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 assumed-stress hybrid 4-node shell element with drilling degrees of freedom

An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or ‘drilling’ degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element by expressing the midside displacement degrees of freedom in terms of displacement and rotational degrees of freedom at corner nodes. The element passes the patch test, is nearly insensitive to mesh distortion, does not ‘lock’, possesses the desirable invariance properties, has no hidden spurious modes, and for the majority of test cases used in this paper produces more accurate results than the other elements employed herein for comparison.     

New strategy for assumed stresses for 4-node hybrid stress membrane element

An oblique (skew) co-ordinate system defined at the origin of the isoparametric co-ordinates is introduced as the reference co-ordinate system for assumed stresses in the formulation of a 4-node hybrid membrane element. The initially uncoupled stresses in these reference co-ordinates can be constrained to satisfy the equilibrium conditions pointwise by using the procedure of Pian and Wu⁷, when suitable incompatible displacements are introduced. Such strategy allows the element stiffness to be formulated by using either the Hellinger-Reissner or the complementary energy functional. Numerical results show that the performances of three elements, constructed by using different sets of incompatible displacements in this paper, are practically identical and are all of slightly higher accuracy than that of the Pian-Sumihara element⁶.     

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.     

An 8-node brick finite element

An 8-node brick element based on the assumed stress hybrid formulation is described. With three additional stress fields, the element stiffness matrix now has the required rank of 18, and the ‘bending’ response is exact with rectangular elements. Surprisingly, the 2 × 2 × 2 Gauss rule suffices for all numerical integrations, to satisfy the constant stress patch test.     

A robust refined quadrilateral plane element

Based on a rational choice of the internal incompatible displacement function and a special formulation of the a priori elimination of the internal non-conforming displacement parameters, a new refined quadrilateral plane element RQ₄ has been developed. The present element can be shown to be computationally efficient, accurate and free from locking, and is better than other elements such as the Plan's element HS, the generalized hybrid element Q_CS6, and the refined hybrid element RGH₄, etc. Several numerical examples are given to show the superior performances of the present element RQ₄.     

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.     

Elastoplastic analysis using the 14-node brick element family

Typical brick-based finite elements in current use are the 8-and 20-node members of the serendipity group. For displacement analyses of solids the 8-node element can be quite stiff in certain deformation modes, while the 20-node element can be quite expensive to use, involving as it does 60 degrees of freedom and a fairly high order of quadrature to avoid spurious eigenmodes of the element stiffness. In this paper a family of intermediate 14-node elements is investigated. Derivation of their properties can be considerably assisted by computer algebra. Performance is evaluated for elastic and elastoplastic problems.     

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.     

The reference coordinates and distortion measures for quadrilateral hybrid stress membrane element

The skew Cartesian coordinate system determined by the Jacobian of the isoparametric transformation evaluated at the origin can be shown to be a geodesic coordinate system at the origin. By using a theory in differential geometry, inverse relations of the isoparametric coordinate transformation can be derived and expressed in terms of these geodesic coordinates. In the formulation of hybrid stress finite elements, it is suggested as a new strategy for assumed stresses that such coordinates be used as the reference coordinates. The theory described is exemplified by its applications to the 4-node hybrid stress membrane elements. A set of new distortion-measuring parameters for the quadrilateral element are also proposed based on such theory.     

The inverse mapping and distortion measures for 8-node hexahedral isoparametric elements

The inverse relations of the isoparametric mapping for the 8-node hexahedra are derived by using the theory of geodesics in differential geometry. Such inverse relations assume the form of infinite power series in the element geodesic coordinates, which are shown to be the skew Cartesian coordinates determined by the Jacobian of the mapping evaluated at the origin. By expressing the geodesic coordinates in turn in terms of the isoparametric coordinates, the coefficients in the resulted polynomials are suggested to be the distortion parameters of the element. These distortion parameters can be used to sompletely describe the inverse relations and the determinant of the Jacobian of the mapping. The meanings of them can also be explained geometrically and mathematically. These methods of defining the distortion measures and deriving the inverse relations of the mapping are completely general, and can be applied to any other two-or three-dimensional isoparametric elements.     

A resultant 8-node solid-shell element for geometrically nonlinear analysis

A new resultant force formulation of 8-node solid element is presented for the linear and nonlinear analysis of thin-walled structures. The global, local and natural coordinate systems were used to accurately model the shell geometry. The assumed natural strain methods with plane stress concept were implemented to remove the various locking problems appearing in thin plates and shells. The correct warping behavior in the very thin twisted beam test was obtained by using an improved Jacobian transformation matrix. The 2 × 2 Gauss integration scheme was used for the calculation of the element stiffness matrix. From the computational viewpoint, the present solid element is very efficient for a large scale of nonlinear modeling. A lot of numerical tests were carried out for the validation of the present 8-node solid-shell element and the results are in good agreement with references.An erratum to this article can be found at     

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.     

Axisymmetric quadrilateral elements for large deformation hyperelastic analysis

In this paper, axisymmetric 8-node and 9-node quadrilateral elements for large deformation hyperelastic analysis are devised. To alleviate the volumetric locking which may be encountered in nearly incompressible materials, a volumetric enhanced assumed strain (EAS) mode is incorporated in the eight-node and nine-node uniformly reduced-integrated (URI) elements. To control the compatible spurious zero energy mode in the 9-node element, a stabilization matrix is attained by using a hybrid-strain formulation and, after some simplification, the matrix can be programmed in the element subroutine without resorting to numerical integration. Numerical examples show the relative efficacy of the proposed elements and other popular eight-node elements. In view of the constraint index count, the two elements are analogous to the Q8/3P and Q9/3P elements based on the u–p hybrid/mixed formulation. However, the former elements are more straight forward than the latter elements in both formulation and programming implementation.     

Hybrid stress reduced-Mindlin elements for thin multilayer plates

A hybrid-stress formulation of isoparametric elements for the analysis of thin multilayer laminated composite plates is presented. The element displacement behaviour is characterized by laminate reference surface inplane and transverse displacements and laminate non-normal cross-section rotations; as a result, simple C^o interpolation of displacement and rotation can be used, and the number of degrees-of-freedom is independent of the number of layers. All components of stress are included and are related to a set of laminate stress parameters, the number of which is independent of the number of layers. Attention is restricted here to thin laminates, for which it is shown that the contributions of transverse shear stress and transverse normal stress to the internal complementary energy can be neglected. As a result of this reduction, a modified stiffness-formation algorithm can be used which provides a significant improvement in computational efficiency. The formulation presented is used to develop an 8-node isoparametric thin multilayer plate element. The resulting element is naturally invariant, of correct rank, and non-locking in the thin plate limit.     

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.     

Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assesment of hybrid stress, hybrid strain and enhanced strain elements

In this paper we discuss and compare three types of 4-node and 9-node finite elements for a recently formulated finite deformation shell theory with seven degrees of freedom. The shell theory takes thickness change into account and circumvents the use of a rotation tensor. It allows for the applicability of three-dimensional constitutive laws and equipes the configuration space with the structure of a vector space. The finite elements themselves are based either on a hybrid stress functional, on a hybrid strain functional, or on a nonlinear version of the enhanced strain concept. As independent variables either the normal and shear resultants, the strain tensor related to the deformation of the midsurface, or the incompatible enhanced strain field are taken as independent variables. The fields of equivalence of these different formulations, their limitations as well as possible improvements are discussed using different numerical examples. Received 10 December 1998     

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

 Expressions for critical timesteps are provided for an explicit finite element method for plane elastodynamic problems in isotropic, linear elastic solids. Both 4-node and 8-node quadrilateral elements are considered. The method involves solving for the eigenvalues directly from the eigenvalue problem at the element level. The characteristic polynomial is of order 8 for 4-node elements and 16 for 8-node elements. Due to the complexity of these equations, direct solution of these polynomials had not been attempted previously. The commonly used critical time-step estimates in the literature were obtained by reducing the characteristic equation for 4-node elements to a second-order equation involving only the normal strain modes of deformation. Furthermore, the results appear to be valid only for lumped-mass 4-node elements. In this paper, the characteristic equations are solved directly for the eigenvalues using <ty>Mathematica<ty> and critical time-step estimates are provided for both lumped and consistent mass matrix formulations. For lumped-mass method, both full and reduced integration are considered. In each case, the natural modes of deformation are obtained and it is shown that when Poisson's ratio is below a certain transition value, either shear-mode or hourglass mode of deformation dominates depending on the formulation. And when Poisson's ratio is above the transition value, in all the cases, the uniform normal strain mode dominates. Consequently, depending on Poisson's ratio the critical time-step also assumes two different expressions. The approach used in this work also has a definite pedagogical merit as the same approach is used in obtaining time-step estimates for simpler problems such as rod and beam elements. Received: 8 January 2002 / Accepted: 12 July 2002 The support of NSF under grant number DMI-9820880 is gratefully acknowledged.     

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 9-node co-rotational quadrilateral shell element

A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems; (3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.