The present thesis investigates the solution of in-plane heterogeneous thin plate problems with edge singularities based on the Kirchhoffs assumptions. As the majority of the numerical approaches employ smooth bases for interpolation, treatment of non-smooth behavior in the vicinity of singularities needs special remedies as either adaptive h-refinement of the DOFs or insertion of pre-eva‎luated enrichment terms consistent with the nature of the singularity. The proposition of the present dissertation is to develop a numerical method that automatically reproduce the singular terms without the need of a-priori knowledge of the singularity order. Equilibrated Singular Basis Functions are formed in the polar coordinate system, consisting from polynomial Chebyshev functions in radial direction and angular trigonometric functions. The treatment of the weak singularity for bi-harmonic problems in the heterogeneous medium is carried out using some special variables resulted from a mapping technique that transmit the singular space to a smooth one. This approach facilitates the solution procedure and improves the behavior of the final matrix equations. Considering the variable coefficients of the PDEs in heterogeneous media, a weak form of the weighted residual integration is applied through the Gauss-Green method to remove differentiation operators from the variable coefficients, in order to simplify the final relations. Construction of singularity induced terms without the need to know the analytical singularity order of the problem is the most important innovation of this research. Another innovation is the considerably easy calculation of generally complicated 2D integrals in the form of a combination of some pre-eva‎luated normalized one-dimensional integrals. It is observed that this study brings a very favorable quality in solving thin plates with edge singularities, especially in the vicinity of singular points.

نيما نورمحمدي

سعيد صرامي، عليرضا عموشاهي