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By Suri J., Farag A. (eds.)

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Once F Γ is computed, it must be extended off the interface to the rest of the domain so that we can use (26). This is accomplished through a velocity extension, discussed later in this section. Since Φ is only defined on the inside of the biofilm, then ∂Φ/∂n must be computed using a one-sided finite-difference approximation at the interface. Suppose x ∈ Γ is where ∂Φ/∂n is to be computed; then it is approximated by Φ(x) − Φ(x − ∇φ) −Φ(x − ∇φ) ∂Φ ≈ = , ∂n ∇φ ∇φ (28) where = ∆x2 + ∆y 2 is a small constant, and we are using the boundary condition on the velocity potential, Φ(x) = 0.

The algorithm terminates when all points have migrated into set A. See (7) for an illustration of sets A, T , and D. The implementation of the fast marching method uses upwind finite differences, where the direction of upwind differences is taken toward smaller values of φ. For example, suppose points (xi−1 , yj ), (xi , yj+1 ) are in set A, then φi−1,j , φi,j+1 are already determined, and we wish to compute a tentative value for φi,j . Equation (32) is discretized using one-sided differences to obtain (D−x φi,j )2 + (D+y φi,j )2 = 1.

It too is also error free. Furthermore, it is also very easy to extend this algorithm to higher dimensions as well as to anisotropic data. Unlike the simple algorithm that does not employ a list, this algorithm is very efficient when the size of the list is small. We subsequently refer to this algorithm as SimpleList. Pseudo code for this algorithm follows. =0) { //only consider non border elements //at this stage, we have a point that is not an element of //the border. iterate over all border elements in the list.

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