Discrete gradient method over polygon mesh

This paper presents a discrete method over domains originally discretized by polygons including triangle, quadrilateral, and general n‐sided polygon elements. In this method, the domain is re‐partitioned into node‐based cells. At each node, the gradient of a physical variable is approximated using a linearly exact discrete operator that involves a set of neighboring nodes. The discrete gradient is subsequently substituted into a weak form to yield a nodal‐integration Galerkin formulation. A unified geometric approach is provided for constructing the gradient operators over an arbitrary polygon mesh. The method does not introduce continuous approximation of the unknown variable; therefore, the numerical computation is very simple. The linear displacement patch test is satisfied by construction. Numerical tests show that the method has comparable accuracy and convergence rate as the displacement finite element method. Examples are also included to illustrate the ability to resist numerical locking in the incompressibility limit and the thin‐element limit. Copyright © 2008 John Wiley & Sons, Ltd.     

Scaled boundary polygons with application to fracture analysis of functionally graded materials

This study presents a framework for the development of polygon elements based on the scaled boundary FEM. Underpinning this study is the development of generalized scaled boundary shape functions valid for any n‐sided polygon. These shape functions are continuous inside each polygon and across adjacent polygons. For uncracked polygons, the shape functions are linearly complete. For cracked polygons, the shape functions reproduce the square‐root singularity and the higher‐order terms in the Williams eigenfunction expansion. This allows the singular stress field in the vicinity of the crack tip to be represented accurately. Using these shape functions, a novel‐scaled boundary polygon formulation that captures the heterogeneous material response observed in functionally graded materials is developed. The stiffness matrix in each polygon is derived from the principle of virtual work using the scaled boundary shape functions. The material heterogeneity is approximated in each polygon by a polynomial surface in scaled boundary coordinates. The intrinsic properties of the scaled boundary shape functions enable accurate computation of stress intensity factors in cracked functionally graded materials directly from their definitions. The new formulation is validated, and its salient features are demonstrated, using five numerical benchmarks. Copyright © 2014 John Wiley & Sons, Ltd.     

A 20‐node hexahedron element with enhanced distortion tolerance

This paper presents a distortion resistant 20‐node hexahedron element that employs two different sets of shape functions for the trial and test functions. The formulation seeks to satisfy the continuity and completeness requirements by exploiting the intrinsic properties of these two sets of shape functions. Several test problems are used to assess the performance of the element under various mesh distortions. The ability of the proposed as well as the classical 20‐node element to maintain solution accuracy under severe mesh distortions has been studied. The proposed element exhibits a very high tolerance to mesh distortions. In particular, for problems involving linear and quadratic displacement fields, the element is capable of reproducing exact solution under all admissible geometrical distortions of the mesh. For test problems involving higher‐order displacement fields, the performance of the present element is in general better than that of the classical element. Copyright © 2004 John Wiley & Sons, Ltd.     

On completeness of shape functions for finite element analysis

Some elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n-dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three-dimensional spaces (e.g. space–time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third-degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher-order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.     

A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

An 8‐node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions, viz., the compatibility enforcing shape functions and completeness enforcing shape functions. The former are chosen to satisfy exactly the minimum inter‐ as well as intra‐element displacement continuity requirements, while the latter are chosen to satisfy all the (linear and higher order) completeness requirements so as to reproduce exactly a quadratic displacement field. Numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions. In this respect, the proposed 8‐node unsymmetric element emerges to be better than the existing symmetric QUAD8, QUAD8/9, QUAD9, QUAD12 and QUAD16 elements, and matches the performance of the quartic element, QUAD25. For test problems involving a cubic or higher order displacement field, the proposed element yields a solution accuracy that is comparable to or better than that of QUAD8, QUAD8/9 and QUAD9 elements. Furthermore, the element maintains a good accuracy even with the reduced 2× 2 numerical integration. Copyright © 2003 John Wiley & Sons, Ltd.     

An unsymmetric 8‐node hexahedral element with high distortion tolerance

Among all 3D 8‐node hexahedral solid elements in current finite element library, the ‘best’ one can produce good results for bending problems using coarse regular meshes. However, once the mesh is distorted, the accuracy will drop dramatically. And how to solve this problem is still a challenge that remains outstanding. This paper develops an 8‐node, 24‐DOF (three conventional DOFs per node) hexahedral element based on the virtual work principle, in which two different sets of displacement fields are employed simultaneously to formulate an unsymmetric element stiffness matrix. The first set simply utilizes the formulations of the traditional 8‐node trilinear isoparametric element, while the second set mainly employs the analytical trial functions in terms of 3D oblique coordinates (R, S, T). The resulting element, denoted by US‐ATFH8, contains no adjustable factor and can be used for both isotropic and anisotropic cases. Numerical examples show it can strictly pass both the first‐order (constant stress/strain) patch test and the second‐order patch test for pure bending, remove the volume locking, and provide the invariance for coordinate rotation. Especially, it is insensitive to various severe mesh distortions. Copyright © 2016 John Wiley & Sons, Ltd.     

Hierarchical high‐order conforming C1 bases for quadrangular and triangular finite elements

We present three new sets of C¹ hierarchical high‐order tensor‐product bases for conforming finite elements. The first basis is a high‐order extension of the Bogner–Fox–Schmit basis. The edge and face functions are constructed using a combination of cubic Hermite and Jacobi polynomials with C¹ global continuity on the common edges of elements. The second basis uses the tensor product of fifth‐order Hermite polynomials and high‐order functions and achieves global C¹ continuity for meshes of quadrilaterals and C² continuity on the element vertices. The third basis for triangles is also constructed using the tensor product of one‐dimensional functions defined in barycentric coordinates. It also has global C¹ continuity on edges and C² continuity on vertices. A patch test is applied to the three considered elements. Projection and plate problems with smooth fabricated solutions are solved, and the performance of the h‐ and p‐refinements are evaluated by comparing the approximation errors in the L₂‐ and energy norms. A plate with singularity is then studied, and h‐ and p‐refinements are analysed. Finally, a transient problem with implicit time integration is considered. The results show exponential convergence rates with increasing polynomial order for the triangular and quadrilateral meshes of non‐distorted and distorted elements. Copyright © 2016 John Wiley & Sons, Ltd.     

A convergence analysis of the performance of the DRM‐MD boundary integral approach

In this article, we study the performance of the dual reciprocity multi‐domains approach (DRM‐MD) when the shape functions of the boundary elements, for both the approximation of the geometry and the surface variables of the governing equations, are quadratic functions. A series of tests are carried out to study the consistency of the proposed boundary integral technique. For this purpose a limiting process of the subdivision of the domain is performed, reporting a comparison of the computed solutions for every refining scheme. Furthermore, the DRM‐MD is solved in its dual reciprocity approximation using two different radial basis interpolation functions, the conical function r plus a constant, i.e. (1+r), and the augmented thin plate spline. Special attention is paid to the contrast between numerical results yielded by the DRM‐MD approach using linear and quadratic boundary elements towards a full understanding of its convergence behaviour. Copyright © 2006 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.     

Universal serendipity elements

Universal serendipity elements (USE) are defined as isoparametric elements having linear, quadratic and cubic node configurations at their edges in an arbitrary manner. Formulation of shape functions and their derivatives for USE is presented. A computer algorithm which allows the efficient application of USE is explained. Only corner nodes are kept transparent to the user. Changing the order of USE is fully automated. It is concluded that the USE concept allows interactive mesh refinement in an efficient manner and eliminates the difficulties encountered in the use of higher order isoparametric serendipity elements.     

含两个分量的四边形单元面积坐标理论

为了便于构造抗畸变的四边形单元,建立了一套新的四边形单元面积坐标理论(QAC-2),并给出了相关的积分和微分公式。该坐标系作为自然坐标,具有明确的物理意义,且只含有两个相互独立的坐标分量,因此易于实现与直角坐标和等参坐标的沟通,便于理解和应用;两个坐标分量与直角坐标之间满足线性变换,在构造单元时易于选择完备的多项式序列,且多项式的完备次数不会随着网格的畸变而下降,因此可以保证单元的精度和抗畸变性能。     

Unsymmetric extensions of Wilson's incompatible four-node quadrilateral and eight-node hexahedral elements

The unsymmetric finite element is based on the virtual work principle with different sets of test and trial functions. In this article, the incompatible four-node quadrilateral element and eight-node hexahedral element originated by Wilson et al. are extended to their unsymmetric forms. The isoparametric shape functions together with Wilson's incompatible functions are chosen as the test functions, while internal nodes at the middle of element sides/edges are added to generate the trial functions with quadratic completeness in the Cartesian coordinate system. A local area/volume coordinate frame is established so that the trial shape functions can be explicitly obtained. The key idea which avoids the matrix inversion is that the trial nodal shape functions are constructed by standard quadratic triangular/tetrahedral elements and then transformed in consistent with the quadrilateral/hexahedral elements. Numerical examples show that the present elements keep the merits of both incompatible and unsymmetric elements, that is, high numerical accuracy, insensitivity to mesh distortion, free of trapezoidal and volumetric locking, and easy implementation.     

An n-sided polygonal finite element for nonlocal nonlinear analysis of plates and laminates

In this study, a locking-free n-sided C¹ polygonal finite element is presented for nonlinear analysis of laminated plates. The plate kinematics is based on Reddy's third-order shear deformation theory (TSDT). The in-plane displacements are approximated using barycentric form of Lagrange shape functions. The weak-form Galerkin formulation based on the kinematics of TSDT requires the C¹ approximation of the transverse displacement over the polygonal element. This is achieved by embedding the C⁰ Lagrange interpolants over a cubic Bernstein-Bezier patch defined over the n-sided polygonal element. Such an approach ensures the continuity of the derivative field at the inter-element edges. In addition, Eringen's stress-gradient nonlocal constitutive equations are used in the present formulation to account for nonlocality. The effect of geometric nonlinearity is taken by considering the von Kármán geometric nonlinearity. Examples are presented to show the effect of nonlocality, geometric nonlinearity, and the lamination scheme on the bending behavior of laminated composite plates. The results are compared with analytical solutions, conventional FEM results, and with those available in the literature. Shear locking is addressed considering reduced integration and consistent interpolation techniques. The patch test is used to check the convergence of the element developed.     

An unsymmetric 4‐node, 8‐DOF plane membrane element perfectly breaking through MacNeal's theorem

Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal's theorem: any 4‐node, 8‐DOF membrane element will either lock in in‐plane bending or fail to pass a C₀ patch test when the element's shape is an isosceles trapezoid. In this paper, a 4‐node plane quadrilateral membrane element is developed following the unsymmetric formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4‐node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates (x,y) and the second form of quadrilateral area coordinates (QACM‐II) (S,T) are applied together. The resulting element US‐ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first‐order patch test and the second‐order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM‐II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well‐known contradiction defined by MacNeal's theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

面积坐标法构造含转角自由度的四结点膜元

以四边形面积坐标作为工具,构造了两个含转角自由度的广义协调四边形单元AQ4和lAQ4。它们通过强式分片检验,与同类单元相比,具有很高的计算精度,能消除梯形闭锁现象,有很强的抗网格畸变的能力。     

Polygon scaled boundary finite elements for crack propagation modelling

An automatic crack propagation modelling technique using polygon elements is presented. A simple algorithm to generate a polygon mesh from a Delaunay triangulated mesh is implemented. The polygon element formulation is constructed from the scaled boundary finite element method (SBFEM), treating each polygon as a SBFEM subdomain and is very efficient in modelling singular stress fields in the vicinity of cracks. Stress intensity factors are computed directly from their definitions without any nodal enrichment functions. An automatic remeshing algorithm capable of handling any n‐sided polygon is developed to accommodate crack propagation. The algorithm is simple yet flexible because remeshing involves minimal changes to the global mesh and is limited to only polygons on the crack paths. The efficiency of the polygon SBFEM in computing accurate stress intensity factors is first demonstrated for a problem with a stationary crack. Four crack propagation benchmarks are then modelled to validate the developed technique and demonstrate its salient features. The predicted crack paths show good agreement with experimental observations and numerical simulations reported in the literature. Copyright © 2012 John Wiley & Sons, Ltd.     

Fully tensorial nodal and modal shape functions for triangles and tetrahedra

This paper presents nodal and modal shape functions for triangle and tetrahedron finite elements. The functions are constructed based on the fully tensorial expansions of one‐dimensional polynomials expressed in barycentric co‐ordinates. The nodal functions obtained from the application of the tensorial procedure are the standard h‐Lagrange shape functions presented in the literature. The modal shape functions use Jacobi polynomials and have a natural global C⁰ inter‐element continuity. An efficient Gauss–Jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices. An example illustrates the approximation properties of the modal functions. Copyright © 2005 John Wiley & Sons, Ltd.     

Patch‐averaged assumed strain finite elements for stress analysis

A finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

Four‐node incompatible plane and axisymmetric elements with quadratic completeness in the physical space

In this paper, four‐node incompatible elements with quadratic completeness in the physical space are developed for plane and axisymmetric problems. Various revision techniques of the incompatible strain matrix or shape function are proposed to guarantee the incompatible elements to pass the constant stress patch test. In contrast with other elements in the same level, the present one gives reasonable stress solutions without any stress recovery such as the bilinear extrapolation and least square approximation. Numerical examples are given to investigate the accuracy, sensitivity to mesh distortion and locking in the incompressible calculation of the present method when irregular coarse meshes are used. Copyright © 2004 John Wiley & Sons, Ltd.