Parallel Solution Techniques for Hybrid Mixed Finite Element Models

Cismaşiu, Ildi; Almeida, J. P. Moitinho de; Castro, Luís Filipe; Harbis, D.C.

doi:10.4203/csets.1.6

Cited by 5 publications

(10 citation statements)

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“…SfS d<C (27) The alternative boundary integral de"nition (28) for the #exibility matrix is obtained integrating equation (27) by parts to exploit constraints (14)}(16) on the stress approximation basis:…”

Section: F"mentioning

confidence: 99%

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Three-dimensional hybrid-Trefftz stress elements

Freitas

Bussamra

2000

Int. J. Numer. Meth. Engng.

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The stress model of the hybrid-Tre!tz "nite element formulation is applied to the linear elastostatic analysis of solids. The stresses are approximated in the domain of the element and displacements on its boundary. Complete, linearly independent, hierarchical polynomial approximation functions are used in both domain and boundary approximations. The displacement basis is de"ned independently on each inter-element surface. Continuity at the edges and on the corners of the elements is not enforced a priori. The stress basis is constrained to solve locally the Beltrami governing di!erential equation. It is derived from the associated Papkovitch}Neuber elastic displacement solution. Generalized variables are used to ensure that the approximations are independent of the geometric description of the elements. The solving system is derived directly from the fundamental relations of elastostatics. The solving system is symmetric, when the same property applies to the local elasticity condition, sparse, described by boundary integral arrays and well suited to p-re"nement and parallel processing. The numerical implementation of these equations is discussed and numerical tests are presented to illustrate the performance of the "nite element formulation. thus obtained can be recognized in the embracing works of Pian and Tong [3] and Brezzi and Fortin [4].The formulation used here was identi"ed later as the stress model of the hybrid-Tre!tz "nite element formulation and applied to di!erent two-dimensional structural analysis problems [5}8]. It is, therefore, directly related with the displacement frame Tre!tz element developed by Jirousek and Stein and their co-workers [9, 10]. The theoretical basis of the formulation is established in [11], where a comparative analysis with the related T-elements is also presented.In the terminology adopted here, a "nite element derived from the direct approximation of the stress "eld is termed a stress "nite element model. The stress "eld is directly related with two fundamental conditions, namely the equilibrium condition in the domain of the element and the di!usivity (Neumann) condition on the associated boundary tractions (Cauchy stresses). Three classes of formulations can be established depending on the equilibrium constraints enforced a priori on the stress basis, namely the hybrid-mixed, the hybrid and the hybrid-Tre!tz formulations of the "nite element stress model [12,13].The hybrid-mixed formulation is the most general. No constraints (besides linear independence and completeness) are placed on the stress approximation basis. Besides the stresses, the displacements are also approximated, and independently, in the domain and on the boundary of the element. These bases are used to enforce on average the equilibrium and the di!usivity conditions, respectively. The hybrid formulation is obtained by constraining a priori the stress approximation to satisfy locally the equilibrium condition. Consequently, the displacement approximation in the domain of the element becomes redundant an...

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Section: F"mentioning

confidence: 99%

“…The sixty generating functions are used to compute the elementary #exibility matrix (28), which is then processed to identify and remove from the approximation basis the linearly dependent terms. The results obtained are shown in Table II.…”

Section: Approximation Basesmentioning

confidence: 99%

Three-dimensional hybrid-Trefftz stress elements

Freitas

Bussamra

2000

Int. J. Numer. Meth. Engng.

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“…To assign approximation (30) to the individual ÿnite elements, it su ces to ensure that the di usivity condition is satisÿed locally along the ÿnite element interfaces, as it is stated below, where m and n identify the side numbers of two ÿnite elements connecting on boundary element p:…”

Section: Boundary Approximation Basesmentioning

confidence: 99%

“…The results above are used to compute matrix T c in the gradient expression (34) of the traction approximation (30).…”

Section: Appendix C: Boundary Approximationmentioning

confidence: 99%

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Shape optimization with hybrid‐Trefftz displacement elements

Freitas

Cismaşiu

2001

Numerical Meth Engineering

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SUMMARYThe displacement model of the hybrid-Tre tz ÿnite element formulation is applied to the shape optimization of linear elastostatic problems. The model is based on the direct approximation of the displacements in the domain of the element and of the tractions on its boundary. The independent description of the geometry is based on the parametric description of the shape of the element sides. The ÿnite element solving system is symmetric and sparse. As the displacement approximation is extracted from the solution set of the governing di erential equations, all structural matrices are deÿned by boundary integral expressions. The approximation bases are compact and naturally p-hierarchical. High-order approximations are implemented to obtain accurate solutions using coarse meshes of macro-elements. The sensitivities are determined analytically and remeshing is not implemented during the optimization procedure because the elements are weakly sensitive to gross shape distortion. Numerical testing is based on a standard set of two-dimensional linear elastostatic problems.

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Elastoplastic dynamic analysis with hybrid stress elements

Freitas¹,

Wang²

2001

Numerical Meth Engineering

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SUMMARYThe stress model of the hybrid ÿnite element formulation is applied to the solution of dynamic elastoplastic structural problems. The stress ÿeld is approximated in the domain of the elements and the displacements on its boundary. The displacement, velocity and acceleration approximations in the domain of the element are implicit, in the sense that they result from a combination of the stress estimate with the time integration procedure that ensures that the equilibrium condition is locally satisÿed. The ÿnite elements are subdivided in plastic cells where a gradient dependent model is implemented using a hybrid formulation based on the approximation of the plastic parameter and the plastic radiation ÿelds in the domain and on the boundary of the plastic cells, respectively. Generalized variables associated with orthogonal and naturally hierarchical bases are used. The resulting solving systems are symmetric, sparse, p-adaptive and well suited to parallel processing. The performance of the element is assessed using a representative set of testing problems.

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Parallel Solution Techniques for Hybrid Mixed Finite Element Models

Cited by 5 publications

References 0 publications

Three-dimensional hybrid-Trefftz stress elements

Three-dimensional hybrid-Trefftz stress elements

Shape optimization with hybrid‐Trefftz displacement elements

Elastoplastic dynamic analysis with hybrid stress elements

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