Hodographs and normals of rational curves and surfaces
Computer Aided Geometric Design
The NURBS book
Geometric Tools for Computer Graphics
Geometric Tools for Computer Graphics
Isogeometric analysis and shape optimization via boundary integral
Computer-Aided Design
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The progress of nanotechnology has made it possible to make miniature electromechanical devices of sub-micrometer scale. This means that we will be in need of design packages that can model the physical properties of objects and their interactions involved down in nanometer scale. Toward this goal, our aim in this paper is to develop a computing procedure for determining molecular interaction forces, i.e. van der Waals force, between objects of arbitrary geometry. Currently there are two types of approaches for calculating van der Waals force. The first type is analytical where analytical force equations are derived for interactions between simple geometries such as spheres and half-spaces. The second type is numerical where volume integrals or surface integrals are conducted over discretized object domains where the object boundaries are approximated by simple mesh geometries. This paper presents a numerical approach that uses non-uniform rational B-spline (NURBS) based surface integrals. The integrals are done on the parametric domains of the NURBS surfaces and Gaussian quadrature points lie exactly on the object surfaces. Salient features of this approach include: 1) Orders of magnitude in accuracy improvement is achieved over other numerical approaches; The fundamental reason for such accuracy improvement is that molecular interaction force is very sensitive to surface geometry since it falls off at the rate of inverse power of 6 ~ 7. Any geometric approximation in object discretization would lead to significant bias in the calculation result. 2) Molecular interactions between arbitrary-shaped objects can be represented and evaluated since the NURBS model can represent exactly common analytical geometries such as spheres in nano-particles and cylinders in nano-rods, and complex geometries such as corrugated sample surfaces. We demonstrate its general shape applicability by calculating van der Waals force between complex geometries such as micro-gears. Further, we give error bounds for NURBS based numerical simulation and develop an adaptive subdivision scheme to improve both calculation accuracy and efficiency.