# Registered Data

## [01107] Efficient methods for Isogeometric Analysis

**Session Date & Time**:- 01107 (1/2) : 5B (Aug.25, 10:40-12:20)
- 01107 (2/2) : 5C (Aug.25, 13:20-15:00)

**Type**: Proposal of Minisymposium**Abstract**: Isogeometric Analysis is a relatively novel technique used to solve PDEs. The same functions that are used to describe the computational domain (typically B-Splines or NURBs) are used also to approximate the solution of the considered PDE. This approach brings several advantages with respect to the classical finite element method, but it also leads to new challenges, in particular from the computational point of view. This mini-symposium aims at gathering researchers that contribute to the improvement of the efficiency for isogeometric methods.**Organizer(s)**: Mattia Tani, John Evans, Angelos Mantzaflaris, Stefan Takacs**Classification**:__65N30__,__65Y20__,__65N38__**Speakers Info**:- Alexandra Bünger (University of British Columbia)
- Michael Barton (BCAM)
- Bernard Kapidani (EPFL)
- Tadej Kanduč (University of Ljubljana)
- Joaquin Cornejo Fuentes (INSA Lyon)
- Melina Merkel (TU Darmstadt)
- Gabriele Loli (University of Pavia)
- Margarita Chasapi (EPFL)

**Talks in Minisymposium**:**[04261] Matrix free weighted quadrature IgA applied to heat transfer problems****Author(s)**:**Joaquin Eduardo Cornejo Fuentes**(INSA Lyon)- Thomas Elguedj (Lamcos)
- Arnaud Duval (Lamcos)
- David Dureisseix (Lamcos)

**Abstract**: IsoGeometric Analysis was introduced as an extension of Finite Element Method to represent the geometry and the solution field. However, the algorithms commonly used in FEM represented a major challenge from a computational point of view in IgA. This communication focuses on novel techniques applied to heat transfer problems which take advantage of the tensor structure of the shape functions, improve computation time of matrix-vector products and enhance the convergence rate of the iterative solver.

**[04410] Solving boundary value problems via the Nystrom method using spline Gauss rules****Author(s)**:**Michael Barton**- Ali Hashemian (BCAM)
- Hanna Sliusarenko (BCAM)
- Sara Remogna (University of Torino)
- Domingo Barrera (University of Granada)

**Abstract**: We propose to use spline Gauss quadrature rules for solving boundary value problems~(BVPs) using the Nystrom method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nystrom method, which involves the use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts.

**[04643] Fast computation of electromagnetic wave propagation with spline differential forms****Author(s)**:**Bernard Kapidani**(Ecole Polytechnique Fédérale Lausanne)- Rafael Vazquez (Ecole Polytechnique Fédérale Lausanne)

**Abstract**: We present a new structure-preserving numerical method for hyperbolic problems which does not rely on the geometric realisation of any dual mesh. We use B-spline based de Rham complexes to construct two exact sequences of discrete differential forms and apply them to solving Maxwell's equations. The method exhibits high order convergence and energy conservation, with computational effort much lower than standard Galerkin. We will also present preliminary results towards extension to multi-patch geometries.

**[04698] Isogeometric Coupling Methods for H(curl) Problems****Author(s)**:**Melina Merkel**(Technische Universität Darmstadt)- Sebastian Schöps (Technische Universität Darmstadt)

**Abstract**: In this work, we present a method for the efficient simulation of electric motors using isogeometric analysis. As these machines include moving parts, conformity of the patches cannot be guaranteed for all rotation angles without modification of the geometry. We therefore use domain decomposition methods, e.g., mortaring or Nitsche-type coupling, for the coupling of stator and rotor. These methods can also be applied to all patches to facilitate patch-parallel computations.

**[05111] Efficient reduced order models for unfitted spline discretizations****Author(s)**:**Margarita Chasapi**(EPFL)- Pablo Antolin (EPFL)
- Annalisa Buffa (EPFL)

**Abstract**: This talk presents a methodology for efficient reduced order modelling of PDEs on unfitted spline discretizations. We are interested in problems formulated on parameterized unfitted geometries and aim to construct efficient reduced basisapproximations. The presented methodology is based on extension of solution snapshots on the background mesh and localization strategies to confine the number of reduced basis functions. Numerical experiments on trimmed spline discretizations show the accuracy and efficiency of the method.

**[05187] Singularity extraction and efficient numerical integration for isogeometric BEM****Author(s)**:**Tadej Kanduc**(University of Ljubljana)

**Abstract**: Quadrature rules to evaluate governing (weakly singular) integrals that appear in boundary integral equations for 3D potential problems are presented. The rules are described by two main features: a higher order isoparametric singularity extraction technique and a spline-based quasi-interpolation technique. Uniform distribution of quadrature nodes is preferable to improve the implementation efficiency. The integration scheme has high order converge rates and since it is tailored for spline integrands, it perfectly fits in the isogeometric framework.

**[05212] Low-rank Tensor Train Methods for IGA with Multiple Patches****Author(s)**:**Alexandra Bünger**(University of British Columbia)- Martin Stoll (Technical University of Chemnitz)
- Tom Christian Riemer (Technical University of Chemnitz)

**Abstract**: In IGA, the equation systems for, e.g., optimization problems may quickly become very costly to assemble and solve. We developed a method exploit the underlying tensor structure with low-rank tensor train approximations. This low-rank formulation can be efficiently used in a block-structured iterative solver to solve challenging PDE problems in a compact format. We recently extended this for multi-patch domains and show how the resulting systems can be treated effectively in the tensor train framework.

**[05302] An efficient solver for space–time isogeometric Galerkin methods for parabolic problems****Author(s)**:**Gabriele Loli**(Università di Pavia )- Monica Montardini (Università di Pavia )
- Giancarlo Sangalli (Università di Pavia )
- Mattia Tani (University of Pavia)

**Abstract**: We present an efficient solver for a Galerkin space–time isogeometric discretization of the heat equation. In particular, we propose a preconditioner that is the sum of Kronecker products of matrices and that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. the polynomial degree of the spline space and the time required for the application is almost proportional to the number of degrees-of-freedom.