A surrogate model for computational homogenization of elastostatics at finite strain using high-dimensional model representation-based neural network

We propose a surrogate model for two-scale computational homogenization of elastostatics at finite strains. The macroscopic constitutive law is made numerically available via an explicit formulation of the associated macroenergy density. This energy density is constructed by using a neural network architecture that mimics a high-dimensional model representation. The database for training this network is assembled through solving a set of microscopic boundary value problems with the prescribed macroscopic deformation gradients (input data) and subsequently retrieving the corresponding averaged energies (output data). Therefore, the two-scale computational procedure for nonlinear elasticity can be broken down into two solvers for microscopic and macroscopic equilibrium equations that work separately in two stages, called the offline and online stages. The finite element method is employed to solve the equilibrium equation at the macroscale. As for microscopic problems, an FFT-based collocation method is applied in tandem with the Newton-Raphson iteration and the conjugate-gradient method. Particularly, we solve the microscopic equilibrium equation in the Lippmann-Schwinger form without resorting to the reference medium. In this manner, the fixed-point iteration that might require quite strict numerical stability conditions in the nonlinear regime is avoided. In addition, we derive the projection operator used in the FFT-based method for homogenization of elasticity at finite strain.     

An FFT-based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture

Cell formulae for the effective crack resistance of a heterogeneous medium obeying Francfort-Marigo's formulation of linear elastic fracture mechanics have been proved recently, both in the context of periodic and stochastic homogenization. This work proposes a numerical strategy for computing the effective, possibly anisotropic, crack resistance of voxelized microstructures using the fast Fourier transform (FFT). Based on Strang's continuous minimum cut—maximum flow duality, we explore a primal-dual hybrid gradient method for computing the effective crack resistance, which may be readily integrated into an existing FFT-based code for homogenizing thermal conductivity. We close with demonstrative numerical experiments.     

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.     

Fast Fourier transform on multipoles (FFTM) algorithm for Laplace equation with direct and indirect boundary element method

In this paper, the fast Fourier transform on multipole (FFTM) algorithm is used to accelerate the matrix-vector product in the boundary element method (BEM) for solving Laplace equation. This is implemented in both the direct and indirect formulations of the BEM. A new formulation for handling the double layer kernel using the direct formulation is presented, and this is shown to be related to the method given by Yoshida (Application of fast multipole method to boundary integral equation method, Kyoto University, Japan, 2001). The FFTM algorithm shows different computational performances in direct and indirect formulations. The direct formulation tends to take more computational time due to the evaluation of an extra integral. The error of FFTM in the direct formulation is smaller than that in the indirect formulation because the direct formulation has the advantage of avoiding the calculations of the free term and the strongly singular integral explicitly. The multipole and local translations introduce approximation errors, but these are not significant compared with the discretization error in the direct or indirect BEM formulation. Several numerical examples are presented to compare the computational efficiency of the FFTM algorithm used with the direct and indirect BEM formulations.     

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation,parallel implementation and scalability properties

An explicit‐dynamics spatially discontinuous Galerkin (DG) formulation for non‐linear solid dynamics is proposed and implemented for parallel computation. DG methods have particular appeal in problems involving complex material response, e.g. non‐local behavior and failure, as, even in the presence of discontinuities, they provide a rigorous means of ensuring both consistency and stability. In the proposed method, these are guaranteed: the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation. The semi‐discrete system of ordinary differential equations is integrated in time using a conventional second‐order central‐difference explicit scheme. A stability criterion for the time integration algorithm, accounting for the influence of the DG discretization stability, is derived for the equivalent linearized system. This approach naturally lends itself to efficient parallel implementation. The resulting DG computational framework is implemented in three dimensions via specialized interface elements. The versatility, robustness and scalability of the overall computational approach are all demonstrated in problems involving stress‐wave propagation and large plastic deformations. Copyright © 2007 John Wiley & Sons, Ltd.     

Implementation of a symmetric boundary element method in transient heat conduction with semi‐analytical integrations

A time‐convolutive variational hypersingular integral formulation of transient heat conduction over a 2‐D homogeneous domain is considered. The adopted discretization leads to a linear equation system, whose coefficient matrix is symmetric, and is generated by double integrations in space and time. Assuming polynomial shape functions for the boundary unknowns, a set of compact formulae for the analytical time integrations is established. The spatial integrations are performed numerically using very efficient formulae just recently proposed. The competitiveness from the computational point of view of the symmetric boundary integral equation approach proposed herein is investigated on the basis of an original computer implementation. Copyright © 1999 John Wiley & Sons, Ltd.     

An Unconditionally Energy Stable Immersed Boundary Method with Application to Vesicle Dynamics

We develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased (the severe time step restriction in an explicit boundary forcing scheme is avoided). As an application, we use our scheme to simulate vesicle dynamics in Navier-Stokes flow.     

An Accelerated Two-Dimensional Unsteady Heat Conduction Calculation Procedure for Thermal-Conductivity Measurement by the Transient Short-Hot-Wire Method

A fast and accurate procedure is proposed for solution of the two-dimensional unsteady heat conduction equation used in the transient short-hot-wire method for measuring thermal conductivity. Finite Fourier transforms are applied analytically in the wire-axis direction to produce a set of one-dimensional ordinary differential equations. After discretization by the finite-volume method in the radial direction, each one-dimensional algebraic equation is solved directly using the tri-diagonal matrix algorithm prior to application of the inverse Fourier transform. The numerical procedure is shown to be very accurate through comparison with an analytical solution, and it is found to be an order of magnitude faster than the usual numerical solution.     

Error analysis of penalty function techniques for constraint definition in linear algebraic systems

The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational discretization of boundary value problems by finite difference or finite element techniques. This paper studies the numerical behaviour of the penalty function method for the special case of individual equation constraints imposed on a symmetric system of linear algebraic equations. Constraint representation and computational roundoff error components are distinguished and asymptotically characterized in terms of the penalty function weight coefficients. On the basis of this study, practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed. Numerical problems encountered in the case of more general constraints are briefly discussed, and procedures for circumventing such difficulties are suggested.     

A damage model for crack prediction in brittle and quasi-brittle materials solved by the FFT method

In this paper, we present a damage model and its numerical solution by means of Fast Fourier Transforms (FFT). The FFT-based formulation initially proposed for linear and non-linear composite homogenization (Moulinec and Suquet in CR Acad Sci Paris Ser II 318:1417–1423 1994; Comput Methods Appl Mech Eng 157:69–94 1998) was adapted to evaluate damage growth in brittle materials. A non-local damage model based on the maximal principal stress criterion was proposed for brittle materials. This non-local model was then connected to the Griffith criterion with the aim of predicting crack growth. By using the proposed model, we carried out several numerical simulations on different specimens in order to assess the fracture process in brittle materials. From these studies, we can conclude that the present FFT-based analysis is capable of dealing with crack initiation and crack growth in brittle materials with high accuracy and efficiency.     

Computation of Dyadic Green's Functions for Electrodynamics in Quasi-Static Approximation with Tensor Conductivity

Homogeneous non-dispersive anisotropic materials, characterized by a positive constant permeability and a symmetric positive definite conductivity tensor, are considered in the paper. In these anisotropic materials, the electric and magnetic dyadic Green's functions are defined as electric and magnetic fields arising from impulsive current dipoles and satisfying the time-dependent Maxwell's equations in quasi-static approximation. A new method of deriving these dyadic Green's functions is suggested in the paper. This method consists of several steps: equations for electric and magnetic dyadic Green's functions are written in terms of the Fourier modes; explicit formulae for the Fourier modes of dyadic Green's functions are derived using the matrix transformations and solutions of some ordinary differential equations depending on the Fourier parameters; the inverse Fourier transform is applied to obtained formulae to find explicit formulae for dyadic Green's functions.     

组合杂交四边形元的多重网格预处理共轭梯度方法

组合杂交元方法是一种求解弹性力学问题的稳定化有限元方法．为了快速求解组合杂交元离散得到的大型、稀疏、对称正定系统,本文研究了多重网格预处理共轭梯度方法．首先,通过选用合适的网格转移算子和光滑策略,得到了有效的多重网格预处理器．其次,通过分析数值试验结果证明所得到的多重网格预处理共轭梯度方法是有效可行的,利用该预处理方法大大降低了系数矩阵的条件数,提高了计算效率．此外,对于一类高性能的组合杂交元,多重网格预处理共轭梯度方法在网格畸变时依然收敛.     

A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator

The homogeneous Dirichlet problem for the biharmonic operator is solved as the variational formulation of two coupled second-order equations. The discretization by a mixed finite element model results in a set of linear equations whose coefficient matrix is sparse, symmetric but indefinite. We describe a class of preconditioned conjugate gradient methods for the numerical solution of this linear system. The precondition matrices correspond to incomplete factorizations of the coefficient matrix. The numerical results show a low computational complexity in both number of computer operations and demand of storage.     

A Fourier–Karhunen–Loève discretization scheme for stationary random material properties in SFEM

In order to overcome the computational difficulties in Karhunen–Loève (K–L) expansions of stationary random material properties in stochastic finite element method (SFEM) analysis, a Fourier–Karhunen–Loève (F–K–L) discretization scheme is developed in this paper, by following the harmonic essence of stationary random material properties and solving a series of specific technical challenges encountered in its development. Three numerical examples are employed to investigate the overall performance of the new discretization scheme and to demonstrate its use in practical SFEM simulations. The proposed F–K–L discretization scheme exhibits a number of advantages over the widely used K–L expansion scheme based on FE meshes, including better computational efficiency in terms of memory and CPU time, convenient a priori error‐control mechanism, better approximation accuracy of random material properties, explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonlinear three-dimensional magnetic field computations usingLagrange finite-element functions and algebraic multigrid

The paper proposes an efficient solution strategy for nonlinear three-dimensional (3-D) magnetic field problems. The spatial discretization of Maxwell's equations uses Lagrange finite-element functions. The paper shows that this discretization is appropriate for the problem class. The nonlinear equation is linearized by the standard fixed-point scheme. The arising sequence of symmetric positive definite matrices is solved by a preconditioned conjugate gradient method, preconditioned by an algebraic multigrid technique. Because of the relatively high setup time of algebraic multigrid, the preconditioner is kept constant as long as possible in order to minimize the overall CPU time. A practical control mechanism keeps the condition number of the overall preconditioned system as small as possible and reduces the total computational costs in terms of CPU time. Numerical studies involving the TEAM 20 and the TEAM 27 problem demonstrate the efficiency of the proposed technique. For comparison, the standard incomplete Cholesky preconditioner is used     

Smart element method I. The Zienkiewicz–Zhu feedback

A new error control finite element formulation is developed and implemented based on the variational multiscale method, the inclusion theory in homogenization, and the Zienkiewicz–Zhu error estimator. By synthesizing variational multiscale method in computational mechanics, the equivalent eigenstrain principle in micromechanics, and the Zienkiewicz–Zhu error estimator in the finite element method (FEM), the new finite element formulation can automatically detect and subsequently homogenize its own discretization errors in a self‐adaptive and a self‐adjusting manner. It is the first finite element formulation that combines an optimal feedback mechanism and a precisely defined homogenization procedure to reduce its own discretization errors and hence to control numerical pollutions. The paper focuses on the following two issues: (1) how to combine a multiscale method with the existing finite element error estimate criterion through a feedback mechanism, and (2) convergence study. It has been shown that by combining the proposed variational multiscale homogenization method with the Zienkiewicz–Zhu error estimator a clear improvement can be made on the coarse scale computation. It is also shown that when the finite element mesh is refined, the solution obtained by the variational eigenstrain multiscale method will converge to the exact solution. Copyright © 2004 John Wiley & Sons, Ltd.     

On least squares approximations to indefinite problems of the mixed type

A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The mehod retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency i.e., the resulting matrix system is always symmetric and positive definite,. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.     

hp Fast multipole boundary element method for 3D acoustics

A fast multipole boundary element method (FMBEM) extended by an adaptive mesh refinement algorithm for solving acoustic problems in three‐dimensional space is presented in this paper. The Collocation method is used, and the Burton–Miller formulation is employed to overcome the fictitious eigenfrequencies arising for exterior domain problems. Because of the application of the combined integral equation, the developed FMBEM is feasible for all positive wave numbers even up to high frequencies. In order to evaluate the hypersingular integral resulting from the Burton–Miller formulation of the boundary integral equation, an integration technique for arbitrary element order is applied. The fast multipole method combined with an arbitrary order h‐p mesh refinement strategy enables accurate computation of large‐scale systems. Numerical examples substantiate the high accuracy attainable by the developed FMBEM, while requiring only moderate computational effort at the same time. Copyright © 2016 John Wiley & Sons, Ltd.     

On Quasi-Newton methods in fast Fourier transform-based micromechanics

This work is devoted to investigating the computational power of Quasi-Newton methods in the context of fast Fourier transform (FFT)-based computational micromechanics. We revisit FFT-based Newton-Krylov solvers as well as modern Quasi-Newton approaches such as the recently introduced Anderson accelerated basic scheme. In this context, we propose two algorithms based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method, one of the most powerful Quasi-Newton schemes. To be specific, we use the BFGS update formula to approximate the global Hessian or, alternatively, the local material tangent stiffness. Both for Newton and Quasi-Newton methods, a globalization technique is necessary to ensure global convergence. Specific to the FFT-based context, we promote a Dong-type line search, avoiding function evaluations altogether. Furthermore, we investigate the influence of the forcing term, that is, the accuracy for solving the linear system, on the overall performance of inexact (Quasi-)Newton methods. This work concludes with numerical experiments, comparing the convergence characteristics and runtime of the proposed techniques for complex microstructures with nonlinear material behavior and finite as well as infinite material contrast.     

APPLICATION OF THE ORDINARY LEAST-SQUARES APPROACH FOR SOLUTION OF COMPLEX VARIABLE BOUNDARY ELEMENT PROBLEMS

Cauchy's theorem is used to generate a Complex Variable Boundary Element Method (CVBEM) formulation for steady, two-dimensional potential problems. CVBEM uses the complex potential, w=ϕ+iψ, to combine the potential function, ϕ, with the stream function, ψ. The CVBEM formulation, using Cauchy's theorem, is shown to be mathematically equivalent to Real Variable BEM which employs Green's second identity and the respective fundamental solution. CVBEM yields an overdetermined system of equations that are commonly solved using implicit and explicit methods that reduce the overdetermined matrix to a square matrix by selectively excluding equations. Alternatively, Ordinary Least Squares (OLS) can be used to minimize the Euclidean norm square of the residual vector that arises due to the approximation of boundary potentials and geometries. OLS uses all equations to form a square matrix that is symmetric, positive definite and diagonally dominant. OLS is more accurate than existing methods and can estimate the approximation error at boundary nodes. The approximation error can be used to determine the adequacy of boundary discretization schemes. CVBEM/OLS provides greater flexibility for boundary conditions by allowing simultaneous specification of both fluid potentials and stream functions, or their derivatives, along boundary elements. © 1997 by John Wiley & Sons, Ltd.