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On the equivalence of various hybrid finite elements and a new orthogonalization method for explicit element stiffness formulation
Abstract
Hybrid finite element methods are superior to conventional displacement-based finite element formulations in many aspects and to date many types of hybrid elements have been proposed based on different assumed stress fields. However, some of them are essentially equivalent to each other and there is no general approach available yet to determine their corresponding relationships. In addition, even though the hybrid techniques have many advantages compared to the displacement-based elements, there is an obvious disadvantage, the additional computational cost of the matrix inversions required in constructing element stiffness matrix. In this paper, we show that a linearly independent transformation of the assumed stress field for a given element leads to an equivalent hybrid finite element. This is verified through the proof of equivalence between various existing hybrid elements proposed by different researchers. To improve the computational efficiency, a new inner product including the compliant matrix of material is introduced. With this new definition of inner product, a simplified orthogonalization procedure for the assumed stress modes is then proposed. Based on the orthonorm stress modes obtained using the proposed orthogonalization, an explicit formulation of the hybrid element stiffness matrix is presented. The effectiveness of the present method is validated by numerical examples.
... A disadvantage of IFMD and HM is the additional computational cost of the inversion of the flexibility matrix required to construct the stiffness matrix. This disadvantage has been solved by Zhang et al. 22,23 by means of the diagonalization of the flexibility matrix. ...
... This aspect can be improved by the diagonalization of this matrix proposed by Zhang. 22,23 Formulation of a plate element ...
... As in the case of interpolation functions of displacements, natural coordinates are used, being n ¼ x/a and ¼ y/b. According to equations (19) and (22), the matrix of the interpolation functions is Replacing [B p ] from equation (11) and [Y] from equation (23) in equation (21) the equilibrium matrix of the element is ...
This paper shows the use of the dual integrated force method in the finite element method and its application to unidirectional composite laminates. This method was developed by SN Patnaik for isotropic materials, considering not only the equilibrium equations but also the compatibility conditions. The method is applied to obtain the stiffness matrix of a four-node rectangular plate element. Then, a three-point flexure test of a unidirectional off-axis composite has been analyzed. The results obtained with this method have been compared with those obtained from a previous analytic approach and with those obtained by commercial software of finite element method, which is based on the stiffness method.
... The complicated natural deformation modes are replaced by the simple basic deformation modes which were derived directly from the displacement field and the energy product was used to avoid the numerical modal analysis. Zhang [37][38][39][40][41] suppressed the zero-energy modes via assumed stress modes with orthogonal basic deformation modes. To get a desired assumed stress field, methods are also required to compare the performance of different elements with different assumed stress fields [37][38][39]. ...
... In [38,40], the following linearly independent transformation of the assumed stress field is further considered as ...
... Consequently, the constants in Eqs. (39) and (40) can only relate to the terms of first-order in displacement field. In other words, the displacements for both rigid body rotations and constant strains can only be included in the terms of first-order which can be separated by using the parameters in Eqs. ...
... En effet, Pian remarque en 2000 [Pian00] que Fröier et al. [Fröi74] démontrent que sa formulation originale [Pian64] est équivalente à celle de l'élément à déplacement incompatible de Wilson et al. [Wils73], mais également à celle de Turner et al. [Turn56]. Gallagher [Zhan07]. Zhang et al. [Zhan07] proposent donc une procédure qui démontre (entre autres) que l'élément de Huang et al. [Huan91] est équivalent à celui de Pian [Pian64], mais aussi à celui de Rubinstein et al. [Rubi83]. ...
... Gallagher [Zhan07]. Zhang et al. [Zhan07] proposent donc une procédure qui démontre (entre autres) que l'élément de Huang et al. [Huan91] est équivalent à celui de Pian [Pian64], mais aussi à celui de Rubinstein et al. [Rubi83]. Zhang et al. [Zhan07] indiquent par ailleurs qu'il a été prouvé que la formulation mixte « enhanced assumed strain » est équivalente à de nombreux éléments hybrides existants (Figure 1.16). ...
... Zhang et al. [Zhan07] proposent donc une procédure qui démontre (entre autres) que l'élément de Huang et al. [Huan91] est équivalent à celui de Pian [Pian64], mais aussi à celui de Rubinstein et al. [Rubi83]. Zhang et al. [Zhan07] indiquent par ailleurs qu'il a été prouvé que la formulation mixte « enhanced assumed strain » est équivalente à de nombreux éléments hybrides existants (Figure 1.16). rr On doit certainement cette dénomination à Felippa [Feli03,Feli06b]. ...
The prediction of rupture initiation and propagation in full-scale aeronautic structures subjected to impact is still nowadays troublesome. One main problem concerns the difference between the scale of the aeronautic structure and the one of the riveted assembly. Indeed, it seems that the modelling of the riveted assembly is not sufficiently refined to model the failure of a sheet metal along rivet lines, while the maximum number of finite elements that can be used in the full-scale model is reached. It is thus proposed to develop a perforated plate super finite element that can localise mechanical fields in a structure subjected to impact loading. The first chapter is dedicated to the literature survey. It introduces numerical methods enabling one to represent discontinuities, and some knowledge necessary to formulate super-elements. It enables to summarize and to choose Piltner's eight-nodded perforated plate finite element formulation. The second chapter highlights the hypotheses used to formulate its interpolation functions, and their link with solid mechanics equations. Moreover, the numerical results provided by these functions (truncated to 4) are considered sufficiently representative to pursue our goal. Then, chapter III highlights the hypotheses linked to the building of the super-element's variational principle. It is concluded that the element is restricted to elastostatic computations, and that it does not allow one to model a riveted assembly. The numerical results of the eight-nodded super-element, obtained thanks to its plugging into ZeBuLoN, are considered satisfactory enough to pursue the developments. Finally, the hypotheses restricting Piltner's element to elastostatics are raised, and the new formulation is then made suitable for crashworthiness in chapter IV.
... Due to its sound solution accuracy, the hybrid stress finite element method [1] has experienced significant developments2345 and has been successfully applied to many scientific and engineering problems6789 . One major issue for the hybrid element formulation is to correctly construct the assumed stress field. ...
... In addition, one can define the following inner product of stress modes [4,7] : ...
A set of basic deformation modes for hybrid stress finite elements are directly derived from the element displacement field.
Subsequently, by employing the so-called united orthogonal conditions, a new orthogonalization method is proposed. The resulting
orthogonal basic deformation modes exhibit simple and clear physical meanings. In addition, they do not involve any material
parameters, and thus can be efficiently used to examine the element performance and serve as a unified tool to assess different
hybrid elements. Thereafter, a convenient approach for the identification of spurious zero-energy modes is presented using
the positive definiteness property of a flexibility matrix. Moreover, based on the orthogonality relationship between the
given initial stress modes and the orthogonal basic deformation modes, an alternative method of assumed stress modes to formulate
a hybrid element free of spurious modes is discussed. It is found that the orthogonality of the basic deformation modes is
the sufficient and necessary condition for the suppression of spurious zero-energy modes. Numerical examples of 2D 4-node
quadrilateral elements and 3D 8-node hexahedral elements are illustrated in detail to demonstrate the efficiency of the proposed
orthogonal basic deformation mode method.
Key wordshybrid stress element–basic deformation mode–assumed stress mode–mode orthogonality–suppression of zero-energy deformation mode
... A number of two-dimensional [25,26,[42][43][44][45] and three-dimensional [30,46] models of concrete materials or structures have been developed to study their mechanical properties. In addition, numerous efficient numerical simulation methods have been developed, for instance, FEM, meshfree method [47,48], discrete element method [49,50], finite difference method [51], extended FEM [32,37,[52][53][54], rigid body spring method [34][35][36], hybrid stress method [55,56], the Alpha FEM [57], and the unsymmetric method [58]. From the point of view of the energy principle, the FEM method can be divided into two types, i.e., methods based on the potential energy principle and methods based on the complementary energy principle. ...
The reuse of recycled aggregate concrete (RAC) is being researched all over the world and lots of works are focused on the notched specimen to study the crack path of RAC. A mathematical algorithm of RAC meshing was presented to explore the failure pattern in RAC. According to this algorithm, the interfacial transition zone can be defined to be an actual thickness at the micron level. Further, a new finite element method (FEM) on the complementary energy principle was introduced to simulate the mechanical behavior of RAC’s mesostructure. The compliance matrix of the element with any shape can be calculated and expressed to be a uniform and explicit expression. Several numerical models of RAC were established, in which the effecting factors of the prenotch size, thickness of ITZ, and the distance from the prenotch to the aggregate were taken into account. Hereafter, these RAC models were subjected to uniaxial tension. The effect of the aforementioned factors on the crack path was simulated. The simulated data manifest that both the mesh mode of RAC and the FEM on complementary energy principle are effective approaches to explore the failure pattern of RAC. The size of the prenotch, thickness of ITZ, and distance from the prenotch to the recycled aggregate have a powerful influence on the path and distribution of the isolated crack, width and length of the crack path, and the shape and path of continuous cracks, respectively.
... Subsequent proposals to inhibit locking in volume elements were based on mixed variational formulations, penalty methods, B-matrix manipulations, enhanced natural and assumed strain concepts, sub-grid methods, and on reduced or selective numerical integration. Some of those methods have been shown to be equivalent or to lead to identical results under certain conditions [Yeo 1996, Djoko et al. 2004, Zhang et al. 2007]. On the other hand, some more recent elements incorporate several of those methods in order to compensate for disadvantages in special cases of geometrical nonlinearity and material behavior, and to provide a reasonable compromise between the contradicting objectives element performance (distortion insensitivity, coarse mesh accuracy, convergence behavior), computational costs, general applicability, and implementational effort. ...
In recent decades, a multitude of concepts and models were developed to understand, assess and predict muscular mechanics in the context of physiological and pathological events. Most of these models are highly specialized and designed to selectively address fields in, e.g., medicine, sports science, forensics, product design or CGI; their data are often not transferable to other ranges of application. A single universal model, which covers the details of biochemical and neural processes, as well as the development of internal and external force and motion patterns and appearance could not be practical with regard to the diversity of the questions to be investigated and the task to find answers efficiently. With reasonable limitations though, a generalized approach is feasible. The objective of the work at hand was to develop a model for muscle simulation which covers the phenomenological aspects, and thus is universally applicable in domains where up until now specialized models were utilized. This includes investigations on active and passive motion, structural interaction of muscles within the body and with external elements, for example in crash scenarios, but also research topics like the verification of in vivo experiments and parameter identification. For this purpose, elements for the simulation of incompressible deformations were studied, adapted and implemented into the finite element code SLang. Various anisotropic, visco-elastic muscle models were developed or enhanced. The applicability was demonstrated on the base of several examples, and a general base for the implementation of further material models was developed and elaborated.
... Many attempts have been made to overcome these shortcomings, and numerous methods have been proposed for solving engineering problems, such as the hybrid stress method, 22,23 the integrated force method, 24,25 the assumed strain method, 26,27 the quasiconforming element method, 28 the Alpha FEM, 29 the unsymmetric method, 30,31 the generalized conforming method, 32 the new natural coordinate methods, 33 the boundary element method, 34 the meshless method, 35,36 and the scaled boundary FEM. 16 However, although many methods have been established in recent decades, little attention has been given to deriving an explicit uniform expression for various elements and without the use of Gaussian integration. ...
The finite element method (FEM) is an effective approach for exploring the failure mechanism of heterogeneous materials. The use of FEM might suffer from several difficulties in terms of keeping the elements and their boundaries balanced, as well as finding interpolation functions. In this study, we introduced an efficient approach to researching the failure mechanism of the material, named Base Force Element Method (BFEM), according to complementary energy. Specifically, the element compliance matrix of an arbitrary shape with four mid‐edge nodes was expressed based on the equilibrium condition. Then, the node displacement was obtained by releasing the governing equation using the Lagrange multiplier method. In addition, both the compliance matrix and the node displacement were represented as explicit expressions without the use of Gaussian integration. A numerical model of the recycled aggregate concrete (RAC) was established according to the Monte Carlo method. A comparative sample of the digital image model was also established using digital image technology. The influences of substituting recycled aggregate and the relative mechanical properties of adhered mortar to those of new mortar on the failure mechanism of RAC were studied. The simulation results indicated that the BFEM is an effective approach to researching the damage mechanism of heterogeneous materials.
... Various finite element models have been proposed and they are robust and insensitive to mesh distortion, such as the equilibrium models (Veubeke 1965(Veubeke , 1972, the hybrid stress method (Pian 1964, Pian and Chen 1982, Pian and Sumihara 1984, Zhang et al. 2007, the integrated force method (Patnaik 1973, Patnaik et al. 1991, the mixed approach (Zienkiewicz 1979), the new natural coordinate methods , Long et al. 2009, Long et al. 2010, the smoothed finite element method (Liu et al. 2007), the incompatible displacement modes (Wilson et al. 1973), the assumed strain method (Simo and Hughes 1986), the enhanced strain modes (Piltner and Taylor 1995), the selectively reduced integration scheme (Hughes 1980), the quasi-conforming element method (Tang et al. 1984), the generalized conforming method (Long and Huang 1988), the Alpha finite element method (Liu et al. 2008), the unsymmetric method (Rajendran and Liew 2003), the new spline finite Corresponding author, Professor E-mail: pengyijiang@bjut.edu.cn element method , the extended finite element method (Moes et al. 1999) and so on. ...
The base force element method (BFEM) is a new finite element method. In this paper, a degenerated 4-mid-node plane element from concave polygonal element of BFEM was proposed. The performance of this quadrilateral element with 4 mid-edge nodes in the BFEM on complementary energy principle is studied. Four examples of linear elastic analysis for plane frame structure are presented. The influence of aspect ratio of the element is analyzed. The feasibility of the 4 mid-edge node element model of BFEM on complementary energy principles researched for plane frame problems. The results using the BFEM are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. It is revealed that the BFEM has better performance compared to the displacement model in the case of large aspect ratio.
... However, the conventional finite element method (FEM) based on the displacement model has some shortcomings, such as large deformation, treatment of incompressible materials, bending of thin plates, and moving boundary problems. In the past decades, numerous efforts techniques have been proposed for developing finite element models which are robust and insensitive to mesh distortion, such as the hybrid stress method [1][2][3][4], the equilibrium models [5,6], the mixed approach [7], the integrated force method [8][9][10][11], the incompatible displacement modes [12,13], the assumed strain method [14][15][16][17], the enhanced strain modes [18,19], the selectively reduced integration scheme [20], the quasiconforming element method [21], the generalized conforming method [22], the Alpha finite element method [23], the new spline finite element method [24,25], the unsymmetric method [26][27][28][29], the new natural coordinate methods [30][31][32][33], the smoothed finite element method [34], and the base force element method [35][36][37][38][39][40][41][42][43]. ...
The four-mid-node plane model of base force element method (BFEM) on complementary energy principle is used to analyze the rock mechanics problems. The method to simulate the crack propagation using the BFEM is proposed. And the calculation method of safety factor for rock mass stability was presented for the BFEM on complementary energy principle. The numerical researches show that the results of the BFEM are consistent with the results of conventional quadrilateral isoparametric element and quadrilateral reduced integration element, and the nonlinear BFEM has some advantages in dealing crack propagation and calculating safety factor of stability.
... Over the past 50 years, numerous efforts techniques have been proposed for developing finite element models which are robust and insensitive to mesh distortion, such as the hybrid stress method proposed by Pian (1964), Pian and Chen (1982), Pian and Sumihara (1984), and Zhang et al. (2007), the equilibrium models in the FEM by Fraeijs de Veubeke (1965, 1972, the mixed approach in the FEM by Taylor and Zienkiewicz (1979), the integrated force method by Patnaik (1973Patnaik ( , 1986a, Patnaik et al. (1991), the incompatible displacement modes proposed by Wilson et al. (1973) and Taylor et al. (1976), the enhanced strain method proposed by Simo and Rifai (1990), the stabilization method proposed by Belytschko and Bachrach (1986), the selectively reduced integration scheme proposed by Hughes (1980), the assumed strain formulations proposed by MacNeal (1982), Piltner and Taylor (1997), the quasi-conforming element method proposed by Tang et al. (1984), the generalized conforming method proposed by Long and Huang (1988), the a FEM proposed by Liu et al. (2008), the new spline FEM proposed by Chen et al. (2010a, b), the unsymmetric method proposed by Rajendran and Liew (2003), Rajendran (2010) and Ooi et al. (2004Ooi et al. ( , 2008, the new natural coordinate methods proposed by , 2010, Chen, et al. (2004Chen, et al. ( , 2008 and Li et al. (2008), and so on. ...
Purpose
– The purpose of this paper is to present a two-dimensional (2D) model of the base force element method (BFEM) based on the complementary energy principle. The study proposes a model of the BFEM for arbitrary mesh problems.
Design/methodology/approach
– The BFEM uses the base forces given by Gao (2003) as fundamental variables to describe the stress state of an elastic system. An explicit expression of element compliance matrix is derived using the concept of base forces. The detailed formulations of governing equations for the BFEM are given using the Lagrange multiplier method. The explicit displacement expression of nodes is given. To verify the 2D model, a program on the BFEM using MATLAB language is made and a number of examples on arbitrary polygonal meshes and aberrant meshes are provided to illustrate the BFEM.
Findings
– A good agreement is obtained between the numerical and theoretical results. Based on the studies, it is found that the 2D formulation of BFEM with complementary energy principle provides reliable predictions for arbitrary mesh problems.
Research limitations/implications
– Due to the use of Lagrange multiplier method, there are more basic unknowns in the control equation. The proposed method will be improved in the future.
Practical implications
– This paper presents a new idea and a new numerical method, and to explore new ways to solve the problem of arbitrary meshes.
Originality/value
– The paper presents a 2D model of the BFEM using the complementary energy principle for arbitrary mesh problems.
... Based on this constitutive relationship various techniques have been developed through the years to predict the stress field. Among them the finite element methods usually are the most popular choice and along this line numerous attempts have also been made to analyze the laminated composites using different hybrid elements [2][3]. In this study, a nonlinear hybrid stress element is presented to investigate the interlaminar free-edge response of the laminated composites with consideration of the material nonlinearity by Hahn-Tsai model. ...
Significant magnitudes of interlaminar normal and shear stresses have been observed near the free edges of composite plates and this free-edge phenomenon has become a benchmark index commonly used to study the composite delamination [1]. It is also noticed that composite materials often exhibit nonlinear behavior to an extent and thus satisfying results related to the free-edge effects can not always be obtained with a linear model. Consequently it is often necessary to take the nonlinearity into consideration in the analysis of laminated composites. A suitable candidate to describe this nonlinear behavior is the Hahn-Tsai model in which the shear stress-strain response is nonlinear but the longitudinal and transverse stress-strain responses are linear. Based on this constitutive relationship various techniques have been developed through the years to predict the stress field. Among them the finite element methods usually are the most popular choice and along this line numerous attempts have also been made to analyze the laminated composites using different hybrid elements [2–3].
In this study, a nonlinear hybrid stress element is presented to investigate the interlaminar free-edge response of the laminated composites with consideration of the material nonlinearity by Hahn-Tsai model. Two different cross-ply laminates, [04/904]s and [904/04]s, and an angle-ply laminate [+454/-454]s are studied under the tension and thermal loading conditions. In order to investigate the effects induced by the material nonlinearity, all six components of the stress tensor near the free-edge and along the interface between two layers with different fiber orientation are examined. It is shown that the material nonlinearity for shear stress-strain response leads to significant reduaction of relevant stresses. On the other hand, the behavior of longitudinal stress is not severely affected with the increase of the interlaminar normal stress and the transverse stress.
... Over the past 50 years, numerous efforts techniques have been proposed for developing finite element models which are robust and insensitive to mesh distortion, such as the hybrid stress method proposed by Pian [4], Pian and Chen [5], Pian and Sumihara [6], and Zhang et al. [7], the incompatible displacement modes proposed by Wilson et al. [8] and Taylor et al. [9], the enhanced strain method proposed by Simo and Rifai [10], the stabilization method proposed by Belytschko and Bachrach [11], the selectively reduced integration scheme proposed by Hughes [12], the assumed strain formulations proposed by MacNeal [13] and Piltner and Taylor [14], the quasi-conforming element method proposed by Tang et al. [15], the generalized conforming method proposed by Long and Huang [16], the Alpha finite element method ( FEM) proposed by Liu et al. [17], the new spline finite element method [18,19] proposed by Chen et al., the unsymmetric method proposed by Rajendran and Liew [20] and Rajendran [21] and Ooi et al. [22], the new natural coordinate methods proposed by Long and Cen et al. [24][25][26][27][28][29][30], the analytical trial function method proposed by Fu et al. [39], and so on. ...
By simply improving the first version of hybrid stress element method proposed by Pian, several 8- and 12-node plane quadrilateral elements, which are immune to severely distorted mesh containing elements with concave shapes, are successfully developed. Firstly, instead of the stresses, the stress function ϕ is regarded as the functional variable and introduced into the complementary energy functional. Then, the fundamental analytical solutions (in global Cartesian coordinates) of ϕ are taken as the trial functions for 2D finite element models, and meanwhile, the corresponding unknown stress-function constants are introduced. Thus, the resulting stress fields must be more reasonable because both the equilibrium and the compatibility relations can be satisfied. Thirdly, by using the principle of minimum complementary energy, these unknown stress-function constants can be expressed in terms of the displacements along element boundaries, which can be interpolated directly by the element nodal displacements. Finally, the complementary energy functional can be rewritten in terms of element nodal displacement vector, and thus, the element stiffness matrix of such hybrid stress-function (HS-F) element is obtained. This technique establishes a universal frame for developing reasonable hybrid stress elements based on the principle of minimum complementary energy. And the first hybrid stress element proposed by Pian is just a special case within this frame. Following above procedure, two 8-node and two 12-node quadrilateral plane elements are constructed by employing different fundamental analytical solutions of Airy stress function. Numerical results show that, the 8-node and 12-node models can produce the exact solutions for pure bending and linear bending problems, respectively, even the element shape degenerates into triangle and concave quadrangle. Furthermore, these elements do not possess any spurious zero energy mode and rotational frame dependence.
This study deals with the comparison of stresses and displacements of a circular ring obtained by two numerical formulations, namely the stiffness method and the hybrid method, applied to isoparametric quadrilateral elements. After explaining the formulation of the hybrid method in finite elements, an orthogonalization method proposed previously for hybrid elements is applied. Then, the computational cost per element of the stiffness method and of the hybrid method with and without orthogonalization has been evaluated for the first time. A circular ring loaded by two opposing forces is analyzed in order to compare the solution obtained by the hybrid method and the stiffness method with experimental and analytical results. The agreement between analytical and experimental results with those numerical is better in the case of the hybrid method than in the case of the stiffness method. It is observed that the element ratio needed to obtain a given relative error is one magnitude order greater in the stiffness method than in the case of the hybrid method.
A new orthogonalization method is presented for the basic deformation modes. The inner products between basic deformation modes both including and excluding the element stiffness matrix are employed and the orthogonality within the orthogonal basic deformation modes is fully independent of the material properties so that they can serve as a uniformed tool to assess a given element. In the proposed approach, the arbitrary displacement field of deformation for different hybrid element can be easily decoupled into a linear combination of a series of basic modes. Thereafter their relating deformation energy, namely the element performance, can be directly studied. In the numerical examples the performances of several hybrid elements that are constructed by different assumed stress fields are provided using the proposed method. The results show that the method is very effective.
The basic deformation modes of a hybrid stress element are obtained by separating the stress modes corresponding to the different deformation modes of a hybrid stress element. Consequently a basic deformation-based approach is presented to investigate the performance for a given hybrid stress element. It is shown that the element deformation modes generally depend upon the assumed stress fields and thus are complex and coupled. In the proposed approach, the coupled stress fields are expressed with the stress parameters such that they can be easily decoupled into the linear combination of a series of basic stress fields. The relevant basic deformation modes are derived and the rigidity of hybrid element for arbitrary deformation modes, namely the element performance can be directly studied. The difficulty associated with the decoupling of various deformation modes is completely avoided. Investigations show that some of these assumed stress fields can not be used to the general hybrid elements.
A set of simple deformation modes that can be used to describe the arbitrary deformation for hybrid elements are directly derived from the element displacement field. It is shown that the nonzero deformation mode appears to be kinematic when it is orthogonal to all the assumed stress modes. Thus the spurious kinematic mode of a given element can be easily determined and suppressed by using the proposed simple deformation modes. Based upon the orthogonality relationship between the initial stress modes and the simple deformation modes, an assumed stress method is presented and the related systematic procedure is illustrated in details. Since the stress modes given by the iso-function method are adopted here to derive the initial stress modes, one can find the necessary stress modes to represent the special stress distribution for different problems and the corresponding hybrid element is guaranteed from hour-glass modes. In the numerical examples several assumed stress fields for 2D-4 node and 3D-8 node elements are constructed by the proposed method and the results show that the method is very effective.
A set of orthogonally assumed stress fields are presented to efficiently formulate the hybrid elements of isotropic and orthotropic materials. In the proposed approach, the inverse of flexibility matrix is avoided, so the computational efficiency is significantly improved. The orthogonal stress field for isotropic material is obtained by using the transformation matrix formed by the eigenvectors of element flexibility matrix. It is noted that there is no numerical computation involved here and the orthogonality within the stress field is fully independent of the material properties. For orthotropic materials, a method of material matrix decomposition is proposed to derive the orthogonal stress field. This orthogonal relationship depends on the ratio between the elastic modulus in the two principal directions but is totally free of the Poison response due to transverse deformation. Several 2-D and 3-D examples show that the proposed method is very straightforward and effective.
This paper shows the use of the dual integrated force method (IFMD) in finite elements method and its application to composite materials. This method was developed by S.N. Patnaik in isotropic materials, considering not only the equilibrium equations but also the compatibility conditions. In the IFMD, the principal unknowns are the displacements, and the structure of governing equation is similar to the stiffness method. It is shown that the governing equation of the IFMD is the same than in the case of the hybrid method of Pian. The method is applied to two examples, a cantilever beam of orthotropic material loaded at the end and one off-axis tensile test in a unidirectional composite specimen. The results of this method have been compared with the ones obtained from the application of the stiffness method and with analytical results. Copyright © 2014 John Wiley & Sons, Ltd.
We introduce a new framework for the development of hybrid stress finite elements for two-dimensional linear elasticity. A family of arbitrarily shaped elements is derived which takes advantage of the special structure of this framework. The key feature is to explicitly approximate, in the parent domain, either the second Piola–Kirchhoff, the first Piola–Kirchhoff, or the Cauchy stresses, and to enforce the divergence-free condition in the physical domain using their corresponding first Piola–Kirchhoff projections. The introduced finite elements may have arbitrary curved edges, and internally satisfy in strong form the equilibrium differential equations. Furthermore, under certain conditions, the new elements may lead to statically admissible stress distributions, by also verifying equilibrium of tractions on the element boundaries. Feasibility and effectiveness of the proposed elements are numerically verified through several benchmark tests.
A novel method is presented to orthogonalize the basic deformation modes for hybord stress elements. It is shown that the resulting orthogonal basic deformation modes are independent of elastic constants and they can be used to investigate the hybrid elements with different assumed stress fields. It is noted that a deformation mode appears to be a zero-energy mode when its strain-driven stress mode is orthogonal to all the assumed stress modes. This provides an easy way to identify the spurious modes of a given element. In addition, based upon the orthogonality relationship between the given initial stress modes and the orthogonal basic deformation modes, the necessary stress modes to formulate a hybrid element free of spurious modes can be straightforwardly selected from the stress matrix including the given initial stress modes. Furthermore, it is found that the orthogonal condition of the basic deformation modes is not only necessary but also sufficient to derive the assumed stress field that can produce a hybrid element free of zero-energy modes. Several numerical examples are illustrated in details to demonstrate the efficacy of the proposed basic deformation mode method.
The following is proved: 1) The linear independence of assumed stress modes is the necessary and sufficient condition for
the nonsingular flexibility matrix; 2) The equivalent assumed stress modes lead to the identical hybrid element. The Hilbert
stress subspace of the assumed stress modes is established. So, it is easy to derive the equivalent orthogonal normal stress
modey by Schmidt's method. Because of the resulting diagonal flexibility matrix, the identical hybrid element is free from
the complex matrix inversion so that the hybrid efficiency is improved greatly. The numerical examples show that the method
is effective.
The hybrid stress method has demonstrated many improvements over conventional displacement-based formulations. A main detraction from the method, however, has been the higher computatational cost in forming element stiffness coefficients due to matrix inversions and manipulations as required by the technique. By utilizing permissible field transformations of initially assumed stresses, a spanning set of orthonormalized stress modes can be generated which simplify the matrix equations and allow explicit expressions for element stiffness coefficients to be derived. The developed methodology is demonstrated using several selected 2-D quadrilateral and 3-D hexahedral elements.
This article compares derivation methods for constructing optimal membrane triangles with corner drilling freedoms. The term “optimal” is used in the sense of exact inplane pure-bending response of rectangular mesh units of arbitrary aspect ratio. Following a comparative summary of element formulation approaches, the construction of an optimal three-node triangle using the ANDES formulation is presented. The construction is based upon techniques developed by 1991 in student term projects, but taking advantage of the more general framework of templates developed since. The optimal element that fits the ANDES template is shown to be unique if energy orthogonality constraints are enforced. Two other formulations are examined and compared with the optimal model. Retrofitting the conventional linear strain triangle element by midpoint-migrating and congruential transformations is shown unable to produce an optimal element, while rank deficiency is inevitable. Use of the quadratic strain field of the 1988 Allman triangle, or linear filtered versions thereof, is also unable to reproduce the optimal element. Moreover these elements exhibit serious aspect ratio lock. These predictions are verified on benchmark examples.
Local infection with necrotizing pathogens induces whole plant immunity to secondary challenge. Pathogenesis-related genes are induced in parallel with this systemic acquired resistance response and thought to be co-regulated. The hypothesis of co-regulation has been challenged by induction of Arabidopsis PR-1 but not systemic acquired resistance in npr1 mutant plants responding to Pseudomonas syringae carrying the avirulence gene avrRpt2. However, experiments with ndr1 mutant plants have revealed major differences between avirulence genes. The ndr1-1 mutation prevents hypersensitive cell death, systemic acquired resistance and PR-1 induction elicited by bacteria carrying avrRpt2. This mutation does not prevent these responses to bacteria carrying avrB.
Systemic acquired resistance, PR-1 induction and PR-5 induction were assessed in comparisons of npr1-2 and ndr1-1 mutant plants, double mutant plants, and wild-type plants. Systemic acquired resistance was displayed by all four plant lines in response to Pseudomonas syringae bacteria carrying avrB. PR-1 induction was partially impaired by either single mutation in response to either bacterial strain, but only fully impaired in the double mutant in response to avrRpt2. PR-5 induction was not fully impaired in any of the mutants in response to either avirulence gene.
Two pathways act additively, rather than in an obligatorily synergistic fashion, to induce systemic acquired resistance, PR-1 and PR-5. One of these pathways is NPR1-independent and depends on signals associated with hypersensitive cell death. The other pathway is dependent on salicylic acid accumulation and acts through NPR1. At least two other pathways also contribute additively to PR-5 induction.
The following two theorems are proved in this paper. (1) The linear independence of assumed stress modes is the necessary and sufficient condition for nonsingular flexibility matrix H. (2) The equivalent assumed stress modes construct the same hybrid element stiffness matrix. Based on the theorems, the Hilbert subspace of the assumed stress modes is established. Therefore, by the Schmidt's method, the equivalent orthogonalized stress modes can be easy to obtain. Because the flexibility matrix is diagonal with the orthogonal stress modes, the complex matrix inverse can be avoided and the hybrid efficiency is improved greatly. More advantages can be found particularly in the analysis for nonlinear material behavior, where the inverse for flexibility matrix cannot be obtained in the explicit form.
This chapter introduces incompatible displacement modes at the element level in order to improve element accuracy. One of the main causes of inaccuracies in lower-order finite elements is their inability to represent certain simple stress gradients. The same basic method of introducing incompatible displacement modes in order to improve the bending properties can be used in three dimensions. The first eight are the standard compatible interpolation functions. The last three are incompatible and are associated with linear shear and normal strains. The nine incompatible modes are eliminated at the element stiffness level by static condensation. As the three-dimensional element degenerates to the same approximation as in the two-dimensional element, the same improvement in accuracy is obtained. This element has been found to be extremely effective in the analysis of massive three-dimensional structures subjected to bending. One element in the thickness direction of arch dams or thick pipe joints has been found to be adequate.
Preface. 1. Introduction. 2. The Hybrid Finite Element Method. 3. Development of Hybrid Element Technique for Analysis of Composites. 4. Partial Hybrid Elements for Analysis of Composites. 5. Numerical Examples of Finite Element Analysis and Global/Local Approach. Index.
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The Dirichlet problem for second order differential equations is chosen as a model problem to show how the finite element method may be implemented to avoid difficulty in fulfilling essential (stable) boundary conditions. The implementation is based on the application of Lagrangian multiplier. The rate of convergence is proved.
A hybrid model, piezoelectric P-S element, is developed for electroelastic crack analysis. The model is used to investigate
coupling crack-tip fields. The obtained numerical solutions recover the 1/√r singularity, and it is in good consistence with the real ones. The results offer a valuable basis for studying the mechanism
of coupling fracture.
A general methodology for deriving explicit element stiffness matrices in hybrid stress formulations is extended to incorporate nonconstant material properties over the element domain for nonlinear elastic analysis. The technique utilizes special stress field transformations to simplify the stiffness definition together with an assumed variation of material properties using isoparametric interpolation functions. The developed technique eliminates the need for numerical integration and matrix inversions, resulting in a substantial increase in computation efficiency. The methodology is demonstrated with the Pian-Sumihara plane quadrilateral element.
A mixed variational principle is suggested for electromechanical analysis, in which the displacements, the transverse stresses and the electric potential are taken as the primary variables, and a three-dimensional hybrid element approach is built on the principle. Two kinds of specific element, namely the bimaterial interface element and the surface element, are developed so that the interlayer traction-continuity and surface traction-free conditions of transverse stresses can be imposed exactly. Two numerical examples of intelligent laminates with piezoelectric layers bonded to the upper and lower surfaces are presented to demonstrate the efficiency of the present approach.
The conventional approaches for stress analysis of composite laminates have encountered discontinuity problems. In this paper, the C 1 continuity of displacements in the in-plane directions and C 0 continuity of displacement along the thickness direction are discussed. Also, the global continuity of transverse stresses and local continuity of in-plane stresses are examined. Then, the formulations of stress analysis in both differential equation form and variational functional form are presented.
A hybrid finite element approach is proposed for the mechanical response of two-dimensional heterogeneous materials with linearly elastic matrix and randomly dispersed rigid circular inclusions of arbitrary sizes. In conventional finite element methods, many elements must be used to represent one inclusion. In this work, each inclusion is embedded inside a polygonal element and only one element is required to represent one inclusion.
In numerically approximating stress and displacement distributions around the inclusion, classical elasticity solutions for a multiply-connected region are employed. A modified hybrid functional is used as the basis of the element formulation where the displacement boundary conditions of the element are automatically considered in a variational sense.
The accuracy and efficiency of the proposed method are demonstrated by two boundary value problems. In one example, the results based on the proposed method with only 64 hybrid elements (450 degrees of freedom) are shown to be almost identical to those based on the traditional method with 2928 conventional elements (5526 degrees of freedom).
One of the first finite elements presented was that by Turner et al.1 Pian2 presented a hybrid plane stress element and Wilson et al.3 presented a non-conforming displacement element. It is shown that the three resulting stiffness matrices are identical.
A partially hybrid stress element for modelling composite laminated plates is developed based on the state space in which only the displacement components and the transverse stress components are assumed to be independent. This formulation satisfies exactly the interlaminar continuity requirements and the surface traction free conditions. It also combines the advantages of both the conventional displacement elements and the fully hybrid stress elements. Numerical examples are illustrated to investigate the accuracy, convergence and shear locking sensitivity of the present method. The nonlinear distributions of the normal and transverse stresses along the thickness direction are also especially studied.
The author gives necessary and sufficient conditions for existence and uniqueness of a class of problems of ″saddle point″ type which are often encountered in applying the method of Lagrangian multipliers. A study of the approximation of such problems by means of ″discrete problems″ (with or without numerical integration) is also given, and sufficient conditions for the convergence and error bounds are obtained.
Three types of partial hybrid finite elements are presented in order to set up a global/local finite element model for analysis of composite laminates. In the global/local model, a composite laminate is divided into three different regions: global, local, and transition regions. These are modeled using three different elements. In the global region, a 4-node degenerated plate/shell element is used to model the overall response of the composite laminate. In the local region, a multilayer element is used to predict detailed stress distribution. In the transition region, a multilayer transition element is used to smoothly connect the two previous elements. The global/local finite element model satisfies the compatibility of displacement at the boundary between the global region and the local region. It also satisfies the continuity of transverse stresses at interlaminar surfaces and traction conditions on the top and bottom surfaces of composite laminates. The global/local finite element model has high accuracy and efficiency for stress analysis of composite laminates. A numerical example of analysis of a laminated strip with free edge is presented to illustrate the accuracy and efficiency of the model.
A classification method is presented to classify stress modes in assumed stress fields of hybrid finite element based on the eigenvalue examination and the concept of natural deformation modes. It is assumed that there only exist m (=n−r) natural deformation modes in a hybrid finite element which has n degrees of freedom and r rigid-body modes. For a hybrid element, stress modes in various assumed stress fields proposed by different researchers can be classified into m stress mode groups corresponding to m natural deformation modes and a zero-energy stress mode group corresponding to rigid-body modes by the m natural deformation modes. It is proved that if the flexibility matrix [H] is a diagonal matrix, the classification of stress modes is unique. Each stress mode group, except the zero-energy stress mode group, contains many stress modes that are interchangeable in an assumed stress field and do not cause any kinematic deformation modes in the element. A necessary and sufficient condition for avoiding kinematic deformation modes in a hybrid element is also presented. By means of the m classified stress mode groups and the necessary and sufficient condition, assumed stress fields with the minimum number of stress modes can be constructed and the resulting elements are free from kinematic deformation modes. Moreover, an assumed stress field can be constructed according to the problem to be solved. As examples, 2-D, 4-node plane element and 3-D, 8-node solid element are discussed. © 1997 John Wiley & Sons, Ltd.
This article discusses two alternative methods of formulating the recently proposed quadrilateral plane element Qcs6, which is based on a generalized functional with multiple stress fields. The first method is derived by an extended Hu-Washizu Principle with the presence of incompatible modes and enforcement of energy orthogonality, while the second one is established on a non-conventional discrete local stress smoothing. Both methods can be applied to other structural element formulations with enhanced efficiency.
The general approach for constituting non-conforming displacement function has been developed for axisymmetric finite element analysis and a pure non-conforming quadrilateral axisymmetric element, from a non-conforming displacement function and without any reduced integration technique, is given.Based on a proposed functional for formulating axisymmetric element and the orthogonal approach, a quadrilateral axisymmetric refined hybrid element has been presented which can be used to achieve superior performances such as higher accuracy and free from locking.
A new approach for choosing the stress terms for a hybrid stress element is based on the condition of vanishing of the virtual work along the element boundary due to the stress terms higher than constant and the additional incompatible displacement. Examples using 4-node plane stress elements have shown that when the incompatible displacements also satisfy the constant strain patch test the resulting elements will provide the most accurate solutions. Advantages of this approach for the formulation of an axisymmetric solid are also indicated.
Formulation and applications of the hybrid-stress finite element model to plane elasticity problems are examined. Conditions for invariance of the element stiffness are established for two-dimensional problems, the results of which are easily extended to three-dimensional cases. Next, the hybrid-stress functional for a 3-D continuum is manipulated into a more convenient form in which the location of optimal stress/strain sampling points can be identified. To illustrate these concepts, 4- and 8-node plane isoparametric hybrid-stress elements which are invariant and of correct rank are developed and compared with existing hybrid-stress elements. For a 4-node element, lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation. Alternative 8-node elements are considered, and the best invariant element is shown to be one in which the stress compatibility equations are invoked. Results are also presented which demonstrate the validity of the optimal sampling points, the effects of reduced orders of numerical integration, and the behaviour of the elements for nearly incompressible materials.
A three-field mixed formulation in terms of displacements, stresses and an enhanced strain field is presented which encompasses, as a particular case, the classical method of incompatible modes. Within this frame-work, incompatible elements arise as particular ‘compatible’ mixed approximations of the enhanced strain field. The conditions that the stress interpolation contain piece-wise constant functions and be L2-ortho-gonal to the enhanced strain interpolation, ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation. The preceding conditions are formulated in a form particularly convenient for element design. As an illustration of the methodology three new elements are developed and shown to exhibit good performance: a plane 3D elastic/plastic QUAD, an axisymmetric element and a thick plate bending QUAD. The formulation described herein is suitable for non-linear analysis.
This convergence criterion is proved for any ‘legalizable’ finite element formulation, i.e. any model for which a conforming displacement version is possible, using the same nodal variables. Engineers may therefore take control of (and responsibility for) convergence, for most of the elements in service today. Stummel's counter-example is explained as a technical misunderstanding, due to inadequate documentation of the patch test. It is suggested, without proof, that the test (properly applied) is universal, provided that it is always combined with an adequate test of stability.
A three-dimensional multilayer composite finite element method has been developed based on a composite variational functional which takes three in-plane strains εx, εx, εxy and three transverse stresses σz, σyz, σxz as the basic variables. The continuity of the transverse stresses σz, σyz, σxz across the laminate thickness is assured a priori by introducing a partial stress field parameter which is associated with the lower and upper surfaces of a lamina in a laminate. A method has been developed to form the partial stress field based on the assumed displacement field. With this method, a three dimensional (3-D) multilayer composite finite element is formulated for stress analysis of composite laminates. A numerical example is given, which shows some advantages of this composite element.
An examination of the variational formulations confirms the similarity between the incompatible displacement model and the assumed stress hybrid model that was pointed out by Irons in 1972. But the basic differences between the two are also identified. For 8-node solid elements the assumed stress terms obtained through a rational procedure also agree with those deduced by Irons purely from his physical insight.
For hybrid finite elements, if an assumed stress field does not contain enough appropriate stress modes, the resulting element will include zero-energy deformation modes and thus can not be used for pratical applications [1–3]. On the other hand, adding extra stress modes will require more computational effort. For a finite element including n degrees of freedom and r rigid body modes, generally m=n−r stress modes are considered to be the optimal choice from the computational point of view. However, so far there still does not exist a universal and rational way to derive the m optimal assumed stress modes that can be used to generate a hybrid element free of zero-energy or kinematic deformation modes.
This paper present a methodology to suppress the zero-energy mode in hybrid element using assumed stress fields. The m basic deformation modes of hybrid element are derived straightforward from the displacement field, which are linearly independent to each other so that they can represent any deformation modes of given element. Their corresponding stress modes are employed to determine the zero-energy modes in the hybrid element. It is shown that a basic deformation mode is a spurious kinematic mode if its corresponding stress mode is orthogonal to all modes in the assumed stress field, and a assumed stress mode is zero-energy mode when it is orthogonal to all basic deformation modes. Based upon the orthogonality relationship between the initial stress modes and the basic stress modes that are corresponding to the basic deformation modes, one can find the necessary m stress modes to formulate a hybrid element free of hour-glass modes. The iso-function method is adopted here to derive the initial stress modes and a related systematic procedure is discussed. Furthermore, it is found that the zero-energy stress modes cannot suppress the zero-energy deformation modes, instead they increase the stiffness associated with the nonzero-energy deformation modes of the element. Thus it is not appropriate to include the zero-energy stress modes into the assumed stress field. The examples of 2-D, 4-node plane element and 3-D, 8-node solid element are provided to illustrate the proposed method.
In this paper a study of the existence of spurious kinematic modes in hybrid-stress finite elements, based on assumed equilibrated stresses and compatible boundary displacements, and the resulting rank-deficiency of the element stiffness matrix, is presented. A method of selection of least-order, stable, invariant, stress fields is developed so as to ensure the prevention of kinematic modes. A 20-node cubic element, a 8-node cubic element and a 4-node square, based on assumed equilibrated stresses within the element and compatible displacements at the boundary of the element, are discussed for purposes of illustration. Comments are made on the generality of the present method, which is based on group theoretical arguments.
In this paper a refined hybrid method based on a new variational formulation and orthogonal approach is developed for plane isoparametric elements. According to this formulation plane refined hybrid elements RGH4 (four-noded model) and RGH8 (eight-noded model), which are typical low- and high-order models, respectively, have been proposed. The element stiffness matrix of the present element can be decomposed into a series of matrices with respect to assumed strain modes and can be derived explicitly or by using lower-order numerical integration. It is shown that the present method is more rational than generalized hybrid methods and hybrid stress method for improving the computational efficiency. A number of numerical examples are used to demonstrate the effectiveness of the proposed elements in terms of convergence, accuracy, sensitivity of distorted mesh and free of locking.
In this paper, a mixed quadrilateral plane element with drilling degrees of freedom using Allman's interpolation scheme is developed. The assumed stress space includes three constant stress modes and four quasi-linear stress modes which are equilibrating for regular element geometry. Owing to the intrinsic orthogonality of the constant and higher order stress modes, the element is particularly efficient. Only a 4 × 4 symmetry matrix is required to be inverted while no incompatible displacement modes are involved. The element has two spurious kinematic modes which, however, can effectively be suppressed by using two very simple stabilization matrices. A number of popular benchmark problems are examined and the accuracy achieved is very satisfactory.
Improved three-dimensional tri-linear elements for finite deformation problems are developed based on an assumed enhanced strain methodology which, in the linear regime, incorporates the classical method of incompatible modes as a particular case. Three crucial modifications of a recently proposed element, which reduces to Wilson's brick in the linear regime, are introduced to prevent locking response in distorted configurations and to maintain proper rank, while preserving excellent performance in bending dominated and localization problems: (i) a modified quadrature rule; (ii) an additional enhancement of the divergence term; and (iii) a modification of the isoparametric shape function derivatives for the three-dimensional problem. Moreover, these modified shape function derivatives are shown to improve the performance of the standard tri-linear brick in distorted configurations. In addition, a strategy is described to circumvent the memory storage requirements in the static condensation procedure of the enhanced strain parameters. The excellent performance of the improved methodology is illustrated in representative numerical simulations.
The extensions of Reissner's two-field (stress and displacement) principle to the cases wherein the displacement field is discontinuous and/or the stress field results in unreciprocated tractions, at a finite number of surfaces (“interelement boundaries”) in a domain (as, for instance, when the domain is discretized into finite elements), is considered. The conditions for the existence, uniqueness, and stability of mixed-hybrid finite element solutions based on such discontinuous fields, are summarized. The reduction of these global conditions to local (“element”) level, and the attendant conditions on the ranks of element matrices, are discussed. Two examples of stable, invariant, least-order elements—a.four-node square planar element and an eight-node cubic element—are discussed in detail.
A new element—the partial hybrid stress element—is extended for vibration analysis of thick laminated composite plates. The variational principle of this element can be derived from the Hellinger-Reissner principle to include inertial effects. Equations of motion can then be formulated through variational manipulation. The element stiffness matrix is established by assuming a stress field only for transverse shear, while other stresses are obtained from an assumed displacement field. Traction-free boundary conditions and interface continuity for transverse shear are satisfied exactly. Both consistent and lumped mass matrices are used to investigate the availability of this new element. Examples are illustrated and compared with other published data to demonstrate the accuracy and efficiency of this proposed partial hybrid stress element for vibration analysis.
A universal template for a 4-node quadrilateral in plane stress is constructed using a combination of old and new techniques. The use of natural quantities, strongly advocated by J.H. Argyris since his Continua and Discontinua 1965 exposition, is further expanded. The qualifier ‘supernatural’ means that all governing equations: kinematic, constitutive and equilibrium, are expressed in both Cartesian and natural forms. These two sets are used to build different components of the template. With timely help from a computer algebra system, a template that include all possible quadrilateral elements that pass the Individual Element Test of Bergan and Hanssen emerges. It yields an infinite number of hitherto undiscovered instances that may be customized to fit special needs. A striking example is the construction of a two-trapezoid macroelement that is bending exact about one direction, for any amount of distortion. This concludes a five decade search that begins with the formulation of the wing-cover rectangular panel in Argyris’ 1954 seminal serial on Energy Methods and Structural Analysis.
In this paper, hybrid variational principles are employed for piezoelectric finite element formulation. Starting from eight-node hexahedral elements with displacement and electric potential as the nodal d.o.f.s, hybrid models with assumed stress and electric displacement are devised. The assumed stress and electric displacement are chosen to be contravariant with the minimal 18 and seven modes respectively. The pertinent coefficients can be condensed at the element level and do not enter the system equation. A number of benchmark tests are exercised. The predicted results indicate that the assumed stress and electric displacements are effective in improving the element accuracy.
Admissible matrix formulation is a patch test approach for efficient construction of multi-field finite element models. In hybrid stress and strain elements, the formulation employs the patch test to identify the constraints on, respectively, the flexibility and stiffness matrices which are most detrimental to the element efficiency. Admissible changes are introduced to the matrices so as to reduce the computational cost while the element accuracy remains virtually intact. In this paper, a comprehensive review of admissible matrix formulation is presented. Finite element techniques seminal to the formulation are also discussed.
This paper investigates the factors relating to element stability, coordinate invariance and optimality in 8- and 20-noded three-dimensional brick elements in the context of hybrid-stress formulations with compatible boundary displacements and both a priori and a posteriori equilibrated assumed stresses. Symmetry group theory is used to guarantee the essential non-orthogonality of the stress and strain fields, resulting in a set of least-order selections of stable invariant stress polynomials. The performance of these elements is examined in numerical examples providing a broad range of analytical stress distributions, and the results are favourably compared to those of the displacement formulation and a stress-function based complete stress approach with regard to displacements, stresses, sampling, convergence and distortion sensitivity.
Four-node axisymmetric solid elements are derived by a new version of hybrid method for which the assumed stresses are expressed in complete polynomials in natural coordinates. The stress equilibrium conditions are introduced through the use of additional displacements as Lagrange multipliers. A rational procedure is to choose the displacement terms such that the resulting strains are also of complete polynomials of the same order. Example problems all indicate that elements obtained by this procedure lead to better results in displacements and stresses than that by other finite elements.
An element stiffness matrix can be derived by the conventional potential energy principle and, indirectly, also by generalized variational principles, such as the Hu-Washizu principle and the Hellinger-Reissner principle. The present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the Hellinger-Reissner principle. It is found that the resulting scheme is equivalent to that considered by Tong (1982) for the construction of hybrid stress elements. In Tong's scheme the inversion of a large flexibility matrix can be avoided. It is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the Hellinger-Reissner principle and the Hu-Washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method.
A new method for the formulation of hybrid elements by the Hellinger-Reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element. The equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co-ordinates. The resulting element possesses all the ideal qualities, i.e. it is invariant, it is less sensitive to geometric distortion, it contains a minimum number of stress parameters and it provides accurate stress calculations. For the formulation of a 4-node plane stress element, a small perturbation method is used to determine the equilibrium constraint equations. The element has been proved to be always rank sufficient.