Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

© EDP Sciences, SMAI 2011This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in Rn (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := Rn\￣Ω. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincar´e-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart- Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory

A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces

International audienceWe investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the mini- mization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented La- grangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algo- rithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented

An adaptation of Nitsche's method to the Tresca friction problem

International audienceWe propose a simple adaptation to the Tresca friction case of the Nitsche-based finite element method given in [Chouly-Hild, 2012, Chouly-Hild-Renard, 2013] 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 H^1(Ω)-norm which is O(h^(1/2+nu)) when the solution lies in H^{3/2+nu}(Ω), 0 < ν ≤ 1/2, in two dimensions and three dimensions, for Lagrange piecewise linear and quadratic finite elements. No additional assumption on the friction set is needed to obtain this proof

Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

International audienceThe 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas

Evaluation of Accuracy and Efficiency of Numerical Methods for Contact Problems

Tato diplomová práce se zabývá variačními metodami, které umožňují formulovat problém kontaktu lineárně pružného tělesa bez tření jako nepodmíněnou variační rovnost, která může být posléze diskretizována a řešena metodou konečných prvků. Hlavní důraz je kladen na Nitscheho metody podle Wriggerse a Zavariseho [56] a podle Fabrého, Pousina a Renarda [15]. V současné době nejrozšířenější konečněprvkové softwarové balíky, jako jsou ANSYS, ABAQUS a COMSOL, využívají pro modelování kontaktu především standardní metody penalty a smíšené metody [57, Kapitola 1.1.1, p.7]. Ukazuje se, že právě Nitscheho metody mají potenciál překonat klasické obtíže spojené se standardními metodami penalty a smíšenými metodami. Na rozdíl od metod penalty jsou Nitscheho metody konzistentní a kontaktní okrajové podmínky jsou vynuceny přesně (na teoretické úrovni). Je také možné využít mnohem menší hodnotu parametru penalty, čímž se lze vyhnout problémům spojeným se špatným podmíněním úlohy, charakteristickým pro metody penalty. Nitscheho metoda ale současně nevyžaduje přidání žádných dalších neznámých (Lagrangeových multiplikátorů) a výsledný diskrétní systém tak není nadbytečně rozšířen, jako je tomu v případě smíšených metod. Oproti smíšeným metodám také není třeba věnovat pozornost splnění Babuškovy-Brezziho podmínky. V této diplomové práci se ukazuje, že analyzované Nitscheho metody úzce souvisejí s metodami penalty a metodou augmentovaného lagrangiánu. V práci jsou prezentovány slabé formulace těchto metod a zkoumají se rozdíly mezi formulací Nitscheho metody podle Wriggerse a podle Fabrého, Pousina a Renarda. Všechny metody jsou implementovány do prostředí FEniCS (výpočetní platforma pro řešení parciálních diferenciálních rovnic metodou konečných prvků) a jejich přesnost a výkonnost se testuje na různých dvourozměrných a trojrozměrných problémech kontaktu lineárně pružného tělesa s dokonale tuhou rovinou. Na jednoduchém dvourozměrném příkladu je ukázáno, že funkce, kterou získáme jako levou stranu diskretizované slabé formy Wriggersovy varianty Nitscheho metody, není spojitá vzhledem k neznámým stupňům volnosti. Tento poznatek vysvětluje problémy s konvergencí Newtonovy metody při řešení Wriggersovou variantou Nitscheho metodou, které jsme zaznamenali při numerických experimentech.This thesis is concerned with various methods that allow us to formulate the frictionless linear elastic contact problems as an unconstrained variational equality, which is then discretised and solved with the finite element method. The main focus is on Nitsche methods in the forms used respectively by Wriggers and Zavarise [56] and Fabré, Pousin and Renard [15]. Currently, standard penalty and mixed methods are dominant in the modern leading finite element software packages such as ANSYS, ABAQUS and COMSOL [57, Chapter 1.1.1, p.7]. Nitsche methods display a potential to overcome classic drawbacks of the penalty and mixed methods. Unlike penalty methods, Nitsche methods are consistent, and contact boundary conditions are enforced precisely (on the theoretical level). Also, a significantly smaller value of the penalty parameter is necessary and the possible ill-conditioning, so characteristic for penalty methods, is thus avoided. At the same time, no additional unknowns (Lagrange multipliers) are introduced; thus, the corresponding discrete system is not enlarged, and one does not have to worry about the Babuška-Brezzi condition. In this thesis was shown that the analysed Nitsche methods are closely related to penalty methods and the augmented Lagrangian method. The weak forms of all these methods are presented, and differences between Wriggers' version and Fabré, Pousin and Renard's version of Nitsche method are investigated. All methods are implemented in FEniCS (the computational platform for solving partial differential equations with the finite element method), and their accuracy and efficiency is tested on various two- and three-dimensional numerical examples of contact of an elastic body with a rigid plane. By means of the simple two-dimensional example it is shown that the function obtained as the left-hand side of the discretised weak form of the Nitsche-Wriggers method is not continuous with respect to the unknown displacement DOFs. This finding explains the convergence problems (of Newton's method) that the Nitsche-Wriggers method suffers from, unlike other investigated methods

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Efficient computation and applications of the Calderón projector

The boundary element method (BEM) is a numerical method for the solution of partial differential equations through the discretisation of associated boundary integral equations.BEM formulations are commonly derived from properties of the Calderón projector, a blocked operator containing four commonly used boundary integral operators. In this thesis, we look in detail at the Calderón projector, derive and analyse a novel use of it to impose a range of boundary conditions, and look at how it can be efficiently computed. Throughout, we present computations made using the open-source software library Bempp, many features of which have been developed as part of this PhD. We derive a method for weakly imposing boundary conditions on BEM, inspired by Nitsche’s method for finite element methods. Formulations for Laplace problems with Dirichlet, Neumann, Robin, and mixed boudary conditions are derived and analysed. For Robin and mixed boundary conditions, the resulting formulations are simpler than standard BEM formulations, and convergence at a similar rate to standard methods is observed. As a more advanced application of this method, we derive a BEM formulation for Laplace’s equation with Signorini contact conditions. Using the weak imposition framework allows us to naturally impose this more complex boundary condition; the ability to do this is a significant advantage of this work. These formulations are derived and analysed, and numerical results are presented. Using properties of the Calderón projector, methods of operator preconditioning for BEM can be derived. These formulations involve the product of boundary operators. We present the details of a discrete operator algebra that allows the easy calculation of these products on the discrete level. This operator algebra allows for the easy implementation of various formulations of Helmholtz and Maxwell problems, including regularised combined field formulations that are immune to ill-conditioning near eigenvalues that are an issue for other formulations. We conclude this thesis by looking at weakly imposing Dirichlet and mixed Dirichlet–Neumann boundary condition on the Helmholtz equation. The theory for Laplace problems is extended to apply to Helmholtz problems, and an application to wave scattering from multiple scatterers is presented