A general Nitsche method, which encompasses symmetric and non-symmetric variants, is proposed for frictionless unilateral contact problems in elasticity. The optimal convergence of the method is ...established both for two- and three-dimensional problems and Lagrange affine and quadratic finite element methods. Two- and three-dimensional numerical experiments illustrate the theory.
A simple skew-symmetric Nitsche’s formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions, ...symmetry conditions for Kirchhoff plates, patch coupling in statics and in modal analysis as well as Signorini contact conditions. For linear boundary or interface conditions, the skew-symmetric formulation is parameter-free. For contact conditions, it remains stable and accurate for a wide range of the stabilization parameter. Several numerical tests are performed to illustrate its accuracy, stability and convergence performance. We investigate particularly the effects introduced by Nitsche’s coupling, including the convergence performance and condition numbers in statics as well as the extra “outlier” frequencies and corresponding eigenmodes in structural dynamics. We present the Hertz test, the block test, and a 3D self-contact example showing that the skew-symmetric Nitsche’s formulation is a suitable approach to simulate contact problems in IGA.
•A simple skew-symmetric Nitsche’s formulation is introduced into IGA.•For linear boundary and interface conditions, the formulation is parameter-free.•For contact conditions, the formulation is stable for the stabilization parameter.•Robustness and accuracy of the method is studied numerically for various problems.
We present a convergence analysis of the penalty method applied to unilateral contact problems in two and three space dimensions. We first consider, under various regularity assumptions on the exact ...solution to the unilateral contact problem, the convergence of the continuous penalty solution as the penalty parameter ε vanishes. Then, the analysis of the finite element discretized penalty method is carried out. Denoting by h the discretization parameter, we show that the error terms we consider give the same estimates as in the case of the constrained problem when the penalty parameter is such that ε=h. We finally extend the results to the case where given (Tresca) friction is taken into account.
We propose a simple adaptation to the Tresca friction case of the Nitsche-based finite element method introduced previously for frictionless unilateral contact. Both cases of unilateral and bilateral ...contact with friction are taken into account, with emphasis on frictional unilateral contact for the numerical analysis. We manage to prove theoretically the fully optimal convergence rate of the method in the H1(Ω)-norm which is O(h12+ν) when the solution lies in H32+ν(Ω), 0<ν⩽k−1/2, in two dimensions and three dimensions, for Lagrange piecewise linear (k=1) and quadratic (k=2) finite elements. No additional assumption on the friction set is needed to obtain this proof.
The aim of the present paper is to study theoretically and numerically the Verlet scheme for the explicit time-integration of elastodynamic problems with a contact condition approximated by Nitsche’s ...method. This is a continuation of papers (Chouly et al. ESAIM Math Model Numer Anal 49(2), 481–502,
2015
; Chouly et al. ESAIM Math Model Numer Anal 49(2), 503–528,
2015
) where some implicit schemes (theta-scheme, Newmark and a new hybrid scheme) were proposed and proved to be well-posed and stable under appropriate conditions. A theoretical study of stability is carried out and then illustrated with both numerical experiments and numerical comparison to other existing discretizations of contact problems.
We introduce a Nitsche-based finite element discretization of the unilateral contact problem in linear elasticity. It features a weak treatment of the nonlinear contact conditions through a ...consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in the H¹ (Ω)-norm for linear finite elements in two dimensions, which is $O({h^{\frac{1}{2} + v}})$ when the solution lies in ${H^{\frac{3}{2} + v}}(\Omega )$ , 0 < v ≤ 1/2. An interest of the formulation is that, as opposed to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf-sup condition needs to be satisfied.
This study is concerned with the elastoplastic torsion problem and its standard finite element approximation using piecewise affine Lagrange finite elements. In the case of a polytopal convex domain ...in dimension n=1,2,3 we obtain an H1-error bound of order h for the solution. For a nonconvex domain, we obtain also an error estimate.
We consider frictional contact problems in small strain elasticity discretized with finite elements and Nitsche method. Both bilateral and unilateral contact problems are taken into account, as well ...as both Tresca and Coulomb friction models. We derive residual a posteriori error estimates for each friction model, following (Chouly et al., in IMA J Numer Anal 38: 921–954, 2018). For the incomplete variant of Nitsche, we prove an upper bound for the dual norm of the residual, for Tresca and Coulomb friction, without any extra regularity and without a saturation assumption. We prove also local lower bounds. Numerical experiments allow to assess the accuracy of the estimates and their interest for adaptive meshing in different situations.
This study is concerned with the elastoplastic torsion problem and its standard finite element approximation using piecewise affine Lagrange finite elements. In the case of a polytopal convex domain ...in dimension n = 1,2,3 we obtain an H-1-error bound of order h for the solution. For a nonconvex domain, we obtain also an error estimate.
This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we apply ...appropriate artificial boundary conditions that yield an equivalent problem, but with bounded subdomains, and we carry out the stability analysis for this coupled problem in truncated domains. Then we devise an optimized Schwarz-in-time (or Schwarz Waveform Relaxation) method for the numerical solving of the coupled equations. Particular emphasis is made on the design of optimized transmission conditions. Notably, for this setting, the optimal transmission conditions can be expressed analytically in a very simple manner. This result is illustrated by some numerical experiments.
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