Mixed finite element approximations based on 3‐D h p ‐adaptive curved meshes with two types of H (div)‐conforming spaces

Computer Methods in Applied Mechanics and Engineering

Durán²,

Gomes³

et al. 2019

Self Cite

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H 1 (Ω), H(curl, Ω), H(div, Ω), and L 2 (Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L 2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.

Section: Approximations With Enriched Space Configurationsmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

High-order composite finite element exact sequences based on tetrahedral–hexahedral–prismatic–pyramidal partitions

Computer Methods in Applied Mechanics and Engineering

Durán²,

Gomes³

et al. 2019

Self Cite

“…However, the price to be paid for extra accuracy in mixed (and also hybrid) methods is higher computational cost for matrix assembly, as shown by the comparison study in Reference . Nevertheless, the effect of combining static condensation and parallelism may reduce CPU times for the mixed methods, as demonstrated in Reference .…”

Section: Discussionmentioning

confidence: 99%

“…We also refer to Reference , for the construction of such stable approximation spaces, and for their application to mixed methods for curvilinear 2D meshes and on manifolds. In Reference , the treatment of 3D meshes is considered based on tetrahedral, affine hexahedral, and affine prismatic elements, and Reference is dedicated to their assembly for 3D curved and hp‐adapted meshes. The methodology used for such constructions is based on appropriate choice of constant vector fields

\hat{v}

, based on the geometry of each master element, which are multiplied by an available set of H 1 hierarchical scalar basis functions

\hat{φ}

to obtain

\hat{Φ} = \hat{v} \hat{φ}

.…”

Section: Notation and General Aspectsmentioning

confidence: 99%

H(div) finite elements based on nonaffine meshes for 3D mixed formulations of flow problems with arbitrary high order accuracy of the divergence of the flux

Numerical Meth Engineering

Durán

Farias

et al. 2020

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Summary Effects of nonaffine elements on the accuracy of 3D H(div)‐conforming finite elements are studied. Instead of convergence order k+1 for the flux and the divergence of the flux obtained with Raviart‐Thomas or Nédélec spaces with normal traces of degree k, based on affine hexahedra or triangular prisms, reduced orders k for the flux and k−1 for the divergence of the flux may occur for distorted elements. To improve this scenario, a hierarchy of enriched flux approximations is considered, by adding internal shape functions up to a higher degree k+n, n>0, while keeping the original normal traces of degree k. The resulting enriched approximations, using multilinear transformations, keep the original flux accuracy (of order k+1 with affine elements or reduced order k otherwise), but enhanced divergence (of order k+n+1, in the affine case, or k+n−1 otherwise) can be reached. The reduced flux accuracy due to quadrilateral face distortions cannot be corrected by including higher order internal functions. The enriched spaces are applied to the mixed finite element formulation of Darcy's model. The computational cost of matrix assembly increases with n, but the condensed system to be solved has the same dimension and structure as the original scheme.

“…As proposed in [11,13], there are other enriched stable spaces {V + , P + } for the Poisson problem that can be obtained by adding to V some appropriate bubble functions to form V + , keeping unchanged the original edge vector functions. Some examples shall be considered in Section 5, as well as their corresponding enriched versions, which are used in the current study.…”

Section: Introductionmentioning

confidence: 99%

Enriched two dimensional mixed finite element models for linear elasticity with weak stress symmetry

Computers & Mathematics with Applications

Gomes

Quinelato

et al. 2020

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The purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry. These approximations are based on the application of enriched versions of classic Poisson-compatible spaces, for stress and displacement variables, and/or on enriched Stokes-compatible space configurations, for the choice of rotation spaces used to weakly enforce stress symmetry. Accordingly, the stress space has to be adapted to restore stability. Such enrichment procedures are done via space increments with extra bubble functions, which have their support on a single element (in the case of H 1-conforming approximations) or with vanishing normal components over element edges (in the case of H(div)-conforming spaces). The advantage of using bubbles as stabilization corrections relies on the fact that all extra degrees of freedom can be condensed, in a way that the number of equations to be solved and the matrix structure are not affected. Enhanced approximations are observed when using the resulting enriched space configurations, which may have different orders of accuracy for the different variables. A general error analysis is derived in order to identify the contribution of each kind of bubble increment on the accuracy of the variables, individually. The use of enriched Poisson spaces improves the rates of convergence of stress divergence and displacement variables. Stokes enhancement by bubbles contributes to equilibrate the accuracy of weak stress symmetry enforcement with the stress approximation order, reaching the maximum rate given by the normal traces (which are not affected).