Louis Kovalevsky

THE VARIATIONAL THEORY OF COMPLEX RAYS FOR THREE-DIMENSIONAL HELMHOLTZ PROBLEMS

This paper proposes an extension of the variational theory of complex rays (VTCR) to three-dimensional linear acoustics, The VTCR is a Trefftz-type approach designed for mid-frequency range problems and has been previously investigated for structural dynamics and 2D acoustics. The proposed 3D formulation is based on a discretization of the amplitude portrait using spherical harmonics expansions. This choice of discretization allows to substantially reduce the numerical integration work by taking advantage of well-known analytical properties of the spherical harmonics. It also permits (like with the previous 2D Fourier version) an effective a priori selection method for the discretization parameter in each sub-region, and allows to estimate the directivity of the pressure field by means of a natural definition of rescaled amplitude portraits. The accuracy and performance of the proposed formulation are demonstrated on a set of numerical examples that include results on an actual case study from the automotive industry.

AN ADAPTIVE NUMERICAL STRATEGY FOR THE MEDIUM-FREQUENCY ANALYSIS OF HELMHOLTZ'S PROBLEM

The variational theory of complex rays (VTCR) is a wave-based predictive numerical tool for medium-frequency problems. In order to describe the dynamic field variables within the substructures, this approach uses wave shape functions which are exact solutions of the governing differential equation. The discretized parameters are the number of substructures (h) and the number of wavebands (p) which describe the amplitude portraits. Its capability to produce an accurate solution with only a few degrees of freedom and the absence of pollution error make the VTCR a suitable numerical strategy for the analysis of vibration problems in the medium-frequency range. This approach has been developed for structural and acoustic vibration problems. In this paper, an error indicator which characterizes the accuracy of the solution is introduced and is used to define an adaptive version of the VTCR. Numerical illustrations are given.

Experimental validation of the Statistical Energy Analysis for coupled reverberant rooms

The Statistical Energy Analysis (SEA) is a statistical approach employed to solve high frequency problems of electromagnetically coupled reverberant cavities at a reduced computational cost. The key aspect of this approach is to avoid solving Maxwells equations inside the cavity, using the power balance principle, which leads directly to the ensemble mean response of the system. In addition, the SEA gives an estimate of the standard deviation of the cavitys energy. Moreover, the diffuse field reciprocity formula is used to calculate the coupling coefficient induced by an aperture, which is neither electrically small nor large. The numerical results obtained for two rooms coupled through a circular aperture are compared with experimental data obtained in the reverberation chamber at IETR. After a brief introduction of the motivation for such a method, the SEA equations are presented, and then the experimental setup is described in section III. Finally, the good agreement between simulation and experiment is presented in section IV.

A new methodology for assessing the global dynamic response of large shell structures under impact loading

Purpose – The determination of the vibration induced by an aircraft impact on an industrial structure requires dynamic studies. The determination of the response by using classical finite element method associated with explicit numerical schemes requires significant calculation time, especially during the transient stage. This kind of calculation requires several load cases to be analyzed in order to consider a wide range of scenarios. Moreover, a large frequency range has to be appropriately considered and therefore the mesh has to be very fine, resulting in a refined time discretization. The purpose of this paper is to develop new ways for calculating the shaking of reinforced concrete structures following a commercial aircraft impact (see Figure 1). The cutoff frequency for this type of loading is typically within the 50-100 Hz range, which would be referred to as the medium-frequency range. Design/methodology/approach – Taking into account this type of problem and assuming that the structure is approp...

A quasi‐optimal coarse problem and an augmented Krylov solver for the variational theory of complex rays

The Variational Theory of Complex Rays (VTCR) is an indirect Trefftz method designed to study systems governed by Helmholtz-like equations. It uses wave functions to represent the solution inside elements, which reduces the dispersion error compared to classical polynomial approaches but the resulting system is prone to be ill conditioned. This paper gives a simple and original presentation of the VTCR using the discontinuous Galerkin framework and it traces back the ill-conditioning to the accumulation of eigenvalues near zero for the formulation written in terms of wave amplitude. The core of this paper presents an efficient solving strategy that overcomes this issue. The key element is the construction of a search subspace where the condition number is controlled at the cost of a limited decrease of attainable precision. An augmented LSQR solver is then proposed to solve efficiently and accurately the complete system. The approach is successfully applied to different examples.

A hybrid approach to calculate mean response and variance in a reverberant environment

In this paper a statistical approach is employed to solve high frequency EMC problems at a reduced computational cost. The case of interest is a system lying within a reverberant cavity that has random or uncertain properties. If a conventional numerical approach is employed for this problem then very significant computational effort is required since Maxwells equations need to be solved for both the cavity and the system. The key aspect of the proposed approach is to avoid solving Maxwells equations inside the cavity by employing a relation known as the diffuse field reciprocity principle, which leads directly to the ensemble mean response of the system and its variance; all that is required is the impedance matrix of the system associated with radiation into infinite space. Theoretical developments leading to the mean and variance are presented. This technique is then applied to a numerical example.