Crack-free joining of $Si_3N_4;and;Al_2O_3$ using 15 layers has been achieved by a unique approach introducing Sialon polytypoids as a functionally graded materials (FGMs) bonding layer. In the past, hot press sintering of multilayered FGMs with 20 layers of thickness $500{mu}m$ each has been fabricated successfully. In this study, the number of layers for FGM was reduced to 15 layers from 20 layers for optimization. For fabrication, model was hot pressed at 38 MPa while heating up to $1700^{circ}$, and it was cooled at $2^{circ}$/min to minimize residual stress during sintering. Initially, FGM with 15 layers had cracks near 90 wt.% 12H / 10 wt.% $Al_2O_3$ and 90 wt.% 12H/10 wt.% $Si_3N_4$ layers. To solve this problem, FEM (finite element method) program based on the maximum tensile stress theory was applied to design optimized FGM layers of crack free joint. The sample is 3-dimensional cylindrical shape where this has been transformed to 2-dimensional axisymmetric mode. Based on the simulation, crack-free FGM sample was obtained by designing axial, hoop and radial stresses less than tensile strength values across all the layers of FGM. Therefore, we were able to predict and prevent the damage by calculating its thermal stress using its elastic modulus and coefficient of thermal expansion. Such analyses are especially useful for FGM samples where the residual stresses are very difficult to measure experimentally.