An approach to combining 3D discrete and finite element methods based on penalty function method

An algorithm combining three-dimensional (3D) discrete and finite element methods is proposed. This new approach is conducted by decomposing the calculation domain into a finite element (FE) calculation domain and a discrete element (DE) calculation domain; the interaction between the two sub-domains is processed by using a penalty function method. Following the established model that combines spherical DEs and FEs, the corresponding numerical code is developed. The vibration process of two cantilever beams under dynamic force is simulated. By comparing the results calculated with different penalty factors set and also with that calculated by the finite element code LS-DYNA, it is found that the calculated results are unanimous and the precision is almost the same as LS-DYNA, as long as the penalty factor is large enough. Moreover, the vibration processes of two plates under impact of rigid spheres are simulated and the accuracy of the model proposed in this paper is further proved in the field of contact mechanics by comparing the simulating results with that calculated by using LS-DYNA. Finally, the impact fracture behavior of a laminated glass plate is simulated, with the influence of model parameters taken into consideration. And the numerical experiments show that the combined model can be used to predict some macroscopical physical quantities, such as the impact force of impactor.     

A finite element solution method for contact problems with friction

A new finite element solution method for the analysis of frictional contact problems is presented. The contact problem is solved by imposing geometric constraints on the pseudo equilibrium configuration, defined as a configuration at which the compatibility conditions are violated. The algorithm does not require any a priori knowledge of the pairs of contactor nodes or segments. The contact condition of sticking, slipping, rolling or tension release is determined from the relative magnitudes of the normal and tangential global nodal forces. Contact iterations are in general found to converge within one or two iterations. The analysis method is applied to selected problems to illustrate the applicability of the solution procedure.     

A finite element method for contact problems related to fracture mechanics

A finite element method for contact problems in crack mechanics is developed on the basis of the penalty function method. The method is successfully applied to three important problems in fracture mechanics: a crack propagated from a pin hole, a two-point supported specimen with an edge crack loaded by a stamp, and a thick plate with a through-wall crack under bending force.     

An adaptive finite element algorithm for contact problems in plasticity

Due to the fact that in contact problems the contact area is not known a priori, a sufficient discretization to obtain a convergent finite element solution cannot be supplied from the outset. Therefore it is necessary to use adaptive finite element methods to adjust automatically the mesh sizes not only in the bodies under consideration but also in the contact zone. In this paper we develop an adaptive method for geometrically linear contact problems, which also includes elastoplastic material behavior. The radial return algorithm is used to derive the error estimator for one time increment of the solution process. The error estimator is based on the Zienkiewicz-Zhu projection scheme, which is extended to account for the special situation in the contact interface.In memoriam of J. C. Simo     

A penalty finite element method for non-Newtonian creeping flows

In this paper we present an iterative penalty finite element method for viscous non-Newtonian creeping flows. The basic idea is solving the equations for the difference between the exact solution and the solution obtained in the last iteration by the penalty method. For the case of Newtonian flows, one can show that for sufficiently small penalty parameters the iterates converge to the incompressible solution. The objective of the present work is to show that the iterative penalization can be coupled with the iterative scheme used to deal with the non-linearity arising from the constitutive law of non-Newtonian fluids. Some numerical experiments are conducted in order to assess the performance of the approach for fluids whose viscosity obeys the power law.     

A contact algorithm for cohesive cracks in the extended finite element method

Cracks with quasibrittle behavior are extremely common in engineering structures. The modeling of cohesive cracks involves strong nonlinearity in the contact, material, and complex transition between contact and cohesive forces. In this article, we propose a novel contact algorithm for cohesive cracks in the framework of the extended finite element method. A cohesive-contact constitutive model is introduced to characterize the complex mechanical behavior of the fracture process zone. To avoid the stress oscillations and ill-conditioned system matrix that often occur in the conventional contact approach, the proposed algorithm employs a special dual Lagrange multiplier to impose the contact constraint. This Lagrange multiplier is constructed by means of the area-weighted average and biorthogonality conditions at the element level. The system matrix can be condensed into a positive definite matrix with an unchanged size at a very low computational cost. In addition, we illustrate solving the cohesive crack contact problem using a novel iteration strategy. Several numerical experiments are performed to illustrate the efficiency and high-quality results of our method in contact analysis of cohesive cracks.     

A novel finite element formulation for frictionless contact problems

This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.     

A finite element method for elasto-plastic structures and contact problems by parametric quadratic programming

In this paper, the stiffness matrix of a contact element is introduced by means of a penalty function expression of the contact pressure and frictional force. The contact condition and the flow rule are expressed by the same form as in a non-associated plastic flow problem. A unified PQP (Parametric Quadratic Programming) model related to contact problems as well as to elasto-plastic structures is constructed. A series of PQP formulae for contact problems and elastic-plastic structures is derived in the text, and some numerical examples are illustrated as well.     

A finite element method for non-self-adjoint problems

This paper presents a general theory and application of the finite element method for some special class of non-self-adjoint problems. The formulation employed here is based on the Galerkin method for linear boundary value and eigenvalue problems described by the partial differential equations of elliptic type, and it can be regarded as an extension of the usual displacement method formulated by the use of the principle of minimum potential energy. In order to illustrate its validity and feasibility, the method is applied to the problems of the two-group neutron diffusion equations and of the stability of a non-conservative system.     

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.     

On a finite element method for dynamic contact/impact problems

This paper addresses the formulation and discrete approximation of dynamic contact/impact initial-value problems. The continuous problem is presented in the context of non-linear kinematics. Standard semi-discrete time integrators are introduced and are shown to be unsuccessful in modelling the kinematic constraints imposed on the interacting bodies during persistent contact. A procedure that bypasses the aforementioned difficulty is proposed by means of a novel variational formulation. Numerical simulations are conducted and the results are reported and discussed.     

Cell optimization for fast contact detection in the discrete element method algorithm

The objective of this paper is to find out a rule of cell optimization for fast contact detection in the discrete element method (DEM) algorithm. The contact detection process was executed by using the conventional cell model or distinct cell model (DCM) under several particle conditions having different particle size ratio and volume ratio, and its time was measured. The conventional cell model is suitable for the conditions that particle size ratio, d_large/d_small, is small and its volume ratio, V_small/V₀, is also small. The optimum cell condition of the conventional cell model is that the mean number of particle in a cell is 1.0–5.0, regardless of the particle size ratio and its volume ratio. Contact detection using DCM, which has several grids for different sized particles, is much faster than that of the conventional model when particles have a large size ratio. The size of the larger cell does not affect the contact detection speed; however, the size of the smaller cell is important for the fast contact detection and its size is determined to be about 3.0 times the particle radius. Therefore, the optimum cell condition is estimated from the particle size ratio and its volume ratio, and a large-scale DEM will be possible.     

A new contact detection method for arbitrary dilated polyhedra with potential function in discrete element method

Contact detection significantly affects the computational efficiency of discrete element simulations, especially for irregularly shaped elements. The dilated polyhedron is constructed by the Minkowski sum of a dilated sphere and a core convex polyhedron. One of the greatest advantages of using the dilated polyhedron in contact detection lies in its ability to be solved by calculating the nearest distance between corresponding core polyhedra. The approximate envelope function (AEF) of a dilated polyhedron is formed by the weighted summation of the second-order dilated function of the polyhedral and spherical functions. The AEF can be used to represent the element in the optimization model for the contact center. Geometric calculations are then employed for the contact points on the core polyhedron, whereupon the contact detection is solved. The accuracy and stability of the proposed method by a 3-D Voronoi tessellation are validated using analytical solutions and previously published simulation results. The efficiency tests show that the speedup of the CPU-based multithread algorithm can reach 14 on a desktop. The direct shear test of the Voronoi shaped ballast is analyzed by this method. The shear stress under different vertical pressure is compared with previously published experimental and simulated results.     

A finite element method for stability problems in finite elasticity

In this paper a finite element method is developed to treat stability problems in finite elasticity. For this purpose the constitutive equations are formulated in principal stretches which allows a general representation of the derivatives of the strain energy function with respect to the principal stretches. These results can then be used to derive an efficient numerical scheme for the computation of singular points.     

A method for rendering 3D finite element vector field solutionsnondivergent

The use of the method of constraints for enforcing the zero divergence condition in vectorial finite-element schemes is discussed. An earlier implementation of the method was shown to produce the correct solution for a 3-D resonant cavity problem modeled by a single finite element. Partial success in extending the method to multielement cases is reported. The reduction in matrix size alone would justify the development of the technique for general multielement grids, but it will require the implementation of a global approach to the method of constraints     

Three-dimensional finite element contact algorithm for metal forming

This work presents a three-dimensional rigid plastic finite element formulation. The workpiece is discretized with eight node hexahedral isoparametric elements. Friction is included in the formulation by means of a shear stress depending on the relative velocity between the workpiece and the tool. Special attention is given to the contact problems, and a three-dimensional contact algorithm based on a discretization of the tool surface with triangular elements is presented. Finally, some selected examples are solved, in order to show the capabilities of the formulation.     

Linear-frictional contact model for 3D discrete element simulations of granular systems

The linear-frictional contact model is the most commonly used contact mechanism for discrete element (DEM) simulations of granular materials. Linear springs with a frictional slider are used for modeling interactions in directions normal and tangential to the contact surface. Although the model is simple in two dimensions, its implementation in 3D faces certain subtle challenges, and the particle interactions that occur within a single time step require careful modeling with a robust algorithm. The paper details a three-dimensional algorithm that accounts for the changing direction of the tangential force within a time step, the transition from elastic to slip behavior within a time step, possible contact sliding during only part of a time step, and twirling and rotation of the tangential force during a time step. Without three of these adjustments, errors are introduced in the incremental stiffness of an assembly. Without the fourth adjustment, the resulting stress tensor is not only incorrect but it is also no longer a tensor. The algorithm also computes the work increments during a time step, both elastic and dissipative.     

A mortared finite element method for frictional contact on arbitrary interfaces

This work concerns finite-element algorithms for imposing frictional contact constraints on intra-element, or embedded surfaces. Existing techniques typically rely on the underlying bulk mesh to implicitly partition the surface, a strategy that can give rise to overconstraint. In the present work, we first apply a mortaring algorithm to the modeling of frictional contact conditions on arbitrary interfaces. The algorithm is based upon a projection of the bulk and surface fields onto independent mortar fields at the interface. We examine the advantages of this approach when combined with extended finite-element approximations to the bulk fields. In particular, the method allows for bulk and surface domains to be partitioned separately, as well as enforce nonlinear contact constraints on surfaces that are not explicitly “fitted” to the bulk mesh. Results from several benchmark problems in frictional contact are provided to demonstrate the accuracy and efficacy of the method, as well as the improvement in robustness compared to existing techniques. We also provide an example that illustrates the effectiveness of the approach in high-speed machining simulation.     

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.     

A mortar‐finite element formulation for frictional contact problems

In this paper a finite element formulation is developed for the solution of frictional contact problems. The novelty of the proposed formulation involves discretizing the contact interface with mortar elements, originally proposed for domain decomposition problems. The mortar element method provides a linear transformation of the displacement field for each boundary of the contacting continua to an intermediate mortar surface. On the mortar surface, contact kinematics are easily evaluated on a single discretized space. The procedure provides variationally consistent contact pressures and assures the contact surface integrals can be evaluated exactly. Copyright © 2000 John Wiley & Sons, Ltd.