Recently, solutions of partial differential equations (PDEs) are under investigation as an effective tool for geometric modeling and shape generation to develop smooth-edge structures. Three-dimensional shape modeling needs the completeness of geometric queries in a design and optimization processes. In this paper, a novel modeling technique using solutions of six-order partial differential equations (PDEs) has been developed to model a 3D shape with a set of boundary conditions. The advantages of using solutions of PDEs for geometry generation over other techniques are PDEs-based solid modeling needs smaller number of parameters and can automatically guarantee some intrinsic smoothness which is not easily achieved in spline-based blending and other conventional techniques. Based on the developed solutions of PDEs, the optimal cell sizes of Body-Centered Cubic (BCC) and Face-Centered Cubic (FCC) lattice structures have been generated which shows smooth structures compared to other conventional techniques. The developed lattice structures have been optimized using fmincon and genetic algorithms. The optimized BCC unit cell width and strut radius are determined to be 0.01 m and 0.0013 m respectively. For the FCC lattice structure, a 0.001 m strut radius and 0.0099 m unit cell width were determined. Finally, these optimal dimensions were used to generate the solid structures and the stresses were evaluated using analytical and ANSYS simulations. The stresses evaluation results indicate that when the aspect ratio becomes 1 the uniform values were obtained.