Combined interior‐point method and semismooth Newton method for frictionless contact problems

In the present paper, a solution scheme is proposed for frictionless contact problems of linear elastic bodies, which are discretized using the finite element method with lower order elements. An approach combining the interior‐point method and the semismooth Newton method is proposed. In this method, an initial active set for the semismooth Newton method is obtained from the approximate optimal solution by the interior‐point method. The simplest node‐to‐node contact model is considered in the present paper, that is, pairs of matching nodes exist on the contact surfaces. However, the discussions can be easily extended to a node‐to‐segment or segment‐to‐segment contact model. In order to evaluate the proposed method, a number of illustrative examples of the frictionless contact problem are shown. The proposed combined method is compared with the interior‐point method and the semismooth Newton method. Two numerical examples that are difficult to solve using the semismooth Newton method are solved effectively using the proposed combined method. It is shown that the proposed method converges within far fewer iterations than the semismooth Newton methods or the interior‐point method. Copyright © 2009 John Wiley & Sons, Ltd.     

An indirect elastodynamic boundary element method with analytic bases

An indirect time‐domain boundary element method (BEM) is presented here for the treatment of 2D elastodynamic problems. The approximated solution in this method is formulated as a linear combination of a set of particular solutions, which are called bases. The displacement and stress fields of a basis are analytically derived by means of solving Lame's displacement potentials. A semi‐collocation method is proposed to be the time‐stepping algorithm. This method is equivalent to a displacement discontinuity method with piecewise linear discontinuities in both space and time. The resulting time‐stepping scheme is explicit. The BEM is implemented to solve three numerical examples, Lamb's problem, half‐plane with a buried crack and Selberg's problem. Though Lamb's problem is considered a difficult problem for numerical methods, the current numerical results for the surface displacements show accurately the characteristics of the Rayleigh wave. This method is efficient and accurate. Copyright © 2003 John Wiley & Sons, Ltd.     

Time‐step constraints in transient coupled finite element analysis

In transient finite element analysis, reducing the time‐step size improves the accuracy of the solution. However, a lower bound to the time‐step size exists, below which the solution may exhibit spatial oscillations at the initial stages of the analysis. This numerical ‘shock’ problem may lead to accumulated errors in coupled analyses. To satisfy the non‐oscillatory criterion, a novel analytical approach is presented in this paper to obtain the time‐step constraints using the θ‐method for the transient coupled analysis, including both heat conduction–convection and coupled consolidation analyses. The expressions of the minimum time‐step size for heat conduction–convection problems with both linear and quadratic elements reduce to those applicable to heat conduction problems if the effect of heat convection is not taken into account. For coupled consolidation analysis, time‐step constraints are obtained for three different types of elements, and the one for composite elements matches that in the literature. Finally, recommendations on how to handle the numerical ‘shock’ issues are suggested. Copyright © 2015 John Wiley & Sons, Ltd.     

An augmented Lagrangian algorithm for contact mechanics based on linear regression

The penalty method for the solution of contact problems yields an approximate satisfaction of the contact constraints. Augmentation schemes, which can be adopted to improve the solution, either include the contact forces as additional unknowns or are strongly affected by the penalty parameter and display a poor convergence rate. In a previous investigation, an unconventional augmentation scheme was proposed, on the basis of estimating the ‘exact’ values of the contact forces through linear interpolation of a database extracted by the previous converged states. An enhanced version of this method is proposed herein. With respect to the original method, the enhanced one eliminates some numerical problems and improves the regularity of the convergence path by carrying out the estimate through linear regression methods. The resulting convergence rate is superlinear, and the method is quite insensitive to the penalty parameter. The main underlying concept is that, within the iterative solution of a non‐linear problem, linear regression techniques may be used as a tool to ‘shoot’ faster to the final solution, on the basis of a set of intermediate data. The generality of this concept makes it potentially applicable to contact problems in more general settings, as well as to other categories of non‐linear problems. Copyright © 2012 John Wiley & Sons, Ltd.     

A new exact,closed‐form a priori global error estimator for second‐ and higher‐order time‐integration methods for linear elastodynamics

In our recent papers, we suggested a new two‐stage time‐integration procedure for linear elastodynamics problems and showed that for long‐term integration, time‐integration methods with zero numerical dissipation are very effective for all linear elastodynamics problems, including structural dynamics, wave propagation and impact problems. In this paper, we have derived a new exact, closed‐form a priori global error estimator for time integration of linear elastodynamics by the trapezoidal rule and the high‐order time continuous Galerkin (TCG) methods with zero numerical dissipation (these methods correspond to the diagonal of the Padé approximation table). The new a priori global error estimator allows the selection of the size (the number) of time increments for the indicated time‐integration methods at the prescribed accuracy as well as the comparison of the effectiveness of the second‐ and high‐order TCG methods at different observation times. A numerical example shows a good agreement between theoretical and numerical results. Copyright © 2011 John Wiley & Sons, Ltd.     

Hierarchical higher‐order dissipative methods for transient analysis

This work focuses on devising an efficient hierarchy of higher‐order methods for linear transient analysis, equipped with an effective dissipative action on the spurious high modes of the response. The proposed strategy stems from the Nørsett idea and is based on a multi‐stage algorithm, designed to hierarchically improve accuracy while retaining the desired dissipative behaviour. Computational efficiency is pursued by requiring that each stage should involve just one set of implicit equations of the size of the problem to be solved (as standard time integration methods) and, in addition, all the stages should share the same coefficient matrix. This target is achieved by rationally formulating the methods based on the discontinuous collocation approach. The resultant procedure is shown to be well suited for adaptive solution strategies. In particular, it embeds two natural tools to effectively control the error propagation. One estimates the local error through the next‐stage solution, which is one‐order more accurate, the other through the solution discontinuity at the beginning of the current time step, which is permitted by the present formulation. The performance of the procedure and the quality of the two error estimators are experimentally verified on different classes of problems. Some typical numerical tests in transient heat conduction and elasto‐dynamics are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints

This article presents asynchronous collision integrators and a simple asynchronous method treating nodal restraints. Asynchronous discretizations allow individual time step sizes for each spatial region, improving the efficiency of explicit time stepping for finite element meshes with heterogeneous element sizes. The article first introduces asynchronous variational integration being expressed by drift and kick operators. Linear nodal restraint conditions are solved by a simple projection of the forces that is shown to be equivalent to RATTLE. Unilateral contact is solved by an asynchronous variant of decomposition contact response. Therein, velocities are modified avoiding penetrations. Although decomposition contact response is solving a large system of linear equations (being critical for the numerical efficiency of explicit time stepping schemes) and is needing special treatment regarding overconstraint and linear dependency of the contact constraints (for example from double‐sided node‐to‐surface contact or self‐contact), the asynchronous strategy handles these situations efficiently and robust. Only a single constraint involving a very small number of degrees of freedom is considered at once leading to a very efficient solution. The treatment of friction is exemplified for the Coulomb model. Special care needs the contact of nodes that are subject to restraints. Together with the aforementioned projection for restraints, a novel efficient solution scheme can be presented. The collision integrator does not influence the critical time step. Hence, the time step can be chosen independently from the underlying time‐stepping scheme. The time step may be fixed or time‐adaptive. New demands on global collision detection are discussed exemplified by position codes and node‐to‐segment integration. Numerical examples illustrate convergence and efficiency of the new contact algorithm. Copyright © 2013 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.     

The maximum principle violations of the mixed‐hybrid finite‐element method applied to diffusion equations

The abundant literature of finite‐element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial–boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time‐dependent solutions for such problems during small time duration obtained by using a non‐conforming mixed‐hybrid finite‐element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite‐difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non‐physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial–boundary value problem will be presented. One of the propositions given to avoid any non‐physical oscillations is to use the mass‐lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd.     

Studies of Newmark method for solving nonlinear systems: (II) Verification and guideline

Abstract

Numerical properties of the Newmark method in the solution of nonlinear systems derived in the accompanying paper are thoroughly confirmed with numerical examples herein. It seems that analytical results can reveal the insight of the Newmark method in the step‐by‐step solution of linear and nonlinear systems. Although the constant average acceleration method is unconditionally stable for linear elastic systems these explorations confirm that it might lead to instability for nonlinear systems. In addition, numerical accuracy for period distortion and amplitude change is also shown to be consistent with the analytical predictions. Therefore, the performance of the Newmark method in the step‐by‐step solution of nonlinear systems is well investigated. As a result, a rough guideline to yield accurate solutions for the use of step‐by‐step integration methods to solve nonlinear systems is proposed.     

A geometrical‐based contact algorithm using a barrier method

Most methods employed in the numerical solution of contact problems in finite element simulations rely on equality‐based optimization methods. Typically, a gap function which is non‐differentiable at the point of contact is used in these kind of approaches. The gap function can be seen as the Macaulay bracket of some distance function, where the latter is differentiable at the point of contact. In this article, we propose to use the distance function directly instead of using the gap function. This will give rise to a formulation involving inequality constraints. This approach eliminates the artificially introduced non‐differentiability. To this end we propose a barrier algorithm as the method of choice to solve the problem. The method originates in optimization literature, where convergence proofs for the method are available. Copyright © 2001 John Wiley & Sons, Ltd.     

A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact,joints, and friction

We present a hard constraint, linear complementarity based, method for the simulation of stiff multibody dynamics with contact, joints and friction. The approach uses a linearization of the modified trapezoidal method, incorporates a Poisson restitution model at collision, and solves only one linear complementarity problem per time step when no collisions are encountered. We prove that, under certain assumptions, the method has order two, a fact that is also demonstrated by our numerical simulations. For the unconstrained (ODE) case, the method achieves second‐order convergence and absolute stability while solving only one linear system per step. When we use a special approximation of the Jacobian matrix for the case where the stiff forces originate in springs and dampers attached to two points in the system, the linear complementarity problem can be solved for any value of the time step and numerical simulation demonstrate that the method is stiffly stable. The method was implemented in UMBRA, an industrial‐grade virtual prototyping software. Copyright © 2005 John Wiley & Sons, Ltd.     

Solving the advection–diffusion equation on unstructured meshes with discontinuous/mixed finite elements and a local time stepping procedure

Explicit schemes are known to provide less numerical diffusion in solving the advection–diffusion equation, especially for advection‐dominated problems. Traditional explicit schemes use fixed time steps restricted by the global CFL condition in order to guarantee stability. This is known to slow down the computation especially for heterogeneous domains and/or unstructured meshes. To avoid this problem, local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local CFL condition have been developed. In this paper, a local time stepping approach is used with a numerical model based on discontinuous Galerkin/mixed finite element methods to solve the advection–diffusion equation. The developments are detailed for general unstructured triangular meshes. Numerical experiments are performed to show the efficiency of the numerical model for the simulation of (i) the transport of a solute on highly unstructured meshes and (ii) density‐driven flow, where the velocity field changes at each time step. The model gives stable results with significant reduction of the computational cost especially for the non‐linear problem. Moreover, numerical diffusion is also reduced for highly advective problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A semi‐implicit time‐stepping model for frictional compliant contact problems

In this paper, we formulate a semi‐implicit time‐stepping model for multibody mechanical systems with frictional, distributed compliant contacts. Employing a polyhedral pyramid model for the friction law and a distributed, linear, viscoelastic model for the contact, we obtain mixed linear complementarity formulations for the discrete‐time, compliant contact problem. We establish the existence and finite multiplicity of solutions, demonstrating that such solutions can be computed by Lemke's algorithm. In addition, we obtain limiting results of the model as the contact stiffness tends to infinity. The limit analysis elucidates the convergence of the dynamic models with compliance to the corresponding dynamic models with rigid contacts within the computational time‐stepping framework. Finally, we report numerical simulation results with an example of a planar mechanical system with a frictional contact that is modelled using a distributed, linear viscoelastic model and Coulomb's frictional law, verifying empirically that the solution trajectories converge to those obtained by the more traditional rigid‐body dynamic model. Copyright © 2004 John Wiley Sons, Ltd.     

Tuned vibration control of overhead line conductors

Linear and non‐linear theoretical and numerical analysis of ultimate response of overhead line conductors is treated in present paper. Interactive linear and non‐linear conditions in ultimate response are considered. Numerical solution of non‐linear problems appearing is made using the updated Lagrangian formulation of motion. Each step of the iteration approaches the solution of linear problem and the feasibility of the parallel processing FETM technique with adaptive mesh refinement and substructuring for non‐linear ultimate wave propagation and ultimate transient dynamic analysis is established. Some numerical results demonstrating current applicabilities and efficiency of procedures suggested are submitted. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical solution of heat and mass impulse transfer problems

Conclusion To solve special classes of stationary and nonstationary two-dimensional problems of momentum, heat, and mass transfer, we developed various packages of applied programs which, after the introduction of specific data, corresponding to a specific statement of the problem, and the development of their program provide its numerical solution. The use of fully implicit approximation methods and of direct solution of the linearized equations provides the possibility of developing effective numerical algorithms, making it possible to obtain the solution in a wide region of Reynolds and Rayleigh numbers.These methods can also be used for numerical investigation of transport processes in turbulent flows and in rheological media. In principle, they can be generalized to solving three-dimensional problems. In this case, however, for a small value of the grid step the number of equations becomes very large, and high-power computers are needed to realize these methods.Translated from German into Russian by E. F. Nogotov.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 49, No. 5, pp. 860–874, November, 1985.     

The bipenalty method for arbitrary multipoint constraints

In finite element (FE) analysis, traditional penalty methods impose constraints by adding virtual stiffness to the FE system. In dynamics, this can decrease the critical time step of the system when conditionally stable time integration schemes are used by introducing spurious modes with high eigenfrequencies. Recent studies have shown that using mass penalties alongside traditional stiffness penalties can mitigate this effect for systems with a one single‐point constraint. In the present work, we extend this finding to include systems with an arbitrary set of multipoint constraints. By analysing the generalised eigenvalue problem, we show that the values of spurious eigenfrequencies may be controlled by the choice of stiffness and mass penalty parameters. The method is demonstrated using numerical examples, including a one‐dimensional contact–impact formulation and a two‐dimensional crack propagation analysis. The results show that constraint imposition using the bipenalty method can be employed such that the critical time step of an analysis is unaffected, whereas also displaying superiority over the mass penalty method in terms of accuracy and versatility. Copyright © 2012 John Wiley & Sons, Ltd.     

RBF meshless modeling of non-Newtonian Hele–Shaw flow

In this paper we consider the problem of injecting a non-Newtonian fluid into a thin cavity. Using the Hele–Shaw approximation the problem reduces to a moving boundary problem in which the pressure is described by a 2D, nonlinear, elliptic equation. Mesh-free methods are very well suited for the numerical solution of moving boundary problems since no remeshing is needed at each time step to correctly represent the boundary. Among these methods, we have chosen the asymmetric RBF collocation method (Kansa's method) to compute the pressure distribution. Kansa's method is truly mesh-free, and is one of the most frequently used methods due to its accuracy and ease of implementation. Once the pressure is known, the velocity at each point in the moving front is computed and, therefore, the location of the front can be updated. To efficiently model the front motion we use the level set method, which is very accurate, and can handle both front collisions and front break-ups without difficulty.     

LATIN: A new view and an extension to wave propagation in nonlinear media

The LATIN (acronym of LArge Time INcrement) method was originally devised as a non‐incremental procedure for the solution of quasi‐static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time‐dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. Copyright © 2017 John Wiley & Sons, Ltd.     

A comparison of direct and preconditioned iterative techniques for sparse,unsymmetric systems of linear equations

In this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric, sparse systems of linear algebraic equations. These problems occur in finite difference and finite element simulations of semiconductor devices, and fluid flow problems. We consider five iterative methods that appear to be the most promising for this class of problems: the biconjugate gradient method, the conjugate gradient squared method, the generalized minimal residual method, the generalized conjugate residual method and the method of orthogonal minimization. Each of these methods was tested using similar preconditioning (incomplete LU factorization) on a set of large, sparse matrices arising from finite element simulation of semiconductor devices. Results are shown where we compare the computation time and memory requirements for each of these methods against one another, as well as against a direct method that uses LU factorization to solve these problems. The results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large, sparse systems of equations, and can offer significant savings in storage and CPU time.     

Numerical integration of the stiff dynamics of geometrically exact shells: an energy‐dissipative momentum‐conserving scheme

This paper presents a new family of time‐stepping algorithms for the integration of the dynamics of non‐linear shells. We consider the geometrically exact shell theory involving an inextensible director field (the so‐called five‐parameter shell model). The main characteristic of this model is the presence of the group of finite rotations in the configuration manifold describing the deformation of the solid. In this context, we develop time‐stepping algorithms whose discrete solutions exhibit the same conservation laws of linear and angular momenta as the underlying physical system, and allow the introduction of a controllable non‐negative energy dissipation to handle the high numerical stiffness characteristic of these problems. A series of algorithmic parameters for the different components of the deformation of the shell (i.e. membrane, bending and transverse shear) fully control this numerical dissipation, recovering existing energy‐momentum schemes as a particular choice of these algorithmic parameters. We present rigorous proofs of the numerical properties of the resulting algorithms in the full non‐linear range. Furthermore, it is argued that the numerical dissipation is introduced in the high‐frequency range by considering the proposed algorithm in the context of a linear problem. The finite element implementation of the resulting methods is described in detail as well as considered in the final arguments proving the aforementioned conservation/dissipation properties. We present several representative numerical simulations illustrating the performance of the newly proposed methods. The robustness gained over existing methods in these stiff problems is confirmed in particular. Copyright © 2002 John Wiley & Sons, Ltd.