An LP empirical quadrature procedure for reduced basis treatment of parametrized nonlinear PDEs
Name
Patera_Computer Methods in Applied Mechanics and Engineering.pdf
Description
Submitted version
Size
532.09 KB
Format
Adobe PDF
Checksum (MD5)
92c7d1af7dd4280e15c1d3d54fcaf959
Author(s)
Yano, Masayuki
Patera, Anthony T
Date Issued
2019
Journal
Computer Methods in Applied Mechanics and Engineering
Publisher
Elsevier BV
Version
Original manuscript
Abstract
© 2018 Elsevier B.V. We present a model reduction formulation for parametrized nonlinear partial differential equations (PDEs). Our approach builds on two ingredients: reduced basis (RB) spaces which provide rapidly convergent approximations to the parametric manifold; sparse empirical quadrature rules which provide rapid evaluation of the nonlinear residual and output forms associated with the RB spaces. We identify both the RB spaces and the sparse quadrature rules in the offline stage through a greedy training procedure over the parameter domain; the procedure requires the dual norm of the finite element (FE) residual at many training points in the parameter domain, but only very few FE solutions—the snapshots retained in the RB space. The quadrature rules are identified by a linear program (LP) empirical quadrature procedure (EQP) which (i) admits efficient solution by a simplex method, and (ii) directly controls the solution error induced by the approximate quadrature. We demonstrate the formulation for a parametrized neo-Hookean beam: the dimension of the approximation space and the number of quadrature points are both reduced by two orders of magnitude relative to FE treatment, with commensurate savings in computational cost.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/J.CMA.2018.02.028
