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 featuresensitive 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.fDOI: 10.1115/1.1881352g 1 Introduction and Motivation Real-world objects are active, responsive, and anisotropic. Oftentimes, they are of arbitrary topology and complex geometry. The fundamental objectives in engineering design are to unambiguously model physical volumetric objects, interactively visualize their geometric and physical attribute properties, and rigorously analyze their kinematic and dynamic natures. With the advent of ever-increasing computing power and more advanced data acquisition technologies, solid geometry has quickly gained popularity as an intuitive and natural paradigm for these purposes. New and powerful solid representations underpin the success of solid modeling and relevant applications. To date, the vast majority of popular solid modeling approaches, as well as commonly used solid modeling systems, are built upon the following geometric foundations: constructive solid geometry sCSGd, boundary representation sB-repsd, and cell decomposition. When our goals are to reconstruct heterogeneous models of physical objects with continuous properties and further model and visualize the objects, prior representations and the current state of the art in engineering design fall short in offering designers an integrated paradigm to represent complex solid geometry, arbitrary topology, and continuously varying material attributes in a single framework simultaneously. To model heterogeneous volumetric objects with high-order continuity, techniques based on splines such as B-splines or NonUniform Rational B-Splines sNURBS df 1‐4g are frequently used. Nonetheless, modeling with B-splines or NURBS has severe shortcomings. Its modeling scope is extremely constrained in terms of geometric, topological, and attribute aspects. First, B-spline and NURBS are defined over a regular, tensor-product domain. A single B-spline or NURBS cannot represent volumes of complex topology without patching or trimming operations. Furthermore, patching multiple B-splines or NURBS to form complex topology is not easy to control. Second, tensor-product splines are essentially smooth everywhere. It is difficult to model high-frequency features. Third, when refining a region of interest in a tensor-product spline patch, it will introduce too many extra degrees of freedom in other less interesting regions nearby in order to retain its regular structure. Attractive properties such as local adaptivity and multiresolution are rather difficult to achieve. To overcome the above difficulties, we propose an integrated approach for representing, modeling, and rendering of multidimensional physical attributes across any volumetric object. Our goal is to demonstrate that the trivariate simplex spline is a promising primitive for both visualization and modeling in engineering design tasks, especially for representing and visualizing heterogeneous models of physical objects and their material properties. Our model makes use of a more general and flexible tetrahedral domain and offers a compact, continuous representation because it is a piecewise polynomial of the lowest possible degree and the highest possible continuity everywhere across the entire tetrahedral domain. It unifies geometry and attribute properties over domains of complex topology. It is possible to represent a complicated heterogeneous object with a single trivariate simplex spline without any additional operations of trimming or patching, while the geometry of the object is explicitly represented by the spline. For example, given a degree n for the trivarite DMS-spline f5g, which uses simplex spline as basis functions, the representation