Piecewise polynomial shape functions for hp-finite element methods

Baitsch, Matthias; Hartmann, Dietrich

doi:10.1016/j.cma.2008.05.016

Cited by 11 publications

(6 citation statements)

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“…Shape functions for two‐dimensional 1‐irregular meshes are proposed by Gupta in 18, and for three dimensions by Morton et al 19. For general k ‐irregular meshes, Legendre‐type shape functions are proposed by Baitsch and Hartmann in 20 and Karniadakis and Sherwin 39. An approach by Cho et al 21, 22 constructs shape functions for non‐matching interface discretization which may also be understood as hanging nodes.…”

Section: Hanging Nodes With Dofsmentioning

confidence: 99%

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Hanging nodes and XFEM

Fries¹,

Byfut

Alizada³

et al. 2010

Numerical Meth Engineering

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SUMMARYThis paper investigates two approaches for the handling of hanging nodes in the framework of extended finite element methods (XFEM). Allowing for hanging nodes, locally refined meshes may be easily generated to improve the resolution of general, i.e. model-independent, steep gradients in the problem under consideration. Hence, a combination of these meshes with XFEM facilitates an appropriate modeling of jumps and kinks within elements that interact with steep gradients. Examples for such an interaction are, e.g. found in stress fields near crack fronts or in boundary layers near internal interfaces between two fluids. The two approaches for XFEM based on locally refined meshes with hanging nodes basically differ in whether (enriched) degrees of freedom are associated with the hanging nodes. Both approaches are applied to problems in linear elasticity and incompressible flows.

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Section: Hanging Nodes With Dofsmentioning

confidence: 99%

“…Other approaches for obtaining conforming shape functions for hanging nodes are e.g. proposed in Baitsch and Hartmann 20 and Cho et al 21, 22. To the best of our knowledge, XFEM has so far not been used in the context of this approach where DOFs are present at the hanging nodes.…”

Section: Introductionmentioning

confidence: 99%

Hanging nodes and XFEM

Fries¹,

Byfut

Alizada³

et al. 2010

Numerical Meth Engineering

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“…For the sake of completeness, other approaches that can be used for local mesh refinement in conjunction with quadrilateral/hexahedral finite elements are briefly introduced in the following. Baitsch & Hartmann also developed a transition element based on piecewise polynomial shape functions [53]. This approach requires subdividing the refined element into a number of quadrilateral subdomains and consequently leads to a comparably large number of internal degrees of freedom within each macroelement.…”

Section: Introductionmentioning

confidence: 99%

High order transition elements: The xy-element concept—Part I: Statics

Duczek

Saputra

Gravenkamp

2020

Computer Methods in Applied Mechanics and Engineering

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Advanced transition elements are of utmost importance in many applications of the finite element method (FEM) where a local mesh refinement is required. Considering problems that exhibit singularities in the solution, an adaptive hp-refinement procedure must be applied. Even today, this is a very demanding task especially if only quadrilateral/hexahedral elements are deployed and consequently the hanging nodes problem is encountered. These element types, are, however, favored in computational mechanics due to the improved accuracy compared to triangular/tetrahedral elements. Therefore, we propose a compatible transition element -xNy-element -which provides the capability of coupling different element types. The adjacent elements can exhibit different element sizes, shape function types, and polynomial orders. Thus, it is possible to combine independently refined h-and p-meshes. The approach is based on the transfinite mapping concept and constitutes an extension/generalization of the pNh-element concept. By means of several numerical examples, the convergence behavior is investigated in detail, and the asymptotic rates of convergence are determined numerically. Overall, it is found that the proposed approach provides very promising results for local mesh refinement procedures.

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“…The third approach consists of weakly imposing the conformity by means of penalty method, Lagrange multipliers, or Nitsche method. The fourth approach is to construct conforming shape functions for elements with hanging nodes and to introduce independent DOFs for the hanging nodes [72][73][74].Such local remeshing schemes have been integrated into the XFEM framework. By constrained approximation, Unger et al [10] developed an adaptive crack growth algorithm using XFEM; Fries et al [11] investigated two approaches, confirming shape functions and constrained approximation, for the handling of hanging nodes in the framework of XFEM.…”

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confidence: 99%

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confidence: 99%

A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

Ling

et al. 2014

Numerical Meth Engineering

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This paper presents a FEM with mesh-separation-based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution-dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi-analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack-tip adaptive mesh refinement strategy to improve the accuracy of crack-tip fields. Both the discontinuities due to intra-element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. due to intra-element cracking and the incompatibility caused by those hanging nodes due to local element refinement at the crack-tip at the elemental level and (ii) a unified FEM framework with some key advantages of other numerical methods such as the Hansbo and Hansbo method, the meshfree methods, the semi-analytical FEM, and the smoothed FEM (SFEM) while retaining the key feature of the standard FEM such as elemental locality.Over the last several decades, there have been rapid developments in advanced numerical methods to cope with the two major challenges mentioned earlier. To broadly categorize, major advances have been made on two dynamic fronts: (i) various meshfree (or meshless) methods and (ii) advanced FEM that can accurately and efficiently account for arbitrary discontinuities. The meshfree methods were developed with the goal of eliminating some difficulties associated with the reliance on a mesh to construct the approximation, and it can be traced back to the earlier work of Lucy [12] and Gingold and Monaghan [13]. The essence of meshfree methods is direct integration of point-wise force interactions among neighboring material points, which can effectively model the arbitrary crack formation and multiple crack interactions in solids. Numerical development along this line has been dynamic ever since the seminal work of Belytschko on the so-called elementfree Galerkin method [14] and the enriched element-free Galerkin of Fleming et al. [15]. Some of the recent fruitful advances are due to Rabczuk and colleagues (e.g., [16][17][18][19]) using extrinsically enriched methods based on the partition of unity (PoU) concept [20,21] and Duflot [22] and Barbieri et al.[23] using weight function enrichment. Recent successful ...

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Piecewise polynomial shape functions for hp-finite element methods

Cited by 11 publications

References 11 publications

Hanging nodes and XFEM

Hanging nodes and XFEM

High order transition elements: The xy-element concept—Part I: Statics

A finite element method with mesh‐separation‐based approximation technique and its application in modeling crack propagation with adaptive mesh refinement

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