Abstract
This paper deals with solving Poisson’s equation with conditions on Dirichlet’s limits in an isosceles trapezium with two cracks. The large singular finite elements method used gives satisfactory results in all the domain of study. Numerical values obtained are very accurate for the constraint function and its first derivatives except at the ends of cracks where major changes were registered.
Highlights
Poisson’s equation is used in many fields of physics
The mode of convergence of the large finite singular elements is exponential as shown in the graph in figure 2 where we see the evolution of the 10-base logarithm of the alignment global error according to the approximation order N (36N being the total number of coefficients akl kept in the series characterizing the solutions to auxiliary problems)
The study of the trapezium made of three equilateral triangles with two cracks using the large finite elements method provides satisfactory results in the entire domain studied
Summary
Poisson’s equation is used in many fields of physics. For example, the study of the elastic deformation of a horizontal membrane applied to distributed load, that of low bar torsion or the flow in a pipe [1,2]. Solving Dirichlet’s problem for Poisson’s equation on a domain with cracks is difficult. At these points i , the series corresponding to the solution of the homogeneous equation associated with the Poisson’s equation are:. Common methods of finite elements or finite differences provide unsatisfactory results when they are used under their standard form. These methods, as shown by various authors [8,9,10,11,12,13], may be slightly improved if they take the analytical form of the solution near singularities into account. The rationale of the method and its convergence properties are discussed by [14,15]
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