David J. Chappell

Solving the stationary Liouville equation via a boundary element method

Energy distributions of linear wave fields are, in the high frequency limit, often approximated in terms of flow or transport equations in phase space. Common techniques for solving the flow equations in both time dependent and stationary problems are ray tracing and level set methods. In the context of predicting the vibro-acoustic response of complex engineering structures, related methods such as Statistical Energy Analysis or variants thereof have found widespread applications. We present a new method for solving the transport equations for complex multi-component structures based on a boundary element formulation of the stationary Liouville equation. The method is an improved version of the Dynamical Energy Analysis technique introduced recently by the authors. It interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. We demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation.

Boundary element dynamical energy analysis: A versatile method for solving two or three dimensional wave problems in the high frequency limit

Dynamical energy analysis was recently introduced as a new method for determining the distribution of mechanical and acoustic wave energy in complex built up structures. The technique interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. As such the applicability of the method is wide ranging and additionally includes the numerical modelling of problems in optics and more generally of linear wave problems in electromagnetics. In this work we consider a new approach to the method with enhanced versatility, enabling three-dimensional problems to be handled in a straightforward manner. The main challenge is the high dimensionality of the problem: we determine the wave energy density both as a function of the spatial coordinate and momentum (or direction) space. The momentum variables are expressed in separable (polar) coordinates facilitating the use of products of univariate basis expansions. However this is not the case for the spatial argument and so we propose to make use of automated mesh generating routines to both localise the approximation, allowing quadrature costs to be kept moderate, and give versatility in the code for different geometric configurations.

Dynamical energy analysis for built-up acoustic systems at high frequencies

Standard methods for describing the intensity distribution of mechanical and acoustic wave fields in the high frequency asymptotic limit are often based on flow transport equations. Common techniques are statistical energy analysis, employed mostly in the context of vibro-acoustics, and ray tracing, a popular tool in architectural acoustics. Dynamical energy analysis makes it possible to interpolate between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. In this work a version of dynamical energy analysis based on a Chebyshev basis expansion of the Perron-Frobenius operator governing the ray dynamics is introduced. It is shown that the technique can efficiently deal with multi-component systems overcoming typical geometrical limitations present in statistical energy analysis. Results are compared with state-of-the-art hp-adaptive discontinuous Galerkin finite element simulations.

Discrete flow mapping: transport of phase space densities on triangulated surfaces

Energy distributions of high-frequency linear wave fields are often modelled in terms of flow or transport equations with ray dynamics given by a Hamiltonian vector field in phase space. Applications arise in underwater and room acoustics, vibroacoustics, seismology, electromagnetics and quantum mechanics. Related flow problems based on general conservation laws are used, for example, in weather forecasting or in molecular dynamics simulations. Solutions to these flow equations are often large-scale, complex and high-dimensional, leading to formidable challenges for numerical approximation methods. This paper presents an efficient and widely applicable method, called discrete flow mapping, for solving such problems on triangulated surfaces. An application in structural dynamics, determining the vibroacoustic response of a cast aluminium car body component, is presented.

A Burton-Miller inverse boundary element method for near-field acoustic holography

An inverse boundary element method based on the Burton-Miller integral equation is proposed for reconstructing the Neumann boundary data from pressure values on a conformal surface in the near-field of an arbitrary radiating object. The accuracy of the reconstruction is compared with that of a method based on the more commonly used Helmholtz integral equation. In particular, the behavior at characteristic frequencies, which are known to be problematic in the Helmholtz integral equation for the forward problem, is examined. The effect of regularization is considered, including the L-curve parameter selection method. Numerical computations are given for noisy data generated from an internal point source.

Transport of phase space densities through tetrahedral meshes using discrete flow mapping

Discrete flow mapping was recently introduced as an efficient ray based method determining wave energy distributions in complex built up structures. Wave energy densities are transported along ray trajectories through polygonal mesh elements using a finite dimensional approximation of a ray transfer operator. In this way the method can be viewed as a smoothed ray tracing method defined over meshed surfaces. Many applications require the resolution of wave energy distributions in three-dimensional domains, such as in room acoustics, underwater acoustics and for electromagnetic cavity problems. In this work we extend discrete flow mapping to three-dimensional domains by propagating wave energy densities through tetrahedral meshes. The geometric simplicity of the tetrahedral mesh elements is utilised to efficiently compute the ray transfer operator using a mixture of analytic and spectrally accurate numerical integration. The important issue of how to choose a suitable basis approximation in phase space whilst maintaining a reasonable computational cost is addressed via low order local approximations on tetrahedral faces in the position coordinate and high order orthogonal polynomial expansions in momentum space.

A boundary integral formalism for stochastic ray tracing in billiards

Determining the flow of rays or non-interacting particles driven by a force or velocity field is fundamental to modelling many physical processes. These include particle flows arising in fluid mechanics and ray flows arising in the geometrical optics limit of linear wave equations. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain.

A boundary integral method for modelling vibroacoustic energy distributions in uncertain built up structures

Abstract A phase-space boundary integral method is developed for modelling stochastic high-frequency acoustic and vibrational energy transport in both single and multi-domain problems. The numerical implementation is carried out using the collocation method in both the position and momentum phase-space variables. One of the major developments of this work is the systematic convergence study, which demonstrates that the proposed numerical schemes exhibit convergence rates that could be expected from theoretical estimates under the right conditions. For the discretisation with respect to the momentum variable, we employ spectrally convergent basis approximations using both Legendre polynomials and Gaussian radial basis functions. The former have the advantage of being simpler to apply in general without the need for preconditioning techniques. The Gaussian basis is introduced with the aim of achieving more efficient computations in the weak noise case with near-deterministic dynamics. Numerical results for a series of coupled domain problems are presented, and demonstrate the potential for future applications to larger scale problems from industry.

Boundary integral models of stochastic ray propagation: Discretisation via the collocation and Nyström methods

A boundary integral framework for stochastic ray tracing in billiards was recently proposed in [1]. In particular, a phase-space boundary integral model for propagating uncertain ray or particle flows was described and shown to interpolate between deterministic and random models of the flow propagation. In this work we describe a discretisation scheme for this class of boundary transfer operators using piecewise constant collocation in the spatial variable and the Nystrom method in the momentum variable. The simplicity of the spatial basis functions means that the corresponding spatial integration can be performed analytically. A range of quadrature rules are compared for the Nystrom method and numerical results with relevance to high-frequency vibro-acoustic wave problems are presented.

Ray and wave scattering in smoothly curved thin shell cylindrical ridges

Abstract We propose wave and ray approaches for modelling mid- and high-frequency structural vibrations through smoothed joints on thin shell cylindrical ridges. The models both emerge from a simplified classical shell theory setting. The ray model is analysed via an appropriate phase-plane analysis, from which the fixed points can be interpreted in terms of the reflection and transmission properties. The corresponding full wave scattering model is studied using the finite difference method to investigate the scattering properties of an incident plane wave. Through both models we uncover the scattering properties of smoothed joints in the interesting mid-frequency region close to the ring frequency, where there is a qualitative change in the dynamics from anisotropic to simple geodesic propagation.