# SSD - Keith Seminar

## Dr. Brendan Keith - The Surrogate Matrix Methodology: Low-cost Assembly for Isogeometric Analysis

Chair for Numerical Mathematics, the Technical University of Munich

### Abstract

This talk will introduce a new class of methods in isogeometric analysis (IGA) based on the surrogate matrix methodology [1,3–5]. The objective is to lower the cost of matrix assembly in IGA without sacrificing accuracy. It is well known that isogeometric methods face a great computational burden at the point of matrix assembly. This is due, in large part, to the quadrature involved in directly computing B-spline or NURBS basis function interactions; see, e.g., [2,6]. Nevertheless, we will show that significant classes of B-splines and NURBS bases have an intrinsic structure which can be easily exploited to avoid most of this quadrature.

The new assembly strategy we will present involves performing quadrature for only a small fraction of the IGA basis function interactions and then approximating the rest through, for example, interpolation. Therefore, most of the quadrature involved in standard IGA assembly is not performed at all and

the predominant expense in most IGA assembly algorithms is avoided. Our strategy, which may be viewed as constructing variable-scale approximations (i.e., surrogates) for each system matrix, retains the accuracy of the standard methods because the structure of our B-spline/NURBS bases allow for a simple correspondence between matrix entries and smooth functions [5].

In this talk, we will summarize the theoretical aspects of the surrogate matrix methodology which, in turn, certify the convergence of new surrogate IGA methods for Poisson’s equation, membrane vibration, plate bending, Stokes’ flow, nonlinear elasticity, and other problems. We will also focus on the implementation of the methodology in existing IGA code, using the open-source GeoPDEs library [7] as an example. For the sake of demonstration, we will show assembly speed-ups of up to fifty times, after only a few small modifications to this software. The capacity for even further speed-ups is clearly possible and similar modifications could be made to other contemporary software libraries.

[1] S. Bauer, M. Mohr, U. Rüde, J. Weismüller, M. Wittmann, and B. Wohlmuth. A two-scale approach for efficient on-the-fly operator assembly in massively parallel high performance multigrid codes. Applied Numerical Mathematics, 122:14–38, 2017.

[2] L. B. Da Veiga, A. Buffa, G. Sangalli, and R. Vázquez. Mathematical analysis of variational isogeometric methods. Acta Numerica, 23:157–287, 2014.

[3] D. Drzisga, B. Keith, and B. Wohlmuth. The surrogate matrix methodology: a priori error estimation. SIAM Journal on Scientific Computing (to appear), 2019.

[4] D. Drzisga, B. Keith, and B. Wohlmuth. The surrogate matrix methodology: A reference implementation for low-cost assembly in isogeometric analysis. arXiv preprint arXiv:1909.04029, 2019.

[5] D. Drzisga, B. Keith, and B. Wohlmuth. The surrogate matrix methodology: Low-cost assembly for isogeometric analysis. arXiv preprint arXiv:1904.06971, 2019.

[6] T. J. Hughes, A. Reali, and G. Sangalli. Efficient quadrature for NURBS-based isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 199(5-8):301–313, 2010.

[7] R. Vázquez. A new design for the implementation of isogeometric analysis in Octave and Matlab: GeoPDEs 3.0. Computers & Mathematics with Applications, 72(3):523–554, 2016.