On the use of the extended finite element method with quadtree/octree meshes

Legrain, Grégory; Allais, Raphaël; Cartraud, Patrice

doi:10.1002/nme.3070

Cited by 50 publications

(41 citation statements)

References 51 publications

(77 reference statements)

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“…A level-set representation of the geometry is typically used: in this case, the level-set is interpolated on the approximation mesh. This couples the geometrical representation to the approximation [20], and prevents the use of higher order approximations due to an insufficient geometrical accuracy. A so-called sub-grid level-set approach has been proposed in [6,7] in order to uncouple geometry and approximation, and thus allow the use of high-order approximations.…”

Section: The Extended Finite Element Methodsmentioning

confidence: 99%

Treatment of nearly-singular problems with the X-FEM

Legrain

Moës

2014

Adv. Model. and Simul. in Eng. Sci.

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Background: In recent years, lot of research have been conducted on fictitious domain approaches in order to simplify the meshing process for computed aided analysis. The behaviour of such non-conforming methods is studied in the case of the approximation of nearly singular solutions. Such solutions appear when problems involve singularities whose center are located outside (but close) of the domain of interest. These solutions are common in industrial structures that usually involve rounded re-entrant corners. Methods: The performance of the finite element method is evaluated in this context by means of a simple unidimensional example. Both numerical and theoretical estimates are considered in order to assess the behaviour of the numerical approximation. It is demonstrated that despite being regular, the convergence of the approximation can be bounded to an algebraic rate that depends on the solution. Reasons for such behaviour are presented, and two complementary strategies are proposed in order to recover optimal convergence rates. The first strategy is based on a proper enrichment of the approximation thanks to the X-FEM, while the second is based on a proper mesh design that follows a geometric progression. Finally, the proposed strategies are extended and validated in 2D. Results: The performance of the two strategies is highlighted for both 1D and 2D examples. Both methods allow to recover proper convergence rates (optimal algebraic rate for h-convergence, exponential for p-convergence) in 1D and 2D. Conclusions: The proposed strategies allow for a very accurate solution for such solutions. The enrichment strategy is valid for both h and p refinement, whereas the mesh-design strategy is only usable for p refinement. However, such enrichment functions can be tedious to derive.

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Section: The Extended Finite Element Methodsmentioning

confidence: 99%

Treatment of nearly-singular problems with the X-FEM

Legrain

Moës

2014

Adv. Model. and Simul. in Eng. Sci.

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“…A balanced quadtree decomposition() is adopted to discretise the images. Quadtree meshes avoid the generation of a large number of elements due to their ability to rapidly transition between geometrical features of different scales() compared with pixel‐ or voxel‐based approaches, eg, the works of Hollister and Kikuchi and Huang and Li based on the FEM. The mesh generation process is fully automatic and significantly reduces the time and human effort required to convert an image into a mesh.…”

Section: Numerical Examplesmentioning

confidence: 99%

A scaled boundary finite element formulation for poroelasticity

Ooi

Song

Natarajan

2018

Numerical Meth Engineering

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Summary This paper develops the scaled boundary finite element formulation for applications in coupled field problems, in particular, to poroelasticity. The salient feature of this formulation is that it can be applied over arbitrary polygons and/or quadtree decomposition, which is widely employed to traverse between small and large scales. Moreover, the formulation can treat singularities of any order. Within this framework, 2 sets of semianalytical, scaled boundary shape functions are used to interpolate the displacement and the pore fluid pressure. These shape functions are obtained from the solution of vector and scalar Laplacian, respectively, which are then used to discretise the unknown field variables similar to that of the finite element method. The resulting system of equations are similar in form as that obtained using standard procedures such as the finite element method and, hence, solved using the standard procedures. The formulation is validated using several numerical benchmarks to demonstrate its accuracy and convergence properties.

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“…The formalism presented above for the weak incorporation of displacement boundary conditions can be generalised to interface problems, see also [2,3,13]. For simplicity, let the reference domain be composed of two subdomains, = 1 ∪ 2 , and let us ignore the Dirichlet boundary conditions on ∂ .…”

Section: Materials Interfacesmentioning

confidence: 99%

“…Moreover, the use Langrange multipliers together with its numerical intricacies, such as the fulfilment of the LBB-condition [8], are avoided. For alternative approaches, see [4,[9][10][11][12][13][14] among others.…”

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confidence: 99%

An unstructured immersed finite element method for nonlinear solid mechanics

Rüberg

Cirak

Aznar

2016

Adv. Model. and Simul. in Eng. Sci.

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We present an immersed finite element technique for boundary-value and interface problems from nonlinear solid mechanics. Its key features are the implicit representation of domain boundaries and interfaces, the use of Nitsche's method for the incorporation of boundary conditions, accurate numerical integration based on marching tetrahedrons and cut-element stabilisation by means of extrapolation. For discretisation structured and unstructured background meshes with Lagrange basis functions are considered. We show numerically and analytically that the introduced cut-element stabilisation technique provides an effective bound on the size of the Nitsche parameters and, in turn, leads to well-conditioned system matrices. In addition, we introduce a novel approach for representing and analysing geometries with sharp features (edges and corners) using an implicit geometry representation. This allows the computation of typical engineering parts composed of solid primitives without the need of boundary-fitted meshes.

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On the use of the extended finite element method with quadtree/octree meshes

Cited by 50 publications

References 51 publications

Treatment of nearly-singular problems with the X-FEM

Treatment of nearly-singular problems with the X-FEM

A scaled boundary finite element formulation for poroelasticity

An unstructured immersed finite element method for nonlinear solid mechanics

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