有限元应力更新中的自动微分
Automatic differentiation in finite element stress updating
AI总结：
该研究将自动微分应用于有限元应力更新，对比解析微分实现岩土工程超塑性临界状态模型的效果，证实AD可简化应力积分实现，同时保持高精度，为计算力学代码开发提供便利。
AI中文摘要：
计算固体力学依赖于微分方程的离散化，需要数值微分与积分，通常通过求积法实现。近来，自动微分（AD）作为支持这些操作的工具受到关注。本文介绍AD在非线性有限元分析中的应用，重点关注应力更新——这是从应变增量计算应力的关键且重复执行的步骤。隐式与显式格式都需要本构模型的导数，而手动推导这些导数既复杂、易出错又耗时。我们对比传统解析微分与自动微分在实现岩土工程中广泛使用的超塑性临界状态模型的效果，采用两种方法模拟不同超固结比的排水三轴压缩试验。结果显示，AD通过消除手动推导导数的需求，显著简化了向后欧拉应力积分的实现；尽管简化了流程，AD仍保持了可与解析方法相媲美的高精度与鲁棒性。研究表明，自动微分可简化非线性材料模型的开发，实现任意复杂度本构定律的高效可靠实现，为构建更灵活、可维护的计算力学代码开辟了道路。
英文摘要：
Computational solid mechanics relies on the discretisation of differential equations, requiring numerical differentiation and integration, often via quadrature. Recently, automatic differentiation (AD) has attracted interest as a tool to support these operations. This paper introduces the use of AD in nonlinear finite element analysis, focusing on stress updating, a key and repeatedly executed step in which stresses are computed from strain increments. Both implicit and explicit schemes require derivatives of constitutive models, which can be complex, error prone, and time consuming to derive analytically. We compare traditional analytical differentiation with automatic differentiation for implementing a hyperplastic Critical State model widely used in geotechnical engineering. Drained triaxial compression tests with varying overconsolidation ratios are simulated using both approaches. The results show that AD significantly simplifies the implementation of backward Euler stress integration by removing the need for manual derivation of derivatives. Despite this simplification, AD maintains high accuracy and robustness comparable to analytical approaches. The findings demonstrate that automatic differentiation can streamline the development of nonlinear material models, enabling efficient and reliable implementation of constitutive laws of arbitrary complexity. This opens the way for more flexible and maintainable computational mechanics codes.
