A node‐to‐node scheme with the aid of variable‐node elements for elasto‐plastic contact analysis

A strategy for a two‐dimensional contact analysis involving finite strain plasticity is developed with the aid of variable‐node elements. The variable‐node elements, in which nodes are added freely where they are needed, make it possible to transform the non‐matching meshes into matching meshes directly. They thereby facilitate an efficient analysis, maintaining node‐to‐node contact during the contact deformation. The contact patch test, wherein the contact patch is constructed out of variable‐node elements, is thus passed, and iterations for equilibrium solutions reach convergence faster in this scheme than in the conventional approach based on the node‐to‐surface contact. The effectiveness and accuracy of the proposed scheme are demonstrated through several numerical examples of elasto‐plastic contact analyses. Copyright © 2015 John Wiley & Sons, Ltd.     

MLS‐based variable‐node elements compatible with quadratic interpolation. Part I: formulation and application for non‐matching meshes

Two‐dimensional variable‐node elements compatible with quadratic interpolation are developed using the moving least‐square (MLS) approximation. The mapping from the parental domain to the physical element domain is implicitly obtained from MLS approximation, with the shape functions and their derivatives calculated and saved only at the numerical integration points. It is shown that the present MLS‐based variable‐node elements meet the patch test if a sufficiently large number of integration points are employed for numerical integration. The cantilever problem with non‐matching meshes is chosen to check the feasibility of the present MLS‐based variable‐node elements, and the result is compared with that from the lower‐order case compatible with linear interpolation. Copyright © 2005 John Wiley & Sons, Ltd.     

Polyhedral elements by means of node/edge‐based smoothed finite element method

The node‐based or edge‐based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces, and so retain a good geometric adaptability. The strain smoothing technique and implicit shape functions based on the linear point interpolation make the element formulation simple and straightforward. The resulting polyhedral elements are free from the excessive zero‐energy modes and yield a robust solution very much insensitive to mesh distortion. Several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior. The smoothed finite element method‐based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.     

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

This paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.     

MLS‐based variable‐node elements compatible with quadratic interpolation. Part II: application for finite crack element

Two‐dimensional finite ‘crack’ elements for simulation of propagating cracks are developed using the moving least‐square (MLS) approximation. The mapping from the parental domain to the physical element domain is implicitly obtained from MLS approximation, with the shape functions and their derivatives calculated and saved only at the numerical integration points. The MLS‐based variable‐node elements are extended to construct the crack elements, which allow the discontinuity of crack faces and the crack‐tip singularity. The accuracy of the crack elements is checked by calculating the stress intensity factor under mode I loading. The crack elements turn out to be very efficient and accurate for simulating crack propagations, only with the minimal amount of element adjustment and node addition as the crack tip moves. Numerical results and comparison to the results from other works demonstrate the effectiveness and accuracy of the present scheme for the crack elements. Copyright © 2005 John Wiley & Sons, Ltd.     

A vertex‐based finite volume method applied to non‐linear material problems in computational solid mechanics

A vertex‐based finite volume (FV) method is presented for the computational solution of quasi‐static solid mechanics problems involving material non‐linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two‐ and three‐dimensional element types. A detailed comparison between the vertex‐based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. Copyright © 2002 John Wiley & Sons, Ltd.     

A new finite element approach for solving three‐dimensional problems using trimmed hexahedral elements

In this paper, a novel finite element approach is presented to solve three‐dimensional problems using trimmed hexahedral elements generated by cutting a simple block consisting of regular hexahedral elements with a computer‐aided design (CAD) surface. Trimmed hexahedral elements, which are polyhedral elements with curved faces, are placed at the boundaries of finite element models, and regular hexahedral elements remain in the interior regions. Shape functions for trimmed hexahedral elements are developed by using moving least square approximation with harmonic weight functions based on an extension of Wachspress coordinates to curved faces. A subdivision of polyhedral domains into tetrahedral sub‐domains is performed to construct shape functions for trimmed hexahedral elements, and numerical integration of the weak form can be carried out consistently over the tetrahedral sub‐domains. Trimmed hexahedral elements have similar properties to conventional finite elements regarding the continuity, the completeness, the node–element connectivity, and the inter‐element compatibility. Numerical examples for three‐dimensional linear elastic problems with complex geometries show the efficiency and effectiveness of the present method. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Enriched contact finite elements for stable peeling computations

During peeling of a soft elastic strip from a substrate, strong adhesional forces act locally inside the peeling zone. It is shown here that when a standard contact finite element (FE) formulation is used to compute the peeling process, a large mesh refinement is required since the numerical solution procedure becomes unstable otherwise. To improve this situation, several different efficient enrichment strategies are presented that provide stable solution algorithms for comparably coarse meshes. The enrichment is based on the introduction of additional unknowns inside the contact elements discretizing the slave surface. These are chosen in order to improve the approximation of the peeling forces, while keeping the overall number of degrees of freedom low. If needed, these additional unknowns can be condensed out locally. The enrichment formulation is developed for both 2D and 3D nonlinear FE formulations. The new enrichment technique is applied to the peeling computation of a gecko spatula. The proposed enriched contact element formulations are also investigated in sliding computations. Copyright © 2011 John Wiley & Sons, Ltd.     

A contact algorithm for the Signorini problem using space–time finite elements

The most common approach in the finite‐element modelling of continuum systems over space and time is to employ the finite‐element discretization over the spatial domain to reduce the problem to a system of ordinary differential equations in time. The desired time integration scheme can then be used to step across the so‐called time slabs, mesh configurations in which every element shares the same degree of time refinement. These techniques may become inefficient when the nature of the initial boundary value problem is such that a high degree of time refinement is required only in specific spatial regions of the mesh. Ideally one would be able to increase the time refinement only in those necessary regions. We achieve this flexibility by employing space–time elements with independent interpolation functions in both space and time. Our method is used to examine the classic contact problem of Signorini and allows us to increase the time refinement only in the spatial region adjacent to the contact interface. We also develop an interface‐tracking algorithm that tracks the contact boundary through the space–time mesh and compare our results with those of Hertz contact theory. Copyright 2004 John Wiley & Sons, Ltd.     

Mid‐node admissible spaces for quadratic triangular arbitrarily curved 2D finite elements

This paper presents the extension of the mid‐node admissible space (MAS) concept for two‐dimensional quadratic triangular element (2DQTE) to the case when more than one edge is curved. The mathematical background for the MAS for 2DQTE with more than one edge curved is developed. A metric, based on the determination of the Jacobian distribution, whose evaluation and implementation is aided through the use of the MAS is developed. The evaluation of the quality of an element based on this metric and procedures to repair an element to produce better results are also presented. Copyright © 2001 John Wiley & Sons, Ltd.     

A novel three‐dimensional contact finite element based on smooth pressure interpolations

This article proposes a new three‐dimensional contact finite element which employs continuous and weakly coupled pressure interpolations on each of the interacting boundaries. The resulting formulation circumvents the geometric bias of one‐pass methods, as well as the surface locking of traditional two‐pass node‐on‐surface methods. A Lagrange multiplier implementation of the proposed element is validated for frictionless quasi‐static contact by a series of numerical simulations. Published in 2001 by John Wiley & Sons, Ltd.     

Distortion‐immune nine‐node displacement‐based quadrilateral thick plate finite elements that satisfy constant‐bending patch test

We employ the linked interpolation concept to develop two higher‐order nine‐node quadrilateral plate finite elements with curved sides that pass the constant bending patch test for arbitrary node positions. The linked interpolation for the plate displacements is expanded with three bubble parameters to get polynomial completeness necessary to satisfy the patch test. In contrast to some other techniques, the elements developed in this way retain a symmetric stiffness matrix at a marginal computational expense at the element level. The new elements generated using this concept are tested on several examples with curved sides or some other kind of geometric distortion. Copyright © 2014 John Wiley & Sons, Ltd.     

A three‐dimensional finite element method with arbitrary polyhedral elements

The ‘variable‐element‐topology finite element method’ (VETFEM) is a finite‐element‐like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement‐based formulation is also given, which effectively treats the case of near‐incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight‐node hex elements in a 3D finite‐deformation elastic–plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter. Copyright © 2006 John Wiley & Sons, Ltd.     

Edge‐based finite element techniques for non‐linear solid mechanics problems

Edge‐based data structures are used to improve computational efficiency of inexact Newton methods for solving finite element non‐linear solid mechanics problems on unstructured meshes. Edge‐based data structures are employed to store the stiffness matrix coefficients and to compute sparse matrix–vector products needed in the inner iterative driver of the inexact Newton method. Numerical experiments on three‐dimensional plasticity problems have shown that memory and computer time are reduced, respectively, by factors of 4 and 6, compared with solutions using element‐by‐element storage and matrix–vector products. Copyright © 2001 John Wiley & Sons, Ltd.     

Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes

Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes are presented. The elements use the linear interpolation functions of the original mesh, but each element is associated with a single node. As a result, a favourable constraint ratio for the volumetric response is obtained for problems in solid mechanics. The uniform strain elements do not require the introduction of additional degrees of freedom and their performance is shown to be significantly better than that of three‐node triangular or four‐node tetrahedral elements. In addition, nodes inside the boundary of the mesh are observed to exhibit superconvergent behaviour for a set of example problems. Published in 2000 by John Wiley & Sons, Ltd.     

Checkerboard‐free topology optimization using non‐conforming finite elements

The objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non‐conforming four‐node finite elements are employed. Since the convergence of the non‐conforming finite element is independent of the Lamé parameters, the stiffness of the non‐conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard‐free property of the non‐conforming element in topology optimization problems and verify it with three typical optimization examples. Copyright © 2003 John Wiley & Sons, Ltd.     

F‐bar aided edge‐based smoothed finite element method using tetrahedral elements for finite deformation analysisof nearly incompressible solids

A new smoothed finite element method (S‐FEM) with tetrahedral elements for finite strain analysis of nearly incompressible solids is proposed. The proposed method is basically a combination of the F‐bar method and edge‐based S‐FEM with tetrahedral elements (ES‐FEM‐T4) and is named ‘F‐barES‐FEM‐T4’. F‐barES‐FEM‐T4 inherits the accuracy and shear locking‐free property of ES‐FEM‐T4. At the same time, it also inherits the volumetric locking‐free property of the F‐bar method. The isovolumetric part of the deformation gradient ( F ^iso) is derived from the F of ES‐FEM‐T4, whereas the volumetric part ( F ^vol) is derived from the cyclic smoothing of J(=det( F )) between elements and nodes. Some demonstration analyses confirm that F‐barES‐FEM‐T4 with a sufficient number of cyclic smoothings suppresses the pressure oscillation in nearly incompressible materials successfully with no increase in DOF. Moreover, they reveal that our method is capable of relaxing the corner locking issue arising at the corner in the cylinder barreling analysis. Copyright © 2016 John Wiley & Sons, Ltd.