Numerical Integration - Numerical Methods in Engineering with MATLAB - page 246

Solution of Part (1) Let the triangle beformedbycollapsing corners 3 and 4ofa
quadri lat eral. T he corner coordinates of thisquadrilateral are x

2] T and

= [ − 1 , − 1 , 2 ,

, − √ 3 [ √ 3

y 0] T . To determine the minimum requiredintegration order foran

exact result, we must examine f [ x (

= , 0 ,

ξ,η ) , y ( ξ,η )] | J ( ξ,η ) | , the integrand in Eq. (6.45).

Since y ) iscubic in x , the integrand is apolynomialof

degree 5in x . Therefore, third-orderintegrationwill suffice. The command used for
the computations issimilar to the one in Example 6.15:

| J ( ξ,η ) | is biquadratic, and f ( x ,

_ >>I=gaussQuad2(@fex6 16,[-1;-1;2;2],...

[sqrt(3);-sqrt(3);0;0],3)

I=
-1.5588

The function that returns z = f ( x , y ) is

_ functionz=fex6 16(x,y)

% Function used in Example 6.16
z = (xˆ2 + yˆ2)/2 - (xˆ3 - 3*x*yˆ2)/6 - 2/3;

Solution of Part (2) The following command executes quadratureover the triangular
element:

>>I=triangleQuad(@fex6

_

16,[-1; -1; 2],[sqrt(3);-sqrt(3); 0])

I=
-1.5588

Since the integrand is a cubic, this result is also exact.
Note thatonly four function evaluations were requiredwhenusing the tri-
angle formulas. In contrast, the function had to beevaluatedat nine points in
Part(1).

EXAMPLE 6.17
The

, , ,

corner coordinates of a triangle are(0 0), (16 10) and (12 20).Compute

A x 2

y 2 dx dy over thistriangle. −