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The effect of the dynamic behavior of fins on the natural convection heat transfer inside a square cavity was studied numerically. Attachment of conductive thin and flexible fins with length equal to 0.2, positioned at 9 locations on both hot and cold wall was examined. The top and the bottom horizontal walls of the cavity were insulated while their left and the right vertical walls were maintained at a constant temperature T_h and T_c. The numerical scheme is based on the finite element method adapted to triangular non-uniform mesh element by a non-linear parametric solution algorithm. Furthermore, the dimensionless equations of flexible parts of the cavity were solved using the Newton-Raphson method. Based on our results, the dynamic behavior of the fins leads to decrease the rate of heat transfer in compared to the rigid fin. It also found the shape of the fins and its positions play an important role in decrease or increase of heat rate inside the cavity.