Thomas Takacs

Analysis-suitable G1 multi-patch parametrizations for C1 isogeometric spaces

One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from p-degree splines (and extensions, such as NURBS), they enjoy up to C p - 1 continuity within each patch. However, global continuity beyond C 0 on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only C 0 at the patch interface. On such domains we study the h-refinement of C 1 -continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the C 1 -continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recently (Kapl et al., 2015b) has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) C 1 splines. This is the starting point of our study. We introduce the class of analysis-suitable G 1 geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of C 1 isogeometric spaces over analysis-suitable G 1 parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of C 1 isogeometric spaces is prevented. We study h-refinement for C 1 continuous isogeometric spaces over multi-patch domains.We introduce analysis-suitable G 1 (AS G 1 ) geometry parametrizations.AS G 1 parametrizations allow optimal approximation properties.For non-AS G 1 geometries the solution may be locked and convergence is prevented.

H 2 regularity properties of singular parameterizations in isogeometric analysis

Graphical abstract Highlights ► We consider the isogeometric method for singularly parameterized domains. ► In this case the underlying function space may not be sufficiently regular. ► We especially focus on H2 regularity for 1-, 2- and 3-dimensional domains. ► We introduce a modification scheme for the test function space to regain regularity.

Approximation error estimates and inverse inequalities for B-splines of maximum smoothness

In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard Sobolev norms and semi-norms. The presented approximation error estimates do not depend on the polynomial degree of the splines but only on the grid size. We will see that the approximation lives in a subspace of the classical B-spline space. We show that for this subspace, there is an inverse inequality which is also independent of the polynomial degree. As the approximation error estimate and the inverse inequality show complementary behavior, the results shown in this paper can be used to construct fast iterative methods for solving problems arising from isogeometric discretizations of partial differential equations.

Dimension and basis construction for analysis-suitable G 1 two-patch parameterizations

Abstract We study the dimension and construct a basis for C 1 -smooth isogeometric function spaces over two-patch domains. In this context, an isogeometric function is a function defined on a B-spline domain, whose graph surface also has a B-spline representation. We consider constructions along one interface between two patches. We restrict ourselves to a special case of planar B-spline patches of bidegree ( p , p ) with p ≥ 3 , so-called analysis-suitable G 1 geometries, which are derived from a specific geometric continuity condition. This class of two-patch geometries is exactly the one which allows, under certain additional assumptions, C 1 isogeometric spaces with optimal approximation properties (cf. Collin et al., 2016 ). Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation, using the isogeometric method. In particular, we analyze the dimension of the C 1 -smooth isogeometric space and present an explicit representation for a basis of this space. Both the dimension of the space and the basis functions along the common interface depend on the considered two-patch parameterization. Such an explicit, geometry dependent basis construction is important for an efficient implementation of the isogeometric method. The stability of the constructed basis is numerically confirmed for an example configuration.

Unstructured spline spaces for isogeometric analysis based on spline manifolds

Based on Grimm and Hughes (1995) we introduce and study a mathematical framework for analysis-suitable unstructured B-spline spaces. In this setting the parameter domain has a manifold structure which allows for the definition of function spaces such as, for instance, B-splines over multi-patch domains with extraordinary points or analysis-suitable unstructured T-splines. Within this framework, we generalize the concept of dual-compatible B-splines (developed for structured T-splines in Beirao da Veiga et al. (2013)). This allows us to prove the key properties that are needed for isogeometric analysis, such as linear independence and optimal approximation properties for h-refined meshes. We introduce a mathematical framework for unstructured B-spline spaces based on manifolds.We generalize the notion of dual-compatibility to manifold domains.We study the linear independence of unstructured B-splines on manifold domains.This allows us to prove optimal approximation properties for h-refined meshes.

Construction of Smooth Isogeometric Function Spaces on Singularly Parameterized Domains

We aim at constructing a smooth basis for isogeometric function spaces on domains of reduced geometric regularity. In this context an isogeometric function is the composition of a piecewise rational function with the inverse of a piecewise rational geometry parameterization. We consider two types of singular parameterizations, domains where a part of the boundary is mapped onto one point and domains where the two parameter lines at a corner of the parameter domain are collinear in the physical domain. We locally map a singular tensor-product patch of arbitrary degree onto a triangular patch, thus splitting the parameterization into a singular bilinear mapping and a regular mapping on a triangular domain. This construction yields an isogeometric function space of prescribed smoothness. Generalizations to higher dimensions are also possible and are briefly discussed in the final section.

Derivatives of isogeometric functions on n-dimensional rational patches in R d

We consider isogeometric functions and their derivatives. Given a geometry mapping, which is defined by an n-dimensional NURBS patch in R d , an isogeometric function is obtained by composing the inverse of the geometry mapping with a NURBS function in the parameter domain. Hence an isogeometric function can be represented by a NURBS parametrization of its graph. We take advantage of the projective representation of the NURBS patch as a B-spline patch in homogeneous coordinates.We derive a closed form representation of the graph of a partial derivative of an isogeometric function. The derivative can be interpreted as an isogeometric function of higher degree and lower smoothness on the same piecewise rational geometry mapping, hence the space of isogeometric functions is closed under differentiation. We distinguish the two cases n = d and n < d , with a focus on n = d - 1 in the latter one.As a first application of the presented formula we derive conditions which guarantee C 1 and C 2 smoothness for isogeometric functions on several singularly parametrized planar and volumetric domains as well as on embedded surfaces. It is interesting to note that the presented conditions depend not only on the general structure of the patch, but on the exact representation of the interior of the given geometry mapping. We derive an exact representation of the derivatives of an isogeometric function.The isogeometric function is defined over a domain of arbitrary dimension.We use the formula to derive smoothness results for functions on singular patches.The conditions depend on the exact representation of the interior of the domain.The derivative formula can be used to derive functionals for regularization.

Construction of analysis-suitable G1 planar multi-patch parameterizations

Isogeometric analysis allows to define shape functions of global

Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis

C^{1}

Approximation properties of multi-patch

continuity (or of higher continuity) over multi-patch geometries. The construction of such