Further Study of Hierarchical Reconstruction Algorithms

层次化重建算法的进一步研究

• 批准号: 0810913
• 负责人: Yingjie Liu
• 金额: $ 16.78万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-15 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0810913&HistoricalAwards=false
• 关键词: Further; Study; Hierarchical; Reconstruction; Algorithms

High resolution capturing schemes for solving conservation laws, e.g., the ENO scheme, smear discontinuities within a few mesh cells and achieve high accuracy where the solution is smooth. A series of works by Cockburn and Shu et al. on discontinuous Galerkin (DG) methods and local DG introduce many new techniques to the DG family and enable it to solve a broader class of equations including conservation laws. Still, the limiting technique for DG is not very mature and is considered to be one of the major open problems in scientific computing. The investigator and his colleagues propose the further study of a new limiting technique, the hierarchical reconstruction (HR). It is a general reconstruction procedure used as a limiter to remove spurious oscillations in the presence of shocks. The HR algorithm, motivated by the moment limiter of Biswas, Devine and Flaherty (1994), involves only a MUSCL, a second order ENO or other piecewise linear reconstructions in each stage of a multi-layer reconstruction process without characteristic decomposition. Therefore it is compact and easy to implement for arbitrary meshes. It does not truncate higher degree terms of a polynomial and actually uses the information from all degree terms. It has been proved that HR does not reduce the approximation order of a polynomial. Moreover, HR can be used for finite volume and central schemes as well without characteristic decomposition, which leads to a new finite volume approach. The investigator and his colleagues also propose the study of the local constant velocity version of the back and forth error compensation and correction method (BFECC) for velocity advections in multi-phase fluid simulation, BFECC for moving meshes and for interpolation between grids. Adapting HR to BFECC wherever necessary could significantly improve the robustness of BFECC for non-smooth solution.This project is on the study and development of new methods for using computers to simulate certain natural phenomena such as airflow passing a wing, shock waves propagating in a body, smokes etc. Computer simulations help scientists and engineers testing various experimental configurations and product designs without conducting costly experiments.Nowadays a lot of special effects in Hollywood movies are made by computer simulation. The BFECC method co-developed by the principal investigator has been used by NVIDIA for smoke simulation, http://developer.download.nvidia.com/SDK/10/direct3d/Source/Smoke/doc/Smoke.wmv.However, computer simulation is a noisy process. Noises constantly come from machine errors, and from the non-smoothness of simulated objects, such as shocks, corners of boundaries, interfaces separating different fluids or tissues in a body etc. Without special techniques, simulation noises caused by shocks can easily destroy a simulation result. In fact, two of the fundamental challenges for developing computational methods are to reduce simulation time and noises from the non-smoothness of simulated objects. The investigator and his colleagues study a new method for removing noise, which is easier to use for complex geometry and less dependent on simulated objects. Preliminary results for simulations of shocks are encouraging. The new idea could be adapted to many other areas and motivate the development of improved computational methods. For example, it could allow a complicated aircraft shape to be simulated more easily, motivate more robust techniques to stabilize simulations of multi-phase fluids, fuel cells etc and provide a black-box de-noising tool for simulations whose underlying physics are more empirical.

用于求解守恒定律的高分辨率捕获方案，例如，ENO格式，在几个网格单元内涂抹不连续性，并在解光滑的地方实现高精度。Cockburn和Shu等人关于间断Galerkin（DG）方法和局部DG的一系列工作为DG族引入了许多新技术，使其能够求解包括守恒律在内的更广泛的一类方程。尽管如此，DG的限制技术还不是很成熟，被认为是科学计算中的主要开放问题之一。研究者和他的同事们建议进一步研究一种新的限制技术，分层重建（HR）。这是一个通用的重建程序，用作限制器，以消除虚假振荡的冲击存在。 由Biswas，Devine和Flaherty（1994）的矩限制器激发的HR算法在多层重建过程的每个阶段仅涉及MUSCL，二阶ENO或其他分段线性重建，而没有特征分解。因此，它是紧凑的，易于实现的任意网格。它不截断多项式的高次项，实际上使用来自所有次项的信息。已证明HR不降低多项式的逼近阶。此外，HR可以用于有限体积和中心格式，以及没有特征分解，这导致了一个新的有限体积方法。研究人员和他的同事还提出了研究多相流体模拟中速度平流的局部恒速版本的来回误差补偿和校正方法（BFECC）、移动网格和网格之间插值的BFECC。在有需要的情况下，将HR调整到BFECC，可以显著提高BFECC对非光滑解的鲁棒性。这个项目是研究和开发使用计算机模拟某些自然现象的新方法，例如气流通过机翼，激波在物体中传播，计算机模拟帮助科学家和工程师测试各种实验配置和产品设计，而无需进行昂贵的现在好莱坞电影中的很多特效都是通过计算机模拟来制作的。由主要研究者共同开发的BFECC方法已被NVIDIA用于烟雾模拟，http：//developer.download.nvidia.com/SDK/10/direct3d/Source/Smoke/doc/Smoke.wmv。然而，计算机模拟是一个嘈杂的过程。噪声经常来自机器错误，以及模拟对象的非平滑性，例如冲击，边界的拐角，分离体内不同流体或组织的界面等。如果没有特殊技术，由冲击引起的模拟噪声很容易破坏模拟结果。事实上，开发计算方法的两个基本挑战是减少模拟时间和来自模拟对象的非平滑性的噪声。研究人员和他的同事们研究了一种去除噪音的新方法，这种方法更容易用于复杂的几何形状，并且对模拟物体的依赖性更小。冲击模拟的初步结果令人鼓舞。这个新想法可以适用于许多其他领域，并推动改进计算方法的发展。例如，它可以允许更容易地模拟复杂的飞机形状，激励更强大的技术来稳定多相流体，燃料电池等的模拟，并为基础物理学更经验的模拟提供黑箱去噪工具。