When: Friday January 23, 2015

Where: RICAM, Austrian Academy of Sciences, Linz, Austria. Science Park 2, room S2 416

What: This mini-workshop is intended to elicit feedback and foster collaboration on topics related to isogeometric analysis. Your active participation in talks and discussions is foreseen. If you are interested in contributing a talk please send an e-mail to Angelos.Mantzaflaris@oeaw.ac.at.

Time for discussions is included in all slots.

The IGA paradigm for the discretisation of Partial Differential Equations (PDEs) leads to a rich interchange between Computer Aided Geometry for Design , Computational Geometry and Numerical Modelling based on PDEs. In this talk we will review briefly the IGA paradigm and detail some of the interactions. More specifically many objects are defined as CSG free form volumes constructs. Hence it is natural to consider Domain Decomposition methods as candidate solvers for ‘real life’ domains. Each atom of the CSG construct can be considered as a simple domain, where IGA is directly implemented. Thus based on CSG trees we can apply, for the solution of large PDEs problems on the global domain, the simplest Schwarz Additive Domain Decomposition Method (SADDM). We suppose that our primitive patches Ωi, i = 1, …, n, are overlapping ( i.e. there is always a pair (i, j) such Ωi ∩ Ωj has a non void interior) and that the respective isoparametric transformations are NON-MATCHING: the reference grid and knots defining each physical domain are not related. 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.

In this talk, discontinuous Galerkin Isogeoemetric Analysis (dG IgA) approximations of elliptic problems with low regularity will be presented. The low regularity comes from the appearance of rough diffusion coefficients of the model problem or from the non-smooth boundary parts of the computational domain. In the proposed method, the domain is subdivided in a union of subdomains T_H(Ω) := {Ωi, i=1..n}, which is compatible with the characteristics of the problem (discontinuities of the diffusion coefficients, location of the boundary singularities e.t.c.). The discrete problem is formulated using dG techniques on the interfaces of the subdomains, that is symmetric numerical fluxes. Initially, the properties of the proposed method will be discussed and error estimates will be shown for the case of having smooth boundary. In the last part of the talk, graded meshes will be constructed in the vicinities of the singular boundary parts in order to recover the optimal convergence rate of the method.

We study the linear space of C^s-smooth isogeometric functions defined on a multi-patch domain Ω of R^2. The C^s-smoothness of these isogeometric functions is found to be equivalent to geometric smoothness of the same order (G^s-smoothness) of their graph surfaces, which motivates us to call them C^s-smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis for multi-patch domains and explain a detailed construction for bicubic and biquartic C^1-smooth isogeometric functions for a particular class of two-patch domains. The resulting functions are used to perform L^2 approximation and to solve the Poisson's equation and the biharmonic equation on two-patch geometries. The numerical results indicate optimal rates of convergence.

New research in PDE-constrained optimization with limited observation suggests that using discretization with higher regularity is needed to get parameter and mesh size independent stability conditions. A robust preconditioner (w.r.t. parameter and mesh size) is also given in this setting. The main idea is that the control space and the Lagrange-multiplier space are the same. Suitable finite elements with high regularity is difficult to find. In this setting, splines and IGA might have a superior advantages compared to classical FEM.