Combination of Newmark method and natural element method for elastodynamics

The natural element method (NEM) is a meshless method. The trial and test functions of the NEM are constructed using natural neighbor interpolations which are based on the Voronoi tessellation of a set of nodes. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which makes imposition of essential boundary conditions easy to implement. We investigate the performance of the NEM combined with the Newmark method for problems of elastodynamics in this article. Applications are considered for a cantilever beam with different initial load conditions. The NEM numerical results are compared with the finite element method. NEM shows promise for these applications.     

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 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.     

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.     

A mortar method for energy‐momentum conserving schemes in frictionless dynamic contact problems

In the present work the mortar method is applied to planar large deformation contact problems without friction. In particular, the proposed form of the mortar contact constraints is invariant under translations and rotations. These invariance properties lay the foundation for the design of energy‐momentum time‐stepping schemes for contact–impact problems. The iterative solution procedure is embedded into an active set algorithm. Lagrange multipliers are used to enforce the mortar contact constraints. The solution of generalized saddle point systems is circumvented by applying the discrete null space method. Numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy‐momentum scheme. Copyright © 2008 John Wiley & Sons, Ltd.     

Modified Newmark family for non‐linear dynamic analysis

A simple modification of the Newmark scheme is suggested, which keeps the total energy of mechanical systems constant and thereby enhances the unconditional stability in non‐linear dynamic analysis. Numerical damping, which is formulated on an energy basis, is also introduced and the dissipative character of the algorithm is guaranteed in the non‐linear regime. Representative examples in support of such a modified Newmark family are presented. Copyright © 2004 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 stabilized meshfree reproducing kernel‐based method for convection–diffusion problems

In this paper, we develop a meshfree particle‐based method for convection–diffusion problems. Discretization is performed by using piecewise constant kernels. The stabilized scheme is based on a new upwind kernel. We show that accurate and stable scheme can be obtained by using purpose‐built kernels. It also shown that under some conditions the classical optimal finite difference scheme can be derived by the new method. Several numerical tests validate the method. Copyright © 2011 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.     

Study on sub‐cycling for flexible multi‐body dynamics based on Newmark method

A sub‐cycling algorithm presented by Belytschko and Mullen (Int. J. Numer. Meth. Engng 1978; 12 (10):1575–1586) has been successfully applied in the finite element method. However, the problem of how to apply the sub‐cycling to flexible multi‐body dynamics (FMD) still lacks investigation. This paper presents a Newmark‐based sub‐cycling, which is suitable for solving condensed FMD models. Common‐step update formulae and sub‐step update formula for the sub‐cycling are established based on the original Newmark integration algorithm. Stability of the procedure is validated by means of energy balance checking during the integral process. Numerical examples indicate that the sub‐cycling is able to enhance the computational efficiency without dropping accuracy greatly. Copyright © 2007 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.     

Development of contact algorithm for three‐dimensional numerical manifold method

This paper customizes a contact detection and enforcing scheme to fit the three‐dimensional (3‐D) numerical manifold method (NMM). A hierarchical contact system is established for efficient contact detection. The mathematical mesh, a unique component in the NMM, is utilized for global searching of possible contact blocks and elements, followed by the local searching to identify primitive hierarchies. All the potential contact pairs are then transformed into one of the two essential entrance modes: point‐to‐plane and crossing‐lines modes, among which real contact pairs are detected through a unified formula. The penalty method is selected to enforce the contact constraints, and a general contact solution procedure in the 3‐D NMM is established. Because of the implicit framework, an open‐close iteration is performed within each time step to determine the correct number of contact pairs among multi‐bodies and to achieve complete convergence of imposed contact force at corresponding position. The proposed contact algorithm extensively utilizes most of the original components of the NMM, namely, the mathematical mesh/cells and the manifold elements, as well as the external components associated with contacts, such as the contact body, the contact facet and the contact vertex. In particular, the utilization of two mutually approaching mathematical cells is efficient in detecting contacting territory, which makes this method particularly effective for both convex and non‐convex bodies. The validity and accuracy of the proposed contact algorithm are verified and demonstrated through three benchmark problems. Copyright © 2013 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.     

Vertex‐to‐face contact searching algorithm for three‐dimensional frictionless contact problems

This article presents a new vertex‐to‐face contact searching algorithm for the three‐dimensional (3‐D) discontinuous deformation analysis (DDA). In this algorithm, topology is applied to the contact rule when any two polyhedrons are close to each other. Attempt is made to expand the original contact searching algorithm from two‐dimensional (2‐D) to 3‐D DDA. Examples are provided to demonstrate the new contact rule for vertex‐to‐face contacts between two polyhedrons with planar boundaries. Copyright © 2005 John Wiley & Sons, Ltd.     

A mortar‐finite element formulation for frictional contact problems

In this paper a finite element formulation is developed for the solution of frictional contact problems. The novelty of the proposed formulation involves discretizing the contact interface with mortar elements, originally proposed for domain decomposition problems. The mortar element method provides a linear transformation of the displacement field for each boundary of the contacting continua to an intermediate mortar surface. On the mortar surface, contact kinematics are easily evaluated on a single discretized space. The procedure provides variationally consistent contact pressures and assures the contact surface integrals can be evaluated exactly. Copyright © 2000 John Wiley & Sons, Ltd.     

A meshless singular boundary method for three‐dimensional elasticity problems

This study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration‐free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

A dual Lagrange method for contact problems with regularized contact conditions

This paper presents an algorithm for solving quasi‐static, non‐linear elasticity contact problems without friction in the context of rough surfaces. Here, we want to model the transition from soft to hard contact in case of rough surfaces on the micro‐scale. The popular dual mortar method is used to enforce the contact constraints in a variationally consistent way without increasing the algebraic system size. The algorithm is deduced from a perturbed Lagrange formulation and combined with mass‐lumping techniques to exploit the full advantages of the duality pairing. This leads to a regularized saddle point problem, for which a non‐linear complementary function and thus a semi‐smooth Newton method can be derived. Numerical examples demonstrate the applicability to industrial problems and show a good agreement to experimentally obtained results. Copyright © 2014 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.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

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.