Solving buckling problems and computing critical points using the multigrid method

Nonlinear systems typically exhibit critical points of instability and bifurcation which create difficulties for computational models. When analyzing structural problems using the finite element method the tangent stiffness operator becomes ill-conditioned and numerical accuracy is reduced near these critical points. The ill-conditioning degrades the performance of iterative equation solvers making it difficult to use these methods to examine critical points. This research describes a method for accurately solving a singular or nearly singular, symmetric system of linear equations and demonstrates algorithms to solve buckling problems and calculate critical states using the multigrid method as an equation solver.;The present research is performed in the context of nonlinear structural mechanics. The quasi-static equilibrium solution of the structure is to be determined and the critical points and bifurcation behavior examined. Three different algorithms are presented for use in the study of buckling problems using multigrid methods: an arc-length method, and two methods for directly computing a critical point. Each of the three algorithms is given in a version appropriate for isolated critical points, and in a version appropriate for compound critical points or isolated critical points in close proximity. The research also details a method of determining the branch directions at a critical point and the quasi static solution tangents at a compound bifurcation point.;The algorithms make use of the eigenmodes of the tangent stiffness operator to create constraints which improve the conditioning of the tangent stiffness operator. The resultant systems are solved using the bordering algorithm and each equation solution is performed using the modified tangent stiffness operator which allows the multigrid method to function properly.