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.     

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 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.     

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.     

Impact analysis of arbitrarily‐shaped bodies using a finite element method and a smoothed particle hydrodynamics method

We propose a new method to obtain contact forces under a non‐smoothed contact problem between arbitrarily‐shaped bodies which are discretized by finite element method. Contact forces are calculated by the specific contact algorithm between two particles of smoothed particle hydrodynamics, which is a meshfree method, and that are applied to each colliding body. This approach has advantages that accurate contact forces can be obtained within an accelerated collision without a jump problem in a discrete time increment. Also, this can be simply applied into any contact problems like a point‐to‐point, a point‐to‐line, and a point‐to‐surface contact for complex shaped and deformable bodies. In order to describe this method, an impulse based method, a unilateral contact method and smoothed particle hydrodynamics method are firstly introduced in this paper. Then, a procedure about the proposed method is handled in great detail. Finally, accuracy of the proposed method is verified by a conservation of momentum through three contact examples. Copyright © 2016 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.     

A modified Newton‐type Koiter‐Newton method for tracing the geometrically nonlinear response of structures

The Koiter‐Newton (KN) method is a combination of local multimode polynomial approximations inspired by Koiter's initial postbuckling theory and global corrections using the standard Newton method. In the original formulation, the local polynomial approximation, called a reduced‐order model, is used to make significantly more accurate predictions compared to the standard linear prediction used in conjunction with Newton method. The correction to the exact equilibrium path relied exclusively on Newton‐Raphson method using the full model. In this paper, we proposed a modified Newton‐type KN method to trace the geometrically nonlinear response of structures. The developed predictor‐corrector strategy is applied to each predicted solution of the reduced‐order model. The reduced‐order model can be used also in the correction phase, and the exact full nonlinear model is applied only to calculate force residuals. Remainder terms in both the displacement expansion and the reduced‐order model are well considered and constantly updated during correction. The same augmented finite element model system is used for both the construction of the reduced‐order model and the iterations for correction. Hence, the developed method can be seen as a particular modified Newton method with a constant iteration matrix over the single KN step. This significantly reduces the computational cost of the method. As a side product, the method has better error control, leading to more robust step size adaptation strategies. Numerical results demonstrate the effectiveness of the method in treating nonlinear buckling problems.     

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.     

Time‐domain vector potential technique for the meshless radial point interpolation method

A time‐domain meshless algorithm based on vector potentials is introduced for the analysis of transient electromagnetic fields. The proposed numerical algorithm is a modification of the radial point interpolation method, where radial basis functions are used for local interpolation of the vector potentials and their derivatives. In the proposed implementation, solving the second‐order vector potential wave equation intrinsically enforces the divergence‐free property of the electric and magnetic fields. Furthermore, the computational effort associated with the generation of a dual node distribution (as required for solving the first‐order Maxwell's equations) is avoided. The proposed method is validated with several examples of 2D waveguides and filters, and the convergence is empirically demonstrated in terms of node density or size of local support domains. It is further shown that inhomogeneous node distributions can provide increased convergence rates, that is, the same accuracy with smaller number of nodes compared with a solution for homogeneous node distribution. A comparison of the magnetic vector potential technique with conventional radial point interpolation method is performed, highlighting the superiority of the divergence‐free formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

An adaptive multi‐scale approach to the modelling of masonry structures

We present an adaptive multi‐scale approach for predicting the mechanical behaviour of masonry structures modelled as dynamic frictional multi‐body contact problems. In this approach, the iterative splitting of the contact problem into normal contact and frictional contact is combined with a semismooth Newton/primal‐dual active‐set procedure to calculate deformations and openings in the model structures. This algorithm is then coupled with a novel adaptive multi‐scale technique involving a macroscopic scale, which is the size of the masonry structure, and a mesoscopic scale, which is the size of the constituents (bricks, stone‐blocks), to predict appearance of dislocations and stress distribution in large‐scale masonry structures. Comparisons of the numerical results with data from experimental tests and from practical observations illustrate the predictive capability of the multi‐scale algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

Assessment of methods for computing the closest point projection,penetration, and gap functions in contact searching problems

In computational contact mechanics problems, local searching requires calculation of the closest point projection of a contactor point onto a given target segment. It is generally supposed that the contact boundary is locally described by a convex region. However, because this assumption is not valid for a general curved segment of a three‐dimensional quadratic serendipity element, an iterative numerical procedure may not converge to the nearest local minimum. To this end, several unconstrained optimization methods are tested: the Newton–Raphson method, the least square projection, the sphere and torus approximation method, the steepest descent method, the Broyden method, the Broyden–Fletcher–Goldfarb–Shanno method, and the simplex method. The effectiveness and robustness of these methods are tested by means of a proposed benchmark problem. It is concluded that the Newton–Raphson method in conjunction with the simplex method significantly increases the robustness of the local contact search procedure of pure penalty contact methods, whereas the torus approximation method can be recommended for contact searching algorithms, which employ the Lagrange method or the augmented Lagrangian method. Copyright © 2015 John Wiley & Sons, Ltd.     

Least‐squares collocation meshless method

A finite point method, least‐squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least‐squares sense. The moving least‐squares interpolant is used to construct the trial functions. The computational effort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution significantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational effort. Copyright © 2001 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.     

An improved iterative method for large strain viscoplastic problems

In this work, some techniques are proposed to improve the usual Newton–Raphson Method (NRM) used in the numerical analysis of large strain viscoplastic problems. These techniques, based on the first‐order perturbation technique, allow to define an adaptive step strategy and to improve the trial solution (guessed solution) for the first iteration at each step. To assess the efficiency of the proposed techniques, a number of numerical examples are presented: necking of a circular bar, plane strain bending, and an axisymmetric hydrostatic bulging process. Compared with the classical Newton–Raphson method, the proposed scheme requires less matrix inversions, less time steps and less CPU time. For a typical axisymmetric hydrostatic bulging process (606 degrees of freedom), the proposed method needs 5 times less matrix inversions, 1·7 times less time steps, and 4 times less CPU time. Copyright © 1999 John Wiley & Sons, Ltd.     

A method for structural dynamic contact problems with friction and wear

A method for structural dynamic contact problems with friction and wear is suggested. The method is obtained by including wear in the non‐smooth contact dynamics method of Moreau. A comparison of the method to the discrete energy‐momentum method of Simo and Tarnow is also outlined briefly. The fully discrete equations are treated using the augmented Lagrangian approach, where a non‐smooth Newton method is used as the equation solver. Two two‐dimensional examples are solved by the method. It is investigated how solutions of contact, friction and wear are influenced by inertia. It is shown that the quasi‐static assumption might be questionable for solving contact problems with friction and wear. Copyright © 2003 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.     

Element‐local level set method for three‐dimensional dynamic crack growth

An approximate level set method for three‐dimensional crack propagation is presented. In this method, the discontinuity surface in each cracked element is defined by element‐local level sets (ELLSs). The local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least‐square sense. A simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements. The discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method (XFEM). The ELLS method and the phantom node technology are combined for the solution of dynamic fracture problems. Numerical examples for three‐dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Crack face contact in X‐FEM using a segment‐to‐segment approach

In this study, we propose a segment‐to‐segment contact formulation (mortar‐based) that uses Lagrange's multipliers to establish the contact between crack faces when modeled with the extended finite element method (X‐FEM) in 2D problems. It is shown that, in general, inaccuracies arise when the contact is formulated following a point‐to‐point approach. This is due to the non‐linear character of the X‐FEM interpolation along the crack faces that leads to crack face interpenetration. However, the segment‐to‐segment approach optimizes the fulfilment of the contact constraints along the whole crack segment, and in practice the contact is modeled precisely. Convergence studies for mesh sequences have been performed, showing the advantages of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

Contact between rolling beams and flat surfaces

This work presents a new approach to model the contact between a circular cross section beam and a flat surface. In a finite element environment, when working with beam elements in contact with surfaces, it is common to consider node or line to surface approaches for describing contact. An offset can be included in normal gap function due to beam cross section dimensions. Such a procedure can give good results in frictionless scenarios, but the friction effects are not usually properly treated. When friction plays a role (e.g., rolling problems or alternating rolling/sliding) more elaboration is necessary. It is proposed here a method that considers an offset not only in normal gap. The basic idea is to modify the classical definition of tangential gap function in order to include the effect of rigid body rotation that occurs in a rolling scenario and, furthermore, consider the moment of friction force. This paper presents the new gap function definition and also its consistent linearization for a direct implementation in a Newton‐Raphson method to solve nonlinear structural problems modeled using beam elements. The methodology can be generalized to any interaction involving elements with rotational degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

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.