High performance of the generalized finite difference method and applications

Resumen

We solve 2D and 3D second-order partial differential equations considering the Generalized Finite Difference Method (GFDM) with third- and fourth-order approximations. First of all, we analyze the influence of the number of points per star and establish some values as references. Secondly, we propose a new strategy to deal with ill-conditioned stars, which are frequent in higher-order approximations. This strategy uses a few points per star in relation to those established as reference and presents excellent results for detecting ill-conditioned stars, increasing the accuracy of the numerical approximation and reducing the computational cost. To implement the algorithm, we use good programming practices together with higher-order approximations in the GFDM to reduce the computational cost at different stages of the calculation. On the other hand, we have developed a strategy to obtain discretizations adapted to the specific problem to be solved. This strategy distributes the points in the domain according to the gradient values, which allows using a discretization with a smaller number of points, reducing the computational cost and maintaining the accuracy that would be achieved with finer discretizations where the points are distributed homogeneously. Furthermore, we develop a 3D adaptive algorithm with fourth-order approximations on irregular initial discretizations. We compare the results with the algorithm of points added halfway. In all applications, we achieve better accuracy with a decrease in the final number of points and computational time. Finally, to test the performance of the algorithm in a real problem, we evaluate the seismic responses in onshore wind turbines using the GFDM coupled with the Newmark method. We compare the history of transversal displacement with a model based on the Finite Element Method using the ABAQUS software. The results are essentially identical and show the validity of the model proposed in the GFDM.