Simplifying FFT-based methods for solid mechanics with automatic differentiation

TL;DR

This paper introduces automatic differentiation into FFT-based solid mechanics methods, simplifying implementation and extending applicability to complex material models, multiscale systems, and multiphysics problems.

Contribution

It integrates automatic differentiation into FFT methods, enabling direct derivation of stresses and tangent stiffness, and broadening their use in complex, multiscale, and multiphysics applications.

Findings

AD enhances derivation of stress and stiffness from energy functionals

Simplifies homogenized tangent stiffness calculations for complex microstructures

Facilitates sensitivity analysis in uncertainty quantification and topology optimization

Abstract

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.