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.

中文翻译：

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通过自动微分简化基于 FFT 的固体力学方法


快速傅里叶变换 （FFT） 方法已广泛用于固体力学中，以解决复杂的均质化问题。然而，当前基于 FFT 的方法面临的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。这些挑战包括手动实现本构定律，以及使用计算成本高昂且复杂的算法将微尺度机制与宏观尺度材料行为耦合。在这里，我们将自动微分 （AD） 整合到 FFT 框架中，以缓解这些挑战。我们证明，基于 FFT 的 AD 增强方法可以直接从能量密度泛函中得出应力和切线刚度，从而促进基于 FFT 的方法扩展到更复杂的材料模型。此外，自动微分简化了具有复杂结构和本构特性的微结构的均质切线刚度计算。这一增强功能使当前基于 FFT 的方法更加模块化，使它们能够解决复杂多尺度系统中的均质化问题，尤其是那些涉及多物理场过程的系统。此外，我们说明了将 AD 增强型 FFT 方法用于超越同质化的问题，例如不确定性量化和拓扑优化，其中自动微分简化了灵敏度的计算。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。