| 1. |
- Pettersson, Klas, 1983, et al.
(author)
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A Feedforward Neural Network for Modeling of Average Pressure Frequency Response
- 2022
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In: Acoustics Australia. - : Springer Science and Business Media LLC. - 0814-6039 .- 1839-2571. ; 50:2, s. 185-201
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Journal article (peer-reviewed)abstract
- The Helmholtz equation has been used for modeling the sound pressure field under a harmonic load. Computing harmonic sound pressure fields by means of solving Helmholtz equation can quickly become unfeasible if one wants to study many different geometries for ranges of frequencies. We propose a machine learning approach, namely a feedforward dense neural network, for computing the average sound pressure over a frequency range. The data are generated with finite elements, by numerically computing the response of the average sound pressure, by an eigenmode decomposition of the pressure. We analyze the accuracy of the approximation and determine how much training data is needed in order to reach a certain accuracy in the predictions of the average pressure response.
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| 2. |
- Aiyappan, S., et al.
(author)
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Homogenization of a Locally Periodic Oscillating Boundary
- 2022
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In: Applied Mathematics and Optimization. - : Springer Science and Business Media LLC. - 1432-0606 .- 0095-4616. ; 86:2
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Journal article (peer-reviewed)abstract
- This paper deals with the homogenization of a mixed boundary value problem for the Laplace operator in a domain with locally periodic oscillating boundary. The Neumann condition is prescribed on the oscillating part of the boundary, and the Dirichlet condition on a separate part. It is shown that the homogenization result holds in the sense of weak L2 convergence of the solutions and their flows, under natural hypothesis on the regularity of the domain. The strong L2 convergence of average preserving extensions of the solutions and their flows is also considered.
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| 3. |
- Aiyappan, S., et al.
(author)
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HOMOGENIZATION OF A NON-PERIODIC OSCILLATING BOUNDARY VIA PERIODIC UNFOLDING
- 2022
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In: DIFFERENTIAL EQUATIONS & APPLICATIONS. - : Element d.o.o.. - 1847-120X .- 1848-9605. ; 14:1, s. 31-47
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Journal article (peer-reviewed)abstract
- This paper deals with the homogenization of an elliptic model problem in a two-dimensional domain with non-periodic oscillating boundary by the method of periodic unfolding. For the non-periodic oscillations, a modulated unfolding is used. The L-2 convergence of the solutions and their fluxes are shown, under natural hypotheses on the domain.
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| 4. |
- Donato, Patrizia, et al.
(author)
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HOMOGENIZATION OF A 2D TWO-COMPONENT DOMAIN WITH AN OSCILLATING THICK INTERFACE
- 2022
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In: Mathematics and Mechanics of Complex Systems. - : Mathematical Sciences Publishers. - 2326-7186 .- 2325-3444. ; 10:2, s. 103-154
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Journal article (peer-reviewed)abstract
- This paper deals with the homogenization of an elliptic boundary value problem in a finite cylindrical domain that consists of two connected components separated by a periodically oscillating interface situated in a band B of positive measure. That is, the amplitude of the oscillating interface is supposed to be fixed, while the period of oscillations is small. On the interface, the flux is assumed to be continuous, and the jump of the solution on the interface is assumed to be proportional to the flux through the interface. Unlike previous works in the literature, here the coefficients are highly oscillating in any directions. For this reason, we need to adapt the periodic unfolding method to our situation, and introduce some related functional spaces. The limit solution is a couple (u1, u2), where u1 is defined in one side Q1 and in B, and u2 is defined in the other side Q2 and in B. We prove that the homogenized problem is a coupled system, where ui solves a homogenized PDE in Qi, with i = 1, 2, while the two limits solve two coupled differential equations B, where only the derivative in one direction appears. We describe also the boundary conditions in each part of the boundaries, and the L2 convergence of the solutions and the fluxes is established. Finally, we prove the convergence of the energies. The main tools when proving these results are a suitable weak compactness result and an accurate study of the limit of the interface integrals on the oscillating boundary. As an illustration of the accuracy of the approximations, a numerical example is provided.
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| 5. |
- Pettersson, Klas, 1983
(author)
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Localization of eigenfunctions in a thin cylinder with a locally periodic oscillating boundary
- 2022
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In: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 511:1
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Journal article (peer-reviewed)abstract
- We study a Dirichlet spectral problem for a second-order elliptic operator with locally periodic coefficients in a thin cylinder. The lateral boundary of the cylinder is assumed to be locally periodic. When the thickness of the cylinder ε tends to zero, the eigenvalues are of order ε−2 and described in terms of the first eigenvalue μ(x1) of an auxiliary spectral cell problem parametrized by x1, while the eigenfunctions localize with rate ε.
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