Abstract
A new approximation technique, called Reference Point Method (RPM), is proposed in order to reduce the computational complexity of algebraic operations for constructing reduced-order models in the case of time dependent and/or parametrized nonlinear partial differential equations. Even though model reduction techniques enableone to decrease the dimension of the initial problem in the sense that far fewer degrees of freedom are needed to represent the solution, the complexity of evaluating the nonlinear terms and assembling the low dimensional operator associated with the reduced-order model still scales with the size of the original high-dimensional model. This point can be critical, especially when the reduced-order basis changes throughout the solution strategy as it is the case for model reduction techniques based on Proper Generalized Decomposition (PGD). Based on the concept of spatial, parameter/time reference points and influence patches, the RPM defines a compressed version of the data from which an approximate low-rank separated representation by patch of the operators can be constructed by explicit formulas at low-cost without resorting to SVD-based techniques. An application of the RPM to PGD-based model reduction for a nonlinear parametrized elliptic PDE previously studied by other authors with reduced-basis method and EIM is proposed. It is shown that computational complexity to construct the reduced-order model can be divided in practice by one order of magnitude compared with the classical PGD approach.
Highlights
Numerical simulation has been playing an increasingly important role in science and engineering due to the need to describe realistic scenarios and derive tools to facilitate the virtual design of new structures while reducing the use of real prototypes
In order to illustrate the bottleneck of nonlinear model reduction and to explain why the construction of the reduced-order model involved during the preliminary step is CPU intensive, let come back to Problem 4 solved at the preliminary step
The expected decrease in computational cost by using reduced-basis approximation is quite modest whatever 215 the dimension reduction k ≪ N is. This bottleneck is clearly not specific to Proper Generalized Decomposition (PGD)-based model reduction but more generally to reducedbasis approximation in nonlinear model reduction. This problem arises due to the fact that: (i) Galerkin projection onto the ROB has to be performed as soon as the reduced-order basis evolves, (ii) non-linear 220 terms (Jacobian and residue) have to be evaluated at each iteration of the solution strategy for each new iterate which prevents pre-computations of operators
Summary
Numerical simulation has been playing an increasingly important role in science and engineering due to the need to describe realistic scenarios and derive tools to facilitate the virtual design of new structures while reducing the use of real prototypes. Another appealing family of model reductions which has received a growing interest during the last decade is based on the Proper Generalized Decomposition (PGD). For any model reduction technique based on the projection of the problem on a given reduced-order basis, the computational complexity of evaluating the nonlinear 65 terms (Jacobian or residue of the solution strategy) and assembling the ROM’s low dimensional operator scales with the size of the original high-dimensional model. This point can be critical especially when the reduced-order basis changes throughout the solution strategy as it is the case for model reduction techniques based on Proper Generalized Decomposition (PGD) This makes the bottleneck of model reduction strategies applied to nonlinear problems.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Computer Methods in Applied Mechanics and Engineering
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
