Editing Fluid Flows with Divergence-Free Biharmonic Vector Field Interpolation

Abstract

Achieving satisfying fluid animation through numerical simulation can be time-consuming because such simulations are computationally expensive to perform and there are few practical post-processing tools for editing of completed simulations – often, the user must modify their scene setup and launch it again from scratch. To address this challenge, we present a divergence-free biharmonic vector field interpolation and extrapolation method for reusing and/or stitching together spatial regions of existing flows. Given velocities and velocity gradients on the boundary of a domain at each timestep, which may be either user-defined or drawn from existing simulations, we fill in the given domain by constructing an optimally smooth, divergence-free, boundary-satisfying vector field. We measure smoothness using the Laplacian energy to allow smooth boundary behavior and enforce divergence constraints through explicit Lagrange multipliers. The prior methods for this problem suffer from non-zero divergence and associated visible compression artifacts, or cannot smoothly match the desired slopes at the domain boundaries. Moreover, we introduce a new extrapolation scheme that can handle unprescribed boundaries by smoothly extending the vector field through the unspecified boundary. In this case, we measure the smoothness using the Hessian energy which provides well-behaved solutions for “free” or natural boundary conditions. We demonstrate that our new interpolation and extrapolation procedures always produce smooth and incompressible flows, as well as enabling a range of natural simulation editing capabilities including hole-filling, copy-pasting, extrapolation, and scene stretching.