arXiv subjects

Pierre Suquet

Publications and source records attributed to Pierre Suquet.

4 recordsLinked to original sources

arXiv2020

A numerical method for computing the overall response of nonlinear composites with complex microstructure

The local and overall responses of nonlinear composites are classically investigated by the Finite Element Method. We propose an alternate method based on Fourier series which avoids meshing and which makes direct use of microstructure images. It is based on the exact expression of the Green function of a linear elastic and homogeneous comparison material. First, the case of elastic nonhomogeneous constituents is considered and an iterative procedure is proposed to solve the Lippman-Schwinger equation which naturally arises in the problem. Then, the method is extended to non-linear constituents by a step-by-step integration in time. The accuracy of the method is assessed by varying the spatial resolution of the microstructures. The flexibility of the method allows it to serve for a large variety of microstructures. (C) 1998 Elsevier Science S.A.

cs.CE↗

arXiv2019

Numerical simulation of model problems in plasticity based on field dislocation mechanics

The aim of this paper is to investigate the numerical implementation of the Field Dislocation Mechanics (FDM) theory for the simulation of dislocation-mediated plasticity. First, the mesoscale FDM theory of Acharya and Roy (2006) is recalled which permits to express the set of equations under the form of a static problem, corresponding to the determination of the local stress field for a given dislocation density distribution, complemented by an evolution problem, corresponding to the transport of the dislocation density. The static problem is solved using FFT-based techniques (Brenner et al., 2014). The main contribution of the present study is an efficient numerical scheme based on high resolution Godunov-type solvers to solve the evolution problem. Model problems of dislocation-mediated plasticity are finally considered in a simplified layer case. First, uncoupled problems with uniform velocity are considered, which permits to reproduce annihilation of dislocations and expansion of dislocation loops. Then, the FDM theory is applied to several problems of dislocation microstructures subjected to a mechanical loading.

cs.CE↗

arXiv2018

Geometric variational principles for computational homogenization

The homogenization of periodic elastic composites is addressed through the reformulation of the local equations of the mechanical problem in a geometric functional setting. This relies on the definition of Hilbert spaces of kinematically and statically admissible tensor fields, whose orthogonality and duality properties are recalled. These are endowed with specific energetic scalar products that make use of a reference and uniform elasticity tensor. The corresponding strain and stress Green's operators are introduced and interpreted as orthogonal projection operators in the admissibility spaces. In this context and as an alternative to classical minimum energy principles, two geometric variational principles are investigated with the introduction of functionals that aim at measuring the discrepancy of arbitrary test fields to the kinematic, static or material admissibility conditions of the problem. By relaxing the corresponding local equations, this study aims in particular at laying the groundwork for the homogenization of composites whose constitutive properties are only partially known or uncertain. The local fields in the composite and their macroscopic responses are computed through the minimization of the proposed geometric functionals. To do so, their gradients are computed using the Green's operators and gradient-based optimization schemes are discussed. A FFT-based implementation of these schemes is proposed and they are assessed numerically on a canonical example for which analytical solutions are available.

math.NA↗

arXiv2017

Convergence of iterative methods based on Neumann series for composite materials: theory and practice

Iterative Fast Fourier Transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential constraints are satisfied in Fourier space, and the constitutive law in real space. The methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli. It is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative Fast Fourier Transform methods. We look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity (Obnosov). Theoretically some of the methods converge when the inclusions have zero or even negative conductivity. However, the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations. There is little point in iterating beyond this. Accuracy improves when the grid size is reduced, showing that the discrepancy is linked to the discretization. Finally, it is shown that none of the three iterative schemes investigated over-performs the others for all possible microstructures and all contrasts.

math.NA↗