arXiv subjects

Vito Vitrih

Publications and source records attributed to Vito Vitrih.

15 recordsLinked to original sources

arXiv2026

An adaptive Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) method for solving the biharmonic equation over planar multi-patch geometries

We present a novel adaptive isogeometric method for solving the biharmonic equation over planar multi-patch domains with possibly extraordinary vertices, parametrized by analysis-suitable G^1 multi-patch geometries. The proposed technique relies on the concept of Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP), which enforces the required C^1-smoothness of the solution across a common edge of two neighboring patches by imposing appropriate continuity conditions by means of Lagrange multipliers. The resulting saddle point problem is solved using a dual-primal formulation, first by a small linear problem for the Lagrange multipliers and then by local, parallelizable linear problems on the individual patches for the coefficients of the numerical solution. While for the local problems on the single patches standard diagonally scaling is used as preconditioner, a suitable preconditioner for the problem of finding the Lagrange multipliers is introduced. To perform adaptive refinement, the solution of the biharmonic equation on the single patches of the multi-patch domain is discretized by employing (truncated) hierarchical B-splines, and an appropriate refinement strategy of the underlying mesh is presented. Finally, the potential of the developed adaptive IETI-DP method for solving the biharmonic equation over planar multi-patch geometries is numerically tested on the basis of several numerical examples. Thereby, the numerical results show on the one hand optimal convergence behavior with respect to adaptive refinement, and on the other hand a good performance of the proposed preconditioner for the linear problem of determining the Lagrange multipliers.

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arXiv2025

An Isogeometric Tearing and Interconnecting (IETI) method for solving high order partial differential equations over planar multi-patch geometries

We present a novel method for solving high-order partial differential equations (PDEs) over planar multi-patch geometries demonstrated on the basis of the polyharmonic equation of order $m$, $m \geq 1$, which is a particular linear elliptic PDE of order $2m$. Our approach is based on the concept of Isogeometric Tearing and Interconnecting (IETI) [43] and allows to couple the numerical solution of the PDE with $C^s$-smoothness, $s \geq m-1$, across the edges of the multi-patch geometry. The proposed technique relies on the use of a particular class of multi-patch geometries, called bilinear-like $G^s$ multi-patch parameterizations [37], to represent the multi-patch domain. The coupling between the neighboring patches is done via the use of Lagrange multipliers and leads to a saddle point problem, which can be solved efficiently first by a small dual problem for a subset of the Lagrange multipliers followed by local, parallelizable problems on the single patches for the coefficients of the numerical solution. Several numerical examples of solving the polyharmonic equation of order $m=2$ and $m=3$, i.e. the biharmonic and triharmonic equation, respectively, over different multi-patch geometries are shown to demonstrate the potential of our IETI method for high-order problems.

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arXiv2024

Isogeometric collocation with smooth mixed degree splines over planar multi-patch domains

We present a novel isogeometric collocation method for solving the Poisson's and the biharmonic equation over planar bilinearly parameterized multi-patch geometries. The proposed approach relies on the use of a modified construction of the C^s-smooth mixed degree isogeometric spline space [20] for s=2 and s=4 in case of the Poisson's and the biharmonic equation, respectively. The adapted spline space possesses the minimal possible degree p=s+1 everywhere on the multi-patch domain except in a small neighborhood of the inner edges and of the vertices of patch valency greater than one where a degree p=2s+1 is required. This allows to solve the PDEs with a much lower number of degrees of freedom compared to employing the C^s-smooth spline space [29] with the same high degree p=2s+1 everywhere. To perform isogeometric collocation with the smooth mixed degree spline functions, we introduce and study two different sets of collocation points, namely first a generalization of the standard Greville points to the set of mixed degree Greville points and second the so-called mixed degree superconvergent points. The collocation method is further extended to the class of bilinear-like G^s multi-patch parameterizations [26], which enables the modeling of multi-patch domains with curved boundaries, and is finally tested on the basis of several numerical examples.

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arXiv2024

A C^s-smooth mixed degree and regularity isogeometric spline space over planar multi-patch domains

We construct over a given bilinear multi-patch domain a novel $C^s$-smooth mixed degree and regularity isogeometric spline space, which possesses the degree $p=2s+1$ and regularity $r=s$ in a small neighborhood around the edges and vertices, and the degree $\widetilde{p} \leq p$ with regularity $\widetilde{r} = \widetilde{p}-1 \geq r$ in all other parts of the domain. Our proposed approach relies on the technique [35], which requires for the $C^s$-smooth isogeometric spline space a degree at least $p=2s+1$ on the entire multi-patch domain. Similar to [35], the $C^s$-smooth mixed degree and regularity spline space is generated as the span of basis functions that correspond to the individual patches, edges and vertices of the domain. The reduction of degrees of freedom for the functions in the interior of the patches is achieved by introducing an appropriate mixed degree and regularity underlying spline space over $[0,1]^2$ to define the functions on the single patches. We further extend our construction with a few examples to the class of bilinear-like $G^s$ multi-patch parameterizations [33,35], which enables the design of multi-patch domains having curved boundaries and interfaces. Finally, the great potential of the $C^s$-smooth mixed degree and regularity isogeometric spline space for performing isogeometric analysis is demonstrated by several numerical examples of solving two particular high order partial differential equations, namely the biharmonic and triharmonic equation, via the isogeometric Galerkin method.

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arXiv2024

Application of a metric for complex polynomials to bounded modification of planar Pythagorean-hodograph curves

By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.

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arXiv2023

A locally based construction of analysis-suitable $G^1$ multi-patch spline surfaces

Analysis-suitable $G^1$ (AS-$G^1$) multi-patch spline surfaces [4] are particular $G^1$-smooth multi-patch spline surfaces, which are needed to ensure the construction of $C^1$-smooth multi-patch spline spaces with optimal polynomial reproduction properties [16]. We present a novel local approach for the design of AS-$G^1$ multi-patch spline surfaces, which is based on the use of Lagrange multipliers. The presented method is simple and generates an AS-$G^1$ multi-patch spline surface by approximating a given $G^1$-smooth but non-AS-$G^1$ multi-patch surface. Several numerical examples demonstrate the potential of the proposed technique for the construction of AS-$G^1$ multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, with optimal rates of convergence over them.

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arXiv2023

Isogeometric collocation for solving the biharmonic equation over planar multi-patch domains

We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally $C^4$-smooth isogeometric spline space [34] to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the $L^2$-norm as well as to equivalents of the $H^s$-seminorms for $1 \leq s \leq 4$. In the studied case of spline degree $p=9$, the numerical results indicate in case of the Greville points a convergence of order $\mathcal{O}(h^{p-3})$ independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order $\mathcal{O}(h^{p-2})$ for all (semi)norms except for the equivalent of the $H^4$-seminorm, where the order $\mathcal{O}(h^{p-3})$ is anyway optimal.

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arXiv2023

$C^1$-smooth isogeometric spline functions of general degree over planar mixed meshes: The case of two quadratic mesh elements

Splines over triangulations and splines over quadrangulations (tensor product splines) are two common ways to extend bivariate polynomials to splines. However, combination of both approaches leads to splines defined over mixed triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes are especially useful for representing complicated geometries obtained e.g. from trimming. As (bi-)linearly parameterized mesh elements are not flexible enough to cover smooth domains, we focus in this work on the case of planar mixed meshes parameterized by (bi-)quadratic geometry mappings. In particular we study in detail the space of $C^1$-smooth isogeometric spline functions of general polynomial degree over two such mixed mesh elements. We present the theoretical framework to analyze the smoothness conditions over the common interface for all possible configurations of mesh elements. This comprises the investigation of the dimension as well as the construction of a basis of the corresponding $C^1$-smooth isogeometric spline space over the domain described by two elements. Several examples of interest are presented in detail.

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arXiv2021

$C^1$ isogeometric spline space for trilinearly parameterized multi-patch volumes

We study the space of $C^1$ isogeometric spline functions defined on trilinearly parameterized multi-patch volumes. Amongst others, we present a general framework for the design of the $C^1$ isogeometric spline space and of an associated basis, which is based on the two-patch construction [7], and which works uniformly for any possible multi-patch configuration. The presented method is demonstrated in more detail on the basis of a particular subclass of trilinear multi-patch volumes, namely for the class of trilinearly parameterized multi-patch volumes with exactly one inner edge. For this specific subclass of trivariate multi-patch parameterizations, we further numerically compute the dimension of the resulting $C^1$ isogeometric spline space and use the constructed $C^1$ isogeometric basis functions to numerically explore the approximation properties of the $C^1$ spline space by performing $L^2$ approximation.

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arXiv2020

A super-smooth $C^1$ spline space over planar mixed triangle and quadrilateral meshes

In this paper we introduce a $C^1$ spline space over mixed meshes composed of triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis. In this context, a mesh is considered to be a partition of a planar polygonal domain into triangles and/or quadrilaterals. The proposed space combines the Argyris triangle, cf. (Argyris, Fried, Scharpf; 1968), with the $C^1$ quadrilateral element introduced in (Brenner, Sung; 2005) and (Kapl, Sangalli, Takacs; 2019) for polynomial degrees $p\geq 5$. The space is assumed to be $C^2$ at all vertices and $C^1$ across edges, and the splines are uniquely determined by $C^2$-data at the vertices, values and normal derivatives at chosen points on the edges, and values at some additional points in the interior of the elements. The motivation for combining the Argyris triangle element with a recent $C^1$ quadrilateral construction, inspired by isogeometric analysis, is two-fold: on one hand, the ability to connect triangle and quadrilateral finite elements in a $C^1$ fashion is non-trivial and of theoretical interest. We provide not only approximation error bounds but also numerical tests verifying the results. On the other hand, the construction facilitates the meshing process by allowing more flexibility while remaining $C^1$ everywhere. This is for instance relevant when trimming of tensor-product B-splines is performed. In the presented construction we assume to have (bi)linear element mappings and piecewise polynomial function spaces of arbitrary degree $p\geq 5$. The basis is simple to implement and the obtained results are optimal with respect to the mesh size for $L^\infty$, $L^2$ as well as Sobolev norms $H^1$ and $H^2$.

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arXiv2020

$C^s$-smooth isogeometric spline spaces over planar multi-patch parameterizations

The design of globally $C^s$-smooth ($s \geq 1$) isogeometric spline spaces over multi-patch geometries is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods [25,28] and [31-33] for the construction of $C^1$-smooth and $C^2$-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of $C^s$-smooth isogeometric multi-patch spline spaces of an arbitrary selected smoothness $s \geq 1$. More precisely, for any $s \geq 1$, we study the space of $C^s$-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular $C^s$-smooth subspace of the entire $C^s$-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this $C^s$-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the $C^s$-smooth spline functions to perform $L^2$ approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed $C^s$-smooth subspace.

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arXiv2019

Isogeometric collocation on planar multi-patch domains

We present an isogeometric framework based on collocation to construct a $C^2$-smooth approximation of the solution of the Poisson's equation over planar bilinearly parameterized multi-patch domains. The construction of the used globally $C^2$-smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi-patch configurations. The basis of the $C^2$-smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [2], which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. [1, 15, 32]) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.

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arXiv2018

Solving the triharmonic equation over multi-patch 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.

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arXiv2017

Dimension and basis construction for $C^{2}$-smooth isogeometric spline spaces over bilinear-like $G^{2}$ 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. [12, 20]). 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 indicate that already the subspace possesses optimal approximation properties.

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arXiv2017

Space of $C^2$-smooth geometrically continuous isogeometric functions on planar multi-patch geometries: Dimension and numerical experiments

We study the space of $C^{2}$-smooth isogeometric functions on bilinearly parameterized multi-patch domains $\Omega \subset \mathbb{R}^{2}$, where the graph of each isogeometric function is a multi-patch spline surface of bidegree $(d,d)$, $d \in \{5,6 \}$. The space is fully characterized by 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; Kapl et al.,2015). This is the reason to call its functions $C^{2}$-smooth geometrically continuous isogeometric functions. In particular, the dimension of this $C^{2}$-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 $C^{2}$-smooth functions for numerical experiments, such as performing $L^{2}$ approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order.

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