Global Journal of Researches in Engineering, E: Civil & Structural, Volume 23 Issue 2

( ) ≈ ( ) +�� ( ) − + − ( ) � 1 where e are defined as: ( ) > 0 ℎ = � ( ) − ( ) � 2 ( ) = 0 ( ) < 0 ℎ = −� ( ) − ( ) � 2 ( ) = 0 For the optimization problem in compliance Eq. (8), it is known that it is satisfied because ( ) < 0 . Then the MMA provides the current design with an approximation of a linear programming problem of the type: (6) (7) (8) (9) with = { |0.9 ( ) + 0.1 ( ) ≤ ≤ 0.9 ( ) + 0.1 ( ) } ∀ = 1,2, … , with ( ) and ( ) being lower and upper asymptotes, respectively, k is the current iteration, n the number of design variables, the design variable and the prescribed volume. The following heuristic rule is used by [44] for updating the asymptotes, for the first two outer iterations, when k =1 and k = 2 are adopted: ( ) + ( ) = 2 ( ) (10) ( ) − ( ) = 1 For ≥ 3 ( ) + ( ) = 2 ( ) ( ) − ( ) = ( ) with ( ) = � � ( ) − ( −1) �� ( −1) − ( −2) � < 0 � ( ) − ( −1) �� ( −1) − ( −2) � > 0 � ( ) − ( −1) �� ( −1) − ( −2) � = 0 (11) where the values of , and were fitted in the respective numerical ranges 0.65 ≤ ≤ 0.75 , 1.15 ≤ ≤ 1.25 and 0.9 ≤ ≤ 1 . It can be seen in Eq. (11) that the MMA saves the signal of three consecutive iterations. Thus, when the signals alternate, the MMA detects that the values of the design variables are oscillating, i.e., � ( ) − approximate the design point ( ) . If the values of the design variables do not oscillate, i.e., � ( ) − ( −1) �� ( −1) − ( −2) � ≥ 0 , then the MMA moves the asymptotes away from the design point in order to accelerate up convergence. There are two approaches to solving subproblems in MMA, the "dual approach" and the "primal-dual interior point approach". The dual approach is based on the dual Lagrangian relaxation corresponding to the subproblem, which seeks the maximization of a concave objective function without other constraints and the non-negativity condition on the variables. This dual problem can be solved by a modified Newton method, and then the dual optimal solution can be translated into a corresponding optimal solution of the primal subproblem, which is used in this paper. III. M ethodology for G eneration 3D S trut-and- T ie M odels and the F inal F lowchart To determine the path load of the 3D bars of the STM from the TO analysis, this paper employs a new procedure to evaluate the struts and ties by the signs of the derivatives of the Von Mises stress components. It is known that for 3D problems they can be written as ( ) ² = 1 2 [( 11 − 22 ) 2 + ( 22 − 33 ) 2 + ( 33 − 11 ) 2 ] + 3( 122 + 232 + 312 ) (12) © 2023 Global Journals Global Journal of Researches in Engineering ( ) E Volume XxXIII Issue II Version I 27 Year 2023 Topology Optimization: Applications of VFLSM and SESO in the Generation of Three-Dimensional Strut-and-Tie Models ( −1) �� ( −1) − ( −2) � , < 0 the asymptotes −� � ( ) − ( ) � 2 − ( ) � � ( ) � 1 � − ≤ 0 ∈