## Section: New Results

### Algebraic representations for geometric modeling

#### A comparison of different notions of ranks of symmetric tensors

Participants : Alessandra Bernardi, Jérome Brachat, Bernard Mourrain.

In [2] , we introduce various notions of rank for a symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications, allow us to describe the hierarchy among all the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.

#### Dimensions and bases of hierarchical tensor-product splines

Participant : Bernard Mourrain.

In [1] , we prove that the dimension of trivariate tensor-product spline space of tri-degree (m,m,m) with maximal order of smoothness over a three-dimensional domain coincides with the number of tensor-product B-spline basis functions acting effectively on the domain considered. A domain is required to belong to a certain class. This enables us to show that, for a certain assumption about the configuration of a hierarchical mesh, hierarchical B-splines span the spline space. This paper presents an extension to three-dimensional hierarchical meshes of results proposed recently by Giannelli and Jüttler for two-dimensional hierarchical meshes.

Joint work with Dmitry Berdinsky, Taiwan Kim, Oh Min-Jae, Sutipong Kiatpanichgij (Department of Naval Architecture and Ocean Engineering, Seoul, South Korea), Cesare Bracco (Dipartimento di Matematica “Giuseppe Peano”, Torino, Italy), Durkbin Cho (Department of Mathematics, Dongguk, South Korea).

#### Bounds on the dimension of trivariate spline spaces: A homological approach

Participant : Bernard Mourrain.

In [8] , we consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by applying homological techniques. We give an insight of different ways of approaching this problem by exploring its connections with the Hilbert series of ideals generated by powers of linear forms, fat points, the so-called Fröberg–Iarrobino conjecture, and the weak Lefschetz property.

Joint work with Nelly Villamizar (RICAM - Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria)

#### High-quality construction of analysis-suitable trivariate NURBS solids by reparameterization methods

Participants : André Galligo, Bernard Mourrain.

High-quality volumetric parameterization of computational domain plays an important role in three-dimensional isogeometric analysis. Reparameterization techniques can improve the distribution of isoparametric curves/surfaces without changing the geometry. In [10] , using the reparameterization method, we investigate the high-quality construction of analysis-suitable NURBS volumetric parameterization. Firstly, we introduce the concept of volumetric reparameterization, and propose an optimal Möbius transformation to improve the quality of the isoparametric structure based on a new uniformity metric. Secondly, from given boundary NURBS surfaces, we present a two-stage scheme to construct the analysis-suitable volumetric parameterization: in the first step, uniformity-improved reparameterization is performed on the boundary surfaces to achieve high-quality isoparametric structure without changing the shape; in the second step, from a new variational harmonic metric and the reparameterized boundary surfaces, we construct the optimal inner control points and weights to achieve an analysis-suitable NURBS solid. Several examples with complicated geometry are presented to illustrate the effectiveness of proposed methods.

Joint work with Gang Xu (College of computer - Hangzhou Dianzi University, China), Timon Rabczuk (Bauhaus-Universität Weimar, Germany).

#### Spline Spaces over Quadrangle Meshes with Complex Topologies

Participants : André Galligo, Bernard Mourrain, Meng Wu.

Motivated by Magneto Hydrodynamic (MHD) simulation with isoparametric elements method, we pursue our work on new types of spline functions defined over a quadrangular mesh, that can follow isobaric curves with node singularities. The practicability of these splines is analyzed for different geometries related to MHD simulation.

This work is done in collaboration with Boniface Nkonga (Inria, EPI CASTOR and University of Nice).

#### Parametric modeling for ship hull deformation

Participant : Elisa Berrini.

The objective of the work is to develop a parametric modeler tool, allowing consistent ship hull deformations with respect to classic naval architecture design constraints. This work will be applied in automatic shape optimization process. Two scientific problematics are addressed : 1) The parametrization of the hull: the numerical representation of the shape from a defined set of parameters; 2) The deformations of curves and surfaces: getting a new shape by modifying chosen parameters from the parameterization set. The consistency with naval architecture constraints is essential.

To produce realistic models, we want to use methods similar to naval architects' ones. The approach under development is based on the extraction and deformation of skeletons curves.