Geometry-based approximation of waves in complex domains

TL;DR

This work introduces a geometry-based surrogate model for time-domain waves in 2D polygonal domains, representing the solution as a sum of field components that encode a free-space propagation plus boundary-induced reflections and diffractions. Reflections are modeled via geometrical optics with a method-of-images approach, while diffractions are incorporated through a time-domain adaptation of the uniform theory of diffraction using a fixed angular modulation, enabling an efficient 1D-in-space reduction. A timetable-driven skeleton automatically determines the number of components, and both a priori and a posteriori error indicators are provided to assess accuracy. Numerical experiments on wedges, cavities, and tall rooms show the surrogate captures key wavefronts with modest errors (often a few percent) and substantial speedups over full FE simulations, even enabling rapid re-evaluation for varied sources or frequencies. The authors also provide open-source code and discuss extensions to 3D, additional physics, and parametrized problems.

Abstract

We consider wave propagation problems over 2-dimensional domains with piecewise-linear boundaries, possibly including scatterers. We assume that the wave speed is constant, and that the initial conditions and forcing terms are radially symmetric and compactly supported. We propose an approximation of the propagating wave as the sum of some special space-time functions. Each term in this sum identifies a particular field component, modeling the result of a single reflection or diffraction effect. We describe an algorithm for identifying such components automatically, based on the domain geometry. To showcase our proposed method, we present several numerical examples, such as waves scattering off wedges and waves propagating through a room in presence of obstacles. Software implementing our numerical algorithm is made available as open-source code.

Figures (15)

• Figure 1: Computation of some timetable entries. The boundary $\partial\Omega$ has 11 sides, so that, e.g., $(\bm a_1)_{14}$ is related to $\bm y_3$ and $(\bm a_1)_{17}$ is related to $\bm y_6$. The shadowed area $\Omega\setminus\Omega_1$ is in darker grey.
• Figure 2: Graphical representation of a reflection off edge $\gamma$. On the left, the law of reflection prescribes $\theta_r=\theta_i$. We show the straight line $\widetilde{\gamma}$ supporting $\gamma$ with a dotted stroke. For a given observation point $\bm x$, $\bm y(\bm x)$ denotes the point of incidence of the reflected field component. On the right, computation of the spatial support $\Omega_n$ (light-grey area) and its complementary shadow zone $\Omega\setminus\Omega_n$ (dark-grey area) for the reflected field component, in the presence of a rectangular obstacle. The dashed portion of edge $\gamma$ denotes the shadow $\gamma\setminus\gamma^{(i)}$. The shadow zone consists of two connected components.
• Figure 3: Example of reflection off an edge in the presence of an obstacle, from \ref{['fig:reflection']}. Neumann conditions are imposed on all edges. Source wave (left), reflected wave (middle), and superimposition of the two (right). Note how the obstacle creates a shadow zone for source and reflected waves. For simplicity, in this plot we are not showing any reflection or diffraction effects due to the rectangular obstacle, since they would be modeled at different stages of the algorithm.
• Figure 4: Initial conditions for the wedge examples, indexed #1 through #4 from left to right. The (dashed) distance between the center of the Gaussian and the boundary vertex is 4 units in all cases.
• Figure 5: Free-space solution $\Psi$. The dashed line denotes the upper bound of the "causality cone" of $\Psi$, i.e., $\rho=t+R$, with $R=1$.

Theorems & Definitions (3)

• Remark 2.1
• Remark 2.2
• Remark 2.3