoff stress tensor. The obtained coupled non‐linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time‐integration algorithm in comparison with the ordinary midpoint rule. Both the quasi‐rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Non-linear vibration of frames by the incremental dynamic stiffness method

The dynamic stiffness method is extended to large amplitude free and forced vibrations of frames. When the steady state vibration is concerned, the time variable is replaced by the frequency parameter in the Fourier series sense and the governing partial differential equations are replaced by a set of ordinary differential equations in the spatial variables alone. The frequency-dependent shape functons are generated approximately for the spatial discretization. These shape functions are the exact solutions of a beam element subjected to mono-frequency excitation and constant axial force to minimize the spatial discretization errors. The system of ordinary differential equations is replaced by a system of non-linear algebraic equations with the Fourier coefficients of the nodal displacements as unknowns. The Fourier nodal coefficients are solved by the Newtonian algorithm in an incremental manner. When an approximate solution is available, an improved solution is obtained by solving a system of linear equations with the Fourier nodal increments as unknowns. The method is very suitable for parametric studies. When the excitation frequency is taken as a parameter, the free vibration response of various resonances can be obtained without actually computing the linear natural modes. For regular points along the response curves, the accuracy of the gradient matrix (Jacobian or tangential stiffness matrix) is secondary (cf. the modified Newtonian method). However, at the critical positions such as the turning points at resonances and the branching points at bifurcations, the gradient matrix becomes important. The minimum number of harmonic terms required is governed by the conditions of completeness and balanceability for predicting physically realistic response curves. The evaluations of the newly introduced mixed geometric matrices and their derivatives are given explicitly for the computation of the gradient matrix.     

Adaptive cross approximation of tensors arising in the discretization of boundary integral operator shape derivatives

The shape derivative of a dense N×N BEM matrix is a sparse three-way tensor with O(N²) non-zero entries, to which standard BEM acceleration techniques such as the adaptive cross approximation (ACA) and FMM cannot be directly applied. The tensor can be used to compute shape sensitivities, or via adjoint equations, the gradient of an objective function. Although for many PDEs, calculation of the tensor can be avoided by expressing the shape derivative of the solution as the solution of a related PDE, this approach is not always easily amenable to BEM. Therefore, the computation of shape derivatives via the sparse three-way tensor is a valuable alternative, provided that efficient acceleration techniques exist. We propose a new algorithm for the approximation of BEM shape derivative tensors based on ACA that achieves the same complexity and error bounds as ACA for the BEM matrix itself. Numerical examples show that despite the much larger amount of data involved, the tensor approximation is only moderately slower than the matrix approximation. We also demonstrate the method on a shape optimization problem from the literature.     

On Using Collocation in Three Dimensions and Solving a Model Semiconductor Problem

A research code has been written to solve an elliptic system of coupled nonlinear partial differential equations of conservation form on a rectangularly shaped three-dimensional domain. The code uses the method of collocation of Gauss points with tricubic Hermite piecewise continuous polynomial basis functions. The system of equations is solved by iteration. The system of nonlinear equations is linearized, and the system of linear equations is solved by iterative methods. When the matrix of the collocation equations is duly modified by using a scaled block-limited partial pivoting procedure of Gauss elimination, it is found that the rate of convergence of the iterative method is significantly improved and that a solution becomes possible. The code is used to solve Poisson’s equation for a model semiconductor problem. The electric potential distribution is calculated in a metal-oxide-semiconductor structure that is important to the fabrication of electron devices.     

APPLICATION OF THE COLLOCATION METHOD IN THREE DIMENSIONS TO A MODEL SEMICONDUCTOR PROBLEM

A research code has been written to solve an elliptic system of coupled non-linear partial differential equations of conservation form on a rectangularly shaped three-dimensional domain. The code uses the method of collocation of Gauss points with tricubic Hermite piecewise continuous polynomial basis functions. The system of equations is solved by iteration. The system of non-linear equations is linearized, and the system of linear equations is solved by iterative methods. When the matrix of the collocation equations is duly modified by using a scaled block-limited partial pivoting procedure of Gauss elimination, it is found that the rate of convergence of the iterative method is significantly improved and that a solution becomes possible. The code is used to solve Poisson's equation for a model semiconductor problem. The electric potential distribution is calculated in a metal-oxide-semiconductor structure that is important to the fabrication of electron devices.     

Second-order iterative approach to inverse scattering: numerical results

We introduce an iterative algorithm for the reconstruction of dielectric profile functions from scattered field data, in which each step corresponds to the solution of a quadratic inversion problem. This means that, at each iteration, we perform a second-order approximation of the scattering operator connecting the unknown profile to the data about a reference profile function. This procedure is then compared with a linear iterative inversion algorithm, and it is pointed out that, within a prescribed class of profile functions, the linear iterative inversion does not converge to the actual solution, whereas the proposed approach does. This feature can be explained by reference not only to the improved approximation provided by the addition of a further term for profile functions of a larger norm but also to the different classes of functions that can be reconstructed by either the linear or the quadratic model. Numerical examples of profile reconstruction in the scalar two-dimensional geometry, with far-zone scattered field data at a fixed frequency, confirm the theoretical analysis.     

Coupled finite element – hierarchical boundary element methods for dynamic soil–structure interaction in the frequency domain

This paper discusses the coupling of finite element and fast boundary element methods for the solution of dynamic soil–structure interaction problems in the frequency domain. The application of hierarchical matrices in the boundary element formulation allows considering much larger problems compared to classical methods. Three coupling methodologies are presented and their computational performance is assessed through numerical examples. It is demonstrated that the use of hierarchical matrices renders a direct coupling approach the least efficient, as it requires the assembly of a dynamic soil stiffness matrix. Iterative solution procedures are presented as well, and it is shown that the application of such schemes to dynamic soil–structure interaction problems in the frequency domain is not trivial, as convergence can hardly be achieved if no relaxation procedure is incorporated. Aitken's Δ²‐method is therefore employed in sequential iterative schemes for the calculation of an optimized interface relaxation parameter, while a novel relaxation technique is proposed for parallel iterative algorithms. It is demonstrated that the efficiency of these algorithms strongly depends on the boundary conditions applied to each subdomain; the fastest convergence is observed if Neumann boundary conditions are imposed on the stiffest subdomain. The use of a dedicated solver for each subdomain hence results in a reduced computational effort. A monolithic coupling strategy, often used for the solution of fluid–structure interaction problems, is also introduced. The governing equations are simultaneously solved in this approach, while the assembly of a dynamic soil stiffness matrix is avoided. Copyright © 2013 John Wiley & Sons, Ltd.     

Fast direct solution of 3-D scattering problems via nonuniform grid-based matrix compression

A fast non-iterative algorithm for the solution of large 3-D acoustic scattering problems is presented. The proposed approach can be used in conjunction with the conventional boundary element discretization of the integral equations of acoustic scattering. The algorithm involves domain decomposition and uses the nonuniform grid (NG) approach for the initial compression of the interactions between each subdomain and the rest of the scatterer. These interactions, represented by the off-diagonal blocks of the boundary element method matrix, are then further compressed while constructing sets of interacting and local basis and testing functions. The compressed matrix is obtained by eliminating the local degrees of freedom through the Schur's complement-based technique procedure applied to the diagonal blocks. In the solution process, the interacting unknowns are first determined by solving the compressed system equations. Subsequently, the local degrees of freedom are determined for each subdomain. The proposed technique effectively reduces the oversampling typically needed when using low-order discretization techniques and provides significant computational savings.     

A meshless method for Kirchhoff plate bending problems

In this work a meshless method for the analysis of bending of thin homogeneous plates is presented. This meshless method is based on the use of radial basis functions to build an approximation of the general solution of the partial differential equations governing the Kirchhoff plate bending problem. In order to obtain a symmetric and non‐singular linear equation system the Hermite collocation method is used. To assess the formulation a series of plates with different boundary conditions are analysed. Comparisons are made with other results available in the literature. Copyright © 2001 John Wiley & Sons, Ltd.     

Efficient preconditioners for boundary element matrices based on grey‐box algebraic multigrid methods

This paper is concerned with the iterative solution of the boundary element equations arising from standard Galerkin boundary element discretizations of first‐kind boundary integral operators of positive and negative order. We construct efficient preconditioners on the basis of so‐called grey‐box algebraic multigrid methods that are well adapted to the treatment of boundary element matrices. In particular, the coarsening is based on an auxiliary matrix that represents the underlying topology in a certain sense. This auxiliary matrix is additionally used for the construction of the smoothers and the transfer operators. Finally, we present the results of some numerical studies that show the efficiency of the proposed algebraic multigrid preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

3D multidomain BEM for solving the Laplace equation

An efficient 3D multidomain boundary element method (BEM) for solving problems governed by the Laplace equation is presented. Integral boundary equations are discretized using mixed boundary elements. The field function is interpolated using a continuous linear function while its derivative in a normal direction is interpolated using a discontinuous constant function over surface boundary elements. Using a multidomain approach, also known as the subdomain technique, sparse system matrices similar to the finite element method (FEM) are obtained. Interface boundary conditions between subdomains leads to an over-determined system matrix, which is solved using a fast iterative linear least square solver. The accuracy and robustness of the developed numerical algorithm is presented on a scalar diffusion problem using simple cube geometry and various types of meshes. Efficiency is demonstrated with potential flow around the complex geometry of a fighter airplane using tetrahedral mesh with over 100,000 subdomains on a personal computer.