arXiv subjects

Fabien Casenave

Publications and source records attributed to Fabien Casenave.

23 records · Page 2Linked to original sources

arXiv2014

A nonintrusive Reduced Basis Method applied to aeroacoustic simulations

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

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arXiv2013

Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012) 539--542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the Empirical Interpolation Method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.

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arXiv2013

Coupled BEM-FEM for the convected Helmholtz equation with non-uniform flow in a bounded domain

We consider the convected Helmholtz equation modeling linear acoustic propagation at a fixed frequency in a subsonic flow around a scattering object. The flow is supposed to be uniform in the exterior domain far from the object, and potential in the interior domain close to the object. Our key idea is the reformulation of the original problem using the Prandtl--Glauert transformation on the whole flow domain, yielding (i) the classical Helmholtz equation in the exterior domain and (ii) an anisotropic diffusive PDE with skew-symmetric first-order perturbation in the interior domain such that its transmission condition at the coupling boundary naturally fits the Neumann condition from the classical Helmholtz equation. Then, efficient off-the-shelf tools can be used to perform the BEM-FEM coupling, leading to two novel variational formulations for the convected Helmholtz equation. The first formulation involves one surface unknown and can be affected by resonant frequencies, while the second formulation avoids resonant frequencies and involves two surface unknowns. Numerical simulations are presented to compare the two formulations.

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arXiv2013

A nonintrusive method to approximate linear systems with nonlinear parameter dependence

We consider a family of linear systems $A_μα=C$ with system matrix $A_μ$ depending on a parameter $μ$ and for simplicity parameter-independent right-hand side $C$. These linear systems typically result from the finite-dimensional approximation of a parameter-dependent boundary-value problem. We derive a procedure based on the Empirical Interpolation Method to obtain a separated representation of the system matrix in the form $A_μ\approx\sum_{m}β_m(μ)A_{μ_m}$ for some selected values of the parameter. Such a separated representation is in particular useful in the Reduced Basis Method. The procedure is called nonintrusive since it only requires to access the matrices $A_{μ_m}$. As such, it offers a crucial advantage over existing approaches that instead derive separated representations requiring to enter the code at the level of assembly. Numerical examples illustrate the performance of our new procedure on a simple one-dimensional boundary-value problem and on three-dimensional acoustic scattering problems solved by a boundary element method.

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arXiv2012

Accurate a posteriori error evaluation in the reduced basis method

In the reduced basis method, the evaluation of the a posteriori estimator can become very sensitive to round-off errors. In this note, the origin of the loss of accuracy is revealed, and a solution to this problem is proposed and illustrated on a simple example.

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