Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element
Résumé
s: In this paper, the Riemann integral and the fundamental of calculus will be used to perform double integrals on polygonal domain surrounded by closed curves. In this context, the double integral with two variables over the domain is transformed into sequences of single integrals with one variable of its primitive. The sequence is arranged anti clockwise starting from the minimum value of the variable of integration. Finally, the integration over the closed curve of the domain is performed using only one variable. The way of integration is illustrated by practical examples in which the area and moments of area are found for arbitrary polygons surrounded by closed, straight lines (triangular, quadrilateral, pentagonal, and hexagonal shapes) and compared with the exact values resulting from dividing the polygon into its standard elementary shapes and the parallel axis theorem. The stiffness matrix is derived for an arbitrary quadrilateral finite element for plate bending. The derived element is a generalization of the first finite element used in the analysis of thin plates known as ACM. The results are tested according to program code written in MATLAB. The method is generally applicable and is valid on a domain surrounded not only by straight edges but by closed curves continuous on the partial intervals of the integration domain. The generalization of this technique to volume integrals over polyhedral domains is possible. keywords: Polygonal element, Riemann integral, moments of area, stiffness matrix 1 Introduction The finite element methods among other numerical methods experienced significant developments after the use of the Voronoi Diagram in partitioning of a plane points into convex polygons [1]. Based on Voronoi Diagram, there were several mesh generator for polygonal and polyhedral elements with topology optimization. These offer a general framework for finite element discretization and analysis, see for example [2] and the mesh generators mentioned therein. In the past fifteen years, many works have been published which operate on polygonal elements in the framework of numerical analysis and used intensively in the fields of applied engineering and physical sciences and even in medical and biological sciences [3, 4, 5, 6, 7, 8, 9, 10,]. An overview of previous developments on conforming polygonal and polyhedral finite elements is included in [9], and an overview on the use of different generalized barycentric coordinates in Galerkin finite element computations is included in [6]. Several other papers that use the polygonal and polyhedral elements in different fields of computational Engineering are listed in [8]. Therefore, it was necessary to use flexible techniques in performing the integrals of the state variables defined on the domains analyzed. These techniques become necessary when changes occur suddenly in the geometry of the domains such as the appearance of cracks or ruptures within them. Some recent publications [11, 12, 13 14, 15] show that the topic is still under study and development; some others [16, 17, 18] reflect its use and wide spreading under different disciplines.
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