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The present paper studies adaptive refinement on multi-patch domains in isogeometric analysis. In particular, we investigate the gluing construction for adaptively refined spline spaces to obtain discretizations that are C 0 smooth across interfaces. We will see that this is closely related to the concept of boundary compatibility of an adaptive spline construction. Given a spline basis (or, more generally, a generating system if linear independence is not guaranteed) on a d -dimensional box domain, there are two possibilities for constructing the spline basis on the domain boundary. Firstly, one can simply restrict the basis functions to the boundary. Secondly, one may restrict the underlying mesh to the boundary and construct the spline basis on the resulting mesh. The two constructions do not necessarily produce the same set of functions. If they do, then the spline bases are said to be compatible . We study this property for hierarchical (HB-) and truncated hierarchical B-splines (THB-splines) and identify sufficient conditions. These conditions are weaker for THB- than for HB-splines. Finally we demonstrate the importance of boundary compatibility for geometric modeling and for adaptive refinement in isogeometric analysis, in particular when considering multi-patch domains. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: Finite Element Tearing and Interconnecting (FETI) methods are a powerful approach to designing solvers for large-scale problems in computational mechanics. The numerical simulation problem is subdivided into a number of independent sub-problems, which are then coupled in appropriate ways. NURBS- (Non-Uniform Rational B-spline) based isogeometric analysis (IGA) applied to complex geometries requires to represent the computational domain as a collection of several NURBS geometries. Since there is a natural decomposition of the computational domain into several subdomains, NURBS-based IGA is particularly well suited for using FETI methods. This paper proposes the new IsogEometric Tearing and Interconnecting (IETI) method, which combines the advanced solver design of FETI with the exact geometry representation of IGA. We describe the IETI framework for two classes of simple model problems (Poisson and linearized elasticity) and discuss the coupling of the subdomains along interfaces (both for matching interfaces and for interfaces with T-joints, i.e. hanging nodes). Special attention is paid to the construction of a suitable preconditioner for the iterative linear solver used for the interface problem. We report several computational experiments to demonstrate the performance of the proposed IETI method. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]