Concurrent algorithm for integrating three-dimensional B-spline functions into machines with shared memory such as GPU

• Published in: Computer methods in applied mechanics and engineering 2022-08, Vol.398, p.115201, Article 115201
• Main Authors: Szyszka, Anna, Woźniak, Maciej, Schaefer, Robert
• Format: Article
• Language: English
• Subjects: Algorithms; B spline functions; Canonical forms; Computer architecture; Concurrency; Finite element method; GPGPU; Isogeometric finite element method; Mathematical analysis; Numerical integration; Quadratures; Scheduling; Trace theory
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2022.115201

The aim of this paper is to analyze the integration for 3D isogeometric finite element method solvers and its effective scheduling on hierarchical computer architecture. Data necessary for concurrency over elements is independent, so computation on this level is trivially concurrent. However, constructing several layers of concurrency for the integration algorithm is challenging. In this work, we propose a multilevel concurrent integration algorithm associated with scheduling that brings one extra degree of possible speedup. Because of one extra degree of possible speedup, we analyze the concurrent integration inside elements. The scheduling algorithm is intended for strongly related hierarchical architectures of a GPU. Using trace theory and Foata Normal Form, we verify integrity of the proposed solution. Summing up, we propose a general method for analyzing concurrency of the integration algorithm. We instantiate this method on a classical element-based integration algorithm, however, this methodology is possible to apply for other integration algorithms, including sum factorization, fast numerical quadrature, or row-wise integration methods. •Multilayer parallel integrating coefficients of IsoGeometric Analysis IGA equations.•Trace theory model used for the formal verification.•Dikert graph based sub-optimal scheduling on cluster with GPU units.•Efficiency and scalability tests of proposed parallel IGA for L2 projection problem.•Application to sum factorization, fast numerical quadrature, row-wise integration.