Three-dimensional 8-node and 20-node refined hybrid isoparametric elements

In this paper a new hybrid variational principle with independent variables of strain, stress and displacement and with a weaker constraint condition of interelement continuity is proposed. Based on this functional, a general formulation of a refined hybrid isoparametric element method has been established by the orthogonal approach. The present formulation is a rational approach to be adopted for deriving high-performance three-dimensional hybrid isoparametric elements even up to the higher-order 20-node element. Several numerical examples are presented to show that the present elements RGH₈(8-node) and RGH₂₀(20-node) have high accuracy, excellent computational efficiency and less sensitivity to mesh distortion.     

Stability of collapsed 8-node 2-D and 20-node 3-D isoparametric finite elements

The collapse of ordinary finite elements to generate the desired strain singularity at the crack tip for fracture mechanics applications can lead to unwanted additional singularities in that area. Although neither the quarter-point nor the half-point 8-node two-dimensional (2-D) and 20-node three-dimensional (3-D) elements exhibit this behaviour, the present article proves that arbitrarily small deviations from the quarter-point element can be constructed which do have additional singularities. Since the general behaviour of the half-point element is not affected by small modifications, this element is better suited to match complex body geometries.     

General behaviour of collapsed 8-node 2-D and 20-node 3-D isoparametric elements

This paper presents a simple and thorough technique to examine the general behaviour of two-dimensional (2-D) 8-node and 3-D 20-node isoparametric elements, collapsed at one side. With this method the conditions for the well-known quarter-point and half-point elements, frequently used in fracture mechanics applications to model the crack tip behaviour, are arrived at in a natural way. It also allows for an in-depth analysis of the Jacobian determinant, and can easily be applied to other types of elements.     

An unsymmetric 8-node hexahedral solid-shell element with high distortion tolerance: Geometric nonlinear formulations

A recent distortion-tolerant unsymmetric 8-node hexahedral solid-shell element US-ATFHS8, which takes the analytical solutions of linear elasticity as the trial functions, is successfully extended to geometric nonlinear analysis. This extension is based on the corotational (CR) approach due to its simplicity and high efficiency, especially for geometric nonlinear analysis where the strain is still small. Based on the assumption that the analytical trial functions can properly work in each increment during the nonlinear analysis, the incremental corotational formulations of the nonlinear solid-shell element US-ATFHS8 are derived within the updated Lagrangian (UL) framework, in which an appropriate updated strategy for linear analytical trial functions is proposed. Numerical examples show that the present nonlinear element US-ATFHS8 possesses excellent performance for various rigorous tests no matter whether regular or distorted mesh is used. Especially, it even performs well in some situations that other conventional elements cannot work.     

Control of spurious mechanisms for 20-node and transition sub-integrated hexahedral elements

In this paper, control of spurious mechanisms for sub-integrated 20-node and transition hexahedral elements is devised by an assumed stress approach. The higher-order stress modes for stabilizing the sub-integrated elements are identified by examining the spurious mechanisms. With an admissible simplification of the flexibility matrix, row vectors of the leverage matrix can be used as stabilization vectors. Numerical examples for the stabilized 20-node element are presented. Accuracy of the derived element is far better than the fully integrated displacement elements. Meanwhile, the former also consumes marginally less CPU time than the latter.     

High order polynomial elements with isoparametric mapping

Finite elements with polynomial interpolation functions of a degree higher than 2 are used comparatively little on large FEM systems, except in shell elements. However, the author has had several years' experience in the use of the so-called ‘isoparametric, reduced Hermitian element’,⁴ Which has behaved excellently in an industrial as well as an educational environment. The reason for the interest in the Hermitian concept is that the overcompatibility of the element reduces the number of unknowns, the solution time and the discontinuities in stresses between elements. Explicit formulae for the family of interpolation polynomials of order q and degree p = 2q + 1 are given and hierarchical Hermite elements are introduced. The families of Hermite and serendipity elements are isomorphic and the latter may thus be extended to arbitrary high order. For some problems the equidistant node configuration in Lagrange elements of degree 3 and higher is not optimal with respect to smoothness, and a new type of element, the ‘Lobatto element’, is introduced. The methods consistently produce results of an accuracy which is above the requirements of usual engineering applications, but in graphics smoothness of curves is important for a convincing representation. The methods are of particular interest in industries working with structures composed of almost linear materials with well-known properties.     

8-node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity

A new C⁰ 8-node 48-DOF hexahedral element is developed for analysis of size-dependent problems in the context of the modified couple stress theory by extending the methodology proposed in our recent work (Shang et al., Int J Numer Methods Eng 119(9): 807-825, 2019) to the three-dimensional (3D) cases. There are two major innovations in the present formulation. First, the independent nodal rotation degrees of freedom (DOFs) are employed to enhance the standard 3D isoparametric interpolation for obtaining the displacement and strain test functions, as well as to approximatively design the physical rotation field for deriving the curvature test function. Second, the equilibrium stress functions instead of the analytical functions are used to formulate the stress trial function whilst the couple stress trial function is directly obtained from the curvature test function by using the constitutive relationship. Besides, the penalty function is introduced into the virtual work principle for enforcing the C¹ continuity condition in weak sense. Several benchmark examples are examined and the numerical results demonstrate that the element can simulate the size-dependent mechanical behaviors well, exhibiting satisfactory accuracy and low susceptibility to mesh distortion.     

Alternative reduced integration avoiding spurious modes for 8-node quadrilateral and 20-node hexahedron finite elements

Integrating the isoparametric 8-node quadrilateral and the 20-node hexahedron elements with Gauss integration based on the 3 point rule produces stiff elements. The excessive stiffness is mainly due to locking phenomenon. One remedy to partly remove locking consists in using reduced integration. Mostly, 2 × 2 or 2 × 2 × 2 integration, respectively, is employed. The lower order integration introduces spurious element modes, however. These modes may deteriorate solutions for finite element models. To overcome this drawback alternative reduced integration procedures are presented. A 5-point rule for the quadrilateral is described. 9-point and 21-point procedures are introduced for the hexahedron. The performance of these procedures is studied by some test problems.     

Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

Inverse mapping and distortion measures for quadrilaterals with curved boundaries

By using the theory of geodesics in differential geometry, inverse relations of the mapping for 8-node quadrilaterals with curved boundaries are derived and expressed in terms of the element geodesic co-ordinates defined at the local origin. The parameters in the expressions of the geodesic co-ordinates in terms of the isoparametric co-ordinates are suggested to be the distortion measures of the element. The mathematical meanings of these parameters can be discussed and explained by using the coefficient vectors in the equations defining the isoparametric co-ordinate transformation. This method of defining the element distortion measures is consistent and general, and can be applied to any other two- or three-dimensional isoparametric finite elements.     

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.     

A drafting program for isoparametric finite elements

Considerable effort has been invested lately in the application of isoparametric finite elements for numerical solution of a wide range of applied mechanics problems. In fact, several general purpose computer programs are now available which are based upon such finite elements. In the present paper, a new application of the isoparametric finite element concept is introduced which significantly extends its usefulness for many practical structural configurations. In this application, final working or architectural drawings of the structure are made from the same (or similar) finite element model as was utilized in a structural integrity analysis. The hardware necessary to produce such drawing, a computer driven plotter or automated drafting machine, is available commercially or through most data centres, and the software concepts required are described herein.     

Use of modified standard 20-node isoparametric brick elements for representing stress/strain fields at a crack tip for elastic and perfectly plastic material

The analysis of defects in engineering structures and components has to take into account the singular strain field at the crack tip. The problems encountered in such analyses have unique geometries, have some non-linear elastic plastic behaviour and are three-dimensional in nature. Their solution calls for the use of the finite element method. Two-dimensional fracture mechanics analysis methods have been developed and proved by other researchers to show that 8-noded collapsed finite elements have the required singular strain fields for both the elastic and perfectly plastic material conditions. This paper proves the conditions under which three-dimensional collapsed elements represent the stress/strain fields at a crack tip required for elastic and perfectly plastic material, including crack tip blunting in the latter case. The collapsed elements presented can be used with confidence and give large savings in computing time, which is an essential point in three-dimensional finite element calculations.

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Résumé L'analyse des défauts dans les constructions et dans les composants doit prendre en considération le champ singulier de déformation à l'extrémité d'une fissure. Les problèmes rencontrés dans une telle analyse présentent des géométries uniques, un comportement élasto-plastique sensiblement non linéaire et sont tri-dimensionnelles par nature. Leur solution fait appel à l'utilisation de la méthode par éléments finis. Des méthodes d'analyse en mécanique de rupture bi-dimensionnelles ont été développées, et d'autres chercheurs ont établi qu'elles montrent que les éléments finis à 8 noeuds présentent les champs de déformation singuliers, requis à la fois pour les conditions de matériaux élastiques et parfaitement plastiques. La présente étude établit les conditions sous lesquelles des éléments tri-dimensionnels représentent les champs contrainte/déformation à l'extrémité d'une fissure, requis dans le cas de matériaux élastiques et parfaitement plastiques, y compris l'arrondissement de l'extrémité de la fissure pour ce dernier cas. Les éléments présentés peuvent être utilisés avec confiance et procurent de grandes économies de temps dans les calculs, ce qui est essentiel dans le cas des éléments finis à 3 dimensions.

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List of symbols x, y, z coordinates in the real space - g, h, s coordinates in the normalized cubic space - , , coordinates in modified normalized cubic space - u, v, w vectors of displacements inx, y andz direction, respectively - vectors of coordinatesx, y andz - vectors of displacementsu, v andw - N vector of shape functions - M vector of reduced shape functions - vector of stress components - vector of strain components - D matrix containing material properties - J determinant of the Jacobian matrix - J Jacobian matrix - r distance from the crack tip     

Analytical integration formulae for linear isoparametric finite elements

Exact and approximate analytical expressions can be derived for integrals arising in finite element methods, employing isoparametric linear quadrilaterals in two space dimensions with bilinear basis functions. The formulae associated with rectangular elements, arbitrarily oriented in space, can be shown to be a special case. The proposed method provides considerable savings in computational effort, in comparison with a numerical method that employs Gaussian quadrature procedures. In addition, the method, when applied to a quadrilateral inscribable in a circle, can be shown to produce better accuracy than the associated (2 × 2) Gaussian quadrature formulae.     

Isoparametric hybrid hexahedral elements for three dimensional stress analysis

Based on a new functional in which displacements, strains and stresses are taken as independent variables, a set of three 8-node hexahedral hybrid elements Q_S11-1, Q_S11-2 and Q_S11-3 is developed. The adoption of separated stress and displacement variables proves to be effective in improving the accuracy of the elements. The new elements are all capable of yielding converging results, and they all possess the properties of having no zero energy deformation modes and of being co-ordinate invariant. From the numerical example of a beam under bending it is concluded that exact solutions can be obtained for right prism elements, while good results are still attainable for severely distorted elements. The relationship between the new hybrid elements and the conventional displacement elements is also explored in this paper.     

A predictor-corrector contouring algorithm for isoparametric 3D elements

This paper describes a new contouring algorithm for isoparametric elements which can be used to map three-dimensional scalar fields. The contours are generated on arbitrary planes intersecting finite element structures. Tracing element contour lines is accomplished by an accurate predictor-corrector technique. A method of finding starting points for the algorithm on the boundary of the elements is also given.     

A transition element for uniform strain hexahedral and tetrahedral finite elements

A transition element is presented for meshes containing uniform strain hexahedral and tetrahedral finite elements. It is shown that the volume of the standard uniform strain hexahedron is identical to that of a polyhedron with 14 vertices and 24 triangular faces. Based on this equivalence, a transition element is developed as a simple modification of the uniform strain hexahedron. The transition element makes use of a general method for hourglass control and satisfies first‐order patch tests. Example problems in linear elasticity are included to demonstrate the application of the element. Copyright © 1999 John Wiley & Sons, Ltd. This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.