Designing subspaces for Reduced Order Modeling (ROM) is crucial for accelerating finite element simulations in graphics and engineering. Unfortunately, it's not always clear which subspace is optimal for arbitrary dynamic simulation. We propose to construct simulation subspaces from force distributions, allowing us to tailor such subspaces to common scene interactions involving constraint penalties, handles-based control, contact and musculoskeletal actuation. To achieve this we adopt a statistical perspective on Reduced Order Modelling, which allows us to push such user-designed force distributions through a linearized simulation to obtain a dual distribution on displacements. To construct our subspace, we then fit a low-rank Gaussian model to this displacement distribution, which we show generalizes Linear Modal Analysis subspaces for uncorrelated unit variance force distributions, as well as Green's Function subspaces for low rank force distributions. We show our framework allows for the construction of subspaces that are optimal both with respect to physical material properties, as well as arbitrary force distributions as observed in handle-based, contact, and musculoskeletal scene interactions.

Eigenanalysis of differential operators, such as the Laplace operator or elastic energy Hessian, is typically restricted to a single shape and its discretization, limiting reduced order modeling (ROM). We introduce the first eigenanalysis method for continuously parameterized shape families. Given a parametric shape, our method constructs spatial neural fields that represent eigen-functions across the entire shape space. It is agnostic to the specific shape representation, requiring only an inside/outside indicator function that depends on shape parameters. Eigenfunctions are computed by minimizing a variational principle over nested spaces with orthogonality constraints. Since eigenvalues may swap dominance at points of multiplicity, we jointly train multiple eigenfunctions while dynamically reordering them based on their eigenvalues at each step. Through causal gradient filtering , this reordering is reflected in backpropagation. Our method enables applications to operate over shape space, providing a single ROM that encapsulates vibration modes for all shapes, including previously unseen ones. Since our eigenanalysis is differentiable with respect to shape parameters, it facilitates eigenfunction-aware shape optimization. We evaluate our approach on shape optimization for sound synthesis and locomotion, as well as reduced-order modeling for elastodynamic simulation.

We present a novel reduced-order fluid simulation technique leveraging Dynamic Mode Decomposition (DMD) to achieve fast, memory-efficient, and user-controllable subspace simulation. We demonstrate that our approach combines the strengths of both spatial reduced order models (ROMs) as well as spectral decompositions. By optimizing for the operator that evolves a system state from one timestep to the next, rather than the system state itself, we gain both the compressive power of spatial ROMs as well as the intuitive physical dynamics of spectral methods. The latter property is of particular interest in graphics applications, where user control of fluid phenomena is of high demand. We demonstrate this in various applications including spatial and temporal modulation tools and fluid upscaling with added turbulence. We adapt DMD for graphics applications by reducing computational overhead, incorporating user-defined force inputs, and optimizing memory usage with randomized SVD. The integration of OptDMD and DMD with Control (DMDc) facilitates noise-robust reconstruction and real-time user interaction. We demonstrate the technique's robustness across diverse simulation scenarios, including artistic editing, time-reversal, and super-resolution. Through experimental validation on challenging scenarios, such as colliding vortex rings and boundary-interacting plumes, our method also exhibits superior performance and fidelity with significantly fewer basis functions compared to existing spatial ROMs. Leveraging the inherent linearity of the DMD formulation, we demonstrate a range of diverse applications. This work establishes another avenue for developing real-time, high-quality fluid simulations, enriching the space of fluid simulation techniques in interactive graphics and animation.

Real-time elastodynamic solvers are well-suited for the rapid simulation of homogeneous elastic materials, with high-rates generally enabled by aggressive early termination of timestep solves. Unfortunately, the introduction of strong domain heterogeneities can make these solvers slow to converge. Stopping the solve short creates visible damping artifacts and rotational errors. To address these challenges we develop a reduced mixed finite element solver that preserves rich rotational motion, even at low-iteration regimes. Specifically, this solver augments time-step solve optimizations with auxillary stretch degrees of freedom at mesh elements, and maintains consistency with the primary positional degrees of freedoms at mesh nodes via explicit constraints. We make use of a Skinning Eigenmode subspace for our positional degrees of freedom. We accelerate integration of non-linear elastic energies with a cubature approximation, placing stretch degrees of freedom at cubature points. Across a wide range of examples we demonstrate that this subspace is particularly well suited for heterogeneous material simulation. Our resulting method is a subspace mixed finite element method completely decoupled from the resolution of the mesh that is well-suited for real-time simulation of heterogeneous domains.

We propose a reduced-space elastodynamic solver that is well suited for augmenting rigged character animations with secondary motion. At the core of our method is a novel deformation subspace based on Linear Blend Skinning that overcomes many of the shortcomings prior subspace methods face. Our skinning subspace is parameterized entirely by a set of scalar weights, which we can obtain through a small, material-aware and rig-sensitive generalized eigenvalue problem. The resulting subspace can easily capture rotational motion and guarantees that the resulting simulation is rotation equivariant. We further propose a simple local-global solver for linear co-rotational elasticity and propose a clustering method to aggregate per-tetrahedra nonlinear energetic quantities. The result is a compact simulation that is fully decoupled from the complexity of the mesh.

We reduce computation time in rigid body simulations by merging collections of bodies when they share a common spatial velocity. Merging relies on monitoring the state of contacts, and a metric that compares the relative linear and angular motion of bodies based on their sizes. Unmerging relies on an inexpensive single iteration projected Gauss-Seidel sweep over contacts between merged bodies, which lets us update internal contact forces over time, and use the same metrics as merging to identify when bodies should unmerge. Furthermore we use a contact ordering for graph traversal refinement of the internal contact forces in collections, which helps to correctly identify all the bodies that must unmerge when there are impacts. The general concept of merging is similar to the common technique of sleeping and waking rigid bodies in the inertial frame, and we exploit this too, but our merging is in moving frames, and unmerging takes place at contacts between bodies rather than at the level of bodies themselves. We discuss the previous relative motion metrics in comparison to ours, and evaluate our method on a variety of scenarios.