Project-Team:GALAAD

Inria | Raweb 2013 | Presentation of the Project-Team GALAAD | GALAAD Web Site


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Section: New Results

Algebraic representations for geometric modeling

Fitting ideals and multiple-points of surface parameterizations

Participant : Laurent Busé.

Parameterized algebraic surfaces are ubiquitous in geometric modeling and the determination of their singular loci is an important problem. Given a birational parameterization $φ$ from $ℙ^{2}$ to $ℙ^{3}$ of a rational algebraic surface $𝒮$ , the purpose of this work is to investigate the sets of points on $𝒮$ whose preimage consists in $k$ or more points, counting multiplicity. In collaboration with Nicolas Botbol (University of Buenos Aires) and Marc Chardin (UMPC), we prove that they can be described in terms of Fitting ideals of some graded parts of the symmetric algebra associated to the parameterization $φ$ . More precisely, we show that the drop of rank of a certain elimination matrix $M (φ)$ at a given point $P \in ℙ^{3}$ is in relation with the fiber of the graph of $φ$ over $P$ . Thus, the Fitting ideals attached to $M (φ)$ provide a filtration of the surface which is in correspondence with the degree and the dimension of the fibers of the graph of the parameterization $φ$ . This property is linked with the double-point formulas that have been extensively studied in the field of intersection theory for finite maps.

This work has been accepted for presentation at the international conference MEGA 2013 and is submitted for publication [33] .

Discriminant of a homogeneous and symmetric polynomial

Participant : Laurent Busé.

Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a well-constrained system of polynomial equations, which expresses the existence of a multiple root. In this work the factorization of a single homogeneous and symmetric polynomial is investigated. Indeed, in this setting the discriminant possesses a lot of symmetries and all of these symmetries produce an independent factor of the global discriminant. The two difficult points here are to prove that each of these factors are irreducible over a nice base ring and to determine its multiplicity in the factorization of the discriminant. This work, in collaboration with Anna Karasoulou (University of Athens) is still under progress.

On the cactus rank of cubic forms

In [14] , we prove that the smallest degree of an apolar 0-dimensional scheme of a general cubic form in $n + 1$ variables is at most $2 n + 2$ , when $n \geq 8$ , and therefore smaller than the rank of the form. For the general reducible cubic form the smallest degree of an apolar subscheme is $n + 2$ , while the rank is at least $2 n$ .

This is a work done by Alessandra Bernardi when she was post-doctorate for DECONSTRUCT IEF project, in collaboration with Kristian Ranestad (University of Oslo).

Grassmann secants and linear systems of tensors

For any irreducible non-degenerate variety $X \subset ℙ^{r}$ , we relate in [11] the dimension of the $s$ -th secant varieties of the Segre embedding of $ℙ^{k} \times X$ to the dimension of the $(k, s)$ -Grassmann secant variety $G S_{X} (k, s)$ of $X$ . We also give a criterion for the $s$ -identifiability of $X$ .

This is a work done by Alessandra Bernardi when she was post-doctorate for DECONSTRUCT IEF project, in collaboration with Edoardo Ballico (University of Trento), Maria Virgina Catalisano (DIPTEM, Genova), Luca Chiantini (University of Sienna).

Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis

Participants : André Galligo, Bernard Mourrain.

In isogeometric analysis framework, computational domain is exactly described using the same representation as that employed in the CAD process. For a CAD object, we can construct various computational domain with same shape but with different parameterization. One basic requirement is that the resulting parameterization should have no self-intersections. In [27] , a linear and easy-to-check sufficient condition for injectivity of trivariate B-spline parameterization is proposed. By an example of 3D thermal conduction problem, we show that different parameterization of computational domain has different impact on the simulation result and efficiency in isogeometric analysis. For problems with exact solutions, we propose a shape optimization method to obtain optimal parameterization of computational domain. The proposed injective condition is used to check the injectivity of initial trivariate B-spline parameterization constructed by discrete Coons volume method, which is the generalization of discrete Coons patch method. Several examples and comparisons are presented to show the effectiveness of the proposed method. Compared with the initial parameterization during refinement, the optimal parameterization can achieve the same accuracy but with less degrees of freedom.

This is a joint work with Régis Duvigneau (Inria, EPI OPALE) and Xu Gang (College of computer - Hangzhou Dianzi University, China).

Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method

Participants : André Galligo, Bernard Mourrain.

In isogeometric analysis, parameterization of computational domain has great effects as mesh generation in finite element analysis. In the paper [26] , based on the concept of harmonic mapping from the computational domain to parametric domain, a variational harmonic approach is proposed to construct analysis-suitable parameterization of computational domain from CAD boundary for 2D and 3D isogeometric applications. Different from the previous elliptic mesh generation method in finite element analysis, the proposed method focuses on isogeometric version, and converts the elliptic PDE into a nonlinear optimization problem, in which a regular term is integrated into the optimization formulation to achieve more uniform and orthogonal iso-parametric structure near convex (concave) parts of the boundary. Several examples are presented to show the efficiency of the proposed method in 2D and 3D isogeometric analysis.

This is a joint work with Régis Duvigneau (Inria, EPI OPALE) and Xu Gang (College of computer - Hangzhou Dianzi University, China).

Spline Spaces over Quadrangle Meshes with Complex Topologies

Participants : Meng Wu, André Galligo, Bernard Mourrain.

We study a new type of spline functions defined over a rectangular mesh equipped with an equivalence relation, in such a way that physical spaces with a complex topology can be represented as an homomorphic image of such meshes. We provide general definitions, a dimension formula for a subclass of these spline spaces, an explicit construction of their bases and also a process for local refinement. These developments, motivated by plane curvilinear mesh constructions are illustrated on several parametrization problems. Our main target in these constructions is to approximate isobaric lines of magnetic fields encountered in MHD (Magnetohydrodynamics) simulation for Tokamaks. Their particularity is that one of the isobaric curve has a node singularity.

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

Lagrangian Curves in Affine Symplectic 4-space

Participant : Evelyne Hubert.

Lagrangian curves in 4-space entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. In [39] we provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian curves with constant symplectic curvatures, to construct a class of Lagrangian tori and determine Lagrangian geodesics.

This is joint work with Emilio Musso, Dipartimento di Scienze Matematiche, Politecnico de Turino (Italy).

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