Eigenfrequency optimized 3D continua, with possibility for cavities

Abstract

Eigenfrequency optimization for 3D continua is formulated and exemplified by the geometry and boundary conditions of a thick plate. Numerical finite element models are based on four node tetrahedra and results from subspace iterations give directly the basis for the continuum redesign. The 3D modeling with a large number of elements has the possibility in optimal design to obtain (as found) not only holes but also cavities in the continuum. Sensitivity analysis is presented on the element level with simple physical interpretation of the involved terms. This general result has general value for control of eigenfrequencies. It is found that in the combination of partial differentiation with the chain rule of differentiation, a specific notation is needed and a suggestion is presented.The optimization method is based on a derived optimality criterion, and as such the maximization problem change to a problem of determining a design with uniform values of this criterion. Non-linear stiffness interpolation may be a physical reality. A two parameter interpolation function is incorporated analytical, also in the sensitivity analysis and the optimality criterion, but without focusing on 1-0 optimal solutions. Two cases of boundary conditions, two cases of total amount of material, and cases of linear and non-linear stiffness interpolation are studied.

Buckling load optimization for 2D continuum models, with alternative formulation for buckling load estimation

“…For other dependence than linear proportionality, see Pedersen and Pedersen (2015) and the application to the 3D buckling design in Pedersen and Pedersen (2018). The simple result (18) is also valid for other finite element models, such as axisymmetric ring elements and 3D tetrahedron elements.…”

Local analytical sensitivity analysis for design of continua with optimized 3D buckling behavior

“…simplified by the stationarity of the Rayleigh quotient with respect to change of the eigenmode and applying a hat notation for gradients with unchanged eigenmode, see Pedersen and Pedersen (2015) with reference to Wittrick (1962). Note for the energies in (11), that the stiffness matrices are based at the solution {D} i for the non-linear elastic problem while the buckling mode {∆} is the eigenmode.…”