Multi-Dimensional Filtering: Reducing the Dimension Through Rotation

Abstract

Over the past few decades there has been a strong effort towards the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters [21] for Discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this post-processor. These advantages can be exploited during flow visualization, for example by applying the SIAC filter to the DG data before streamline computations [Steffan et al., IEEE-TVCG 14(3): 680-692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [Walfisch et al., JOMP 38(2);164-184] proposed a univariate filter implemented along the streamline curves. Until now, this technique remained a numerical experiment. In this paper we introduce the SIAC line filter and explore how the orientation, structure and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization community.

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

“…The authors in [5] proposed that one-dimensional SIAC filtering need not necessarily only be done in a way that is aligned with the coordinate axes, but rather it could be implemented as…”

“…In addition, introducing a rotation in the kernel leads to the problem of determining the optimal orientation. The authors in [5] link the rotation angle directly to the mesh, hence the optimal rotation problem inherits all the limitations encountered by the optimal characteristic length problem. In this paper, we demonstrate the impact of the characteristic length and rotation choices and provide a framework that, although not necessarily optimal, leads to satisfactory results.…”

“…The properties of the SIAC filter that are of importance to this work that we inherit when moving to LSIAC are as follows [5]:…”

“…Unlike the traditional sampling schemes (seen in Sections 4.1.2 and 4.1.3) the results using the LSIAC filter, if properly utilized, can preserve fluid properties [5]. This is due to the explicit accuracy conserving nature of the post-processing scheme on cG and dG simulation data.…”

“…2) When moving to general domains representing complex geometries, it is often impossible to filter at all points within the domain using the traditional SIAC filter due to its filter width overlap requirements generated by the tensor-product footprint. These two limitations of traditional SIAC filtering motivated the development of the line-SIAC (LSIAC) filter [5], a family of one-dimensional SIAC filters (symmetric and one-sided) that can filter along a segment in any direction within an FEM or FVM field. In this paper, we argue that the LSIAC filter allows us to transform arbitrary, unstructured high-order data over complex geometries in a way that is consistent with the underlying simulation data (and hence in a verifiable way) while remaining computationally tractable.…”