A 3D beam‐to‐beam contact finite element for coupled electric–mechanical fields

In this paper the formulation of an electric–mechanical beam‐to‐beam contact element is presented. Beams with circular cross‐sections are assumed to get in contact in a point‐wise manner and with clean metallic surfaces. The voltage distribution is influenced by the contact mechanics, since the current flow is constricted to small contacting spots. Therefore, the solution is governed by the contacting areas and hence by the contact forces. As a consequence the problem is semi‐coupled with the mechanical field influencing the electric one. The electric–mechanical contact constraints are enforced with the penalty method within the finite element technique. The virtual work equations for the mechanical and electric fields are written and consistently linearized to achieve a good level of computational efficiency with the finite element method. The set of equations is solved with a monolithic approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A novel contact interaction formulation for voxel‐based micro‐finite‐element models of bone

Voxel‐based micro‐finite‐element (μFE) models are used extensively in bone mechanics research. A major disadvantage of voxel‐based μFE models is that voxel surface jaggedness causes distortion of contact‐induced stresses. Past efforts in resolving this problem have only been partially successful, ie, mesh smoothing failed to preserve uniformity of the stiffness matrix, resulting in (excessively) larger solution times, whereas reducing contact to a bonded interface introduced spurious tensile stresses at the contact surface. This paper introduces a novel “smooth” contact formulation that defines gap distances based on an artificial smooth surface representation while using the conventional penalty contact framework. Detailed analyses of a sphere under compression demonstrated that the smooth formulation predicts contact‐induced stresses more accurately than the bonded contact formulation. When applied to a realistic bone contact problem, errors in the smooth contact result were under 2%, whereas errors in the bonded contact result were up to 42.2%. We conclude that the novel smooth contact formulation presents a memory‐efficient method for contact problems in voxel‐based μFE models. It presents the first method that allows modeling finite slip in large‐scale voxel meshes common to high‐resolution image‐based models of bone while keeping the benefits of a fast and efficient voxel‐based solution scheme.     

A three‐dimensional surface stress tensor formulation for simulation of adhesive contact in finite deformation

A three‐dimensional surface adhesive contact formulation is proposed to simulate macroscale adhesive contact interaction characterized by the van der Waals interaction between arbitrarily shaped deformable continua under finite deformation. The proposed adhesive contact formulation uses a double‐layer surface integral to replace the conventional double volume integration to compute the adhesive contact force vector. Considering nonlinear finite deformation, we have derived the surface stress tensor and the corresponding tangent stiffness matrix in a Galerkin weak formulation. With the surface stress formulation, the adhesive contact problems are solved in the framework of nonlinear continuum mechanics by using the standard Lagrange finite element method. Surface stress tensors are formulated for both interacting bodies. Numerical examples show that the proposed surface contact algorithm is accurate, efficient, and reliable for three‐dimensional adhesive contact problems of large deformations for both quasi‐static and dynamic simulations. Copyright © 2015 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.     

Adaptive methods for thermomechanical coupled contact problems

In this paper an adaptive method for the analysis of thermomechanical coupled multi‐body contact problems is presented. The method is applied to non‐linear elastic solids undergoing finite (thermal) deformations. The contact model considers non‐linear pressure‐dependent heat flux as well as frictional heating in the interface. A time–space‐finite element discretization of the governing equations is formulated including unilateral constraints due to contact. A staggered solution algorithm has been constructed that allows an independent spatial discretization of the coupled subproblems. A posteriori projection‐based error estimators, which enforce implicitly the special boundary conditions due to thermal contact, are used to control the spatial discretization as well as the adaptive time stepping. Numerical examples are presented to corroborate the applicability of the adaptive algorithm to the considered problem type. Copyright © 2004 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.     

Frame‐indifferent beam finite elements based upon the geometrically exact beam theory

This paper describes a new beam finite element formulation based upon the geometrically exact beam theory. In contrast to many previously proposed beam finite element formulations the present discretization approach retains the frame‐indifference (or objectivity) of the underlying beam theory. The space interpolation of rotational degrees‐of‐freedom is circumvented by the introduction of a reparameterization of the weak form corresponding to the equations of motion of the geometrically exact beam theory. Copyright © 2002 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.     

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 spatially curved‐beam element with warping and Wagner effects

This paper presents a new spatially curved‐beam element with warping and Wagner effects that can be used for the non‐linear large displacement analysis of members that are curved in space. The non‐linear behaviour of members curved in space shows that the Wagner effects are substantial in the large twist rotation analysis. Most existing finite beam element models, such as ABAQUS and ANSYS cannot predict the non‐linear large displacement response of members curved in space correctly because the Wagner effects, viz. the Wagner moment and the corresponding finite strain terms, have not been considered in these finite beam elements. As a consequence, these finite beam elements do not provide correct predictions for the out‐of‐plane buckling and postbuckling behaviour of arches as well. In this paper, the symmetric tangent stiffness matrix has been derived based on the finite rotations parameterized by the conventional displacements. The warping and Wagner effects: both the Wagner moment and the corresponding finite strain terms and their constitutive relationship, are included in the spatially curved‐beam element. Two components of the initial curvature, the initial twist and their interactions with the displacements are also considered in the spatially curved‐beam element. This ensures that the large twist rotation analysis for the members curved in space is accurate. Comparisons with existing experimental, analytical and numerical results show that the spatially curved‐beam element is accurate and efficient for the non‐linear elastic analysis of curved members, buckling and postbuckling analysis of arches, and in its ability to predict large deflections and twist rotations in more arbitrarily curved members. Copyright © 2005 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.     

Non‐linear spatial Timoshenko beam element with curvature interpolation

The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid‐body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two‐node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N‐node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second‐order stress resultants are identified and the section response takes into account non‐linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite rotation quadrilateral element for multi‐layer beams

The paper presents the finite rotations beam equations derived on use of the generalized Reissner hypothesis with a scalar parameter for the transverse extension. The beam strain and change of curvature measures are obtained from the right stretch strain, and the virtual work is given for Biot‐type stress and couple resultants. The strain energy for the first‐order isotropic elastic material is assumed in terms of the right stretch strain, and constitutive equations for the beam stress and couple resultants are derived. Two finite rotation elements are developed from the derived beam equations: a beam element with the transverse stretch and a quadrilateral element. First, the beam element with the uniformly under‐integrated tangent operator is developed. Next, the formula linking the middle‐line variables and the interface variables of the beam is introduced consistently with the generalized Reissner kinematics. Linearization of this formula is performed, and the derived tangent operator is used to convert the two‐node beam element to a four‐node quadrilateral. Both the finite elements have been tested on several numerical examples, some of highly non‐linear characteristics, and their accuracy is very good. It has been established that the quadrilateral element, which is intended for applications to multi‐layer beams, performs very well for high elemental aspect ratios, and can therefore be applied to modelling of very thin layers. Copyright © 1999 John Wiley & Sons, Ltd.     

A hierarchical sequential ALE poromechanics model for tire‐soil‐water interaction on fluid‐infiltrated roads

This paper introduces a hierarchical sequential arbitrary Lagrangian‐Eulerian (ALE) model for predicting the tire‐soil‐water interaction at finite deformations. Using the ALE framework, the interaction between a rolling pneumatic tire and the fluid‐infiltrated soil underneath will be captured numerically. The road is assumed to be a fully saturated two‐phase porous medium. The constitutive response of the tire and the solid skeleton of the porous medium is idealized as hyperelastic. Meanwhile, the interaction between tire, soil, and water will be simulated via a hierarchical operator‐split algorithm. A salient feature of the proposed framework is the steady state rolling framework. While the finite element mesh of the soil is fixed to a reference frame and moves with the tire, the solid and fluid constituents of the soil are flowing through the mesh in the ALE model according to the rolling speed of the tire. This treatment leads to an elegant and computationally efficient formulation to investigate the tire‐soil‐water interaction both close to the contact and in the far field. The presented ALE model for tire‐soil‐water interaction provides the essential basis for future applications, for example, to a path‐dependent frictional‐cohesive response of the consolidating soil and unsaturated soil, respectively. Copyright © 2017 John Wiley & Sons, Ltd.     

A mortar finite element approach for point,line, and surface contact

An approach for investigating finite deformation contact problems with frictional effects with a special emphasis on nonsmooth geometries such as sharp corners and edges is proposed in this contribution. The contact conditions are separately enforced for point contact, line contact, and surface contact by employing 3 different sets of Lagrange multipliers and, as far as possible, a variationally consistent discretization approach based on mortar finite element methods. The discrete unknowns due to the Lagrange multiplier approach are eliminated from the system of equations by employing so‐called dual or biorthogonal shape functions. For the combined algorithm, no transition parameters are required, and the decision between point contact, line contact, and surface contact is implicitly made by the variationally consistent framework. The algorithm is supported by a penalty regularization for the special scenario of nonparallel edge‐to‐edge contact. The robustness and applicability of the proposed algorithms are demonstrated with several numerical examples.     

An accurate and robust contact detection algorithm for particle‐solid interaction in combined finite‐discrete element analysis

When applying the combined finite‐discrete element method for analysis of dynamic problems, contact is often encountered between the finite elements and discrete elements, and thus an effective contact treatment is essential. In this paper, an accurate and robust contact detection algorithm is proposed to resolve contact problems between spherical particles, which represent rigid discrete elements, and convex quadrilateral mesh facets, which represent finite element boundaries of structural components. Different contact scenarios between particles and mesh facets, or edges, or vertices have been taken into account. For each potential contact pair, the contact search is performed in an hierarchical way starting from mesh facets, possibly going to edges and even further to vertices. The invalid contact pairs can be removed by means of two reasonable priorities defined in terms of geometric primitives and facet identifications. This hierarchical contact searching scheme is effective, and its implementation is straightforward. Numerical examples demonstrated the accuracy and robustness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.