We propose an algorithm for the computational homogenization of locally periodic hyperelastic structures undergoing large deformations due to external quasi-static loading. The algorithm performs clustering of macroscopic deformations into subsets called "centroids", and, as a new ingredient, approximates the homogenized coefficients using sensitivity analysis of micro-configurations with respect to the macroscopic deformation. The novel "model-order reduction" approach significantly reduces the number of microscopic problems that must be solved in nonlinear simulations, thereby accelerating the overall computational process. The degree of reduction can be controlled by a user-defined error tolerance parameter. The algorithm is implemented in the finite element framework SfePy, and its performance effectiveness is demonstrated using two-dimensional test examples, when compared with solutions obtained by the proper orthogonal decomposition method, and by the full "FE-square" simulations. Extensions beyond the present implementations and the scope of tractable problems are discussed.
In this paper we address three aspects of nonlinear computational homogenization of elastic solids by two-scale finite element methods. First, we present a nonlinear formulation of the finite element heterogeneous multiscale method FE-HMM in a Lagrangean formulation that covers geometrical nonlinearity and, more general, hyperelasticity. Second, a-priori estimates of FE-HMM, which exist so far only for the fully linear elastic case in solid mechanics, are assessed in the regime of nonlinear elasticity. The measured convergence rates agree fairly well with those of the fully linear regime. Thir…
Fluid-flow devices with low dissipation, but high contact area, are of importance in many applications. A well-known strategy to design such devices is multi-scale topology optimization (MTO), where optimal microstructures are designed within each cell of a discretized domain. Unfortunately, MTO is computationally very expensive since one must perform homogenization of the evolving microstructures, during each step of the homogenization process. As an alternate, we propose here a graded multiscale topology optimization (GMTO) for designing fluid-flow devices. In the proposed method, several pr…
We investigate volume-element sampling strategies for the stochastic homogenization of particle-reinforced composites and show, via computational experiments, that an improper treatment of particles intersecting the boundary of the computational cell may affect the accuracy of the computed effective properties. Motivated by recent results on a superior convergence rate of the systematic error for periodized ensembles compared to taking snapshots of ensembles, we conduct computational experiments for microstructures with circular, spherical and cylindrical inclusions and monitor the systematic …
The paper presents two-scale numerical algorithms for stress-strain analysis of porous media featured by self-contact at pore level. The porosity is constituted as a periodic lattice generated by a representative cell consisting of elastic skeleton, rigid inclusion and a void pore. Unilateral frictionless contact is considered between opposing surfaces of the pore. For the homogenized model derived in our previous work, we justify incremental formulations and propose several variants of two-scale algorithms which commute iteratively solving of the micro- and the macro-level contact subproblems…
A cornerstone of computational solid mechanics in the context of digital transformation are databases for microstructures obtained from advanced tomography techniques. Uniform discretizations of pixelized images in 2D are the raw-data point of departure for simulation analyses. This paper proposes the concept of a unified error analysis for image-based microstructure representations in uniform resolution along with adaptively coarsened discretizations. The analysis distinguishes between a modeling error due to finite, possibly coarsened image resolution and a discretization error, investigates…
Efficient Fine-Scale Simulation of Nonlinear Hyperelastic Lattice Structures
Concace, Inria Center at the University of Bordeaux（孔萨塞，波尔多大学Inria中心）；Sorbonne Université, LIP6, CNRS（索邦大学，LIP6，法国国家科学研究中心）；Université Marie et Louis Pasteur, UTBM, CNRS, ICB UMR 6303（玛丽和路易·巴斯德大学，UTBM，法国国家科学研究中心，ICB UMR 6303）；Ecole Polytechnique Federale de Lausanne, Institute of Mathematics（洛桑联邦理工学院，数学研究所）；Univ. Toulouse, INSA-Toulouse, IMT-Albi, ISAE-SUPAERO, CNRS UMR 5312, ICA（图卢兹大学，图卢兹国立应用科学学院，IMT阿尔比，ISAE-SUPAERO，法国国家科学研究中心UMR 5312，ICA）；Univ. Toulouse, INSA-Toulouse, CNRS UMR 5219, IMT（图卢兹大学，图卢兹国立应用科学学院，法国国家科学研究中心UMR 5219，IMT）；Institut Universitaire de France (IUF)（法兰西学院）
The focus of our work is dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g.~functionally-graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in $\mathbb{R}^d$, $d\geqslant 1$ within the framework of multiple scales expansion. When either $d=1$ or $d=2$, this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbat…
For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration, and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-…
