Overlapping radial basis function interpolants for spectrally accurate approximation of functions of eigenvalues with application to buckling of composite plates

Many physical simulations involve eigenvalue computations: natural frequencies, stability, …Engineering optimization requires many evaluations of such eigenvalues leading to excessive computational times, especially for global optimization techniques. Accurate approximations of eigenvalues with respect to optimization variables that often are natural parameters of the physical systems are then required to alleviate the computational burden. Dependence of the critical eigenvalue with respect to these natural parameters is often complex; part of the reason is that the critical eigenvalue is the minimum of several eigenvalues, resulting in a loss of differentiability for physical parameters where the critical eigenvalue becomes multiple. This discontinuous derivative prevents from accurate approximation whenever the approximation model is smooth such as most of the standard approximation techniques (kriging, artificial neural network, …). In this work, we present an original strategy that takes into account this discontinuous behavior by dividing the input space through the clustering of the gradient space. Radial basis interpolants are then constructed over each region and a patch region is defined that encompasses the non differentiable region. This allows us to retrieve over the whole domain of definition excellent convergence rates and numerical experiments over realistic numerical simulation show that it is possible to achieve spectrally accurate approximations.