A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering

Microstructure is a feature of crystals with multiple symmetry-related energy-minimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetry-related states on a fine scale to minimize energy. This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.

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On the computation of crystalline microstructure

In recent years the FFT-based homogenization method of Moulinec and Suquet has been established as a fast, accurate and robust tool for obtaining effective properties in linear elasticity and conductivity problems. In this work we discuss FFT-based homogenization for elastic problems at large deformations, with a focus on the following improvements. Firstly, we exhibit the fixed point method introduced by Moulinec and Suquet for small deformations as a gradient descent method. Secondly, we propose a Newton---Krylov method for large deformations. We give an example for which this methods needs approximately 20 times less iterations than Newton's method using linear fixed point solvers and roughly $$100$$100 times less iterations than the nonlinear fixed point method. However, the Newton---Krylov method requires 4 times more storage than the nonlinear fixed point scheme. Exploiting the special structure we introduce a memory-efficient version with 40 % memory saving. Thirdly, we give an analytical solution for the micromechanical solution field of a two-phase isotropic St.Venant---Kirchhoff laminate. We use this solution for comparison and validation, but it is of independent interest. As an example for a microstructure relevant in engineering we discuss finally the application of the FFT-based method to glass fiber reinforced polymer structures.

Efficient fixed point and Newton---Krylov solvers for FFT-based homogenization of elasticity at large deformations

We derive a nonlinear partial differential equation for the convex envelope of a given function. The solution is interpreted as the value function of an optimal stochastic control problem. The equation is solved numerically using a convergent finite difference scheme.

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The convex envelope is the solution of a nonlinear obstacle problem

The direct numerical solution of a non-convex variational problem (P) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by numerical solution of the relaxed problem (RP) leading to a (degenerate) convex minimisation problem. The problem (RP) has a minimiser u and a related stress field σ = DW**(⊇u) which is known to coincide with the stress field obtained by solving (P) in a generalised sense involving Young measures. If u h is a finite element solution, σ h := DW**(⊇u h ) is the related discrete stress field. We prove a priori and a posteriori estimates for σ-σ h in L 4/3 (Ω) and weaker weighted estimates for ⊇u - ⊇u h . The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments.

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Numerical solution of the scalar double-well problem allowing microstructure

Free convection along a vertical flat plate embedded in a porous medium is considered, within the framework of boundary layer approximations. In some cases, similarity solutions can be obtained by solving a boundary value problem involving an autonomous third-order nonlinear equation, depending on a parameter related to the temperature on the wall. The paper deals with existence and uniqueness questions to this problem, for every value of the parameter.

On a family of differential equations for boundary layer approximations in porous media

The goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope. Such a problem arises in various physical situations where the function considered is some energy that has to be minimized.This study is a first step toward understanding how to approximate the quasi-convex envelope of a function. The importance of this issue is due to the various applications that are encountered, in particular, in the field of material science. ©1994 (Copyright) Society for Industrial and Applied Mathematics

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Approximated convex envelope of a function

http://www.math.uha.fr/ps/200501hoernel.pdf

On the concave and convex solutions of a mixed convection boundary layer approximation in a porous medium

The Blasius problem $f'''+ff''=0$ on $[0,+\infty[$, $f(0)=-a, f'(0)=b, f'(+\infty)=\l$ is exhaustively investigated. In particular the difficult and scarcely studied case $b<0\leq\l$ is analyzed in details, in which the shape and the number of solutions is determined. The method is first to reduce to the Crocco equation $u''=-\frac su$ and then to use an associated autonomous planar vector field. The most useful properties of Crocco solutions appear to be related to canard solutions of a slow fast vector field.

On the Blasius problem

We introduce blowing-up coordinates to study the autonomous third order nonlinear
differential equation : $f'''+\frac{m+1}{2}ff''-m f'^2=0$ on $(0,\infty)$, subject
to the boundary conditions $f(0)=a\in\mathbb R$, $f'(0)=1$ and $f'(t)\to 0$ as $t\to\infty$.
This problem arises when looking for similarity solutions to problems of
boundary-layer theory in some contexts of fluids mechanics, as free convection in porous
medium or flow adjacent to a stretching wall. We study the corresponding plane dynamical
systems and apply the results obtained to the original boundary value problem, in order to
solve questions for which direct approach fails.

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Bernard Brighi

Papers

On a family of differential equations for boundary layer approximations in porous media

Approximated convex envelope of a function

On the concave and convex solutions of a mixed convection boundary layer approximation in a porous medium

On the Blasius problem

Blowing-up coordinates for a similarity boundary layer equation