Drawing a quadrangular surface

Question

Is there a way to draw a quadrangular surface in Mathematica so that it shows the bilinear term (not two triangles)? The quadrangular polygon is given by the expression

$$\phi = \sum_i^4 N_i U_i$$

where the $N_i$ are the Lagrangian shape functions (from finite element analysis), and the $U_i$ are the degrees of freedom. The shape functions are given by

N1[xi_, eta_] := (1 - eta) (1 - xi)/4;
N2[xi_, eta_] := (1 + eta) (1 - xi)/4;
N3[xi_, eta_] := (1 + eta) (1 + xi)/4;
N4[xi_, eta_] := (1 - eta) (1 + xi)/4;

So there is a bilinear term xi*eta that gives the curvature to the element.

Answer 1

A polygon is a section of a plane, so it has no curvature, and using Polygon in Graphics3D won't work. You need a ParametricPlot3D.

N1[xi_, eta_] := (1 - eta) (1 - xi)/4;
N2[xi_, eta_] := (1 + eta) (1 - xi)/4;
N3[xi_, eta_] := (1 + eta) (1 + xi)/4;
N4[xi_, eta_] := (1 - eta) (1 + xi)/4;
shapeFuncs = {N1[xi, eta], N2[xi, eta], N3[xi, eta], N4[xi, eta]};
U = RandomReal[{0, 1}, 4];
phi[xi_, eta_] = shapeFuncs.U;
ParametricPlot3D[{{xi, eta, phi[xi, eta]}}, {xi, -1, 1}, {eta, -1, 1}]