Generation of the unstructured fe-grids for complex 2d objects/Nestruktūrinių be tinklų generavimas sudėtingiems dvimačiams objektams
For the numerical simulation of engineering problems, the finite element method (FEM) is among the most popular approaches. One of the main concerns in a finite element analysis is the adequacy of the finite element grid. The accuracy of the FEM depends on the size, shape and placement of the elements. On the other hand, the total computational cost is determined by the total number of elements in FE model. An increased accuracy can be obtained by the global reduction of the element size, but this can be characterised by drastically increased computational cost. Thus, in many engineering applications it is desirable to generate not regular FE mesh with finer grid in the regions where accuracy of numerical simulation is of most importance and with more coarse grid in the other regions.
In this paper we present a new approach to the grid generation of the multimaterial or multidomain engineering systems by the advancing front technique. This technique has proved successful in generating unstructured meshes in two and three dimensions [1–9]. The algorithm of the technique is summarised in section 2. Common for all approaches of advancing front mesh generation is that the generation problem is divided into three parts. First, the specification of the mesh size attributes, second, the discretisation of the boundaries, and, third, the discretisation of the interior of the domain. In the advancing front technique the front is defined as the boundary between the gridded and ungridded region. The key algorithmic step that must be addressed to advancing front methods is the proper introduction of new elements into the ungridded region. For triangular and tetrahedral grids the elements are introduced sequentially one at a time. The most obvious advantage of the advancing front method is that it directly incorporates free form geometry.
Direct implementation of the advancing front technique for multimaterial or multidomain engineering applications is still challenging. Grid generation in the place of few materials or domain contact must ensure the compatibility of nodes on common boundary segments (nodes on common boundary segments must be in the same positions). The advancing front technique does not include non-convex domain, so at the first step non-convex domain of discretisation is decomposed into few convex subdomains. The subdomain of interest must be defined by describing a course background mesh of triangle elements, covering the entire multidomain region, which forms the input for finite element analysis.
In this work, a black box architecture expert system has been developed which incorporates the information about the object geometry as well as the boundary and loading conditions, distribution of materials characteristics to generate an a priori (before the finite element analysis is carried out) mesh which is more refined around the critical regions (singularities, re-entrant corners, regions with high-stress concentration, etc) of the problem domain. This system uses a new concept of subtracting to locate the critical regions in the domain and to assign priority and mesh size to them. This involves the decomposition of the original structure into substructures (or primitives) for which an initial and approximate analysis can be performed by using analytical solutions and heuristics. When incorporated into and compared with the traditional approach to the adaptive finite element analysis, it is expected that the proposed approach, which starts the process with near optimal meshes, will be more accurate and efficient.
Several numerical examples are presented and discussed. Examples demonstrate that our approach enables to generate the compatible meshes for multimaterial or multidomain problems. The quality of meshes is good, there are no ill-shaped elements. By the proposed expert system we can generate the mesh for any complex structure. The generation of 2D meshes is only the first step using the proposed expert system; in future we shall extend it for 3D meshes.
During the last decade a lot of research has been devoted to extension of the advancing front technique to the parallel computers [8, 10, 11]. But the application of the technique to parallel processors is still challenging. In fact, we have to solve how to minimise inter-processor communication during mesh generation of subdomains. The proposed expert system for complex structures grid generation enables to use it with parallel computers. At the first step the domain of discretisation is decomposed into subdomains and all the surfaces defining the boundaries of subdomains to be gridded are triangulated. Later all subdomains can be meshed concurrently and no more inter-processor communication is required. The master task sends to workers tasks information about dividing common boundaries and information of each subdomain. The workers tasks receive their subdomain data and mesh their subdomain. Later the master receives the information from the workers tasks and joins gridded subdomains to one structure, ensuring the compatibility of nodes on common boundaries. So this suggested expert system enables to minimise the communications and costs of computations. The implementation of the expert system to parallel processors is to be done in the future.
First Published Online: 26 Jul 2012
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