Quelques contributions à la modélisation numérique de structures élancées pour l'informatique graphique

It is interesting to observe that many of the deformable objects around us are characterized by a slender structure: either in one dimension, like hair, plants, strands, or in two dimensions, such as paper, the leaves of trees or clothes. Simulating the mechanical behavior of such structures numerically is useful to predict their static shape, their dynamics, or the stress they undergo. However, to perform these simulations, several problems need to be addressed. First, the model (mechanical, numerical) should be adapted to the phenomena which it is aimed at reproducing. Then, the chosen mechanical model should be discretized consistently. Finally, it is necessary to identify the parameters of the model in order to reproduce a specific instance of the phenomenon. In this thesis we shall discuss these three points, in the context of the simulation of slender structures.

In the first part, we propose a discrete dynamic Kirchhoff rod model of high degree, based on elements with piecewise affine curvature and twist: the Super-Space-Clothoids. This spatial discretization is computed accurately through a dedicated method, adapted to floating-point arithmetic, using power series expansions. The use of curvature and twist as degrees of freedom allows us to make elastic forces implicit in the integration scheme. The model has been used successfully to simulate the growth of climbing plants or hair motion. Our comparisons with two reference models have shown that in the case of curly rods, our approach offers the best trade-off in terms of spatial accuracy, richness of motion and computational efficiency.

In the second part, we focus on identifying the undeformed configuration of a shell in the presence of frictional contact forces, knowing its shape at equilibrium and the physical parameters of the material. Such a method is of utmost interest in Computer Graphics when, for example, a user often wishes to model a virtual garment under gravity and contact with other objects regardless of physics. The goal is then to interpret the shape and provide the right ingredients to the cloth simulator, so that the cloth is actually at equilibrium when matching the input shape. To tackle such an inverse problem, we propose a least squares formulation which can be optimized using the adjoint method. However, the multiplicity of equilibria, which makes our problem ill-posed, leads us to "guide" the optimization by penalizing shapes that are far from the target shape. Finally, we show how it is possible to consider frictional contact in the inversion process by reformulating the computation of equilibrium as an optimization problem subject to conical constraints. The adjoint method is also adjusted to this non-regular case. The results we obtain are very encouraging and have allowed us to solve complex cases where the algorithm behaves intuitively.

Directors:

• Mme Florence Bertails-Descoubes (Chargée de Recherche - INRIA Rhône-Alpes )
• Mr Bernard Brogliato (Directeur de Recherche - INRIA Rhône-Alpes )

Raporteurs:

• Mr Eitan Grinspun (Associate professeur - Columbia University )
• Mr Loïc Barthe (Maître de conférence - Université Paul Sabatier )

Examinators:

• Mr Basile Audoly (Directeur de recherche - CNRS Paris 6 )
• Mme Evelyne Hubert (Professeur - INRIA Sophia-Antipolis )
• Mr Eric Blayo (Professeur - Université Joseph Fourier )