Imposing Dirichlet boundary conditions directly for FFT-based computational micromechanics.

Area of Science:

• Computational Mechanics
• Materials Science
• Numerical Analysis

Background:

• Computational homogenization typically uses periodic boundary conditions with fast Fourier transform (FFT) methods.
• Imposing Dirichlet or Neumann boundary conditions is crucial for specific mechanical and thermal applications.
• Existing methods for Dirichlet boundary conditions in mechanical homogenization (Lagrange multipliers, buffer zones) compromise computational efficiency.

Purpose of the Study:

• To develop a direct method for imposing Dirichlet boundary conditions in FFT-based computational homogenization for mechanics.
• To overcome limitations of existing methods that do not leverage FFT's computational advantages.
• To integrate the new method into existing computational homogenization frameworks.

Main Methods:

• Development of the Moulinec-Suquet discretization tailored for Dirichlet boundary conditions on rectangular domains.
• Utilizing a formulation based on the deformation gradient and the Green's operator of the vector Laplacian.
• Employing carefully selected weights at boundary points for direct imposition of conditions.

Main Results:

• A novel technique for directly imposing Dirichlet boundary conditions without indefinite systems is presented.
• The method is shown to be compatible with discrete sine/cosine transforms, enabling seamless integration with FFT-based codes.
• Numerical examples validate the effectiveness and capabilities of the proposed approach.

Conclusions:

• The introduced technique offers an efficient and direct way to handle Dirichlet boundary conditions in mechanical computational homogenization.
• This advancement preserves the computational benefits of FFT-based methods, making them more versatile.
• The method is ready for integration into existing computational homogenization software.