Dinh Bao Phuong Huynh

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi’s and Babuška’s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi’s saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuška’s inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.

Real-Time Reliable Simulation of Heat Transfer Phenomena

In this paper we discuss the application of the certified reduced basis method and the associated software package rbMIT c to “worked problems” in steady and unsteady conduction. Each worked problem is characterized by an input parameter vector — material properties, boundary conditions and sources, and geometry — and desired outputs — selected fluxes and temperatures. The methodology and associated rbMIT c software, as well as the educational worked problem framework, consists of two distinct stages: an Offline (or “Instructor”) stage in which a new heat transfer worked problem is first created; and an Online (or “Lecturer”/“Student”) stage in which the worked problem is subsequently invoked in (say) various in‐ class, project, or homework settings. In the very inexpensive Online stage, given an input parameter value, the software returns both (i) an accurate reduced basis output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element dis

A Two-Step Certified Reduced Basis Method

In this paper we introduce a two-step Certified Reduced Basis (RB) method. In the first step we construct from an expensive finite element “truth” discretization of dimension

Multifidelity Uncertainty Propagation for Optimization Under Uncertainty

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Adaptive Port Reduction in Static Condensation

an intermediate RB model of dimension

A STATIC CONDENSATION REDUCED BASIS ELEMENT APPROXIMATION: APPLICATION TO THREE-DIMENSIONAL ACOUSTIC MUFFLER ANALYSIS

N\ll {\mathcal{N}}

Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings

. In the second step we construct from this intermediate RB model a derived RB (DRB) model of dimension M≤N. The construction of the DRB model is effected at cost

Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics

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A Successive Constraint Linear Optimization Method for Lower Bounds of Parametric Coercivity and Inf-Sup Stability Constants

and in particular at cost independent of

The extended finite element method for fracture in composite materials

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