Fraeijs De Veubeke Research Articles

Two families of two‐dimensional finite element interpolation functions of general degree with general continuity

AbstractThis work presents a generalization of the functions used in Fraeijs de Veubeke's quadrilateral (FVQ) element and in Hsieh, Clough and Tocher's (HCT) triangle, allowing for an arbitrary degree of the approximation, , and for arbitrary order of continuity, . These functions can be used to define finite elements that lead by direct assembly to approximations. Their determination for each element is procedural, generalizing the reasoning used by Zienkiewicz to explain the FVQ element. The completeness of each basis is confirmed by comparing its dimension with that of a broken basis, defined primitive element wise, upon which the continuity constraints are successively imposed. The order of continuity at the vertices that is required to enable the independent definition of the normal derivatives at the sides is confirmed numerically. The corresponding interpolation functions are symbolically derived using Mathematica, and simple matlab codes using the data thus obtained are made available, to illustrate their application.

A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity

This paper introduces a new computational framework for the analysis of large strain fast solid dynamics. The paper builds upon previous work published by the authors (Gil et al., 2014) [48], where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables. In this work, the formulation is further enhanced with four key novelties. First, the use of a new geometric conservation law for the co-factor of the deformation leads to an enhanced mixed formulation, advantageous in those scenarios where the co-factor plays a dominant role. Second, the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems. Moreover, the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations, dual of that expressed in terms of conservation variables. Third, the new use of a tensor cross product [61] greatly facilitates the algebraic manipulations of expressions involving the co-factor of the deformation. Fourth, the development of a stabilised Petrov–Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. As an example, a polyconvex Mooney–Rivlin material is used and, for completeness, the eigen-structure of the resulting system of equations is studied to guarantee the existence of real wave speeds. Finally, a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology, benchmarking it against an ample spectrum of alternative numerical strategies, including implicit multi-field Fraeijs de Veubeke–Hu–Washizu variational type approaches and explicit cell and vertex centred Finite Volume schemes.

Fraeijs de Veubeke variational principle‐based cylindrical shell finite element model

AbstractIn this paper a dimensionally reduced cylindrical shell model based on the dual‐mixed variational principle of Fraeijs de Veubeke will be presented. The fundamental variables of this variational principle are the not a priori symmetric stress tensor and the skew‐symmetric rotation tensor. The tensor of first‐order stress functions is applied to satisfy translational equilibrium. A shell model derived in this way makes the application of the classical kinematical hypotheses unnecessary, and enables us to use unmodified three‐dimensional constitutive equations. On the basis of this shell model, a new dual‐mixed cylindrical shell finite elements capable of both h‐ and p‐approximation can be derived. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Comparison of dual-mixed h- and p-version finite element models for axisymmetric problems of cylindrical shells

Dimensionally reduced cylindrical shell models using complementary energy-based variational formulations of a priori non-symmetric stresses are compared. One of them is based on the three-field dual-mixed Hellinger–Reissner variational principle, the fundamental variables of which are the stress tensor, the rotation and displacement vectors. The other one is derived from the two-field dual-mixed Fraeijs de Veubeke variational principle in terms of the self-equilibrated stress field and rotations. The most characteristic properties of the shell models are that the kinematical hypotheses used in the classical shell theories are not applied and the unmodified three-dimensional constitutive equations are employed. Our investigations are restricted to the axisymmetric case. The developed dual-mixed hp finite element models with C0 continuous tractions and with discontinuous rotations and displacements are presented for bending–shearing (including tension–compression) problems. The computational performance of the constructed shell elements is compared through two representative model problems. It is numerically proven that no significant differences can be experienced between the two well-performing shell elements in the convergence rates.

Efficient Hybrid-EAS solid element for accurate stress prediction in thick laminated beams, plates, and shells

In this paper, we present a novel and efficient 3-D Hybrid-EAS solid element formulation to accurately predict interlaminar stresses in thick laminated beams, plates, and shells. The element formulation is based on the mixed three-field Fraeijs de Veubeke–Hu–Washizu (FHW) functional principle. The element is designed to have quadratic C0 transverse shear stress field through the thickness direction, while the displacement field remains linear. The continuity of the transverse shear stress at the layer interfaces, together with the vanishing transverse shear stresses at the outer surfaces of the composite structure, are satisfied exactly by using the transverse-shear-stress degrees of freedom (dofs). This method is more elegant than adding the above stress constraints on the FHW functional via a penalty or Lagrange multipliers, and is amenable for implementation in general purpose finite-element codes. Numerical examples show the excellent agreement with Pagano’s exact solution for composite beam/plate problems. The present element satisfies the interlaminar stress continuity, and accurately captures the quadratic variation of the transverse shear stresses in the thickness when compared to other element formulations, in particular the well-known hybrid formulation by Mau, Tong, and Pian (1972) [23]. In addition, the present approach is readily generalizable to the case with large deformation and nonlinear materials.

A novel versatile multilayer hybrid stress solid-shell element

This paper presents a versatile multilayer locking free hybrid stress solid-shell element that can be readily employed for a wide range of geometrically linear elastic structural analyses, i.e. from shell-like isotropic structures to multilayer anisotropic composites. This solid-shell element has eight nodes with only displacement degrees of freedom and a few internal parameters that provide the locking free behavior and accurate interlaminar stress resolution through the element thickness. These elements can be stacked on top of each other to model multilayer structures, fulfilling the interlaminar stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of composite laminates. The element formulation is based on the modified form of the well-known Fraeijs de Veubeke---Hu---Washizu (FHW) multifield variational principle with enhanced assumed strains (EAS formulation) and assumed natural strains (ANS formulation) to alleviate the different types of locking phenomena in solid-shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar stress field in multilayer structures, which is achieved by incorporating an assumed stress field in a standard EAS formulation based on the FHW principle. To assess the present formulation's accuracy, a variety of popular numerical benchmark examples related to element patch tests, convergence, mesh distortion, shell and laminated composite analyses are investigated and the results are compared with those available in the literature. This assessment reveals that the proposed solid-shell formulation provides very accurate results for a wide range of structural analyses.

An optimal versatile partial hybrid stress solid-shell element for the analysis of multilayer composites

SUMMARY In the present contribution we propose an optimal low-order versatile partial hybrid stress solid-shell element that can be readily employed for a wide range of geometrically linear elastic structural analyses, that is, from shell-like isotropic structures to multilayer anisotropic composites. This solid-shell element has eight nodes with only displacement degrees of freedom and only a few internal parameters that provide the locking-free behavior and accurate interlaminar shear stress resolution through the element thickness. These elements can be stacked on top of each other to model multilayer composite structures, fulfilling the interlaminar shear stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of composite laminates. The element formulation is based on the modified form of the well-known Fraeijs de Veubeke–Hu–Washizu multifield variational principle with enhanced assumed strains formulation and assumed natural strains formulation to alleviate the different types of locking phenomena in solid-shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar shear stress field in multilayer structures, which is achieved by the introduction of the assumed interlaminar shear stress field in a standard enhanced assumed strains formulation based on the Fraeijs de Veubeke–Hu–Washizu principle. The numerical testing of the present formulation, employing a variety of popular numerical benchmark examples related to element patch test, convergence, mesh distortion, shell and laminated composite analyses, proves its accuracy for a wide range of structural analyses.Copyright © 2012 John Wiley & Sons, Ltd.

Optimal low-order fully integrated solid-shell elements

This paper presents three optimal low-order fully integrated geometrically nonlinear solid-shell elements based on the enhanced assumed strain (EAS) method and the assumed natural strain method for different types of structural analyses, e.g. analysis of thin homogeneous isotropic and multilayer anisotropic composite shell-like structures and the analysis of (near) incompressible materials. The proposed solid-shell elements possess eight nodes with only displacement degrees of freedom and a few internal EAS parameters. Due to the 3D geometric description of the proposed elements, 3D constitutive laws can directly be employed in these formulations. The present formulations are based on the well-known Fraeijs de Veubeke---Hu---Washizu multifield variational principle. In terms of accuracy as well as efficiency point of view, the choice of the optimal EAS parameters plays a very critical role in the EAS method, therefore a systematic numerical study has been carried out to find out the optimal EAS parameters to alleviate different locking phenomena for the proposed solid-shell formulations. To assess the accuracy of the proposed solid-shell elements, a variety of popular numerical benchmark examples related to element convergence, mesh distortions, element aspect ratios and different locking phenomena are investigated and the results are compared with the well-known solid-shell formulations available in the literature. The results of our numerical assessment show that the proposed solid-shell formulations provide very accurate results, without showing any numerical problems, for a variety of geometrically linear and nonlinear structural problems.

The Best L<sup>2</sup> Norm Error Estimate of Lower Order Finite Element Methods for the Fourth Order Problem

In the paper, we analyze the L 2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L 2 norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including: the Powell-Sabin C 1 P2 macro element, the nonconforming Morley element, the C 1 Q2 macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming WangXu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature: can the L 2 norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm?

A mixed solid-shell element for the analysis of laminated composites

SUMMARY This paper presents a versatile low order locking-free mixed solid-shell element that can be readily employed for a wide range of linear elastic structural analyses, that is, from thick isotropic structures to multilayer anisotropic composites. This solid-shell element has eight nodes with only displacement degrees of freedom and few assumed stress parameters that provide very accurate interlaminar stress calculations through the element thickness. These elements can be stacked on top of each other to model multilayer structures, fulfilling the interlaminar stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of the laminate. The element formulation is based on the well-known Fraeijs de Veubeke–Hu–Washizu mixed variational principle with enhanced assumed strains formulation and assumed natural strains formulation to alleviate the different types of locking phenomena in solid-shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar stress field in multilayer structures, which is achieved by the introduction of a constraint equation on the interlaminar stresses in the Fraeijs de Veubeke–Hu–Washizu principle-based enhanced assumed strains formulation. The intelligent computer coding of the present formulation makes the present element appropriate for a wide range of structural analyses. To assess the present formulation's accuracy, a variety of popular numerical benchmark examples related to element convergence, mesh distortion, and shell and laminated composite analyses are investigated and the results are compared with those available in the literature. These benchmark examples reveal that the proposed formulation provides very good results for the structural analysis of shells and multilayer composites. Copyright © 2011 John Wiley & Sons, Ltd.

Derivation of a dual-mixed hp-finite element model for axisymmetrically loaded cylindrical shells

The two-field dual-mixed Fraeijs de Veubeke variational formulation of three-dimensional elasticity serves as the starting point of the derivation of a dimensionally reduced shell model presented in this paper. The fundamental variables of this complementary energy-based variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium, while the rotation tensor plays the role of a Lagrange multiplier to ensure rotational equilibrium. The volumetric locking-free shell model uses unmodified three-dimensional constitutive equations, and no classical kinematical hypotheses are employed during the derivation. The numerical performance of the related low-order h-, and higher-order p-version finite elements developed for axisymmetrically loaded cylindrical shells is investigated by two representative model problems. It is numerically proven that no negative effect can be experienced when the thickness is small and tends to zero.

Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis

AbstractThis study presents a general procedure of creating pure equilibrium tetrahedral finite elements for use under the elastostatic hypothesis. These pure equilibrium elements are of the Fraeijs de Veubeke type and the degree of the polynomial approximation functions of their internal stress field is the parameter generating this new elements family. The spurious kinematic modes (SKM), inherent in the equilibrium approach, are eliminated at the element level by converting each tetrahedron into a super‐element defined as an assembly of four tetrahedral primitive elements. A mathematical discussion on the number of SKM of the primitive elements as well as their elimination by the super‐element technique has been carried out. The development of first and second degree elements is presented here in detail and their efficiency in the frame of global error estimation by dual analysis is emphasized by two numerical applications. The main attribute of the error estimation by dual analysis is that it provides an upper bound on the global discretization error. Copyright © 2009 John Wiley & Sons, Ltd.

A natural neighbour method based on Fraeijs de Veubeke variational principle for materially non-linear problems

The natural neighbour method can be considered as one of many variants of the meshless methods. In the present paper, a new approach based on the Fraeijs de Veubeke (FdV) functional, which is initially developed for linear elasticity, is extended to the case of geometrically linear but materially non-linear solids. The new approach provides an original treatment to two classical problems: the numerical evaluation of the integrals over the domain A and the enforcement of boundary conditions of the type ui = ũi on Su. In the absence of body forces (Fi = 0), it will be shown that the calculation of integrals of the type \({\int\nolimits_A \cdot {\rm d}A}\) can be avoided and that boundary conditions of the type ui = ũi on Su can be imposed in the average sense in general and exactly if ũi is linear between two contour nodes, which is obviously the case for ũi = 0.

A natural neighbour method for linear elastic problems based on Fraeijs de Veubeke variational principle

AbstractThe natural neighbour method can be considered as belonging to the meshless methods. Classically, the development of this method is based on the virtual work principle. In the present paper, we use the natural neighbour method for 2D domains starting from the Fraeijs de Veubeke variational principle and we approximate separately the displacement field, the stress field and the strain field: the assumed strains and the assumed stresses are constant over each Voronoi cell, the assumed surface reactions are constant along the edge where the displacements are imposed, while the assumed displacements are interpolated by Laplace interpolants. In the absence of body forces, it is shown that the calculation of integrals on the area of the solid domain can be avoided: only integrals on the edges of the Voronoi cells are necessary. On the other hand, displacements can be imposed in the average sense on some boundaries of the domain. Patch tests and some applications in the elastic domain are given in the paper and show the effectiveness of the method. Copyright © 2007 John Wiley & Sons, Ltd.

Optimal solid shell element for large deformable composite structures with piezoelectric layers and active vibration control

In this paper, we present an optimal low-order accurate piezoelectric solid-shell element formulation to model active composite shell structures that can undergo large deformation and large overall motion. This element has only displacement and electric degrees of freedom (dofs), with no rotational dofs, and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out-of-plane bending). The combination of the present optimal piezoelectric solid-shell element and the optimal solid-shell element previously developed allows for efficient and accurate analyses of large deformable composite multilayer shell structures with piezoelectric layers. To make the 3-D analysis of active composite shells containing discrete piezoelectric sensors and actuators even more efficient, the composite solid-shell element is further developed here. Based on the mixed Fraeijs de Veubeke–Hu–Washizu (FHW) variational principle, the in-plane and out-of-plane bending behaviours are improved via a new and efficient enhancement of the strain tensor. Shear-locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. We also present an optimal-control design for vibration suppression of a large deformable structure based on the general finite element approach. The linear-quadratic regulator control scheme with output feedback is used as a control law on the basis of the state space model of the system. Numerical examples involving static analyses and dynamic analyses of active shell structures having a large range of element aspect ratios are presented. Active vibration control of a composite multilayer shell with distributed piezoelectric sensors and actuators is performed to test the present element and the control design procedure. Copyright © 2005 John Wiley & Sons, Ltd.

Optimal solid shells for non-linear analyses of multilayer composites. I. Statics

We present in this paper a simple low-order solid-shell element formulation – having only displacement degrees of freedom (dofs), i.e., without rotational dofs – that has an optimal number of parameters to pass the patch tests, and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio. The formulation of this element is based on the mixed Fraeijs de Veubeke–Hu–Washizu (FHW) variational principle leading to a novel enhancing strain tensor (EAS method) that renders the computation particularly efficient, with improved in-plane and out-of-plane bending behavior (Poisson thickness locking), especially in refined analyses of composite structures involving a large number of high aspect-ratio layers. We also review the equivalence between various choices of the enhancing strains in tensor form, and point out the relative efficiency of these choices. We discuss the enhanced assumed strain (EAS) formulations based on both the Green–Lagrange strain and the displacement gradient (the companion paper), point out the pitfalls in each approach, e.g., not passing the patch test, and the possibility and the method to remedy the problem. Shear locking and curvature thickness locking are treated using the assumed natural strain (ANS) method. The element passes the patch tests (both membrane and out-of-plane bending). We provide an optimal combination of the ANS method and the minimal number of EAS parameters required to pass the out-of-plane bending patch test. Numerical examples involving static analyses of multilayer shell structures having a large range of element aspect ratios are presented. Finally, we note that the topic in this paper is a fitting dedication to Professor Ekkehard Ramm, who has made important pioneering contributions in this research direction.

Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, Pages 145–197 of Stress Analysis, Edited by O. C. Zienkiewicz and G. S. Holister, Published by John Wiley & Sons, 1965

International Journal for Numerical Methods in EngineeringVolume 52, Issue 3 p. 287-342 Classic ReprintFree Access Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, Pages 145–197 of Stress Analysis, Edited by O. C. Zienkiewicz and G. S. Holister, Published by John Wiley & Sons, 1965 O. C. Zienkiewicz, Corresponding Author O. C. Zienkiewicz [email protected] Department of Civil Engineering, University of Wales Swansea, Swansea SA2 8PP, U.K.Department of Civil Engineering, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, U.K.Search for more papers by this author O. C. Zienkiewicz, Corresponding Author O. C. Zienkiewicz [email protected] Department of Civil Engineering, University of Wales Swansea, Swansea SA2 8PP, U.K.Department of Civil Engineering, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, U.K.Search for more papers by this author First published: 24 July 2001 https://doi.org/10.1002/nme.339Citations: 41AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL REFERENCES 1 de Veubeke B Fraeijs, Zienkiewicz OC. Strain energy bounds in finite element analysis. Journal of Strain Analysis 1967; 2: 265– 271. Citing Literature Volume52, Issue330 September 2001Pages 287-342 ReferencesRelatedInformation

On the convexity of C 1 surfaces associated with some quadrilateral finite elements

We analyse the convexity property of two classical finite elements and of the associated piecewise polynomial C 1 surfaces. The first one is the piecewise cubic quadrilateral of Fraeijs de Veubeke. The second one is a piecewise quadratic rectangle introduced by Sibson and Thomson and generalized by Sablonniere and Zedek. In both cases, we first study the local problem and then we extend our results to the associated C 1 surfaces.

On the Original Publication of the General Canonical Functional of Linear Elasticity

The general canonical functional of linear elastostatics is associated with the names of Hu and Washizu, who published it independently in 1955. This note discusses how that functional, in a generalized four-field form, had been derived by B. M. Fraeijs de Veubeke in a 1951 technical report. This report presents five of the seven canonical functionals of elasticity. In addition to the general functional, it exhibits what is likely the first derivation of the strain-displacement dual of the Hellinger-Reissner functional. The tour of five variational principles takes only a relatively small portion of the report: 8 pages out of 56. The bulk is devoted to the use of energy methods for analysis of wing structures. The title, technology focus, and limited dissemination may account for the subsequent neglect of this original contribution to variational mechanics. [S0021-8936(00)00401-3]

Duality in nonlinear theory of shells

The computational problem of shell structures in large displacements and large rotations has prompted numerous theories and approximations. The introduction of the finite rotation was successfully realized in 1972 by Fraeijs de Veubeke in three-dimension, and led to the creation of a dual nonlinear variational principle where the stresses as new unknowns were statically admissible. Following this line, its extension to shells, though not so easy, allows to provide statically admissible surface stresses. Now stability problems are directly governed by accurate stresses, and particularly in case of shells. After a recall of the dual variational principle for shells with large displacements, finite rotations and transverse shear deformations, a dual theory of stability is developed, namely the stability criterion and the post-buckling in the dual theory.