TY - JOUR

T1 - A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

AU - Mika, Michal L.

AU - Hughes, Thomas J.R.

AU - Schillinger, Dominik

AU - Wriggers, Peter

AU - Hiemstra, René R.

N1 - Funding Information: M.L. Mika, R.R. Hiemstra and D. Schillinger gratefully acknowledge funding from the German Research Foundation through the DFG Emmy Noether Award SCH 1249/2-1. T.J.R. Hughes and R.R. Hiemstra were partially supported by the National Science Foundation Industry/University Cooperative Research Center (IUCRC) for Efficient Vehicles and Sustainable Transportation Systems (EV-STS), and the United States Army CCDC Ground Vehicle Systems Center (TARDEC/NSF Project # 1650483 AMD 2 ). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors thank Mona Dannert and Udo Nackenhorst for very helpful discussions and comments.

PY - 2021/6/1

Y1 - 2021/6/1

N2 - The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L2-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2d dimensional domain, where d, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C0-conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.

AB - The Karhunen–Loève series expansion (KLE) decomposes a stochastic process into an infinite series of pairwise uncorrelated random variables and pairwise L2-orthogonal functions. For any given truncation order of the infinite series the basis is optimal in the sense that the total mean squared error is minimized. The orthogonal basis functions are determined as the solution of an eigenvalue problem corresponding to the homogeneous Fredholm integral equation of the second kind, which is computationally challenging for several reasons. Firstly, a Galerkin discretization requires numerical integration over a 2d dimensional domain, where d, in this work, denotes the spatial dimension. Secondly, the main system matrix of the discretized weak-form is dense. Consequently, the computational complexity of classical finite element formation and assembly procedures as well as the memory requirements of direct solution techniques become quickly computationally intractable with increasing polynomial degree, number of elements and degrees of freedom. The objective of this work is to significantly reduce several of the computational bottlenecks associated with numerical solution of the KLE. We present a matrix-free solution strategy, which is embarrassingly parallel and scales favorably with problem size and polynomial degree. Our approach is based on (1) an interpolation based quadrature that minimizes the required number of quadrature points; (2) an inexpensive reformulation of the generalized eigenvalue problem into a standard eigenvalue problem; and (3) a matrix-free and parallel matrix–vector product for iterative eigenvalue solvers. Two higher-order three-dimensional C0-conforming multipatch benchmarks illustrate exceptional computational performance combined with high accuracy and robustness.

KW - Fredholm integral eigenvalue problem

KW - Isogeometric analysis

KW - Kronecker products

KW - Matrix-free solver

KW - Random fields

UR - http://www.scopus.com/inward/record.url?scp=85102649189&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2021.113730

DO - 10.1016/j.cma.2021.113730

M3 - Article

AN - SCOPUS:85102649189

VL - 379

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 113730

ER -