## Robust Eulerian-On-Lagrangian Rods

### Abstract

This paper introduces a method to simulate complex rod assemblies and stacked layers with implicit contact handling, through Eulerian-on-Lagrangian (EoL) discretizations. Previous EoL methods fail to handle such complex situations, due to ubiquitous and intrinsic degeneracies in the contact geometry, which prevent the use of remeshing and make simulations unstable. We propose a novel mixed Eulerian-Lagrangian discretization that supports accurate and efficient contact as in EoL methods, but is transparent to internal rod forces, and hence insensitive to degeneracies. By combining the standard and novel EoL discretizations as appropriate, we derive mixed statics-dynamics equations of motion that can be solved in a unified manner with standard solvers. Our solution is simple and elegant in practice, and produces robust simulations on large-scale scenarios with complex rod arrangements and pervasive degeneracies. We demonstrate our method on multi-layer yarn-level cloth simulations, with implicit handling of both intra and inter-layer contacts.

### Citation

@article {SRBO20,
author  = {Sánchez-Banderas, Rosa M. and Rodríguez, Alejandro and Barreiro, Héctor and Otaduy, Miguel A.},
title   = {{Robust Eulerian-on-Lagrangian Rods}},
number  = "4",
volume  = "39",
journal = {ACM Transactions on Graphics (Proc. of ACM SIGGRAPH)},
year    = {2020}
}

### Description

In this work we introduce a method to simulate complex rod assemblies and stacked layers with implicit contact handling through Eulerian-on-Lagrangian discretizations.

Rods in contact tend to concentrate large local bending due to the low bending stiffness of the rod and the action of external forces. Then, it is interesting to introduce discretization nodes at the contact points to accurately represent rod bending. Moreover, if contacts are considered to be persistent, the overall computation can be simplified by avoiding explicit collision detection. Eulerian-on-Lagrangian (EoL) discretizations allow such efficient representation while enabling sliding motion in the material domain by introducing a set of generalized coordinates ($$\mathbf{x}, u$$) storing both the Lagrangian (spatial) coordinates $$\mathbf{x}$$ and the Eulerian (material) coordinates $$\mathbf{u}$$ at the contact.

However, while EoL is beneficial for the efficient simulation of rods in contact, it is not free of difficulties. As shown in the figure below, elastic forces become infinitely stiff when two sliding rod nodes get arbitrarily close to each other $$(\Delta u = u_b - u_ a \approx 0)$$, thus leading to numerical instabilities. Usual approaches like remeshing (node collapse) for solving such degeneracies are not viable when the nodes represent two sliding contacts, as it is paramount to retain the Eulerian coordinates of both nodes in order to determine how the contacts continue sliding.

Our solution to robustly handle degenerate situations under sliding contact is to introduce another type of mixed Eulerian-Lagrangian node in the discretization. This node has only a free Eulerian coordinate, as this property is key to capture sliding, while its Lagrangian coordinate is defined through the interpolation of adjacent nodes. We call this node Eulerian with Interpolated Lagrangian node (EIL). Linear interpolation of the Lagrangian coordinates produces geometric properties that make the EIL node transparent to internal rod forces, hence making the simulation robust against degenerate discretizations. Nodes slide and cross each other robustly.

### Results

In our results, we show the application of our solution to challenging simulations of yarn-level cloth. Beyond single woven or knit fabrics, we extend the power of EoL methods to stacked layers of fabrics, by handling implicitly both intra-fabric as well as inter-fabric contacts. We demonstrate results on several familiar settings: tablecloth layers, pant pockets, and shirt tags.

We also show that our method enables scalable EoL-based simulation of complex knit fabrics where multiple yarns slide and cross each other. To date, these fabrics could be handled only through traditional Lagrangian methods with explicit contact handling or assuming periodicity to predict the relaxed pattern shapes.

The combined simplicity and effectiveness of our solution leads to an elegant implementation and robust results, despite ubiquitous discretization degeneracies. While we only demonstrate our ideas on rods, we believe that the core concepts can also be extended to other EoL domains such as thin shells.

### Contact

Rosa M. Sánchez-Banderas – rosanban@gmail.com
Alejandro Rodríguez – alejandro.rodriguez@seddi.com
Héctor Barreiro – hecbarcab@gmail.com