Galerkin Method of Regularized Stokeslets for Procedural Fluid Flow with Control Curves

TL;DR

This work addresses the challenge of designing incompressible velocity fields that follow user-defined control curves by introducing a curve-based workflow built on the Method of Regularized Stokeslets (MRS) with a Galerkin discretization. Velocities (and optionally angular velocities) along polylines are enforced via a small dense linear system, producing a global velocity field in unbounded domains through a linear superposition $\mathbf{u}(\mathbf{x}) = \mathbf{S}^\epsilon(\mathbf{x}, \mathbf{y}) \mathbf{f}$, while allowing spatially varying regularization $\epsilon(\mathbf{y})$ and torque contributions. The approach yields intuitive, artist-friendly control with provable incompressibility, supports angular-velocity constraints via Kelvinlets, and achieves efficient evaluation with GPU acceleration in typical curve-based workflows, exemplified by integration in Houdini. This method broadens practical fluid authoring for graphics by combining curve-based constraints with physically grounded Stokes flow, reducing volumetric discretization requirements and enabling robust performance for small to moderate problem sizes in visuals pipelines.

Abstract

We present a new procedural incompressible velocity field authoring tool, which lets users design a volumetric flow by directly specifying velocity along control curves. Our method combines the Method of Regularized Stokeslets with Galerkin discretization. Based on the highly viscous Stokes flow assumption, we find the force along a given set of curves that satisfies the velocity constraints along them. We can then evaluate the velocity anywhere inside the surrounding infinite 2D or 3D domain. We also show the extension of our method to control the angular velocity along control curves. Compared to a collocation discretization, our method is not very sensitive to the vertex sampling rate along control curves and only requires a small linear system solve.

Figures (4)

• Figure 1: We compare the velocity field produced by Galerkin (left), collocation (middle), and point-based (right) discretizations with curves having different sampling rates. From the top, we have 512, 11, and 7 vertices along the control curve. For a dense set of points, all three methods perform equally well, but for a sparse set of points, the velocity field from the Galerkin method best follows the control curve. In the bottom row, we can observe that the only velocity we get with Galerkin discretization can respect the horizontal velocity along the long horizontal edge of the input control curve.
• Figure 2: We specify a velocity of constant magnitude, an angular velocity of constant magnitude, or both, along the control curve (a). We visualize the resulting velocity field on the vertical and horizontal planes by projecting the colored vectors, which represent the velocities, to the two planes. We can specify the velocity and angular velocity independently (b, c) or at the same time (d, e). When we perform the coupled solve of velocity and angular velocity (d), the result slightly differs from the decoupled one (e). While the coupled solve (d) respects the input constraints better, we found that the result of the decoupled solve (e), which outputs the sum of the velocity due to velocity and angular velocity control, is more intuitive given (b) and (c).
• Figure 3: We specify a constant magnitude velocity along the bunny curve with three different constant $\epsilon$ values. Changing the $\epsilon$ value effectively changes the influence distance.
• Figure 4: Given four control curves with constant magnitude velocity constraints pointing toward the center (left), we use our 2D version (middle) and 3D version (right) of our method to get an incompressible velocity field, respectively. With the 3D version, the field is not incompressible in the 2D slice.