Optimal Point Placement for Mesh Smoothing

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smoothing problems that do not fit into the generalized linear programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This is the final version, and will appear in a special issue of J. Algorithms for papers from SODA '9

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

Geometric Reasoning with polymake

The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of polymake so far and a short description of the technical background

Efficient discretization of signed distance fields

A Signed distance field (SDF) is an implicit function that returns the distance to the surface of a volume given a point in the space. The sign of the field indicates if the point is inside or outside the volume. These fields are usually used to accelerate computer graphics algorithms in different areas, such as rendering or collision detection. There are many well-defined primitives and operators to model objects using these functions. For example, SDFs allow applying smooth boolean operations between primitives. Applying these operators to triangles meshes can require complex algorithms susceptible to precision problems. Even though SDFs allow modelling objects, they currently are not a used format, and not many modelling tools use it. Most of the time, we want to calculate this field from triangle meshes. If the mesh is two-manifold, the easiest way to calculate the signed distance from a point is by searching for the minimum distance at all the mesh triangles. This strategy requires iterating all the triangles for each query to the signed distance field. There are methods based on different strategies that accelerate this nearest triangle search. If the user does not require getting exact distances to the object, other strategies exist that discretize the space in some fixed sample points. Then, the queries to arbitrary points are calculated using an interpolation of the precalculated discretization. This project presents a new approach based on an octree-like subdivision to accelerate the computation of these signed distance fields queries from a triangle mesh. The main idea is to construct an octree structure in which each leaf will contain only the nearest triangles for all the points in that region. Therefore, when the user wants to calculate the distance from an arbitrary point in the space, it will only compare the triangles influencing that region. Moreover, we present a method to calculate approximated distances based on the discretization approach mentioned before. We designed and developed an octree discretization strategy and explored different interpolation techniques. The distance computation of this discretization is accelerated by the strategy developed in the project

An hybrid system approach to nonlinear optimal control problems

We consider a nonlinear ordinary differential equation and want to control its behavior so that it reaches a target by minimizing a cost function. Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given a sequence of states, we introduce an hybrid approximation of the nonlinear controllable domain and propose a new algorithm computing a controllable, piecewise convex approximation. The same way the nonlinear optimal control problem is replaced by an hybrid piecewise affine one. Stating a hybrid maximum principle suitable to our hybrid model, we deduce the global structure of the hybrid optimal control steering the system to the target

A Systematic Review of Algorithms with Linear-time Behaviour to Generate Delaunay and Voronoi Tessellations

Triangulations and tetrahedrizations are important geometrical discretization procedures applied to several areas, such as the reconstruction of surfaces and data visualization. Delaunay and Voronoi tessellations are discretization structures of domains with desirable geometrical properties. In this work, a systematic review of algorithms with linear-time behaviour to generate 2D/3D Delaunay and/or Voronoi tessellations is presented