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

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

On the appearance of internal wave attractors due to an initial or parametrically excited disturbance

In this paper we solve two initial value problems for two-dimensional internal gravity waves. The waves are contained in a uniformly stratified, square-shaped domain whose sidewalls are tilted with respect to the direction of gravity. We consider several disturbances of the initial stream function field and solve both for its free evolution and for its evolution under parametric excitation. We do this by developing a structure-preserving numerical method for internal gravity waves in a two-dimensional stratified fluid domain. We recall the linearized, inviscid Euler-Boussinesq model, identify its Hamiltonian structure, and derive a staggered finite difference scheme that preserves this structure. For the discretized model, the initial condition can be projected onto normal modes whose dynamics is described by independent harmonic oscillators. This fact is used to explain the persistence of various classes of wave attractors in a freely evolving (i.e. unforced) flow. Under parametric forcing, the discrete dynamics can likewise be decoupled into Mathieu equations. The most unstable resonant modes dominate the solution, forming wave attractors

On discretisation schemes for a boundary integral model of stochastic ray propagation

A boundary integral operator method 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 discretisation schemes for this class of boundary integral operators using piecewise constant collocation in the spatial variable and either the Nyström method or the collocation method in the momentum variable. The simplicity of the spatial basis means that the corresponding spatial integration can be performed analytically. Convergence properties of the discretisation schemes and strategies for numerical implementation are presented and discussed

Two-dimensional mobile breather scattering in a hexagonal crystal lattice

We describe, for the first time, the full 2D scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated to the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60$^\circ$ scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.Comment: 4 pages, 5 figure

Improved approximation of phase-space densities on triangulated domains using Discrete Flow Mapping with p-refinement

We consider the approximation of the phase-space flow of a dynamical system on a triangulated surface using an approach known as Discrete Flow Mapping. Such flows are of interest throughout statistical mechanics, but the focus here is on flows arising from ray tracing approximations of linear wave equations. An orthogonal polynomial basis approximation of the phase-space density is applied in both the position and direction coordinates, in contrast with previous studies where piecewise constant functions have typically been applied for the spatial approximation. In order to improve the tractability of an orthogonal polynomial approximation in both phase-space coordinates, we propose a careful strategy for computing the propagation operator. For the favourable case of a Legendre polynomial basis we show that the integrals in the definition of the propagation operator may be evaluated analytically with respect to position and via a spectrally convergent quadrature rule for the direction coordinate. A generally applicable spectral quadrature scheme for integration with respect to both coordinates is also detailed for completeness. Finally, we provide numerical results that motivate the use of p-refinement in the orthogonal polynomial basis

Mass recovery of carbonated fabrics of glass fibres after isothermal heating

Acknowledgement: Authors acknowledge financial support from Latvian National Program IMIS2Leaching of Na+ ions in sodium oxide (Na2O) and silica (SiO2) containing glass is well investigated mainly due to its weak weathering. The object of this study was naturally (at room conditions) leached, steady state product on surface of sodium oxide-silica-alumina (Al2O3) glass fibers (in fabric) in a form of shell of "glyed" trona crystals as a result of interaction of leached Na+ ions and H2O and CO2 from atmosphere. There are presented results of continued former investigation of mass loss by isothermal heating of fabric and mass recovery in different atmospheres during the first phase of adsorption (at least 0.25h) without changes of state of crystals obtained during preheating at different temperatures. There are observed two ways of decomposition of trona (Na3H (CO3)2•2H2O) with its beginning at about 55-570C and 73-750C. The regression analysis of mass restoring in different atmospheres indicates to simultaneous and exponential mass increase by physical adsorption of CO2 and H2O having the different parameters of exponents vs time. Decomposition of trona is discussed in terms of parameters of exponent vs preheating temperature.Institute of Solid State Physics, University of Latvia as the Center of Excellence has received funding from the European Union’s Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-TeamingPhase2 under grant agreement No. 739508, project CAMART

Uncertainty quantification for phase-space boundary integral models of ray propagation

Vibrational and acoustic energy distributions of wave fields in the high-frequency regime are often modeled using flow transport equations. This study concerns the case when the flow of rays or non-interacting particles is driven by an uncertain force or velocity field and the dynamics are determined only up to a degree of uncertainty. A boundary integral equation description of wave energy flow along uncertain trajectories in finite two-dimensional domains is presented, which is based on the truncated normal distribution, and interpolates between a deterministic and a completely random description of the trajectory propagation. The properties of the Gaussian probability density function appearing in the model are applied to derive expressions for the variance of a propagated initial Gaussian density in the weak noise case. Numerical experiments are performed to illustrate these findings and to study the properties of the stationary density, which is obtained in the limit of infinitely many reflections at the boundary

Weakly coupled heat bath models for Gibbs-like invariant states in nonlinear wave equations

We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.Comment: 17 pages, 1 figur