Computational Homogenization of Heterogeneous Materials with Finite Elements

Serie: Solid Mechanics and Its Applications 258

This monograph provides a concise overview of the main theoretical and numerical tools to solve homogenization problems in solids with finite elements. Starting from simple cases (linear thermal case) the problems are progressively complexified to finish with nonlinear problems. Les mer
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This monograph provides a concise overview of the main theoretical and numerical tools to solve homogenization problems in solids with finite elements. Starting from simple cases (linear thermal case) the problems are progressively complexified to finish with nonlinear problems. The book is not an overview of current research in that field, but a course book, and summarizes established knowledge in this area such that students or researchers who would like to start working on this subject will acquire the basics without any preliminary knowledge about homogenization. More specifically, the book is written with the objective of practical implementation of the methodologies in simple programs such as Matlab. The presentation is kept at a level where no deep mathematics are required.

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1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Why computational homogenization? . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Brief historical and recent advances . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Industrial applications and use in commercial softwares . . . . . . . . . . 31.4 Position of the present monograph as compared to available otherbooks on that topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Overview and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Review of classical FEM formulations and discretizations . . . . . . . . . . . 52.1 Steady-state thermal problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Strong form of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Weak forms of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 2D FEM discretization with linear elements . . . . . . . . . . . . . . 72.1.4 Assembly of the elementary systems . . . . . . . . . . . . . . . . . . . . 122.1.5 Prescribing Dirichlet conditions . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Strong form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 2D discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.4 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Conduction properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 The notion of RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Localization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Averaged quantities and Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . 313.3.1 Averaging theorem: temperature gradient . . . . . . . . . . . . . . . . 313.3.2 Averaging theorem: heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.3 Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Computation of the effective conductivity tensor . . . . . . . . . . . . . . . . 333.4.1 The superposition principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.2 Definition of the effective conductivity tensor . . . . . . . . . . . . 34vvi Contents3.5 Periodic boundary conditions for the thermal problem: numericalimplementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Numerical calculation of effective conductivity with 2D FEM . . . . . 393.6.1 Transverse effective conductivity . . . . . . . . . . . . . . . . . . . . . . . 393.6.2 Computation of the out-of plane properties using a 2D RVE 413.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Elasticity and thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1 Localization problem for elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Averaged quantities and Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . 504.2.1 Averaging theorem: strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.2 Averaging theorem : stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Definition of the effective elastic tensor . . . . . . . . . . . . . . . . . . . . . . . . 524.3.1 Strain approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.2 Stress approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4 Computations of the effective properties with FEM . . . . . . . . . . . . . . 554.4.1 2D case: transverse effective properties . . . . . . . . . . . . . . . . . . 554.4.2 Computation of out-of-plane elastic properties using a 2DRVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4.3 Full 3D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.5 Periodic boundary conditions for 2D elastic problem: practicalimplementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.6 Extension to thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 Porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757 Second-order linear homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 Filter-based homogenization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 Nonlinear Computational Homogenization . . . . . . . . . . . . . . . . . . . . . . . . 81References