DESIGN OF ENERGY CONSERVING ALGORITHMS FOR FRICTIONLESS DYNAMIC CONTACT PROBLEMS

This paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum, angular momentum, and energy in finite element simulations. It is seen that a Lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties. A related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized. This penalty method sacrifices the energy conservation property, but is dissipative under all conditions of changing contact so that the global algorithm remains stable. Notably, it is also shown that augmented Lagrangian iteration utilizing this penalty kernel reproduces the energy conserving (i.e. Lagrange multiplier) solution to any desired degree of accuracy. The result is a robust, stable method even in the context of large deformations, as is shown by some representative numerical examples. In particular, the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations. © 1997 by 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.     

Locally constrained projections on grids

A technique is formulated for projecting vector fields from one unstructured computational grid to another grid so that a constraint condition such as a conservation property holds at the cell or element level on the ‘receiving’ grid. The approach is based on ideas from constrained optimization and certain mixed or multiplier‐type finite element methods in which Lagrange multipliers are introduced on the elements to enforce the constraint. A theoretical analysis and estimates for the associated saddle‐point problem are developed and a new algorithm is proposed for efficient solution of the resulting discretized problem. In the algorithm a reduced Schur's complement problem is constructed for the multipliers and the projected velocity computation reduces to a post‐processing calculation. In some instances the reduced system matrix can be factored so that repeated projections involve little more than forward and backward substitution sweeps. Numerical tests with an element of practical interest demonstrate optimal rate of convergence for the projected velocities and verify the local conservation property to expected machine precision. A practical demonstration for environmental simulation of Florida Bay concludes the study. Copyright © 2001 John Wiley & Sons, Ltd.     

Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method

In this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy‐momentum–preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty‐based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time‐stepping algorithm is shown.     

A domain coupling method for finite element digital image correlation with mechanical regularization: Application to multiscale measurements and parallel computing

A general method is proposed to couple two subregions analyzed with finite element digital image correlation even when using a mechanical regularization (regularized digital image correlation). A Lagrange multiplier is introduced to stitch both displacements fields in order to recover continuity over the full region of interest. Another interface unknown is introduced to ensure, additionally, the equilibrium of the mechanical models used for regularization. As a first application, the method is used to perform a single measurement from images at two different resolutions. Secondly, the method is also extended to parallel computing in regularized digital image correlation. The problem is formulated at the interface and solved with a Krylov‐type algorithm. A dedicated preconditioner is proposed to significantly accelerate convergence. The resulting method is a good candidate for the analysis of large data sets. Copyright © 2016 John Wiley & Sons, Ltd.     

Improved implicit integrators for transient impact problems—geometric admissibility within the conserving framework

The value of energy and momentum conserving algorithms has been well established for the analysis of highly non‐linear systems, including those characterized by the nonsmooth non‐linearities of an impact event. This work proposes an improved integration scheme for frictionless dynamic contact, seeking to preserve the stability properties of exact energy and momentum conservation without the heretofore unavoidable compromise of violating geometric admissibility as established by the contact constraints. The physically motivated introduction of a discrete contact velocity provides an algorithmic framework that ensures exact conservation locally while remaining independent of the choice of constraint treatment, thus making full conservation equally possible in conjunction with a penalty regularization as with an exact Lagrange multiplier enforcement. The discrete velocity effects are incorporated as a post‐convergence update to the system velocities, and thus have no direct effect on the non‐linear solution of the displacement equilibrium equation. The result is a robust implicit algorithmic treatment of dynamic frictionless impact, appropriate for large deformations and fully conservative for a range of geometric constraints. Copyright © 2001 John Wiley & Sons, Ltd.     

Methods for connecting dissimilar three‐dimensional finite element meshes

Two methods are presented for connecting dissimilar three‐dimensional finite element meshes. The first method combines the concept of master and slave surfaces with the uniform strain approach for finite elements. By modifying the boundaries of elements on a slave surface, corrections are made to element formulations such that first‐order patch tests are passed. The second method is based entirely on constraint equations, but only passes a weaker form of the patch test for non‐planar surfaces. Both methods can be used to connect meshes with different element types. In addition, master and slave surfaces can be designated independently of relative mesh resolutions. Example problems in three‐dimensional linear elasticity are presented. Published in 2000 by John Wiley & Sons, Ltd.     

Reliable second‐order hexahedral elements for explicit methods in nonlinear solid dynamics

Second‐order hexahedral elements are common in static and implicit dynamic finite element codes for nonlinear solid mechanics. Although probably not as popular as first‐order elements, they can perform better in many circumstances, particularly for modeling curved shapes and bending without artificial hourglass control or incompatible modes. Nevertheless, second‐order brick elements are not contained in typical explicit solid dynamic programs and unsuccessful attempts to develop reliable ones have been reported. In this paper, 27‐node formulations, one for compressible and one for nearly incompressible materials, are presented and evaluated using non‐uniform row summation mass lumping in a wide range of nonlinear example problems. The performance is assessed in standard benchmark problems and in large practical applications using various hyperelastic and inelastic material models and involving very large strains/deformations, severe distortions, and contact‐impact. Comparisons are also made with several first‐order elements and other second‐order hexahedral formulations. The offered elements are the only second‐order ones that performed satisfactorily in all examples, and performed generally at least as well as mass lumped first‐order bricks. It is shown that the row summation lumping is vital for robust performance and selection of Lagrange over serendipity elements and high‐order quadrature rules are more crucial with explicit than with static/implicit methods. Whereas the reliable performance is frequently attained at significant computational expense compared with some first‐order brick types, these elements are shown to be computationally competitive in flexure and with other first‐order elements. These second‐order elements are shown to be viable for large practical applications, especially using today's parallel computers. Published in 2010 by 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 regularized force‐based beam element with a damage–plastic section constitutive law

A new beam finite element is presented, with a generalized section constitutive law based on damage mechanics and plasticity, to analyse the cyclic structural response of plane frames. Both displacement‐based and force‐based (FB) approaches are used and compared, to demonstrate the significant advantages of the FB formulation in the presence of material non‐linearity. In order to overcome the analytical problems and the pathological mesh dependency of the numerical response in the presence of strain‐softening post‐peak behaviour, a classical non‐local regularization procedure is adopted first, based on the integral definition of the associated variable governing the damaging evolution process. Subsequently, for the FB element a new simple regularization technique is proposed based on a selected integration procedure along the element length, which predefines the location of the Gauss points in the beam region, where the localization phenomena take place. As for the other computational aspects, an iterative element state determination is adopted for the FB formulation and a local predictor–corrector algorithm is used to solve the incremental evolution problems of the damage and plastic internal variables. Finally, some examples are shown on simple beams and frames, subjected to monotonically increasing and cyclic loading conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Elimination of spurious static modes in meshes of hybrid compatible triangular and tetrahedral finite elements

In the assumed displacement, or primal, hybrid finite element method, the requirements of continuity of displacements across the sides are regarded as constraints, imposed using Lagrange multipliers. In this paper, such a formulation for linear elasticity, in which the polynomial approximation functions are not associated with nodes, is presented. Elements with any number of sides may be easily used to create meshes with irregular vertices, when performing a non‐uniform h‐refinement. Meshes of non‐uniform degree may be easily created, when performing an hp‐refinement. The occurrence of spurious static modes in meshes of triangular elements, when compatibility is strongly enforced, is discussed. An algorithm for the automatic selection, based on the topology of a mesh of triangular elements, of the sides in which to decrease the degree of the approximation functions, in order to eliminate all these spurious modes and preserve compatibility, is presented. A similar discussion is presented for the occurrence of spurious static modes in meshes of tetrahedral elements. An algorithm, based on heuristic criteria, that succeeded in eliminating these spurious modes and preserving compatibility in all the meshes of tetrahedral elements of uniform degree that were tested, is also presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Three‐dimensional quasi‐static frictional contact by using second‐order cone linear complementarity problem

A new formulation is presented for the three‐dimensional incremental quasi‐static problems with unilateral frictional contact. Under the assumptions of small rotations and small strains, a second‐order cone linear complementarity problem is formulated, which consists of complementarity conditions defined by bilinear functions and second‐order cone constraints. The equilibrium configurations are obtained by using a combined smoothing and regularization method for the second‐order cone complementarity problem. Copyright © 2005 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.     

A modified least‐squares mixed finite element with improved momentum balance

The main goal of this contribution is to provide an improved mixed finite element for quasi‐incompressible linear elasticity. Based on a classical least‐squares formulation, a modified weak form with displacements and stresses as process variables is derived. This weak form is the basis for a finite element with an advanced fulfillment of the momentum balance and therefore with a better performance. For the continuous approximation of stresses and displacements on the triangular and tetrahedral elements, lowest‐order Raviart–Thomas and linear standard Lagrange interpolations can be used. It is shown that coercivity and continuity of the resulting asymmetric bilinear form could be established with respect to appropriate norms. Further on, details about the implementation of the least‐squares mixed finite elements are given and some numerical examples are presented in order to demonstrate the performance of the proposed formulation. Copyright © 2009 John Wiley & Sons, Ltd.     

A density field parametrization for topology optimization using Bernstein elements

A new way of describing the density field in density‐based topology optimization is introduced. The new method uses finite elements constructed from Bernstein polynomials rather than the more common Lagrange polynomials. Use of the Bernstein finite elements allows higher‐order elements to be used in the density‐field interpolation without producing unrealistic density values, ie, values lower than zero or higher than one. Results on several test problems indicate that using the higher‐order Bernstein elements produces optimal designs with sharper estimates of the optimal boundary on coarse design meshes. However, higher‐order elements are also required in the structural analysis to prevent the appearance of unrealistic material distributions. The Bernstein element density interpolation can be combined with adaptive mesh refinement to further improve design accuracy even on design domains with complex geometry.     

Development of a finite element contact analysis algorithm for charged‐hydrated soft tissues with large sliding

This paper focuses on a finite element analysis of contact phenomena with large sliding between charged‐hydrated biological soft tissues, such as articular cartilages, based on the triphasic theory. The impenetrability constraint between the contacting bodies and the continuity of the interstitial fluid and ion phases at the contact surfaces are imposed by applying a Lagrange multiplier approach with the contact pressure, chemical potential of the fluid and electrochemical potentials of ions as Lagrange multipliers. A node‐to‐segment one‐pass approach is adopted to cope with large deformations and sliding between the contact surfaces. To pass the contact patch test, contact boundary integrations are performed on both the master and slave contact surfaces. On the other hand, the degrees of freedom of the multipliers at the master nodes are eliminated by projecting the master nodes onto the slave surface to avoid overconstraint. The effectiveness of the proposed algorithm is verified by a couple of numerical examples, in which continuous distributions of displacement, fluid flow, ionic molar flow and Lagrange multipliers on or across the contact surface are confirmed. Copyright © 2008 John Wiley & Sons, Ltd.     

Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes

Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes are presented. The elements use the linear interpolation functions of the original mesh, but each element is associated with a single node. As a result, a favourable constraint ratio for the volumetric response is obtained for problems in solid mechanics. The uniform strain elements do not require the introduction of additional degrees of freedom and their performance is shown to be significantly better than that of three‐node triangular or four‐node tetrahedral elements. In addition, nodes inside the boundary of the mesh are observed to exhibit superconvergent behaviour for a set of example problems. Published in 2000 by John Wiley & Sons, Ltd.     

On the performance of high‐order finite elements with respect to maximum principles and the nonnegative constraint for diffusion‐type equations

The main aim of this paper is to document the performance of p‐refinement with respect to maximum principles and the nonnegative constraint. The model problem is steady‐state anisotropic diffusion with decay (which is a second‐order elliptic partial differential equation). We consider the standard single‐field formulation based on the Galerkin formalism and two least squares‐based mixed formulations. We employ Lagrange polynomials with unequal‐spaced points, and polynomials of order p = 1 to p = 10 are used. It is shown that the violation of the nonnegative constraint cannot be overcome with p‐refinement alone for anisotropic diffusion. We shall illustrate the performance of p‐refinement by using several representative problems. The intended outcomes of the paper are twofold. First, this study will caution the users of high‐order approximations about their performance with respect to maximum principles and the nonnegative constraint. Second, this study will help researchers develop new methodologies for enforcing maximum principles and the nonnegative constraint under high‐order approximations. Copyright © 2012 John Wiley & Sons, Ltd.     

A multiplicative finite strain finite element framework for the modelling of semicrystalline polymers and polycarbonates

This paper presents a phenomenological model for the simulation and analysis of stress‐induced orientational hardening in semicrystalline polymers and polycarbonates at finite strains. The notion of intermediate (local) stress‐free configuration is used to develop a set of constitutive equations, and its relation to the multiple natural (stress‐free) configurations in the class of materials being considered here is discussed. A hyperelastic stored energy function, written with respect to the intermediate stress‐free configuration is presented to model the finite elastic response. It is then combined with the J₂‐flow theory to model the finite inelastic response. The isochoric constraint during inelastic deformation is treated via an exact multiplicative decomposition of the deformation gradient into volume‐preserving and spherical parts. The numerical solution algorithm is based on the use of operator splitting technique that results in a product formula algorithm with elastic‐predictor/inelastic‐corrector components. Numerical results are presented to show the behaviour of the model. Copyright © 2000 John Wiley & Sons, Ltd.     

Accurate and consistent FE modelling of soft docking of micro/nano paired-satellites using variational inequalities

Time-dependent variational inequality-based formulations describing the behavior of elastodynamic contact problems have been used to model the soft docking of micro/nano paired-satellites. The solution strategy is based upon the iterative use of two subproblems. In the first subproblem, the technique of Quadratic Programming was used to predict the contact surface and the stresses acting on it. In the second subproblem, the technique of Lagrange multipliers was used to impose the boundary conditions and contact constraint obtained in step I. The solution accounts for the effect of friction through the use of an appropriate regularization technique in the virtual work expressions. The accuracy of the proposed variational inequality (VI) model of the soft docking process is validated by comparing the VI results with commercially available software. Finally, the effects of the bending stiffness of the docking probe, the friction between the candidate contact surfaces are investigated and analyzed using the newly developed model.