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Creators/Authors contains: "Aanjaneya, Mridul"

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1. This paper proposes a novel method to efficiently solve infeasible low-dimensional linear programs (LDLPs) with billions of constraints and a small number of unknown variables, where all the constraints cannot be satisfied simultaneously. We focus on infeasible linear programs generated in the RLibm project for creating correctly rounded math libraries. Specifically, we are interested in generating a floating point solution that satisfies the maximum number of constraints. None of the existing methods can solve such large linear programs while producing floating point solutions.

We observe that the convex hull can serve as an intermediate representation (IR) for solving infeasible LDLPs using the geometric duality between linear programs and convex hulls. Specifically, some of the constraints that correspond to points on the convex hull are precisely those constraints that make the linear program infeasible. Our key idea is to split the entire set of constraints into two subsets using the convex hull IR: (a) a set X of feasible constraints and (b) a superset V of infeasible constraints. Using the special structure of the RLibm constraints and the presence of a method to check whether a system is feasible or not, we identify a superset of infeasible constraints by computing the convex hull in 2-dimensions. Subsequently, we identify the key constraints (i.e., basis constraints) in the set of feasible constraints X and use them to create a new linear program whose solution identifies the maximum set of constraints satisfiable in V while satisfying all the constraints in X. This new solver enabled us to improve the performance of the resulting RLibm polynomials while solving the corresponding linear programs significantly faster.

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Free, publicly-accessible full text available June 20, 2025
2. Abstract

We present a novel method for faster physics simulations of elastic solids. Our key idea is to reorder the unknown variables according to the Fiedler vector (i.e., the second-smallest eigenvector) of the combinatorial Laplacian. It is well known in the geometry processing community that the Fiedler vector brings together vertices that are geometrically nearby, causing fewer cache misses when computing differential operators. However, to the best of our knowledge, this idea has not been exploited to accelerate simulations of elastic solids which require an expensive linear (or non-linear) system solve at every time step. The cost of computing the Fiedler vector is negligible, thanks to an algebraic Multigrid-preconditioned Conjugate Gradients (AMGPCG) solver. We observe that our AMGPCG solver requires approximately 1 s for computing the Fiedler vector for a mesh with approximately 50Kvertices or 100Ktetrahedra. Our method provides a speed-up between$$10\%$$$10%$$$30\%$$$30%$at every time step, which can lead to considerable savings, noting that even modest simulations of elastic solids require at least 240 time steps. Our method is easy to implement and can be used as a plugin for speeding up existing physics simulators for elastic solids, as we demonstrate through our experiments using the Vega library and the ADMM solver, which use different algorithms for elasticity.

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3. Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in$\mathbb {R}^2$or$\mathbb {R}^3$, such as curves or surfaces. Suchmanifold PDEsoften involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold’s interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of theclosest point method(CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM’s finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver that can scale to tens of millions of degrees of freedom, allowing PDEs to be solved on more complex manifolds. We demonstrate our method’s convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

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Free, publicly-accessible full text available June 17, 2025
4. We propose a height-field-based real-time simulation method for sand and water mixtures. Inspired by the shallow-water assumption, our approach extends the governing equations to handle two-phase flows of sand and water using height fields. Our depth-integrated governing equations can model the elastoplastic behavior of sand, as well as sand-water-mixing phenomena such as friction, diffusion, saturation, and momentum exchange. We further propose an operator-splitting time integrator that is both GPU-friendly and stable under moderate time step sizes. We have evaluated our method on a set of benchmark scenarios involving large bodies of heterogeneous materials, where our GPU-based algorithm runs at real-time frame rates. Our method achieves a desirable trade-off between fidelity and performance, bringing an unprecedentedly immersive experience for real-time applications.
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Free, publicly-accessible full text available December 10, 2024
5. We present a generalized constitutive model for versatile physics simulation of inviscid fluids, Newtonian viscosity, hyperelasticity, viscoplasticity, elastoplasticity, and other physical effects that arise due to a mixture of these behaviors. The key ideas behind our formulation are the design of a generalized Kirchhoff stress tensor that can describe hyperelasticity, Newtonian viscosity and inviscid fluids, and the use of pre-projection and post-correction rules for simulating material behaviors that involve plasticity, including elastoplasticity and viscoplasticity. We show how our generalized Kirchhoff stress tensor can be coupled together into a generalized constitutive model that allows the simulation of diverse material behaviors by only changing parameter values. We present several side-by-side comparisons with physics simulations for specific constitutive models to show that our generalized model produces visually similar results. More notably, our formulation allows for inverse learning of unknown material properties directly from data using differentiable physics simulations. We present several 3D simulations to highlight the robustness of our method, even with multiple different materials. To the best of our knowledge, our approach is the first to recover the knowledge of unknown material properties without making explicit assumptions about the data.

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Free, publicly-accessible full text available August 16, 2024
6. Free, publicly-accessible full text available October 1, 2024
7. We present an end-to-end method for capturing the dynamics of 3D human characters and translating them for synthesizing new, visually-realistic motion sequences. Conventional methods employ sophisticated, but generic, control approaches for driving the joints of articulated characters, paying little attention to the distinct dynamics of human joint movements. In contrast, our approach attempts to synthesize human-like joint movements by exploiting a biologically-plausible, compact network of spiking neurons that drive joint control in primates and rodents. We adapt the controller architecture by introducing learnable components and propose an evolutionary algorithm for training the spiking neural network architectures and capturing diverse joint dynamics. Our method requires only a few samples for capturing the dynamic properties of a joint's motion and exploits the biologically-inspired, trained controller for its reconstruction. More importantly, it can transfer the captured dynamics to new visually-plausible motion sequences. To enable user-dependent tailoring of the resulting motion sequences, we develop an interactive framework that allows for editing and real-time visualization of the controlled 3D character. We also demonstrate the applicability of our method to real human motion capture data by learning the hand joint dynamics from a gesture dataset and using our framework to reconstruct the gestures with our 3D animated character. The compact architecture of our joint controller emerging from its biologically-realistic design, and the inherent capacity of our evolutionary learning algorithm for parallelization, suggest that our approach could provide an efficient and scalable alternative for synthesizing 3D character animations with diverse and visually-realistic motion dynamics.

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8. This paper proposes fast polynomial evaluation methods for correctly rounded elementary functions generated using our RLibm approach. The resulting functions produce correct results for all inputs with multiple representations and rounding modes. Given an oracle, the RLibm approach approximates the correctly rounded result rather than the real value of an elementary function. A key observation is that there is an interval of real values around the correctly rounded result such that any real value in it rounds to the correct result. This interval is the maximum freedom available to RLibm’s polynomial generation procedure. Subsequently, the problem of generating correctly rounded elementary functions using these intervals can be structured as a linear programming problem. Our prior work on the RLibm approach uses Horner’s method for polynomial evaluation. This paper explores polynomial evaluation techniques such as Knuth’s coefficient adaptation procedure, parallel execution of operations using Estrin’s procedure, and the use of fused multiply-add operations in the context of the RLibm approach. If we take the polynomial generated by the RLibm approach and subsequently perform polynomial evaluation optimizations, it results in incorrect results due to rounding errors during polynomial evaluation. Hence, we propose to integrate the fast polynomial evaluation procedure in the RLibm’s polynomial generation process. Our new polynomial evaluation procedure that combines parallel execution with fused multiply-add operations outperforms the Horner’s method used by RLibm’s correctly rounded functions. We show the resulting polynomials for 32-bit float are not only correct but also faster than prior functions in RLibm by 24%
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9. This study presents a new method for modeling the interaction between compressible flow, shock waves, and deformable structures, emphasizing destructive dynamics. Extending advances in time-splitting compressible flow and the Material Point Methods (MPM), we develop a hybrid Eulerian and Lagrangian/Eulerian scheme for monolithic flow-structure interactions. We adopt the second-order WENO scheme to advance the continuity equation. To stably resolve deforming boundaries with sub-cell particles, we propose a blending treatment of reflective and passable boundary conditions inspired by the theory of porous media. The strongly coupled velocity-pressure system is discretized with a new mixed-order finite element formulation employing B-spline shape functions. Shock wave propagation, temperature/density-induced buoyancy effects, and topology changes in solids are unitedly captured.
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