Parameterization of planar curves immersed in triangulations with application to finite elements

We construct a method for the parameterization of a class of planar piecewise C²‐curves over a collection of edges in an ambient triangulation. The map from the collection of edges to the curve is the closest‐point projection. A distinguishing feature of the method is that edges in the ambient triangulation need not interpolate the curve. We formulate conditions on the ambient triangulations so that the resulting parameterization over its selected edges is (i) bijective, (ii) maps simple, connected collection of edges to simple, connected components of the curve, and (iii) is C¹ within each edge of the collection. These properties of the parameterization make it particularly useful in the construction of high‐order finite element approximation spaces on planar curves immersed in triangulations. We discuss this application and illustrate it with numerical examples. The parameterization method applies to a large class of planar curves, including most ones of interest in engineering and computer graphics applications, and to a large family of triangulations, including acute‐angled triangulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Crypto-DOF finite elements

New finite elements, called crypto-DOF elements, are introduced, utilizing degrees of freedom that satisfy the differential equation but do not appear explicitly in the discrete equations. The crypto-DOF elements can be advantageous in adaptive finite element techniques, and to approximate singular-type eigensolutions (e.g. in convective diffusion or stress concentration problems) without altering the original mesh or introducing new degrees of freedom. In this paper the crypto-DOF elements, the construction of crypto-DOF element families and selected numerical results are presented.     

Polytope finite elements

In this paper, finite elements based on arbitrary convex and non‐convex polytopes are introduced. Polytopes in combination with natural element coordinates (NECs) permit a uniform element formulation of interpolation functions that are independent of the dimension of space, localization and the number of vertices. NECs based on the natural neighbor interpolation are restricted to the polytope and can be understood as an extension of the barycentric coordinates on simplexes. The differentiation and integration of these interpolation functions on the basis of NECs is essential for finite element approximations. The accuracy of the finite element interpolation or approximation can be controlled by either applying the h‐version or by utilizing the p‐version of the finite element method (FEM). Advantages in the handling of hanging nodes are discussed. Furthermore, we present construction methods for Lagrangian as well as for hierarchical interpolation functions based on NECs. Numerical experiments on different convex and non‐convex decompositions will show the usability, accuracy and convergence of the developed polytope FEM. Copyright © 2007 John Wiley & Sons, Ltd.     

Polygonal finite elements for finite elasticity

Nonlinear elastic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise because nonlinear elastic materials are characterized by non‐convex stored‐energy functions as a result of their ability to undergo large reversible deformations, are incompressible or nearly incompressible, and often times possess complex microstructures. In this work, we propose and explore an alternative approach to model finite elasticity problems in two dimensions by using polygonal discretizations. We present both lower order displacement‐based and mixed polygonal finite element approximations, the latter of which consist of a piecewise constant pressure field and a linearly‐complete displacement field at the element level. Through numerical studies, the mixed polygonal finite elements are shown to be stable and convergent. For demonstration purposes, we deploy the proposed polygonal discretization to study the nonlinear elastic response of rubber filled with random and periodic distributions of rigid particles, as well as the development of cavitation instabilities in elastomers containing vacuous defects. These physically‐based examples illustrate the potential of polygonal finite elements in studying and modeling nonlinear elastic materials with complex microstructures under finite deformations. Copyright © 2014 John Wiley & Sons, Ltd.     

Hermite infinite elements and graded quadratic B-spline finite elements

Quadratic B-spline finite elements are defined for a graded mesh. Hermite infinite elements are proposed to extend the applicability of these finite elements to unbounded regions. Test problems used to compare this technique with published procedures show that the quadratic B-spline finite element solution has, as expected, lower error bounds than a linear element solution. These experiments also demonstrate that the Hermite infinite elements used to close the B-spline finite element arrays lead to error norms comparable in size with other infinite element formulations. The generation of solitary waves in a semi-infinite shallow channel by boundary forcing is modelled by the Korteweg-de Vries equation using an array of graded elements closed by a zero pole infinite element. The resulting simulation of solitary wave motion across a non-uniform mesh confirms existing work and illustrates the effectiveness of the present formulation.     

Trends in finite elements

Some recent trends in the mathematics of finite elements and practical experience are outlined. The reliability, robustness and effectiveness of the method are discussed. Although only linear stationary problems are discussed, the general trends described are also typical for nonlinear and time-dependent problems. In general, the analysis of elliptic problems seems to be the most advanced     

High-order curved finite elements

Recent finite element analysis of adaptive refinement has focused attention on high-order approximation within elements. High-order approximation may be attained over curved elements with either rational basis functions expressed directly in the global co-ordinates or with a particular class of isoparametric basis functions. Theory for construction of high-order rational bases yields modifications to the blending method for a special class of elements in order to achieve higher order approximation within these elements.     

Random field finite elements

The probabilistic finite element method (PFEM) is formulated for linear and non-linear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in non-linear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem and a two-dimensional plane-stress beam bending problem. The moments calculated compare favourably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.     

Continuously deforming finite elements

A general class of time-dependent co-ordinate transformations is introduced in a variational formulation for evolution problems. The variational problem is posed with respect to both solution and transformation field variables. An approximate analysis using finite elements is developed from the continuous variational form. Modified forms of the variational functional are considered to ensure the deforming mesh is not top irregular. ODE system integrators are utilized to integrate the resulting semidiscrete systems. In Part I we consider the formulation for problems in one spatial dimension and time, including, in particular, convection-dominated flows described by the convection-diffusion, Burgers' and Buckley–Leverett equations. In Part II the extension of the method to two dimensions and supporting numerical experiments are presented.     

Simple cylindrical and spherical finite elements

Finite element approximations are developed for three‐dimensional domains naturally represented in either cylindrical or spherical coordinates. Lines of constant radius, axial length, or angle are used to represent the domain and cast approximations that are natural for these geometries. As opposed to general isoparametric three‐dimensional elements generated in conventional parent space, these elements can be evaluated analytically and do not generate geometric discretization error. They also allow for anisotropic material coefficients that are frequently aligned in either cylindrical or spherical coordinates. Several examples are provided that show convergence properties and comparison with analytical solutions of the Poisson equation.     

Hierarchal finite elements and precomputed arrays

Two-dimensional C° finite elements of arbitrary polynomial order are developed for a general functional which includes a least squares or potential energy functional as special cases. The elements are ‘hierarchal’ in the sense that the nodal variables for polynomial order p constitute a subset of the nodal variables for order p+1. It is shown that the elemental arrays for high polynomial order may be efficiently computed by using hierarchal elements together with precomputed arrays—i.e., arrays which are computed once, stored on permanent file, and then re-used in all subsequent applications of the program. A number of example problems are solved. Comparisons are made of the relative efficiency of finite element convergergence with mesh refinement (polynomial order held fixed) and with increasing polynomial order (mesh held fixed). The latter approach is found superior.     

On quadrature and singular finite elements

Special quadrature rules are described for elastic finite elements that have r^q behaviour (0 < q < 1) directly induced in natural element co-ordinates. In general, the quadrature points and weights can vary with the exponent q. For two-dimensional problems with a square-root singularity (q = 1/2), the use of special quadrature results in significant improvements over regular Gauss quadrature. The development of special quadrature rules for three-dimensional elements is shown to be a difficult task. Several special case rules are developed and tested for a line-type singular element, and a precise rule is given for a point-type singular element in three dimensions.     

A sensitive method to determine the bonding strength between two flexible non-compatible polymers

The influence of different surface morphologies and compatibilizers on the bonding strength between high-density polyethylene (HDPE) and isotactic polypropylene (iPP) is determined by T-peel tests. To optimize the relative comparison, the polymer foils are prepared to have distinct areas with different bond strengths. The advantage of this comparative method is the observation of significant differences of the bonding strength within a single sample, eliminating such sensitive parameters as the dimensions and peel angle.     

Multiple quadrature underintegrated finite elements

New multiple-quadrature-point underintegrated finite elements with hourglass control are developed. The elements are selectively underintegrated to avoid volumetric and shear locking and save computational time. An approach for hourglass control is proposed such that the stabilization operators are obtained simply by taking the partial derivatives of the generalized strain rate vector with respect to the natural co-ordinates so that the elements require no stabilization parameter. To improve accuracy over the traditional one-point-quadrature elements, several quadrature points are used to integrate the internal forces, especially for tracing the plastic fronts in the mesh during loading and unloading in elastic–plastic analysis. Two- and four-point-quadrature elements are proposed for use in the two- and three-dimensional elements, respectively. Other multiple-quadrature points can also be employed. Several numerical examples such as thin beam, plate and shell problems are presented to demonstrate the applicability of the proposed elements.     

Comment on hybrid finite elements

It is shown that an apparently sound selection of trial functions can give rise to hybrid elements with rank deficient stiffness matrices and unused interior fields. An account is given of modifications which overcome these problems for a particular plate bending element.     

Arbitrary discontinuities in finite elements

A technique for modelling arbitrary discontinuities in finite elements is presented. Both discontinuities in the function and its derivatives are considered. Methods for intersecting and branching discontinuities are given. In all cases, the discontinuous approximation is constructed in terms of a signed distance functions, so level sets can be used to update the position of the discontinuities. A standard displacement Galerkin method is used for developing the discrete equations. Examples of the following applications are given: crack growth, a journal bearing, a non‐bonded circular inclusion and a jointed rock mass. Copyright © 2001 John Wiley & Sons, Ltd.     

Extrapolated galerkin time finite elements

A new time finite-element method based on the extrapolation technique and the Galerkin time finite-element method is presented. In this method, the second-order governing differential equations of motion for dynamic problems are rewritten as a set of first order differential equations in state space. The standard Galerkin method is then employed for the temporal discretization. The algorithm is first-order accurate only. Based on the first-order Galerkin time finite-element formulation, the extrapolation technique is introduced to improve the order of accuracy. It is achieved by expressing the numerical amplification matrix of higher-order algorithm as a linear combination of the basic amplification matrices evaluated at selected instances of time. The matrices are combined with different weighting factors. The pairs of the selected instance of time and the corresponding weighting factors are free parameters. Unconditionally stable higher-order accurate formulations can be derived by properly choosing the free parameters. Algorithms up to fourth-order accurate are presented in this paper. Detailed analyses on stability, numerical dissipation and numerical dispersion are also given. Comparisons of the present algorithms with some well-known time-integration methods are presented to demonstrate the versatility of the present method, in particular its accuracy in the higher-order formulations.     

Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering

Classical finite-element and boundary-element formulations for the Helmholtz equation are presented, and their limitations with respect to the number of variables needed to model a wavelength are explained. A new type of approximation for the potential is described in which the usual finite-element and boundary-element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions evenly distributed on the unit sphere. Compared with standard piecewise polynomial approximation, the plane-wave basis is shown to give considerable reduction in computational complexity. In practical terms, it is concluded that the frequency for which accurate results can be obtained, using these new techniques, can be up to 60 times higher than that of the conventional finite-element method, and 10 to 15 times higher than that of the conventional boundary-element method.     

Linear smoothed polygonal and polyhedral finite elements

The strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method. In this work, we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes. The main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain. This new strain is then used in the computation of the stiffness matrix. The convergence properties and accuracy of the proposed scheme are discussed by solving a few benchmark problems. Numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the same optimal convergence rate as traditional quadrilateral and hexahedral approximations. The accuracy is also improved, and all the methods tested pass the patch test to machine precision. Copyright © 2016 John Wiley & Sons, Ltd.