Systematic Closure Approximations for Multiscale Simulations Qiang Du Department of Mathematics/Materials Sciences Penn State University http://www.math.psu.edu/qdu Joint work with C. Liu, Y. Hyon and P. Yu
Motivation multiscale modeling/simulation@psu Longqing Chen, Zikui Liu, Padma Raghvan, Lei Zhang, Jian Zhang, W-M Feng, Maria Emelianenko (Carnegie Mellon), Chris Wolverton (Ford), Steve Langer (NIST), Shenyang Hu (Los Alamos), Materials simulations/design NSF-DMR-ITR, NSF-IUCRC Chun Liu, Jiakou Wang, Cheng Dong, Meghan Henty, Y-K Hyon, Maggie Slattery, Rob Kunz, Sue Gilmor, Manlin Li, Rolf Ryham (Rice), Xiaoqiang Wang (FSU), Peng Yu (G-S), Complex and biological fluids NSF-DMS Hongyuan Zha, Jorge Sofo, Richard Li, Tianjiang Li, Combined Ab-initio/Manifold learning, NSF-CISE Bridging scales
Different Approaches/Philosophies: Two philosophies for bridging scales: online offline Multiscale simulations via different paths: A hand-shake approach conventional A manifold learning approach data-centric A systematic closure approach math/phys 3
Bridging scales: MatCASE as an example First-principles calculations Bulk thermodynamic data Interfacial energies, lattice parameters and elastic constants Kinetic data Experimental data CALPHAD Bulk thermodynamic database Database for lattice parameters, elastic constants and interfacial energies Kinetic database Phase-field simulation Plasticity of phases Microstructure in 2D and 3D Elasticity of phases Models on individual scales OOF: Object-oriented finite element analysis are given, but model parameters are passed between scales Mechanical responses of simulated microstructures 4
Bridging scales: online coupling Coupled simulations to link small and large scales HMM, Quasi-Continuum Multiscale FEM MD Mesh-free, Eqn-free, Ω A data-centric online approach CAMLET : Combined ab-initio manifold learning toolbox via on-line data mining (Du-Li-Sofo-Zha) intrinsic dimension estimation local coordinate charts global coordinates aligning interpolating energy landscape continued validation and learning 5
Bridging scales: micro-macro coupling Micro-macro models arise in the study of soft matter that consists of large, massive particles (cells, vesicles, polymers) in a sea of small, light particles (water molecules, ) original system micro-macro continuum Large scale quantities are first expressed by quantities defined on small scales (micro-macro coupling), the system is then closed with only large scale quantities Simple Complex Simple Closure 6
Micro-Macro Models (for dilute polymer solutions) Micro: stochastic particle dynamics Macro: almost Navier-Stokes continuum Bead-Spring Dumbbell Kramers Chain 7
Micro-Macro Model of Complex fluid : fluid velocity, f : molecular configuration PDF Polymeric contribution to stress tensor: Ψ: spring potential ζ friction coefficient, k Boltzmann constant, λ polymer density, T temperature 8
Examples of spring potential Hookean dumbbell: Oldroyd-B fluid (effect of f can be integrated out) FENE dumbbell: => no closed-form macro constitutive equation. 9
Micro-Macro Multiscale Simulation Monte-Carlo is expensive (10 3-10 4 dumbbells at each macroscopic point). Fokker-Planck is also expensive to solve (high-dim) and the FENE force is singular. Still a computational challenge for the full resolution at microscopic level. Moment closure: derive effective macroscopic equations from the microscopic model. 10
Closure: 2-d illustration From Fokker-Planck, derive effective macroscopic equations via moments Eg: Hookean dumbbell Closed system if expressed by the same moments Oldroyd-B constitutive equation 11
Closure: 2-d illustration FENE potential: Requires higher moments for nonlinear potential Some ad hoc closures: Quadratic closure (Doi-Edward 1988): FENE-P (Bird et al 1977): 12
Systematic Closure Tradition: pick a closure relation to close system, such closure rule effectively constrains the PDF Questions: is there a PDF that satisfies the closure? how well does it capture the microscopic behavior? Why not do it the other way around? Assume the PDF f is limited to an ansatz Yu-Du-Liu, 2005 Consistency! 13
FENE-S closure (YDL SIAM MMS 2005) Ansatz: observing the equilibrium PDF Construct an ansatz containing such PDF b,β,γ :determined from second moments; used to close the moment equations. 14
A Closure System: FENE-S N-S equation with polymeric stress 15
FENE-S closure Closure based on an ansatz which approximates the equilibrium PDF => Reduction of 4 equations in 4+1 dimension to 6 equations in 2+1 dimension Energy law Theorem (YDL05, MMS): local existence of classical solution; global existence for data around equilibrium Long time stability results also studied (DLY05) 16
Numerical Validation: driving cavity FENE: Navier-Stokes + Monte Carlo FENE-S: Navier-Stokes + Closure 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Contour plots of polymeric normal and shear stresses in FENE & FENE-S Relative Error Stream function Normal Stress Shear Stress FENE 2.7% 20.2% 25.8% FENE-S 1.1% 1.5% 2.7% 17
Numerical Validation: driving cavity Comparison of stream functions in FENE & FENE-S and in Newtonian 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 FENE & FENE-S Newtonian 18
Numerical validation: Constant Shear Flow Compare stress (FENE vs FENE-S): 2.5 2 Fokker Planck Moment Closure 7 6 Fokker Planck Moment Closure 5 Shear Stress 1.5 1 Normal Stress 4 3 2 0.5 1 0 0 2 4 6 8 10 12 14 16 18 20 κ 0 0 2 4 6 8 10 12 14 16 18 20 Agreement is excellent for small shear rates, but deteriorates as shear rate k increases. κ 19
Enhancement: higher order closure A high order PDF ansatz (DLY05, MMS) Parameters {C k,l } are determined by the moments and in turn used to close the system Linear closure system I, J, K : constant matrices 20
Numerical Test: Comparison with Other Closure The proposed higher-order closure approximations outperform the FENE-P and FENE-L in these cases (moderate flow rates) Limitation: unstable for large flow rates increased computational costs with additional moments 21
Enhancement: new ansatz FENE-S captures equilibrium PDF, but for large flow rate, the PDF tends to be singular, thus FENE-S is no longer adequate New ansatz with double peaks (HDL06) scaling factor : peak variable : 22
New Closure: FENE-DS Stationary Navier-Stokes + 3 moment equations Derivation and model are similar to FENE-S FENE-DS = FENE-S 23
New Closures: FENE-D Dynamic, 3 moment equations + 2 equations for By observing the symmetry of f, the latter ones are determined via integration over a half-disk The integration may be evaluated via numerical quadrature or analytic approximation 24
New Closures: FENE-D An analytic approximation for uniform flow rate 25
New Closures: FENE-D An analytic approximation for polymeric stress 26
Numerical Test: simple extension flow FENE Via C-K FENE-D 27
Numerical Test : peak position in simple constant extension flow vs. extension rate 28
Numerical Test: Comparison with Other Closure Normal stress in simple extension flow: FENE-D provides much better accuracy for much wider range of flow rates 29
Numerical Test: Comparison with Other Closure Shear stress for general gradient: FENE-D provides much better accuracy 30
Secrete of Success: capture of PDF FENE FENE-D 31
Closure in multiscale modeling Importance of closure MD Micro-Macro Continum Simple Complex Simple Closure Coupled simulation is expensive. Leads to more efficient simulations Leads to better understanding Deriving new one is challenging, but worthwhile A good ansatz is crucial for a successful closure More validation works are underway References available: http://www.math.psu.edu/qdu 32