Abstract The main objective of this dissertation is to develop the methods for the analysis of complex domains, free from the difficulties associated with geometry-conforming meshing. To this end, the proposed approach in this study involves discretizing the domain using the fixed grid approach, which encloses the computational domain. This approach requires two key steps: The first step is to determine an appropriate strategy to correctly incorporate the boundary conditions of the problem into the calculations of each element. The second step is to calculate the domain integrals without the need for geometry-conforming discretization. The method proposed in this study for the first step is the addition of singular functions to approximate the solution near the boundary. In this case, if singular functions are used that satisfy the governing equation (the fundamental solution of the equation), the domain integration only requires calculating integrals over regular, rectangular-shaped elements of the grid, without the need to eliminate empty regions. In the second step, a method is proposed that performs numerical integration over irregular and relatively complex geometric domains, without the need for geometry-conforming discretization. For this purpose, a generic function is approximated through a set of Legendre polynomials whose coefficients are determined by numerical integration. The information needed for the points falling outside the physical domain is re-generated through the polynomials and the unknown coefficients. The integration of the generic function is then summarized into a set of integrals (of the Legendre polynomials) multiplied by the function’s discrete values. The Legendre polynomials are integrated through the divergence theorem. By incorporating this integration method into the first step, the proposed strategy allows for the analysis of a broader range of problems, without the limitations of relying on the fundamental solution of the governing equation. This approach has been eva‎luated in this study. To assess the accuracy and efficiency of the proposed methods, a variety of problems were solved, and the results were compared to analytical values or those obtained from commonly used software. Keywords: Complex geometry, Fixed grid approach, Singular functions, Numerical integration

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محمد مهدي سعادتپور , بشير موحديان عطار , حسين عموشاهي