Bilinear interpolation: Difference between revisions

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@@ Line 105: / Line 105: @@
 <lang J>
 Note 'FEA'
+   Here we develop a general method to generate isoparametric interpolants.
-   The interpolant is the dot product of the four function values with the values at the nodes.
-   Sum of four linear functions of two variables (xi, eta) is one at each of 4 nodes.
+   The interpolant is the dot product of the four shape function values evaluated
-   Let the base element have nodal coordinates (corners) (x,y) with at +/-1.
+   at the coordinates within the element with the known values at the nodes.
+   The sum of four shape functions of two variables (xi, eta) is 1 at each of four nodes.
+   Let the base element have nodal coordinates (xi, eta) of +/-1.
@@ Line 115: / Line 118: @@
    |               |
    |        (0,0)  |
-   |       *       |   (The usual finite element node order
+   |       *       |
-   |               |   is counter clockwise.  My order to
+   |               |
-   |               |   me is more reasonable.)
+   |               |
    |               |
    +---------------+
@@ Line 126: / Line 129: @@
    f1( 1,-1) = 1, f1(all other corners) is 0.
    ...
-   The coordinates (xi,eta) are a vector length 2 as argument y
-   Supply to interpolate the function values at nodes 0 1 2 and 3 as the x argument.
-)
+   Choose a shape function.
-f0 =:  1r4 * [: */  1  1&-  NB. f0(xi,eta) = (1-xi)(1-eta)/4
-f1 =: _1r4 * [: */ _1  1&-  NB. f1(xi,eta) = (1+xi)(1-eta)/4
+   Use shape functions C0 + C1*xi + C2*eta + C3*xi*eta .
+   Given (xi,eta) as the vector y form a vector of the
-f2 =: _1r4 * [: */  1 _1&-  NB. ...
+   coefficients of the constants (1, xi, eta, and their product)
-f3 =:  1r4 * [: */ _1 _1&-
+      shape_function =: 1 , {. , {: , */
-functions =: f0`f1`f2`f3`:0
+      CORNERS NB. are the ordered coordinates of the corners
-CORNERS =: 21 A. -.+:#:i.4
+   _1 _1
-NB. prove the function values at the corners are correct.
+_1
-assert (=i.4) -: functions"1 CORNERS    NB. 4 by 4 identity matrix
+   _1  1
+1
+      (=i.4)  NB. rows of the identity matrix are the values of each shape functions at each corner
+0 0 0
+1 0 0
+0 1 0
+0 0 1
+      (=i.4x) %. shape_function"1 x: CORNERS  NB. Compute the values of the constants as rational numbers.
+r4  1r4  1r4 1r4
+   _1r4  1r4 _1r4 1r4
+   _1r4 _1r4  1r4 1r4
+r4 _1r4 _1r4 1r4
+   This method extends to higher order interpolants having more nodes or to other dimensions.
-interpolate =: (+/ .*) functions
+)
+mp =: +/ .*  NB. matrix product
+CORNERS =: 21 A.-.+:#:i.4
+shape_function =: 1 , {. , {: , */
+COEFFICIENTS =: (=i.4) %. shape_function"1 CORNERS
+shape_functions =: COEFFICIENTS mp shape_function
+interpolate =: mp shape_functions
 </lang>
 <pre>
-Note. 'demonstrate the interpolant with a saddle'
+Note 'demonstrate the interpolant with a saddle'
    lower left has value 1,
    lower right: 2
    upper left: 2.2
    upper right: 0.7
-   GRID is an array of coordinate pairs (hence rank 3)
-   ordered so that _1 _1 is in the lower left of the picture,
-   directed with 1 _1 to lower right.
 )
@@ Line 158: / Line 178: @@
 viewmat SADDLE
 </pre>
-[[Image:J_bilinear_interpolant.jpg|right]]
+[[Image:J_bilinear_interpolant.jpg]]
 =={{header|Racket}}==