Hivel2D users manual
vector from a corner node on that side to the target point. If the cross products for all
sides are greater than zero, then the node lies within that element.
Interpolation Scheme for Quadrilateral Elements
For convenience in interpolating, a local coordinate system (x,h) is used where -1
≤ ξ ≤ 1 and -1 ≤ η ≤ 1 (Figure E1) to derive the bilinear interpolation functions. First the
target point (xp,yp) is written in (x,h) coordinates.
Figure E1. Local coordinates for quadrilateral and triangular elements
This is done using Newton-Raphson iteration to solve the following equations:
4
x = ∑ φi xi
(E1)
i=1
and
4
y = ∑ φi yi
(E2)
i=1
for the local coordinates x and h where the bilinear shape functions for a quadrilateral
element are:
1
φ1 = (1 - ξ)(1 - η)
4
1
φ2 = (1 + ξ)(1 - η)
4
54
Appendix E Interpolation Program
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