The Variational Theory of Complex Rays to solve Mid-Frequency acoustical problem 

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High-Frequency approaches extended to Mid-Frequency

High frequency analysis are strongly dominated by a tool called Statistical Energy Analysis (SEA) [Lyon and Maidanik, 1962]. Its simple theoretical frame, its inexpensive computational cost and its efficacy (when basic hypothesis are respected) make this tool a obliged choice when high-frequency is to tackle. SEA brings, due to its statistical nature, global previsions in term of energy over a frequency band. No spacial informations are provided. This is not a a problem when, as in high frequency, the diffuse field hypothesis is respected. Unlikely, in mid-frequency this hypothesis is not fully respected. SEA prevision in terms of average energy are still valid but less useful due to a still strong localization of the response. In this section SEA principles will be introduced. As a second step some SEA-based techniques modified to keep into account mid-frequency are presented.

The Statistical Energy Analysis

Since its introduction in [Lyon and Maidanik, 1962] SEA becomes the preferential tool for high-frequency analysis. Nowadays it’s commonly used both as a research tool and as an industrial tool [Troclet, 1995]. The SEA basic idea is to divide the structure in subsystem and then study the energy transmission between the structure to find the vibrational response. This response has to be thought as an average in terms of energy on a wave band. To calculate the energy a power balance on substructures i is to consider: Pi in = Pi diss+åj Pi j coup.

Statistical modal Energy distribution Analysis

A very interesting extension of SEA to mid-frequency is the Statistical modal Energy distribution Analysis also known as SmEdA [Guyader et al., 1988]. SmEdA relies on the SEA energetic approach but adds a kinetic energy calculation in every subsystem in order improve the information quality and so approach the mid-frequency range. For a vibroacoustic problem a decouping between the acoustic nature problem and the structural dynamic problem. Evaluation of the energy transfer is based on an a priori calculation of all coupled system eigenmodes. A FEM based snapshot technique allows then to find SEA needed coefficient. Successful applications of this method can be found in [Maxit and Guyader, 2001b, Maxit and Guyader, 2001a,Totaro et al., 2009]. This approach provides satisfactory results for plate assemblies. Moreover it allows to understand the influence of geometry and damping on the energy exchanges between each component of the system. The principal weak point of the method is the necessity of a FEM calculation. This is in contrast with SEA simplicity and efficacy, in some case can result very expensive.

Ray Tracing Method

Derived from linear optic the Ray Tracing Method was first introduced in [Krokstad et al., 1968] to predict acoustic performances in rooms. It proposes to calculate the vibrational field by a set of rays (which are plane waves propagating) whose path is followed until fully dumped. Transmissions and  reflections of these rays are computed by classical Snell law. If frequency and damping are high enough, RTM is a inexpensive and accuratemethod. If not calculation time increases making this technique a non optimal choice. An other difficulty concerns the choice of imposed initial direction for the wave propagation. If the geometry is complex making the optimal choice is an hard task. Exemples of application of this method could be found in [Allen and Berkley, 1979] for acoustic and in [Chae and Ih, 2001] for thin plate.

Quasi-SEA method

As stated in sub-section 3.1, SEA is limited in high frequency by its strong hypothesis. The basic idea of Quasi-SEA is to introduce a modal energy approach. In [Mace, 2003] SEA hypothesis are analyzed in details and a technique to relax these hypothesis in order to extend the lower limit of validity is proposed. Expressions for the energy influence coefficients of a built-up structure are found in terms of the modes of the whole structure.
These coefficients relate frequency average energies of the subsystems to the subsystem input powers. Rain-on-the-roof excitation over a frequency band W is assumed. It is then seen that the system can be described by an SEA model only if a particular condition involving the mode shapes of the system is satisfied. Broadly, the condition holds if the mode shapes of the modes in the frequency band of excitation are, on average, typical enough of all the modes of the system in terms of the distribution of energy throughout the system. If this condition is satisfied then the system can be modeled using an quasi-SEA approach, irrespective of the level of damping or of the strength of coupling. However, the resulting model need not be of a proper SEA form, and in particular the indirect coupling loss factors may not be negligible.

Asymptotical Scaled Modal Analysis

The Asymptotical Scaled Modal Analysis (ASMA) was first introduced for plates in [De Rosa and Franco, 2008,De Rosa and Franco, 2010]. The main hypothesis of ASMA is that the modal frequency response of a system can be approximated by the modal frequency response of a scaled system, when approaching mid-frequency. The main system and the scaled system are linked by an equivalence in terms of quadratic velocity. Quadratic velocity is a global characteristic of the system that can be driven from an inexpensive SEA analysis. If the original system is properly scaled in terms of side lengths and damping, the square velocity is the same. This means that the scaled system keeps energetic properties of the original unscaled system. The new scaled properties makes the system easily solvable by a standard FEM with few modes. The techniques leads to an important reduction of computational costs and can be directly applied to any finite element solver. ASMA has been applied to metallic and composite plates.

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A Trefftz methods category in order to solve Mid-Frequency  problems

Trefftz methods born from the idea to impose an a priori solution of the approximation space. These function verifies the equilibrium equation but not the boundary condition. The desired solution in this space converge towards the exact solution under the hypothesis of complete metric space [Herrera, 1984]. This property is called Tcompletion. [Zienkiewicz, 1997] and [Kita and Kamiya, 1995] provides a review of Trefftz methods applied in different context. Common characteristics of Trefftz methods are a fast convergency in terms of DOF and a ill-conditioning derived from the chosen approximation space.

UltraWeak Variational Formulation

Ultra Weak Variational Formulation is based on a domain discretization in elements. Every element is connected to others by an interface and on this interface a variable is added. This variable must satisfy a weak formulation over the boundary. Pressure is the associated primal variable and it must satisfy Helmotz equation. The vibrational field is approximated by a set of plane waves and a Galerkin routine leads to resolution of a system matrix whose solution is the interface variables’ vector. Continuity between the elements is taken into account in the formulation by the intermediary of a dual variable (normal speed) way similar to the method FETI (see subsection 2.1.4). Once the interface variables are computed, the field inside each element can be reconstructed by solving a problem locally.
When large problems are solved, the method typically uses a pre-conditioner. Although UWVF allows to make little effort computations, it can present numerical instabilities and remains, in principle, an h-method. It also suffers from poor condition number (as the totality of Trefftz methods). In [Cessenat and Despres, 1998], it is demonstrated that a uniform distributed propagation directions maximizes the determinant of the UWVF matrix, and therefore the distribution is leading to the best possible condition number. A comparison of the method with PUFEM (see section 2.3.1) is presented in [Huttunen et al., 2006] on the problem solving Helmholtz 2D irregular meshes. The authors conclude that both methods provide accurate results with coarse meshes. A general trend is that UWVF seems more efficient than the PUFEM in the high frequencies, while PUFEM is better in low frequencies (which is not surprising being PUFEM a low-frequency derived method). On the other hand, the PUFEM condition number is much better than UWVF (even preconditioned) at higher frequencies. Note also that the work presented in [Gittelson et al., 2009] show that UWVF is a special case of Discontinuous Galerkin approaches (see [Farhat et al., 2003]) using plane waves.

Table of contents :

Contents
List of Figures
Introduction
1 The Mid-Frequency problem in acoustic 
1 Reference problem and notation
2 Low-Frequency approaches extended to Mid-Frequency
2.1 The finite element method and its improvements to tackle midfrequency
2.2 The Boundary Element Method
2.3 The enrichment methods
2.4 The meshless methods
3 High-Frequency approaches extended to Mid-Frequency
3.1 The Statistical Energy Analysis
3.2 Energy Flow Analysis
3.3 Wave Intensity Analysis
3.4 Statistical modal Energy distribution Analysis
3.5 Ray Tracing Method
3.6 Quasi-SEA method
3.7 Asymptotical Scaled Modal Analysis
4 A Trefftz methods category in order to solve Mid-Frequency problems
4.1 Ultra Weak Variational Formulation
4.2 T-elements
4.3 The Discontinuous Enrichment Method
4.4 Wave Boundary Element Method
4.5 Wave Based Method
5 Hybrid methods for Mid-Frequency
5.1 The Hybrid Finite Element/Statistical Energy Analysis method .
6 Conclusion
2 The Variational Theory of Complex Rays to solve Mid-Frequency acoustical problem 
1 An introduction to VTCR
2 The basic concepts of the VTCR
3 Properties of the VTCR
4 Illustration of the performance of the VTCR in acoustical and structural
4.1 Two-dimensional Helmholtz problems in acoustics
4.2 Structural vibration problems
4.3 Structures containing holes
4.4 Structural vibrations of shells
4.5 The Fourier Version of the VTCR
4.6 Error estimator
4.7 3D acoustics
5 VTCR, still an open question?
3 The Reduced Order Model techniques 
1 Reduced Order Model techniques
2 The Proper Orthogonal Decomposition
2.1 Reduced Order Model based on POD
3 The Proper Generalized Decomposition
4 PGD-VTCR, an alliance to tackle mid-frequency broad band 
1 Proper Generalized Decomposition and VTCR: a ROM technique to perform efficient frequency band calculations
1.1 The reference problem and the choice of the approximation .
2 A simple algorithm to introduce PGD-VTCR
2.1 A numerical example
2.2 A surprising numerical problem
3 A Petrov-Galerkin algorithm
3.1 A numerical example
3.2 How to proper dimension matrices
3.3 Petrov-Galerkin algorithm’s limitations
4 Conclusion
5 A new class of algorithms 
1 A dedicated pre-conditioner for VTCR
2 An improved algorithm for PGD-VTCR
3 Minimal residue direction algorithm: the state of the art for PGD-VTCR on frequency bands
3.1 A numerical example
4 Conclusion
6 PGD applied to acoustic with uncertainty parameters 
1 Uncertainty in Mid-Frequency acoustic
2 A PGD technique to take account for uncertainty in VTCR for Mid-
Frequency analysis
2.1 1D validation example
2.2 A low cost solution towards high-frequency
2.3 2D preliminary results
3 Conclusion
Conclusion 
Bibliography 

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