通过自动微分简化基于 FFT 的力学方法

Mohit Pundir, David S. Kammer
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引用次数: 0

摘要

快速傅立叶变换(FFT)方法已广泛应用于固体力学领域,以解决复杂的均质化问题。然而,目前基于 FFT 的方法面临着一些挑战,限制了它们对复杂材料模型或复杂力学问题的适用性。这些挑战包括手工实现构成规律,以及使用计算昂贵的复杂算法将微观机理与宏观材料行为结合起来。我们证明,基于 AD 增强的 FFT 方法可以直接从能量密度函数推导出应力和切线刚度,从而促进基于 FFT 的方法扩展到更复杂的材料模型。此外,AD 简化了具有复杂结构和构造特性的微结构的均质化切线刚度计算。这一改进使当前基于 FFT 的方法更加模块化,使它们能够处理复杂多尺度系统中的均质化问题,特别是那些涉及多物理过程的问题。我们的工作将简化复杂固体力学问题中基于 FFT 方法的数值实现。
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Simplifying FFT-based methods for mechanics with automatic differentiation
Fast-Fourier Transform (FFT) methods have been widely used in solid mechanics to address complex homogenization problems. However, current FFT-based methods face challenges that limit their applicability to intricate material models or complex mechanical problems. These challenges include the manual implementation of constitutive laws and the use of computationally expensive and complex algorithms to couple microscale mechanisms to macroscale material behavior. Here, we incorporate automatic differentiation (AD) within the FFT framework to mitigate these challenges. We demonstrate that AD-enhanced FFT-based methods can derive stress and tangent stiffness directly from energy density functionals, facilitating the extension of FFT-based methods to more intricate material models. Additionally, AD simplifies the calculation of homogenized tangent stiffness for microstructures with complex architectures and constitutive properties. This enhancement renders current FFT-based methods more modular, enabling them to tackle homogenization in complex multiscale systems, especially those involving multiphysics processes. Our work will simplify the numerical implementation of FFT-based methods for complex solid mechanics problems.
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