Patrick Laborde

Bio: Patrick Laborde is an academic researcher from Institut de Mathématiques de Toulouse. The author has contributed to research in topics: Finite element method & Rate of convergence. The author has an hindex of 16, co-authored 38 publications receiving 1242 citations. Previous affiliations of Patrick Laborde include Paul Sabatier University & University of Bordeaux.

High-order extended finite element method for cracked domains

Abstract: The aim of the paper is to study the capabilities of the Extended Finite Element Method (XFEM) to achieve accurate computations in non smooth situations such as crack problems. Although the XFEM method ensures a weaker error than classical finite element methods, the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity. The difficulty can be overcome by modifying the enrichment of the finite element basis with the asymptotic crack tip displacement solutions as well as with the Heaviside function. Numerical simulations show that the modified XFEM method achieves an optimal rate of convergence (i.e. like in a standard finite element method for a smooth problem)

Crack tip enrichment in the XFEM using a cutoff function

Abstract: We consider a variant of the eXtended Finite Element Method (XFEM) in which a cutoff function is used to localize the singular enrichment surface. The goal of this variant is to obtain numerically an optimal convergence rate while reducing the computational cost of the classical XFEM with a fixed enrichment area. We give a mathematical result of quasi-optimal error estimate. One of the key points of this paper is to prove the optimality of the coupling between the singular and the discontinuous enrichments. Finally, we present some numerical computations validating the theoretical result. These computations are compared with those of the classical XFEM and a non-enriched method. Copyright © 2008 John Wiley & Sons, Ltd.

Mass redistribution method for finite element contact problems in elastodynamics

Abstract: This paper is devoted to a new method dealing with the semi-discretized finite element unilateral contact problem in elastodynamics. This problem is ill-posed mainly because the nodes on the contact surface have their own inertia. We introduce a method based on an equivalent redistribution of the mass matrix such that there is no inertia on the contact boundary. This leads to a mathematically well-posed and energy conserving problem. Finally, some numerical tests are presented.

Extension of the mortar finite element method to a variational inequality modeling unilateral contact

Abstract: The purpose of this paper is to extend the mortar finite element method to handle the unilateral contact model between two deformable bodies. The corresponding variational inequality is approximated using finite element meshes which do not fit on the contact zone. The mortar technique allows one to match these independent discretizations of each solid and takes into account the unilateral contact conditions in a convenient way. By using an adaptation of Falk's lemma and a bootstrap argument, we give an upper bound of the convergence rate similar to the one already obtained for compatible meshes.

Computational Contact Mechanics

Abstract: This paper describes modern techniques used to solve contact problems within Computational Mechanics. On the basis of a continuum description of contact, the mathematical structure of the contact formulation for finite element methods is derived. Emphasis is also placed on the constitutive behavior at the contact interface for normal and tangential (frictional) contact. Furthermore, different discretization schemes currently applied to solve engineering problems are formulated for small and finite strain problems. These include isoparametric interpolations, node-to-segment discretizations and also mortar and Nitsche techniques. Furthermore, several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed. Here, especially the penalty and Lagrange multiplier schemes are considered and also SQP- and linear-programming methods are reviewed. Keywords: contact mechanics; friction; penalty method; Lagrange multiplier method; contact algorithms; finite element method; finite deformations; discretization methods

The extended/generalized finite element method: An overview of the method and its applications

Abstract: An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non-smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi-field problems. Copyright © 2010 John Wiley & Sons, Ltd.

A review of extended/generalized finite element methods for material modeling

Abstract: The extended and generalized finite element methods are reviewed with an emphasis on their applications to problems in material science: (1) fracture, (2) dislocations, (3) grain boundaries and (4) phases interfaces. These methods facilitate the modeling of complicated geometries and the evolution of such geometries, particularly when combined with level set methods, as for example in the simulation growing cracks or moving phase interfaces. The state of the art for these problems is described along with the history of developments.