Efficient energy evaluations for active B-Spline/NURBS surfaces

• Efficient Active B-Spline/NURBS energy evaluations based on an Analytic Solution.
• Efficient algorithm for computing the stiffness of an Active B-Spline/NURBS Surface.
• Accurate and stable algorithms for computing integrals of B-Spline Basis Functions.
• Detailed analysis of computational efficiency, accuracy and stability.
• Up to four times faster than standard Gaussian Quadrature for practical cases.

The construction of the stiffness matrix associated with an Active B-Spline/NURBS surface is one of the most important but time consuming operations performed in CAD/CAM/CAE. This paper aims to address this problem and presents a novel, computationally efficient, generalised mathematical framework and accompanying algorithms based on an analytic solution to the problem. The approach is shown to extend seamlessly to the problem of computing mass, damping and forcing matrices, and importantly, can handle variable mass, damping, and stiffness coefficients. The capabilities of the algorithms are illustrated and their respective performances verified through detailed analysis of the computational efficiency, accuracy and stability in several practical case studies. The main benefit of the proposed approach is a reduction in computation times required for the evaluation of the stiffness matrix by up to a factor of 4, over the standard Gaussian Quadrature approach, for the practical cases considered, while preserving a high degree of accuracy and stability. Additionally, no assumptions regarding the problem complexity, degree, or regularity of the knot vector are imposed upon the solution in order to achieve the computational saving.