Uniform meshes may be appropriate for some problems

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Unformatted text preview: form meshes may be appropriate for some problems having simple geometric shapes, but, even there, nonuniform meshes might provide better performance when solutions vary rapidly, e.g., in boundary layers. Finite element techniques and software have always been associated with unstructured and nonuniform meshes. Early software left it to users to generate meshes manually. This required the entry of the coordinates of all element vertices. Node and element indexing, typically, was also done manually. This is a tedious and error prone process that has largely been automated, at least in two dimensions. Adaptive solution-based mesh re nement procedures concentrate meshes in regions of rapid solution variation and attempt to automate the task of modifying (re ning/coarsening) an existing mesh 1, 5, 6, 9, 11]. While we will not attempt a thorough treatment of all approaches, we will discuss the essential ideas of mesh generation by (i) mapping techniques where a complex domain is transformed into a simpler one where a mesh may be easily generated and (ii) direct techniques where a mesh is generated on the original domain. 5.2.1 Mesh Generation by Coordinate Mapping Scientists and engineers have used coordinate mappings for some time to simplify geometric di culties. The mappings can either employ analytical functions or piecewise polynomials as presented in Chapter 4. The procedure begins with mappings x = f1 ( ) y = f2( ) 5.2. Mesh Generation 3 that relate the problem domain in physical (x y) space to its image in the simpler ( ) space. A simply connected region and its computational counterpart appear in Figure 5.2.1. It will be convenient to introduce the vectors x = x y] T f ( ) = f1( ) f2( )] T (5.2.1a) and write the coordinate transformation as x = f( ) f ( ξ ,1) 111 000 1,2 11 00 η 2,2 1,2 f ( 0, η ) y f ( 1, η ) 111 000 111 000 111 000 111 000 1,1 x (5.2.1b) 2,1 f ( ξ ,0) 1 0 1 0 1 0 1 0 11111111111111111111111111 00000000000000000000000000 1 0 1 0 1 0 1 0 1 0 1 0 1,1 2,2 ξ 2,1 Figure 5.2.1: Mapping of a simply...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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