Optimization of the cross-section of hollow bars under torsion using the method of fundamental solutions

The method of fundamental solutions is a boundary-type mesh-free method, which is very suitable for problems with unknown or moving boundaries. In this paper, the method of fundamental solutions is employed for shape optimization in torsion problems. The objective of this work has been to find optimum corners radii of hollow cross-sections under torsion for minimizing the maximum stress. First, it is shown that for the optimum value of the corner radius, the maximum shearing stress on the outer boundary should be equal to the shearing stress at internal corner. Considering this fact, a suitable objective function is defined and then it is minimized using the Levenberg-Marquardt method, which is a gradient-based optimization method. The configuration of collocation and source points has a very important effect on the accuracy of the solution in the method of fundamental solutions. Here, a two-constraint method is used for proper configuration of source and collocation points. To verify the accuracy of the developed code for torsion analysis of hollow members using the method of fundamental solutions, an example with a hollow elliptical domain is presented. The obtained numerical results are compared with the results of exact solution, which show a very good agreement. The optimum values of corners radii for members with square, rectangle and trapezoid cross-sections and different thicknesses have been successfully found. Then, using the obtained results, a formula for the optimum value of the radius of internal corners of hollow rectangle cross sections is constructed.