Topology optimization of frequency and dynamic stiffness for continuum structure using smooth bi-directional evolutionary structural optimization

The classic bi-directional evolutionary structural optimization (BESO), which is known as “hard kill” in nature, discretely removes the inefficient material in the continuum structure. Using BESO method, structural stiffness and mass matrix of the elements under low sensitivity and inefficiently utilized suddenly reduce. It means that when solving structural topological optimization, highly efficient elements are possibly incorrectly deleted by original BESO. As a result, the smooth bi-directional evolutionary structural optimization (SBESO), as a variant of the BESO procedure, is proposed to overcome this disadvantage. The introduced SBESO was based on the philosophy that if an element was not really necessary for the structure, its contribution to the structural stiffness and mass would gradually diminish until it no longer influenced the structure. This removal process was thus performed smoothly. This procedure was known as ‘‘soft-kill’’ in nature; where not all of the elements removed from the structure domain using the original BESO criterion were rejected. The weighted function was introduced to regulate element’s mass and stiffness matrix, combined with controlling the element deletion rate to make inefficient elements gradually deleted. This method provided good conditioning for the new system of equations that could be resolved in the next iteration because these elements were important to the structure. In this paper, the proposed SBESO method was applied to resolve the structural dynamic topology optimization including frequency optimization and dynamic stiffness optimization of continuum structure under harmonic force excitation. The frequency optimization model was developed to increase the fundamental natural frequency of continuum structure which was away from the frequency of the external force. The dynamic stiffness model was derived to perform structural optimal design according to the structural strain energy of the continuum structure under dynamic load. Aiming at maximizing the natural frequency of continuum structure, the SBESO procedure was employed to solve it. Subsequently the effect of various weighted function including constant, linear and trigonometric function on topology optimization was compared and analyzed. Performing the structural topology optimization of dynamic stiffness in the time domain was quite expensive due to heavy computational time for function and sensitivity. Thus the equivalent static loads (ESL) method in the time domain was used to overcome these disadvantages. Combining ESL and SBESO, structural topology optimization of dynamic stiffness was resolved at ease for the continuum structure under dynamic load. Numerical results showed that SBESO could inhibit the incorrect element deletion, by regulating the element deletion rate and weighted function. Therefore the inefficient elements would gradually diminish. It was also found that the employed linear and trigonometric functions more contributed to obtaining the optimal solution of frequency for the continuum structure compared with constant function. In addition, decreasing element deletion rate aroused increasingly smooth boundary of optimal configuration in structural topology which converged to identical configuration by SBESO. Consequently the optimization method of SBESO improves the original optimization criterion of BESO, which is significantly theoretical to address the dynamic optimization design of continuum structures.