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International audience; In isogeometric analysis (IGA) the domain of interest is usually represented by B-spline or NURBS patches, as they are present in standard CAD models. Complex domains can often be represented as a union of simple overlapping subdomains, parameterized by (tensor-product) spline patches. Numerical simulation on such overlapping multi-patch domains is a serious challenge in IGA. To obtain non-overlapping subdomains one would usually reparameterize the domain or trim some of the patches. Alternatively, one may use methods that can handle overlapping subdomains. In this paper, we propose a non-iterative, robust and efficient method defined directly on overlapping multi-patch domains. Consequently, the problem is divided into several sub-problems, which are coupled in an appropriate way. The resulting system can be solved directly in a single step. We compare the proposed method with iterative Schwarz domain decomposition approaches and observe that our method reduces the computational cost significantly, especially when handling subdomains with small overlaps. Summing up, our method significantly simplifies the domain parameterization problem, since we can represent any domain of interest as a union of overlapping patches without the need to introduce trimming curves/surfaces. The performance of the proposed method is demonstrated by several numerical experiments for the Poisson problem and linear elasticity in two and three dimensions.

We present functional-type a posteriori error estimates in isogeometric analysis. These estimates, derived on functional grounds, provide guaranteed and sharp upper bounds of the exact error in the energy norm. Moreover, since these estimates do not contain any unknown/generic constants, they are fully computable, and thus provide quantitative information on the error. By exploiting the properties of non-uniform rational B-splines, we present efficient computation of these error estimates. The numerical realization and the quality of the computed error distribution are addressed. The potential and the limitations of the proposed approach are illustrated using several computational examples.

International audience; Local refinement with hierarchical B-spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis. By exploiting a multilevel control structure, we show that truncated hierarchical B-spline (THB-spline) representations support interactive modeling tools, while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis. A selection of illustrative 2D and 3D numerical examples demonstrates the potential of the hierarchical framework.

While isogeometric analysis has the potential to close the gap between computer aided design and finite element methods, the underlying structure of NURBS (non-uniform rational B -splines) is a weakness when it comes to local refinement. We propose a hybrid method that combines a globally C 1 -continuous, piecewise polynomial finite element basis with rational NURBS-mappings in such a way that an isoparametric setting and exact geometry representation are preserved. We define this basis over T -meshes with a hierarchical structure that allows locally restricted refinement. Combined with a state-of-the-art a posteriori error estimator, we present an adaptive refinement procedure. This concept is successfully demonstrated with the Laplace equation, advection–diffusion problems and linear elasticity problems.

We present two-sided a posteriori error estimates for isogeometric discretization of elliptic problems. These estimates, derived on functional grounds, provide robust, guaranteed and sharp two-sided bounds of the exact error in the energy norm. Moreover, since these estimates do not contain any unknown/generic constants, they are fully computable, and thus provide quantitative information on the error. The numerical realization and the quality of the computed error distribution are addressed. The potential of the proposed estimates are illustrated using several computational examples.