Finite prism method based topology optimization of beam cross section for buckling load maximization

The use of the finite element method (FEM) for buckling topology optimization of a beam cross section requires large numerical cost due to the discretization in the length direction of the beam. This investigation employs the finite prism method (FPM) as a tool for linear buckling analysis, reducing degrees of freedom of three-dimensional nodes of FEM to those of two-dimensional nodes with the help of harmonic basis functions in the length direction. The optimization problem is defined as the maximization problem of the lowest eigenvalue, for which a bound variable is introduced and set as the design objective to treat mode switching phenomena of multiple eigenvalues. The use of the bound formulation also helps the proposed optimization to treat beams having local plate buckling modes as the fundamental modes as well as beams having global buckling modes. The axial stress is calculated according to the distribution of material modulus which is interpolated using the SIMP approach. Optimization problems finding cross-section layouts from rectangular, L-shaped and generally-shaped design domains are solved for various beam lengths to ascertain the effectiveness of the proposed method.