Fundamental solutions for water wave animation

Abstract

This paper investigates the use of fundamental solutions for animating detailed linear water surface waves. We first propose an analytical solution for efficiently animating circular ripples in closed form. We then show how to adapt the method of fundamental solutions (MFS) to create ambient waves interacting with complex obstacles. Subsequently, we present a novel wavelet-based discretization which outperforms the state of the art MFS approach for simulating time-varying water surface waves with moving obstacles. Our results feature high-resolution spatial details, interactions with complex boundaries, and large open ocean domains. Our method compares favorably with previous work as well as known analytical solutions. We also present comparisons between our method and real world examples.

Making Procedural Water Waves Boundary‐aware

“…SWE discretizations are usually Eulerian [CL95, LvdP02, HHL*05, WMT07, ATBG08], but some researchers have proposed Lagrangian 2D SPH discretizations as well [LH10, SBC*11]. [JSMF*18] use a wavelet representation that discretizes waves as functions of space, frequency, and direction to handle local interactions with wind and obstacles, and [SHW19] use fundamental solutions to animate wave‐obstacle interactions. A recent method computes Lagrangian waves on top a moving water surface to add detail to an existing simulation [SSJ*20].…”

Coupling 3D Liquid Simulation with 2D Wave Propagation for Large Scale Water Surface Animation Using the Equivalent Sources Method

“…Our method couples the 2D wave solver of Schreck et al [SHW19] to the boundary of a 3D fluid simulation. This particular surface wave solver has a number of unique advantages over ones used in the past: First of all, unlike the constant‐speed waves produced by SWE and Lattice Boltzmann method wave simulations, [SHW19] is based on Airy wave theory and produces waves with physical behaviors closer to those of the 3D fluid simulation. Secondly, contrary to the methods that use 2D grids or meshes to represent the surrounding water surface waves, our combined method allows fluid domains that are literally infinitely large.…”

“…In Section 5, we explain how to ensure a seamless transition between 𝒟 2 d and 𝒟 3 d for linear waves radiating from 𝒟 3 d. We build upon [SHW19] and use the state given by the 3D liquid simulation as boundary conditions for the 2D solver. We also discuss how to modify the boundary conditions of the 3D fluid simulation to prevent artificial internal reflections, and to send in new waves from the 2D simulation.…”

Kelvin transformations for simulations on infinite domains

“…Assuming the fundamental solution (Green's function) to the PDE on the free space is known, one may directly construct the solution on an infinite domain by integrating the Green's function over the source. Integral methods have been applied to surface reconstructions [Barill et al 2018], water wave animations [Schreck et al 2019], and evaluation of fluid velocities from vortex filaments [Weißmann and Pinkall 2010], vortex sheets [Brochu et al 2012;Da et al 2015Da et al , 2016, vortex particles [Golas et al 2012;Zhang and Bridson 2014], or a vorticity field [Zhang et al 2015], just to name a few. These integrations are straightforward in a free space.…”