Automatic generation of anisotropic quadrilateral meshes on three‐dimensional surfaces using metric specifications

A new algorithm for constructing full quadrilateral anisotropic meshes on 3D surfaces is proposed in this paper. The proposed method is based on the advancing front and the systemic merging techniques. Full quadrilateral meshes are constructed by systemically converting triangular elements in the background meshes into quadrilateral elements.By using the metric specifications to describe the element characteristics, the proposed algorithm is applicable to convert both isotropic and anisotropic triangular meshes into full quadrilateral meshes. Special techniques for generating anisotropic quadrilaterals such as new selection criteria of base segment for merging, new approaches for the modifications of the background mesh and construction of quadrilateral elements, are investigated and proposed in this study. Since the final quadrilateral mesh is constructed from a background triangular mesh and the merging procedure is carried out in the parametric space, the mesh generator is robust and no expensive geometrical computation that is commonly associated with direct quadrilateral mesh generation schemes is needed. Copyright © 2002 John Wiley & Sons, Ltd.     

Q‐Morph: an indirect approach to advancing front quad meshing

Q‐Morph is a new algorithm for generating all‐quadrilateral meshes on bounded three‐dimensional surfaces. After first triangulating the surface, the triangles are systematically transformed to create an all‐quadrilateral mesh. An advancing front algorithm determines the sequence of triangle transformations. Quadrilaterals are formed by using existing edges in the triangulation, by inserting additional nodes, or by performing local transformations to the triangles. A method typically used for recovering the boundary of a Delaunay mesh is used on interior triangles to recover quadrilateral edges. Any number of triangles may be merged to form a single quadrilateral. Topological clean‐up and smoothing are used to improve final element quality. Q‐Morph generates well‐aligned rows of quadrilaterals parallel to the boundary of the domain while maintaining a limited number of irregular internal nodes. The proposed method also offers the advantage of avoiding expensive intersection calculations commonly associated with advancing front procedures. A series of examples of Q‐Morph meshes are also presented to demonstrate the versatility of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

Fast and accurate 4‐node quadrilateral

An accurate and variationally consistent 4‐node quadrilateral element is introduced where high coarse mesh accuracy and low mesh distortion sensitivity are characteristic qualities, even when incompressibility is approached for plane strain. One‐point quadrature integration procedure is adopted and a new improved stabilization technique is developed. Orthogonality conditions are utilized so that the patch test is satisfied for arbitrary quadrilaterals. Several numerical examples including a convergence rate study are presented which confirm the excellent performance of this element. Copyright © 2004 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.     

Verification of three‐dimensional anisotropic adaptive processes

A verification methodology for adaptive processes is devised. The mathematical claims made during the process are identified and measures are presented in order to verify that the mathematical equations are solved correctly. The analysis is based on a formal definition of the optimality of the adaptive process in the case of the control of the L^∞‐norm of the interpolation error. The process requires a reconstruction that is verified using a proper norm. The process also depends on mesh adaptation toolkits in order to generate adapted meshes. In this case, the non‐conformity measure is used to evaluate how well the adapted meshes conform to the size specification map at each iteration. Finally, the adaptive process should converge toward an optimal mesh. The optimality of the mesh is measured using the standard deviation of the element‐wise value of the L^∞‐norm of the interpolation error. The results compare the optimality of an anisotropic process to an isotropic process and to uniform refinement on highly anisotropic 2D and 3D test cases. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

An L∞–Lp mesh‐adaptive method for computing unsteady bi‐fluid flows

This paper discusses the contribution of mesh adaptation to high‐order convergence of unsteady multi‐fluid flow simulations on complex geometries. The mesh adaptation relies on a metric‐based method controlling the L ^p‐norm of the interpolation error and on a mesh generation algorithm based on an anisotropic Delaunay kernel. The mesh‐adaptive time advancing is achieved, thanks to a transient fixed‐point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure. In the time direction, we enforce the equidistribution of the error, i.e. the error minimization in L ^∞ norm. This adaptive approach is applied to an incompressible Navier–Stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes. Applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows. Copyright © 2010 John Wiley & Sons, Ltd.     

A new automatic adaptive 3D solid mesh generation scheme for thin‐walled structures

A new algorithm to generate three‐dimensional (3D) mesh for thin‐walled structures is proposed. In the proposed algorithm, the mesh generation procedure is divided into two distinct phases. In the first phase, a surface mesh generator is employed to generate a surface mesh for the mid‐surface of the thin‐walled structure. The surface mesh generator used will control the element size properties of the final mesh along the surface direction. In the second phase, specially designed algorithms are used to convert the surface mesh to a 3D solid mesh by extrusion in the surface normal direction of the surface. The extrusion procedure will control the refinement levels of the final mesh along the surface normal direction. If the input surface mesh is a pure quadrilateral mesh and refinement level in the surface normal direction is uniform along the whole surface, all hex‐meshes will be produced. Otherwise, the final 3D meshes generated will eventually consist of four types of solid elements, namely, tetrahedron, prism, pyramid and hexahedron. The presented algorithm is highly flexible in the sense that, in the first phase, any existing surface mesh generator can be employed while in the second phase, the extrusion procedure can accept either a triangular or a quadrilateral or even a mixed mesh as input and there is virtually no constraint on the grading of the input mesh. In addition, the extrusion procedure development is able to handle structural joints formed by the intersections of different surfaces. Numerical experiments indicate that the present algorithm is applicable to most practical situations and well‐shaped elements are generated. Copyright © 2004 John Wiley & Sons, Ltd.     

A novel four‐node quadrilateral smoothing element for stress enhancement and error estimation

A four‐node, quadrilateral smoothing element is developed based upon a penalized‐discrete‐least‐squares variational formulation. The smoothing methodology recovers C¹‐continuous stresses, thus enabling effective a posteriori error estimation and automatic adaptive mesh refinement. The element formulation is originated with a five‐node macro‐element configuration consisting of four triangular anisoparametric smoothing elements in a cross‐diagonal pattern. This element pattern enables a convenient closed‐form solution for the degrees of freedom of the interior node, resulting from enforcing explicitly a set of natural edge‐wise penalty constraints. The degree‐of‐freedom reduction scheme leads to a very efficient formulation of a four‐node quadrilateral smoothing element without any compromise in robustness and accuracy of the smoothing analysis. The application examples include stress recovery and error estimation in adaptive mesh refinement solutions for an elasticity problem and an aerospace structural component. Copyright © 1999 John Wiley & Sons, Ltd.     

Two simple and efficient displacement‐based quadrilateral elements for the analysis of composite laminated plates

Two simple 4‐node 20‐DOF and 4‐node 24‐DOF displacement‐based quadrilateral elements named RDKQ‐L20 and RDKQ‐L24 are developed in this paper based on the first‐order shear deformation theory (FSDT) for linear analysis of thin to moderately thick laminates. The deflection and rotation functions of the element sides are obtained from Timoshenko's laminated composite beam functions. Linear displacement interpolation functions of the standard 4‐node quadrilateral isoparametric plane element and displacement functions of a quadrilateral plane element with drilling degrees of freedom are taken as in‐plane displacements of the proposed elements RDKQ‐L20 and RDKQ‐L24, respectively. Due to the application of Timoshenko's laminated composite beam functions, convergence can be ensured theoretically for very thin laminates. The elements are simple in formulation, and shear‐locking free for extremely thin laminates even with full integration. A hybrid‐enhanced procedure is employed to improve the accuracy of stress analysis, especially for transverse shear stresses. Numerical tests show that the new elements are convergent, not sensitive to mesh distortion, accurate and efficient for analysis of thin to moderately thick laminates. Copyright © 2004 John Wiley & Sons, Ltd.     

Analysis of a rotation‐free 4‐node shell element

In this paper, a shell element for small and large deformations is presented based on the extension of the methodology to derive triangular shell element without rotational degrees of freedom (so‐called rotation‐free). As in our original triangular S3 element, the curvatures are computed resorting to the surrounding elements. However, the extension to a quadrilateral element requires internal curvatures in order to avoid singular bending stiffness. The quadrilateral area co‐ordinates interpolation is used to establish the required expressions between the rigid‐body modes of normal nodal translations and the normal through thickness bending strains at mid‐side. In order to propose an attractive low‐cost shell element, the one‐point quadrature is achieved at the centre for the membrane strains, which are superposed to the bending strains in the centred co‐rotational local frame. The membrane hourglass control is obtained by the perturbation stabilization procedure. Free, simply supported and clamped edges are considered without introducing virtual nodes or elements. Several numerical examples with regular and irregular meshes are performed to show the convergence, accuracy and the reasonable little sensitivity to geometric distortion. Based on an updated Lagrangian formulation and Newton iterations, the large displacements of the pinched hemispherical shell show the effectiveness of the proposed simplified element (S4). Finally, the deep drawing of a square box including large plastic strains with contact and friction completes the ability of the rotation‐free quadrilateral element for sheet‐metal‐forming simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

A new crack tip element for the phantom‐node method with arbitrary cohesive cracks

We have developed a new crack tip element for the phantom‐node method. In this method, a crack tip can be placed inside an element. Therefore, cracks can propagate almost independent of the finite element mesh. We developed two different formulations for the three‐node triangular element and four‐node quadrilateral element, respectively. Although this method is well suited for the one‐point quadrature scheme, it can be used with other general quadrature schemes. We provide some numerical examples for some static and dynamic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved four‐node hybrid‐mixed element based upon Mindlin's plate theory

Based on the mixed shear projected (MiSP) approach (Reference [54]: Int. J. Numer. Meth. Engng 1998; 42 :1149–1179), an enhanced bending approximation for homogeneous isotropic plates is presented. Some hard benchmark tests, such as the skew plate (30°) problem, have often shown poor convergence when low‐order elements (3‐ or 4‐node element) are developed using linear approximations for kinematic variables. To put right this weakness, we propose a high‐order interpolation for rotational dofs which results in more rich bending curvatures. The mid‐side rotational nodes are eliminated using a combination of local discrete kinematic and constitutive Mindlin hypotheses. The derived 4‐node quadrilateral element, called MiSP4+, is free of shear locking and passes all patch tests for thick and thin plates in an arbitrary mesh. Copyright © 2002 John Wiley & Sons, Ltd.     

Quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method (QACM‐II)

A novel strategy for developing low‐order membrane elements with analytical element stiffness matrices is proposed. First, some complete low‐order basic analytical solutions for plane stress problems are given in terms of the new quadrilateral area coordinates method (QACM‐II). Then, these solutions are taken as the trial functions for developing new membrane elements. Thus, the interpolation formulae for displacement fields naturally possess second‐order completeness in physical space (Cartesian coordinates). Finally, by introducing nodal conforming conditions, new 4‐node and 5‐node membrane elements with analytical element stiffness matrices are successfully constructed. The resulting models, denoted as QAC‐ATF4 and QAC‐ATF5, have high computational efficiency since the element stiffness matrices are formulated explicitly and no internal parameter is added. These two elements exhibit excellent performance in various bending problems with mesh distortion. It is demonstrated that the proposed strategy possesses advantages of both the analytical and the discrete method, and the QACM‐II is a powerful tool for constructing high‐performance quadrilateral finite element models. Copyright © 2008 John Wiley & Sons, Ltd.     

A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates

A new discrete Kirchhoff quadrilateral element based on the refined third-order theory is developed for the analysis of composite plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid-surface. The inplane displacements and the shear strains are interpolated using bilinear interpolation functions and the mid-surface rotations are interpolated using bi-quadratic functions based on the discrete Kirchhoff technique. The element stiffness matrix and the consistent load vector are developed using the principle of virtual work. The finite element formulation is validated by comparing the results for simply-supported plate with the analytical Navier solution. Comparison of the present results with those using other available elements based on the TOT establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The element is free from shear locking     

A hybrid‐mixed four‐node quadrilateral plate element based on sampling surfaces method for 3D stress analysis

The hybrid‐mixed assumed natural strain four‐node quadrilateral element using the sampling surfaces (SaS) technique is developed. The SaS formulation is based on choosing inside the plate body N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the thickness direction permits the presentation of the plate formulation in a very compact form. The SaS are located at Chebyshev polynomial nodes that allow one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and have no spurious zero energy modes, the assumed natural strain concept is employed. The developed hybrid‐mixed four‐node quadrilateral plate element passes patch tests and exhibits a superior performance in the case of coarse distorted mesh configurations. It can be useful for the 3D stress analysis of thin and thick plates because the SaS formulation gives the possibility to obtain solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity. Copyright © 2016 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 four‐node,shear‐deformable shell element developed via explicit Kirchhoff constraints

An efficient, four‐node quadrilateral shell element is formulated using a linear, first‐order shear deformation theory. The bending part of the formulation is constructed from a cross‐diagonal assembly of four three‐node anisoparametric triangular plate elements, referred to as MIN3. Closed‐form constraint equations, which arise from the Kirchhoff constraints in the thin‐plate limit, are derived and used to eliminate the degrees‐of‐freedom associated with the ‘internal’ node of the cross‐diagonal assembly. The membrane displacement field employs an Allman‐type, drilling degrees‐of‐freedom formulation. The result is a displacement‐based, fully integrated, four‐node quadrilateral element, MIN4T, possessing six degrees‐of‐freedom at each node. Results for a set of validation plate problems demonstrate that the four‐node MIN4T has similar robustness and accuracy characteristics as the original cross‐diagonal assembly of MIN3 elements involving five nodes. The element performs well in both moderately thick and thin regimes, and it is free of shear locking. Shell validation results demonstrate superior performance of MIN4T over MIN3, possibly as a result of its higher‐order interpolation of the membrane displacements. It is also noted that the bending formulation of MIN4T is kinematically compatible with the existing anisoparametric elements of the same order of approximation, which include a two‐node Timoshenko beam element and a three‐node plate element, MIN3. Copyright © 2000 John Wiley & Sons, Ltd.     

Two improvements in formulation of nine‐node element MITC9

The paper concerns a well‐known two‐dimensional nine‐node quadrilateral element MITC9, which is based on two‐level approximations of strains (assumed strain method). The element has good accuracy, but does not pass the patch test. As the first improvement, we propose a modification of the element's transformations, partly resolving the problem with the patch test. The source of the problem is the use of covariant components in a (local) natural co‐basis, different at each sampling point. As the second improvement, we use the corrected shape functions of Celia MA, Gray WG. An improved isoparametric transformation for finite element analysis. International Journal for Numerical Methods in Engineering 1984; 20 :1447–1459, extending their applicability to the nine‐node element for plane elasticity and the 3 × 3 integration. Originally, they are tested for an eight‐node element for the heat conduction equation and the 4 × 4 integration. The improved element, designated as MITC9i, is based on the Green strain and derived from the potential energy for the plane stress condition. It is subjected to a range of tests, to confirm that it passes the patch test for several types of mesh distortions, to prove its coarse mesh accuracy and the absence of locking as well as to establish its sensitivity to mesh distortions. The improved element MITC9i performs substantially better than the MITC9 element, QUAD9** element, and our previous 9‐AS element.Copyright © 2012 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.