Heterogeneous Stochastic Multiscale Method
Ianyqui Falcão Costa1; Liliane de Allan Fonseca2; Ézio da Rocha Araújo1
1 Federal University of Pernambuco; 2 Federal University of Pernabuco
doi:10.20906/CPS/CILAMCE2017-1243
Resumo
We generalize a theoretical and computational framework for upscaling in random heterogeneous micro-properties for solid mechanics published elsewhere. By heterogeneous we mean both multi-physics models at different scales and random parameters at distinct locations in the physical domain, probably correlated, e.g. Stokes-Brinkman and fractures at microscale and heterogeneous Darcy at macroscale when the target problem is hydrocarbon reservoir simulation (hyperbolic problem) or a matriz with aleatoric inclusions at microscale and homogeneous matriz at macroscale when the target problem is solid and structural mechanics (elliptic problem). The approach uses a mixing of information theory, thermodynamics science and statistical mechanics in order to extract relevant probabilistic information from microscale and adaptively control the stochastic error between the macroscale and microscale responses. We then define a goal-oriented upscaling procedure to guarantee equivalence between micro and macroscale target to specific output, and it generalization to a mathematical sound multi-goal-oriented response where users control distinct accuracies of specific target responses. Preliminary applications in one and two-dimensions domain for elliptic and hyperbolic equations are presented. Applications use a number of realizations (samples) from microscale parameters distributions producing a consistently reduced number of equivalent macroscale realizations, as required by user desired accuracy. The full formulation requires solving a difficult optimization problem for an extended Lagrangian formulation that is solved with a new Parallel Deterministic Annealing, and alternative way to solve by Homotopy Method is traced. Mathematical formulation is such that parallelization is done over realizations so that the overall computation cost of the stochastic upscaling is on the same order of a deterministic one.
Palavras-chave: multiscale; heterogeneous; stochastic; multiphysics