The Pseudo-Direct Numerical Simulation method for multi-scale problems in mechanics

We present an overview of the Pseudo-Direct Numerical Simulation (P-DNS in short) method for the solution of multi-scale phenomena. The method can be seen as an adaptation of the variational multi-scale (VMS) method, where the fine solution is solved numerically instead of analytically. Also, from the point of view of homogenization methods it can be seen as an evolution of Finite Element square (FE2) methods, where the most expensive part of the computations is performed offline. The name of P-DNS arises from the premise that in all multi-scale simulations the numerical result obtained with a very fine discretization is correct without the need to introduce any additional model (i.e. turbulence models) or stabilization procedures for transport equation terms (as in residual-based VMS methods). What is intended here is that the P-DNS solution tends to the DNS solution of the problem, accepting as a premise that the DNS solution is a reliable result. In this paper we present first an overview of the P-DNS methodology in the context of an abstract Dirichlet problem involving a second order differential operator that could be non-symmetric and non-necessarily positive definite. Next, the P-DNS approach is applied to the fluid mechanics equations accounting for turbulent phenomena. Examples showing the applicability of the P-DNS method for solving transport problems involving different scales are then presented.