Improved Approximation of PhaseSpace Densities on Triangulated Domains Using Discrete Flow Mapping with pRefinement
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Abstract
We consider the approximation of the phasespace 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 phasespace 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 phasespace 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 prefinement in the orthogonal polynomial basis.
Keywords
High frequency wave asymptotics Ray tracing Frobenius–Perron operator Liouville equation Geometrical optics VibroacousticsMathematics Subject Classification
MSC 35Q82 MSC 37C30 MSC 65R201 Introduction
In this work we consider the case where \(\mathbf {V}\) describes a Hamiltonian system, and hence the divergence operator in the generalised Liouville equation (1) reduces to the Poisson bracket as \(\nabla \cdot (\mathbf {V}\rho )=\{\rho ,H\}\), where H is the associated Hamiltonian. This class of problems is particularly interesting since it includes, as an important case, short wavelength asymptotic approximations for solving linear wave equations in terms of their underlying ray dynamics. Note that for homogeneous domains with a constant wave speed, and thus a constant and conserved phasespace volume, then the right side of Eq. (3) evaluates to \(\rho (\phi ^{\tau }(X),0)\). In this context, direct “deterministic” methods based on directly tracking swarms of trajectories in phasespace have proved very popular. Such methods are often collectively referred to as ray tracing, see for example [6]. Methods related to ray tracing but tracking the timedynamics of interfaces in phasespace, such as moment methods and level set methods, have been developed in [14, 28, 32, 43, 44] amongst others. Direct physicspreserving finite difference discretisations of the Liouville equation have also been proposed in Refs. [19, 42]. These methods are best suited to problems with relatively low reflection orders (or problems in one space dimension) and have found applications in acoustics, seismology and optical illumination problems.
A number of approaches have been suggested in order to obtain efficient discretizations of the boundary operator in DEA. A boundary element method for the stationary Liouville equation (4) was proposed in [9, 10], which extended the DEA approach to larger multicomponent structures and threedimensional applications. It is worth emphasizing here that since the density \(\rho \) lives in phasespace, it also needs to be discretized in the direction (or momentum) coordinate. A global orthogonal polynomial basis approximation has typically been used for this purpose. One advantage of a full phasespace formulation such as DEA is that problems due to caustics where ray trajectories focus on a single point in position space are avoided. In phasespace 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 postprocessing step to compute the density distribution within the solution domain.
A major advantage of DEA is that the choice of subsystem division is no longer critical, since the SEA assumptions related to this criticality have been removed. In fact, one can use the elements of a mesh as substructures in DEA providing huge flexibility and a wealth of potential applications. Applying DEA on the same meshes used for finite element models in industrial applications means that the enduser can access midtohigh frequencies without the need for extensive remodelling, as would be required for SEA. 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 semianalytic techniques, which exploit the local geometric simplicity. In this socalled Discrete Flow Mapping (DFM) approach, the trajectory flow is approximated by a discretized flow across a mesh. The technique has since been extended to general convex polygonal mesh elements and to industrial scale applications [8].
Whilst DFM employs orthogonal polynomial approximations of the density in the direction coordinate, the spatial approximation has thus far been limited to piecewiseconstant local approximations on the edges of the mesh elements, or subdivisions of those edges, in order to compute the boundary operator using fast semianalytic methods. This can lead to poor accuracy, particularly in problems with high dissipation where the solutions are dominated by the initial density. In this work, we discuss a prefinement strategy in both position and direction to gain an improved approximation of the phasespace density on a prescribed mesh. To facilitate the efficient implementation of such a strategy we have derived semianalytic integration formulae for the projection of the boundary integral operator onto a higher order orthogonal polynomial basis. Here, the integration with respect to the spatial coordinate is performed analytically and the integration with respect to the direction coordinate is done numerically. In fact, a carefully designed strategy for computing the arising multidimensional integrals is crucial for the efficient implementation of the method on large meshes. We show in this paper that a direct approach to the analytic spatial integration using the repeated application of the integration by parts rule leads to highly unstable results. However, an indirect approach using the recurrence relations of the Legendre polynomials can be shown to give a stable representation for the integral in terms of modified Bessel functions. Furthermore, an efficient computation of the integrals, independently of the choice of orthogonal polynomial basis functions, is possible using a 2D adaptive and spectrally convergent quadrature method. In this case one can exploit the smoothness of the integrands in the boundary operator over appropriately defined subsets.
The remainder of the paper is structured as follows: in Sect. 2 we detail the boundary version of the Frobenius–Perron operator (3) as a model for propagating an initial phasespace density to give a solution to the stationary Liouville equation (4). In Sect. 3 we describe the finite dimensional approximation of the phasespace density in both the position and direction coordinates. The computational procedure for obtaining the discretized boundary integral operator is described, including three approaches for computing the spatial integrals: an unstable direct formula, a stable recursion formula and a spectral quadrature method. Finally, in Sect. 4 we give a selection of numerical results; we reproduce the wellknown decay rate for propagation into free space and perform computations on several bounded domain configurations with varying degrees of symmetry and complexity.
2 Propagating PhaseSpace Densities via Integral Operators
2.1 Boundary Integral Operators
Consider a polygonal domain \(\varOmega \subset \mathbb {R}^2\) with boundary \(\varGamma \) and the phasespace coordinates \(Y_s=(s,p_s)\) on \(\varGamma \) as illustrated in the left plot of Fig. 1. Here s parametrizes \(\varGamma \) and \(p_s\) denotes the direction component tangential to \(\varGamma \) at s. Figure 1 also depicts the boundary map \(\phi (s,p_s)=(s',p'_{s})\), which takes \((s,p_s)\) to a point \(s'\in \varGamma \) with direction \(p'_{s}\). We may write this simply as \(X_s=\phi (Y_s)\). In general, the form of the boundary map \(\phi \) depends on the specific problem. In our considerations, \(\phi \) obeys the law of (specular) reflection.
2.2 Source Terms
2.3 Multidomain Problems
A generalization to multidomain problems such as triangle meshes with subdomains \(\varOmega _{j}\), \(j=1,\dots ,N\), is straightforward by introducing a multidomain boundary map \(\phi _{i,j}\) and a weight function \(w_{i,j}\) describing the flow from the boundary of the domain \(\varOmega _j\) to the boundary of the domain \(\varOmega _i\), see the right plot of Fig. 1. Note that \(\varOmega =\cup _{j=1}^{N}\varOmega _{j}\) becomes the union of all subdomains and \(\varGamma \) becomes the union of all subdomain boundaries thus \(\varGamma =\cup _{j=1}^{N}\partial \varOmega _{j}\). Each subdomain \(\varOmega _j\) has its own phasespace boundary coordinates \((s_j,p_j)\). We then define the boundary integral operator \(\mathscr {B}_{i,j}\), which transports the phasespace density \(\rho ^{\varGamma }\) from the boundary phasespace of \(\varOmega _j\) to the boundary phasespace of \(\varOmega _i\) as illustrated in the right plot of Fig. 1. If the properties of two neighboring domains \(\varOmega _j\) and \(\varOmega _i\) are different, for example if \(c_i\ne c_j\), where \(c_i\) is the propagation speed in \(\varOmega _i\) (likewise for \(c_j\) in \(\varOmega _j\)), then the weight function \(w_{i,j}\) will account for the probability of transmission or reflection at the common edge. Note that the case of transporting the density within a subdomain is included in this formulation when \(i=j\). The operator \(\mathscr {B}\) is then constructed from the set of interdomain operators \(\mathscr {B}_{i,j}\). In order to obtain a discrete representation of the phasespace density \(\rho ^{\varGamma }\) and the operator \(\mathscr {B}_{i,j}\), we consider a finite basis approximation as described in the next section.
3 Discretization
In this section we discuss the discretization of the boundary operator as well as computational issues associated with the efficient and fast implementation of DFM on triangle meshes. We detail how the twodimensional integral in space and direction can be separated and provide analytical formulae for the spatial integral. We note that a direct approach to evaluating the spatial integral analytically is unstable in general for nonconstant spatial basis functions. An alternative stable iterative approach is presented instead and compared with an efficient, and widely applicable, spectrally convergent quadrature strategy. We conclude this section by outlining a computational algorithm for DFM with prefinement on triangle meshes.
3.1 DFM with pRefinement on Triangle Meshes
3.2 Exact Spatial Integration
We illustrate the numerical instability of the direct formula (23) by computing \(\mathfrak {I}_{m,m'}(\theta )\) for \(\theta \in (1,1)\) on a unit sided equilateral triangle with a specified initial edge (due to symmetry the result will be independent of this choice). In Fig. 3, the direct formula (23) is compared against the recursive solution (25) and a numerical solution using Clenshaw–Curtis quadrature [39] for different combinations of the polynomial degrees m and \(m'\). In both examples we set the dissipation rate to be \(\mu =0.005\) and the propagation speed as \(c=1\).
The left plot of Fig. 3 shows the case \(m=2\) and \(m'=0\), where all three approaches compute the same values for the integral to double precision. The numerical integration was performed using three integration subregions \(\theta _j\in (\pi /2,\pi /6]\cup (\pi /6,\pi /6)\cup [\pi /6,\pi /2)\) as described in (20) since \(\varphi _{\pm }=\pi /3\). The lack of smoothness of the integral \(\mathfrak {I}_{m,m'}(\theta )\) at the numerical integration subdivision points \(\theta =2\theta _j/\pi =\pm 1/3\) is also clear in Fig. 3, emphasizing the motivation for performing this subdivision. The right plot of Fig. 3 shows the case when \(m=m'=2\), which gives rise to large deviations between the direct formula and other two methods. Note that, in general, the discrepancy increases when the degree of the basis functions is increased and/or the dissipation factor \(\mu \) is decreased. The right plot of Fig. 3 shows that the direct formula (23) is already numerically unstable for relatively low degree basis functions. For the MATLAB implementation here, we find that using the recursive formula (25) is typically between two and three times faster than applying the spectral quadrature method. In order to demonstrate the flexibility of the method in accommodating general smooth orthonormal basis approximations in (12), we propose an efficient adaptive quadrature strategy in the following section.
3.3 Efficient Adaptive Quadrature Rule
In this section we describe the spectrally converging adaptive quadrature strategy that we use to compute the integral entries of the matrix \(\mathbf {B}\) (21) for general smooth orthonormal basis functions \(\hat{P}_{m}^{l}(s_j)\) and \(\tilde{P}_{n}(p_j)\). Note that whenever possible, it is preferable to adopt analytic spatial integration strategies for reasons of computational efficiency. However, in order to demonstrate the wider applicability of the proposed methodology, and for completeness, we detail a rapidly converging fully numerical strategy. We propose an adaptive Clenshaw–Curtis method [39, 40] and note that spectral convergence property depends crucially on the regularity of the integrand. Preservation of the rapid convergence property is the main motivation for the integral subdivision strategy discussed in Sect. 3.1.
 1.
Let \(\alpha =0\) and fix \(M_0+1\) as an initial number of scaled Clenshaw–Curtis quadrature nodes. Define a prescribed tolerance level "tol" for the accuracy of any quadrature rules.
 2.Calculate the \(M_\alpha +1\) values of \(\theta _j\) prescribed by the Clenshaw–Curtis quadrature nodes, then perform the following operations for each value of \(\theta _j\):

compute the reflection/transmission angle \(\theta '_i\) as described in the text immediately after Eq. (21);

compute the basis functions \(\tilde{P}_{n}(p_j)\) and \(\tilde{P}_{n'}(p'_i)\). For a Legendre polynomial basis this can be done using Eq. (13) and the recurrence (24);

compute the probability function \(\lambda _{i,j}(\theta _j)\) as described in the text immediately after Eq. (15);

find the admissible ranges \((s_{min}(\theta _j),s_{max}(\theta _j))\) for the integration with respect to \(s_j\) using Eqs. (18) and (19).

 3.If \(\hat{P}_{m}^{l}(s_j)\) are scaled Legendre polynomials, then compute the spatial integral analytically as described in Sect. 3.2. Otherwise:
 (a)
Let \(\beta =0\) and fix \(m_0+1\) as an initial number of scaled Clenshaw–Curtis quadrature nodes.
 (b)Calculate the \(m_\beta +1\) values of \(s_j\) prescribed by the Clenshaw–Curtis quadrature nodes, then perform the following operations for each value of \(s_j\):

compute the point \(s'_i\) and the distance \(d(s_i',s_j)\) using the geometry of the mesh and the vector prescribed by \((s_j,p_j)\). On a triangle mesh one can directly compute d as a linear function of \(s_j\) using the sine rule;

compute the orthogonal polynomial basis functions \(\hat{P}_{m}^{l}(s_j)\) and \(\hat{P}_{m'}^{l'}(s'_{i})\). The computation will depend on the chosen basis.

 (c)
Compute the integral over \(s_j\) appearing in Eq. (21) using \(m_\beta \) quadrature nodes. If the tolerance "tol" has not been reached, then add one to \(\beta \), let \(m_{\beta }=2^\beta m_0\) and return to step (b). If "tol" has been reached then proceed to step 4.
 (a)
 4.
Compute the integral over \(\theta _j\) appearing in Eq. (21) using \(M_\alpha \) quadrature nodes. If the tolerance "tol" has not been reached, then add one to \(\alpha \), let \(M_{\alpha }=2^\alpha M_0\) and return to step 2. If "tol" has been reached then use the computed value as an entry of the matrix \(\mathbf {B}\).
4 Numerical Results
4.1 Free Space Propagation
In the right plot of Fig. 4 we show the averaged relative error for the energy density computed by DFM with different orders of approximation in space, and a fixed order in direction. The relative error values are averaged over annular segments with radial width \(\varDelta _R=1\), as indicated by the white dashed lines in the left plot of Fig. 4. Note that we have excluded the average error in the circular segment containing the source point since the DFM result here is just the direct (and exact) source contribution. Clearly, as the order of the spatial approximation \(N_s\) increases, the error is correspondingly decreasing. The approximation in direction is fixed with a relatively large choice of \(N_p=300\), since the freespace problem only gives propagation in a single direction, radially outward from the source. As such, the analytic solution in phasespace will behave as a deltadistribution in the direction coordinate, making it challenging to model using a polynomial basis. An identical value for the direction basis order was used in all cases for consistency, and to clearly show the convergence related to increasing the spatial approximation order only.
The example shown in Fig. 4 has a relatively small and simple mesh. A basis order of \(N_p=300\) would be computationally infeasible for complex builtup engineering applications. However, the directional dependence of the energy density in complex structures is often relatively smooth due to multiple reflections and irregular geometry, which makes such a large choice of \(N_p\) unnecessary. In general, engineering applications will also require the study of dissipative problems and hence we now change the dissipation factor to be \(\mu =1\) in the freespace radiation problem considered above. Figure 5 shows the numerical solution for this problem with \(N_p=12\) and with either a constant or a quadratic spatial basis.
The lower right plot of Fig. 5 shows the ray tracing solution (26) with an additional dissipation factor for consistency with the DFM calculation. The energy density predicted by a DFM simulation with \(N_s=0\) and \(N_p=12\) is shown in the upper left plot. Spurious localization and shadowing effects arise due to the piecewise constant approximation in space introducing discontinuities at the vertices of the mesh. The upper right plot of Fig. 5 shows another DFM simulation on the same mesh, but here with \(N_s=2\) and \(N_p=12\). The circular symmetry of the energy density shown in the exact solution has been preserved showing a distinct qualitative improvement from the piecewise constant approximation. The higher order simulation leads to an approximate doubling of the computation time, but even so, the simulations presented here took less than 30 s.
4.2 Closed Cavity Simulation
We perform numerical simulations on a closed polygonal cavity taken from Ref. [22]. In Fig. 6, we have placed a source point inside the cavity at (10, 5); whereas in Fig. 7, the source point is located on the boundary at (2.5, 7). The total number of mesh elements is 538 and we take \(\omega =100\pi \), \(c=1\) and \(\rho ^f=1\) for all computations.
Figure 6 shows the predicted energy density for different approximation orders \(N_s\) and \(N_p\), and with dissipation rate \(\mu =0.5\). The upper left plot of Fig. 6 shows the result with a piecewise constant approximation in space and \(N_p=12\) in direction. Localized energy stripes near the source point are again evident, as they were in the free space propagation problem. The upper right plot of Fig. 6 shows the same calculation, but now with \(N_s=2\). We notice that the circular symmetry around the source has been restored for only a modest increase in the computational time from around 30–45 s. The lower plots of Fig. 6 show the results of higher order simulations in both the space and the direction basis approximations with a maximum computational time of 6 min. The results are visually similar to those with \(N_s=2\) and \(N_p=12\). The sequence of plots in Fig. 6 demonstrate the convergence of DFM for increasing \(N_s\) and the gain in accuracy that can be achieved by applying spatial approximations with \(N_s\geqslant 2\) close to a source point.
The density distribution inside a cavity will depend on a number of factors including the geometry, the source point location and the damping. Figure 7 shows the result of moving the source point to the boundary and increasing the damping to \(\mu =1\). Notice the clear difference between the energy density in the upper and lower parts of the cavity. The result in the left plot was computed using a piecewise constant approximation in space, and shows a sudden jump between the energy densities in the upper and lower regions. In the right plot we observe a smoother decay from the lower region to the upper region as a result of increasing the spatial approximation order. Therefore the right plot better captures the wave energy transmitted indirectly from the source to the upper part of the cavity due to reflections at the lower boundary. Hence, a higher order spatial approximation can restore symmetry and reduce spurious shadowing effects in both the near and farfield of a point source excitation.
4.3 Coupled TwoCavity System
The right plot of Fig. 8 shows the ratio \(\mathrm {P}_\text {left}/\mathrm {P}_\text {right}\) for five different frequency values. We compare the results of the DFM algorithm presented here for various orders of approximation with the DEA results given in [7]. The spectral convergence of the integrals arising in the DFM computation for this example is demonstrated in “Appendix”. The DEA results use a \(6^{\mathrm {th}}\) order Chebyshev basis approximation along each edge of the polygonal cavity, and the same order of approximation globally in direction. The right plot of Fig. 8 also shows the results of 21 finite element method (FEM) simulations for the associated Helmholtz equation wave problem at an equispaced range of frequencies within \(\pm 5\) Hz of the center frequencies used in the DEA and DFM computations. The FEM computations are performed using discontinuous Galerkin methods as reported in [7].
The results shown in the right plot of Fig. 8 demonstrate a good agreement between the DFM and the DEA results, particularly for lower damping (i.e. lower frequencies). We note that for finer triangle meshes than the one employed here (larger N), the results of the three DFM simulations become indistinguishable. For the mesh in the computations here, however, there is a considerable improvement when \(N_s\) is increased from 0 to 2. The DFM results with \(N_s=2\) and \(N_p=12\) or \(N_p=24\) agree well with the DEA result and lie towards the center of the range of FEM wave problem results. The computational times for these results were approximately 95 s per frequency when \(N_s=0\), 150 s per frequency when \(N_s=2\), \(N_p=12\) and 550 s per frequency when \(N_s=2\), \(N_p=24\). Note that this compares favourably with the computational times for the DEA method reported in [7]; the 6\({\text {th}}\) order DEA result shown in Fig. 8 took around 3000 s per frequency to compute.
4.4 Discussion: Computational Costs and Scaling
The computational costs of running a MATLAB based DFM code for the numerical experiments in this paper have been reported in the previous subsections. The times quoted are for nonoptimised code running on a 3.4 GHz processor and without taking advantage of the fact that DFM is embarrassingly parallel with respect to the number of mesh elements. The cost of the algorithm as a whole is dominated by the cost of computing the entries of the matrix \(\mathbf {B}\) and so scales with the number of nonzero entries of this matrix. As the problem size grows, the costs associated with solving the linear system become more significant, particularly in the case of low dissipation, but this still plays a relatively minor role for the examples presented here. An important advantage of the method over conventional numerical solvers for the Helmholtz equation when the wavenumber k becomes large is that the computational costs scale independently of k.
The approximate scaling of the algorithm as the number of degrees of freedom is increased is linear with respect to the number of mesh cells, since each triangular mesh element only transmits rays to a maximum of three other cells with which it shares a direct physical connection (as well as possible reflections into the same element). The scaling is less favourable with respect to the basis approximation order, and approximately scales as \(O(N_s^2 N_p^2)\). However, we note that our results demonstrate a significant increase in accuracy for a moderate increase in the order of the spatial basis approximation from \(N_s=0\) to \(N_s=2\). In order to achieve the quadratic scaling of the algorithm with respect to the basis order in practice, then optimised integration routines such as the semianalytic spectral methods described in Sect. 3.3 are crucial. Otherwise, the computational costs associated with the quadrature would also grow with the basis order in a suboptimal manner. The results shown in Sect. 4 therefore demonstrate that DFM with prefinement using semianalytic integration and spectrally convergent quadrature is an efficient method for predicting energy distributions in complex structures.
5 Conclusions
A Discrete Flow Mapping algorithm with prefinement in phasespace has been presented for approximating phasespace densities on triangulated domains. In particular, this is the first study incorporating orthogonal polynomial basis approximations of any specified order \(N_s\ge 0\) in position space. The additional computational cost resulting from the higher order spatial approximations has been minimised by a careful evaluation of the propagation operator using semianalytic integration methods for a Legendre polynomial basis, or full spectral quadrature in general. The numerical results presented verify the approach and demonstrate that for practical applications, a moderate increase in the order of the spatial approximation from \(N_s=0\) to \(N_s=2\) can be sufficient to improve the qualitative nature of the numerical solution.
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