# High order contact detection between fibres

Emile Hohnadel, Octave Crespel, Thibaut Métivet, Florence Bertails-Descoubes

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## To cite this version:

Emile Hohnadel, Octave Crespel, Thibaut Métivet, Florence Bertails-Descoubes. High order contact detection between fibres. RNL 2024 - 27e Rencontre du Non-Linéaire, Mar 2024, Paris, France. pp.1-1, 2024. [hal-04885297](https://hal.science/hal-04885297v1)

HAL Id: hal-04885297

https://hal.science/hal-04885297v1

Submitted on 14 Jan 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# High order contact detection between fibres

Emile Hohnadel

Octave Crespel

Thibaut Métivet

Florence Bertails-Descoubes

Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble, France

emile.hohnadel@inria.fr

## 1. Motivation and challenges

Our objective is to correctly evaluate contact forces in a fibrous assembly by studying the effects of detection methods and curvature at contact.

We compare a low order model, **Discrete Elastic Rods** [3], and a high order one, **Super-Helix** [2], to simulate inextensible Kirchhoff rods.

The contact response is computed using the non-regular solver **so-bogus** [4].

In the Computer Graphics community, low order proxy (typically segments for lines), are generally used during the contact detection step.

We show that using a low order proxy is a source of spurious jumps in the force response, with the example of the curved three point test, and propose an alternative algorithm to compute the closest point between a pair of curves, which we apply in the simulation of the combing of hair wisps.

<figure><img src="image_3.png" /><figcaption>**Figure:** Geometrically discontinuous contact detection linked to segment proxies</figcaption></figure>

## 2. Curved three point test

<figure><img src="image_4.png" /><figcaption>Indented state</figcaption></figure>

At contact, the curvature of the fibre is **constant** and equals its natural curvature $κ_0$.

<figure><img src="image_5.png" /><figcaption>**Figure:** Normalised force $\bar{F}$ as a function of normalised indentation $\bar{\delta}$ for the curved three point test. Comparison of the analytic curve (in black) with the simulation results. Only our method (in **red**) completely removes the jumps.</figcaption></figure>

## 3. Role of curvature at contact

Jumps appear in the curved three point test with segment proxy as can be seen in the figure on the left. To study the **amplitude of the jumps**, the force is normalised with an analytic solution. The jumps are bigger with **DER** [3] but remain present with **SH** [2] as can be seen in the plots below. **The curvier the fibre, the bigger the jumps.**

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

## 4. Exact contact detection

<figure><img src="image_8.png" /></figure>

**Figure:** Schema of the **Branch and Bound** approach used in our method [1] to find the **closest points** between two curves in 3D. The upper branch underlines the **early stop** of the method for distant curves. The bounding volumes are obtained with the **Taylor-Lagrange inequality**. They rely on an **upper bound of the second derivative** of the curve.

## 5. Hair wisp combing

<figure><img src="image_9.png" /><figcaption>**Figure:** Combing of a wavy 2025 hair wisp with natural curvature $κ_0 = 0.6 \text{ cm}^{-1}$.</figcaption></figure>

<figure><img src="image_10.png" /></figure>

**Figure:** Plot of the force of the wavy wisp on the comb during its passage with two contact detection methods. We observe a shift between the forces obtained with the segment proxy (in black) and our exact method (in **yellow**). Our method integrate less noise, hence increasing the accuracy of force prediction.

## Bibliography

* O. Crespel, E. Hohnadel, T. Metivet, F. Bertails-Descoubes. Contact detection between fibres: high order makes a difference, submitted.
* F. Bertails, B. Audoly, M.-P. Cani, B. Querleux, F. Leroy, J.-L. Lévêque. Super-Helices for Predicting the Dynamics of Natural Hair, ACM Trans. Graph., 25 (2006).
* M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, E. Grinspun. Discrete Elastic Rods, ACM Trans. Graph., 27 (2008) 63:1-63:12
* G. Daviet, F. Bertails-Descoubes, L. Boissieux. A hybrid iterative solver for robustly capturing coulomb friction in hair dynamics, ACM Trans. Graph., 139 (2011).

<figure><img src="image_11.png" /></figure>