An unbiased Nitsche's approximation of the frictional contact between two elastic structures

International audienceMost of the numerical methods dedicated to the contact problem involving two elastic bodies are based on the master/slave paradigm. It results in important detection difficulties in the case of self-contact and multi-body contact, where it may be impractical, if not impossible , to a priori nominate a master surface and a slave one. In this work we introduce an unbiased finite element method for the finite element approximation of frictional contact between two elastic bodies in the small deformation framework. In the proposed method the two bodies expected to come into contact are treated in the same way (no master and slave surfaces). The key ingredient is a Nitsche-based formulation of contact conditions. We carry out the numerical analysis of the method, and prove its well-posedness and optimal convergence in the H 1-norm. Numerical experiments are performed to illustrate the theoretical results and the performance of the method

Introduction of a segment-to-segment penalty contact formulation

The modeling of contact problems in solid mechanics using the finite element method is a challenging and complicated task. Stable, efficient and accurate algorithms are required for finding an effective solution for general contact problems. This thesis introduces a transition from a node-to-segment penalty contact formulation to an effective segment-to-segment penalty contact formulation. The main issues of the node-to-segment approach are convergence problems and inaccurate results in case of nonconforming meshes. These disadvantages are caused by the fact that the contact constraints are satisfied only at the nodal locations. The relatively new segment-to-segment formulations provide a way to apply the constraint conditions along the entire boundary in a weak integral sense. This usually results in better stability and accuracy. Several different discretization schemes for the proposed segment-to-segment formulation are presented. Moreover, additional complexities caused by the new formulation are discussed and the solutions for these problems are introduced. The different discretization schemes are compared by simple examples as well as big and complicated real models. Additionally, the widely accepted numerical benchmark test known as “patch test” is conducted to compare the results given by the different contact implementations. The results obtained by the comparisons support the expected outcome of the segment-to-segment approach. The advantages of the segment-to-segment formulations become clear and a much better contact algorithm is introduced into an open source finite element analysis program

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described