1. Isogeometric collocation with smooth mixed degree splines over planar multi-patch domainsMario Kapl, Aljaž Kosmač, Vito Vitrih, 2026, original scientific article Abstract: 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 Cs-smooth mixed degree isogeometric spline space [1] 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 Cs-smooth spline space [2] 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 Grevile 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 Gs multi-patch parameterizations [3], which enables the modeling of multi-patch domains with curved boundaries, and is finally tested on the basis of several numerical examples.
Keywords: isogeometric analysis, collocation, C^s-smoothness, mixed degree spline space, multi-patch surface
Published in RUP: 05.03.2026; Views: 75; Downloads: 2
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2. O prostorih gladkih zlepkov in izogeometričnih metodah za reševanje parcialnih diferencialnih enačb višjih redov nad večdelnimi domenami : doktorska disertacijaAljaž Kosmač, 2026, doctoral dissertation Keywords: isogeometric analysis, partial differential equations, multi-patch domains, splines, Cs-smoothnes, Galerkin method, collocation, mixed degree and regularity spline spaces
Published in RUP: 09.02.2026; Views: 243; Downloads: 5
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3. An Isogeometric Tearing and Interconnecting (IETI) method for solving high order partial differential equations over planar multi-patch geometriesMario Kapl, Aljaž Kosmač, Vito Vitrih, 2026, original scientific article Abstract: We present a novel method for solving high-order partial differential equations (PDEs) over planar multi-patch geometries with
possibly extraordinary vertices demonstrated on the basis of the polyharmonic equation of order m, m ≥ 1, which is a particular linear elliptic PDE of order 2m. Our approach is based on the concept of Isogeometric Tearing and Interconnecting (IETI) and allows to couple the numerical solution of the PDE with Cs-smoothness, , 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 Gs multi-patch parameterizations, 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 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 for the polyharmonic equation of order m = 1, m = 2 and m = 3, i.e. for the Poisson’s, the biharmonic and the triharmonic equation, respectively, are shown to demonstrate the potential of our IETI method for solving high-order problems over planar multi-patch geometries with possibly extraordinary vertices.
Keywords: isogeometric analysis, Galerkin method, C^s-smoothness, Tearing and Interconnecting, multi-patch domain, polyharmonic equation
Published in RUP: 02.02.2026; Views: 238; Downloads: 4
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4. A ▫$C^s$▫-smooth mixed degree and regularity isogeometric spline space over planar multi-patch domainsMario Kapl, Aljaž Kosmač, Vito Vitrih, 2026, original scientific article Abstract: 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 Kapl and Vitrih (2021), 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 Kapl and Vitrih (2021), 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 $G^8$ multi-patch parameterizations (Kapl and Vitrih (2018); Kapl and Vitrih (2021)), which enables the design of multi-patch domains having curved boundaries and interfaces. Finally, the great potential of the $C^8$-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.
Keywords: isogeometric analysis, Galerkin method, C^s-smoothness, mixed degree and regularity spline space, multi-patch domain
Published in RUP: 01.07.2025; Views: 762; Downloads: 5
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