Hu Washizu Research Articles

Comparison of Three-Dimensional Flexible Beam Elements for Dynamic Analysis: Classical Finite Element Formulation and Absolute Nodal Coordinate Formulation

Three formulations for a flexible spatial beam element for dynamic analysis are compared: a Timoshenko beam with large displacements and rotations, a fully parametrized element according to the absolute nodal coordinate formulation (ANCF), and an ANCF element based on an elastic line approach. In the last formulation, the shear locking of the antisymmetric bending mode is avoided by the application of either the two-field Hellinger–Reissner or the three-field Hu–Washizu variational principle. The comparison is made by means of linear static deflection and eigenfrequency analyses on stylized problems. It is shown that the ANCF fully parametrized element yields too large torsional and flexural rigidities, and shear locking effectively suppresses the antisymmetric bending mode. The presented ANCF formulation with the elastic line approach resolves most of these problems.

From the Hu–Washizu formulation to the average nodal strain formulation

We present a stabilized finite element method for the Hu–Washizu formulation of linear elasticity based on simplicial meshes leading to the stabilized nodal strain formulation or node-based uniform strain elements. We show that the finite element approximation converges uniformly to the exact solution for the nearly incompressible case.

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Geometrically linear problems

AbstractIn this paper a new reduced integration eight‐node solid‐shell finite element is presented. The enhanced assumed strain (EAS) concept based on the Hu–Washizu variational principle requires only one EAS degree‐of‐freedom to cure volumetric and Poisson thickness locking. One key point of the derivation is the Taylor expansion of the inverse Jacobian with respect to the element center, which closely approximates the element shape and allows us to implement the assumed natural strain (ANS) concept to eliminate the curvature thickness and the transverse shear locking. The second crucial point is a combined Taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking‐free hourglass stabilization without rank deficiency. Hence, the element requires only a single integration point in the shell plane and at least two integration points in thickness direction. The formulation fulfills both the membrane and the bending patch test exactly, which has, to the authors' knowledge, not yet been achieved for reduced integration eight‐node solid‐shell elements in the literature. Owing to the three‐dimensional modeling of the structure, fully three‐dimensional material models can be implemented without additional assumptions. Copyright © 2009 John Wiley & Sons, Ltd.

Anisotropic thin-walled beam models: A rational deduction from three-dimensional elasticity

In this paper anisotropic thin-walled beam models are rationally deduced from three-dimensional elasticity by means of a constrained approach. Consistent frictionless internal constraints on both stress and strain dual fields are enforced through a modified Hu–Washizu functional obtained by a nonstandard application of Lagrange multipliers. Beam theories accounting for different shear refinement levels are justified, showing that this variational approach enables the development of new refined models, including high-order nonconventional effects and enhancing standard treatments of shear deformation effects. In agreement with the constrained problem, a locally equilibrated approximation of the stress field acting on beam cross-section is recovered in closed form. Finally, cases of laminated thin-walled beams as well as of unilateral conewise constitutive behavior (with special reference to bimodular materials) are investigated.

Improved 4-node Hu–Washizu elements based on skew coordinates

Mixed 4-node elements based on the Hu–Washizu (HW) functional are developed for stress and strain representations in various coordinates, including the skew, natural and Cartesian ones. The HW functional is used in incremental form, suitable for non-linear materials. The key features of our approach are as follows. (1) The representations of stress and strain are assumed in skew coordinates associated with the natural basis at the element’s center, which implies that, for a linear elastic case, the homogenous equilibrium equations and the compatibility condition are satisfied point-wise. For stress, the same 5- and 7-parameter representations as for the Hellinger–Reissner (HR) elements by Wisniewski and Turska [Wisniewski K, Turska E. Improved four-node Hellinger–Reissner elements based on skew coordinates. Int J Numer Methods Eng 2008;76:798–836] are used. For strain, a 9-parameter linear representation is selected. (2) A mixed element HW14-S using a 5-parameter representation of stresses assumed in skew coordinates is developed from the non-enhanced HW functional. This element is equally accurate as our HR5-S element of Wisniewski and Turska (1998), the HR element by Yuan et al. [Yuan K-Y, Huang Y-S, Pian THH. New strategy for assumed stress for 4-node hybrid stress membrane element. Int J Numer Methods Eng 1993;36:1747–63], and the HW elements by Piltner and Taylor [Piltner R, Taylor RL. A quadrilateral mixed finite element with two enhanced strain modes. Int J Numer Methods Eng 1995;38:1783–808; Piltner R, Taylor RL. A systematic construction of B-bar functions for linear and non-linear mixed-enhanced finite elements for plane elasticity problems. Int J Numer Methods Eng 1999;44:615–39], and Piltner [Piltner R. An implementation of mixed enhanced finite elements with strains assumed in Cartesian and natural element coordinates using sparse B ¯ -matrices. Eng Comput 2000;17(8):933–49]. Compared to these HW elements, our element uses a smaller number of parameters. (3) A mixed/enhanced element HW18 using a 7-parameter representation of stress is developed from the enhanced HW functional. For the elements based on this stress representation, the strain representation has to be enriched; we use a 2-parameter EADG enhancement. Various combinations of the natural, skew and Cartesian coordinates are tested, and these for which this element performs best are selected. (4) A specific modification of the F T F product, consisting of the expansion of F and the selection of meaningful terms in the product, was applied to selected elements. With this modification, the element HW14-S performs better for coarse distorted meshes than the HW elements described in the literature. The developed elements are based on the Green strain, and are tested for linear and non-linear constitutive laws modified by the zero normal stress condition, because they will be used as a membrane part of a shell element. Several numerical tests show their performance, in particular, their robustness to the element’s shape distortion for coarse meshes.

A Piezoelectric Finite Shell Element for the Geometrically Nonlinear Analysis of Sensors

AbstractA geometrically nonlinear finite element formulation to analyze piezoelectric shell structures is presented. The formulation is based on the mixed field variational functional of Hu–Washizu. Within this variational principle the independent fields are displacements, electric potential, strains, electric field, stresses and dielectric displacements. The mixed formulation allows an interpolation of the strains and the electric field through the shell thickness, which is an essential advantage when using a three dimensional material law. It is remarked that no simplification regarding the constitutive relation is assumed. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent resultant stress and resultant dielectric displacement fields. The shell structure is modeled by a reference surface with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electrical degree of freedom, which is the difference of the electric potential through the shell thickness. The developed mixed hybrid shell element fulfills the in–plane, bending and shear patch tests, which have been adopted for coupled field problems. A numerical investigation of a smart antenna demonstrates the applicability of the piezoelectric shell element under the consideration of geometrical nonlinearity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua

The structural boundary-value problem in the context of nonlocal (integral) elasticity and quasi-static loads is considered in a geometrically linear range. The nonlocal elastic behaviour is described by the so-called Eringen model in which the nonlocality lies in the constitutive relation. The diffusion processes of the nonlocality are governed by an integral relation containing a recently proposed symmetric spatial weight function expressed in terms of an attenuation function. A firm variational basis to the nonlocal model is given by providing the complete set of variational formulations, composed by ten functionals with different combinations of the state variables. In particular the nonlocal counterpart of the classical principles of the total potential energy, complementary energy and mixed Hu–Washizu principle and Hellinger–Reissner functional are recovered. Some examples concerning a piecewise bar in tension are provided by using the Fredholm integral equation and the proposed nonlocal FEM.

The Hu–Washizu variational principle for the identification of imperfections in beams

AbstractThis paper presents a procedure for the identification of imperfections of structural parameters based on displacement measurements by static tests. The proposed procedure is based on the well‐known Hu–Washizu variational principle, suitably modified to account for the response measurements, which is able to provide closed‐form solutions to some inverse problems for the identification of structural parameter imperfections in beams. Copyright © 2008 John Wiley & Sons, Ltd.

Smooth finite element methods: Convergence, accuracy and properties

AbstractA stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed.We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi‐equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one‐subcell element with a quasi‐equilibrium finite element, leading to a global a posteriori error estimate.We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost.It is shown numerically that the one‐cell smoothed four‐noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.

On the derivation of structural models with general thermomechanical prestress

The vibrating behaviour of thin structures is affected by prestress states. Hence, the effects of thermal prestress are important research subjects in view of ambient vibration monitoring of civil structures. The interaction between prestress, geometrically non-linear behaviour, as well as damping and its coupling with the aforementioned phenomena has to be taken into account for a comprehensive understanding of the structural behaviour. Since the literature on this subject lacks a clear procedure to derive models of thin prestressed and damped structures from 3D continuum mechanics, this paper presents a new derivation of models for thin structures accounting for generic prestress, moderate rotations and viscous damping. Although inspired by classical approaches, the proposed procedure is quite different, because of (i) the definition of a modified Hu-Washizu (H-W) functional, accounting for stress constraints associated with Lagrange multipliers, in order to derive lower-dimensional models in a convenient way; (ii) an original definition of a (mechanical and thermal) strain measure and a rotation measure enabling one to identify the main terms in the strain energy and to derive a cascade of lower-dimensional models (iii) a new definition of strain-rotation domains providing a clear interpretation of the classical assumptions of small perturbations and small strains and moderate rotations; (iv) the introduction of a pseudo-potential with stress constraints to account for viscous damping. The proposed procedure is applied to thin beams.

A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications

The paper is focused on a piezoelectric solid shell finite element formulation. A geometrically nonlinear theory allows large deformations and includes stability problems. The formulation is based on a variational principle of the Hu–Washizu type including six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. The element has eight nodes with displacements and the electric potential as nodal degrees of freedom. A bilinear distribution through the thickness of the electric field is assumed to obtain correct results in bending dominated situations. The presented element is able to model arbitrary curved shells and incorporates a 3D-material law. Numerical examples demonstrate the ability of the proposed model to analyze piezoelectric devices.

Locking-free finite element methods for linear and nonlinear elasticity in 2D and 3D

The uniform convergence of finite element approximations based on a modified Hu–Washizu formulation for the nearly incompressible linear elasticity is analyzed. We show the optimal and robust convergence of the displacement-based discrete formulation in the nearly incompressible case with the choice of approximations based on quadrilateral and hexahedral elements. These choices include bases that are well known, as well as newly constructed bases. Starting from a suitable three-field problem, we extend our α-dependent three-field formulation to geometrically nonlinear elasticity with Saint-Venant Kirchhoff law. Additionally, an α-dependent three-field formulation for a general hyperelastic material model is proposed. A range of numerical examples using different material laws for small and large strain elasticity is presented.

Torsional vibration of a cracked rod by variational formulation and numerical analysis

The torsional vibration of a circumferentially cracked cylindrical shaft is studied through an “exact” analytical solution and a numerical finite element (FE) analysis. The Hu–Washizu–Barr variational formulation is used to develop the differential equation and the boundary conditions of the cracked rod. The equations of motion for a uniform cracked rod in torsional vibration are derived and solved, and the Rayleigh quotient is used to further approximate the natural frequencies of the cracked rod. Results for the problem of the torsional vibration of a cylindrical shaft with a peripheral crack are provided through an analytical solution based on variational formulation to derive the equation of motion and a numerical analysis utilizing a parametric three-dimensional (3D) solid FE model of the cracked rod. The crack is modelled as a continuous flexibility based on fracture mechanics principles. The variational formulation results are compared with the FE alternative. The sensitivity of the FE discretization with respect to the analytical results is assessed.

Theoretical aspects of the smoothed finite element method (SFEM)

AbstractThis paper examines the theoretical bases for the smoothed finite element method (SFEM), which was formulated by incorporating cell‐wise strain smoothing operation into standard compatible finite element method (FEM). The weak form of SFEM can be derived from the Hu–Washizu three‐field variational principle. For elastic problems, it is proved that 1D linear element and 2D linear triangle element in SFEM are identical to their counterparts in FEM, while 2D bilinear quadrilateral elements in SFEM are different from that of FEM: when the number of smoothing cells (SCs) of the elements equals 1, the SFEM solution is proved to be ‘variationally consistent’ and has the same properties with those of FEM using reduced integration; when SC approaches infinity, the SFEM solution will approach the solution of the standard displacement compatible FEM model; when SC is a finite number larger than 1, the SFEM solutions are not ‘variationally consistent’ but ‘energy consistent’, and will change monotonously from the solution of SFEM (SC = 1) to that of SFEM (SC → ∞). It is suggested that there exists an optimal number of SC such that the SFEM solution is closest to the exact solution. The properties of SFEM are confirmed by numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.

Convergence in the incompressible limit of finite element approximations based on the Hu-Washizu formulation

The classical Hu–Washizu mixed formulation for plane problems in elasticity is examined afresh, with the emphasis on behavior in the incompressible limit. The classical continuous problem is embedded in a family of Hu–Washizu problems parametrized by a scalar α for which $$\alpha = \lambda \big/ \mu$$ corresponds to the classical formulation, with λ and μ being the Lame parameters. Uniform well- posedness in the incompressible limit of the continuous problem is established for α ≠ − 1. Finite element approximations are based on the choice of piecewise bilinear approximations for the displacements on quadrilateral meshes. Conditions for uniform convergence are made explicit. These conditions are shown to be met by particular choices of bases for stresses and strains, and include bases that are well known, as well as newly constructed bases. Though a discrete version of the spherical part of the stress exhibits checkerboard modes, it is shown that a λ-independent a priori error estimate for the displacement can be established. Furthermore, a λ-independent estimate is established for the post-processed stress. The theoretical results are explored further through selected numerical examples.

Four‐node semi‐EAS element in six‐field nonlinear theory of shells

AbstractWe propose a new four‐node C0 finite element for shell structures undergoing unlimited translations and rotations. The considerations concern the general six‐field theory of shells with asymmetric strain measures in geometrically nonlinear static problems. The shell kinematics is of the two‐dimensional Cosserat continuum type and is described by two independent fields: the vector field for translations and the proper orthogonal tensor field for rotations. All three rotational parameters are treated here as independent. Hence, as a consequence of the shell theory, the proposed element has naturally six engineering degrees of freedom at each node, with the so‐called drilling rotation. This property makes the element suitable for analysis of shell structures containing folds, branches or intersections. To avoid locking phenomena we use the enhanced assumed strain (EAS) concept. We derive and linearize the modified Hu–Washizu principle for six‐field theory of shells. What makes the present approach original is the combination of EAS method with asymmetric membrane strain measures. Based on literature, we propose new enhancing field and specify the transformation matrix that accounts for the lack of symmetry. To gain knowledge about the suitability of this field for asymmetric strain measures and to assess the performance of the element, we solve typical benchmark examples with smooth geometry and examples involving orthogonal intersections of shell branches. Copyright © 2006 John Wiley & Sons, Ltd.

Development of a discrete Reissner–Mindlin element on Winkler foundation

Over the years several plate-bending elements have been developed by several researchers. Most of the developed elements have been either for thin plates or thick plates. Recently, a set of robust triangular and quadrilateral elements, which showed excellent performance for both thin and thick plates, had been developed. Among the series of the triangular elements developed is the Discrete Reissner–Mindlin (DRM) element. The main features of the DRM element are its mixed interpolation formulation and its discrete collocation constraints. In this paper the DRM element has been extended to solve problems of plates on elastic foundations. The Hu–Washizu variational principle is used to formulate the element stiffness matrix. Deflection, bending moment and shear force responses of plates on Winkler-type foundation and with various thicknesses, loading conditions and foundation elastic constants, using both the DRM finite element and the classical approach, are presented. It has been demonstrated that the DRM finite element solutions agree very well with the classical solutions, specifically, for deflection response in the case of both thin and thick plates. Therefore, it can be concluded that, the DRM plate element is both effective and reliable for solving problems of thin and thick plates on elastic foundations.

A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications

A discontinuous Galerkin formulation of the boundary value problem of finite-deformation elasticity is presented. The primary purpose is to establish a discontinuous Galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the DG concept can show its superiority compared to the continuous formulation. The method is based on a general Hu–Washizu–de Veubeke functional allowing for displacement and stress discontinuities in the domain interior. It is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation. The consistency and linearized stability of the method in the non-linear range as well as its convergence rate are proven. An implementation in three dimensions is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner. In order to demonstrate the versatility, accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided. Copyright © 2006 John Wiley & Sons, Ltd.

Structural analysis of composite laminates using a mixed hybrid shell element

The structural analysis of thin composite structures requires robust and effective shell elements. In this paper the variational formulation is based on a Hu–Washizu functional with independent displacements, stress resultants and shell strains. For the independent shell strains an additional interpolation part is introduced. This yields an improved convergence behaviour especially for laminated shells with coupled membrane and bending stiffness. The developed mixed hybrid shell element possesses the correct rank and fulfills the in–plane and bending patch test. The formulation is tested by several nonlinear examples including bifurcation and post–buckling response. The essential feature of the new element is the robustness in nonlinear computations with large rigid body motions. It allows very large load steps in comparison to standard displacement models.

Conditions for equivalence between the Hu–Washizu and related formulations, and computational behavior in the incompressible limit

The relationship of the Hu–Washizu mixed formulation to other mixed and enhanced formulations is examined in detail, in the context of linear elasticity, with a view to presenting a unified framework for such formulations. The Hu–Washizu formulation is considered in both its classical form and in a modified form that is suited to establishing well-posedness in the incompressible limit. Recently established theoretical results on uniform well-posedness are studied computationally by considering a range of numerical examples, which illustrate where appropriate the good behavior predicted by the theory, as well as the locking behavior that is evident in those problems in which the conditions for stability and convergence are not satisfied.