164 research outputs found
An Efficient Reduced-Order Approach for Nonaffine and Nonlinear Partial Differential Equations
In the presence of nonaffine and highly nonlinear terms in parametrized partial differential equations, the standard Galerkin reduced-order approach is no longer efficient, because the evaluation of these terms involves high computational complexity. An efficient reduced-order approach is developed to deal with “nonaffineness” and nonlinearity. The efficiency and accuracy of the approach are demonstrated on several test cases, which show significant computational savings relative to classical numerical methods and relative to the standard Galerkin reduced-order approach.Singapore-MIT Alliance (SMA
Efficient and accurate nonlinear model reduction via first-order empirical interpolation
We present a model reduction approach that extends the original empirical
interpolation method to enable accurate and efficient reduced basis
approximation of parametrized nonlinear partial differential equations (PDEs).
In the presence of nonlinearity, the Galerkin reduced basis approximation
remains computationally expensive due to the high complexity of evaluating the
nonlinear terms, which depends on the dimension of the truth approximation. The
empirical interpolation method (EIM) was proposed as a nonlinear model
reduction technique to render the complexity of evaluating the nonlinear terms
independent of the dimension of the truth approximation. We introduce a
first-order empirical interpolation method (FOEIM) that makes use of the
partial derivative information to construct an inexpensive and stable
interpolation of the nonlinear terms. We propose two different FOEIM algorithms
to generate interpolation points and basis functions. We apply the FOEIM to
nonlinear elliptic PDEs and compare it to the Galerkin reduced basis
approximation and the EIM. Numerical results are presented to demonstrate the
performance of the three reduced basis approaches.Comment: 38 pages, 6 figures, 6 table
Reduced formulation of a steady fluid-structure interaction problem with parametric coupling
We propose a two-fold approach to model reduction of fluid-structure
interaction. The state equations for the fluid are solved with reduced basis
methods. These are model reduction methods for parametric partial differential
equations using well-chosen snapshot solutions in order to build a set of
global basis functions. The other reduction is in terms of the geometric
complexity of the moving fluid-structure interface. We use free-form
deformations to parameterize the perturbation of the flow channel at rest
configuration. As a computational example we consider a steady fluid-structure
interaction problem: an incmpressible Stokes flow in a channel that has a
flexible wall.Comment: 10 pages, 3 figure
Parametric free-form shape design with PDE models and reduced basis method
We present a coupling of the reduced basis methods and free-form deformations for shape optimization and design of systems modelled by elliptic PDEs. The free-form deformations give a parameterization of the shape that is independent of the mesh, the initial geometry, and the underlying PDE model. The resulting parametric PDEs are solved by reduced basis methods. An important role in our implementation is played by the recently proposed empirical interpolation method, which allows approximating the non-affinely parameterized deformations with affinely parameterized ones. These ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization. The proposed approach is demonstrated on an airfoil inverse design problem. © 2010 Elsevier B.V
Model reduction and level set methods for shape optimization problems
In this work two topics related to mathematical shape optimization are considered. Topological optimization methods need not know the correct topology (number of connected components and possible holes) of the optimal shape beforehand. Shape optimization can be performed by a topological gradient descent algorithm. Computational applications of topological optimization and level set based shape optimization involve the optimal damping of vibrating structures and an inverse problem of reconstructing a shape based on noisy interferogram measurements.
For parametric shape optimization with partial differential constraints the model reduction approach of reduced basis methods is considered. In the reduced basis method a basis of snapshot solutions is used to construct a problem-dependent approximation space that has much smaller dimension than the underlying finite element approximations. The state constraints for optimization are then replaced with their reduced basis approximation, leading to efficient shape optimization methods. Computational examples involve the optimal engineering design of airfoils in potential and thermal flow
Model order reduction with novel discrete empirical interpolation methods in space-time
This work proposes novel techniques for the efficient numerical simulation of
parameterized, unsteady partial differential equations. Projection-based
reduced order models (ROMs) such as the reduced basis method employ a
(Petrov-)Galerkin projection onto a linear low-dimensional subspace. In
unsteady applications, space-time reduced basis (ST-RB) methods have been
developed to achieve a dimension reduction both in space and time, eliminating
the computational burden of time marching schemes. However, nonaffine
parameterizations dilute any computational speedup achievable by traditional
ROMs. Computational efficiency can be recovered by linearizing the nonaffine
operators via hyper-reduction, such as the empirical interpolation method in
matrix form. In this work, we implement new hyper-reduction techniques
explicitly tailored to deal with unsteady problems and embed them in a ST-RB
framework. For each of the proposed methods, we develop a posteriori error
bounds. We run numerical tests to compare the performance of the proposed ROMs
against high-fidelity simulations, in which we combine the finite element
method for space discretization on 3D geometries and the Backward Euler time
integrator. In particular, we consider a heat equation and an unsteady Stokes
equation. The numerical experiments demonstrate the accuracy and computational
efficiency our methods retain with respect to the high-fidelity simulations
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