Finite deformation frictional mortar contact using a semi‐smooth Newton method with consistent linearization

A two‐dimensional, finite deformation frictional contact formulation with Coulomb's law is presented. The approach considers multibody contact and is based on a mortar formulation. The enforcement of contact constraints is realized with dual Lagrange multipliers. These alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase. Friction kinematic variables are formulated in an objective way and enter non‐smooth complementarity functions for expressing the contact constraints. An active set strategy is derived by applying a semi‐smooth Newton method, which treats contact nonlinearities, material and geometrical nonlinearities in one single iterative scheme. By further carrying out a consistent linearization for both normal and frictional contact forces and constraints, a robust and highly efficient algorithm for linear and higher‐order (quadratic) interpolation is achieved. Efficiency of the proposed method and quality of results are demonstrated in several examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm for large deformation inelasticity

The present paper focuses on the consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm. A general and modular algorithmic setting, suited for almost any rate constitutive equations, is presented where the finite deformation consistent tangent modulus is obtained as a by‐product of the integration algorithm. Numerical examples illustrate the good performance of the proposed formulation, especially for large deformation increments with noteworthy superimposed rotation, where the consistent formulation converges quadratically in a reasonable number of iterations. Copyright © 2008 John Wiley & Sons, Ltd.     

A variational formulation for finite elasticity with independent rotation and Biot-axial fields

The fundamentals of the geometrically nonlinear mechanics of the three-dimensional elastic continuum are derived, starting from a general variational framework established for the polar model and passing through a constitutive definition of the non-polar medium itself. A constrained variational setting follows, having as unknown vector fields the displacement, the rotation vector and the axial of the Biot stress. It embraces both the rotational equilibrium and the characterization of the rotation as Euler-Lagrange equations. These conditions can then be satisfied in a weak sense within discrete approximations. It is also shown that the classical approach of the non-polar continuum can be accomodated as a particular case of the present formulation. A consistent linearization is then proposed and a simple solid finite element developed to test the computational viability of the formulation. A few examples assess the capability of the element to represent large three-dimensional rotations. Communicated by S. N. Atluri, 2 August 1996     

A finite deformation mortar contact formulation using a primal–dual active set strategy

In recent years, nonconforming domain decomposition techniques and, in particular, the mortar method have become popular in developing new contact algorithms. Here, we present an approach for 2D frictionless multibody contact based on a mortar formulation and using a primal–dual active set strategy for contact constraint enforcement. We consider linear and higher‐order (quadratic) interpolations throughout this work. So‐called dual Lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation, thus avoiding an increase in system size. For this type of contact formulation, we provide a full linearization of both contact forces and normal (non‐penetration) and tangential (frictionless sliding) contact constraints in the finite deformation frame. The necessity of such a linearization in order to obtain a consistent Newton scheme is demonstrated. By further interpreting the active set search as a semi‐smooth Newton method, contact nonlinearity and geometrical and material nonlinearity can be resolved within one single iterative scheme. This yields a robust and highly efficient algorithm for frictionless finite deformation contact problems. Numerical examples illustrate the efficiency of our method and the high quality of results. Copyright © 2009 John Wiley & Sons, Ltd.     

Complementarity methods for multibody friction contact problems in finite deformations

This paper deals with the frictional contact occurring between deformable elastoplastic bodies subjected to large displacements and finite deformations. Starting from a standard slave/master formulation we have developed a symmetrical formulation with which the unilateral contact conditions and the friction law are satisfied for each body. From the continuum equations, the discretized frictional contact problem is set as a complementarity problem and solved using Lemke's mathematical programming method. The efficiency of the method is illustrated in the case of several examples. Copyright © 2001 John Wiley & Sons, Ltd.     

A formulation of arbitrarily shaped surface elements for three-dimensional large deformation contact with friction

The paper introduces a general theory for the numerical simulation of large deformation contact problems. The contacting bodies under consideration may be of two- or three-dimensional shape modelled by finite elements. A contact finite element which can be applied to handle multi-body contact as well as contact with rigid bodies is developed. The element is universal in the sense that it can be used as a surface element for any known finite element model and includes friction. The frictional behaviour of the model obeys Coulomb's law of friction distinguishing between sticking and sliding contact. The algorithmic treatment is based on a penalty formulation for the normal and sticking contact. The corresponding consistent tangential stiffness matrices are derived, leading to an overall quadratic convergence behaviour for the method. This feature is demonstrated in a number of representative examples. © 1997 by John Wiley & Sons, Ltd.     

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.     

Finite deformation formulation for embedded frictional crack with the extended finite element method

We reformulate an extended finite element (FE) framework for embedded frictional cracks in elastoplastic solids to accommodate finite deformation, including finite stretching and rotation. For the FE representation, we consider a Galerkin approximation in which both the trial and weighting functions adapt to the current contact configuration. Contact and frictional constraints employ two Kuhn–Tucker conditions, a contact/separation constraint nesting over a stick/slip constraint for the case when the crack faces are in frictional sliding mode. We integrate finite deformation bulk plasticity into the formulation using the multiplicative decomposition technique of nonlinear continuum mechanics. We then present plane strain simulations demonstrating various aspects of the extended FE solutions. The mechanisms considered include combined opening and frictional sliding in initially straight, curved, and S‐shaped cracks, with and without bulk plasticity. To gain further insight into the extended FE solutions, we perform mesh convergence studies focusing on both the global and the local responses of structures with cracks, including the distribution of the normal component of traction on the crack faces. Copyright © 2009 John Wiley & Sons, Ltd.     

Frictionless 2D Contact formulations for finite deformations based on the mortar method

In this paper two different finite element formulations for frictionless large deformation contact problems with non-matching meshes are presented. Both are based on the mortar method. The first formulation introduces the contact constraints via Lagrange multipliers, the other employs the penalty method. Both formulations differ in size and the way of fulfilling the contact constraints, thus different strategies to determine the permanently changing contact area are required. Starting from the contact potential energy, the variational formulation, the linearization and finally the matrix formulation of both methods are derived. In combination with different contact detection methods the global solution algorithm is applied to different two-dimensional examples.     

A contact algorithm for frictional crack propagation with the extended finite element method

We present an incremental quasi‐static contact algorithm for path‐dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non‐linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Genuinely resultant shell finite elements accounting for geometric and material non-linearity

A general theoretical framework is presented for the fully non-linear analysis of shells by the finite element method. The governing equations are derived exclusively in terms of resulting quantities through a logical and straightforward descent from three-dimensional continuum mechanics without appealing to simplifying assumptions (hence the name genuinely resultant). As a result, the underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. The underlying mathematical structure and the variational formulation of the theory are examined. This appears to be crucial for the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method. The treatment of finite rotations through an arbitrary parametrization of the rotation group and the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis are other issues studied in this paper. A numerical analysis is presented in order to assess the effectiveness of the proposed formulation. Small strain problems as well as finite strain deformation of rubber-like shells undergoing finite rotations are considered. Special attention is devoted to the assessment of the relevance of the drilling degree-of-freedom and highly non-uniform through-the-thickness deformation in the case of shells made of incompressible material.     

Wrinkling of nonlinear membranes

 Membranes are stiff under tension but switch over to wrinkling when compression occurs. Roddeman proposed a kinematic model to handle this phenomenon under finite deformation conditions. The wrinkling conditions of Roddeman are transformed into the reference configuration. This results in a more simple nonlinear formulation. For application in a finite element code a consistent linearization was carried out. Numerical examples for linear and nonlinear orthotropic constitutive equations are discussed. Received 21 December 2001 / Accepted 10 April 2002     

Finite element method used in contact problems with dry friction

In this paper, we present a formulation of numerical approximations of the frictional quasi-contact problem with dry friction between a deformable body and a foundation with possibility to consider the case of two deformable bodies. We consider numerical approximations of 3D static contact problem with dry friction, using finite contact elements. Saddle-point algorithm, Lagrange incremental multipliers method and penalty functions are used to enforce finite element surface contact constrains for incremental formulation of the frictional quasi-static problem. Some typical examples in the elastic contact problems with dry friction are presented.     

Direct differentiation method for sensitivity analysis based on transfer matrix method for multibody systems

On one hand, the new version of transfer matrix method for multibody systems (NV‐MSTMM), has been proposed by formulating transfer equations of elements in acceleration level instead of position level as in the original discrete time transfer matrix method of multibody systems to study multibody system dynamics. This new formulation avoids local linearization and allows using any integration algorithms. On the other hand, sensitivity analysis is an important way to improve the optimization efficiency of multibody system dynamics. In this paper, a totally novel direct differentiation method based on NV‐MSTMM for sensitivity analysis of multibody systems is developed. Based on direct differentiation method, sensitivity analysis matrix for each kind of element is established. By assembling transfer matrices and sensitivity analysis matrices based on differentiation law of multiplication, the sensitivity analysis equation of overall transfer equation is deduced. The computing procedure of the proposed method is also presented. All these improvements as well as three numerical examples show that the direct differentiation method based on NV‐MSTMM is suitable for optimizing the dynamic sensitivity in multi–rigid‐body systems.     

Analysis of 3D problems using a new enhanced strain hexahedral element

The now classical enhanced strain technique, employed with success for more than 10 years in solid, both 2D and 3D and shell finite elements, is here explored in a versatile 3D low‐order element which is identified as HIS. The quest for accurate results in a wide range of problems, from solid analysis including near‐incompressibility to the analysis of locking‐prone beam and shell bending problems leads to a general 3D element. This element, put here to test in various contexts, is found to be suitable in the analysis of both linear problems and general non‐linear problems including finite strain plasticity. The formulation is based on the enrichment of the deformation gradient and approximations to the shape function material derivatives. Both the equilibrium equations and their variation are completely exposed and deduced, from which internal forces and consistent tangent stiffness follow. A stabilizing term is included, in a simple and natural form. Two sets of examples are detailed: the accuracy tests in the linear elastic regime and several finite strain tests. Some examples involve finite strain plasticity. In both sets the element behaves very well, as is illustrated in numerous examples. Copyright © 2003 John Wiley & Sons, Ltd.     

A TORSIONAL SPRING-LIKE BEAM ELEMENT FOR THE DYNAMIC ANALYSIS OF FLEXIBLE MULTIBODY SYSTEMS

A new finite element beam formulation for modelling flexible multibody systems undergoing large rigid-body motion and large deflections is developed. In this formulation, the motion of the ‘nodes’ is referred to a global inertial reference frame. Only Cartesian position co-ordinates are used as degrees of freedom. The beam element is divided into two subelements. The first element is a truss element which gives the axial response. The second element is a torsional spring-like bending element which gives the transverse bending response. D'Alembert principle is directly used to derive the system's equations of motion by invoking the equilibrium, at the nodes, of inertia forces, structural (internal) forces and externally applied forces. Structural forces on a node are calculated from the state of deformation of the elements surrounding that node. Each element has a convected frame which translates and rotates with it. This frame is used to determine the flexible deformations of the element and to extract those deformations from the total element motion. The equations of motion are solved along with constraint equations using a direct iterative integration scheme. Two numerical examples which were presented in earlier literature are solved to demonstrate the features and accuracy of the new method.     

A computational method for frictional contact problem using finite element method

Using the finite element method a numerical procedure is developed for the solution of the two-dimensional frictional contact problems with Coulomb's law of friction. The formulation for this procedure is reduced to a complementarity problem. The contact region is separated into stick and slip regions and the contact stress can be solved systematically by applying the solution technique of the complementarity problem. Several examples are given to demonstrate the validity of the present formulation.     

Least squares finite element analysis of laminar boundary layer flows

Based on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer flow problems. The method is essentially a discrete element-wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non-linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders non-linear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous flow along a flat plate are presented to demonstrate the applicability of the method as well as to illuminate discussions on the theoretical aspects of the method.