Nestor Calvo

Polyhedrization of an arbitrary 3D point set

Given a 3D point set, the problem of defining the volume associated, dividing it into a set of regions (elements) and defining a boundary surface is tackled. Several physical problems need to define volume domains, boundary surfaces and approximating functions from a given point distribution. This is for instance the case of particle methods, in which all the information is the particle positions and there are not boundary surfaces definition. Until recently, all the FEM mesh generators were limited to the generation of simple elements as tetrahedral or hexahedral elements (or triangular and quadrangular in 2D problems). The reason of this limitation was the lack of any acceptable shape function to be used in other kind of geometrical elements. Nowadays, there are several acceptable shape functions for a very large class of polyhedra. These new shape functions, together with a generalization of the Delaunay tessellation presented in this paper, gives an optimal marriage and a powerful tool to solve a large variety of physical problems by numerical methods. The domain partition into polyhedra presented here is not a standard mesh generation. The problem here is: for a given node distribution to find a suitable boundary surface and a suitable mesh to be used in the solution of a physical problem by a numerical method. To include new nodes or change their positions is not allowed.

All-hexahedral element meshing: Generation of the dual mesh by recurrent subdivision

Abstract The domain geometry is defined by means of a closed all-quadrilateral mesh. The outer mesh imposes very strong restrictions on the possible connectivities between the inner hexahedral elements. Following the guidelines of the outer topology, the inner one is almost entirely defined. Several ways may be decided for certain configurations, some of them requiring special considerations in order to achieve a valid FEM mesh. The process is entirely performed by constructing the (graph theoretical) dual of the hexahedral mesh, this means no metric information is handled until the final (positioning and smoothing) steps. The essential steps of this scheme are described by means of examples.

The extended Delaunay tessellation

The extended Delaunay tessellation (EDT) is presented in this paper as the unique partition of a node set into polyhedral regions defined by nodes lying on the nearby Voronoi spheres. Until recently, all the FEM mesh generators were limited to the generation of tetrahedral or hexahedral elements (or triangular and quadrangular in 2D problems). The reason for this limitation was the lack of any acceptable shape function to be used in other kind of geometrical elements. Nowadays, there are several acceptable shape functions for a very large class of polyhedra. These new shape functions, together with the EDT, gives an optimal combination and a powerful tool to solve a large variety of physical problems by numerical methods. The domain partition into polyhedra presented here does not introduce any new node nor change any node position. This makes this process suitable for Lagrangian problems and meshless methods in which only the connectivity information is used and there is no need for any expensive smoothing process.

Topology optimization of three-dimensional load-bearing structures using boundary elements

A numerical procedure for the topological optimization of three-dimensional linear elastic problems using boundary elements and the Topological Derivative (TD) is presented in this work. The TD is a function which characterizes the sensitivity of a given cost function to the change of the domain topology, like opening a small hole in a continuum. In particular, for this work the total potential strain energy is selected as cost function, and the TD is computed from the stress field by means of the Topological-Shape Sensitivity Method. The optimization problem is solved incrementally. In every step small portions of the model domain are removed by deleting small portions of material associated with the internal points. The new geometry is then remeshed and the resulting boundary element discretization checked (and if necessary fixed) in order to avoid geometrically invalid models. This procedure is repeated until a given stopping criterion is satisfied. The proposed strategy proves to be flexible and robust. A Furthermore, a number of examples are solved and the results discussed and compared to those available in the literature.

Topology Optimization of Potential and Elasticity Problems Using Boundary Elements

Topological Optimization provides a powerful framework to obtain the optimal domain topology for several engineering problems. The topological derivative is a function which characterizes the sensitivity of a given problem to the change of its topology, like opening a small hole in a continuum or changing the connectivity of rods in a truss. A numerical approach for the topological optimization of 2D potential and linear elastic problems and 3D linear elastic problems using Boundary Elements is presented in this work. The topological derivative is computed from strain and stress results which are solved by means of a standard Boundary Element analysis. Models are discretized using linear or constant elements and a periodic distribution of internal points over the domain. The total potential energy is selected as cost function. The evaluation of the topological derivative is performed as a post-processing procedure. Afterwards, material is removed from the model by deleting the internal points and boundary nodes with the lowest or highest values of the topological derivate. The new geometry is then remeshed using algorithms capable of detecting “holes” at those positions where internal points and boundary points have been removed. The procedure is repeated until a given stopping criterion is satisfied. The proposed strategy proved to be flexible and robust. A number of examples are solved and results are compared to those available in the literature.