Macroscopically consistent non-local modelling of heterogeneous media

International audienceWithin the framework of the homogenization of heterogeneous media, a non local model is proposed. A field of non-local filtered stiffness tensor is introduced by filtering the solution to the homogenization problem. The filtered stiffness tensor, depending on the filter to heterogeneity size ratio, provides a continuous transition from the actual micro-scale heterogeneous stiffness field to the macro-scale homogenized stiffness tensor. For any intermediate filter size, the homogenization of the filtered stiffness yields exactly the homogenized stiffness, therefore it is called macroscopically consistent. The non-local stiffness tensor is intrinsically non symmetric, but its spatial fluctuations are smoothed, allowing for a less refined discretization in numerical methods. As a by-product, a two step heterogeneous multiscale method is proposed to reduce memory and computational time requirements of existing direct schemes while controlling the accuracy of the result. The first step is the estimation of the filtered stiffness at sampling points by means of an oversampling strategy to reduce boundary effects. The second step is the numerical homogenization of the obtained sampled filtered stiffness

On the effectiveness of the Moulinec–Suquet discretization for composite materials

Moulinec and Suquet introduced a method for computational homogenization based on the fast Fourier transform which turned out to be rather computationally efficient. The underlying discretization scheme was subsequently identified as an approach based on trigonometric polynomials, coupled to the trapezoidal rule to substitute full integration. For problems with smooth solutions, the power of spectral methods is well-known. However, for heterogeneous microstructures, there are jumps in the coefficients, and the solution fields are not smooth enough due to discontinuities across material interfaces. Previous convergence results only provided convergence of the discretization per se, that is, without explicit rates, and could not explain the effectiveness of the discretization observed in practice. In this work, we provide such explicit convergence rates for the local strain as well as the stress field and the effective stresses based on more refined techniques. More precisely, we consider a class of industrially relevant, discontinuous elastic moduli separated by sufficiently smooth interfaces and show rates which are known to be sharp from numerical experiments. The applied techniques are of independent interest, that is, we employ a local smoothing strategy, utilize Féjer means as well as Bernstein estimates and rely upon recently established superconvergence results for the effective elastic energy in the Galerkin setting. The presented results shed theoretical light on the effectiveness of the Moulinec–Suquet discretization in practice. Indeed, the obtained convergence rates coincide with those obtained for voxel finite element methods, which typically require higher computational effort

Accounting for weak interfaces in computing the effective crack energy of heterogeneous materials using the composite voxel technique

We establish a computational methodology to incorporate interfaces with lower crack energy than the surrounding phases when computing the effective crack energy of brittle composite materials. Recent homogenization results for free discontinuity problems are directly applicable to the time-discretized Francfort-Marigo model of brittle fracture in the anti-plane shear case, and computational tools were introduced to evaluate the effective crack energy on complex microstructures using FFT-based solvers and a discretization scheme based on a combinatorially consistent grid. However, this approach only accounts for the crack resistance per volume and is insensitive to the crack resistance of the interface which is expected to play a significant role by considerations from materials science. In this work we introduce a remedy exploiting laminate composite voxels. The latter were originally introduced to enhance the accuracy of solutions for elasticity problems on regular voxel grids. We propose an accurate approximation of the effective crack energy of a laminate with weak interface where an explicit solution is available. We incorporate this insight into an efficient algorithmic framework. Finally, we demonstrate the capabilities of our approach on complex microstructures with weak interfaces between different constituents

Lamination-based efficient treatment of weak discontinuities for non-conforming finite element meshes

When modelling discontinuities (interfaces) using the finite element method, the standard approach is to use a conforming finite-element mesh in which the mesh matches the interfaces. However, this approach can prove cumbersome if the geometry is complex, in particular in 3D. In this work, we develop an efficient technique for a non-conforming finite-element treatment of weak discontinuities by using laminated microstructures. The approach is inspired by the so-called composite voxel technique that has been developed for FFT-based spectral solvers in computational homogenization. The idea behind the method is rather simple. Each finite element that is cut by an interface is treated as a simple laminate with the volume fraction of the phases and the lamination orientation determined in terms of the actual geometrical arrangement of the interface within the element. The approach is illustrated by several computational examples relevant to the micromechanics of heterogeneous materials. Elastic and elastic-plastic materials at small and finite strain are considered in the examples. The performance of the proposed method is compared to two alternative, simple methods showing that the new approach is in most cases superior to them while maintaining the simplicity

Stress-strain analysis of aortic aneurysms

Tato práce se zabývá problematikou aneurysmat břišní aorty a možností využít konečnoprvkovou deformačně-napěťovou analýzu těchto aneurysmat ke stanovení rizika ruptury. První část práce je věnována úvodu do problematiky, popisu kardiovaskulární soustavy člověka s důrazem na abdominální aortu, anatomii, fyziologii a patologii stěny tepny s důrazem na procesy vedoucí ke vzniku aneurysmatu. Dále se práce věnuje rizikovým faktorům přispívajících ke vzniku aneurysmat spolu s analýzou současných klinických postupů ke stanovení rizika ruptury spolu se srovnáním navrhovaného kritéria maximálního napětí. Dominantní část této disertace je věnována identifikaci faktorů ovlivňujících napjatost a deformaci stěny aneurysmatu spolu s návrhem nových postupů, prezentací vlastních poznatků vedoucích ke zpřesnění určení rizika ruptury pomocí deformačně- napěťové analýzy a metody konečných prvků. Nejprve je analyzován vliv geometrie, vedoucí k závěru, že je nezbytné používání individuálních geometrií pacienta. Dále je pozornost zaměřena na odbočující tepny, které ve stěně působí jako koncentrátor napětí a mohou tedy ovlivňovat napjatost v ní. Jako další podstatný faktor byl identifikován vliv nezatížené geometrie a bylo napsáno makro pro její nalezení, které bylo opět zahrnuto jako standardní součást do výpočtového modelu. Mechanické vlastnosti jak stěny aneurysmatu, tak intraluminálního trombu jsou experimentálně testovány pomocí dvouosých zkoušek. Také je zde analyzován vliv modelu materiálu, kde je ukázáno, že srovnávání maximálních napětí u jednotlivých modelů materiálu není vhodné díky zcela rozdílným gradientům napětí ve stěně aneurysmatu. Dále je zdůrazněna potřeba znalosti distribuce kolagenních vláken ve stěně a navržen program k jejímu získání. Intraluminální trombus je analyzován ve dvou souvislostech. Jednak je ukázán vliv jeho ruptury na napětí ve stěně a jednak je analyzován vliv jeho poroelastické struktury na totéž. Posledním identifikovaným podstatným faktorem je zbytková napjatost ve stěně. Její významnost je demonstrována na několika aneurysmatech a i tato je zahrnuta jako integrální součást do našeho výpočtového modelu.Na závěr jsou pak navrženy další možné směry výzkumu.This thesis deals with abdominal aortic aneurysms and the possibility of using finite element method in assessment of their rupture risk. First part of the thesis is dedicated to an introduction into the problem, description of human cardiovascular system where the abdominal aorta, its anatomy, physiology and pathology is emphasized. There Processes leading to formationing of abdominal aortic aneurysms are also discussed. Risk factors contributing to creation of aneurysms are discussed next. Finally, an analysis of current clinical criteria which determine rupture risk of an abdominal aortic aneurysm is presented and compared with the new maximum stress criterion being currently in development. Main part of the thesis deals with the identification of relevant factors which affect stress and deformation of aneurysmal wall. This is connected with proposals of new approaches leading to predicting the rupture risk more accurately by using finite element stress-strain analysis. The impact of geometry is analyzed first with the conclusion that patient-specific geometry is a crucial input in the computational model. Therefore its routine reconstruction has been managed. Attention is then paid to the branching arteries which were neglected so far although they cause a stress concentration in arterial wall. The necessity of knowing the unloaded geometry of aneurysm is then emphasized. Therefore a macro has been written in order to be able to find the unloaded geometry for any patient-specific geometry of aneurysm. Mechanical properties of both aneurysmal wall and intraluminal thrombus were also experimentally tested and their results were fitted by an isotropic material model. The effect of the material model itself has been also investigated by comparing whole stress fields of several aneurysms. It has been shown that different models predict completely different stresses due to different stress gradients in the aneurysmal wall. The necessity of known collagen fiber distribution in arterial wall is also emphasized. A special program is then presented enabling us to obtain this information. Effect of intraluminal thrombus on the computed wall stress is analyzed in two perspectives. First the effect of its failure on wall stress is shown and also the impact of its poroelastic structure is analyzed. Finally the residual stresses were identified as an important factor influencing the computed wall stress in aneurysmal wall and they were included into patient-specific finite element analysis of aneurysms. Further possible regions of investigation are mentioned as the last part of the thesis.

A computational multi-scale approach for brittle materials

Materials of industrial interest often show a complex microstructure which directly influences their macroscopic material behavior. For simulations on the component scale, multi-scale methods may exploit this microstructural information. This work is devoted to a multi-scale approach for brittle materials. Based on a homogenization result for free discontinuity problems, we present FFT-based methods to compute the effective crack energy of heterogeneous materials with complex microstructures

A review of nonlinear FFT-based computational homogenization methods

Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform