# The surrogate matrix methodology: Low-cost assembly for isogeometric analysis

## TL;DR

This paper introduces a low-cost surrogate matrix methodology for isogeometric analysis that significantly reduces assembly time by performing limited quadrature and interpolating remaining matrix entries, with negligible impact on accuracy.

## Contribution

The paper presents a novel surrogate matrix approach that reduces computational cost in IGA by combining selective quadrature with B-spline interpolation, easily integrable into existing software.

## Key findings

- Over fifty-fold reduction in assembly time.
- Negligible impact on solution accuracy.
- Applicable to various PDE problems.

## Abstract

A new methodology in isogeometric analysis (IGA) is presented. This methodology delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed from element-scale quadrature formulas. To generate surrogate matrices, quadrature must only be performed on certain elements in the computational domain. This, in turn, determines only a subset of the entries in the final matrix. The remaining matrix entries are computed by a simple B-spline interpolation procedure. Poisson's equation, membrane vibration, plate bending, and Stokes' flow problems are studied. In these problems, the use of surrogate matrices has a negligible impact on solution accuracy. Because only a small fraction of the original quadrature must be performed, we are able to report beyond a fifty-fold reduction in overall assembly time in the same software. The capacity for even further speed-ups is clearly demonstrated. The implementation used here was achieved by a small number of modifications to the open-source IGA software library GeoPDEs. Similar modifications could be made to other present-day software libraries.