Citation Context

...generally the IB method is only first-order accurate for nonsmooth but continuous quantities. Different variations of IB method have been proposed to improve the accuracy. For example, Lai and Peskin =-=[18]-=- proposed a new formally secondorder IB method with reduced numerical viscosity and applied the new method to simulate FIG. 1. A diagram of the geometry for the interface problems discussed in this pa...

Citation Context

...e [23], cell and tissue deformation under shear flow [24–26]. At present there exist several versions of the immersed boundary method. The version we use here is different from most existing versions =-=[5, 7, 9, 11, 27, 28]-=- in two aspects. (i) The discretization of the Navier–Stokes equations is different; the fractionalstep-projection scheme is applied and the skew-symmetric scheme is used for the nonlinear term instea...

Citation Context

...body geometry as the no-slip boundary condition is enforced at the Lagrangian points by adding appropriate boundary forces. The boundary forces that exist as singular functions along the surface in the continuous equations are described by discrete delta functions that smear (regularize) the forcing effect over the neighboring Eulerian cells. Peskin originally used the IBM to simulate blood flow inside a heart with flexible valves, where the forcing function was computed by Hooke’s law [31,32]. This technique was later extended to rigid bodies by taking the spring constant to be a large value [3,22]. Goldstein et al. [17] applied the concept of feedback control to compute the force on the rigid immersed surface. The difference between the velocity solution and the boundary velocity is used in a proportional-integral controller. For the aforementioned techniques to model flow over rigid bodies, the choice of gain (stiffness) remains ad hoc and large gain results in stiff equations. Our intention is to remove all tuning parameters and formulate the IBM in a general framework for rigid bodies (as well as bodies with prescribed surface motion). In our formulation, we treat the boundary force...

Citation Context

...12, 13, 36], the deformation of red blood cell in a shear flow [10], the swimming of bacterial organisms and others [9, 11]. This method has also been applied to handle problems with rigid boundaries =-=[17, 34]-=-. In order to deal with rigid boundaries, Lai and Peskin [17] proposed to evaluate the force density using an expression of the form, f(s, t) = κ(X e (s) − X(s, t)), (5) where κ is a constant, κ ≫ 1, ...

Citation Context

...to the method described in that paper. The operator u⋅� in the nonlinear term of the Navier–Stokes equations was discretized as a skew symmetric operator to remove the effects of numerical viscosity (=-=Lai and Peskin, 2000-=-). Essentially, the fluid equations are discretized on a regular rectangular grid in the physical space of the position variable x, and the boundary equations are discretized in a one-dimensional spac...

Citation Context

... (16) because there it only amounts to a redefinition of the pressure.) 3 Numerical Implementation of the pIB method We now describe a formally second-order IB method to solve the equations of motion =-=[15,20]-=-. The word ‘formally’ is used as a reminder that this scheme is only second-order accurate for problems with smooth solutions. Even though our solutions are not smooth (the velocity has jumps in deriv...