Reduced Basis Approximation for Shape Optimization in Thermal Flows with a Parametrized Polynomial Geometric Map

Reduced basis approximations for geometrically parametrized advection-diffusion equations are investigated. The parametric domains are assumed to be images of a reference domain through a piecewise polynomial map; this may lead to nonaffinely parametrized diffusion tensors that are treated with an empirical interpolation method. An a posteriori error bound including a correction term due to this approximation is given. Results concerning the applied methodology and the rigor of the corrected error estimator are shown for a shape optimization problem in a thermal flow.

[1]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[2]  Annalisa Quaini,et al.  Reduced basis methods for optimal control of advection-diffusion problems ∗ , 2007 .

[3]  Luca Dedè,et al.  Adaptive and Reduced Basis methods for optimal control problems in environmental applications , 2008 .

[4]  G. Rozza,et al.  Parametric free-form shape design with PDE models and reduced basis method , 2010 .

[5]  Jan S. Hesthaven,et al.  A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations , 2008 .

[6]  Karsten Urban,et al.  A Reduced-Basis Method for solving parameter-dependent convection-diffusion problems around rigid bodies , 2006 .

[7]  Anthony T. Patera,et al.  A natural-norm Successive Constraint Method for inf-sup lower bounds , 2010 .

[8]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[9]  Thomas W. Sederberg,et al.  Free-form deformation of solid geometric models , 1986, SIGGRAPH.

[10]  Ngoc Cuong Nguyen,et al.  A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations , 2007, J. Comput. Phys..

[11]  A. Patera,et al.  A Successive Constraint Linear Optimization Method for Lower Bounds of Parametric Coercivity and Inf-Sup Stability Constants , 2007 .

[12]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[13]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[14]  Gianluigi Rozza,et al.  Reduced basis methods for Stokes equations in domains with non-affine parameter dependence , 2009 .

[15]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .