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

Tóth, Balázs; Kocsán, Lajos György

doi:10.1016/j.finel.2012.11.002

Cited by 13 publications

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“…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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“…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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“…This is based on the extended version of the three-field dual-mixed variational formulation of elastostatics [1,2] to linear elastodynamics, the independent fields of which are the non-symmetric stress tensor, the displacement-and the rotation vector. An important property of the related shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell middle surface are not used, i.e., unmodified three-dimensional constitutive equations are applied.…”

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“…We assume axisymmetric boundary conditions, as well as homogeneous and isotropic material properties, in this case the variables depend only on the meridian coordinate ξ 1 , the thickness coordinate −d/2 ≤ ξ 3 ≤ d/2 and the time t. The shell-thickness d is considered to be constant. The fundamental variables, i.e., the stresses σ kλ , σ k3 , the displacements u k and the rotations φ s are approximated by polynomials of first-and second-degree in ξ 3 :see the details in [1,2,6]. It is important to note here that this is the only hypothesis used in the derivation of the dimensionally reduced shell model.…”

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confidence: 99%

“…Thus the number of the unknown functions will be 10 including 6 stress-and 4 displacement components. The degrees of the polynomial space applied to the h-and pversion finite element modeling can be found in [1,2,6]. …”

mentioning

confidence: 99%

“…see the details in [1,2,6]. It is important to note here that this is the only hypothesis used in the derivation of the dimensionally reduced shell model.…”

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confidence: 99%

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Dual‐mixed variational formulation and hp finite element method for axisymmetric shell problems in elastodynamics

Tóth

2013

Proc Appl Math and Mech

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A new dimensionally reduced axisymmetric shell model is presented briefly for modeling time-dependent problems. This is based on the extended version of the three-field dual-mixed variational formulation of elastostatics [1,2] to linear elastodynamics, the independent fields of which are the non-symmetric stress tensor, the displacement-and the rotation vector. An important property of the related shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell middle surface are not used, i.e., unmodified three-dimensional constitutive equations are applied. The computational performance of the new h-and p-version axisymmetric shell finite elements is tested through a representative cylindrical shell problems. The development presented in this paper has been motivated by the fact that efficient dual-mixed hp plate and shell finite elements were managed previously to be developed for elastostatics by [1][2][3][4][5]. the fundamental variables of which are the a priori non-symmetric stress tensor σ k , the displacement vector u k and the rotation vector φ s . Furthermore V denotes the volume of the body in the undeformed configuration, S = S σ ∪S u (S σ ∩S u = ∅) defines the bounding surface of V , k s is the covariant permutation tensor and u k is the prescribed displacement vector on the surface part S u with outward unit normal n , as well as b k and ρ stand for, respectively, the density of the body forces and the material, and t ∈ [t 0 , t 1 ] defines a closed time interval. The fourth-order tensor C pqk with symmetry properties C pqk = C pq k = C k pq is the elastic compliance tensor. The solution of the linear elastodynamic problem can be sought as the stationary point of functional (1). The subsidiary conditions to (1) are the stress boundary conditionswhere p k are prescribed surface tractions on S σ , as well as the initial conditions2 Basic assumption of the shell modelLet us consider an axisymmetric shell of length L as a three-dimensional body. We assume axisymmetric boundary conditions, as well as homogeneous and isotropic material properties, in this case the variables depend only on the meridian coordinate ξ 1 , the thickness coordinate −d/2 ≤ ξ 3 ≤ d/2 and the time t. The shell-thickness d is considered to be constant. The fundamental variables, i.e., the stresses σ kλ , σ k3 , the displacements u k and the rotations φ s are approximated by polynomials of first-and second-degree in ξ 3 :see the details in [1,2,6]. It is important to note here that this is the only hypothesis used in the derivation of the dimensionally reduced shell model. Then the number of the independent stress components is reduced by a priori satisfaction of the timedependent prescribed surface loads on the inner and outer surfaces of the shell and by the elimination of the rotations. Our investigations is restricted to bending-shearing problems of the axisymmetic shell. Thus the number of the unknown functions will be 10 including 6 stress-and 4 displacement components. The degree...

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Two-field mixed hp-finite elements for time-dependent problems in the refined theories of thermodynamics

Tóth,

Molnár,

Kovács

2024

Continuum Mech. Thermodyn.

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Modern manufacturing technologies allow heterogeneous materials with complex inner structures (e.g., foams) to be easily produced. However, their utilization is not straightforward, as the classical constitutive laws are not necessarily valid. According to various experimental observations, the Guyer–Krumhansl equation is a promising candidate for modeling such complex structures. However, practical applications need a reliable and efficient algorithm capable of handling both complex geometries and advanced heat equations. In the present paper, we derive new two-field variational formulations which treat the temperature and the heat flux as independent field variables, and we develop new, advanced hp-type mixed finite element methods, which can be reliably applied. We investigate their convergence properties for various situations, challenging in relation to stability and the treatment of fast propagation speeds. That algorithm is also proved to be outstandingly efficient, providing solutions four magnitudes faster than commercial algorithms.

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

Cited by 13 publications

References 45 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 variational formulation and hp finite element method for axisymmetric shell problems in elastodynamics

Two-field mixed hp-finite elements for time-dependent problems in the refined theories of thermodynamics

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