A structural discrete size and topology optimization method with extended approximation concepts
Jiayi Fu, Hai Huang
Abstract
Abstract This work focused on the discrete size and topology optimization problems for the structures that could be composed of bars, beams, plates, or a combination of them. A corresponding optimization method was put forward based on the approximate concepts that were extended from the one applied in the previous engineering method. In the proposed method, the primal problem was firstly transformed into a series of explicitly approximate problems involving both 0/1 topology and discrete size variables with the extended approximate concepts. 0/1 topology variables were determined with a genetic algorithm (GA), and the corresponding discrete size variables were optimized after determining 0/1 topology variables. In size optimization, the variables were firstly supposed as continuous and determined, as $${\overline{\mathbf{X}}}^{{\left( {\mathbf{p}} \right)}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mfenced> <mml:mi>p</mml:mi> </mml:mfenced> </mml:msup> </mml:math> , with a dual method through the second-level approximate problems. Then the available discrete size values adjacent to $${\overline{\mathbf{X}}}^{{\left( {\mathbf{p}} \right)}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mfenced> <mml:mi>p</mml:mi> </mml:mfenced> </mml:msup> </mml:math> were selected from the original discrete set, and the size variables could be further determined from these values with a GA. Structural analyses were only conducted before establishing the approximate problems in iteration cycles. Numerical examples were given to illustrate the performance of this method, and the results indicate that this method is quite efficient for the discrete size and topology optimization problem.