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

( , ) = { ( , ) > 0 ∀ ∈ \ ; ( , ) = 0 ∀ ∈ ; ( , ) < 0 ∀ ∈ \ } with ⊂ {( , ) 2 } is any point in the design domain D and ∂Ω is the solid domain boundary as shown in Fig. 3 for a 2D case. (2) In classical LSM for TO, such as [30] and [31], the design evolution is based on the solution of the Hamilton-Jacobi partial differential Eq. (4). Thus, it needs an appropriate choice of finite difference methods on a fixed cartesian mesh. In general, the design update involves differentiation, resetting and velocity extension. (3) (4) ∅ − | ∅| = 0 = . �− ∅ | ∅| � with ∅ denotes the gradient of a function, is the pseudo time that represents the evolution of the function ( , ) defined, ( , ) is the normal velocity vector (pointing outwards) based on the derivatives of the shape functions in the TO problem. Fig. 3: Evolution history of the 2D geometry to a 3D level Recently, Wang and Kang [34, 35] proposed a 100-line Velocity Field Level Set (VFLS), implemented in Matlab code. The structural shape and topology are updated by a velocity field constructed with the base function and velocity design variables defined throughout the domain. Then, the velocity field determines the search direction of the shape and the topological evolution can be obtained by a generic mathematical programming algorithm, which makes it more convenient and efficient to deal with multiple constraints and types of design variables. For VFLS, we have: (5) ( ) = � ( ) =1 with ( = 1,2, … , ) are the velocity design variables at N velocity points distributed throughout the main design, and ( ) are the basic functions. It is observed that when satisfies the properties of the Kronecker delta it has = of Eq.(3). II. O ptimization A lgorithm - M oving A symptotes M ethod To accelerate and stabilize the present 3D STM in this paper, MMA is employed, which is a mathematical programming algorithm suitable for TO, capable of handling optimization of many constraints and design variables. At each step of the algorithm's iterative process, a convex approximation subproblem is generated and solved. The generation of these subproblems is controlled by the moving asymptotes, which can both stabilize and accelerate the convergence of the overall process, [44]. The optimal solution of the subproblem may or may not be accepted: if so, the outer iteration is completed; if not, a new inner iteration is performed, in which a new subproblem is generated and solved. The iterations are repeated until the values of the approximations of the objective function and the constraints become greater than or equal to the values of the original function when evaluated in the optimal solution of the subproblem, that is, until the conservative condition is satisfied for the functions involved. The approximations that characterize the MMA are rational functions whose asymptotes are updated at each iteration. It is noteworthy that the use of rational approximations is justified by the fact that in several structural engineering problems where reciprocal variables arise, that is, interaction and mutual effort, given the objective function or a constraint ( ) , the approximation functions are given by: Global Journal of Researches in Engineering © 2023 Global Journals ( ) E Volume XxXIII Issue II Version I 26 Year 2023 Topology Optimization: Applications of VFLSM and SESO in the Generation of Three-Dimensional Strut-and-Tie Models