2018/05/23 by Maria Vasilyeva, Vasilyeva, Maria, Eric T. Chung +7 · 1 citation
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #Advanced Numerical Methods in Computational Mathematics
paper · pdf · doi:10.48550/arxiv.1805.09420
In this paper, we present an upscaling method for problems in perforated\ndomains with non-homogeneous boundary conditions on perforations. Our\nmethodology is based on the recently developed Non-local multicontinuum method\n(NLMC). The main ingredient of the method is the construction of suitable local\nbasis functions with the capability of capturing multiscale features and\nnon-local effects. We will construct multiscale basis functions for the coarse\nregions and additional multiscale basis functions for perforations, with the\naim of handling non-homogeneous boundary conditions on perforations. We start\nwith describing our method for the Laplace equation, and then extending the\nframework for the elasticity problem and parabolic equations. The resulting\nupscaled model has minimal size and the solution has physical meaning on the\ncoarse grid. We will present numerical results (1) for steady and unsteady\nproblems, (2) for Laplace and Elastic operators, and (3) for Neumann and Robin\nnon-homogeneous boundary conditions on perforations. Numerical results show\nthat the proposed method can provide good accuracy and provide significant\nreduction on the degrees of freedom.\n