Topology optimization employing a condensation method for nonlinear structural frames with supplemental mass

Abstract

A topology optimization schema employing a condensation method for nonlinear frame structures with supplemental mass subjected to time‐varying excitation is presented. In the context of the design of structural frames, in certain applications, the supplemental mass can be order(s) of magnitude larger than the mass of the system itself. Thus, condensing the system of governing equations to only those associated with the supplemental mass, reduces the complexity and computational cost of the dynamic analysis and thus the optimization process. In addition to considering material nonlinearity, distributed plasticity, and multiaxial interactions, the hysteretic beam finite element (FE) model employed in this study has constant elastic stiffness and hysteretic matrices facilitating a Guyan‐type condensation of the nonlinear dynamic equations. Furthermore, the nonlinear dynamic equations and element evolution equations of hysteretic FE modeling are concisely presented as a system of first‐order nonlinear ordinary differential equations (ODEs) that can be solved using general ODE solvers. The schema is demonstrated on various design problems considering different arrangements of supplemental mass, where the objective is to minimize the volume of the structure subject to maximum displacement constraint using a single p‐norm.

Topology optimization of structural frames based on a nonlinear Timoshenko beam finite element considering full interaction

“…Recently, an alternative approach to topology optimization of frame structures considering material nonlinearity has been suggested 24‐26 . With this approach, a hysteretic FE modeling scheme 27‐31 is employed, whereby nonlinearity is modeled through hysteretic degrees‐of‐freedom (DOF) that evolve according to nonlinear first‐order ordinary differential equations (ODEs) (evolution equations), while distributed plasticity is considered through appropriate hysteretic interpolation functions.…”

“…Extension to a general nonlinear Timoshenko element would thus, enable appropriate material allocation based upon the element's capacities while automatically simulating the dominant response based on the updated element's properties during the optimization process. Consequently, due to the constant stiffness and hysteretic matrices with the hysteretic FE modeling approach, it was shown that condensation could be employed when the mass of the structure (i.e., the consistent mass) is insignificant relative to the supported mass (concentrated) 26 as is the case for many structural engineering design applications.…”

“…Recently, an alternative approach to topology optimization of frame structures considering material nonlinearity has been suggested. [24][25][26] With this approach, a hysteretic FE modeling scheme [27][28][29][30][31] is employed, whereby nonlinearity is modeled through hysteretic degrees-of-freedom (DOF) that evolve according to nonlinear first-order ordinary differential equations (ODEs) (evolution equations), while distributed plasticity is considered through appropriate hysteretic interpolation functions. Owing to the first-order mathematical form of the hysteretic FE model, the entire system of governing equations, that is, the dynamic equilibrium and evolution equations, can thus be straightforwardly expressed as a set of nonlinear ODEs and solved using general ODE solution methods.…”