利用自动微分简化固体力学中基于FFT的方法
Simplifying FFT-based methods for solid mechanics with automatic differentiation
AI总结：
该研究将自动微分纳入FFT框架，直接从能量密度泛函推导应力和切线刚度，简化了复杂微观结构的均匀化切线刚度计算，使基于FFT的方法更模块化，并能应用于多物理场过程、不确定度量化和拓扑优化等复杂固体力学问题。
摘要：
快速傅里叶变换（FFT）方法已被广泛应用于固体力学中以解决复杂的均匀化问题。然而，当前的基于FFT的方法面临一些挑战，限制了其在复杂材料模型或复杂力学问题中的适用性。这些挑战包括本构定律的手动实现，以及使用计算成本高昂且复杂的算法将微观机制耦合到宏观材料行为中。在此，我们将自动微分（AD）纳入FFT框架中以缓解这些挑战。我们证明，AD增强的基于FFT的方法可以直接从能量密度泛函推导应力和切线刚度，从而促进基于FFT的方法向更复杂材料模型的扩展。此外，自动微分简化了具有复杂架构和本构特性的微观结构的均匀化切线刚度计算。这一增强使当前的基于FFT的方法更加模块化，使其能够处理复杂多尺度系统中的均匀化，特别是涉及多物理场过程的系统。此外，我们展示了AD增强的FFT方法在均匀化之外问题中的应用，例如不确定度量化和拓扑优化，其中自动微分简化了敏感性的计算。我们的工作将简化基于FFT的方法在复杂固体力学问题中的数值实现。
英文摘要：
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, automatic differentiation 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. Furthermore, we illustrate the use of the AD-enhanced FFT method for problems that extend beyond homogenization, such as uncertainty quantification and topology optimization where automatic differentiation simplifies the computation of sensitivities. Our work will simplify the numerical implementation of FFT-based methods for complex solid mechanics problems.
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