Computational strategies toward the modelling of the intervertebral disc

Lumbar back pain has considerable socio-economical impacts, motivating a recently increasing interest from the research community. Yet, mechanisms triggering pain are not fully understood and this considerably hinders the development of efficient treatments and therapies. The objective of this thesis is to participate to the general understanding of the biomechanics of the spine through the development of computational strategies for the intervertebral disc. The intervertebral disc is a complex structure mainly comprised of the nucleus pulposus and the annulus fibrosus. The nucleus pulposus is the gelatinous core of the disc, which consists of a charged and hydrated extra-cellular matrix and an ionised interstitial fluid. It is enclosed in the annulus fibrosus which is formed by concentric layers of aligned collagen fibre sheets, oriented in an alternating fashion. A biphasic swelling model has been derived using mixture theory for soft, hydrated and charged tissues in order to capture the salient characteristics of the disc's behaviour. The model fully couples the solid matrix under finite deformations with the ionised interstitial fluid. The nucleus is assumed to behave isotropically while the effects of the collagen fibres in the annulus fibrosus are accounted for with a transversely isotropic model. The fixed negative charges of the proteoglycans, which induce an osmotic pressure responsible for the swelling capabilities of the disc, are constitutively modelled under the simplifying Lanir hypothesis. A Newton-Raphson solver was specifically built to solve the resulting nonlinear system of equations, together with a verification procedure to ensure successful implementation of the code. This was first reduced to the one dimensional case in order to demonstrate the appropriateness of the biphasic swelling model. The three dimensional model exhibited numerical instabilities, manifesting in the form of non-physical oscillations in the pressure field near boundaries, when loads and free-draining boundary conditions are simultaneously applied. As an alternative to considerable mesh refinement, these spurious instabilities have been addressed using a Galerkin Least-Square formulation, which has been extended for finite deformations. The performance and limitations of the GLS framework, which drastically reduces the pressure discrepancies and prevents the oscillations from propagating through the continuum, are demonstrated on numerical examples. Finally, the current state of the model's development is assessed, and recommendations for further improvements are proposed

Stabilized MorteX method for mesh tying along embedded interfaces

We present a unified framework to tie overlapping meshes in solid mechanics applications. This framework is a combination of the X-FEM method and the mortar method, which uses Lagrange multipliers to fulfill the tying constraints. As known, mixed formulations are prone to mesh locking which manifests itself by the emergence of spurious oscillations in the vicinity of the tying interface. To overcome this inherent difficulty, we suggest a new coarse-grained interpolation of Lagrange multipliers. This technique consists in selective assignment of Lagrange multipliers on nodes of the mortar side and in non-local interpolation of the associated traction field. The optimal choice of the coarse-graining spacing is guided solely by the mesh-density contrast between the mesh of the mortar side and the number of blending elements of the host mesh. The method is tested on two patch tests (compression and bending) for different interpolations and element types as well as for different material and mesh contrasts. The optimal mesh convergence and removal of spurious oscillations is also demonstrated on the Eshelby inclusion problem for high contrasts of inclusion/matrix materials. Few additional examples confirm the performance of the elaborated framework

Powell–Sabin B-splines for smeared and discrete approaches to fracture in quasi-brittle materials

AbstractNon-Uniform Rational B-splines (NURBS) and T-splines can have some drawbacks when modelling damage and fracture. The use of Powell–Sabin B-splines, which are based on triangles, can by-pass these drawbacks. Herein, smeared as well as discrete approaches to fracture in quasi-brittle materials using Powell–Sabin B-splines are considered.For the smeared formulation, an implicit fourth-order gradient damage model is adopted. Since quadratic Powell–Sabin B-splines employ C1-continuous basis functions throughout the domain, they are well-suited for solving the fourth order partial differential equation that emerges in this higher order damage model. Moreover, they can be generated from an arbitrary triangulation without user intervention. Since Powell–Sabin B-splines are generated from a classical triangulation, they are not necessarily boundary-fitting and in that case they are not isogeometric in the strict sense.For discrete fracture approaches, the degree of continuity of T-splines is reduced to C0 at the crack tip. Hence, stresses need to be evaluated and weighted at the integration points in the vicinity of the crack tip in order to decide when the critical stress is reached. In practice, stress fields are highly irregular around crack tips. Furthermore, aligning a T-spline mesh with the new crack segment can be difficult. Powell–Sabin B-splines also remedy these drawbacks as they are C1-continuous at the crack tip and stresses can be directly computed, which vastly increases the accuracy and simplifies the implementation. Moreover, re-meshing is more straightforward using Powell–Sabin B-splines. A current limitation is that, in three dimensions, there is no procedure (yet) for constructing Powell–Sabin B-splines on arbitrary tetrahedral meshes

Phase-field models for brittle and cohesive fracture

In this paper we first recapitulate some basic notions of brittle and cohesive fracture models, as well as the phase-field approximation to fracture. Next, a critical assessment is made of the sensitivity of the phase-field approach to brittle fracture, in particular the degradation function, and the use of monolithic versus partitioned solution schemes. The last part of the paper makes extensions to a recently developed phase-field model for cohesive fracture, in particular for propagating cracks. Using some simple examples the current state of the cohesive phase-field model is shown

Méthode MorteX pour des problèmes du couplage des maillages, du contact et de l'usure

International audienceUne nouvelle méthode MorteX est proposée ; elle combine la méthode mortar pour des interfaces non-conformes et la méthode X-FEM afin d’assurer la continuité de contraintes à travers d’une interface entre une surface conforme au maillage et une autre placée au sein d’un autre maillage. Afin d’éviter la surcontrainte des interfaces et des oscillations émergeant dans le cas de forts contrastes de la densité des maillages et de leur raideurs, une méthode de régularisation a été proposée. La méthode permet d’assembler des maillages recouvrant et traiter le problème du contact frottant et de l’usure

MorteX method for contact along real and embedded surfaces: coupling X-FEM with the Mortar method

A method to treat frictional contact problems along embedded surfaces in the finite element framework is developed. Arbitrarily shaped embedded surfaces, cutting through finite element meshes, are handled by the X-FEM. The fric-tional contact problem is solved using the monolithic augmented Lagrangian method within the mortar framework which was adapted for handling embedded surfaces. We report that the resulting mixed formulation is prone to mesh locking in case of high elastic and mesh density contrasts across the contact interface. The mesh locking manifests itself in spurious stress oscillations in the vicinity of the contact interface. We demonstrate that in the classical patch test, these oscillations can be removed simply by using triangular blending elements. In a more general case, the triangulation is shown inefficient, therefore stabilization of the problem is achieved by adopting a recently proposed coarse-graining interpolation of Lagrange multipliers. Moreover, we demonstrate that the coarse-graining is also beneficial for the classical mortar method to avoid spurious oscillations for contact interfaces with high elastic contrast. The performance of this novel method, called MorteX, is demonstrated on several examples which show as accurate treatment of frictional contact along embedded surfaces as the classical mortar method along boundary fitted surfaces

Domain tying across virtual interfaces : coupling X-FEM with the Mortar method

International audienceIn this paper we propose a unified framework of the mortar domain decomposition method and extended finite element method (X-FEM). This framework allows to deal in an efficient manner with two cumbersome aspects of the finite element methods, namely incompatible interface discretizations and internal discontinuities. Features of mortar methods in the context of mesh tying, and of X-FEM in the context of void/inclusion treatment are exploited to formulate the weak coupling along an inclusion's surface and the virtual surface of the host mesh. It has a potential to address a multitude of problems from accurate substructuring to efficient wear simulation in contact problems

Computational framework for monolithic coupling for thin fluid flow in contact interfaces

We developed a computational framework for simulating thin fluid flow in narrow interfaces between contacting solids, which is relevant for a range of engineering, biological and geophysical applications. The treatment of this problem requires coupling between fluid and solid mechanics equations, further complicated by contact constraints and potentially complex geometrical features of contacting surfaces. We developed a monolithic finite-element framework for handling mechanical contact, thin incompressible viscous flow and fluid-induced tractions on the surface of the solid, suitable for both one- and two-way coupling approaches. Additionally, we consider the possibility of fluid entrapment in ”pools” delimited by contact patches and its pressurization following a non-linear compressibility constitutive law. Image analysis algorithms were adapted to identify the local status of each interface element (i.e. distinguish between contact, fluid flow and trapped fluid zones) within the Newton–Raphson loop. First, an application of the proposed framework for a problem with a model geometry is given, and the robustness is demonstrated by the residual-wise and status-wise convergence. The full capability of the developed two-way coupling framework is demonstrated on a problem of a fluid flow in contact interface between a solid with representative rough surface and a rigid flat. The evolution of the contact pressure, fluid flow pattern and the morphology of trapped fluid zones under increasing external load until the complete sealing of the interface is displayed. Additionally, we demonstrated an almost mesh-independent result of a refined post-processing approach to the real contact-area computation. The developed framework permits not only to study the evolution of effective properties of contact interfaces, such as transmissivity and real contact area, but also to highlight the difference between one- and two-way coupling approaches, and, in particular, to quantify the effect of trapped fluid ”pools” on the coupled problem

Résistance thermique et électrique des interfaces de contact entre des surfaces rugueuses

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