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In the process of structural analysis we often come to structures that can be analyzed with simpler methods than the standard approaches. For these structures, known as regular structures, the matrices involved are in canonical forms and their eigen-solution can be performed in a simple manner. However, by adding or removing some elements or nodes, such methods cannot be utilized. Here, an efficient method is developed for the analysis of irregular structures in the form a regular structure with additional or missing nodes or with additional or missing supports. In this method, the saving in computational time is considerable. The power of the method becomes more apparent when the analysis should be repeated very many times as it is the case in optimal design or non-linear analysis.

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In this paper using the eigenvalues and eigenvectors symmetric block diagonal matrices with infinite dimension and numerical method of finite difference a closed form solution for exact solving of Laplace equation is presented. Moreover, the method of this paper has applications in different states of boundary conditions like Newman, Dirichlet and other mixed boundary conditions. Moreover, with the method of this paper, a mathematical model for the exact solution of the Poisson equation is derived. Since these equations have many applications in engineering problems, in each part examples like water seepage problem through the soil and torsion of prismatic bars are presented. Finally the method is provided for torsion problem of prismatic bars with non-circular and non-rectangular cross sections by using of conformal mapping.