High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation

Legrain, Grégory; Chevaugeon, Nicolas; Dréau, Kristell

doi:10.1016/j.cma.2012.06.001

Cited by 75 publications

(72 citation statements)

References 49 publications

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“…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. Alternatively, the use of the so-called Nurbs-Enhanced X-FEM [10] allows to consider the exact geometrical representation independently of the approximation mesh, see Figure 5 for a comparison of both http://www.amses-journal.com/content/2/1/13 approaches.…”

Section: The Extended Finite Element Methodsmentioning

confidence: 99%

“…All the numerical examples are conducted on a geometrical domain which consists in the bounding-box of the physical domain shown in Figure 18, and the X-FEM is used thanks to the definition of the geometry in terms of a level-set function. A sub-grid level-set [6,7] approach is used in order to be able to represent the geometry accurately on coarse Figure 18 Blunt re-entrant corner convergence problem. http://www.amses-journal.com/content/2/1/13 computational meshes.…”

Section: D Numerical Examplesmentioning

confidence: 99%

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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%

Section: D Numerical Examplesmentioning

confidence: 99%

Treatment of nearly-singular problems with the X-FEM

Legrain

Moës

2014

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

View full text Add to dashboard Cite

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“…A large number of researchers have investigated a variety of concepts do not require the generation of a conforming mesh and modelling geometrical features independently of the finite element mesh used for analysis. These concepts [1], [2], [3], [4] differ from each other on the following points :…”

Section: Introductionmentioning

confidence: 99%

“…Techniques to represent boundaries : Explicit surface representations [4], Level set representation [1], [3], parametric function to Level set representation [1], Medical image modalities [2], [3].…”

Section: Introductionmentioning

confidence: 99%

Analysis using higher-order XFEM: implicit representation of geometrical features from a given parametric representation

Moumnassi

Bordas

Figueredo

et al. 2014

Mechanics & Industry

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Résumé :Nous présentons une approche prometteuse afin de réduire les difficultés liées aux maillages de géométries avec frontières courbes pour l'analyse avec deséléments finis d'ordre supérieur. Une analyse par XFEM d'ordre supérieur dans le cas de la modélisation des interfaces matériau-vide est testée sur un ensemble représentatif de problèmes d'élasticité linéaire. Les frontières implicites courbes sont approximéesà l'intérieur d'un maillage grossier non structuré en utilisant les informations paramétriques extraites de la représentation paramétrique (la plus populaire en conception CAO). Cette approximation génère un sous-maillage gradué (SMG)à l'intérieur deséléments traversés par la frontière qui sera utiliséà des fins d'intégrations numérique. Exemples de géométries et des expériences numériques illustrent la précision et la robustesse de l'approche proposée. Abstract :We present a promising approach to reduce the difficulties associated with meshing complex curved domain boundaries for higher-order finite elements. In this work, higher-order XFEM analyses for strong discontinuity in the case of linear elasticity problems are presented. Curved implicit boundaries are approximated inside an unstructured coarse mesh by using parametric information extracted from the parametric representation (the most common in Computer Aided Design CAD). This approximation provides local graded sub-mesh (GSM) inside boundary elements (i.e. an element split by the curved boundary) which will be used for integration purpose. Sample geometries and numerical experiments illustrate the accuracy and robustness of the proposed approach.

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“…The LSM has also been used in [36] to represent the geometry in non-conforming meshes. Other authors also combine the LSM for boundary representation with the XFEM to represent the solution gradient discontinuities into an element containing more than one material [37,38] Since the mesh is not conforming with the geometry, these methods require the information of the problem domain to be available during the evaluation of element integrals. The accuracy of the results provided by these techniques depends on the accuracy of the integration process.…”

Section: Introduction and Review Motivationmentioning

confidence: 99%

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

Soriano¹

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High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation

Cited by 75 publications

References 49 publications

Treatment of nearly-singular problems with the X-FEM

Treatment of nearly-singular problems with the X-FEM

Analysis using higher-order XFEM: implicit representation of geometrical features from a given parametric representation

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

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