A monolithic smoothing‐gap algorithm for contact‐impact based on the signed distance function

A new algorithm has been developed for smoothing the surfaces in finite element formulations of contact‐impact. A key feature of this method is that the smoothing is done implicitly by constructing smooth signed distance functions for the bodies. These functions are then employed for the computation of the gap and other variables needed for implementation of contact‐impact. The smoothed signed distance functions are constructed by a moving least‐squares approximation with a polynomial basis. Results show that when nodes are placed on a surface, the surface can be reproduced with an error of about one per cent or less with either a quadratic or a linear basis. With a quadratic basis, the method exactly reproduces a circle or a sphere even for coarse meshes. Results are presented for contact problems involving the contact of circular bodies. Copyright © 2002 John Wiley & Sons, Ltd.     

On the modelling of smooth contact surfaces using cubic splines

In this paper, a new strategy for the smooth representation of 2D contact surfaces is developed and implemented. The contact surfaces are modelled using cubic splines which interpolate the finite element nodes. These splines provide a unique surface normal vector and do not require prior knowledge of surface tangents and normals. C²‐continuous cubic splines are suitable for representing rigid contact surfaces, while C¹‐continuous Overhauser splines are shown to be most suitable for representing flexible contact surfaces. A consistent linearization of the kinematic contact constraints, based on the spline interpolation, is derived. The new spline‐based contact surface interpolation scheme does not influence the element calculations. Consequently, it can be easily implemented in standard FE codes. Several numerical examples are used to illustrate the advantages of the proposed smooth representation of contact surfaces. The results show a significantimprovement in accuracy compared to traditional piecewise element‐based surface interpolation. The predicted contact stresses are also less sensitive to the mismatch in the meshes of the different contacting bodies. This property is useful for problems where the contact area is unknown a priori and when there is significant tangential slip. 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.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 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 mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method

NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems. The contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding segmentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order NURBS and Lagrange polynomial elements. The contact constraints are satisfied exactly with the augmented Lagrangian formulation proposed by Alart and Curnier, whereby a Newton-like solution scheme is applied to solve the saddle point problem simultaneously for displacements and Lagrange multipliers. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. In both small and large deformation cases, the quality of the contact pressures is shown to improve significantly over that achieved with Lagrange discretizations. In large deformation and large sliding examples, the NURBS discretization provides an improved smoothness of the traction history curves. In both cases, increasing the order of the discretization is either beneficial or not influential when using isogeometric analysis, whereas it affects results negatively for Lagrange discretizations.     

An hp‐adaptive finite element method for scattering problems in computational electromagnetics

An hp‐adaptive finite element (FE) approach is presented for a reliable, efficient and accurate solution of 3D electromagnetic scattering problems. The radiation condition in the far field is satisfied automatically by approximation with infinite elements (IE). Near optimal discretizations that can effectively resolve local rapid variations in the scattered field are sought adaptively by mesh refinements blended with graded polynomial enrichments. The p‐enrichments need not be spatially isotropic. The discretization error can be controlled by a self‐adaptive process, which is driven by implicit or explicit a posteriori error estimates. The error may be estimated in the energy norm or in a quantity of interest. A radar cross section (RCS) related linear functional is used in the latter case. Adaptively constructed solutions are compared to pure uniform p approximations. Numerical, highly accurate, and fairly converged solutions for a number of generic problems are given and compared to previously published results. Copyright © 2004 John Wiley & Sons, Ltd.     

A high‐order generalized FEM for through‐the‐thickness branched cracks

This paper presents high‐order implementations of a generalized finite element method for through‐the‐thickness three‐dimensional branched cracks. This approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks, independently of the background finite element mesh. Representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies. The combination of local refinement at crack fronts and high‐order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

This paper focuses on the application of NURBS‐based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar‐based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C⁰‐continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 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.     

Hydrostatic fluid loading in non‐linear finite element analysis

Deformation dependent pressure loading on solid structures is created by the interaction of fluids with the deformable surface of a structure. Such fairly simple load models are valid for static and quasi‐static analyses and they are a very efficient tool to represent the influence of a fluid on the behaviour of structures. Completing previous studies on the deformation dependence of the loading with the assumption of finite gas volumes, the current contribution focuses on the influence of modifications of the size and shape of a finite structure containing an incompressible fluid with a free surface. The linearization of the corresponding virtual work expression necessary for a Newton‐type solution leads to additional terms for the volume dependence. Investigating these terms the conservativeness of the problem can be proven by the symmetry of the linearized form. Similarly to gas loading, the discretization of the structure with finite elements leads to standard stiffness matrix forms plus the so‐called load stiffness matrices and a rank‐one update for each filled structure part, if the loaded surface segments are identical with element surfaces. Some numerical examples show the effectiveness of the approach and the necessity to take the corresponding terms in the variational expression and the following linearization into account. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Parameter‐free,weak imposition of Dirichlet boundary conditions and coupling of trimmed and non‐conforming patches

We present a parameter‐free domain sewing approach for low‐order as well as high‐order finite elements. Its final form contains only primal unknowns; that is, the approach does not introduce additional unknowns at the interface. Additionally, it does not involve problem‐dependent parameters, which require an estimation. The presented approach is symmetry preserving; that is, the resulting discrete form of an elliptic equation will remain symmetric and positive definite. It preserves the order of the underlying discretization, and we demonstrate high‐order accuracy for problems of non‐matching discretizations concerning the mesh size h as well as the polynomial degree of the order of discretization p. We also demonstrate how the method may be used to model material interfaces, which may be curved and for which the interface does not coincide with the underlying mesh. This novel approach is presented in the context of the p‐version and B‐spline version of the finite cell method, an embedded domain method of high order, and compared with more classical methods such as the penalty method or Nitsche's method. Copyright © 2014 John Wiley & Sons, Ltd.     

A 3D contact smoothing method using Gregory patches

In this work, a method is developed for smoothing three‐dimensional contact surfaces. The method can be applied to both regular and irregular meshes. The algorithm employs Gregory patches to interpolate finite element nodes and provide tangent plane continuity between adjacent patches. The resulting surface interpolation is used to calculate gaps and contact forces, in a variationally consistent way, such that contact forces due to normal and frictional contact vary smoothly as slave nodes transition from one patch to the next. This eliminates the ‘chatter’ which typically occurs in a standard contact algorithm when a slave node is situated near a master facet edge. The elimination of this chatter provides a significant improvement in convergence behaviour, which is illustrated by a number of numerical examples. Furthermore, smoothed surfaces also provide a more accurate representation of the actual surface, such that resulting stresses and forces can be more accurately computed with coarse meshes in many problems. This fact is also demonstrated by the numerical examples. Published in 2002 by John Wiley & Sons, Ltd.     

Different approaches for mixed LSFEMs in hyperelasticity: Application of logarithmic deformation measures

We present geometrically nonlinear formulations based on a mixed least‐squares finite element method. The L²‐norm minimization of the residuals of the given first‐order system of differential equations leads to a functional, which is a two‐field formulation dependent on displacements and stresses. Based thereon, we discuss and investigate two mixed formulations. Both approaches make use of the fact that the stress symmetry condition is not fulfilled a priori due to the row‐wise stress approximation with vector‐valued functions belonging to a Raviart‐Thomas space, which guarantees a conforming discretization of H(div). In general, the advantages of using the least‐squares finite element method lie, for example, in an a posteriori error estimator without additional costs or in the fact that the choice of the polynomial interpolation order is not restricted by the Ladyzhenskaya‐Babu?ka‐Brezzi condition (inf‐sup condition). We apply a hyperelastic material model with logarithmic deformation measures and investigate various benchmark problems, adaptive mesh refinement, computational costs, and accuracy.     

An energy‐momentum‐conserving temporal discretization scheme for adhesive contact problems

Numerical solution of dynamic problems requires accurate temporal discretization schemes. So far, to the best of the authors’ knowledge, none have been proposed for adhesive contact problems. In this work, an energy‐momentum‐conserving temporal discretization scheme for adhesive contact problems is proposed. A contact criterion is also proposed to distinguish between adhesion‐dominated and impact‐dominated contact behaviors. An adhesion formulation is considered, which is suitable to describe a large class of interaction mechanisms including van der Waals adhesion and cohesive zone modeling. The current formulation is frictionless, and no dissipation is considered. Performance of the proposed scheme is compared with other schemes. The proposed scheme involves very little extra computational overhead. It is shown that the proposed new temporal discretization scheme leads to major accuracy gains both for single‐degree‐of‐freedom and multi‐degree‐of‐freedom systems. The single‐degree‐of‐freedom system is critically analyzed for various parameters affecting the response. For the multi‐degree‐of‐freedom system, the effect of the time step and mesh discretization on the solution is also studied using the proposed scheme. It is further shown that a temporal discretization scheme based on the principle of energy conservation is not sufficient to obtain a convergent solution. Results with higher order contact finite elements for discretizing the contact area are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

A marching cubes based failure surface propagation concept for three‐dimensional finite elements with non‐planar embedded strong discontinuities of higher‐order kinematics

This paper presents an advanced failure surface propagation concept based on the marching cubes algorithm initially proposed in the field of computer graphics and applies it to the embedded finite element method. When modeling three‐dimensional (3D) solids at failure, the propagation of the failure surface representing a crack or shear band should not exhibit a strong sensitivity to the details of the finite element discretization. This results in the need for a propagation of the discrete failure zone through the individual finite elements, which is possible for finite elements with embedded strong discontinuities. Whereas for two‐dimensional calculations the failure zone propagation location is easily predicted by the maximal principal stress direction, more advanced strategies are needed to achieve a smooth failure surface in 3D simulations. An example for such method is the global tracking algorithm, which predicts the crack path by a scalar level set function computed on the basis of the solution of a simplified heat conduction like problem. Its prediction may though lead to various scenarios on how the failure surface may propagate through the individual finite elements. In particular, for a hexahedral eight‐node finite element, 256 such cases exist. To capture all those possibilities, the marching cubes algorithm is combined with the global tracking algorithm and the finite elements with embedded strong discontinuities in this work. In addition, because many of the possible cases result in non‐planar failure surfaces within a single finite element and because the local quantities used to describe the kinematics of the embedded strong discontinuities are physically meaningful in a strict sense only for planar failure surfaces, a remedy for such scenarios is proposed. Various 3D failure propagation simulations outline the performance of the proposed concept. Copyright © 2013 John Wiley & Sons, Ltd.