We study the convergences of several FFT-based schemes that are widely applied in computational homogenization for deriving effective coefficients, and the term "convergence" here means the limiting behaviors as spatial resolutions going to infinity. Those schemes include Moulinec-Suquent's scheme [Comput Method Appl M, 157 (1998), pp. 69-94], Willot's scheme [Comptes Rendus Mécanique, 343 (2015), pp. 232-245], and the FEM scheme [Int J Numer Meth Eng, 109 (2017), pp. 1461-1489]. Under some reasonable assumptions, we prove that the effective coefficients obtained by those schemes are all convergent to the theoretical ones. Moreover, for the FEM scheme, we can present several convergence rate estimates under additional regularity assumptions.
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…
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 Fourier-acceleration 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 …
A stable and accurate X-FFT solver for linear elastic homogenization problems in 3D
University of Duisburg-Essen（杜伊斯堡-埃森大学）；Institute of Engineering Mathematics（工程数学研究所）；Fraunhofer Institute for Industrial Mathematics ITWM（弗劳恩霍夫工业数学研究所）；Center for Nanointegration Duisburg-Essen (CENIDE)（杜伊斯堡-埃森纳米集成中心）
Efficient numerical characterization is a key problem in composite material analysis. To follow accuracy improvement in image tomography, memory efficient methods of numerical characterization have been developed. Among them, an FFT based solver has been proposed by Moulinec and Suquet (1994,1998) bringing down numerical characterization complexity to the FFT complexity. Nevertheless, recent development of tomography sensors made memory requirement and calculation time reached another level. To avoid this bottleneck, the new leaps in the field of Quantum Computing have been used. This paper wi…
Computational homogenization is the gold standard for concurrent multi-scale simulations (e.g., FE2) in scale-bridging applications. Experimental and synthetic material microstructures are often represented by 3D image data. The computational complexity of simulations operating on such three-dimensional high-resolution voxel data comprising billions of unknowns induces the need for algorithmically and numerically efficient solvers. The inability of voxelized 3D geometries to capture smooth material interfaces accurately, along with the necessity for complexity reduction, motivates a special lo…
FFT-based homogenization on periodic anisotropic translation invariant spaces
University of Kaiserslautern（凯泽斯劳滕大学）
本文推导了在各向异性平移不变空间中均质化准静态弹性方程的离散形式，采用变分公式和所谓的Lippmann-Schwinger方程，并统一和扩展了截断傅里叶级数方法、常数有限元近似和各向异性晶格推导。所得Lippmann-Schwinger方程在各向异性平移不变空间中的形式首次将傅里叶方法和有限元方法统一到一个数学框架中。进一步定义并刻画了所得到的周期化Green算子。该算子在Dirichlet核对应于对角矩阵的情况下，与来自截断傅里叶级数方法和各向异性晶格推导的Galerkin投影算子相同，并且对于其他Dirichlet核也相同。此外，证明了周期化Green算子的有界性。该算子仅当平移空间由Dirichlet核生成时才构成投影。de la Vallée Poussin均值和Box样条的数值示例展示了该框架的灵活性。
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 micro…
Computational homogenization of time-harmonic Maxwell's equations
在本文中，我们考虑一个数值均匀化技术的卷曲卷曲问题，这是基于框架的局部正交分解，并提出了在[D。作者声明：A.弗菲尔特SIAM J. Numer. Anal. 56-3：1570-1596，2018]对于具有本质边界条件的问题。上述工作的结果建立了时间谐波麦克斯韦方程组的定量均匀化结果，该结果超出了周期性的假设，但是，该方法的实际实现仍然是开放的。在本文中，我们转移的结果，从本质边界条件的自然边界条件，我们表明，该方法产生一个可计算的数值方法。我们还研究了源项的边界值如何影响计算复杂性和精度。我们的研究结果将支持各种数值实验，无论是在$2D$和$3D$。
Computational Realization of a Non-Equidistant Grid Sampling in Photoacoustics with a Non-Uniform FFT
University of Vienna（维也纳大学）；Medical University of Vienna（维也纳医科大学）；Radon Institute of Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院应用计算数学拉德诺研究所）
We present and analyze a methodology for numerical homogenization of spatial networks, modelling e.g. diffusion processes and deformation of mechanical structures. The aim is to construct an accurate coarse model of the network. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data at a low computational cost. We prove optimal convergence of the proposed …
Numerical Homogenization of Landau-Lifshitz Equation with Rough Coefficients
Shanghai Jiao Tong University（上海交通大学）；University of Science and Technology of China（中国科学技术大学）；Suzhou Institute for Advanced Research, University of Science and Technology of China（中国科学技术大学苏州高等研究院）；Soochow University（苏州大学）
Direct numerical simulation of diffusion through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to homogenize the model by replacing the spatially-varying fine-scale diffusivity with an effective diffusivity calculated from the solution of an appropriate boundary value problem. In this paper, we present a new semi-analytical method for solving this boundary value problem and computing the effective diffusivity for pixellated, locally-isotropic, heterogeneous media. We compare our new solution…
A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients requires solving local corrector problems followed by upscaling relevant local data. The most naive way of computing homogenized coefficients is by solving a local elliptic problem, which is known to suffer from the so-called resonance error dominating all other errors inherent in multiscale computations. A far more efficient modelling strategy, based on adding …
