  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.
