Helmholtz equation

Summary

In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where ∇2 is the Laplace operator, k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example, consider the wave equation \left(\nabla^2-\frac{1}{c^2}\fra

Official source

https://en.wikipedia.org/wiki/Helmholtz_equation

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EPFL1996

We study a PDE-constrained optimization problem, where the shape and liner material of the nacelle of an aircraft engine are optimized in order to minimize the noise radiated by the engine. More precisely, the acoustic problem is modeled by the Helmholtz equation with varying wavenumber k on an exterior domain. A model reduction strategy is employed to alleviate the cost of the design optimization: the minimal rational interpolation technique is used to construct a surrogate (w.r.t. k) for the quantity of interest at fixed shape/material parameter values, and a parametric model order reduction approach is employed to combine surrogates at different shape/material designs, resulting in a non-intrusive methodology. Numerical experiments for shape and shape/material optimization are provided, to showcase the effectiveness of the presented methodology.

2021

We introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives on a different discrete space that resolves the local singularities of the solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least-squares or an interpolatory approach, yielding the standard rational interpolation method (SRI), a vector- or function-valued version of it (

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-SRI), and the minimal rational interpolation method (MRI). In the context of building an approximation for linear or quadratic functionals of the Helmholtz solution, we perform several numerical experiments to compare the proposed methodologies. Our simulations show that, for interior resonant problems (whose singularities are encoded by poles on the real axis), the spatially adaptive

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-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the

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-SRI method seems to be the best-performing one.

2021

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-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the

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-SRI method seems to be the best-performing one.

2021