Dual-mixed hp finite element methods using first-order stress functions and rotations

Bertóti, E.

doi:10.1007/s004660000148

Cited by 11 publications

(9 citation statements)

References 17 publications

(22 reference statements)

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“…The developments of the new dimensionally reduced cylindrical shell model and related dual-mixed finite elements are motivated by the results of the first-order stress function-based finite element formulations addressed in [1,2]. Cylindrical shell models using the equilibrium hypothesis (the definition is given in Section 3) have been developed and investigated assuming axisymmetric problems in [3][4][5].…”

Section: The Fraeijs De Veubeke Variational Principlementioning

confidence: 99%

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

Kocsán

2013

Proc Appl Math and Mech

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In 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. The Fraeijs de Veubeke variational principleThe developments of the new dimensionally reduced cylindrical shell model and related dual-mixed finite elements are motivated by the results of the first-order stress function-based finite element formulations addressed in [1,2]. Cylindrical shell models using the equilibrium hypothesis (the definition is given in Section 3) have been developed and investigated assuming axisymmetric problems in [3][4][5].The functional of the two-field dual-mixed Fraeijs de Veubeke variational principle [6] can be derived from the complementary energy functional by adding a Lagrange multiplier term which ensures the symmetry of the stress tensor:where σ pq , D k pq , ϕ,ũ, ∈ qpc and n q are, respectively, the stress tensor, the fourth-order elastic compliance tensor, the rotation vector, the prescribed displacement, the covariant permutation tensor and the outward normal of the bounding surface of the three-dimensional elastic body. Applicability of (1) requires that the stress tensor σ pq satisfies a priori the translational equilibrium equationsand the stress boundary conditions σ k n =p k on S p , where q k is the density of the body forces andp k is the prescribed surface tractions on the surface S p . The displacement boundary condition on surface S u is imposed weakly in (1).By introducing the first-order stress function tensor Ψ k .p , a stress field that fulfills (2) can be obtained:whereσ k is a particular solution to (2). Geometric descriptionIn the Cartesian frame of reference x a , we consider a three-dimensional cylindrical shell with curvilinear coordinate system ξ k . The parametric form of the middle surface is defined by the set of equations, where R is the radius of the middle surface. Let L denote the length and d be the uniform thickness of the shell. Then it occupies the region V = {r(ξ2 }, and the inner and outer surfaces3 About the new cylindrical shell modelIn the development of the dimensionally reduced cylindrical shell model, all the variables are expanded into power series with respect to the thickness coordinate around the shell middle surface. For example:

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Section: The Fraeijs De Veubeke Variational Principlementioning

confidence: 99%

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

Kocsán

2013

Proc Appl Math and Mech

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“…Making use of Eqs. (13)- (14) and expression (15) obtained for the Christoffel symbols, the three-dimensional scalar equilibrium equations (22) of the cylindrical shell can be written as…”

Section: Equilibrium Equations For Cylindrical Shellsmentioning

confidence: 99%

Dual‐mixed hp finite element model for elastic cylindrical shells

Tóth¹

2011

Z Angew Math Mech

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Key words Dimensional reduction, cylindrical shell model, axisymmetric load, dual-mixed locking-free hp finite element.A dimensionally reduced cylindrical shell model using a three-field complementary energy-based Hellinger-Reissner's variational principle of non-symmetric stresses, rotations, and displacements is presented. An important property of the shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell mid-surface are not applied. A dual-mixed hp finite element model with stable polynomial stress-and displacement interpolation and C 0 continuous normal components of stresses is constructed and presented for the bending-shearing problem, using unmodified three-dimensional inverse stress-strain relations for linearly elastic materials. It is shown through an example that the convergences in the energy norm as well as in the maximum norm of stresses and displacements are rapid for both h-extension and p-approximation, not only for thin but also for moderately thick shells loaded axisymmetrically, even if the Poisson ratio is close to the incompressibility limit of 0.5.

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“…After that, some researchers combined the SFE method with the stochastic perturbation approach [26,27] and the boundary element method [28]. In order to gain the advantages of the DFE method, the hybrid method was developed, which uses assumed displacement and stress functions [27,[29][30][31]. In addition, there are various books providing details about the SFE method [32,33].…”

Section: Introductionmentioning

confidence: 99%

Curvature‐ and displacement‐based finite element analyses of flexible slider crank mechanisms

Kuo

Cleghorn

2010

Numer Methods Biomed Eng

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SUMMARYThe paper presents the applications of the curvature-and displacement-based finite element methods to flexible slider crank mechanisms. The displacement-based method usually needs more elements or highdegree polynomials to obtain highly accurate solutions. The curvature-based method assumes a polynomial to approximate a curvature distribution, and the expressions are investigated to obtain the displacement and rotation distributions. During the process, the boundary conditions associated with displacement, rotation, and curvature are imposed, which leads the great reduction of the number of degrees of freedom that are required. The numerical results demonstrate that the errors obtained by applying the curvature-based method are much smaller than those by applying the displacement-based method, based on the comparison of the same number of degrees of freedom.

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Dual-mixed hp finite element methods using first-order stress functions and rotations

Cited by 11 publications

References 17 publications

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

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

Dual‐mixed hp finite element model for elastic cylindrical shells

Curvature‐ and displacement‐based finite element analyses of flexible slider crank mechanisms

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