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.     

An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented‐Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.     

An algorithm for the matrix‐free solution of quasistatic frictional contact problems

A contact enforcement algorithm has been developed for matrix‐free quasistatic finite element techniques. Matrix‐free (iterative) solution algorithms such as non‐linear conjugate gradients (CG) and dynamic relaxation (DR) are desirable for large solid mechanics applications where direct linear equation solving is prohibitively expensive, but in contrast to more traditional Newton–Raphson and quasi‐Newton iteration strategies, the number of iterations required for convergence is typically of the same order as the number of degrees of freedom of the model. It is therefore crucial that each of these iterations be inexpensive to per‐form, which is of course the essence of a matrix free method. In applying such methods to contact problems we emphasize here two requirements: first, that the treatment of the contact should not make an average equilibrium iteration considerably more expensive; and second, that the contact constraints should be imposed in such a way that they do not introduce spurious energy that acts against the iterative solver. These practical concerns are utilized to develop an iterative technique for accurate constraint enforcement that is suitable for non‐linear conjugate gradient and dynamic relaxation iterative schemes. Copyright © 1999 John Wiley & Sons, Ltd.     

Path‐following analysis of thin‐walled structures and comparison with asymptotic post‐critical solutions

A path‐following non‐linear elastic analysis for structures composed of assemblages of flat slender elastic panels is presented. The proposed path‐following method employs FEM technology and a kinematical model to analyse these structures using a Koiter asymptotic approach. As a result it is possible to verify the accuracy achieved by the asymptotic method. The proposed mixed path‐following formulation is both efficient and robust with regards to the locking extrapolation phenomenon that strongly affects compatible formulations. The use of an HC finite element makes it possible to avoid the problem of the finite rotations in the space, maintaining a high degree of continuity and making the numeric formulation simple and efficient. Copyright © 2002 John Wiley & Sons, Ltd.     

Genetic approach to optimal topology/controller design of adaptive structures

Integrated optimization of structural topology and control system is considered. The problem is formulated as mixed discrete-continuous multi-objective programming with a linear quadratic regulator cost index, and measures of robustness and controllability as objectives. The Genetic Algorithm (GA), a guided random search technique, is adopted for the problem-solving. A member elimination strategy that allows deleted members to be recovered is suggested in the search procedure. As verification for the proposed method, optimum layout and actuator placement for a 45-bar truss is illustrated. Numerical results indicate that the genetic algorithm can converge to optimum solutions by searching only a minor fraction of the solution space. Discussions on the algorithm are presented. © 1998 John Wiley & Sons, Ltd.     

Adaptive damping for dynamic relaxation problems with non‐monotonic spectral response

Non‐inertial transients may be effectively solved using explicit time integration with arbitrary inertial and damping properties. The usual approach relies on a diagonal lumped mass on which strictly mass proportional damping is based. While the damping coefficient can be adaptively determined on the basis of the estimated frequency of the predominant response, local changes in stiffness, often associated with changing contact conditions, can cause abrupt changes in the damping coefficient. This can substantially impede the progress of the solution, particularly when deformations involve large translations or rotations of parts of the system. A modification to the mass proportional damping is developed and implemented to avoid the deleterious effects of sudden changes in the damping coefficient. A new procedure is thus implemented so that the usual dynamic relaxation method will automatically adapt to changing conditions of response in such a way as to avoid overdamping low modes due to subsequent higher frequency events. Copyright © 2002 John Wiley & Sons, Ltd.     

Variational approach to the analysis of steady‐state thermo‐elastic wear regimes

A transient wear process on frictional interface of two thermo‐elastic bodies in a relative steady sliding motion induces shape evolution of contact interface and tends to a steady state for which the wear process occurs at fixed contact stress and strain distribution. The temperature field generated by frictional and wear dissipation on the contact surface is assumed to reach a steady state. This state is assumed to correspond to minimum of the wear dissipation power and the temperature field corresponds to maximum of the heat entropy production. The stationarity conditions of the response functionals provide the contact pressure distribution and the corresponding temperature field. The present approach extends the authors previous analyses of optimal or steady‐state contact shapes by accounting for coupled wear and thermal distortion effects. The wear rule is assumed as a non‐linear relation of wear rate to shear stress and relative sliding velocity. The analysis of disk and drum brakes is presented with account for thermal distortion effect. It is shown that the contact shape in a steady thermo‐elastic state essentially differs from that specified for mechanical loading with neglect of thermal effects. The thermal instability regimes are not considered in the paper. Copyright © 2009 John Wiley & Sons, Ltd.     

An adaptive hp-version of the finite element method applied to flame propagation problems

This paper describes an adaptive hp-version mesh refinement strategy and its application to the finite element solution of one-dimensional flame propagation problems. The aim is to control the spatial and time discretization errors below a prescribed error tolerance at all time levels. In the algorithm, the optimal time step is first determined in an adaptive manner by considering the variation of the computable error in the reaction zone. Later, the method uses a p-version refinement till the computable a posteriori error is brought down below the tolerance. During the p-version, if the maximum allowable degree of approximation is reached in some elements of the mesh without satisfying the global error tolerance criterion, then conversion from p- to h-version is performed. In the conversion procedure, a gradient based non-uniform h-version refinement has been introduced in the elements of higher degree approximation. In this way, p-version and h-version approaches are used alternately till the a posteriori error criteria are satisfied. The mesh refinement is based on the element error indicators, according to a statistical error equi-distribution procedure. Numerical simulations have been carried out for a linear parabolic problem and premixed flame propagation in one-space dimension. © 1997 John Wiley & Sons, Ltd.     

Three‐dimensional adaptive meshing by subdivision and edge‐collapse in finite‐deformation dynamic–plasticity problems with application to adiabatic shear banding

This paper is concerned with the development of a general framework for adaptive mesh refinement and coarsening in three‐dimensional finite‐deformation dynamic–plasticity problems. Mesh adaption is driven by a posteriori global error bounds derived on the basis of a variational formulation of the incremental problem. The particular mesh‐refinement strategy adopted is based on Rivara's longest‐edge propagation path (LEPP) bisection algorithm. Our strategy for mesh coarsening, or unrefinement, is based on the elimination of elements by edge‐collapse. The convergence characteristics of the method in the presence of strong elastic singularities are tested numerically. An application to the three‐dimensional simulation of adiabatic shear bands in dynamically loaded tantalum is also presented which demonstrates the robustness and versatility of the method. Copyright © 2001 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.     

Adaptive Kronrod–Patterson integration of non‐linear finite element matrices

Efficient simulation of unsaturated moisture flow in porous media is of great importance in many engineering fields. The highly non‐linear character of unsaturated flow typically gives sharp moving moisture fronts during wetting and drying of materials with strong local moisture permeability and capacity variations as result. It is shown that these strong variations conflict with the common preference for low‐order numerical integration in finite element simulations of unsaturated moisture flow: inaccurate numerical integration leads to errors that are often far more important than errors from inappropriate discretization. In response, this article develops adaptive integration, based on nested Kronrod–Patterson–Gauss integration schemes: basically, the integration order is adapted to the locally observed grade of non‐linearity. Adaptive integration is developed based on a standard infiltration problem, and it is demonstrated that serious reductions in the numbers of required integration points and discretization nodes can be obtained, thus significantly increasing computational efficiency. The multi‐dimensional applicability is exemplified with two‐dimensional wetting and drying applications. While developed for finite element unsaturated moisture transfer simulation, adaptive integration is similarly applicable for other non‐linear problems and other discretization methods, and whereas perhaps outperformed by mesh‐adaptive techniques, adaptive integration requires much less implementation and computation. Both techniques can moreover be easily combined. Copyright © 2009 John Wiley & Sons, Ltd.     

A mixed FEM approach to stress‐constrained topology optimization

We present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion. The main idea is to adopt a mixed finite‐element discretization scheme wherein not only displacements (as usual) but also stresses are the variables entering the formulation. By doing so, any stress constraint may be handled within the optimization procedure without resorting to post‐processing operation typical of displacement‐based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed. Two dual variational principles of Hellinger–Reissner type are presented in continuous and discrete form that, which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes (Int. J. Numer. Meth. Engng. 1984; 24 (3):359–373). Extensive numerical simulations are performed and ongoing extensions outlined, including the optimization of elastoplastic and incompressible media. Copyright © 2007 John Wiley & Sons, Ltd.     

Interface dynamic stiffness matrix approach for three‐dimensional transient multi‐region boundary element analysis

A novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three‐dimensional multi‐region elastodynamic problems in the time domain. The proposed procedure is based on the interface stiffness matrix approach for static multi‐region problems using variational principles together with the concept of Duhamel integrals. Unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the Duhamel (convolution) integrals. Although the method is not restricted to a special discretization technique, the regions are discretized using the boundary element method combined with the convolution quadrature method. This results in a time‐domain methodology with the advantages of performing computations in the Laplace domain, which produces very accurate and stable results as verified on test examples. In addition, the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient. Finally, practical applications in the area of soil–structure interaction and tunneling problems are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

Thermal flutter analysis of large‐scale space structures based on finite element method

During the orbital day–night crossing period, the suddenly applied thermal loading is apt to introducing vibration on flexible appendages of large‐scale space structures. This kind of thermally‐induced vibration is a typical failure of modern spacecrafts. However, owing to the complexity of this problem, many earlier researches study only the vibration of simplified beam models, which can hardly describe the performance of practical structures. This paper aims at using the finite element method to analyse the non‐linear vibration of practical thin‐walled large‐scale space structures subjected to suddenly applied thermal loading. In this study, the coupling effect between structural deformations and the incident normal solar heat flux is considered; the necessary condition of thermally‐induced vibration is verified; and the criterion of thermal flutter is established. Copyright © 2006 John Wiley & Sons, Ltd.     

A novel method for integrating first‐ and second‐order differential equations in elastohydrodynamic lubrication for the solution of smooth isothermal,line contact problems

This paper contains details of recent developments in the analysis of elastohydrodynamic lubrication problems using the finite element method. A steady state isothermal finite element formulation of the smooth line contact problem with Newtonian lubricant behaviour is presented containing both first‐ and second‐order formulations of the hydrodynamic equation. Previous problems with the limited range of applicability of both first‐ and second‐order finite difference solutions have been overcome by summing both the first‐ and second‐order equations' weighted contributions. Application of the method to a range of problems unattainable by either single first‐ or second‐order formulations is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

A Class of Mathematical Programs with Equilibrium Constraints: A Smooth Algorithm and Applications to Contact Problems

We discuss a special mathematical programming problem with equilibrium constraints (MPEC), that arises in material and shape optimization problems involving the contact of a rod or a plate with a rigid obstacle. This MPEC can be reduced to a nonlinear programming problem with independent variables and some dependent variables implicity defined by the solution of a mixed linear complementarity problem (MLCP). A projected-gradient algorithm including a complementarity method is proposed to solve this optimization problem. Several numerical examples are reported to illustrate the efficiency of this methodology in practice.     

A new approach to solving problems of multi‐state system reliability optimization

Usually engineers try to achieve the required reliability level with minimal cost. The problem of total investment cost minimization, subject to reliability constraints, is well known as the reliability optimization problem. When applied to multi‐state systems (MSS), the system has many performance levels, and reliability is considered as a measure of the ability of the system to meet the demand (required performance). In this case, the outage effect will be essentially different for units with different performance rate. Therefore, the performance of system components, as well as the demand, should be taken into account. In this paper, we present a technique for solving a family of MSS reliability optimization problems, such as structure optimization, optimal expansion, maintenance optimization and optimal multistage modernization. This technique combines a universal generating function (UGF) method used for fast reliability estimation of MSS and a genetic algorithm (GA) used as an optimization engine. The UGF method provides the ability to estimate relatively quickly different MSS reliability indices for series‐parallel and bridge structures. It can be applied to MSS with different physical nature of system performance measure. The GA is a robust, universal optimization tool that uses only estimates of solution quality to determine the direction of search. Copyright © 2001 John Wiley & Sons, Ltd.     

A computational approach for multi‐scale analysis of heterogeneous elasto‐plastic media subjected to short duration loads

The asymptotic expansion homogenization (AEH) approach has found wide acceptance for the study of heterogeneous structures due to its ability to account for multi‐scale features. The emphasis of the present study is to develop consistent AEH numerical formulations to address elasto‐plastic material response of structures subjected to short‐duration transient loading. A second‐order accurate velocity‐based explicit time integration method, in conjunction with the AEH approach, is currently developed that accounts for large deformation non‐linear material response. The approach is verified under degenerate homogeneous conditions using existing experimental data in the literature and its ability to account for heterogeneous conditions is demonstrated for a number of test problems. Copyright © 2004 John Wiley & Sons, Ltd.