Construct a laminar coordinate system at the blue point. (Use equations 9.3.16 and 9.3.17 from the book chapter). Assume that the blue point is at the center of element 5.
The and coordinates of the master and slave nodes are
Since there are two master nodes in an element, the parent element is a four-noded quad with shape functions
The isoparametric map is
Plugging in the numbers, we get
Therefore, the derivatives with respect to are
Since the blue point is at the center of the element, the values of and at that point are zero. Therefore, the values of the derivatives at that point are
Therefore,
The local laminar basis vector is given by
The laminar basis vector is given by
A plot of the vectors is shown in Figure 13.
The Maple code for this calculation is shown below.
> #
> # Shape functions
> #
> N1m := 1/4*(1-xi)*(1-eta):
> N2m := 1/4*(1+xi)*(1-eta):
> N1p := 1/4*(1-xi)*(1+eta):
> N2p := 1/4*(1+xi)*(1+eta):
> N := linalg[matrix](1,4,[N1m,N2m,N1p,N2p]);
> #
> # Compute local laminar basis vectors at the blue point
> #
> # Isoparametric map
> #
> x := x1m*N1m + x2m*N2m + x1p*N1p + x2p*N2p;
> y := y1m*N1m + y2m*N2m + y1p*N1p + y2p*N2p;
> #
> # Derivative of Isoparametric map
> #
> dx_dxi := diff(x, xi);
> dy_dxi := diff(y, xi);
> #
> # Jacobian evluated at blue point
> #
> dx_dxi_cen := subs(eta=0, dx_dxi);
> dy_dxi_cen := subs(eta=0, dy_dxi);
> #
> # Local laminar basis vectors at the blue point
> #
> norm_dx_dxi_cen := simplify(sqrt(dx_dxi_cen^2 + dy_dxi_cen^2));
> ehat_x := vector([simplify(dx_dxi_cen/norm_dx_dxi_cen),
> simplify(dy_dxi_cen/norm_dx_dxi_cen), 0]);
> ehat_z := vector([0, 0, 1]);
> ehat_y := crossprod(ehat_z, ehat_x);