Renormalization group approach to multiscale modelling in materials science

Dendritic growth, and the formation of material microstructure in general, necessarily involves a wide range of length scales from the atomic up to sample dimensions. The phase field approach of Langer, enhanced by optimal asymptotic methods and adaptive mesh refinement, copes with this range of scales, and provides an effective way to move phase boundaries. However, it fails to preserve memory of the underlying crystallographic anisotropy, and thus is ill-suited for problems involving defects or elasticity. The phase field crystal (PFC) equation-- a conserving analogue of the Hohenberg-Swift equation --is a phase field equation with periodic solutions that represent the atomic density. It can natively model elasticity, the formation of solid phases, and accurately reproduces the nonequilibrium dynamics of phase transitions in real materials. However, the PFC models matter at the atomic scale, rendering it unsuitable for coping with the range of length scales in problems of serious interest. Here, we show that a computationally-efficient multiscale approach to the PFC can be developed systematically by using the renormalization group or equivalent techniques to derive appropriate coarse-grained coupled phase and amplitude equations, which are suitable for solution by adaptive mesh refinement algorithms

Algorithm Refinement for Fluctuating Hydrodynamics

This paper introduces an adaptive mesh and algorithm refinement method for fluctuating hydrodynamics. This particle-continuum hybrid simulates the dynamics of a compressible fluid with thermal fluctuations. The particle algorithm is direct simulation Monte Carlo (DSMC), a molecular-level scheme based on the Boltzmann equation. The continuum algorithm is based on the Landau–Lifshitz Navier–Stokes (LLNS) equations, which incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. It uses a recently developed solver for the LLNS equations based on third-order Runge–Kutta. We present numerical tests of systems in and out of equilibrium, including time-dependent systems, and demonstrate dynamic adaptive refinement by the computation of a moving shock wave. Mean system behavior and second moment statistics of our simulations match theoretical values and benchmarks well. We find that particular attention should be paid to the spectrum of the flux at the interface between the particle and continuum methods, specifically for the nonhydrodynamic (kinetic) time scales

Variable high-order multiblock overlapping grid methods for mixed steady and unsteady multiscale viscous flows, part II: hypersonic nonequilibrium flows

The variable high-order multiblock overlapping (overset) grids method of Sjogreen & Yee (CiCP, Vol.5, 2008) for a perfect gas has been extended to nonequilibrium flows. This work makes use of the recently developed high-order well-balanced shock-capturing schemes and their filter counterparts (Wang et al., J. Comput. Phys., 2009, 2010) that exactly preserve certain non-trivial steady state solutions of the chemical nonequilibrium governing equations. Multiscale turbulence with strong shocks and flows containing both steady and unsteady components is best treated by mixing of numerical methods and switching on the appropriate scheme in the appropriate subdomains of the flow fields, even under the multiblock grid or adaptive grid refinement framework. While low dissipative sixth- or higher-order shock-capturing filter methods are appropriate for unsteady turbulence with shocklets, second- and third-order shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence. It is anticipated that our variable high-order overset grid framework capability with its highly modular design will allow an optimum synthesis of these new algorithms in such a way that the most appropriate spatial discretizations can be tailored for each particular region of the flow. In this paper some of the latest developments in single block high-order filter schemes for chemical nonequilibrium flows are applied to overset grid geometries. The numerical approach is validated on a number of test cases characterized by hypersonic conditions with strong shocks, including the reentry flow surrounding a 3D Apollo-like NASA Crew Exploration Vehicle that might contain mixed steady and unsteady components, depending on the flow conditions

A Hybrid Particle Scheme for Simulating Multiscale Gas Flows with Internal Energy Nonequilibrium

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/83650/1/AIAA-2010-820-828.pd

Assessment of an All-Particle Hybrid Method for Hypersonic Rarefied Flow

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106450/1/AIAA2013-1203.pd

Evaluation of a Hybrid Boltzmann-Continuum Method for High Speed Nonequilibrium Flows

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/83556/1/AIAA-2010-1569-683.pd

Algorithm Refinement for Fluctuating Hydrodynamics

This paper introduces an adaptive mesh and algorithm refinement method for fluctuating hydrodynamics. This particle-continuum hybrid simulates the dynamics of a compressible fluid with thermal fluctuations. The particle al