Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations
TL;DR: In this paper, the authors extended the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in volving.
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Abstract: In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in- volving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numeri- cal results are presented to assess our approach.
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Citations
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References
The Reduced Basis Method for Incompressible Viscous Flow Calculations
TL;DR: In this article, the reduced basis method is used in conjunction with a standard continuation technique to approximate the solution curve for the nonlinear equations resulting from discretizing the Navier-Stokes equations by finite element methods.
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Certified Real-Time Solution of Parametrized Partial Differential Equations
Nguyen Ngoc Cuong,Karen Veroy,Anthony T. Patera +2 more
- 01 Jan 2005
TL;DR: This work assumes that μ is a P-vector of parameters in a prescribed closed input domain D ⊂ ℝp, and thus encapsulates the behavior relevant to the desired engineering context.
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Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems
TL;DR: The procedure is introduced; the asymptotic bounding properties and optimal convergence rate of the error estimator are proved; computational considerations are discussed; and, finally, corroborating numerical results are presented.
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Parametric families of reduced finite element models. theory and applications
TL;DR: In this paper, a plate element model of a cantilevered box beam with varying rib stiffnesses is used to demonstrate efficiency and highlight practical difficulties of the proposed approaches for predictions of static responses, modal frequencies, modeshapes, and sensitivities of those quantities to design parameters.
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