A multibody dynamics model of bacterial

Transcription

1 A multibody dynamics model of bacterial Martin Servin1 1 UMT Research Lab - Department of Physics Umea University August 26, 2015 (1 : 17)

2 Nonsmooth multidomain dynamics What has this to do with bacterias? (2 : 17)

3 Bacterial aggregate of microorganisms attachment, growth and dispersion microbial infections, dental plaque, organic waste in pipes (3 : 17)

4 Bacterial adhesion - E. coli pili = surface organelles, 1µm & 1k subunit macromolecules in helix structure strong and weak bonds (4 : 17)

5 Pili force Laser tweezer measurement [7] Multibody simulation [2] (5 : 17)

6 Biofilm model rigid bacterias contact constraints pili constraints hydrodynamics (6 : 17)

7 Nonsmooth multidomain dynamics Heterogeneous multibody dynamics Rigid, flexible, fluid, hydraulics, electronics,... Nonsmooth contact dynamics - large fixed time-step Variational stepper with constraint regularization Numerical solvers for heterogeneous subsystems (7 : 17)

8 Smooth versus nonsmooth dynamics Smooth dynamics smooth trajectories smooth forces & constraints small time-step linear or nonlinear solver Nonsmooth dynamics velocity discontinuities + impulses & inequalities large time-step QP or MLCP solver Nonsmooth contact dynamics (Moreau [6], Jean [3], Acary [1], Servin [8]) (8 : 17)

9 Multibody dynamics Multibody system (q, q ) on descriptor form (q, q, λ, λ ) Mq + M q G(q)T λ G (q)t λ = fe (1) ελ + g(q) = 0 (2) γλ + G (q)q = w(t) (3) constrained by g(q) = 0 and G (q)q = w(t) with compliance ε and damping γ. (9 : 17)

10 Multibody dynamics - numerical solver Linearized varational time stepper SPOOK (Lacoursie re [4, 5]) qn+1 = qn + hq n+1 M GT G Σ G 0 {z (4) Mq n + hfn G T q n+1 0 λ = h4 Υg + ΥGq n (5) Σ λ ωn } {z } {z } z H r Diagonal regularization and stabilization matrices Σ= 4 diag h2 εi τhi Constraint potential and dissipation 1 diag (γi ) h 1 Υ = diag τi 1+4 h U = 12 gt ε 1 g Σ = R = 21 (G v)t γ 1 G v (10 : 17)

11 Nonsmooth MBD - numerical solver ncluding frictional contacts, impacts, joint ant motor limits lead to limits and complementarity conditions on the solution variables Hz + r = w+ w (6) 0 6 w+ z l > w u z > 0 The problem transforms from linear system to a mixed linear complementarity condition (MLCP) (11 : 17)

12 Nonsmooth MBD - numerical solver Direct MLCP solver block pivoting method block sparse LDLT factorization fill-reducing reordering BLAS3 optimized terative MLCP solver block sparse projected Gauss-Seidel (PGS) Hybrid direct-iterative split solver vehicle vs granular normals vs tangents Research prototype code and AgX Dynamics [4] (12 : 17)

13 Biofilm model rigid bacterias contact constraints pili constraints hydrodynamics (13 : 17)

14 Piecewise linear constraint - extension δ - Jacobian G = g δ g = δ k2 max(0, δ δ2 ) κθ(δ δ2 ) (7) G = 1 k2 Θ(δ δ2 ) (8) Constraint regularization and multiplier limit λ = k1 g, δ < δ 1, δ > δ2 λ = λ1, δ1 6 δ 6 δ2 (14 : 17) (9) (10)

15 Biofilm simulation A first mockup model - with cartoon physics parameters (15 : 17)

16 and conclusions Bacterial biofilm as a multibody system A piecewise linear pili constraint was introduced Future work Nonsmooth dynamics at discontinuity Dynamic generation of pili constraints Simple hydrodynamics Numerical experiments and validation (16 : 17)

17 References [1] V. Acary and B. Brogliato. Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Springer Verlag, [2] J. Zakrisson et al, Rigid multibody simulation of a helix-like structure: the dynamics of bacterial adhesion pili, European Biophysics Journal, (2015). [3] M. Jean, The non-smooth contact dynamics method, Comput. Methods Appl. Mech. Eng., 177, (1999) [4] C. Lacoursière, M. Linde, SPOOK: a variational time-stepping scheme for rigid multibody systems subject to dry frictional contacts, submitted (2013). [5] C. Lacoursière, Ghosts and Machines: Regularized Variational Methods for nteractive Simulations of Multibodies with Dry Frictional Contacts, PhD thesis, UmeåUniversity, Sweden, (2007) [6] Moreau J.J. Unilateral Contact and Dry Friction in Finite Freedom Dynamics, volume 302 of Non-smooth Mechanics and Applications, CSM Courses and Lectures. Springer, Wien, 1988 [7] Andersson M, et al, Dynamic force spectroscopy of E. coli P pili. Biophys J 91: (2006). [8] M. Servin, K. Bodin, C. Lacoursière, D. Wang, Examining the smooth and nonsmooth discrete element approach to granular matter, nt. J. Numer. Meth. Engng (2014). Martin Servin 1, August 26, 2015 (17 : 17)