Well-suited and adaptive post-processing for the visualization of hp simulation results

Abstract

. Wellsuited and adaptive post-processing for the visualization of hp simulation results. Journal of Computational Physics, Elsevier, 2018, 375, pp.1179-1204. 10.1016/j.jcp.2018 Well-suited and adaptive post-processing for the visualization of hp simulation results AbstractWhile high order methods became very popular as they allow to perform very accurate solutions with low computational time and memory cost, there is a lack of tools to visualize and post-treat the solutions given by these methods. Originally, visualization softwares were developed to post-process results from methods such that finite differences or usual finite elements and therefore process linear primitives. In this paper, we present a methodology to visualize results of high order methods. Our approach is based on the construction of an optimized affine approximation of the high order solution which can therefore be handled by any visualization software. A representation mesh is constructed and the process is guided by an a posteriori estimate which control the error between the numerical solution and its representation pointwise. This point by point control is crucial as under their picture form, data correspond to values mapped on elements where anyone can pick up a pointwise information. A strategy is established to ensure that discontinuities are well represented. These discontinuities come either from the physical problem (material change) or the numerical method (discontinuous Galerkin method) and are pictured accurately. Several numerical examples are presented to demonstrate the potential of the method.

Accurate, Automatic and Compressed Visualization of Radiated Helmholtz Fields from Boundary Element Solutions

“…In this section, we present a method developed by the authors [8] to visualize a hp solution noted f num . Standard visualization softwares are unable to visualize and post-treat such solutions.…”

“…We introduce a quantification of the error between f num and f vis . To this purpose, we introduce Hausdorff distance on X × Y allowing to measure the gap between the graph of two functions [8]. As Hausdorff distance involves two embedded optimizations, it is expensive to employ.…”

“…In this case, the basis functions are chosen such that φ j (x) = L j (x)e ik c,x−ξj where x ∈ V. In order to verify the assumption (8), the angle between c and (x − y)/ x − y must be small. As previously, the direction c is defined as c = (x c − y c )/ x c − y c where x c and y c are defined as "average" of V and Γ .…”

“…In each element E V , a local field will be therefore reconstructed with local directions and basis functions. Furthermore, a subdivision of Γ can be established to respect the assumption on the directions (8). The boundary Γ is decomposed in Γ = ∪ Γn Γ n such that there is no overlapping between each part Γ n .…”

“…Furthermore, as an asymptotic behaviour of the kernel is known, the use of a combination of high order and plane waves can be used to improve the approximation of the solution [3,6,9]. Finally, as standard visualization softwares are unable to visualize such high order polynomials nor plane waves functions, we combine this with an algorithm to visualize high order functions in any visualization software [8].…”