Level-set-based Topological Derivative Optimisation Framework for 3D Metamaterials Design

This repository implements a 3D level-set based topology optimisation framework for metamaterial design. The overall idea is to combine:

• a periodic homogenisation procedure to compute the effective stiffness tensor of a unit-cell microstructure;
• a material objective defined from the homogenised stiffness, which drives the design toward a target mechanical response;
• a topological-derivative / level-set update scheme that iteratively modifies the material distribution.

The implementation is organised around three core files:

• main_3d_UTI.py: the main optimisation loop and the outer update logic;
• definition_3d_UTI.py: PDE definitions, homogenisation, material modelling, and the cost function / gradient;
• init_3d_UTI.py: numerical parameters, optimisation settings, and material / solver configuration.

1. Overall algorithm framework

The workflow can be summarised as follows:

1. Initialise a 3D periodic unit cell and a level-set field.
2. Convert the level-set field into a two-phase material distribution (solid / void).
3. Solve periodic cell problems for several macroscopic strain loading cases to obtain the homogenised stiffness tensor $C_{\mathrm{hom}}$.
4. Evaluate a scalar cost function from $C_{\mathrm{hom}}$ and compute its gradient with respect to the stiffness tensor.
5. Use the chain-rule-based topological derivative to obtain a sensitivity field on the design domain.
6. Update the level-set field through a level-set / line-search based optimisation step.
7. Repeat until convergence, optionally with mesh refinement and volume-control continuation.

In other words, the method is a topology optimisation loop in which the design variable is the level-set field, while the physical response is evaluated through periodical homogenisation.

2. Main components of the code

2.1 Periodic homogenisation

In definition_3d_UTI.py, the code builds a periodic 3D unit-cell setting and solves representative cell problems under different macroscopic strain states. The resulting homogenised stiffness tensor is then used as the basis for the objective evaluation.

2.2 Level-set based design update

In main_3d_UTI.py, the optimisation loop uses the sensitivities obtained from the cost function, updates the level-set field, and performs a line-search / step-size control strategy to improve stability.

2.3 Configuration and numerical parameters

In init_3d_UTI.py, the user can tune:

• material properties such as Young's modulus and Poisson ratio;
• optimisation and line-search settings;
• volume-continuation / stage-control behaviour;
• mesh resolution and refinement strategy.

3. Current cost function setting

The current implementation is specifically set up for an uncoupled transverse isotropic (UTI) design objective.

The objective is constructed from representative stiffness components in a tetragonal / z-rot4 design space, including quantities such as:

• $C_{11}^{t}$, $C_{12}^{t}$, $C_{13}^{t}$, $C_{33}^{t}$, $C_{44}^{t}$, $C_{66}^{t}$;
• the UTI-related invariants $h_b$, $h_a$, $H$;
• a transverse-isotropy penalty term based on the residual $R_{TI}$.

The current objective is implemented in definition_3d_UTI.py through functions such as:

• uti_ratio_term
• grad_uti_ratio_term
• ti_penalty_term
• grad_ti_penalty_term
• Phi_uti
• grad_Phi_uti

In its current form, the total cost is approximately:

$$ J = J_{hb} + J_{ti} $$

where

$$ J_{hb} = \beta_a h_b^2 + \beta_b / (h_a^2 + H^2 + \varepsilon) $$

and $J_{ti}$ is a penalty term enforcing the transverse-isotropy relation.

4. Important note for other cases

This framework is currently configured for the UTI objective. If you want to optimise for a different material class or a different target property, the main optimisation machinery does not need to be rewritten. Instead, you should replace the cost function and its gradient in definition_3d_UTI.py, especially the functions listed above.

In practice, for a new case you would:

1. define the new scalar objective from the homogenised stiffness tensor;
2. derive or implement the corresponding gradient with respect to the stiffness components;
3. replace the current UTI-related functions in definition_3d_UTI.py;
4. keep the rest of the level-set update and homogenisation pipeline unchanged.

5. How to run

The main entry point is main_3d_UTI.py. After installing the required dependencies (especially FEniCS), run the script to start the optimisation loop.

6. Summary

This project provides a 3D topology optimisation framework for metamaterial design based on:

• periodic homogenisation;
• topological derivative / level-set update;
• a currently implemented UTI-specific cost function.

It is therefore well suited for UTI-type design targets, while remaining extensible to other material objectives by replacing the cost function module.