A full 3D finite element analysis using adaptive refinement and PCG solver with back interpolation

Abstract

In this paper, adaptive refinement finite element analyses were carried out for full 3D problems. In order to achieve an optimal computation cost and to eliminate the effects of singular points, the adaptive refinement procedure was applied in conjunction with a back interpolation for the construction of a good initial guess for the solution of the linear equations system. The combined use of the adaptive refinement procedure, the back interpolation scheme and the preconditioned conjugate gradient (PCG) solver lead to a significant reduction in the operations for the solution of the simultaneous linear equations in the adaptive refinement analysis. In many cases the number of iterations needed by the PCG solver to reach a converged solution is independent of the number of equations in the global system. The numerical results obtained indicated that for a series of carefully designed adaptive meshes, the computational cost required to solve the linear system could be made only proportional to the number of degrees of freedom in the mesh. To our knowledge, this is the first time that the global stiffness equations in 3D stress analysis have been solved with this efficiency. The same set of numerical results also showed that, in general, the total computational cost of the adaptive refinement procedure can usually be minimized by gradually reducing the target relative error of the solution during successive refinements rather than employing a constant target relative error throughout the whole adaptive analysis.