Isogeometric Functions Research Articles

An approximate [formula omitted] multi-patch space for isogeometric analysis with a comparison to Nitsche’s method

We present an approximately C1-smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). The construction extends the one presented in [1] for two-patch domains. A key property of IGA is that it is simple to achieve high order smoothness within a single patch. However, to represent more complex geometries one often uses a multi-patch construction. In this case, the global continuity for the basis functions is in general only C0. Therefore, to obtain C1-smooth isogeometric functions, a special construction for the basis is needed. Such spaces are of interest when solving numerically fourth-order problems, such as the biharmonic equation or Kirchhoff–Love plate/shell formulations, using an isogeometric Galerkin method.Isogeometric spaces that are globally C1 over multi-patch domains can be constructed as in [2–6]. The constructions require geometry parametrizations that satisfy certain constraints along the interfaces, so-called analysis-suitable G1 parametrizations. To allow C1 spaces over more general multi-patch parametrizations, one needs to increase the polynomial degree and/or to relax the C1 conditions. Thus, we define function spaces that are not exactly C1 but only approximately. We adopt the construction for two-patch domains, as developed in [1], and extend it to more general multi-patch domains.We employ the construction for a biharmonic model problem and compare the results with Nitsche’s method. We compare both methods over complex multi-patch domains with non-trivial interfaces. The numerical tests indicate that the proposed construction converges optimally under h-refinement, comparable to the solution using Nitsche’s method. In contrast to weakly imposing coupling conditions, the approximate C1 construction is explicit and no additional terms need to be introduced to stabilize the method/penalize the jump of the derivative at the interface. Thus, the new proposed method can be used more easily as no parameters need to be estimated.

Construction of approximate [formula omitted] bases for isogeometric analysis on two-patch domains

In this paper, we develop and study approximately smooth basis constructions for isogeometric analysis over two-patch domains. One key element of isogeometric analysis is that it allows high order smoothness within one patch. However, for representing complex geometries, a multi-patch construction is needed. In this case, a C0-smooth basis is easy to obtain, whereas C1-smooth isogeometric functions require a special construction. Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation and the Kirchhoff–Love plate or shell formulation, using an isogeometric Galerkin method.With the construction of so-called analysis-suitable G1 (in short, AS-G1) parametrizations, as introduced in Collin et al. (2016), it is possible to construct C1 isogeometric spaces which possess optimal approximation properties, cf. (Kapl et al., 2019). These geometries need to satisfy certain constraints along the interfaces and additionally require that the regularity r and degree p of the underlying spline space satisfy 1≤r≤p−2. The problem is that most complex geometries are not AS-G1 geometries. Therefore, we define basis functions for isogeometric spaces by enforcing approximate C1 conditions following the basis construction from Kapl et al. (2017). For this reason, the defined function spaces are not exactly C1 but only approximately.We study the convergence behavior and define function spaces that converge optimally under h-refinement, by locally introducing functions of higher polynomial degree and lower regularity. The convergence rate is optimal in several numerical tests performed on domains with non-trivial interfaces. While an extension to more general multi-patch domains is possible, we restrict ourselves to the two-patch case and focus on the construction over a single interface.

Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations

The design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des. 52–53:75–89, 2017, Kapl et al. Comput. Aided Geom. Des. 69:55–75, 2019 and Kapl and Vitrih J. Comput. Appl. Math. 335:289–311, 2018, Kapl and Vitrih J. Comput. Appl. Math. 358:385–404, 2019 and Kapl and Vitrih Comput. Methods Appl. Mech. Engrg. 360:112684, 2020 for the construction of C1-smooth and C2-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of Cs-smooth isogeometric multi-patch spline spaces of degree p, inner regularity r and of a smoothness s ≥ 1, with p ≥ 2s + 1 and s ≤ r ≤ p − s − 1. More precisely, we study for s ≥ 1 the space of Cs-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular Cs-smooth subspace of the entire Cs-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this Cs-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the Cs-smooth spline functions to perform L2 approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed Cs-smooth subspace.

A new stochastic isogeometric analysis method based on reduced basis vectors for engineering structures with random field uncertainties

A new stochastic isogeometric analysis method based on reduced basis vectors (SRBIGA) is proposed for engineering structures with random field material properties and external loads. Based on the Galerkin isogeometric functions, the proposed SRBIGA applies the Karhunen–Loève expansion to discretize the random field uncertainties. Inspired by the stochastic Krylov subspace theory, the structural responses of linear elasticity structures with random field uncertainties are represented based on the reduced basis vectors. The tremendous advantage of SRBIGA over the spectral stochastic isogeometric analysis (SSIGA) in terms of the computational efficiency is disclosed through the comparison analysis in theoretical aspects. Three illustrative examples demonstrate that the proposed SRBIGA has not only significantly higher efficiency but also higher accuracy and better robustness than the SSIGA and that it can provide a novel and expedient stochastic structural analysis method for practical large-scale complex engineering structures when both material properties and external loads are spatially random.

An isogeometric boundary element method for liquid sloshing in the horizontal eccentric annular tanks with multiple porous baffles

Boundary element method (BEM) coupled with the isogeometric boundary based on Non-Uniform Rational B-Splines (NURBS), which is termed IGABEM, is a new numerical method and has attracted wide attention in recent years. In this paper, IGABEM is first applied to the liquid sloshing problem in the horizontal rectangular, circular or elliptical annular tanks with multiple porous baffles, assuming incompressible, irrotational and inviscid flows. As a key improvement compared with the conventional polynomial shape function in BEM, the IGABEM for the analysis of liquid sloshing can exactly represent the geometry no matter how coarse discretize the studied boundary, which then produces no geometrical errors. A zoning method is carried out to study arrangement of the multiple porous baffles in the tanks. Neighboring zones is joined together by the boundary conditions of the porous baffles and virtual interface which correlate the flux and velocity potential. By discretizing boundaries of each zone using the isogeometric function, the governing integral equation on the boundary by using BEM can be formulated into the dynamic equation in frequency domain. Three configurations of baffles are taken into account such as the top-mounted baffles, bottom-mounted baffles and annular baffles. The results obtained by the proposed method are verified by comparing with the reported solutions and very good agreement is obtained. The effects of tank's geometry, liquid filling level and the baffle's parameters with various lengths, submergence depths, number of baffles, and porosities on the natural sloshing frequency, wave surface elevation, sloshing force and velocity field are investigated and several conclusions are achieved.

Solving the triharmonic equation over multi-patch planar domains using isogeometric analysis

We present a framework for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. Our approach is based on the construction of a globally C 2 -smooth isogeometric spline space which is used as discretization space. The generated C 2 -smooth space consists of three different types of isogeometric functions called patch, edge and vertex functions. All functions are entirely local with a small support, and numerical examples indicate that they are well-conditioned. The construction of the functions is simple and works uniformly for all multi-patch configurations. While the patch and edge functions are given by a closed form representation, the vertex functions are obtained by computing the null space of a small system of linear equations. Several examples demonstrate the potential of our approach for solving the triharmonic equation.

Bases and dimensions of C1-smooth isogeometric splines on volumetric two-patch domains

We analyze the spaces of trivariate C1–smooth isogeometric functions on two-patch domains. Our aim is to generalize the corresponding results from the bivariate (Kapl et al. (2015) [25]) to the trivariate case. In the first part of the paper, we introduce the notion of gluing data and use it to define glued spline functions on two-patch domains. Applying the fundamental observation that “matched Gk–constructions always yield Ck–continuous isogeometric elements”, see Groisser and Peters (2015) [14], to graph hypersurfaces in four-dimensional space, allows us to characterize C1–smooth geometrically continuous isogeometric functions as the push-forwards of these functions for suitable gluing data. The second part of the paper is devoted to various special classes of gluing data. We analyze how the generic dimensions depend on the number of knot spans (elements) and on the spline degree. Finally we show how to construct locally supported basis functions in specific situations.

Construction of analysis-suitable [formula omitted] planar multi-patch parameterizations

The construction of smooth surfaces of complex shapes is at the heart of computer-aided design (CAD). Many different approaches generating C1-smooth surfaces are available and well-studied. Isogeometric analysis (IGA) has sparked new interest in these methods, since it allows to incorporate CAD based parameterizations into numerical simulations. In IGA one can utilize shape functions of global C1 continuity (or of higher continuity) over multi-patch geometries. Such functions can then be used to discretize high order partial differential equations, such as the biharmonic equation. However, the requirements posed by the IGA simulation are often different from the requirements in CAD. The construction ofC1-smooth isogeometric functions is a non-trivial task and requires particular multi-patch parameterizations to ensure that the resulting C1 isogeometric spaces possess optimal approximation properties. For this purpose, we select so-called analysis-suitable G1 (AS-G1) parameterizations, proposed in Collin et al. (2016).In this work, we show through examples that it is possible to construct AS-G1 multi-patch parameterizations of planar domains, given their boundary. More precisely, given a generic multi-patch geometry, we generate an AS-G1 multi-patch parameterization possessing the same boundary, the same vertices and the same first derivatives at the vertices, and which is as close as possible to this initial geometry. Our algorithm is based on a quadratic optimization problem with linear side constraints. Numerical tests also confirm that C1 isogeometric spaces over AS-G1 multi-patch parameterized domains converge optimally under mesh refinement, while for generic parameterizations the convergence order is severely reduced.

Dimension and basis construction for C2-smooth isogeometric spline spaces over bilinear-like G2 two-patch parameterizations

A particular class of planar two-patch geometries, called bilinear-like G 2 two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like G 2 two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of C 2 -smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. Groisser and Peters (2015) and Kapl et al. (2015). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like G 2 two-patch parameterizations. Numerical results obtained by performing L 2 -approximation and solving Poisson’s equation indicate that already the subspace possesses optimal approximation properties. • Introduction of bilinear-like G 2 two-patch parameterizations . • Investigation of the C 2 smooth isogeometric space over this specific geometry. • Computation of dimension of the space. • Construction of a simple explicitly given and well-conditioned basis of the space. • Numerical experiments show optimal rates of convergence.

Space of [formula omitted]-smooth geometrically continuous isogeometric functions on planar multi-patch geometries: Dimension and numerical experiments

We study the space of C2-smooth isogeometric functions on bilinearly parameterized multi-patch domains Ω⊂R2, where the graph of each isogeometric function is a multi-patch spline surface of bidegree (d,d), d∈{5,6}. The space is fully characterized by the equivalence of the C2-smoothness of an isogeometric function and the G2-smoothness of its graph surface (cf. Groisser and Peters (2015), Kapl et al. (2015)). This is the reason to call its functions C2-smooth geometrically continuous isogeometric functions.In particular, the dimension of this C2-smooth isogeometric space is investigated. The study is based on the decomposition of the space into three subspaces and is an extension of the work Kapl and Vitrih (2017) to the multi-patch case. In addition, we present an algorithm for the construction of a basis, and use the resulting globally C2-smooth functions for numerical experiments, such as performing L2 approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order.

Dimension and basis construction for analysis-suitable G1 two-patch parameterizations

We study the dimension and construct a basis for C1-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 G1 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, C1 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 C1-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.

Space of [formula omitted]-smooth geometrically continuous isogeometric functions on two-patch geometries

The space of C2-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is considered. The investigation of the dimension of the spaces of biquintic and bisixtic C2-smooth geometrically continuous isogeometric functions on such domains is presented. In addition, C2-smooth isogeometric functions are constructed to be used for performing L2-approximation and for solving triharmonic equation on different two-patch geometries. The numerical results indicate optimal approximation order.

Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries

We generate a basis of the space of bicubic and biquartic C1-smooth geometrically continuous isogeometric functions on bilinear multi-patch domains Ω⊂R2. The basis functions are obtained by suitably combining C1-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains (cf. [18]). They are described by simple explicit formulas for their spline coefficients.These C1-smooth isogeometric functions possess potential for applications in isogeometric analysis, which is demonstrated by several examples (such as the biharmonic equation). In particular, the numerical results indicate optimal approximation power.

Isogeometric analysis with geometrically continuous functions on two-patch geometries

We study the linear space of Cs-smooth isogeometric functions defined on a multi-patch domain Ω⊂R2. We show that the construction of these functions is closely related to the concept of geometric continuity of surfaces, which has originated in geometric design. More precisely, the Cs-smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order (Gs-smoothness) of their graph surfaces. This motivates us to call them Cs-smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of C1-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is analyzed in more detail. Numerical experiments with bicubic and biquartic functions for performing L2 approximation and for solving Poisson’s equation and the biharmonic equation on two-patch geometries are presented and indicate optimal rates of convergence.

Derivatives of isogeometric functions on n-dimensional rational patches in [formula omitted

We consider isogeometric functions and their derivatives. Given a geometry mapping, which is defined by an n-dimensional NURBS patch in Rd, 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 C1 and C2 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.