We present a numerical procedure for computing guaranteed two-sided bounds on the effective coefficients of elliptic partial differential operators in a three-dimensional setting. The upper bounds are obtained via a standard variational formulation, discretized using the finite element method. To derive the lower bounds, we formulate the corresponding dual variational problem and construct suitable approximation spaces within the finite element framework. To reduce the computational overhead associated with evaluating the lower bounds, we propose an efficient estimation technique based on a single FFT-based projection. Theoretical justification of the proposed procedure is provided, and its performance is demonstrated through illustrative numerical examples.

The computational efficiency and rapid convergence of fast Fourier transform (FFT)-based solvers render them a powerful numerical tool for periodic cell problems in multiscale modeling. On regular grids, they tend to outperform traditional numerical methods. However, we show that their convergence slows down significantly when applied to microstructures with smooth, highly-contrasted coefficients. To address this loss of performance, we introduce a Green-Jacobi preconditioner, an enhanced successor to the standard discrete Green preconditioner that preserves the quasilinear complexity, $\mathcal{O}(N \log N)$, of conventional FFT-based solvers. Through numerical experiments, we demonstrate the effectiveness of the Jacobi-accelerated FFT (J-FFT) solver within a linear elastic framework. For problems characterized by smooth data and high material contrast, J-FFT significantly reduces the iteration count of the conjugate gradient method compared to the standard Green preconditioner. These findings are particularly relevant for phase-field fracture simulations, density-based topology optimization, and solvers that use adaption of the grid, which all introduce smooth variations in the material properties that challenge conventional FFT-based solvers.

Computational homogenization accelerated by Green function preconditioning with Fast Fourier transforms (FFT) is classically performed on a uniform grid, which hinders the discretization accuracy. In this work, we consider a more accurate geometry representation obtained by transforming the uniform computational grid into a boundary-conforming one. The mechanical problem is discretized using the finite element method (FEM) with isoparametric transformation of elements. Boundary adaptation can require large localized geometrical transformations of the grid, which is naturally accounted for in the FEM discretization. Rigorous bounds on the spectrum of eigenvalues of the resulting discrete system with Green preconditioner are provided. For grid transformations with projection of the nearest nodes to boundary, the modified eigenvalues correspond to eigenvectors localized at the material phases boundaries, so that the effective spectrum remains favorable for the preconditioned conjugate gradient solver. Numerical investigations confirm that the accuracy of the homogenized properties and the local fields obtained on boundary-conforming grids are greatly improved over uniform grid ones, at the expense of a moderate increase in computational cost.

This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of non-symmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection-)diffusion-reaction problems.

We generalize and provide a linear algebra-based perspective on a finite element (FE) ho-mogenization scheme, pioneered by Schneider et al. (2017)[1] and Leuschner and Fritzen (2018)[2]. The efficiency of the scheme is based on a preconditioned, well-scaled refor-mulation allowing for the use of the conjugate gradient or similar iterative solvers. The geometrically-optimal preconditioner-a discretized Green's function of a periodic homo-geneous reference problem-has a block-diagonal structure in the Fourier space which per-mits its efficient inversion using fast Fourier transform (FFT) techniques for generic regular meshes. This implies that the scheme scales as O(n log(n)), like FFT, rendering it equiva-lent to spectral solvers in terms of computational efficiency. However, in contrast to clas-sical spectral solvers, the proposed scheme works with FE shape functions with local sup-ports and does not exhibit the Fourier ringing phenomenon. We show that the scheme achieves a number of iterations that are almost independent of spatial discretization. The scheme also scales mildly with phase contrast. We also discuss the equivalence between our displacement-based scheme and the recently proposed strain-based homogenization technique with finite-element projection. (c) 2023 Published by Elsevier Inc.

We present a numerical scheme for obtaining guaranteed (reliable) and arbitrarily close two sided bounds to effective (homogenized) parameters of the linear elasticity problem. For the upper bounds, we use standard finite element (FE) discretization of the so‐called primal problem with preconditioning based on the fast discrete Fourier transformation (FFT). For the lower bounds, we use the dual formulation and some smoother FE approximation spaces. Moreover, instead of solving the discretized dual problem, we can only compute an L 2 ‐orthogonal projection of an auxiliary field built from the primal solution. The projection can be computed easily by FFT and provides a lower bound of almost the same quality as that obtained as the exact solution of the discretized dual problem. In addition, a simple low‐dimensional optimization improves the projected solution. Numerical examples are presented to support the theoretical developments.

We propose a matrix-free finite element (FE) homogenization scheme that is considerably more efficient than generic FE implementations. The efficiency of our scheme follows from a preconditioned well-scaled reformulation allowing for the use of the conjugate gradient or similar iterative solvers. The geometrically-optimal preconditioner -- a discretized Green's function of a periodic homogeneous reference problem -- has a block-diagonal structure in the Fourier space which permits its efficient inversion using the fast Fourier transform (FFT) techniques for generic regular meshes. This implies that the scheme scales as $\mathcal{O}(n \log(n))$ like FFT, rendering it equivalent to spectral solvers in terms of computational efficiency. However, in contrast to classical spectral solvers, the proposed scheme works with FE shape functions with local supports and is free of the Fourier ringing phenomenon. We showcase that the scheme achieves the number of iterations that are almost independent of spatial discretisation and scales mildly with the phase contrast. Additionally, we discuss the equivalence between our displacement-based scheme and the recently proposed strain-based homogenization technique with finite-element projection.

A numerical procedure providing guaranteed two-sided bounds on the effective coefficients of elliptic partial differential operators is presented. The upper bounds are obtained in a standard manner through the variational formulation of the problem and by applying the finite element method. To obtain the lower bounds we formulate the dual variational problem and introduce appropriate approximation spaces employing the finite element method as well. We deal with the 3D setting, which has been rarely considered in the literature so far. The theoretical justification of the procedure is presented and supported with illustrative examples.

Micromechanical homogenization is often carried out with Fourier-accelerated methods that are prone to ringing artifacts. We here generalize the compatibility projection introduced by Vondřejc, Zeman & Marek [Comput. Math. Appl. 68, 156 (2014)] beyond the Fourier basis. In particular, we formulate the compatibility projection for linear finite elements while maintaining Fourieracceleration and the fast convergence properties of the original method. We demonstrate that this eliminates ringing artifacts and yields an efficient computational homogenization scheme that is equivalent to canonical finite-element formulations on fully structured grids.

Fourier-accelerated micromechanical homogenization has been developed and applied to a variety of problems, despite being prone to ringing artifacts. In addition, the majority of Fourier-accelerated solvers applied to FFT-accelerated schemes only apply to convex problems. We here introduce a that allows to employ modern efficient and non-convex iterative solvers, such as trust-region solvers or LBFGS in a FFT-accelerated scheme. These solvers need the explicit energy functional of the system in their standard form. We develop a modified trust region solver, capable of handling non-convex micromechanical homogenization problems such as continuum damage employing the approximate incremental energy functional. We use the developed solver as the solver of a ringing-free FFT-accelerated solution scheme, namely the projection based scheme with finite element discretization.

Numerical methods for elliptic partial differential equations usually lead to systems of linear equations with sparse, symmetric, and positive definite matrices. In many methods, these matrices can be obtained as sums of local symmetric positive semidefinite matrices. In this article, we use this assumption and introduce a method that provides guaranteed lower and upper bounds to all individual eigenvalues of the preconditioned matrices. We apply the method for preconditioners arising from the same discretization problem but with simplified coefficients. The method uses solely the data over the solution domain and local connections between the degrees of freedom defined by the discretization.

A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, and preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data.

Fast Fourier transform (FFT) based methods have turned out to be an effective computational approach for numerical homogenisation. In particular, Fourier-Galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials. Their computational effectiveness benefits from efficient FFT based algorithms as well as a favourable condition number. Here these kind of methods are accelerated by low-rank tensor approximation techniques for a solution field using canonical polyadic, Tucker, and tensor train formats. This reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem. It significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation. The advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients. This approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material.

AbstractWe enhanced the efficiency of Fast Fourier transform (FFT) based Galerkin methods on numerical homogenisation problems by exploiting low‐rank tensor approximations in canonical, Tucker, and tensor train formats. This leads to a significant reduction in computational complexity and memory requirement. The advantages of the approach are demonstrated in a numerical example of a model homogenisation problem with stochastic heterogeneous material coefficients.

Homogenization problems represent one important part of large scale modeling. Reducing the data describing material properties via one or a few constants may reduce memory and computational demands of the computational process of many real life problems. We consider the elliptic partial differential equation with a oscillating anizotropic material data. Using the classical homogenization theory [1, 4], we aim to find the effective related parameter. For numerical solution, instead of finite element basis, we use Fourier spectral basis. The resulting matrix-vector form of the problem is described. It was already shown that this system of equations can be solved using the conjugate gradient method [10]. In this paper, we introduce a certain class of preconditioning of this problem. Our approach is based on exact incorporating of one or more frequencies of the data into the preconditioner in such manner that the preconditioning matrix remains easy to invert.

Problems of finding effective material properties discussed in the classical works on homogenisation [1, 2] originate from asymptotic analysis of elliptic partial differential equations with oscillating anisotropic data. In 1994, Moulinec and Suquet [3] introduced an iterative algorithm for efficient determination of effective properties. In spite of its popularity and a deep research interest in this method, there are still some opportunities for improving its numerical effectiveness. We focus on preconditioning of related linear systems, which is necessary for solving large problems by iterative solvers. We introduce a general scheme for construction of preconditioning matrices based on some appropriate approximation of original material data.