We evaluate the integral expressions for the matrices (12), (14), and (16) numerically using Gaussian quadrature [27]. We shall explain the computation of the element stiffness matrix; the computation of the mass and damping matrices follow suit. Assuming the element's parametric domain is , the expression for entry of the stiffness matrix of a D-NURBS surface element takes the integral form

where, according to (16),

Here, the are the columns of the Jacobian matrix for the D-NURBS surface element.

We apply Gaussian quadrature to compute the above integral approximately. The integral is obtained by applying Gaussian quadrature on the 1-D interval twice. Given integer and , we can find Gauss weights , and abscissas , in two directions of the parametric domain such that can be approximated by ([27])

We apply the de Boor algorithm [9] to evaluate .

Generally speaking, for integrands that are polynomial of degree
2*N*-1 or less, Gaussian quadrature evaluates the integral exactly
with *N* weights and abscissas. For D-NURBS, is not
polynomial unless the model is reduced to a B-spline. In our system,
we choose and to be integers between 4 and 7. Our
experiments reveal that matrices computed in this way lead to stable,
convergent solutions.

Demetri Terzopoulos |