Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena

Project Information

CoDiNA

Grant agreement ID: 101045083

DOI

10.3030/101045083

EC signature date 11 May 2022

Start date 1 June 2022

End date 31 May 2027

Funded under

• European Research Council (ERC)

Total cost

€ 1 936 503,00

EU contribution

€ 1 936 503,00

Coordinated by

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA
Austria

Periodic Reporting for period 1 - CoDiNA (Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena)

Reporting period: 2022-06-01 to 2024-11-30

Summary of the context and overall objectives of the project

Modern numerical algorithms and computing systems are capable of simulating physical systems far more efficiently than those from just a few years ago. However, this recent increase in computational power comes with an insatiable demand for complexity and scale. Digital artists now wish to model textiles at the level of individual fibers and oceans at all possible scales: from kilometer-scale tsunamis, to meter-scale waves that break against the shore, down to millimeter-scale ripples. This complexity is exacerbated when solving inverse physics problems, such as optimal control, or inverse design. Efficient simulation is essential for these inverse problems, because looping over hundreds of simulations is infeasible if the computations are too expensive.

Advances in computing hardware, high-performance computing systems, and parallel computation can only carry the field forward so far. Simulating every individual cloth fiber or water droplet is infeasible even with the best hardware imaginable, so we really need advances in numerical algorithms to drive the state of the art in the simulation of multi-scale natural phenomena. Particularly clever algorithms can speed up simulations by orders of magnitude, transforming simulations that used to run for days on a super computer into ones that run in real-time on a laptop. Furthermore, the nature of the algorithmic speed-up can provide scientific insights: an efficient approximation or mathematical change of variables can teach us about hidden structures that we might have overlooked before. Finally, optimized algorithms save computation time and energy resources compared to naïve computations on powerful machines.

We will redefine the state of the art in numerical simulation and animation by combining three different approaches: First, we will develop analytical tools customized for large scales; we will derive optimal numerical algorithms by viewing physics through the lens of computational complexity, and by observing limiting behaviors as problems increase in size. Next, we will gain physical insights by simulating huge numbers of smaller simulations and generating large data sets. By discovering trends in this data, we will faithfully approximate aggregate behaviors in systems that are too complex for analytical techniques to penetrate. Finally, by framing the derivation of numerical algorithms as a constrained optimization problem, we will be able to deliver provably optimal code for a given piece of hardware and precisely control accuracy/efficiency trade-offs.

The combination of these research directions will enable efficient simulations of complex systems that are currently unfeasible to compute. Due to the timely and groundbreaking nature of these proposed directions, we also expect to develop entirely unprecedented methods for physics simulation and discover a number of theoretical insights and scientific advances along the way.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Regarding coupling simulations, we developed a physical theory for merging the two most powerful and popular numerical algorithms for simulating water surface waves: the shallow water equations and Airy wave theory. This work finally closes a long-standing gap in computer animation and natural phenomena simulation.

Our research on simulation control and phenomena-specific modeling progressed in two directions. First, we improved vortex-based fluid simulation by reducing the simulation degrees of freedom from three spatial dimensions down to one-dimensional vortex filaments and replacing non-differentiable contact and topological operations with smooth operations based on differentiable implicit functions. We also introduced a theoretical reduction of all topological optimization problems related to physical moments. All prior methods relied on brute force mechanisms for discovering optimal mass distributions. Our new algorithm replaces these inefficient algorithms (which require thousands of computational degrees of freedom and only yield an approximate solution when converged) with a theoretically equivalent problem based geometric surfaces that has less than ten computational degrees of freedom and produces the exact solution. This work substantially advances the world’s interpretation of these problems by placing them within a unified theoretical framework and uses the new theory to solve them far more efficiently than any previous approach.

To succeed in the numerical homogenization of more complex microstructures, we first need a robust numerical algorithm for handling structures with non-manifold surfaces separating individual spatial cells. We recently developed a new surface tracking algorithm that beats the state-of-the-art in terms of robustness and scalability, and we developed the first method for simulating efficient and reliable static friction forces at large scales. These new methods open the door to deriving stress forces from soap films and large collections of rigid bodies in contact, which should substantially advance the state of the art in virtual granular materials and foams.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

[Hafner et al. 2024] generalizes all previous work in the field of topology optimization for the computation of shapes that balance, float, and spin in controllable ways. Previous methods require thousands of computational variables, and took several hours to compute. Our method makes a large theoretical contribution by offering the novel insight that all of these applications can be translated into a narrow class of mathematical constraints. Next, given the simpler problem, we offer a nearly analytical solution which reduces the number of computational variables by several orders of magnitude. While the original problem is inherently discrete and suffers from difficult translation into fast algorithms, the new problem formulation is inherently continuous and lends itself to off-the-shelf fast optimization techniques like Newton’s method. This paper received an honourable mention at SIGGRAPH 2024.

The PhD thesis of Georg Sperl [Sperl 2022] accumulates a wealth of novel information on the simulation of cloth using numerical homogenization. The method works well for the special case of cloth made of woven or knitted elastic fibers, and the thesis shows how to use similar ideas for impressive visual rendering in real-time graphics applications, and how similar techniques can be used to discover new yarn patterns in real-world practical applications. This work lays the groundwork for future projects in the homogenization of other difficult physical systems like granular materials and foams.

Our mesh-based surface tracking algorithm [Heiss-Synak* and Kalinov* et al. 2024] lays a new foundation for surface tracking with non-manifold surfaces. The method is suitable for difficult simulations involving multiple immiscible phases of materials, like multi-phase fluids (oil, water, air), soap foams, and simulations of multicellular biological tissues. Our approach emphasizes numerical robustness at all levels, and we show that it vastly outperforms previous works on novel statistical benchmarks. It is also time- and memory-efficient compared to the state of the art.

Fluid simulation with implicit vortex filaments

Accurate and efficient simulation of static friction

Coupling shallow water equations with Airy wave theory

Mesh-based surface tracking of multiple materials