topax

Differentiable multidisciplinary topology optimization

A gallery of topology optimization examples with differentiable finite element analysis implemented in JAX and JAX-FEM.

Gallery

Minimum compliance

Andreassen, Erik, et al. “Efficient topology optimization in MATLAB using 88 lines of code.” Structural and Multidisciplinary Optimization 43.1 (2011): 1–16.

Example	Optimization history	MATLAB reference	Compliance
example_top88.ipynb		top88(60,20,0.5,3,2.4,1)	216.8137
example_top71.ipynb		top71(60,20,0.5,3,2.4,2)	233.7146
example_top82.ipynb		top82(150,50,0.5,3,6,1)	217.8814
example_top110.ipynb		top110(60,20,0.5,3,1.8,3)	189.1405

Heat conduction

Bendsøe, M. P., and Sigmund, O. Topology Optimization: Theory, Methods, and Applications. Springer, 2004, Section 5.1.6.

Example	Optimization history	MATLAB reference	Objective
example_topopt_heat.ipynb		toph(40,40,0.4,3.0,1.2)	447.9944

Compliant mechanism

Bendsøe, M. P., and Sigmund, O. Topology Optimization: Theory, Methods, and Applications. Springer, 2004, Section 5.1.5.

Example	Optimization history	MATLAB reference	Objective
example_topopt_mems.ipynb		topm(40,20,0.3,3.0,1.2)	-1.1131886

Geometrical nonlinearity

Wang, F., Lazarov, B. S., Sigmund, O., and Jensen, J. S. “Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems.” Computer Methods in Applied Mechanics and Engineering 276 (2014): 453–472.

C-shape fictitious-domain validation

Load	Deformed topology history	Reference	Result
144 kN		21.2747 kJ	20.8721 kJ
240 kN		56.9401 kJ	58.1354 kJ
300 kN		84.9415 kJ	84.4505 kJ

Stress constraints

Yang, D., Liu, H., Zhang, W., and Li, S. “Stress-constrained topology optimization based on maximum stress measures.” Computers & Structures 198 (2018): 23–39.

Example	Optimization history	Reference	Result
example_topopt_stress.ipynb		$V/V_0=26.5%$, $\sigma_{\max}=69.9$	$V/V_0=26.36%$, $\sigma_{\max}=69.999$

Multiple material phases

Bendsøe, M. P., and Sigmund, O. Topology Optimization: Theory, Methods, and Applications. Springer, 2004, Section 2.9.3.

Example	Optimization history	Materials	Compliance
example_topopt_multimaterial.ipynb		$E_1=1$, $E_2=0.2$, $V_1/V_0=V_2/V_0=25%$	118.7094

Negative Poisson's ratio

Xia, L., and Breitkopf, P. “Design of materials using topology optimization and energy-based homogenization approach in Matlab.” Structural and Multidisciplinary Optimization 52 (2015): 1229–1241.

Example	Optimization history	Uniaxial deformation	Reference	Result
example_topopt_auxetic.ipynb			$V/V_0=50%$, $\nu^H=-0.448$	$V/V_0=50%$, $\nu^H=-0.4480$

Citation

@software{xu2026topax,
  author = {Weipeng Xu and Tianju Xue},
  title = {topax: Topology optimization with differentiable finite element analysis},
  url = {https://github.com/xwpken/topax},
  year = {2026},
}