Development of a method for triangulation of inhomogeneous regions represented by functions
DOI:
https://doi.org/10.15587/1729-4061.2019.174010Keywords:
triangulation, discrete model, functional representation, implicit function, triangle, heterogeneous domainAbstract
In the process of designing structures from inhomogeneous materials, there is the need to build discrete models that consider the peculiarities of the geometrical shape of subdomains from different materials. The first stage in the modeling of such structures is the construction of a geometric model. In order to describe the shapes of inhomogeneous structures, we have proposed a functional approach, based on the use of systems of implicit functions and R-functions. The first implicit function defines the shape of a structure. The implicit functions starting from a second one determine the shapes of the subdomains whose boundaries must be considered when building a discrete model. Each implicit function within the system exceeds zero at inner points of the respective domain or subdomain, is equal to zero at the border, and is less than zero at outer points. The result is a possibility to describe the shapes of domains and subdomains of arbitrary complexity.
We have constructed a method for the triangulation of structures from inhomogeneous materials, whose shape is assigned functionally. The devised method makes it possible to consider the shape of subdomains from different materials used in the structure. The basic idea of the method implies consistent correction of coordinates for the nodes from the primary triangulation of the domain. Primary triangulation can be arbitrary, but it must fully capture the structure. At each step, the boundary of the structure or a subdomain from a particular material is approached by the node closest to the respective boundary. Following the displacement of each node, the coordinates of neighboring nodes are computed by minimizing the functional of exponents in the planes of incident elements. At the same time, for the elements that are incident at nodes and whose coordinates were changed, meeting the Delaunay condition is checked; if necessary, the operation «flip» is performed to change the diagonal. Upon removing the outer nodes, one would obtain a discrete model in which the boundaries of the structure and subdomains from different materials are approximated by nodes and edges of elements.
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Copyright (c) 2019 Serhii Choporov, Serhii Homeniuk, Sergii Grebenyuk, Oleksii Kudin
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