Derived data structure algorithms for unstructured finite element meshes

A set of derived data structure algorithms for unstructured finite element meshes is presented. Both serial and parallel algorithms are described for each data structure. Colouring groups for the elements are used to facilitate parallelization on shared memory architectures. Scaling studies indicate that the parallel algorithms are most efficient when the number of elements per processor is on the order of 10⁶ or higher, and overall efficiencies of 60–70% are achieved down to 0.5×10⁶ elements per processor. Although the meshes under consideration are tetrahedral, the algorithms are general in nature and can be extended to arbitrary element types with minimal effort. Copyright © 2002 John Wiley & Sons, Ltd.     

Finite elements for materials with strain gradient effects

A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first‐order strain gradients and the associated work‐conjugate higher‐order stresses. In conventional displacement‐based approaches, the interpolation of displacement requires C¹‐continuity in order to ensure convergence of the finite element procedure for higher‐order theories. Mixed‐type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C⁰‐continuous shape functions and can achieve the same convergence as C¹ elements. These C⁰ elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

H(div) finite elements based on nonaffine meshes for 3D mixed formulations of flow problems with arbitrary high order accuracy of the divergence of the flux

Effects of nonaffine elements on the accuracy of 3D H(div)-conforming finite elements are studied. Instead of convergence order k+1 for the flux and the divergence of the flux obtained with Raviart-Thomas or Nédélec spaces with normal traces of degree k, based on affine hexahedra or triangular prisms, reduced orders k for the flux and k−1 for the divergence of the flux may occur for distorted elements. To improve this scenario, a hierarchy of enriched flux approximations is considered, by adding internal shape functions up to a higher degree k+n, n>0, while keeping the original normal traces of degree k. The resulting enriched approximations, using multilinear transformations, keep the original flux accuracy (of order k+1 with affine elements or reduced order k otherwise), but enhanced divergence (of order k+n+1, in the affine case, or k+n−1 otherwise) can be reached. The reduced flux accuracy due to quadrilateral face distortions cannot be corrected by including higher order internal functions. The enriched spaces are applied to the mixed finite element formulation of Darcy's model. The computational cost of matrix assembly increases with n, but the condensed system to be solved has the same dimension and structure as the original scheme.     

The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes

The finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.     

A new mass lumping scheme for the mixed hybrid finite element method

Diffusion‐type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M‐matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M‐matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M‐matrix for rectangular elements and can be an M‐matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than π/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright © 2006 John Wiley & Sons, Ltd.     

Centroidal Voronoi tessellation‐based finite element superconvergence

In this article, a finding on finite element superconvergence is reported. The Laplacian operator with Dirichlet boundary condition is considered. The linear finite element solutions have an O(h^2+α)(α≈0.5)‐superconvergence in l₂ norm at nodes on an almost equilateral triangular mesh generated based on centroidal Voronoi tessellation, for an arbitrary 2D bounded domain. Extensive numerical examples are presented to demonstrate the superconvergence property. Copyright © 2008 John Wiley & Sons, Ltd.     

Improving the modified XFEM for optimal high-order approximation

This paper investigates the accuracy of high-order extended finite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modified XFEM, a specific form of the stable generalised finite element method, is found to offer advantages in cost and complexity over other approaches, but suffers from suboptimal rates of convergence due to spurious higher-order contributions to the approximation space. An improved modified XFEM is presented, with basis functions “corrected” by projecting out higher-order contributions that cannot be represented by the standard finite element basis. The resulting corrections are independent of the equations being solved and need be pre-computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order p ≤ 4. These are the first truly optimal convergence results achieved using the XFEM for a curved weak discontinuity and are also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order p>1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order p ≤ 4.     

Modal validation of sandwich shell finite elements based on a p‐order shear deformation theory including zigzag terms

This paper describes a set of improved C⁰‐compatible composite shell finite elements for evaluating the global dynamic response (natural frequencies and mode shapes) of sandwich structures. Combining a through‐the‐thickness displacement approximation of variable high order with a first‐order zigzag function, the proposed finite elements are suited for modelling sandwich plates and doubly curved shells with a non‐uniform thickness and are more accurate than conventional models based on the first‐ and third‐order shear deformation theories, especially in sandwich panels with highly heterogeneous properties. The new finite element model is then validated by a comparison with the standard shell and 3D solid models. From these investigations, it can be concluded that adding a zigzag function even to high‐order polynomial approximations of the through‐the‐thickness displacement is a useful tool for refining the modelling of sandwich structures. In addition, the proposed formulation is sufficiently versatile to represent with the same level of accuracy the behaviour of thin‐to‐thick laminated shells as well as of strongly heterogeneous sandwich structures. Copyright © 2008 John Wiley & Sons, Ltd.     

Moving particle finite element method with global smoothness

We describe a new version of the moving particle finite element method (MPFEM) that provides solutions within a C⁰ finite element framework. The finite elements determine the weighting for the moving partition of unity. A concept of ‘General Shape Function’ is proposed which extends regular finite element shape functions to a larger domain. These are combined with Shepard functions to obtain a smooth approximation. The Moving Particle Finite Element Method combines desirable features of finite element and meshfree methods. The proposed approach, in fact, can be interpreted as a ‘moving partition of unity finite element method’ or ‘moving kernel finite element method’. This method possesses the robustness and efficiency of the C⁰ finite element method while providing at least C¹ continuity. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 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.     

Convergence analysis of the Q1-finite element method for elliptic problems with non-boundary-fitted meshes

The aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain's boundary. This problematic of the last century (e.g. the finite difference methodology) returns to topical studies with the huge development of domain embedding, fictitious domain or Cartesian-grid methods. These methods use regular structured meshes (most often Cartesian) for non-aligned domains. Although non-boundary-fitted approaches become more and more applied, very few studies are devoted to theoretical error estimates. In this paper, the convergence of a Q₁-non-conforming finite element method is analyzed for second-order elliptic problems with Dirichlet, Robin or Neumann boundary conditions. The finite element method uses standard Q₁-rectangular finite elements. As the finite element approximate space is not contained in the original solution space, this method is referred to as non-conforming. A stair-step boundary defined from the Cartesian mesh approximates the original domain's boundary. The convergence analysis of the finite element method for such a kind of non-boundary-fitted stair-stepped approximation is not treated in the literature. The study of Dirichlet problems is based on similar techniques as those classically used with boundary-fitted linear triangular finite elements. The estimates obtained for Robin problems are novel and use some more technical arguments. The rate of convergence is proved to be in 𝒪(h^1/2) for the H¹-norm for all general boundary conditions, and classical duality arguments allow one to obtain an 𝒪(h) error estimate in the L²-norm for Dirichlet problems. Numerical results obtained with fictitious domain techniques, which impose original boundary conditions on a non-boundary-fitted approximate immersed interface, are presented. These results confirm the theoretical rates of convergence. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order layerwise finite element for the damped free‐vibration response of thick composite and sandwich composite plates

A high‐order layerwise finite element methodology is presented, which enables prediction of the damped dynamic characteristics of thick composite and sandwich composite plates. The through‐thickness displacement field in each discrete layer of the laminate includes quadratic and cubic polynomial distributions of the in‐plane displacements, in addition to the linear approximations assumed by linear layerwise theories. Stiffness, mass and damping matrices are formulated from ply to structural level. Interlaminar shear stress compatibility conditions are imposed on the discrete layer matrices, leading to both size reduction and prediction of interlaminar shear stresses at the laminate interfaces. The C¹ continuous finite element implemented yields an element damping matrix in addition to element stiffness and mass matrices. Application cases include thick [0/90/0], [±θ]_S and [±θ] composite plates with interlaminar damping layers and sandwich plates with composite faces and foam core. In the latter case, modal frequencies and damping were also experimentally determined and compared with the finite element predictions. Copyright © 2008 John Wiley & Sons, Ltd.     

A method for creating a class of triangular C1 finite elements

Finite elements providing a C¹ continuous interpolation are useful in the numerical solution of problems where the underlying partial differential equation is of fourth order, such as beam and plate bending and deformation of strain‐gradient‐dependent materials. Although a few C¹ elements have been presented in the literature, their development has largely been heuristic, rather than the result of a rational design to a predetermined set of desirable element properties. Therefore, a general procedure for developing C¹ elements with particular desired properties is still lacking. This paper presents a methodology by which C¹ elements, such as the TUBA 3 element proposed by Argyris et al., can be constructed. In this method (which, to the best of our knowledge, is the first one of its kind), a class of finite elements is first constructed by requiring a polynomial interpolation and prescribing the geometry, the location of the nodes and the possible types of nodal DOFs. A set of necessary conditions is then imposed to obtain appropriate interpolations. Generic procedures are presented, which determine whether a given potential member of the element class meets the necessary conditions. The behaviour of the resulting elements is checked numerically using a benchmark problem in strain‐gradient elasticity. Copyright © 2011 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

On the performance of non‐conforming finite elements for the upper bound limit analysis of plates

In this paper, the upper bound limit analysis of thin plates in bending is addressed using various types of triangular finite elements for the generation of velocity fields and second‐order cone programming for the minimization problem. Three different C¹‐discontinuous finite elements are considered: the quadratic six‐node Lagrange triangle (T6), an enhanced T6 element with a cubic bubble function at centroid (T6b) and the cubic Hermite triangle (H3). Through numerical examples involving Johansen and von Mises yield criteria, it is shown that cubic elements (H3) give far better results in terms of convergence rate and precision than fully conforming elements found in the literature, especially for problems involving clamped boundaries. Copyright © 2013 John Wiley & Sons, Ltd.     

Bounds for element size in a variable stiffness cohesive finite element model

The cohesive finite element method (CFEM) allows explicit modelling of fracture processes. One form of CFEM models integrates cohesive surfaces along all finite element boundaries, facilitating the explicit resolution of arbitrary fracture paths and fracture patterns. This framework also permits explicit account of arbitrary microstructures with multiple length scales, allowing the effects of material heterogeneity, phase morphology, phase size and phase distribution to be quantified. However, use of this form of CFEM with cohesive traction–separation laws with finite initial stiffness imposes two competing requirements on the finite element size. On one hand, an upper bound is needed to ensure that fields within crack‐tip cohesive zones are accurately described. On the other hand, a lower bound is also required to ensure that the discrete model closely approximates the physical problem at hand. Both issues are analysed in this paper within the context of fracture in multi‐phase composite microstructures and a variable stiffness bilinear cohesive model. The resulting criterion for solution convergence is given for meshes with uniform, cross‐triangle elements. A series of calculations is carried out to illustrate the issues discussed and to verify the criterion given. These simulations concern dynamic crack growth in an Al₂O₃ ceramic and in an Al₂O₃/TiB₂ ceramic composite whose phases are modelled as being hyperelastic in constitutive behaviour. Copyright © 2004 John Wiley & Sons, Ltd.     

Converting a tetrahedral mesh to a prism–tetrahedral hybrid mesh for FEM accuracy and efficiency

This paper presents a computational method for converting a tetrahedral mesh to a prism–tetrahedral hybrid mesh for improved solution accuracy and computational efficiency of finite element analysis. The proposed method performs this conversion by inserting layers of prism elements and deleting tetrahedral elements in sweepable sub‐domains, in which cross‐sections remain topologically identical and geometrically similar along a certain sweeping path. The total number of finite elements is reduced because roughly three tetrahedral elements are converted to one prism element. The solution accuracy of the finite element analysis improves since a prism element yields a more accurate solution than a tetrahedral element due to the presence of higher‐order terms in the shape function. Only previously known method for creating such a prism–tetrahedral hybrid mesh was to manually decompose a target volume into sweepable and non‐sweepable sub‐volumes and mesh each of the sub‐volumes separately. Unlike the previous method, the proposed method starts from a cross‐section of a tetrahedral mesh and replaces the tetrahedral elements with layers of prism elements until prescribed quality criteria can no longer be satisfied. A series of computational fluid dynamics simulations and structural analyses have been conducted, and the results verified a better performance of prism–tetrahedral hybrid mesh. Copyright © 2009 John Wiley & Sons, Ltd.     

Three dimensional analysis of compressible potential flows with the finite element method

Solutions of elliptic boundary value problems are obtained with variational formulations by minimization of functionals. Conventional tools of the finite element strategy are emphasized; linear or quadratic approximations on triangular elements, curved iso-parametric elements, N_i shape funtions, L_i homogeneous co-ordinates. High speed of computation of the final assembled matrices require the use of the Formal–FORMAC calculus. Illusrations of the finite element method are chosen in the class of the 2^d–3^d compressible subsonic irrotational flows in a nozzle and around one or several lifting bodies of arbitary shape.