(300a) A Novel Method of Grid Generation for Finite Elements

The finite-element method is a powerful and general approach widely used in Computer Aided Engineering (CAE) for solving partial differential and integral equations to obtain approximate solutions to a wide variety of engineering problems where the variables are related by means of algebraic, differential and integral equations. Finite elements has become the defacto industry standard for solving multi-disciplinary engineering problems that can be described by equations of calculus. Applications cut across several industries by virtue of the applications – solid mechanics (civil, aerospace, automotive, mechanical, biomedical, electronics), fluid mechanics (geotechnical, aerospace, electronic, environmental, hydraulics, biomedical, chemical), heat transfer (automotive, aerospace, electronic, chemical), acoustics (automotive, mechanical, aerospace), electromagnetics (electronic, aerospace) and many, many more. The first step in finite element analysis is the grid generation which can be defined as the process of breaking up a physical domain (control volume) into smaller sub-domains (elements) in order to facilitate the numerical solution of a partial differential equation. Grid generation is a critical step in finite element analysis on which the accuracy of the predicted distributions largely rests on. A loose grid would lead to a considerable error in predictions while an unnecessarily tight grid would make the calculations computationally expensive without discernible improvement in the results. This paper presents a novel mesh generation approach based on an efficient sampling technique called the Hammersley Sequence Sampling (HSS) technique based on a quasi-random number generator. It has been shown [1,2,3] that this sampling technique based on Hammersley points provides an optimal design scheme for placing the n points on a k-dimensional hypercube. This scheme ensures that the sample set is more representative of the population, showing uniformity properties in multi-dimensions. One of the main advantages of HSS is that the number of samples required to obtain a given accuracy of estimates does not scale exponentially with the number of dimensions. We capitalized on these properties of the Hammersley Sequence sampling technique by utilizing it for grid generation in the control region for the partial differentiation solver in Fluent computational fluid dynamics (CFD) solver. Some preliminary results have been presented in the following paragraphs

Preliminary Results:

As an initial demonstration, we considered a problem in which a 2-D 4X4 square surface is heated to different levels at the 4 edges and hence has different boundary temperatures on each side as shown in figure 1. The objective is to find the steady-state surface temperature distribution.

Figure 1: 2D slab at different temperatures on each side

The problem was solved using various types and number of grids. For example, figure 2 shows three different grid designs: 1) uniform grid with 169 internal nodes (UNI1), 2) grid formed with similar nodes generated using HSS (HSS1) 3) uniform grid with 1024 internal nodes (UNI2). The comparison between the surface temperature distribution computed is shown in figure 2. It has been found that the results with 1024 internal nodes (design 3) is closer to realistic results. One can see that HSS with much smaller number of grid points is a closer match to the realistic results. However, HSS surface is jagged and not as smooth as the other 2 surfaces is because Fluent does not include the capability to draw the mesh automatically given the internal points ( in this case HSS points). Hence a 164 node grid with triangular elements was first constructed and each individual node point was manually moved to HSS coordinates. The 2 sets of connected circles are points of comparison at the same position on each surface. If we look closely at the surfaces, HSS1 contours are closer to those of UNI2 than of UNI1 .

Figure 2: Comparison of temp. distribution in a heated slab with different grids

This is evident in the first set of circles in the middle of each surface, C1, C2 and C3. Around 90% of circle C3 consists of blue shaded region while 40% of circle C2 and only only 5% or less of circle C1. is blue shaded. Another aspect of comparison is the second set of circles C4, C5 and C6 at the top region of each surface. In C6 and C5, only around 10% and 15% respectively are shaded yellow, while C4 it is 30%. Having demonstrated that the HSS method of grid generation is more efficient than the existing methods for simple surfaces, we proceed to apply this method to large scale and higher dimensional systems. For the preliminary analysis, the HSS grid was constructed manually which was time consuming and in the further this process would be automated. The CFD results computed using the HSS grid would be analyzed and recommendations put forth.

References:

1. Diwekar, U. M.; ‘Introduction to applied optimization (2003)', Kluwer Academic Publishers, Netherlands.

2. Kalagnanam, J.R. and Diwekar, U.M.; ‘ An efficient sampling technique for off-line quality control', Technometrics, 39:308, 1997.

3. Wang, R., Diwekar, U.M., and Gregoire-Padro, C.;,' Latin Hypercube Hammersley Sampling for risk and uncertainty analysis', Environmental Progress, 23(2):141, 2004.