On the Treatment of Field Quantities and Elemental Continuity in FEM Solutions.

As the finite element method (FEM) and the finite volume method (FVM), both traditional and high-order variants, continue their proliferation into various applied engineering disciplines, it is important that the visualization techniques and corresponding data analysis tools that act on the results produced by these methods faithfully represent the underlying data. To state this in another way: the interpretation of data generated by simulation needs to be consistent with the numerical schemes that underpin the specific solver technology. As the verifiable visualization literature has demonstrated: visual artifacts produced by the introduction of either explicit or implicit data transformations, such as data resampling, can sometimes distort or even obfuscate key scientific features in the data. In this paper, we focus on the handling of elemental continuity, which is often only continuous or piecewise discontinuous, when visualizing primary or derived fields from FEM or FVM simulations. We demonstrate that traditional data handling and visualization of these fields introduce visual errors. In addition, we show how the use of the recently proposed line-SIAC filter provides a way of handling elemental continuity issues in an accuracy-conserving manner with the added benefit of casting the data in a smooth context even if the representation is element discontinuous.

GPU-based volume visualization from high-order finite element fields.

This paper describes a new volume rendering system for spectral/hp finite-element methods that has as its goal to be both accurate and interactive. Even though high-order finite element methods are commonly used by scientists and engineers, there are few visualization methods designed to display this data directly. Consequently, visualizations of high-order data are generally created by first sampling the high-order field onto a regular grid and then generating the visualization via traditional methods based on linear interpolation. This approach, however, introduces error into the visualization pipeline and requires the user to balance image quality, interactivity, and resource consumption. We first show that evaluation of the volume rendering integral, when applied to the composition of piecewise-smooth transfer functions with the high-order scalar field, typically exhibits second-order convergence for a wide range of high-order quadrature schemes, and has worst case first-order convergence. This result provides bounds on the ability to achieve high-order convergence to the volume rendering integral. We then develop an algorithm for optimized evaluation of the volume rendering integral, based on the categorization of each ray according to the local behavior of the field and transfer function. We demonstrate the effectiveness of our system by running performance benchmarks on several high-order fluid-flow simulations.

GPU-based interactive cut-surface extraction from high-order finite element fields.

We present a GPU-based ray-tracing system for the accurate and interactive visualization of cut-surfaces through 3D simulations of physical processes created from spectral/hp high-order finite element methods. When used by the numerical analyst to debug the solver, the ability for the imagery to precisely reflect the data is critical. In practice, the investigator interactively selects from a palette of visualization tools to construct a scene that can answer a query of the data. This is effective as long as the implicit contract of image quality between the individual and the visualization system is upheld. OpenGL rendering of scientific visualizations has worked remarkably well for exploratory visualization for most solver results. This is due to the consistency between the use of first-order representations in the simulation and the linear assumptions inherent in OpenGL (planar fragments and color-space interpolation). Unfortunately, the contract is broken when the solver discretization is of higher-order. There have been attempts to mitigate this through the use of spatial adaptation and/or texture mapping. These methods do a better job of approximating what the imagery should be but are not exact and tend to be view-dependent. This paper introduces new rendering mechanisms that specifically deal with the kinds of native data generated by high-order finite element solvers. The exploratory visualization tools are reassessed and cast in this system with the focus on image accuracy. This is accomplished in a GPU setting to ensure interactivity.