Michel Bercovier | Searcharxiv

arXiv2015

Overlapping non Matching Meshes Domain Decomposition Method in Isogeometric Analysis

One of the important aspects of IsoGeometric Analysis (IGA) is the strong link between Computer Aided Design and analysis. Two of IGA'a major challenge are the assembly of patches (Constructive Solid Geometry geometries made of Boolean primitives) and local refinement. In the present work we apply the Additive Schwarz Domain Decomposition on overlapping domains. To leverage the power of IGA we consider non matching meshes to solve PDEs on CSG assemblies. As a side effect also we deal with local refinement . As a first step we study a collection of simple two domains problems, using the software package GeoPDEs, all of them converged rapidly, even on very distorted domains. For local refinement we implement a Chimera type of zooming together with ASDDM iterations, and here again we get very attractive results. A last effect is that the additive methods are naturally parallelizable and thus can extend to large 3D problems. We give some examples of computations on multi-patch bodies. This new implementation of the DD methods brings many interesting problems such as: the definition of boundary conditions on trimmed patches, choice of blending operators on the intersections , optimization of the overlap size, choice of preconditioning to guarantee convergence ( as we do not have a maximum principle here.)

math.NA↗

arXiv2014

Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes

We solve the following problem: given a polynomial of order $n$ and the corresponding $B\'ezier$ tensor product patches over an unstructured regular quadrilateral mesh of any valence, find a solution to the $G^{1}1$ or $C^{1}1$ approximation (resp. interpolation) problem ! Constraints defining regularity conditions across patches have to be satisfied. The resulting number of free degrees of freedom must be such that for instance the interpolation problem has a solution. This is similar to studying the minimal determining set (MDS) for a $C^{1}$ continuity construction. The givenunstructured quadrilateral mesh can include a cubic boundary curve. The final surface approximation or PDE solution is obtained by energy methods. We completely solve the problem and show that there is always a solution for $n\ge 5$ and under some mesh restrictions for $n=4$. From a practical point of view, the present paper provides a way to build first order smooth interpolation/approximation and solutions to partial differential equations for arbitrary structures of quadrilateral meshes.

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