Geometric multigrid (GMG) methods are a fundamental tool for efficiently solving large sparse linear systems. A requirement for GMG is a hierarchy of grids; however, many practical volumetric domains are available only as single, irregular tetrahedral meshes, making the construction of a multigrid hierarchy necessary. Existing approaches often trade off speed against hierarchy quality: remeshing- or coarsening-based methods can be expensive to construct, whereas graph-based techniques are fast but often yield weaker multigrid performance. We introduce GravoTet, which bridges this gap by combining geometric structure with graph-based efficiency to construct fast and effective multigrid hierarchies. GravoTet builds a vertex hierarchy and then generates graph-Voronoi diagrams whose dual cells define coarse tetrahedra, enabling rapid construction of multigrid levels. Boundary elements are explicitly prioritized during both sampling and tet generation to preserve boundary. In our evaluation, we solve Poisson and biharmonic problems on irregular tetrahedral meshes and compare GravoTet against state-of-the-art geometric multigrid, algebraic multigrid and direct solvers, demonstrating superior performance, particularly on large meshes.

We present a computational framework for designing geometric metamaterials capable of approximating freeform 3D surfaces via rotationally deployable kirigami patterns. While prior inverse design methods typically rely on standard, well-studied patterns, such as equilateral triangles or quadrilaterals, we step back to examine the broader design space of the patterns themselves. Specifically, we derive principled rules to determine whether a given planar tiling can be cut into a rotationally deployable hinged kirigami structure with possible curvature adaptation. These insights allow us to generate and validate a broad family of novel tiling patterns beyond traditional examples. We further analyze two key deployment states of a general pattern: the commonly used maximal area expansion, and the maximal rotation angle reached just before face collisions occur, which we adopt as the default for inverse design as it allows for simple deployment in practice, i.e., rotating the faces to their natural limit. Finally, we solve the inverse problem: given a target 3D surface, we compute a planar tiling that, when cut and deployed to its maximal rotation angle, approximates the input geometry. We show that for a subset of patterns, the deployed configurations are hole-free, demonstrating that curvature can be achieved from planar sheets through local combinatorial changes. Our experiments, including physical fabrications, demonstrate the effectiveness of our approach and validate a wide range of previously unexplored patterns that are both physically realizable and geometrically expressive.

Given a sequence of poses of a body we study the motion resulting when the body is immersed in a (possibly) moving, incompressible medium. With the poses given, say, by an animator, the governing second-order ordinary differential equations are those of a rigid body with time-dependent inertia acted upon by various forces. Some of these forces, like lift and drag, depend on the motion of the body in the surrounding medium. Additionally, the inertia must encode the effect of the medium through its added mass. We derive the corresponding dynamics equations which generalize the standard rigid body dynamics equations. All forces are based on local computations using only physical parameters such as mass density. Notably, we approximate the effect of the medium on the body through local computations avoiding any global simulation of the medium. Consequently, the system of equations we must integrate in time is only 6 dimensional (rotation and translation). Our proposed algorithm displays linear complexity and captures intricate natural phenomena that depend on body-fluid interactions.

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the [Formula: see text] metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces; and (4) the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

We consider motion effected by shape change. Such motions are ubiquitous in nature and the human made environment, ranging from single cells to platform divers and jellyfish. The shapes may be immersed in various media ranging from the very viscous to air and nearly inviscid fluids. In the absence of external forces these settings are characterized by constant momentum. We exploit this in an algorithm which takes a sequence of changing shapes, say, as modeled by an animator, as input and produces corresponding motion in world coordinates. Our method is based on the geometry of shape change and an appropriate variational principle. The corresponding Euler-Lagrange equations are first order ODEs in the unknown rotations and translations and the resulting time stepping algorithm applies to all these settings without modification as we demonstrate with a broad set of examples.

Simulation of stellar atmospheres, such as that of our own sun, is a common task in CGI for scientific visualization, movies and games. A fibrous volumetric texture is a visually dominant feature of the solar corona---the plasma that extends from the solar surface into space. These coronal fibers can be modeled as magnetic filaments whose shape is governed by the magnetohydrostatic equation. The magnetic filaments provide a Lagrangian curve representation and their initial configuration can be prescribed by an artist or generated from magnetic flux given as a scalar texture on the sun's surface. Subsequently, the shape of the filaments is determined based on a variational formulation. The output is a visual rendering of the whole sun. We demonstrate the fidelity of our method by comparing the resulting renderings with actual images of our sun's corona.

We introduce variable thickness, viscous vortex filaments. These can model such varied phenomena as underwater bubble rings or the intricate "chandeliers" formed by ink dropping into fluid. Treating the evolution of such filaments as an instance of Newtonian dynamics on a Riemannian configuration manifold we are able to extend classical work in the dynamics of vortex filaments through inclusion of viscous drag forces. The latter must be accounted for in low Reynolds number flows where they lead to significant variations in filament thickness and form an essential part of the observed dynamics. We develop and document both the underlying theory and associated practical numerical algorithms.