Numerical Model Reduction with error estimation for computational homogenization of non-linear consolidation

Fredrik Ekre,Fredrik Larsson,Kenneth Runesson,Ralf Jänicke

doi:10.1016/j.cma.2021.114334

Abstract

Numerical Model Reduction (NMR) is adopted for solving the non-linear microscale problem that arises from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. A reduced basis is obtained for the pressure field using Proper Orthogonal Decomposition and the pertinent displacement basis is obtained using Nonuniform Transformation Field Analysis. An explicit, fully computable, a posteriori error estimator is derived based on the linearized error equation for quantification of the NMR error in terms of a suitably chosen energy norm. The performance of the error estimates is demonstrated via a set of numerical examples with varying load amplitudes.

Highlights

Computational homogenization using the so-called “Finite Element squared” (FE2) procedure, cf. Feyel et al [1], is a known approach to multiscale modeling where the constitutive relation is replaced with subscale computations carried out on Representative Volume Elements (RVE)
Computer Methods in Applied Mechanics and Engineering 389 (2022) 114334 homogenization” technique introduced by Fish and coworkers [4,5], which relies on the concept of Transformation Field Analysis (TFA) [6]
Janicke et al [13] applied this approach to poroelasticity, whereby the pore pressure acts similar to inelastic strains in the Nonuniform Transformation Field Analysis (NTFA) framework

Summary

Introduction

Computational homogenization using the so-called “Finite Element squared” (FE2) procedure, cf. Feyel et al [1], is a known approach to multiscale modeling where the constitutive relation is replaced with subscale computations carried out on Representative Volume Elements (RVE). The main advantage of FE2, compared to a fully resolved solution, is the reduced computational cost, while still taking small scale processes or structures into account. It is of interest to investigate methods to reduce the computational cost of solving the individual RVE problems. A number of Numerical Model Reduction (NMR) methods have been proposed for reducing the solution space of a discrete RVE problem. Waseem et al [2] and Aggestam et al [3] presented reduced models for computational homogenization of linear transient heat flow based on Spectral Decomposition (SD).

Objectives

Methods

Conclusion