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首页> 外文期刊>Computer Methods in Applied Mechanics and Engineering >Variational inequality approach to enforcing the non-negative constraint for advection-diffusion equations
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Variational inequality approach to enforcing the non-negative constraint for advection-diffusion equations

机译：对流扩散方程实施非负约束的变分不等式方法

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摘要

Predictive simulations are crucial for the success of many subsurface applications, and it is highly desirable to obtain accurate non-negative solutions for transport equations in these numerical simulations. To this end, optimization-based methodologies based on quadratic programming (QP) have been shown to be a viable approach to ensuring discrete maximum principles and the non-negative constraint for anisotropic diffusion equations. In this paper, we propose a computational framework based on the variational inequality (VI) which can also be used to enforce important mathematical properties (e.g., maximum principles) and physical constraints (e.g., the non-negative constraint). We demonstrate that this framework is not only applicable to diffusion equations but also to non-symmetric advection diffusion equations. An attractive feature of the proposed framework is that it works with any weak formulation for the advection diffusion equations, including single-field formulations, which are computationally attractive. A particular emphasis is placed on the parallel and algorithmic performance of the VI approach across large-scale and heterogeneous problems. It is also shown that QP and VI are equivalent under certain conditions. State-of-the-art QP and VI solvers available from the PETSc library are used on a variety of steady-state 2D and 3D benchmarks, and a comparative study on the scalability between the QP and VI solvers is presented. We then extend the proposed framework to transient problems by simulating the miscible displacement of fluids in a heterogeneous porous medium and illustrate the importance of enforcing maximum principles for these types of coupled problems. Our numerical experiments indicate that VIs are indeed a viable approach for enforcing the maximum principles and the non-negative constraint in a large-scale computing environment. Also provided are Firedrake project files as well as a discussion on the computer implementation to help facilitate readers in understanding the proposed framework. (C) 2017 Elsevier B.V. All rights reserved.
机译：预测模拟对于许多地下应用的成功至关重要，因此非常希望在这些数值模拟中获得用于输运方程的精确非负解。为此，基于二次规划（QP）的基于优化的方法已被证明是确保离散最大原理和各向异性扩散方程的非负约束的可行方法。在本文中，我们提出了一个基于变分不等式（VI）的计算框架，该框架还可以用于强制执行重要的数学属性（例如，最大原理）和物理约束（例如，非负约束）。我们证明了该框架不仅适用于扩散方程，而且适用于非对称对流扩散方程。所提出的框架的一个吸引人的特征是，它可以与对流扩散方程的任何弱公式一起使用，包括具有计算吸引力的单场公式。特别强调的是VI方法在大规模和异构问题上的并行和算法性能。还表明，QP和VI在某些条件下是等效的。 PETSc库中提供的最先进的QP和VI求解器可用于各种稳态2D和3D基准，并且对QP和VI求解器之间的可伸缩性进行了比较研究。然后，我们通过模拟非均质多孔介质中流体的混溶位移，将提出的框架扩展到瞬态问题，并说明了针对这些类型的耦合问题实施最大原则的重要性。我们的数值实验表明，VI确实是在大规模计算环境中执行最大原则和非负约束的可行方法。还提供了Firedrake项目文件以及有关计算机实现的讨论，以帮助促进读者理解建议的框架。 （C）2017 Elsevier B.V.保留所有权利。

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