BIEM-based variational principles for elastoplasticity with unilateral contact boundary conditions

The structural step problem for elastic-plastic internal-variable materials is addressed in the presence of frictionless unilateral contact conditions. Basing on the BIEM (boundary integral equation method) and making use of deformation-theory plasticity (through the backward-difference method of computational plasticity), two variational principles are shown to characterize the solution to the step problem: one is a stationarity principle having as unknowns all the problem variables, the other is a saddle-point principle having as unknowns the increments of the boundary tractions and displacements, along with the plastic strain increments in the domain. The discretization by boundary and interior elements transforms the above principles into well-posed mathematical programming formulations belonging to the symmetric Galerkin BEM formulations (with features such as a symmetric sign-definite coefficient matrix, double integrations, and hypersingular integrals).     

A contact formulation based on localized Lagrange multipliers: formulation and application to two‐dimensional problems

The non‐penetration condition in contact problems is traditionally based on the classical Lagrange multiplier method. This method makes extensive use of modelling details of the contacting bodies for contact enforcement as the contact surface meshes are in general non‐matching. To deal with this problem we introduce a novel element in the Lagrange multiplier approach of contact modelling, namely, a contact frame placed in between contacting bodies. It acts as a medium through which contact forces are transferred without violating equilibrium in the contact domain for discrete contact models. Only nodal information of the contacting bodies is required which makes the proposed contact enforcement generic. The contact frame has its own independent freedoms, which allows the formulation to pass contact patch tests by design. Copyright © 2002 John Wiley & Sons, Ltd.     

A new strategy for the solution of frictional contact problems

This article is devoted to the development of a new heuristic algorithm for the solution of the general variational inequality arising in frictional contact problems. The existing algorithms devised for the treatment of the variational inequality representing frictional contact rely on the decomposition of the physical problem into two sub-problems which are then solved iteratively. In addition, the penalty function method and/or the regularization techniques are typically used in the solution of these reduced sub-problems. These techniques introduce user-defined parameters which could influence the convergence and accuracy of the solution. The new method presented in this article overcomes these difficulties by providing a solution for the general variational inequality without decomposition into sub-problems. This is accomplished using a new heuristic algorithm which utilizes mathematical programming techniques, and thus avoids the use of penalty or regularization methods. The versatility and reliability of the developed algorithm were demonstrated through implementation to the case of frictional contact of an elastic hollow cylinder with a rigid foundation. © 1998 John Wiley & Sons, Ltd.     

A new computational approach to contact mechanics using variable‐node finite elements

In this paper, a new computational strategy for two‐dimensional contact problems is developed with the aid of variable‐node finite elements within the range of infinitesimal deformations. The variable‐node elements, which are among MLS (moving least square)‐based finite elements, enable us to transform node‐to‐surface contact problems into node‐to‐node contact problems. This contact formulation with variable‐node elements leads to an accurate and effective solution procedure, needless to mention that the contact patch test is passed without any additional treatment. Through several numerical examples, we demonstrate its simplicity and the effectiveness of the proposed scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

A variational approach with embedded roughness for adhesive contact problems

Abstract

A new variational formulation is proposed for the solution of contact problems for profiles of arbitrary shape indenting a deformable half-plane, with special focus on rough indenters. The method exploits the new idea of accounting for the shape of roughness as a correction to the normal gap function. The resulting interface finite element with eMbedded Profile for Joint Roughness (MPJR interface finite element) is derived and its implementation is comprehensively described. The method is applied to a range of contact problems very challenging for traditional methods, opening new perspectives for the solution of contact problems with roughness and adhesion.     

A hybrid variational‐collocation immersed method for fluid‐structure interaction using unstructured T‐splines

We present a hybrid variational‐collocation, immersed, and fully‐implicit formulation for fluid‐structure interaction (FSI) using unstructured T‐splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational‐collocation method. We use T‐splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T‐splines offer us two main advantages against non‐uniform rational B‐splines: they can be locally refined and they are unstructured. The generalized‐α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T‐junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems

This paper presents a general method for dynamic particle refinement in smoothed particle hydrodynamics (SPH). Candidate particles are split into several ‘daughter’ particles according to a given refinement pattern centred about the original particle. Through the solution of a non‐linear minimization problem the optimal mass distribution of the daughter particles is obtained so as to reduce the errors introduced to the underlying density field. This procedure necessarily conserves the mass of the system. Conservation of energy and momentum results are also discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

A general boundary element method approach to the solution of three-dimensional frictionless contact problems

A direct boundary element method is presented for three-dimensional stress analysis of frictionless contact problems. The isoparametric formulation of the boundary element method is implemented for the general case of contact in the absence of friction, which is limited to linear elastic homogeneous and isotropic materials. An iterative procedure is employed to determine the correct size of the contact zone by finding a boundary solution compatible with the contact condition. The applicability of the procedure is tested by application to three problems of advancing and conforming contact. The computed results are compared with numerical and analytical solutions where possible.     

A new finite‐element formulation for electromechanical boundary value problems

In this paper, a new finite‐element formulation for the solution of electromechanical boundary value problems is presented. As opposed to the standard formulation that uses scalar electric potential as nodal variables, this new formulation implements a vector potential from which components of electric displacement are derived. For linear piezoelectric materials with positive definite material moduli, the resulting finite‐element stiffness matrix from the vector potential formulation is also positive definite. If the material is non‐linear in a fashion characteristic of ferroelectric materials, it is demonstrated that a straightforward iterative solution procedure is unstable for the standard scalar potential formulation, but stable for the new vector potential formulation. Finally, the method is used to compute fields around a crack tip in an idealized non‐linear ferroelectric material, and results are compared to an analytical solution. Copyright © 2002 John Wiley & Sons, Ltd.     

A new fixed‐point algorithm for hardening plasticity based on non‐linear mixed variational inequalities

Moving from the seminal papers of Han and Reddy, we propose a fixed‐point algorithm for the solution of hardening plasticity two‐dimensional problems. The continuous problem may be classified as a mixed non‐linear non‐differentiable variational inequality of the second type and is discretized by means of a truly mixed finite‐element scheme. One of the main peculiarities of our approach is the use of the composite triangular element of Johnson and Mercier for the approximation of the stress field. The non‐differentiability is coped with via regularization whereas the non‐linearity is approached with a fixed‐point iterative strategy. Numerical results are proposed that investigate the sensitivity of the approach with respect to the mesh size and the regularization parameter ε. The simplicity of the proposed fixed‐point scheme with respect to more classical return mapping approaches seems to represent one of the key features of our algorithm. Copyright © 2003 John Wiley & Sons, Ltd.     

Smooth C1‐interpolations for two‐dimensional frictional contact problems

Finite deformation contact problems are associated with large sliding in the contact area. Thus, in the discrete problem a slave node can slide over several master segments. Standard contact formulations of surfaces discretized by low order finite elements leads to sudden changes in the surface normal field. This can cause loss of convergence properties in the solution procedure and furthermore may initiate jumps in the velocity field in dynamic solutions. Furthermore non‐smooth contact discretizations can lead to incorrect results in special cases where a good approximation of the contacting surfaces is needed. In this paper a smooth contact discretization is developed which circumvents most of the aformentioned problems. A smooth deformed surface with no slope discontinuities between segments is obtained by a C¹‐continuous interpolation of the master surface. Different forms of discretizations are possible. Among these are Bézier, Hermitian or other types of spline interpolations. In this paper we compare two formulations which can be used to obtain smooth normal and tangent fields for frictional contact of deformable bodies. The formulation is developed for two‐dimensional applications and includes finite deformation behaviour. Examples show the performance of the new discretization technique for contact. Copyright © 2001 John Wiley & Sons, Ltd.     

A general conjugate‐gradient‐based predictor–corrector solver for explicit finite‐element contact

A Lagrange‐multiplier based approach is presented for the general solution of multi‐body contact within an explicit finite element framework. The technique employs an explicit predictor step to permit the detection of interpenetration and then utilizes a corrector step, whose solution is obtained with a pre‐conditioned matrix‐free conjugate gradient projection method, to determine the Lagrange multipliers necessary to eliminate the predicted penetration. The predictor–corrector algorithm is developed for deformable bodies based upon the central difference method, and for rigid bodies from momentum and energy conserving approaches. Both frictionless and Coulomb‐based frictional contact idealizations are addressed. The technique imposes no time‐step constraints and quickly mitigates velocity discontinuities across closed interfaces. Special attention is directed toward contact between rigid bodies. Algorithmic moment arms conserve the translational and angular momentums of the system in the absence of external loads. Elastic collisions are captured with a two‐phase predictor–corrector approach and a geometrically approximate velocity jump criterion. The first step solves the inelastic contact problem and identifies inactive constraints between rigid bodies, while the second step generates the necessary velocity jump condition on the active constraints. The velocity criterion is shown to algorithmically preserve the system kinetic energy for two unconstrained rigid bodies. Copyright © 1999 John Wiley & Sons, Ltd. This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.     

An energy‐momentum‐conserving temporal discretization scheme for adhesive contact problems

Numerical solution of dynamic problems requires accurate temporal discretization schemes. So far, to the best of the authors’ knowledge, none have been proposed for adhesive contact problems. In this work, an energy‐momentum‐conserving temporal discretization scheme for adhesive contact problems is proposed. A contact criterion is also proposed to distinguish between adhesion‐dominated and impact‐dominated contact behaviors. An adhesion formulation is considered, which is suitable to describe a large class of interaction mechanisms including van der Waals adhesion and cohesive zone modeling. The current formulation is frictionless, and no dissipation is considered. Performance of the proposed scheme is compared with other schemes. The proposed scheme involves very little extra computational overhead. It is shown that the proposed new temporal discretization scheme leads to major accuracy gains both for single‐degree‐of‐freedom and multi‐degree‐of‐freedom systems. The single‐degree‐of‐freedom system is critically analyzed for various parameters affecting the response. For the multi‐degree‐of‐freedom system, the effect of the time step and mesh discretization on the solution is also studied using the proposed scheme. It is further shown that a temporal discretization scheme based on the principle of energy conservation is not sufficient to obtain a convergent solution. Results with higher order contact finite elements for discretizing the contact area are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient numerical methods for the large‐scale,parallel solution of elastoplastic contact problems

Quasi‐static elastoplastic contact problems are ubiquitous in many industrial processes and other contexts, and their numerical simulation is consequently of great interest in accurately describing and optimizing production processes. The key component in these simulations is the solution of a single load step of a time iteration. From a mathematical perspective, the problems to be solved in each time step are characterized by the difficulties of variational inequalities for both the plastic behavior and the contact problem. Computationally, they also often lead to very large problems. In this paper, we present and evaluate a complete set of methods that are (1) designed to work well together and (2) allow for the efficient solution of such problems. In particular, we use adaptive finite element meshes with linear and quadratic elements, a Newton linearization of the plasticity, active set methods for the contact problem, and multigrid‐preconditioned linear solvers. Through a sequence of numerical experiments, we show the performance of these methods. This includes highly accurate solutions of a three‐dimensional benchmark problem and scaling our methods in parallel to 1024 cores and more than a billion unknowns. Copyright © 2015 John Wiley & Sons, Ltd.     

Rate‐dependent projection operators for frictional contact constraints

We develop rate‐dependent regularization approaches for three‐dimensional frictional contact constraints based on the Kelvin and Maxwell viscoelastic constitutive models. With the present regularization schemes, we aim to provide a basis to better model friction and to stabilize the contact analysis while keeping the contact model as simple as possible. The key feature of the regularization approaches, implemented using an implicit time integrator, is that one can recover in the limit the widely used rate‐independent elastoplastic regularization framework without encountering numerical difficulties. Intermediate contact tractions are defined in terms of the relative displacement, the relative velocity, and the regularization parameters. The projection operators operate on the intermediate tractions and yield contact tractions that satisfy all the discretized contact constraints. The use of projection operators allows a systematic implementation of the present regularization schemes. Through numerical simulations, we observed that the Maxwell‐type regularization effectively avoids convergence problems, even for relatively large time step sizes, while the Kelvin‐type regularization does not. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization

We consider the numerical simulation of non‐linear multi‐body contact problems in elasticity on complex three‐dimensional geometries. In the case of warped contact boundaries and non‐matching finite element meshes, particular emphasis has to be put on the discretization of the transmission of forces and the non‐penetration conditions at the contact interface. We enforce the discrete contact constraints by means of a non‐conforming domain decomposition method, which allows for optimal error estimates. Here, we develop an efficient method to assemble the discrete coupling operator by computing the triangulated intersection of opposite element faces in a locally adjusted projection plane but carrying out the required quadrature on the faces directly. Our new element‐based algorithm does not use any boundary parameterizations and is also suitable for isoparametric elements. The emerging non‐linear system is solved by a monotone multigrid method of optimal complexity. Several numerical examples in 3D illustrate the effectiveness of our approach. Copyright © 2008 John Wiley & Sons, Ltd.     

A direct constraint‐Trefftz FEM for analysing elastic contact problems

A direct constraint technique, based on the hybrid‐Trefftz finite element method, is first presented to solve elastic contact problems without friction. For efficiency, static condensation is employed to condense a large model down to a smaller one which involves nodes within the potential contact surfaces only. This model can remarkably reduce computational time and effort. Subsequently, the contact interface equation is constructed by introducing the contact conditions of compatibility and equilibrium. Based on the formulation developed, a general solution strategy, which is applicable to the well‐known three classical situations (receding, conforming and advancing) is developed. Finally, three typical examples related to the three situations mentioned are provided to verify the reliability and applicability of the approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A formulation for electrostatic‐mechanical contact and its numerical solution

The progress in advanced technology fields requires more and more sophisticated formulations to consider contact problems properly. This paper is devoted to the development of a new constitutive model for electrostatic‐mechanical contacts, based on a micro–macro approach to describe the contact behaviour. The electric‐mechanical contact constitutive law is obtained considering the real microscopic shape of the contacting surfaces, the microscopic behaviour of force transmission and current flow. Some thermo‐mechanical macroscopic models based on microscopic characterizations have already been developed to compute the normal and tangential contact stiffness and the thermal contact resistance. On the basis of such macroscopic models, a similar model, suitable for the electric‐mechanical field, is developed. With reference to the thermal constriction resistance the electric contact resistance is studied, assuming a flux tube around each contacting asperity, and choosing a suitable geometry for its narrowing at the contact zone. The contact element geometry is based on well known theoretical and experimental micro‐mechanical laws, suitably adapted for the FEM formulation. The macroscopic stiffness matrix is calculated on the basis of the microscopic laws and it is continuously updated as a function of the changes in the mechanical and electric significant parameters. A consistent linearization of the set of equations is developed to improve the computational speed, within the framework of implicit methods. Copyright © 2005 John Wiley & Sons, Ltd.     

A yield‐limited Lagrange multiplier formulation for frictional contact

An analogy with rigid plasticity is used to develop a constitutive framework for quasi‐static frictional contact between finitely deforming solids. Within this setting, a Lagrange multiplier method is used to impose a sharp distinction between stick and slip. The scope of the multipliers is limited by a constitutively defined ‘yield’ function and a finite element‐based predictor–corrector scheme is employed to efficiently determine the regions of stick and slip and the associated tractions. Selected simulations of planar quasi‐static problems are presented to validate the method and illustrate its capabilities. Copyright © 2000 John Wiley & Sons, Ltd.     

A dual mortar approach for 3D finite deformation contact with consistent linearization

In this paper, an approach for three‐dimensional frictionless contact based on a dual mortar formulation and using a primal–dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two‐dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi‐smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis. Copyright © 2010 John Wiley & Sons, Ltd.