Comparative Study of Unstructured Meshes Made of Triangles and Quadrilaterals

Extensive studies have been conducted on the generation of unstructured meshes of triangles for the purpose of nite element analysis. Delaunay triangulation has been the basis of several such methods, and has lead to algorithms that can produce unstructured meshes of triangles for any planar domain. However, there have not been many robust methods for producing unstructured meshes of quadrilaterals. This is unfortunate, because for plain stress and plain strain problems solved using nite element analysis, 4-noded quadrilateral elements perform much better than 3-noded triangular elements, especially when the discretisation is not dense. In this paper, we present an extremely simple and guaranteed method to generate meshes of quadri-laterals from meshes of triangles generated using a robust Delaunay triangulation algorithm. The elements thus obtained have less than optimal aspect ratios. In order to determine if the less than optimal aspect ratios of the elements aaect the quality of results when these meshes are used for nite element modeling, we applied these meshes to the solution of a problem. We are of the opinion that a mesh of quadrilaterals generated in the inexpensive manner presented in this paper produces results that compare well even with those from mapped meshes. Thus, the presented method provides an eecient, simple and eeective way to generate unstructured meshes of quadrilaterals of reasonable quality. 1 Background In the past, a number of algorithms for generation of unstructured triangular meshes have been reported Chew 89, Cavendish 83, Ruppert 92] (Structured or mapped meshes are developed through mappings of a mesh deened in a logical domain into a geometric domain, whereas unstructured meshes do not depend upon a topologically similar logical domain. However, it has been the analysts' experience that in nite element analyses, for a given number of degrees of freedom (DOF), 4-noded quadrilateral elements provide better results than 3-noded triangular elements Cook et al 74]. The 3-noded constant strain triangle behaves poorly in bending because the stresses and strains are constant within the element. However, the results improve when the mesh becomes ner Cook et al 74]. There have been attempts to come up with algorithms yielding good-quality meshes of quadrilaterals. The Paving algorithm is one of them Blacker 90]. There are a number of algorithms that produce mapped (structured) meshes of quadrilaterals. However, a disadvantage of mapped meshes is that they cannot be adapted to t any region. There have been quite a few algorithms …