Symbolic homogenization and structure optimization for a periodically perforated cylindrical shell
The main focus of this paper lies in the drastic model reduction for a complex multi-scale problem of linear elasticity. The core of the work lies in the structural optimization of periodic perforated cylindrical shells under a given load on a small portion of the surface. Periodic structure of the shell is a frame of beams. Algorithm presented in the paper utilizes two our recent analysis works on homogenization and dimension reduction of the problem in the perforated 3D shell to two-dimensional homogenized shell, and then dimension reduction w.r.t. the beam thickness in auxiliary cell-problems to problem on a frame of beams, by asymptotic method. In this paper, we found the analytical solution for the orthotropic cylindrical shell, obtained in the limit, by variable separation and Fourier analysis. The solution depends explicitly on the effective properties, which are computed symbolically, and on the design variables of the structure. Cell-problems are solved by symbolic means and the structural design for this type of the loading has been optimized. Moreover, this yields a semi-analytic optimization problem and the practical usage of the underlying theoretical derivation. We stress that the homogenization and dimension reduction of a shell with holes and the analytic solution to the corresponding macroscopic problem are new.