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

Chappell, David J.; Tanner, Gregor; Löchel, Dominik; Søndergaard, Niels

doi:10.1098/rspa.2013.0153

Cited by 35 publications

(64 citation statements)

References 45 publications

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“…A phase-space density ρ is transported from the phase-space on the boundary to the next boundary intersection via the boundary integral operator [11] …”

Section: Boundary Integral Operatorsmentioning

confidence: 99%

“…One advantage of a full phase-space formulation such as DEA is that problems due to caustics where ray trajectories focus on a single point in position space are avoided. In phase-space the rays do not intersect since their momentum coordinates are distinct, and it is only after projecting down onto position space that the caustics become apparent; for an example of a DEA simulation including caustics see [11]. Hence caustics do not affect the convergence of the boundary integral model itself, only the post-processing step to compute the density distribution within the solution domain.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. [11] it was shown that local, piecewise constant spatial approximations of the density on the edges of a triangulated surface could be efficiently computed using semi-analytic techniques, which exploit the local geometric simplicity. In this so-called Discrete Flow Mapping (DFM) approach, the trajectory flow is approximated by a discretized flow across a mesh.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Improved Approximation of Phase-Space Densities on Triangulated Domains Using Discrete Flow Mapping with p-Refinement

et al. 2017

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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.

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“…A phase-space density ρ is transported from the phase-space on the boundary to the next boundary intersection via the boundary integral operator [11] …”

Section: Boundary Integral Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Improved Approximation of Phase-Space Densities on Triangulated Domains Using Discrete Flow Mapping with p-Refinement

et al. 2017

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“…Frequency We introduce a weight function w ij in (5), which contains (in addition to the usual damping term) reflection and transmission coefficients characterising the coupling between subsystems i and j at the interface. When restricting the implementation to planar sub-domains of simple geometric shape such as for triangulated surfaces, there is a very efficient way of evaluating the operators B ij in terms of the so-called discrete flow mapping (DFM) technique as described in [1].…”

Section: A Basis Representation For the Ray Tracing Operator Bmentioning

confidence: 99%

“…Due to the geometric simplicity of typical (planar) mesh elements, DEA can be modified using the method of Discrete Flow Mapping (DFM) [1] which gives rise to huge efficiency gains. DEA provides then detailed resolution of the energy density variation throughout the structure under consideration.…”

Section: Introductionmentioning

confidence: 99%

Dynamical energy analysis on mesh grids: A new tool for describing the vibro-acoustic response of complex mechanical structures

et al. 2014

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We present a new approach for modelling noise and vibration in complex mechanical structures in the mid-to-high frequency regime. It is based on a dynamical energy analysis (DEA) formulation which extends standard techniques such as statistical energy analysis (SEA) towards non-diffusive wave fields. DEA takes into account the full directionality of the wave field and makes sub-structuring obsolete. It can thus be implemented on mesh grids commonly used, for example, in the finite element method (FEM). The resulting mesh based formulation of DEA can be implemented very efficiently using discrete flow mapping (DFM) as detailed in [1] and described here for applications in vibro-acoustics. A mid-to-high frequency vibro-acoustic response can be obtained over the whole modelled structure. Abrupt changes of material parameter at interfaces are described in terms of reflection/transmission ma- systems are considered: a double-hull structure used in the ship-building industry and a cast aluminium shock tower from a Range Rover. We demonstrate that DEA with DFM implementation can handle multi-mode wave propagation effectively, taking into account mode conversion between shear, pressure and bending waves at interfaces, and on curved surfaces.

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On hybrid convolution quadrature approaches for modeling time‐domain wave problems with broadband frequency content

Rowbottom

Chappell

2021

Numerical Meth Engineering

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We propose two hybrid convolution quadrature based discretizations of the wave equation on interior domains with broadband Neumann boundary data or source terms. The convolution quadrature method transforms the time-domain wave problem into a series of Helmholtz problems with complex-valued wavenumbers, in which the boundary data and solutions are connected to those of the original problem through the -transform. The hybrid method terminology refers specifically to the use of different approximations of these Helmholtz problems, depending on the frequency. For lower frequencies, we employ the boundary element method, while for more oscillatory problems, we develop two alternative high frequency approximations based on plane wave decompositions of the acoustic field on the boundary. In the first approach, we apply dynamical energy analysis to numerically approximate the plane wave amplitudes. The phases will then be reconstructed using a novel approach based on matching the boundary element solution to the plane wave ansatz in the frequency region where we switch between the low and high frequency methods. The second high frequency method is based on applying the Neumann-to-Dirichlet map for plane waves to the given boundary data. Finally, we investigate the effectiveness of both hybrid approaches across a range of numerical experiments.

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

Cited by 35 publications

References 45 publications

Improved Approximation of Phase-Space Densities on Triangulated Domains Using Discrete Flow Mapping with p-Refinement

Improved Approximation of Phase-Space Densities on Triangulated Domains Using Discrete Flow Mapping with p-Refinement

Dynamical energy analysis on mesh grids: A new tool for describing the vibro-acoustic response of complex mechanical structures

On hybrid convolution quadrature approaches for modeling time‐domain wave problems with broadband frequency content

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