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Template Mapping Using Adaptive Splines and Optimization of the Parameterization

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Advanced Methods for Geometric Modeling and Numerical Simulation

Part of the book series: Springer INdAM Series ((SINDAMS,volume 35))

Abstract

We consider the construction of a spline map (a volumetric deformation) that transforms a template, which is given in the domain, into a target shape. More precisely, the domain is equipped with a set of surface patches (the template skeleton) and target patches for some of them (which are called the constraining patches) are specified. The constructed spline map approximately transforms the constraining patches into the associated target patches. Possible applications include isogeometric segmentation and parameterization of the computational domain. In particular, the approach should be useful when performing isogeometric segmentation and parameterization for a large class of computational domains possessing similar shapes. We present a solution approach, which is based on least-squares fitting. In order to deal with the influence of the parameterization, the well-established approaches of point and tangent distance minimization are employed for the iterative solution of the resulting nonlinear optimization problems. Additionally, we enrich the approach with spline space refinement. The efficiency and performance of the approach are investigated experimentally. We demonstrate that the optimization of the parameterization, which is used in the point or tangent distance minimization, is an essential step of the procedure. In addition, we use adaptive spline refinement in order to save computational resources. The proposed template mapping approach is also applied to a case of industrial interest, as well as to a volumetric example.

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Notes

  1. 1.

    Different constraints are present at patch boundaries.

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Acknowledgements

Supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 678727 (the MOTOR project), as well as by the ERC advanced grant CHANGE (GA no. 694515) and by the Austrian Science Fund (FWF NFN S117). We are grateful to our project partners from Dortmund for providing the data for the industrial example.

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Authors and Affiliations

  1. Institute of Applied Geometry, Johannes Kepler University, Linz, Austria

    Svajūnas Sajavičius, Bert Jüttler & Jaka Špeh

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  1. Svajūnas Sajavičius
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  2. Bert Jüttler
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  3. Jaka Špeh
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Correspondence to Bert Jüttler .

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Editors and Affiliations

  1. Department of Mathematics and Computer Science "U. Dini", University of Florence, Florence, Italy

    Carlotta Giannelli

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Hendrik Speleers

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Sajavičius, S., Jüttler, B., Špeh, J. (2019). Template Mapping Using Adaptive Splines and Optimization of the Parameterization. In: Giannelli, C., Speleers, H. (eds) Advanced Methods for Geometric Modeling and Numerical Simulation. Springer INdAM Series, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-030-27331-6_9

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