39 research outputs found
Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations
We introduce a new method, the Local Monge Parametrizations (LMP) method, to
approximate tensor fields on general surfaces given by a collection of local
parametrizations, e.g.~as in finite element or NURBS surface representations.
Our goal is to use this method to solve numerically tensor-valued partial
differential equations (PDE) on surfaces. Previous methods use scalar
potentials to numerically describe vector fields on surfaces, at the expense of
requiring higher-order derivatives of the approximated fields and limited to
simply connected surfaces, or represent tangential tensor fields as tensor
fields in 3D subjected to constraints, thus increasing the essential number of
degrees of freedom. In contrast, the LMP method uses an optimal number of
degrees of freedom to represent a tensor, is general with regards to the
topology of the surface, and does not increase the order of the PDEs governing
the tensor fields. The main idea is to construct maps between the element
parametrizations and a local Monge parametrization around each node. We test
the LMP method by approximating in a least-squares sense different vector and
tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply
the LMP method to two physical models on surfaces, involving a tension-driven
flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP
method thus solves the long-standing problem of the interpolation of tensors on
general surfaces with an optimal number of degrees of freedom.Comment: 16 pages, 6 figure
A PDE based approach to multi-domain partitioning and quadrilateral meshing
International audienceIn this paper, we present an algorithm for partitioning any given 2d domain into regions suitable for quadrilateral meshing. It can deal with multi-domain geometries with ease, and is able to preserve the symmetry of the domain. Moreover, this method keeps the number of singularities at the junctions of the regions to a minimum. Each part of the domain, being four-sided, can then be meshed using a structured method. The partitioning stage is achieved by solving a PDE constrained problem based on the geometric properties of the domain boundaries
Practical 3D frame field generation
International audienceFigure 1: Our algorithm produces smooth frame fields in volumes. Frames (a) are represented by spherical harmonic functions (b), attached to each vertex of a tetrahedral mesh. Streamlines and singularities of the field are shown in yellow and red, respectively. Abstract Given a tetrahedral mesh, the algorithm described in this article produces a smooth 3D frame field, i.e. a set of three orthogonal directions associated with each vertex of the input mesh. The field varies smoothly inside the volume, and matches the normals of the volume boundary. Such a 3D frame field is a key component for some hexahedral meshing algorithms, where it is used to steer the placement of the generated elements. We improve the state-of-the art in terms of quality, efficiency and reproducibility. Our main contribution is a non-trivial extension in 3D of the existing least-squares approach used for optimizing a 2D frame field. Our algorithm is inspired by the method proposed by Huang et al. [2011], improved with an initialization that directly enforces boundary conditions. Our initialization alone is a fast and easy way to generate frames fields that are suitable for remeshing applications. For better robustness and quality, the field can be further optimized using nonlinear optimization as in Li et al [2012]. We make the remark that sampling the field on vertices instead of tetrahedra significantly improves both performance and quality