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6.2 Calculation of Element Matrices


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 tex2html_wrap_inline2199 , the expression for entry tex2html_wrap_inline2201 of the stiffness matrix of a D-NURBS surface element takes the integral form


where, according to (16),


Here, the tex2html_wrap_inline2025 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 tex2html_wrap_inline2205 and tex2html_wrap_inline2207 , we can find Gauss weights tex2html_wrap_inline2209 , tex2html_wrap_inline2211 and abscissas tex2html_wrap_inline2213 , tex2html_wrap_inline2215 in two directions of the parametric domain such that tex2html_wrap_inline2201 can be approximated by ([27])


We apply the de Boor algorithm [9] to evaluate tex2html_wrap_inline2219 .gif

Generally speaking, for integrands that are polynomial of degree 2N-1 or less, Gaussian quadrature evaluates the integral exactly with N weights and abscissas. For D-NURBS, tex2html_wrap_inline2239 is not polynomial unless the model is reduced to a B-spline. In our system, we choose tex2html_wrap_inline2205 and tex2html_wrap_inline2207 to be integers between 4 and 7. Our experiments reveal that matrices computed in this way lead to stable, convergent solutions.

Demetri Terzopoulos | Source Reference