Adaptive finite element mesh triangulation using self-organizing neural networks

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Abstract

The finite element method is a computationally intensive method. Effective use of the method requires setting up the computational framework in an appropriate manner, which typically requires expertise. The computational cost of generating the mesh may be much lower, comparable, or in some cases higher than the cost associated with the numeric solver of the partial differential equations, depending on the application and the specific numeric scheme at hand.

The aim of this paper is to present a mesh generation approach using the application of self-organizing artificial neural networks through adaptive finite element computations. The problem domain is initially constructed using the self-organizing neural networks. This domain is used as the background mesh which forms the input for finite element analysis and from which adaptive parameters are calculated through adaptivity analysis. Subsequently, self-organizing neural network is used again to adjust the location of randomly selected mesh nodes as is the coordinates of all nodes within a certain neighborhood of the chosen node. The adjustment is a movement of the selected nodes toward a specific input point on the mesh. Thus, based on the results obtained from the adaptivity analysis, the movement of nodal points adjusts the element sizes in a way that the concentration of elements will occur in the regions of high stresses. The methods and experiments developed here are for two-dimensional triangular elements but seem naturally extendible to quadrilateral elements.

Introduction

Almost any finite element (FE) analysis initially requires the domain under consideration to be discretized into an appropriate FE mesh. This process is a tedious task to undertake manually and the quality of the resulting mesh depends upon the skills of the engineer. A poor quality FE mesh, where there are large differences between the stresses of adjacent elements, will ultimately reduce the accuracy of the analysis results. Adaptive mesh generation can be used to produce FE meshes having acceptable quality and accuracy. The objective of adaptivity and error estimation is to remove the human factors in mesh design and automate the FE procedure [1], [2], [3], [4], [5].

For any adaptive FE technique to be implemented, the availability of a mesh generator which works in accordance with the information received from the adaptive calculations of the posteriori error estimator using FE solution is necessary [2]. A limit on the overall error for the FE problem is fixed and the procedure is repeated until a solution mesh is obtained with the error over the domain being within acceptable limits.

Subsequent to the adaptive computations, mesh generation schemes may be used to increase the population of elements of the coarse mesh based on adaptivity results. Another way to create meshes based on adaptivity analysis is to populate a mesh with the required number of elements and subsequently reconfigure the element sizes in order to comply with the FE and adaptivity analysis conducted on the initial mesh.

FE meshes must have certain properties in order to be acceptable for computation. In this paper structural mechanics related FE has been considered. The following guidelines are considered standard [2]:

  • 1.

    The mesh should be finer in regions where the solution is believed to be changing rapidly or to have large gradients. Thus smaller elements should be used near singularity points such as reentrant corners or cracks, near holes, near small features of the boundary, near the location of rapidly changing boundary data, at and near in-homogeneities, etc.

  • 2.

    All elements should be well proportioned. The aspect ratio of the element (namely, the ratio between its largest and smallest dimensions) should be close to unity. Square elements are the best quadrilaterals, but even an aspect ratio of 1.5 or 2 is acceptable.

  • 3.

    All interior angles of the element must be significantly smaller than 180°. For example, a quadrilateral with three of its vertices lying on a nearly straight line is usually unacceptable.

  • 4.

    Transition from large elements to small elements must be made gradually. The ratio between the sizes of two neighboring elements may be 1.5 or 2 but should not be much greater than this.

Manevitz et al. [6] conducted mesh generation using a density function to generate finer elements in the regions where the solution is believed to be changing rapidly or to have large gradients. This density function was not based on adaptivity analysis.

In this paper the concept of self-organizing neural networks is used to generate an initial mesh. The same approach is used once adaptive FE has been carried out on the initial mesh in order to reconfigure the size and location of the elements of the initial mesh to comply with the adaptivity analysis.

The following sections describe the concepts of self-organizing neural network, finite element and adaptivity methods used in this paper, in addition to the application of self-organizing neural networks to the mesh generation and the adjustment of FE domains. Examples are presented to illustrate the performance of the proposed approach and finally concluding remarks are given.

Section snippets

Self-organizing neural networks

A self-organizing neural network, as described by Kohonen [7], [8] is a system of neurons linked by a topology. Such a network can then learn to adjust its weights’ parameters based on the input in such a way as to automatically create a map of responsive neurons that topologically resembles the input data. Such networks are often referred to as self-organizing maps (SOM). A SOM consists of neurons organized on an array or two-dimensional grid. The number of neurons may vary from a few dozen up

Finite element and adaptivity

In the case of FE idealization based upon simple constant strain triangular elements the accuracy of the results depends on the topology of the mesh chosen for FE analysis [2], [6]. Adaptive remeshing may be used to determine an efficient mesh by taking into account the domain error. An efficient mesh is defined as the one in which the error is equally distributed over the domain [9].

Through adaptive remeshing the domain error is reduced as well as uniformly distributed over the domain until

FE meshing using self-organizing neural networks

When applying the FE method to a given domain, one has to divide the domain into a finite number of non-overlapping subdomains (elements). (In two dimensions, the elements are usually triangles or quadrilaterals.) One also has to define a finite number of nodes, which are the vertices of the elements, and possibly other points as well. The collections of elements and nodes (and the connections among them), constitutes the FE mesh, whose quality is an essential ingredient in achieving accurate

Examples

The first example is a simply supported beam loaded on the center. The number of elements in the mesh is 2242. Fig. 7 shows the domain after the initial mesh generation using SOM. Fig. 8 shows the domain after reconfiguration of elements using FE, adaptivity and SOM methods.

The second example represents a dam loaded on the top left hand side and restrained at the bottom edges. The number of elements in the domain is 2376. Fig. 9, Fig. 10 show the initial and reconfigured mesh domain after FE,

Conclusions

This paper presented the use of self-organizing map (SOM) neural networks based on Kohonen algorithm for generating finite element meshes. The initial mesh is generated using the SOM concepts based on two different domain criteria, physical domain and computational domain. A mesh is generated inside a computational domain and is mapped on a one-to-one basis to a physical domain with a different mesh topology. Using SOM concepts random points are selected within the computational domain and

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