On the efficient numerical simulation of kinetic theory models of complex fluids and flows Francisco (Paco) Chinesta & Amine Ammar LMSP UMR CNRS –
Download ReportTranscript On the efficient numerical simulation of kinetic theory models of complex fluids and flows Francisco (Paco) Chinesta & Amine Ammar LMSP UMR CNRS –
On the efficient numerical simulation of kinetic theory models of complex fluids and flows Francisco (Paco) Chinesta & Amine Ammar LMSP UMR CNRS – ENSAM PARIS, France [email protected] Laboratoire de Rhéologie GRENOBLE, France In collaboration with: R. Keunings Polymer solutions and melts M. Laso LCP M. Mackley & A. Ma Suspensions of CNT Molecular dynamics The Brownian dynamics different Kinetic theory: scales: q R q1 2 Fokker-Planck Eq. r1 ψ( x, y, z, t, q1,, qN ) r2 (3 1 3N ) D Deterministic, Stochastic & BCF solvers x, y , z , t rN+1 qN Constitutive Eq. General Micro-Macro approach Div X A Div v 0 pId 2N D g ( q) ( q) d q C d A D dt q q q Solving the deterministic Fokker-Planck equation New efficient solvers for: I. Reducing the simulation time of grid discretizations. II. Computing multidimensional solutions where grid methods don’t run. I. Reducing the simulation time The idea … U u( xi , t p ) p Model: PDE AU F p n U j p j 1 p j p 1 N N Model: PDE a f p + Karhunen-Loève decomposition i 1, , N , p 1, , P p 1 R AU F n n 1 n1 R P P1 n n n N 1. FENE Model 1D q 3D 1 H(q) q2 1 2 b H(q) H( q ) 1 1 q 2 b2 300.000 FEM dof ~10 dof ~10 functions (1D, 2D or 3D) 2. Non-Linear Models: Doi LCP With only 6 d.o.f. !! Larson & Ottinger (Macromolecules, 1991) (u, v, w) (0,0, x ) II. Computing multidimensional solutions q1 q2 It is time for dreaming! r1 1 (q ) A t q 4 q q r2 q (q1, q2 , , qN ) (q1 , q2 ,, q N , x, y, z, t ) rN+1 qN For N springs, the model is defined in a 3N+3+1 dimensional space !! ~ 10 approximation functions are p 1 enough a ~1010 p ~101 ~101 BUT ~10 ( x1, x2 , , x3N 3 , t ) i (t ) i ( x1, x2 , , x3N 3 ) i 1 How defining those high-dimensional functions ? Natural answer: with a nodal description 10 nodes = 10 function values 1D q1 1D q2 10 dof 10x10 dof 2D 80D r1 r2 1080 dof rN+1 1080 ~ presumed number of elementary particles in the universe !! qN >1000D No function can be defined in a such space from a computational point of view !! F.E.M. Computing multidimensional solutions The idea … n n 1 n u( x, y ) j Fj ( x )G j ( y ) j 1 Model: PDE Fn 1 ( x ) Gn 1 ( y ) n 1 FEM ( x1, GRID DOF N DIM 100010 1030 n , x10 ) j Fj1 ( x1 ) j 1 Fj10 ( x10 ) DOF N DIM 1000 10 104 1. MBS-FENE q2 q1 (q1, q2 ) 1F1 (q1 )G1(q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1 (q1 )G1 (q2 ) 2 F2 (q1 )G2 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1(q1 )G1 (q2 ) 2 F2 (q1 )G2 (q2 ) 3F3 (q1 )G3 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1(q1 )G1(q2 ) 2 F2 (q1 )G2 (q2 ) 4 F4 (q1 )G4 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1(q1 )G1(q2 ) 2 F2 (q1 )G2 (q2 ) 5F5 (q1 )G5 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1(q1 )G1(q2 ) 2 F2 (q1 )G2 (q2 ) 6 F6 (q1 )G6 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1(q1 )G1(q2 ) 2 F2 (q1 )G2 (q2 ) 7 F7 (q1 )G7 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1 (q1 )G1 (q2 ) 2 F2 (q1 )G2 (q2 ) 8F8 (q1 )G8 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1(q1 )G1(q2 ) 2 F2 (q1 )G2 (q2 ) 9 F9 (q1 )G9 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1 (q1 )G1 (q2 ) 2 F2 (q1 )G2 (q2 ) 10 F10 (q1 )G10 (q2 ) Solution EF q2 q1 G q2 F q1 (q1, q2 ) 1F1(q1 )G1 (q2 ) 2 F2 (q1 )G2 (q2 ) 11F11(q1 )G11 (q2 ) Solution EF q2 q1 G q2 F q1 1D/9D q1 q2 809 ~ 1016 FEM dof 2D/10D 1040 FEM dof q9 80x9 RM dof 100.000 RM dof 2. Complex Flows n ( x, q, t ) j Fj ( x) G j ( q) H j (t ) j 1 Example: Flow involving short fiber suspensions Kinematics: FEM-DVESS 3. Entangled polymer models based on reptation motion s u s=0 s=1 Doi-Edwards Model 2 D du Dr Dt u dt s 2 n Ottinger Model: double reptation, CCR, chain stretching, … u, s j Fj u G j ( s) j 1 Ongoing works : (I) Stochastic models can be also reduced ! y=1 1 Reduced Brownian Configurations Fields Discretization B ( I G) Ba T n 1 B F T n 4x4 1000x1000 1. Solve i=1 and computed the reduced approximation basis 2. Solve for all i>1 the reduced problem: Epoxy-A 0.025% MWNT in epoxy-A Apparent Viscosity [Pa-s] Ongoing works: (II) Suspensions of CNT: Aggregation/Orientati on model 10000 0.05% MWNT in epoxy-A 1000 0.1% MWNT in epoxy-A 0.25% MWNT in epoxy-A 0.5% MWNT in epoxy-A 100 10 1 0,1 1 10 Shear rate [s-1] Enhanced modeling: ψ( x, y, z, t, n, p) + The associated Fokker-Planck equation 100 1000 Perspectives • Enhanced kinetic model for CNT suspensions taking into account orientation and aggregation effects: FP & BD simulations. Collaboration with M. Mackley • Reduction of Stochastic, Brownian and molecular dynamics simulations. • Fast micro-macro simulations of complex flows: Lattice-Boltzmann & Reduced-FP; and many others mathematical topics (stabilization, wavelet bases, mixed formulations, enhanced particles methods, …). Collaboration with T. Phillips.