arXiv subjects

Aljaž Kosmač

Publications and source records attributed to Aljaž Kosmač.

6 recordsLinked to original sources

arXiv2026

An adaptive Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) method for multi-patch structured Kirchhoff--Love plates

We present a novel adaptive isogeometric method for solving the Kirchhoff-Love plate problem for plates whose mid-surface is defined by a planar multi-patch domain with possibly extraordinary vertices, parametrized by an analysis-suitable $G^1$ multi-patch geometry. 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 Kirchhoff-Love plate problem 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 Kirchhoff-Love plate problem on 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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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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