Trivariate Simplex Splines for Inhomogeneous Solid Modeling in Engineering Design

This paper presents a new inhomogeneous solid modeling paradigm for engineering design. The proposed paradigm can represent, model, and render multidimensional physical attributes across any volumetric objects of complicated geometry and topology. A modeled object is formulated with a trivariate simplex spline defined over a tetrahedral decomposition of its three-dimensional domain. Heterogeneous material attributes associated with solid geometry can be easily modeled and edited by manipulating the control vectors and∕or associated knots of trivariate simplex splines. We also develop a feature-sensitive fitting algorithm that can reconstruct a compact, continuous trivariate simplex spline from measured, structured, or unstructured volumetric grids of real-world inhomogeneous objects. In addition, we propose a fast direct rendering algorithm for interactive data analysis and visualization of the simplex spline volumes. Our experiments demonstrate that the proposed paradigm augments the current engineering design techniques with new and unique advantages.