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: 741; Downloads: 41
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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: 852; Downloads: 39
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3. 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: 1259; Downloads: 8
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4. On ▫$L^2$▫ approximation by planar Pythagorean-hodograph curvesRida T. Farouki, Marjetka Knez, Vito Vitrih, Emil Žagar, 2025, original scientific article Abstract: The ▫$L^2$▫ approximation of planar curves by Pythagorean-hodograph (PH) polynomial curves is addressed, based on the distance defined by a metric for planar curves represented as complex valued functions of a real parameter. Because of the nonlinear nature of polynomial PH curves, constructing ▫$L^2$▫ approximants involves solving a nonlinear optimization problem. However, a simplified method that requires only the solution of a linear system may be developed by formulating the ▫$L^2$▫ approximation in the preimage space. The extension of the methodology to approximation by PH B-spline curves is also addressed, and several examples are provided to illustrate its implementation and potential.
Keywords: ▫$L^2$▫ approximation, complex polynomial, Pythagorean-hodograph curve, Pythagorean-hodograph spline, preimage
Published in RUP: 30.05.2025; Views: 2946; Downloads: 30
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7. Hermite interpolation by rational G [sup] k motions of low degreeGašper Jaklič, Bert Jüttler, Marjetka Knez, Vito Vitrih, Emil Žagar, 2013, original scientific article Abstract: Interpolation by rational spline motions is an important issue in robotics and related fields. In this paper a new approach to rational spline motion design is described by using techniques of geometric interpolation. This enables us to reduce the discrepancy in the number of degrees of freedom of the trajectory of the origin and of the rotational part of the motion. A general approach to geometric interpolation by rational spline motions is presented and two particularly important cases are analyzed, i.e., geometric continuous quartic rational motions and second order geometrically continuous rational spline motions of degree six. In both cases sufficient conditions on the given Hermite data are found which guarantee the uniqueness of the solution. If the given data do not fulfill the solvability conditions, a method to perturb them slightly is described. Numerical examples are presented which confirm the theoretical results and provide an evidence that the obtained motions have nice shapes.
Keywords: mathematics, numerical analysis, motion design, geometric interpolation, rational spline motion, geometric continuity
Published in RUP: 03.04.2017; Views: 4056; Downloads: 50
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8. Construction of G[sup]3 rational motion of degree eightKarla Ferjančič, Marjetka Knez, Vito Vitrih, 2015, original scientific article Abstract: The paper presents a construction of a rigid body motion with point trajectories being rational spline curves of degree eight joining together with ▫$G^3$▫ smoothness. The motion is determined through interpolation of positions and derivative data up to order three in the geometric sense. Nonlinearity in the spherical part of construction results in a single univariate quartic equation which yields solutions in a closed form. Sufficient conditions on the regions for the curvature data are derived, implying the existence of a real admissible solution. The algorithm how to choose appropriate data is proposed too. The theoretical results are substantiated with numerical examples.
Keywords: motion design, geometric interpolation, rational spline motion, geometric continuity
Published in RUP: 15.10.2015; Views: 5774; Downloads: 147
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