Analysis of three-dimensional grids: cubes and cubic coefficients

Symbolic form of a cubic equation for the four-point rectangle

“…Prepare a routine that develops the cubic equation for the eight-point cube [3,4]. Let the operator u(P) transform its argument into P 3 .…”

“…Prepare a routine that develops the cubic equation for the eight-point cube [3,4]. Let the operator u(P) transform its argument into P 3 . Apply the routine in [3] to arguments on the bottom plane of the cube that are A=u(A), B=u(B), C=u(C), and D=u(D).…”

“…Let the operator u(P) transform its argument into P 3 . Apply the routine in [3] to arguments on the bottom plane of the cube that are A=u(A), B=u(B), C=u(C), and D=u(D). The arguments on the top plane of the cube are F=u(A+t), G=u(B+t), H=u(C+t), I=u(D+t).…”

“…It was first derived by operational methods [2]. That method can be used to derive a cubic equation for the eight-point cube [3,4]. The eight-point equation can be used to develop a cubic equation for the four-point rectangle [3][4][5].…”

“…That method can be used to derive a cubic equation for the eight-point cube [3,4]. The eight-point equation can be used to develop a cubic equation for the four-point rectangle [3][4][5]. This approach is used to recover the analytical forms of the ten coefficients in a new cubic interpolating equation for the four-point rectangle.…”