We summarize recent achievements in applying Nitsche's method to some contact and friction problems. We recall the setting of Nitsche's method in the case of unilateral contact with Tresca friction in linear elasticity. Main results of the numerical analysis are detailed: consistency, well-posedness, fully optimal convergence in H 1 (Ω)-norm, residual-based a posteriori error estimation. Some numerics and some recent extensions to multibody contact, contact in large transformations and contact in elastodynamics are presented as well.
Abstract. In this paper, we develop and analyze a finite element fictitious domain approach based on Nitsche's method for the approximation of frictionless contact problems of two deformable elastic bodies. In the proposed method, the geometry of the bodies and the boundary conditions, including the contact condition between the two bodies, are described independently of the mesh of the fictitious domain. We prove that the optimal convergence is preserved. Numerical experiments are provided which confirm the correct behavior of the proposed method.Math. classification. 65N85, 35M85, 74M15.
In this paper, we consider unilateral contact problem without friction between a rigid body and deformable one in the framework of isogeometric analysis. We present the theoretical analysis of the mixed problem. For the displacement, we use the pushforward of a NURBS space of degree p and for the Lagrange multiplier, the pushforward of a B-Spline space of degree p − 2. These chooses of space ensure to prove an inf − sup condition and so on, the stability of the method. An active set strategy is used in order to avoid of geometrical hypothesis of the contact set. An optimal a priori error estimate is demonstrated without assumption on the unknown contact set. Several numerical examples in two-and three-dimensions and in small and large deformation demonstrate the accuracy of the proposed method. * EPFL SB MATHICSE MNS (Bât. MA), station 8, CH 1015 Lausanne (Switzerland).
We introduce a residual-based a posteriori error estimator for contact problems in two and three dimensional linear elasticity, discretized with linear and quadratic finite elements and Nitsche's method. Efficiency and reliability of the estimator are proved under a saturation assumption. Numerical experiments illustrate the theoretical properties and the good performance of the estimator.
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