A truly meshless approach to structural topology optimization based on the Direct Meshless Local Petrov–Galerkin (DMLPG) method

With a wide applicability in several types of engineering problems, topology optimization is one of the most interesting fields of structural optimization. Many meshless methods have been developed, however, they were less explored in topology optimization compared to other methods, as the Finite Element Method (FEM). The Direct Meshless Local Petrov–Galerkin (DMLPG) is characterized as a truly meshless method since it does not use a mesh at any stage of its development. It has been applied to solve many boundary value problems, achieving results with good precision and computational efficiency. Instead of performing the numerical integral of complicated shape functions, the DMLPG considers low-degree polynomials. The new topology optimization approach proposed in this work couples the DMLPG method with a Bi-Directional Evolutionary Structural Optimization (BESO) method. DMLPG is used to obtain smooth nodal displacements, strains, and stresses, and BESO updates the structural geometry based on design sensitivity values. Numerical examples performed were compared with the results obtained with FEM and with other works in the literature, showing the applicability and validity of the technique.