We therefore develop an adaptive surrogate model in this chapter using snapshot POD and diffuse approximation for the solution of the microscopic problems.
With the presence of instability phenomena at both scales, the arc-length path following technique is adopted to solve both macroscopic and microscopic problems.
A corresponding three-scale numerical method is then derived using the finite element discretization with numerical quadrature, where the macroscopic and mesoscopic permeability is upscaled at quadrature points from mesoscopic and microscopic problems, respectively.
We derive the steady state formulations and discuss several topics related to the numerical implementation of the model, namely the solution procedure of the discretized microscopic problems, evaluation of the homogenized coefficients and an approximation of the convolution integrals of the macroscopic model, so that the fading memory effects are computationally tractable.
All of it is an infomercial for being better than you are now, better than you were yesterday, better tomorrow than you could have fathomed, working up a sweat as you thunder through your microscopic problems and emerge with the corpses of your SoulCycle class and his ambiguous response to your dinner invite in your rearview mirror.
The obtained uniqueness and stability conditions associated with a continuum rate formulation of the microscopic equilibrium problem, contain unusual nonlinear contributions arising from crack interface self-contact mechanisms whose significance for an accurate evaluation of macroscopic critical loads at the onset of instability and bifurcation is then investigated in an innovative way.
Microscopic cell problems are solved for active nodes only; microscopic data in inactive nodes is approximated from microscopic data of active nodes with a similar temperature evolution.
The computational cost of solving numerous mesoscopic and microscopic flow problems is further reduced by applying a Petrov Galerkin reduced basis method at the mesoscopic and microscopic scales.
It consists of a macroscopic heat equation and a family of microscopic cell problems that model the phase transition of the mixture.
By considering uncertainty in the geometric properties of the microstructure, a significant computational cost is added to the solution of a large set of microscopic equilibrium problems.
