A parameterized non-intrusive reduced order model and
error analysis for general time-dependent nonlinear partial
differential equations and its applications
error analysis for general time-dependent nonlinear partial
differential equations and its applications
File(s)
Accepted version
Author(s)
Xiao, D
Fang, F
Pain, C
Navon, I
Type
Journal Article
Abstract
A novel parameterized non-intrusive reduced order model (P
-NIROM) based on
proper orthogonal decomposition (POD) has been developed.
This P-NIROM is a
generic and e
ffi
cient approach for model reduction of parameterized partia
l di
ff
eren-
tial equations (P-PDEs). Over existing parameterized redu
ced order models (P-ROM)
(most of them are based on the reduced basis method), it is non
-intrusive and inde-
pendent on partial di
ff
erential equations and computational codes. During the tra
ining
process, the Smolyak sparse grid method is used to select a se
t of parameters over a
specific parameterized space (
Ω
p
∈ R
P
). For each selected parameter, the reduced ba-
sis functions are generated from the snapshots derived from
a run of the high fidelity
model. More generally, the snapshots and basis function set
s for any parameters over
Ω
p
can be obtained using an interpolation method. The P-NIROM c
an then be con-
structed by using our recently developed technique [
50
,
53
] where either the Smolyak
or radial basis function (RBF) methods are used to generate a
set of hyper-surfaces
representing the underlying dynamical system over the redu
ced space.
The new P-NIROM technique has been applied to parameterized
Navier-Stokes
equations and implemented with an unstructured mesh finite e
lement model. The ca-
pability of this P-NIROM has been illustrated numerically b
y two test cases: flow past
a cylinder and lock exchange case. The prediction capabilit
ies of the P-NIROM have
been evaluated by varying the viscosity, initial and bounda
ry conditions. The results
show that this P-NIROM has captured the quasi-totality of th
e details of the flow with
CPU speedup of three orders of magnitude. An error analysis f
or the P-NIROM has
been carried out.
-NIROM) based on
proper orthogonal decomposition (POD) has been developed.
This P-NIROM is a
generic and e
ffi
cient approach for model reduction of parameterized partia
l di
ff
eren-
tial equations (P-PDEs). Over existing parameterized redu
ced order models (P-ROM)
(most of them are based on the reduced basis method), it is non
-intrusive and inde-
pendent on partial di
ff
erential equations and computational codes. During the tra
ining
process, the Smolyak sparse grid method is used to select a se
t of parameters over a
specific parameterized space (
Ω
p
∈ R
P
). For each selected parameter, the reduced ba-
sis functions are generated from the snapshots derived from
a run of the high fidelity
model. More generally, the snapshots and basis function set
s for any parameters over
Ω
p
can be obtained using an interpolation method. The P-NIROM c
an then be con-
structed by using our recently developed technique [
50
,
53
] where either the Smolyak
or radial basis function (RBF) methods are used to generate a
set of hyper-surfaces
representing the underlying dynamical system over the redu
ced space.
The new P-NIROM technique has been applied to parameterized
Navier-Stokes
equations and implemented with an unstructured mesh finite e
lement model. The ca-
pability of this P-NIROM has been illustrated numerically b
y two test cases: flow past
a cylinder and lock exchange case. The prediction capabilit
ies of the P-NIROM have
been evaluated by varying the viscosity, initial and bounda
ry conditions. The results
show that this P-NIROM has captured the quasi-totality of th
e details of the flow with
CPU speedup of three orders of magnitude. An error analysis f
or the P-NIROM has
been carried out.
Date Issued
2017-01-16
Date Acceptance
2016-12-24
Citation
Computer Methods in Applied Mechanics and Engineering, 2017, 317, pp.868-889
ISSN
0045-7825
Publisher
Elsevier
Start Page
868
End Page
889
Journal / Book Title
Computer Methods in Applied Mechanics and Engineering
Volume
317
Copyright Statement
© 2016 Elsevier B.V. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (E
Grant Number
EP/K003976/1
RG80519
Subjects
Science & Technology
Technology
Physical Sciences
Engineering, Multidisciplinary
Mathematics, Interdisciplinary Applications
Mechanics
Engineering
Mathematics
Parameterized
Non-intrusive ROM
PDE
RBF
POD
Smolyak sparse grid
PROPER ORTHOGONAL DECOMPOSITION
VARIATIONAL DATA ASSIMILATION
SHALLOW-WATER EQUATIONS
PETROV-GALERKIN METHODS
FINITE-ELEMENT METHODS
EMPIRICAL INTERPOLATION
SPARSE GRIDS
FLUID-FLOW
REDUCTION
STRATEGIES
Applied Mathematics
01 Mathematical Sciences
09 Engineering
Publication Status
Published