Contact Problems In Linear Elasticity Research Articles

Rapid Methods for the Resolution of Contact Problems in Static Linear Elasticity

In this paper, the two‐dimensional Signorini static contact problem in linear elasticity is presented. We present the weak formulation of the frictional contact problems, and the boundary integral operators are used to propose a boundary variational formulation whose resolution by the generalized Newton method is presented. Moreover, a particular formulation by the fixed point method associated with the augmented Lagrangian is proposed for efficient analysis of contact problems with Coulomb friction, and powerful algorithms are constructed. The discretization is carried over by using the Galerkin method. The resulting linear system is solved by using a preconditioned conjugate gradient (CG) iterative solver.

A physics‐based model reduction approach for node‐to‐segment contact problems in linear elasticity

AbstractThe article presents a novel model order reduction method for mechanical problems in linear elasticity with nonlinear contact conditions. Recently, we have proposed an efficient reduction scheme for the node‐to‐node formulation [Manvelyan et al., Comput Mech 68, 1283–1295 (2021)] that leads to linear complementarity problems (LCPs). Here, we enhance the underlying contact problem to a node‐to‐segment formulation, which leads to quadratic inequalities as constraints. The adjoint system corresponds to a nonlinear complementarity problem that describes the Lagrange multiplier. The latter is solved by a Newton‐type iteration based on a LCP solver in each time step. Since the maximal set of potential contact nodes is predefined, an additional substructuring by Craig–Bampton can be performed. This contact treatment turns out to be necessary and allows exclusion of the Lagrange multipliers and the nodal displacements at contact from reduction. The numerical solutions of the reduced contact problem achieve high accuracy and the dynamic contact carries over the behavior of the full order model. Moreover, if the contact area is small compared to the overall structure, the reduction scheme performs very efficiently. The performance of the resulting reduction method is assessed on two 2D computational examples from linear elasticity.

A new finite element paradigm to solve contact problems with roughness

This article’s main scope is the presentation of a computational method for the simulation of contact problems within the finite element method involving complex and rough surfaces. The approach relies on the MPJR (eMbedded Profile for Joint Roughness) interface finite element proposed in [Paggi, M., Reinoso, J., 2020. Mech. Adv. Mater. Struct. 27:1731–1747], which is nominally flat but can embed at the nodal level any arbitrary height to reconstruct the displacement field due to contact in the presence of roughness. Here, the formulation is generalized to handle 3D surface height fields and any arbitrary nonlinear interface constitutive relation, including friction and adhesion. The methodology is herein validated with BEM solutions for linear elastic contact problems. Then, a selection of nonlinear contact problems prohibitive to be simulated by BEM and by standard contact algorithms in FEM are detailed, to highlight the promising aspects of the proposed method for tribology.

An efficient model order reduction scheme for dynamic contact in linear elasticity

The paper proposes an approach for the efficient model order reduction of dynamic contact problems in linear elasticity. Instead of the augmented Lagrangian method that is widely used for mechanical contact problems, we prefer here the Linear Complementarity Programming (LCP) method as basic methodology. It has the advantage of resulting in the much smaller dual problem that is associated with the governing variational principle and that turns out to be beneficial for the model order reduction. Since the shape of the contact zone depends strongly on the acting outer forces, the LCP for the Lagrange multipliers has to be solved in each time step. The model order reduction scheme, on the other hand, is applied to the large linear system for the displacements and computed in advance by means of an Arnoldi process. In terms of computational effort the reduction scheme is very appealing because the contact constraints are fully satisfied while the reduction acts only on the displacements. As an extension of our approach, we furthermore take up the idea of the Craig-Bampton method in order to distinguish between interior nodes and the nodes in the contact zone. A careful performance analysis closes the paper.

Material stiffness optimization for contact stress distribution in frictional elastic contact problems with multiple load cases

This paper concerns the heuristic-based material stiffness optimization of frictional linear elastic contact problems for having control over the contact stress distribution, aiming to extend the material stiffness optimization to multiple loading conditions, in which each of the loadings acts solely on the structures. A decrease level of the variance of the contact stress is introduced and a weighted sum of the decrease levels under all load cases is constructed as the objective function. The individual criterion for contact problems with multiple contact regions is addressed. The worst case design is adopted for multiple load cases, and an extreme reference stress, which is the highest stress level of the subdomain under all load cases, is defined to control the Young’s modulus modification process in a finite element framework. Through three numerical examples, it is demonstrated how an even distribution of the contact stress can be obtained for contact problems subjected to multiple load cases with single or multiple contact regions. Some new features of the material stiffness optimization with multiple loading conditions are also illustrated.

Topology optimization for compliance and contact pressure distribution in structural problems with friction

This paper concerns density-based topology optimization of linear elastic contact problems, aiming to present robust and practically realizable designs for different objective functions. First we revisit a compliance minimization with frictionless contact problem from the literature and present crisp solid–void designs, based on the so-called modified robust topology optimization formulation. An adaptation of this problem to frictional contact is then solved for various friction coefficients and it is checked that the optimization algorithm indeed exploits the presence of friction for lowering the objective further. Secondly, we propose and demonstrate the use of a p-norm based objective function to control the distribution and variation of contact pressure, on an a priori unknown area of contact, between a body of unknown topology and an obstacle. To have control over the contact pressure, a Lagrange multiplier based contact formulation is used within a coupled Newton solution, for imposing impenetrability, friction, and the corresponding complementarity conditions. The adjoint method is employed for deriving consistent design sensitivities for the mixed formulation involving both displacements and contact Lagrange multipliers. Through a series of numerical examples, it is demonstrated how an even distribution of contact pressure and crisp solid–void designs can be obtained for problems with and without friction.

Comparison of two projection methods for the solution of frictional contact problems

Frictional contact problems in linear elasticity are considered in this paper. The contact constraint is imposed in the weak sense using the fixed point method, which leads to a variational equation problem. For solving such a nonlinear variational problem, we study two projection methods using different self-adaptive rules. Based on the self-adaptive projection method, we propose a modified self-adaptive rule which is more effective to update the parameter. The methods can be implemented easily in conjunction with the boundary element method for the solution. Numerical experiments are reported to illustrate theoretical results.

A self-adaptive projection method for contact problems with the BEM

• A new and simple computational method is developed for the contact problem. • We use the variational method to analyze the convergence of proposed method. • We give the self-adaptive rule and the boundary element approximation for the method. • The performance of the method is demonstrated by the numerical results. A self-adaptive algorithm, based on the projection and boundary integral methods, is designed and analyzed for frictionless contact problems in linear elasticity. Using the equivalence between the contact problem and a variational formulation with a projection fixed point problem of infinite dimensions, we develop an iterative algorithm that formulates the contact boundary condition into a sequence of Robin boundary conditions. In order to improve the performance of the method, we propose a self-adaptive rule which updates the penalty parameter automatically. As the iteration process is given by the displacement and the stress on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments show that the method presented is efficient and robust.

An adaptive least‐squares mixed finite element method for the Signorini problem

We present and analyze a least squares formulation for contact problems in linear elasticity which employs both, displacements and stresses, as independent variables. As a consequence, we obtain stability and high accuracy of our discretization also in the incompressible limit. Moreover, our formulation gives rise to a reliable and efficient a posteriori error estimator. To incorporate the contact constraints, the first‐order system least squares functional is augmented by a contact boundary functional which implements the associated complementarity condition. The bilinear form related to the augmented functional is shown to be coercive and therefore constitutes an upper bound, up to a constant, for the error in displacements and stresses in . This implies the reliability of the functional to be used as an a posteriori error estimator in an adaptive framework. The efficiency of the use of the functional as an a posteriori error estimator is monitored by the local proportion of the boundary functional term with respect to the overall functional. Computational results using standard conforming linear finite elements for the displacement approximation combined with lowest‐order Raviart‐Thomas elements for the stress tensor show the effectiveness of our approach in an adaptive framework for two‐dimensional and three‐dimensional Hertzian contact problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 276–289, 2017

Stabilized mixed hp-BEM for frictional contact problems in linear elasticity

We investigate hp-stabilization for variational inequalities and boundary element methods based on the approach introduced by Barbosa and Hughes for finite elements. Convergence of a stabilized mixed boundary element method is shown for unilateral frictional contact problems for the Lame equation. Without stabilization, this may not converge because the inf-sup constant need not be bounded away from zero for natural discretizations, even for fixed h and p. Both a priori and a posteriori error estimates are presented in the case of Tresca friction, for discretizations based on Bernstein or Gauss-Lobatto-Lagrange polynomials as test and trial functions. We also consider an extension of the a posteriori estimate to Coulomb friction. Numerical experiments underline our theoretical results.

Combined symmetric BEM and semi-smooth Newton method for the contact problems in linear elasticity of Yukawa type

The semi-smooth Newton method and the boundary element method are developed and analyzed for the solution of 2-D Signorini contact problems in linear elasticity of Yukawa type. First we consider the contact problem with Tresca friction. This leads to a constrained non-differentiable minimization problem where the solvability is in general problematic. But, by utilizing the Fenchel duality theory, the dual formulation in terms of contact stresses turns out to be a quadratic optimization problem with a smooth functional. The regularization of the dual problem motivated by the augmented Lagrangian is suitable for the application of the generalized Newton method. Applying the boundary integral equation method, the problem is reduced to the boundary curve. The corresponding boundary integral equations are approximated by using a Galerkin method with the help of B-splines on the boundary curve (BEM). This yields an algebraic system of linear equations with dense stiffness matrix but which is symmetric. The symmetry property of the stiffness matrix enables the application of efficient iterative solution strategies for the linear systems at each Newton step. Second, the methods are carried over to the Coulomb friction problem by means of a fixed point approach. In particular, the analysis of the algorithm is presented and some numerical examples, which show a remarkable efficiency and reliability of the semi-smooth Newton method, are given.

Boundary augmented Lagrangian method for contact problems in linear elasticity

An augmented Lagrangian method, based on the fixed point method and boundary variational formulations, is designed and analysed for frictionless contact problems in linear elasticity. Using the equivalence between the contact boundary condition and a fixed point problem, we develop a new iterative algorithm that formulates the contact problem into a sequence of corresponding linear variational equations with the Steklov–Poincaré operator. Both theoretical results and numerical experiments show that the method presented is efficient.

On [formula omitted]-adaptive BEM for frictional contact problems in linear elasticity

A mixed formulation for a Tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations, which is later extended to Coulomb friction. The discrete Lagrange multiplier, an approximation of the surface traction on the contact boundary part, is a linear combination of biorthogonal basis functions. In case of curved (isoparametric) elements, these are the solutions of local problems. In particular, the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems. Thus, enabling an efficient application of a semi-smooth Newton solver for the discrete mixed problem, converging locally super-linearly in the frictional case and quadratically in the frictionless case. Typically, the solution of frictional contact problems is of reduced regularity at the interface between contact to non-contact and from stick to slip. To identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced. In a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation. For the latter a residual and a bubble error estimation are considered explicitly. The numerical experiments show the applicability of the derived error estimations and the superiority of hp-adaptivity compared to low order uniform and adaptive approaches.

Mixed finite element methods for two-body contact problems

This paper presents mixed finite element methods of higher-order for two-body contact problems of linear elasticity. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The main focus is on the convergence of the scheme and on a priori estimates for the h- and the p-method. For this purpose, a discrete inf–sup condition is proven which guarantees the stability of the mixed method. Numerical results confirm the theoretical findings.

3D reconstruction of TMJ after resection of the cyst and the stress–strain analyses

The aim of this article is 3D analysis of the temporomandibular joint (TMJ) patient, who underwent surgery, during which the right TMJ was resected along with the ramus of mandible and consequently the joint was reconstructed with subtotal replacement. The main goal is to give a suitable formulation of mathematical model, which describes the changes of stresses in TMJ incurred after the surgery. The TMJ is a complex, sensitive and highly mobile joint which works bilaterally so each side influences the contralateral joint and because of this the distribution of the stresses is changed in the healthy joint as well. Detailed knowledge about function these are necessary for clinical application of temporomandibular joint prosthesis and also help us estimate the lifetime of the prosthesis a possibilities of alteration in the contra lateral joint components. The geometry for the 3D models is taken from the CT scan date and its numerical solution is based on the theory of semi-coercive unilateral contact problems in linear elasticity. This article provides medical part with case report, discretion of treatment, than the methods of mathematical modeling and his possibilities are described and finally results are reported.

A Nitsche-Based Method for Unilateral Contact Problems: Numerical Analysis

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 $H^1(\Omega)$-norm for linear finite elements in two dimensions, which is $O(h^{\frac{1}{2}+\nu})$ when the solution lies in $H^{\frac{3}{2}+\nu}(\Omega)$, $0<\nu\leq 1/2$. 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.

An interior-point algorithm for the minimization arising from 3D contact problems with friction

The paper deals with a variant of the interior-point method for the minimization of strictly quadratic objective function subject to simple bounds and separable quadratic inequality constraints. Such minimizations arise from the finite element approximation of contact problems of linear elasticity with friction in three space dimensions. The main goal of the paper is the convergence analysis of the algorithm and its implementation. The optimal preconditioners for solving ill-conditioned inner linear systems are proposed. Numerical experiments illustrate the computational efficiency for large-scale problems.

Steady Sliding Frictional Contact Problems in Linear Elasticity

The existence and uniqueness of an equilibrium solution to frictional contact problems involving a class of moving rigid obstacles is studied. At low friction coefficient values, the steady sliding frictional contact problem is uniquely solvable, thanks to the Lions-Stampacchia theorem on variational inequalities associated with a nonsymmetric coercive bilinear form. It is proved that the coerciveness of the bilinear form can be lost at some positive critical value of the friction coefficient, depending only on the geometry and the elastic properties of the body. An example presented here, shows that infinitely many solutions can be obtained when the friction coefficient is larger than the critical value. This result is paving the road towards a theory of jamming in terms of bifurcation in variational inequality. The particular case where the elastic body is an isotropic half-space is studied. The corresponding value of the critical friction coefficient is proved to be infinite in this case. In the particular case of the frictionless situation, our analysis incidentally unifies the approaches developed by Lions-Stampacchia (variational inequalities) and Hertz (harmonic analysis on the half-space) to contact problems in linear elasticity.

Mixed FEM of higher‐order for a frictional contact problem

AbstractThis paper presents mixed finite element methods of higher‐order for an idealized frictional contact problem in linear elasticity. The approach relies on a saddle point formulation where the frictional contact condition is captured by a Lagrange multiplier. The convergence of the mixed scheme is proven and some a priori estimates for the h‐ and p‐method are derived. Furthermore, a posteriori error estimates are presented which rely on the estimation of the discretization error of an auxiliary problem and some further terms capturing the error in the friction and complementary conditions. Numerical results confirm the applicability of the a posteriori error estimates within h‐ and hp‐adaptive schemes. (© 2011 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact

We present and analyze subspace correction methods for the solution of variational inequalities of the second kind and apply these theoretical results to non smooth contact problems in linear elasticity with Tresca and non-local Coulomb friction. We introduce these methods in a reflexive Banach space, prove that they are globally convergent and give error estimates. In the context of finite element discretizations, where our methods turn out to be one- and two-level Schwarz methods, we specify their convergence rate and its dependence on the discretization parameters and conclude that our methods converge optimally. Transferring this results to frictional contact problems, we thus can overcome the mesh dependence of some fixed-point schemes which are commonly employed for contact problems with Coulomb friction.