Eigendecomposition-based convergence analysis of the Neumann series for laminated composites and discretization error estimation

In computational homogenization for periodic composites, the Lippmann-Schwinger integral equation constitutes a convenient formulation to devise numerical methods to compute local fields and their macroscopic responses. Among them, the iterative scheme based on the Neumann series is simple and efficient. For such schemes, a priori global error estimates on local fields and effective property are not available, and this is the concern of this article, which focuses on the simple, but illustrative, conductivity problem in laminated composites. The global error is split into an iteration error, associated with the Neumann series expansion, and a discretization error. The featured nonlocal Green's operator is expressed in terms of the averaging operator, which circumvents the use of the Fourier transform. The Neumann series is formulated in a discrete setting, and the eigendecomposition of the iterated matrix is performed. The ensuing analysis shows that the local fields are computed using a particular subset of eigenvectors, the iteration error being governed by the associated eigenvalues. Quadratic error bounds on the effective property are also discussed. The discretization error is shown to be related to the accuracy of the trapezoidal quadrature scheme. These results are illustrated numerically, and their extension to other configurations is discussed.     

Adaptive multivlevel methods in three space dimensions

We consider the approximate solution of self-adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. As the performance of hierarchical bases preconditioners deteriorates in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy.     

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.     

Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincaré–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.     

On the contact mechanics of a rolling cylinder on a graded coating. Part 1: Analytical formulation

In this paper, the fully coupled rolling contact problem of a graded coating/substrate system under the action of a rigid cylinder is investigated. Using the singular integral equation approach, the governing equations of the rolling contact problem are constructed for all possible stick/slip regimes. Applying the Gauss–Chebyshev numerical integration method, the governing equations are converted to systems of algebraic equations. A new numerical algorithm is proposed to solve these systems of equations. Both the coupled and the uncoupled solutions to the problem are found through an implemented iterative procedure. In Part I of this paper, the analytical formulation of the rolling contact problem and the discretization of the governing equations are introduced for all assumed stick/slip regimes. A detailed discussion of the proposed numerical algorithm, the iteration procedure and the numerical results, obtained using the analytical formulation, are given in Part II.     

A meshless method for conjugate heat transfer problems

Mesh reduction methods such as boundary element methods, method of fundamental solutions, and spectral methods all lead to fully populated matrices. This poses serious challenges for large-scale three-dimensional problems due to storage requirements and iterative solution of a large set of non-symmetric equations. Researchers have developed several approaches to address this issue including the class of fast-multipole techniques, use of wavelet transforms, and matrix decomposition. In this paper, we develop a domain decomposition, or the artificial sub-sectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address the coefficient matrix issue. The meshless method we employ is based on expansions using radial-basis functions (RBFs).An efficient physically based procedure provides an effective initial guess of the temperatures along the sub-domain interfaces. The iteration process converges very efficiently, offers substantial savings in memory, and features superior computational efficiency. The meshless iterative domain decomposition technique is ideally suited for parallel computation. We discuss its implementation under MPI standards on a small Windows XP PC cluster. Numerical results reveal the domain decomposition meshless methods produce accurate temperature predictions while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods.     

Strict lower bounds with separation of sources of error in non‐overlapping domain decomposition methods

This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element and domain decomposition methods. In addition to a fully parallel computation, the proposed lower bounds separate the algebraic error (due to the use of a domain decomposition iterative solver) from the discretization error (due to the finite element), which enables the steering of the iterative solver by the discretization error. These lower bounds are also used to improve the goal‐oriented error estimation in a substructured context. Assessments on 2D static linear mechanic problems illustrate the relevance of the separation of sources of error and the lower bounds' independence from the substructuring. We also steer the iterative solver by an objective of precision on a quantity of interest. This strategy consists in a sequence of solvings and takes advantage of adaptive remeshing and recycling of search directions. Copyright © 2016 John Wiley & Sons, Ltd.     

Three‐level BDDC in two dimensions

Balancing Domain Decomposition by Constraints (BDDC) methods are non‐overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems. They are similar to the balancing Neumann–Neumann algorithm. However, in BDDC methods, a small number of continuity constraints are enforced across the interface, and these constraints form a new coarse, global component. An important advantage of using such constraints is that the Schur complements that arise in the computation will all be strictly positive definite. The matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation. However, this problem can ultimately become a bottleneck, if the number of subdomains is very large. In this paper, two three‐level BDDC methods are introduced for solving the coarse problem approximately in two‐dimensional space, while still maintaining a good convergence rate. Estimates of the condition numbers are provided for the two three‐level BDDC methods and numerical experiments are also discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical analysis of a three‐dimensional super‐elastic constitutive model

This note deals with the efficient approximation of a non‐linear constitutive relation arising in the study of the three‐dimensional mechanical behaviour of shape memory alloys at constant temperature. In particular, a variable time‐step discretization is investigated. For such an algorithm we prove sharp error estimates of optimal order and exactness for a class of experimentally relevant situations. We also report numerical results relative to proportional and non‐proportional loading tests which fully confirm the theoretical analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

求解非定常N-S方程的稳定化分数步长法研究

本文研究了求解非定常Navier-Stokes方程的稳定化分数步长法.首先,通过一阶精度的算子分裂,将非线性项和不可压缩条件分裂到两个不同的子问题中,并对非线性项采用Oseen迭代.格式分为两步:第一步求解一个线性椭圆问题;第二步求解一个广义的Stokes问题.这两个子问题关于速度都满足齐次Dilichlet边界条件.同时,在格式的第二步添加了局部稳定化项,使用等阶序对来加强数值解的稳定性.通过能量估计方法,对速度与压力做了收敛性分析和误差估计.最后,数值实验验证了方法的有效性.     

Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

We present new iterative solvers for large‐scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second‐order accuracy can be obtained at very small overcost with respect to first‐order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2 × 2 block symmetric indefinite linear system arising from mixed (displacement‐pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

A penalty finite element method for non-Newtonian creeping flows

In this paper we present an iterative penalty finite element method for viscous non-Newtonian creeping flows. The basic idea is solving the equations for the difference between the exact solution and the solution obtained in the last iteration by the penalty method. For the case of Newtonian flows, one can show that for sufficiently small penalty parameters the iterates converge to the incompressible solution. The objective of the present work is to show that the iterative penalization can be coupled with the iterative scheme used to deal with the non-linearity arising from the constitutive law of non-Newtonian fluids. Some numerical experiments are conducted in order to assess the performance of the approach for fluids whose viscosity obeys the power law.     

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.     

Estimation of model errors in the calibration of viscoelastic material models

In this paper we discuss a posteriori error estimation in the context of parameter identification problems. In particular, we pay attention to the errors arising from the discretization of the parameters, which is perceived as a model error. A previously developed method for goal‐oriented a posteriori error estimation is employed for two fundamentally different types of model hierarchies. The numerical results show rather good agreement between estimated and actual errors. Copyright © 2008 John Wiley & Sons, Ltd.     

Time‐space PGD for the rapid solution of 3D nonlinear parametrized problems in the many‐query context

This work deals with the question of the resolution of nonlinear problems for many different configurations in order to build a ‘virtual chart’ of solutions. The targeted problems are three‐dimensional structures driven by Chaboche‐type elastic‐viscoplastic constitutive laws. In this context, parametric analysis can lead to highly expensive computations when using a direct treatment. As an alternative, we present a technique based on the use of the time‐space proper generalized decomposition in the framework of the LATIN method. To speed up the calculations in the parametrized context, we use the fact that at each iteration of the LATIN method, an approximation over the entire time‐space domain is available. Then, a global reduced‐order basis is generated, reused and eventually enriched, by treating, one‐by‐one, all the various parameter sets. The novelty of the current paper is to develop a strategy that uses the reduced‐order basis for any new set of parameters as an initialization for the iterative procedure. The reduced‐order basis, which has been built for a set of parameters, is reused to build a first approximation of the solution for another set of parameters. An error indicator allows adding new functions to the basis only if necessary. The gain of this strategy for studying the influence of material or loading variability reaches the order of 25 in the industrial examples that are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical verification of solutions for Signorini problems using Newton‐like method

We proposed a numerical method to verify the existence of solutions for a simplified Signorini problem (Comput. Math. Appl. 2000; 40 :1003–1013). Using sequential iteration method, we numerically constructed a set containing solutions that satisfies the hypothesis of Schauder's fixed point theorem in a certain Sobolev space. It is difficult to apply this method to the problem of which associated operator is not retractive in a neighborhood of the solution. In this paper, in order to overcome such a difficulty, we describe an alternative approach to this problem. Numerical examples are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

抛物问题的质量集中非协调有限元法

主要讨论了一类抛物问题的质量集中非协调有限元方法。首先,我们给出了所讨论问题的质量集中非协调有限元Crank-Nicolson全离散逼近格式。其次,对讨论问题的解与所给出逼近格式的解之间的误差估计进行了分析研究。最后利用椭圆投影算子,我们得到了关于L2模和能量模方面的一些误差估计式。     

A semi‐local spectral/hp element solver for linear elasticity problems

We develop an efficient semi‐local method for speeding up the solution of linear systems arising in spectral/hp element discretization of the linear elasticity equations. The main idea is to approximate the element‐wise residual distribution with a localization operator we introduce in this paper, and subsequently solve the local linear system. Additionally, we decouple the three directions of displacement in the localization operator, hence enabling the use of an efficient low energy preconditioner for the conjugate gradient solver. This approach is effective for both nodal and modal bases in the spectral/hp element method, but here, we focus on the modal hierarchical basis. In numerical tests, we verify that there is no loss of accuracy in the semi‐local method, and we obtain good parallel scalability and substantial speed‐up compared to the original formulation. In particular, our tests include both structure‐only and fluid‐structure interaction problems, with the latter modeling a 3D patient‐specific brain aneurysm. Copyright © 2014 John Wiley & Sons, Ltd.     

A recipe to detect the error in discretization schemes

The necessity for a reliable measure of the discretization error arises in adaptive mesh refinement and in moving mesh adaptation. The present work discusses a detector of the discretization error based on the interpolation reconstruction of the operators. The technique presented here is named operator recovery error source detector (ORESD). Its main features are: First, the technique is based on the operators being discretized and does not require any user intervention or any a priori knowledge of the solution or its properties. Second, the ORESD is an a posteriori error indicator, but it is shown to be consistent with the a priori error provided by the modified equation approach. Third, the technique is based on the operators being solved and is tailored to the specific problem at hand. Four, the technique is simple and is based on a small stencil, resulting in a very inexpensive error detection. In the present work, the ORESD is derived and applied to two tutorial examples: divergence and gradient. With the aid of the two examples and using the general derivation, the ORESD is then applied to the gas dynamics equations. Two benchmarks are used to test the performance. First, a shock tube problem is solved (Sod's benchmark) in a Lagrangian and in a Eulerian frame. Second, the Colella's wedge problem is solved using CLAWPACK. Finally, the ORESD is applied to the 2D Poisson equation on a uniform and on a non‐uniform grid to test the application to elliptic problems. In all examples the operator recovery error source detector succeeds in detecting the real sources of error. Copyright © 2004 John Wiley & Sons, Ltd.     

An explicit residual‐type error estimator for ‐quadrilateral extended finite element method in two‐dimensional linear elastic fracture mechanics

This paper deals with explicit residual a posteriori error estimation analysis for ‐quadrilateral extended finite element method (XFEM) discretizations applied to the two‐dimensional problem of linear elastic fracture mechanics. The result is twofold. First, to enable estimation procedures with application to XFEM, a specific quasi‐interpolation operator of averaging type is constructed. The main challenge here arises from the different types of enrichments implemented, and hence, to impose the constant‐preserving property of the interpolation operator on an element, we use the idea of an extension operator. An upper bound on the discretization error measured in the energy norm and associated local error indicators are then constructed and analyzed. The second result follows from the error analysis and concerns an alternative choice of branch functions used in XFEM applications. In particular, the branch functions have to be chosen to fulfill the divergence‐free conditions within the crack tip element and traction‐free boundary conditions on the crack faces. Then, the corresponding XFEM solution gains a better accuracy with less degrees of freedom. Finally, numerical examples are provided with comparative results. Copyright © 2012 John Wiley & Sons, Ltd.