Domain decomposition methods with nonlinear localization for the buckling and post-buckling analyses of large structures

• Nonlinear localization can deal with larger increments than the classical NKS methods.
• Mixed DDM can use larger increments and pass limit points, with respect to NKS methods.
• Important reduction on the number of global iterations.
• The amount of exchanged data between processors is reduced.
• Sensitivity study for the Robin operator.

The paper is focussed on the robustness of parallel computation in the case of buckling and post-buckling analyses. In the nonlinear context, domain decomposition methods are mainly used as a solver for the tangent problem solved at each iteration of a Newton–Raphson algorithm. In case of strongly nonlinear and heterogeneous problems as those encountered in buckling and post-buckling, this procedure may lead to severe difficulties regarding convergence and efficiency. The problem of convergence is regarded as the most critical issue at the industrial level. Indeed if a method, which can show efficiency for some problems, is not robust with respect to convergence the method will not be implemented by industrial end-users. Therefore, two paths are explored to gain robustness when making use of domain decomposition methods: (1) a nonlinear localization strategy which may also improve the robustness by treating the nonlinearity at the subdomain level; and (2) a mixed framework allowing to circumvent the problem of local divergence (i.e. at the subdomain level). It is to be noted that those two ingredients may also be used to improve the numerical efficiency of the method but this is not the main focus of the paper. Simple structures are first considered to illustrate the method performances. Results obtained in the case of a boxed structure and of a stiffened panel are then discussed.