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

with respect to Von Mises stresses. The volume constraints are considered in the analyses, as the implementation of a spatial filter and the conjugate gradient method with the incomplete Cholesky preconditioner to speed up the solution of the linear system of each step of the search. a) Problem Formulation Considering the classical topology problem for the maximum stiffness of statically loaded linear elastic structures, a TO mathematical formulation for continuum structure can be discussed. Considering the TO problem as minimizing the deformation energy of a given structure considering the equilibrium, it follows that W=2U. The problem can then be defined as: (1) with being the element's elasticity matrix, is the element's strain vector, is the volume of an element, NE is the number of finite elements of the mesh, is the stiffness matrix, = is the equilibrium equation, is the vector of loads applied to the structure, is the design variable of the i - th element, is the vector of design variables. b) Smooth Evolutionary Structural Optimization (SESO) The ESO method, which heuristic is based on the gradual and systematic removal of elements whose contribution to the stiffness of the structure are insignificant, was proposed by Xie and Steven [42]. The SESO method proposed by Simonetti et al. [43] is based on the ESO philosophy and applies a weighting to the constitutive matrix so that the element that would be eliminated is maintained and receives a smoothing characteristic. This treatment procedure applies a degradation in the value of its initial stiffness in such, during the removal process, its influence can contribute and determine its permanence or its definitive withdrawal from the design domain. Thus, the elements located near the limit to the left of this maximum strain energy are kept in the structure, defining a smoother heuristic removal. In Fig. 2, ( ) is the constitutive matrix of element j, = + is the domain of elements that can be withdrawn, is the domain of elements that must be effectively removed, is the domain of elements that are returned to the structure, 0 ≤ � � ≤ 1 is a weighted function. Fig. 2: Classic procedure in strain energy: (a) SESO and (b) ESO c) The Level Set Method (LSM) LSM is a technique for representing moving interfaces or boundaries using a fixed mesh. The dynamics of the interfaces can be formulated as the evolution of the level function defined by ( ( ), ) , which is continuous Lipschitz and is usually defined as follows © 2023 Global Journals Global Journal of Researches in Engineering ( ) E Volume XxXIII Issue II Version I 25 Year 2023 Topology Optimization: Applications of VFLSM and SESO in the Generation of Three-Dimensional Strut-and-Tie Models Minimize: ( ) = 1 2 = ∑ 1 2 ∫ ( ) 1 Subject to: = ( ) = ∑ − ≤ 0 =1 = { 1 2 3 … }, = 1 = 0