Séance Séminaire

Séminaire ACSIOM

Domain Decomposition and non matching meshes in Isogeometric Analysis

Isogeometric analysis (IGA) is a rapidly developing paradigm for the discretization of Partial Differential Equations (PDEs). The basic idea detailed i consists in defining the same global isoparametric transformation for the ?exact? computational domain using B-Splines or NURBS and for the approximation functions for the PDEs solution. One of the aims is to avoid the costly steps of mesh generation and CAD interchange. The IGA paradigm gives rise to a rich interchange between Computer Aided Geometry for Design , Computational Geometry and Numerical Modelling based on PDEs. Geometric methods can lead to new algorithms in unilateral (contact) methods,to multi grid subdivision solvers and more. In this talk we will review briefly the IGA paradigm and detail some the above mentioned interactions , such as CSG free from volumes construction and solution of IGA problems by Domain Decomposition methods based on such constructions, medial axis and contact problems, subdivision and multi grid methods. Hence based on CSG trees we can apply, for the solution of large PDEs problems on such domains, the simplest Schwarz Additive Domain Decomposition Method (SADDM). For a simple example consider that $\Omega_i, i=1,...,n$ , are overlapping (that is : there is always a pair $(i,j)$ such that $\Omega_i \cap \Omega_j $ has a non void interior) and that the respective isoparametric transformations are \emph{non matching}: the pair of reference grid and knots defining each physical domain are not related. The respective inverse isogeometric mapping from $\Omega_i $ to $\hat{\Omega}_i $ defines in $\hat{\Omega}_j $ a \textit{trimming line} ( resp. a \textit{trimming surface} in 3D), and the corresponding (partial) boundary $\Gamma_{j,i}$ of $\Omega_j $. In SADDM we need to compute at each iteration the trace of the solution $u_i$ obtained in the sub-domain $\Omega_i$, on the boundary $\Gamma_{j,i}$ of the sub-domain $\Omega_j $. It gives rise to a non homogeneous boundary condition (BC) problem on each sub domain. Application of this condition in IGA is not straightforward. We analyse different approaches such as approximation, least square and others, and compare them. We give several examples illustrating the power of this approach: direct use of CGS primitives, local zooming instead of refinements, and parallelization for large problems. We show that there is no degradation of the powerful approximation properties of IGA when using non matching meshes. http://www.cs.huji.ac.il/~berco/