Structural optimization of trusses with the L1 and L2 Penalty Method

With several applications in civil engineering, such as in roofing, transmission towers and bridge support, the truss stands out as a fundamental structure. The importance of the truss has led the research community to devote great attention to its optimal design, seeking efficient and innovative solutions. This paper aims to present the formulation and algorithm of the hybrid L1 and L2 Penalty Method, which is applied in the structural optimization of planar trusses. This method consists of using an auxiliary function in which the constraints of the problem are introduced into the objective function by means of a penalty parameter, which penalizes the objective function if the constraints are violated. The penalty parameter is updated at each iteration by two approaches, namely, one based on the multiplicative function and the other on the logarithmic function. The objective of the optimization problems studied in this research is to determine the cross-sectional areas that will minimize the total mass of the structure, imposing constraints on the axial stresses and forces and nodal displacements. A computational code is developed with the open-source Scilab program which is used to solve three structural optimization problems. The fminsearch function is used in the iterative cycle to find the minimum of the new objective function. The numerical results obtained with the developed program are compared with those obtained by the fmincon function. The structures are discretized by the Finite Element Method, considering the constitutive relation of the linear and elastic material and infinitesimal deformations.