Bi-linear interpolation

Depending on the type of elements used appropriate interpolation functions are used to obtain the elemental discretizations of the unknown variables. In the present derivation a mixed formulation consisting of nine-node bi-quadratic shape functions for velocity and the corresponding bi-linear interpolation for the pressure is adopted. To approximate stres.ses a 3 x 3 subdivision of the velocity-pressure element is considered and within these sub-elements the stresses are interpolated using bi-linear shape functions. This arrangement is shown in Edgure 3.1. 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

The marker points move at the loeal flow veloeity interpolated from the Eulerian grid. The interpolation seheme is important sinee interpolation error may result in a violation of mass eonservation. Ideally, the interpolation scheme must satisfy the mass eonservation at the diserete level in the same way as done in the flow solver [19]. However, the Peskin distribution function or simple bi-linear interpolation is usually used in the fronttracking method [3]. Note that none of these interpolation sehemes satisfies the mass conservation at diserete level. This is in fact an important factor for the change in drop volume espeeially for long simulations. 

In the bi-hnear interpolation, a simplistic rationale is followed, i.e., for calculating the pixel value of a particular position X,Y), four adjacent pixel values are used [21]. The closer the pixel is to the position (X,Y), the more influence (weight) it will carry. The method is not merely a falhng function of distance from the pixel. Rather it considers a weighted approach based on its spatial locations in a two-dimensional space [47]. The derivation of bi-linear interpolation weights can be expressed as follows [48, 49] ... 

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