Automatic mesh-free boundary analysis: Multi-objective optimization

The paper is concerned with the numerical solution of two-dimensional potential problems, through a mesh-free boundary model in a multi-objective optimization framework that automatically generates Pareto-optimal meshfree discretization arrangements. This robust new strategy of analysis allows for simultaneously improving the solution accuracy, the conditioning of the numerical solver, the stability and efficiency of the mesh-free analysis. The boundary mesh-free model (BMFM) is built on the boundary integral equation of the Laplace potential, with a moving least squares (MLS) approximation of variables. The model considers independent MLS approximations in each boundary segment and performs integration with standard numerical quadrature. The main novelty of the paper is the automatic generation of Pareto-optimal nodal arrangements and corresponding compact supports of the mesh-free boundary model, by means of an evolutionary multi-objective optimization process, based on genetic algorithms, which uses reliable very efficient objective functions. A benchmark problem is presented to assess the accuracy and efficiency of the modeling strategy. The remarkably accurate results obtained, in perfect agreement with those of analytical solutions, make very reliable this robust new strategy of automatic mesh-free boundary analysis in a multi-objective optimization framework.