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The finite cell method for polygonal meshes: poly-FCM. (English) Zbl 1398.65296

Summary: In the current article, we extend the two-dimensional version of the finite cell method (FCM), which has so far only been used for structured quadrilateral meshes, to unstructured polygonal discretizations. Therefore, the adaptive quadtree-based numerical integration technique is reformulated and the notion of generalized barycentric coordinates is introduced. We show that the resulting polygonal (poly-)FCM approach retains the optimal rates of convergence if and only if the geometry of the structure is adequately resolved. The main advantage of the proposed method is that it inherits the ability of polygonal finite elements for local mesh refinement and for the construction of transition elements (e.g. conforming quadtree meshes without hanging nodes). These properties along with the performance of the poly-FCM are illustrated by means of several benchmark problems for both static and dynamic cases.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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