arXiv subjects

Fabian Löschner

Publications and source records attributed to Fabian Löschner.

3 recordsLinked to original sources

arXiv2026

Progressively Projected Newton's Method

Newton's Method is widely used to find the solution of complex non-linear simulation problems in Computer Graphics. To guarantee a descent direction, it is common practice to clamp the negative eigenvalues of each element Hessian prior to assembly - a strategy known as Projected Newton (PN) - but this perturbation often hinders convergence. In this work, we observe that projecting only a small subset of element Hessians is sufficient to secure a descent direction. Building on this insight, we introduce Progressively Projected Newton (PPN), a novel variant of Newton's Method that uses the current iterate residual to cheaply determine the subset of element Hessians to project. The global Hessian thus remains closer to its original form, reducing both the number of Newton iterations and the amount of required eigen-decompositions. We compare PPN with PN and Project-on-Demand Newton (PDN) in a comprehensive set of experiments covering contact-free and contact-rich deformables (including large stiffness and mass ratios), co-dimensional, and rigid-body simulations, and a range of time step sizes, tolerances and resolutions. PPN consistently performs fewer than 10% of the projections required by PN or PDN and, in the vast majority of cases, converges in fewer Newton iterations, which makes PPN the fastest solver in our benchmark. The most notable exceptions are simulations with very large time steps and quasistatics, where PN remains a better choice.

cs.GR↗

arXiv2025

SymX: Energy-based Simulation from Symbolic Expressions

Optimization time integrators are effective at solving complex multi-physics problems including deformable solids with non-linear material models, contact with friction, strain limiting, etc. For challenging problems, Newton-type optimizers are often used, which necessitates first- and second-order derivatives of the global non-linear objective function. Manually differentiating, implementing, testing, optimizing, and maintaining the resulting code is extremely time-consuming, error-prone, and precludes quick changes to the model, even when using tools that assist with parts of such pipeline. We present SymX, an open source framework that computes the required derivatives of the different energy contributions by symbolic differentiation, generates optimized code, compiles it on-the-fly, and performs the global assembly. The user only has to provide the symbolic expression of each energy for a single representative element in its corresponding discretization and our system will determine the assembled derivatives for the whole simulation. We demonstrate the versatility of SymX in complex simulations featuring different non-linear materials, high-order finite elements, rigid body systems, adaptive discretizations, frictional contact, and coupling of multiple interacting physical systems. SymX's derivatives offer performance on par with SymPy, an established off-the-shelf symbolic engine, and produces simulations at least one order of magnitude faster than TinyAD, an alternative state-of-the-art integral solution.

cs.CE↗

arXiv2023

Pitfalls of Projection: A study of Newton-type solvers for incremental potentials

Nonlinear systems arising from time integrators like Backward Euler can sometimes be reformulated as optimization problems, known as incremental potentials. We show through a comprehensive experimental analysis that the widely used Projected Newton method, which relies on unconditional semidefinite projection of Hessian contributions, typically exhibits a reduced convergence rate compared to classical Newton's method. We demonstrate how factors like resolution, element order, projection method, material model and boundary handling impact convergence of Projected Newton and Newton. Drawing on these findings, we propose the hybrid method Project-on-Demand Newton, which projects only conditionally, and show that it enjoys both the robustness of Projected Newton and convergence rate of Newton. We additionally introduce Kinetic Newton, a regularization-based method that takes advantage of the structure of incremental potentials and avoids projection altogether. We compare the four solvers on hyperelasticity and contact problems. We also present a nuanced discussion of convergence criteria, and propose a new acceleration-based criterion that avoids problems associated with existing residual norm criteria and is easier to interpret. We finally address a fundamental limitation of the Armijo backtracking line search that occasionally blocks convergence, especially for stiff problems. We propose a novel parameter-free, robust line search technique to eliminate this issue.

cs.GR↗