Transcript ppt
The weak immersed boundary method for large-scale fluid-structure simulation E. Givelberg Department of Physics and Astronomy IDIES, JHU The immersed boundary method A general method for simulating systems containing elastic and possibly active tissue immersed in a viscous, incompressible fluid Peskin AND McQueen – blood flow in the heart – – – – – – – swimming organisms (L. Fauci) insect flight (J. Wang) platelet aggregation during blood clotting (A. Fogelson) valveless pumping (E. Jung) whirling instability of an elastic filament (S. Lim) flow in collapsible tubes (M. Rozar) flapping of a flexible filament in a flowing soap film (L. Zhu) C. S. Peskin, Acta Numerica, 11:479-517,2002. Immersed Boundary Method: Lagrangian-Eulerian Formulation Navier-Stokes equations discretized on a periodic rectangular 3-d grid Hooke’s spring law Hooke’s spring law discretize d on a 1-d grid viscous incompressible fluid Fourth order PDE discretized on a 2-d grid Immersed Boundary Equations Navier-Stokes equations viscous, incompressible fluid No-slip boundary condition Newton’s law using Dirac’s delta function An additional equation specifying F(q, t) is needed to complete the system. Immersed Boundary Equations • Fluid: – Periodic boundary conditions • To enable the use of FFT solvers • Immersed Boundary: – Fibers z q1 – Elastic plate q2 Discretization of the Equations (in 2d) • Fluid: – rectangular lattice with mesh width h – Eulerian quantities: indexed by k = (k1, k2) • For example: – Volume element: • Immersed material: – Lagrangean grid with mesh width comparable to h – Lagrangean quantities: indexed by q • The index set of q can be of dimension = 0, 1 or 2 – Volume element: • Time: Fluid-structure interaction using the discrete Dirac delta function Immersed Boundary Method: Time Step n + 1 Compute the immersed material force Fn. Spread the force to the fluid grid: Solve the Navier-Stokes equations: Move the immersed material: New spatial discretization: the weak immersed boundary method A Wavelet Basis for Fluid Velocity and Pressure Fluid Velocity Basis Function Weak solutions of the immersed boundary equations • We are looking for weak solutions of the immersed boundary equations of the form • The velocity of the immersed boundary is naturally determined by the no-slip boundary condition: • The fluid force is determined from the immersed boundary force by “spreading” Mass Matrix, Gradient, Laplacian The inner product of basis functions depends only on the difference We have Similarly, define Weak solutions of the discretized Navier-Stokes Equations Time discretized Navier-Stokes equations: Here and The weak form is: Solution of the equations using the Discrete Fourier Transform The DFT of a sequence aj is: Because the discretized Navier-Stokes equations are in convolution form, we obtain: The solution is: The Gibbs Phenomenon One possible natural representation of the immersed boundary force is where Unfortunately, this f(x, t) does not belong to the span of the velocity basis functions When the immersed boundary force is discontinuous or singular, projection onto leads to Gibbs-like phenomena. Force Spreading (smoothing) While the immersed boundary velocity is naturally determined by the fluid velocity field, the force F(q, t) of the immersed boundary does not have an obvious natural representation on the fluid domain. In the weak immersed boundary method this force is represented by: where i.e. Weak Immersed Boundary Method: Time Step n + 1 Compute the immersed material force Fn. Spread the force to the fluid grid: Solve the weak Navier-Stokes equations using FFT to obtain Compute immersed material velocity: Move the immersed material: Volume conservation test Fluid: [0, 1] x [0, 1] cm2 Immersed boundary: initially, a stretched ellipse 200 Lagrangian grid points K = 1000 Time step: QuickTime™ and a decompressor are needed to see this picture. Volume Conservation Weak I.B. Peskin-Printz Immersed Boundary method Volume Conservation Peskin-Printz The weak IB method: Conclusion • Improves volume conservation. • Is a new framework that unifies several basic methods: – – – – Fast Fourier transform based methods Wavelets Finite Elements Smoothed Particle Hydrodynamics • Provides a new framework for theoretical analysis and leads to new numerical methods Thank you!