MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint

Low Rank Tensor Methods in Galerkin-based Isogeometric Analysis

Angelos Mantzaflaris, Bert Jüttler, Boris N. Khoromskij and Ulrich Langer


The global (patch-wise) geometry map, which describes the computational domain, is a new feature in isogeometric analysis. This map has a global tensor structure, inherited from the parametric spline geometry representation.

The use of this global structure in the discretization of partial differential equations may be regarded as a drawback at first glance, as opposed to the purely local nature of (high-order) classical finite elements. In this work we demonstrate that it is possible to exploit the regularity of this structure and to identify the great potential for the efficient implementation of isogeometric discretizations. First, we formulate tensor-product B-spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure.

Second, we derive an algorithm for the the separation of variables in the integrands arising in the discretization. This is possible by means of low rank approximation of the integral kernels. We arrive at a compact, separated representation of the integrals. The separated form implies an expression of Galerkin matrices as Kronecker products of matrix factors with small dimensions. This representation is very appealing, due to the reduction in both memory consumption and computation times.

Our benchmarks, performed using the C++ library G+Smo, demonstrate that the use of tensor methods in isogeometric analysis possesses significant advantages.

Jun 14, 2016
Jun 15, 2016
MSC Codes:
65F30, 65F50, 65N35, 65F10
isogeometric analysis, low rank approximation, stiffness matrix, tensor decomposition, kronecker product, numerical quadrature

Related publications

2017 Repository Open Access
Angelos Mantzaflaris, Bert Jüttler, Boris N. Khoromskij and Ulrich Langer

Low rank tensor methods in Galerkin-based isogeometric analysis

In: Computer methods in applied mechanics and engineering, 316 (2017), pp. 1062-1085