Abstract The adjoint method is used for high-fidelity aerodynamic shape optimization and is an efficient approach for computing the derivatives of a function of interest with respect to a large number of design variables. Over the past few decades, various approaches have been used to implement the adjoint method in computational fluid dynamics solvers. However, further advances in the field are hindered by the lack of performance assessments that compare the various adjoint implementations. Therefore, we propose open benchmarks and report a comprehensive evaluation of the various approaches to adjoint implementation. We also make recommendations on effective approaches, that is, approaches that are efficient, accurate, and have a low implementation cost. We focus on the discrete adjoint method and describe adjoint implementations for two computational fluid dynamics solvers by using various methods for computing the partial derivatives in the adjoint equations and for solving those equations. Both source code transformation and operator-overloading algorithmic differentiation tools are used to compute the partial derivatives, along with finite differencing. We also examine the use of explicit Jacobian and Jacobian-free solution methods. We quantitatively evaluate the speed, scalability, memory usage, and accuracy of the various implementations by running cases that cover a wide range of Mach numbers, Reynolds numbers, mesh topologies, mesh sizes, and number of CPU cores. We conclude that the Jacobian-free method using source code transformation algorithmic differentiation to compute the partial derivatives is the best option because it computes exact derivatives with the lowest CPU time and the lowest memory requirements, and it also scales well up to 10 million cells and over one thousand CPU cores. The superior performance of this approach is primarily due to its Jacobian-free adjoint strategy. The cases presented herein are publicly available and represent platform-independent benchmarks for comparing other current and future adjoint implementations. Our results and discussion provide a guide for discrete adjoint implementations, not only for computational fluid dynamics but also for a wide range of other partial differential equation solvers.

中文翻译：

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计算流体动力学的有效伴随方法

摘要 伴随法用于高保真空气动力学形状优化，是计算感兴趣函数相对于大量设计变量的导数的有效方法。在过去的几十年中，已使用各种方法在计算流体动力学求解器中实现伴随方法。然而，由于缺乏比较各种伴随实施的绩效评估，该领域的进一步进展受到阻碍。因此，我们提出开放基准并报告对各种伴随实施方法的综合评估。我们还就有效方法提出建议，即高效、准确且实施成本低的方法。我们专注于离散伴随方法，并通过使用各种方法计算伴随方程中的偏导数和求解这些方程来描述两个计算流体动力学求解器的伴随实现。源代码转换和运算符重载算法微分工具都用于计算偏导数，以及有限差分。我们还研究了显式 Jacobian 和 Jacobian 解法的使用。我们通过运行涵盖广泛的马赫数、雷诺数、网格拓扑、网格大小和 CPU 内核数量的案例，定量评估各种实现的速度、可扩展性、内存使用和准确性。我们得出结论，使用源代码转换算法微分计算偏导数的 Jacobian-free 方法是最好的选择，因为它以最低的 CPU 时间和最低的内存要求计算精确的导数，并且它也可以很好地扩展到 1000 万个单元和超过一千个 CPU 内核。这种方法的优越性能主要归功于其无雅可比伴随策略。此处介绍的案例是公开可用的，代表了与平台无关的基准，用于比较其他当前和未来的伴随实现。我们的结果和讨论为离散伴随实现提供了指导，不仅适用于计算流体动力学，还适用于各种其他偏微分方程求解器。