The Effect of Data Transformations on Scalar Field Topological Analysis of High-Order FEM Solutions

Abstract

a) Sampled vorticity. (b) Subdivided vorticity.(c) L-SIAC vorticity. Fig. 1: Topological segmentation of counter-rotating vortex sampled using different methodologies discussed in the paper and by filtering the contour tree for segments that resemble vortex-like structures. The number, shape, and boundaries of the segments are different for the three techniques.Abstract-High-order finite element methods (HO-FEM) are gaining popularity in the simulation community due to their success in solving complex flow dynamics. There is an increasing need to analyze the data produced as output by these simulations.Simultaneously, topological analysis tools are emerging as powerful methods for investigating simulation data. However, most of the current approaches to topological analysis have had limited application to HO-FEM simulation data for two reasons. First, the current topological tools are designed for linear data (polynomial degree one), but the polynomial degree of the data output by these simulations is typically higher (routinely up to polynomial degree six). Second, the simulation data and derived quantities of the simulation data have discontinuities at element boundaries, and these discontinuities do not match the input requirements for the topological tools. One solution to both issues is to transform the high-order data to achieve low-order, continuous inputs for topological analysis. Nevertheless, there has been little work evaluating the possible transformation choices and their downstream effect on the topological analysis. We perform an empirical study to evaluate two commonly used data transformation methodologies along with the recently introduced L-SIAC filter for processing high-order simulation data. Our results show diverse behaviors are possible. We offer some guidance about how best to consider a pipeline of topological analysis of HO-FEM simulations with the currently available implementations of topological analysis.

Fast Barycentric-Based Evaluation Over Spectral/hp Elements

“…In this tensorial case, computing the barycentric weights is simpler since we need to only compute the k j univariate weights for dimension j associated with the grid z q,j for q ∈ [k j ]. Thus, we need to only compute d j=1 k j weights, as opposed to the full set of d j=1 k j multivariate weights associated with (9). Evaluating η → p(η) is likewise faster in this case: once the univariate weights are computed, then each univariate barycentric evaluation η j → p j (η j ) requires O(k j ) complexity.…”

“…However, in the case of history points (positions in the field at which one wants to track a particular quantity of interest over time) [16], pathlines/streamlines [17], isosurface evaluation, [9] or refinement and mortaring [12], evaluation of solutions expansions at arbitrary point locations is required. From the perspective of point evaluation over the entire domain, these operations require two phases: given a point in the domain, first finding the element in which that point resides, and then a fast evaluation of the solution expansion on an individual element.…”

Fast Barycentric-Based Evaluation Over Spectral/hp Elements

“…However, in the case of history points (positions in the field at which one wants to track a particular quantity of interest over time) [16], pathlines/streamlines [17], isosurface evaluation, [9] or refinement and mortaring [12], evaluation of solution expansions at arbitrary point locations is required. From the perspective of point evaluation over the entire domain, these operations require two phases: given a point in the domain, first finding the element in which that point resides, and then a fast evaluation of the solution expansion on an individual element.…”