Ray-tracing polymorphic multidomain spectralhp elements for isosurface rendering

Manuscript received 30 Nov. 2004; revised 8 June 2005; accepted 20 June 2005; published online 9 Nov. 2005. For information on obtaining reprints of this article, please send e-mail to: tvcg@, and reference IEEECS Log Number TVCG-0150-1104.
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methods based on the a priori knowledge that the data was produced by a high-order finite element simulation (notable exceptions are Coppola et al. [6] with respect to particle tracking, Wiley et al. [7], [8], [9] with respect to ray-casting of curved-quadratic elements, and Brasher and Haimes [10] with respect to cut-planes). Current visualization methods are typically based on lower-order representations of the data (lower in the sense of what is customarily used in high-order numerical simulations)—for instance, using linear interpolation for determining the values of data in areas not at grid locations. Accurate low-order approximations of high-order data require far more degrees of freedom than the original data set and may introduce data artifacts due to mismatches in the approximation properties of the low-order representation and the original high-order discretization.
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 12, NO. 1, JANUARY/FEBRUARY 2006
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Ray-Tracing Polymorphic Multidomain Spectral/hp Elements for Isosurface Rendering
Blake Nelson and Robert M. Kirby, Member,pose of this paper is to present a ray-tracing isosurface rendering algorithm for spectral/hp (high-order finite) element methods in which the visualization error is both quantified and minimized. Determination of the ray-isosurface intersection is accomplished by classic polynomial root-finding applied to a polynomial approximation obtained by projecting the finite element solution over element-partitioned segments along the ray. Combining the smoothness properties of spectral/hp elements with classic orthogonal polynomial approximation theory, we devise an adaptive scheme which allows the polynomial approximation along a raysegment to be arbitrarily close to the true solution. The resulting images converge toward a pixel-exact image at a rate far faster than sampling the spectral/hp element solution and applying classic low-order visualization techniques such as marching cubes.
THE use of simulation science [1] as a means of scientific inquiry is increasing at a tremendous rate. The process of mathematically modeling physical phenomena, experimentally estimating important key modeling parameters, numerically approximating the solution of the mathematical model, and computationally solving the resulting algorithm has inundated the scientific and engineering worlds, allowing for rapid advances in our modern understanding and utilization of the world around us.
. The authors are with the School of Computing and Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT, 84112. E-mail: {bnelson, kirby}@.
Quantification and, ideally, elimination of error is a critical component of the scientific process; it allows scientists to judiciously evaluate which component of the simulation science process (e.g., modeling, numerical approximation, parameter estimation) requires refinement in light of comparison with the real phenomenon of interest. Over the last 40 years, tremendous effort has been exerted in the pursuit of numerical methods which are both flexible and accurate, hence providing sufficient fidelity to be employed in the numerical solution of a large number of models and sufficient quantification of accuracy to allow researchers to focus their attention on model refinement and uncertainty quantification. High-order finite element methods (a variant of which is the spectral/hp element methods) using either the continuous Galerkin or discontinuous Galerkin formulation have reached a level of sophistication such that they are now commonly applied to a diverse set of real-life engineering problems in computational solid mechanics, fluid mechanics, acoustics, and electromagnetics (e.g., [2], [3], [4], [5]).
In this paper, we seek to develop a ray-tracing isosurface rendering algorithm which exploits the properties of spectral/hp element data [11], and for which the visualization error is both quantified and minimized. This is accomplished by reducing the determination of the rayisosurface intersection of spectral/hp element data to classic polynomial root-finding applied to a polynomial approximation obtained by projecting the finite element solution over element-partitioned segments along the ray. Combining the smoothness properties of spectral/hp finite elements with classic orthogonal polynomial approximation theory, we devise an adaptive scheme which allows the polynomial approximation along a ray-segment to be arbitrary close to the true solution. The methodology presented herein has broader application than merely spectral/hp elements; this methodology can be adapted to any volumetric function which meets the smoothness properties of the spectral/hp element discretizations exploited in this work.