Bounce-back scheme

As with other numerical approaches, appropriate boundary conditions (BCs) are necessary for meaningful simulations. Since the principal variables in LBM are the density distributions LBM BCs are implemented through specifying the unknown / entering the simulation domain across boundaries, instead of the macroscopic fluid properties such as velocity and pressure. This feature poses both conveniences and difficulties. For example, to model a no-slip boundary over a solid surface, one can simply reverse the particle directions toward the boundary back to their original locations, the so-called bounce-back scheme. Periodic boundaries are even easier to implement all particles which leave the domain across a periodic boundary will reenter... 

Yin X, Zhang J (2012) An improved bounce-back scheme for complex boundary conditions in lattice Boltzmann method. J Comput Phys 231 4295-4303... 

The halfway bounce-back scheme with interpolation is used to deal with the curved surfaces [22]. This boundary treatment has a superlinear accuracy when the wall surface varies between two adjacent nodes and approximately second-order accuracy if the wall surface is in the middle. This method is easy to implement for complicated boundary conditions without special considerations for the comers. 

Ziegler [121] showed that if the rigid boundary was located midway between the nearest lattice sites, the bounce-back scheme would produce second-order accuracy. Therefore the physical boundary is assumed to lie midway between the closest lattice points in the flows and the closest boundary point (i.e., a point that lies inside the solid surface). 

In the frozen-layer scheme, a continuum limit is applied to the group of frozen-layer particles to analytically obtain the effective form of the dissipative and random forces of interactions between the wall particles and the DPD fluid particle. An explicit rule is employed to maintain the impermeability of solid walls. Three possible rules for achieving this are illustrated in Fig. 3. For free-slip, specular reflection instead of bounce-back reflection for particles that cross the free-slip boundaries is employed. 

The density fluctuations arising near the boundary from the implementation of the conservative force at the waU is a cause of concern when implementing the no-sUp boundary condition. To address this, a no-sUp scheme based on the equivalent force between the waU and fluid particles may he introduced. In this method, layers of frozen particles are chosen as in the frozen-layer scheme, but a key difference is that the coefficient for the conservative force is adjusted in a manner so as to incorporate wall and fluid particle interactions. This leads to a correct no-slip implementation without fluctuations in density. Another proposal to reduce density fluctuations near the walls represents the boundary with two layers of frozen particles. The particles that penetrate the boundary are made to bounce back into the computational domain. No-sUp at the wall is produced by using a twin image of the system being simulated. The second image has the same configurations and dynamics as the first, but operates with a different... 

More sophisticated boundary conditions have been developed using finite-volume methods [149,150] and interpolation [151-153]. A simple, physically motivated interpolation scheme has been proposed [151,154], which both improves the accuracy of the bounce-back rule and is unconditionally stable for all boundary positions the scheme has both linear and quadratic versions. A more general framework for this class of interpolation schemes has been extensively analyzed in a comprehensive and seminal paper [113] the Multi-Reflection Rule proposed in [113] is the most... 

For SFM, maintaining a constant separation between the tip and the sample means that the deflection of the cantilever must be measured accurately. The first SFM used an STM tip to tunnel to the back of the cantilever to measure its vertical deflection. However, this technique was sensitive to contaminants on the cantilever." Optical methods proved more reliable. The most common method for monitoring the defection is with an optical-lever or beam-bounce detection system. In this scheme, light from a laser diode is reflected from the back of the cantilever into a position-sensitive photodiode. A given cantilever deflection will then correspond to a specific position of the laser beam on the position-sensitive photodiode. Because the position-sensitive photodiode is very sensitive (about 0.1 A), the vertical resolution of SFM is sub-A. 

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