. 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.

With the increasing use of high-order methods and high-order meshes, scientific visualization software need to adapt themselves to reliably render the associated meshes and numerical solutions. In this paper, a novel approach, based on OpenGL 4 framework, enables a GPU-based rendering of high-order meshes as well as an almost pixel-exact rendering of high-order solutions. Several aspects of the OpenGL Shading Language and in particular the use of dedicated shaders (GPU programs) allows to answer this visualization challenge. Fragment shaders are used to compute the exact solution for each pixel, made possible by the transfer of degrees of freedom and shape functions to the GPU with textures. Tessellation shaders, combined with geometric error estimates, allow us to render high-order curved meshes by providing an adaptive subdivision of elements on the GPU directly. A convenient way to compute bounds for high-order solutions is described. The interest of using Bézier basis instead of Lagrange functions lies in the existence of fast and robust evaluation of polynomial functions with de Casteljau algorithm. A technique to plot highly nonlinear isolines and wire frames with a desired thickness is derived. It is based on a finite difference scheme performed on GPU. In comparison with standard techniques, we remove the use of any linear interpolation step and the need to generate a priori a fixed subdivided mesh. This reduces the memory footprint, improves the accuracy and the speed of the rendering. Finally, the method is illustrated with various 3D examples.

No abstract

No abstract

We propose a methodology to generate an accurate and efficient reconstruction of radiated fields based on high order interpolation. As the solution is obtained with the convolution by a smooth but potentially high frequency oscillatory kernel, our basis functions therefore incorporate plane waves. Directional interpolation is shown to be efficient for smart directions. An adaptive subdivision of the domain is established to limit the oscillations of the kernel in each element. The new basis functions, combining high order polynomials and plane waves, provide much better accuracy than low order ones. Finally, as standard visualization softwares are generally unable to represent such fields, a method to have a well-suited visualization of high order functions is used. Several numerical results confirm the potential of the method.