## Locating elements

The most time-consuming part of the flow line construction is in locating the elements that contain points on the flow line, since there may be several such searches required for each flow line. These searches are required to establish the starting conditions.

At each layer, Gu et al  isolate the intervals in the direction for that layer that might contain the query point. For example, in the first (x) layer, with the query point q = (qx, qv, qz) and with

vertex Vi, let xmm. be the minimum x-coordinate of an element stored

in the vertex Vj and let xlenRthj be the largest length in the x-direction

of all elements in the branches at Vl. Then, in searching for the element containing q, we search only those vertices and branches with *mi„, ^ 4X and xmin. +xlemth, > qx. This process is then repeated

for the y - and z-coordinates in the lower layers of the structure.

Finally, the possible elements containing a point (X0,Y0,Z0) will be reduced to a small number for which precise inclusion tests must be done. This is done by solving the nonlinear equations for r, s, and t.

Sx =X0-Nj(r, s,t)Xi =0

8у = Yo ~ Ni (r> s, t)Yi=0 (4-23)

gz=ZQ-Ni(r, s,t)Zj=<d Using Newton's method :

Jk(£k*-4k) = - gk (4-24)

where J is the Jacobian matrix, q is the vector (r, s, t) in the curvilinear coordinate system, and g is functionsg = (gx, gy, gz) .If

we write dr = rk+l —rk, dX = gx and so on for s, t, у and z, Newton's method can be written in the form convenient in FEM:

 dx dr dy = Jk ds dz dt

To ensure the convergence and improve the solution speed, we project the point first onto the coordinate planes and check whether it is inside the projections of the element bounding box.

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