[Amath-seminars] Boeing Lecture on Thursday Oct. 30

Joel Zylberberg joelzy at uw.edu
Mon Oct 27 16:57:11 PDT 2014


Hi All,

Reminder that our next Boeing lecture will be Thurs Oct 30, by Charbel
Farhat from Stanford (details below).

Interested faculty / staff should please sign up on this google document
<https://docs.google.com/document/d/19r6lKXjN9Y2JYjtuJL-zayndX8J-rav17-SuZpdKecE/edit>
to
meet with the speaker.

I am out of town this week, so anyone requiring logistical assistance, etc.
should please contact Braden Brinkman (bradenb at uw.ed).



*Energy-Conserving Sampling and Weighting for the Model Reductionof
2nd-Order Nonlinear Dynamical Systems*

*Charbel Farhat, Stanford University*
*Thurs, Oct. 30, 4pm*
*Smith 205*

The computational efficiency of a typical projection-based nonlinear model
reduction method hinges on the efficient approximation, for
explicit computations, of the projection onto a subspace of a residual
vector. For implicit computations, it also hinges on the additional
efficient approximation of the projection of the Jacobian of this residual
with respect to the solution. The Empirical Interpolation
Method (EIM), its discrete counterpart DEIM, the Gauss-Newton with
Approximated Tensors (GNAT) method, and the Gappy Proper
Orthogonal Decomposition method (Gappy POD) are popular methods for
performing such approximations. They differ in several aspects such as
their applicability at the continuous, semi-discrete, or fully discrete
level(s), their underlying left and right projectors, and their suitability
for explicit and/or implicit schemes. However, they all share accuracy as
the primary driver for their algorithmic design. They have also all
demonstrated various forms of success for the reduction of nonlinear
computational models emanating from elliptic, parabolic, and first-order
hyperbolic partial differential equations. The first objective of this talk
however is to show that they all lack robustness for second-order
nonlinear dynamical systems because they do not necessarily preserve the
numerical stability properties of the computational model they reduce.

Consequently, the second objective of this talk is to present ECSW, an
Energy-Conserving Sampling and Weighting method for the model reduction
of second-order nonlinear dynamical systems such as those arising, for
example, in structural dynamics, solid mechanics, wave propagation, and
device analysis. This proposed hyper reduction method is physics-based and
natural for finite element semi-discretizations. It is applicable at both
the semi-discrete and discrete levels. Unlike all aforementioned reduction
methods, it preserves the Lagrangian structure associated with Hamilton's
principle and therefore is guaranteed to preserve the numerical stability
properties of the nonlinear system it reduces. The error committed by ECSW
during an online approximation is bounded by the error committed during the
offline approximation of the training samples. Therefore, the online error
can be estimated a priori and is controllable. The performance of ECSW
will be first demonstrated for a set of academic but nevertheless
challenging nonlinear dynamic response problems taken from the literature,
and compared to that of DEIM and its unassembled variant recently
introduced for finite element computations under the name UDEIM. Next, the
potential of ECSW for complex second-order dynamical systems with strong
nonlinearities will be highlighted with the realistic simulation of the
transient response of a generic V-hull vehicle to an underbody blast event.
For this hihgly nonlinear time-dependent problem, ECSW will be shown to
deliver an excellent level of accuracy while enabling the reduction of CPU
time by more than four orders of magnitude.

--
Joel Zylberberg

Acting Assistant Professor
Department of Applied Mathematics
University of Washington
Seattle, WA

http://faculty.washington.edu/joelzy/

~ The neuroscience is theoretical, but the fun is real
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