Optimized Implementation of Recursive Sub-Division Technique for Higher-Order Finite-Element Isosurface and Streamline Visualization

The authors propose that recursive subdivision techniques be used for some of these tasks and for some special subclasses. Specifically, they advocate the use of interval computation methods that treat the parametric range as an interval. These techniques can be extended to evaluating piecewise continuously differentiable functions over intervals. Also presented is a general schema for designing algorithms using interval evaluation of functions. The power of this methodology is illustrated through the design of algorithms for commonly required processing tasks on geometric objects.

We propose a parameter optimization technique using the response surface methodology (RSM) for the accurate biological cell simulation which calculates cell behavior by numerically integrating the differential equation and generates action potentials, intracellular Ca transient, contraction, and intracellular ATP consumption by using parameter values of major ion permeability and amplitude of Ko dependency. Since most of these parameters cannot be directly measured by experiment, these are searched for by applying an optimization technique, that is, by minimizing an objective function defined as a difference between a measured and a calculated waveform of action potentials in a cardiac myocyte. We employ the RSM as the optimization technique. In the RSM, a quadratic polynomial is used in general, and a mathematical technique is used to calculate the extreme points. Because our parameter space cannot be approximated by a single quadratic polynomial surface, we adopt a recursive subdivision technique and use the coefficient of multiple determination to representing a response surface in each subspace for a criteria of the subdivision. We confirmed the effectiveness of the proposed technique by searching for parameters which are determined in advance.

We present algorithms for rendering the intersection curves of implicit surfaces using octree-based recursive space subdivision techniques. Interval techniques used during the subdivision process increase the algorithms' robustness.

In this paper, we present a novel isosurface visualization technique that guarantees the accurate visualization of isosurfaces with complex attribute data defined on (un)structured (curvi)linear hexahedral grids. Isosurfaces of high-order hexahedral-based finite element solutions on both uniform grids (including MRI and CT scans) and more complex geometry representing a domain of interest that can be rendered using our algorithm. Additionally, our technique can be used to directly visualize solutions and attributes in isogeometric analysis, an area based on trivariate high-order NURBS (Non-Uniform Rational B-splines) geometry and attribute representations for the analysis. Furthermore, our technique can be used to visualize isosurfaces of algebraic functions. Our approach combines subdivision and numerical root finding to form a robust and efficient isosurface visualization algorithm that does not miss surface features, while finding all intersections between a view frustum and desired isosurfaces. This allows the use of view-independent transparency in the rendering process. We demonstrate our technique through a straightforward CPU implementation on both complex-structured and complex-unstructured geometries with high-order simulation solutions, isosurfaces of medical data sets, and isosurfaces of algebraic functions.

The purpose 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 ray-segment 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.