Interactive High‐Quality Visualization of Higher‐Order Finite Elements

Abstract

Higher-order finite element methods have emerged as an important discretization scheme for simulation. They are increasingly used in contemporary numerical solvers, generating a new class of data that must be analyzed by scientists and engineers. Currently available visualization tools for this type of data are either batch oriented or limited to certain cell types and polynomial degrees. Other approaches approximate higher-order data by resampling resulting in trade-offs in interactivity and quality. To overcome these limitations, we have developed a distributed visualization system which allows for interactive exploration of non-conforming unstructured grids, resulting from space-time discontinuous Galerkin simulations, in which each cell has its own higher-order polynomial solution. Our system employs GPU-based raycasting for direct volume rendering of complex grids which feature non-convex, curvilinear cells with varying polynomial degree. Frequency-based adaptive sampling accounts for the high variations along rays. For distribution across a GPU cluster, the initial object-space partitioning is determined by cell characteristics like the polynomial degree and is adapted at runtime by a load balancing mechanism. The performance and utility of our system is evaluated for different aeroacoustic simulations involving the propagation of shock fronts.

Interactive Volume Visualization of General Polyhedral Grids

“…This comparison is difficult since to our knowledge there exist only four other GPU volume renderers capable of directly rendering convex polyhedral cells [10,26,32,39]. Three of those [10,26,39] can process non-convex cells without requiring subdivision.…”

“…Three of those [10,26,39] can process non-convex cells without requiring subdivision. The HAVS level-of-detail extension has been included in this list since it can cope with polyhedral cells on a basic level.…”

“…Space-time discontinuous Galerkin simulations are based on even more complex grids containing non-convex curvilinear cells. Uffinger et al [39] have proposed a volume rendering algorithm for such grids and the corresponding simulations, which not only contain a simple scalar value but a polynomial of varying degree per cell. This polynomial representation allows for using fewer cells to resolve spatially small features.…”

“…The per-tetrahedra memory consumption for CIEL has been estimated based on the overall memory consumption stated in the corresponding publication. Scalar data are represented as polynomials in the case of the higher-order finite-elements visualization technique (HOFEV) [39]. Therefore a direct comparison to the other techniques in Table 1 would be biased toward techniques dealing with conventional CFD data.…”

Data-Adapted Volume Rendering for Scattered Point Data

“…Existing reconstruction methods for scattered point data can broadly be divided into 2 categories: grid based and non-grid based. The grid based methods are to resample the data on a uniform grid [3], or tetrahedralize it to create an unstructured grid, and then reconstruct the value at an arbitrary point in the resulting grid with spline-based methods [4], [5] or high order tetrahedral methods [6], [7]. In non-grid based methods, arguably the most prominent methods are radial basis functions (RBFs) that were introduced to volume rendering by Jang et al [8].…”