🧠 Scikit Topt

A lightweight, flexible Python library for topology optimization built on top of scikit-fem.

Features

To contribute to the open-source community and education—which I’ve always benefited from—I decided to start this project.

The currently supported features are as follows:

• Coding with Python
• Implement FEA on unstructured mesh using scikit-fem
• Topology optimization using the density method and its optimization algorithm
  • Optimality Criteria (OC) Method
  • Modified OC Method
  • Lagrange Method
• Multiple objective functions (forces)
• High-performance computation using sparse matrices with Scipy and PyAMG
• easy installation with pip/poetry

ToDo

• Make workflow for multi-load cases efficient
• Set break point from the optimization loop
• Organize documentation
• Add LevelSet

Install Package

pip install scitopt
poetry add scitopt

Optimize Toy Problem with Python Script

import scitopt

tsk = scitopt.mesh.toy_problem.toy1()
cfg = scitopt.core.KKT_Config()

optimizer = scitopt.core.KKT_Optimizer(cfg, tsk)

optimizer.parameterize()
optimizer.optimize()

Optimize Toy Problem with command line.

OMP_NUM_THREADS=3 OPENBLAS_NUM_THREADS=3  MKL_NUM_THREADS=3 PYTHONPATH=./ python ./scitopt/core/optimizer/kkt.py \
 --dst_path ./result/test1_kkt1 \
 --interpolation SIMP \
 --p_init 1.0 \
 --p 3.0 \
 --p_step -4 \
 --filter_radius_init 0.2 \
 --filter_radius 0.08 \
 --filter_radius_step 2 \
 --move_limit_init 0.20 \
 --move_limit 0.02 \
 --move_limit_step 2 \
 --vol_frac_init 0.60 \
 --vol_frac 0.40 \
 --vol_frac_step 2 \
 --beta_init 1.0 \
 --beta 2.0 \
 --beta_step 2 \
 --beta_curvature 2.0 \
 --percentile_init 70 \
 --percentile 90 \
 --percentile_step -4 \
 --eta 0.8 \
 --record_times 100 \
 --max_iters 100 \
 --lambda_v 0.01 \
 --lambda_decay  0.8 \
 --mu_p 2.50 \
 --export_img true \
 --sensitivity_filter false \
 --task_name down_box \
 --mesh_path box-down.msh \
 --solver_option pyamg \
 --rho_min 1e-2 \
 --E0 210e9 \
 --E_min 210e5 \
 --design_dirichlet true

Optiization Algorithm

Density Method

Optimality Criteria (OC) Method

The OC method is a widely used algorithm for compliance minimization problems in structural topology optimization. It updates the material distribution (density field) based on a set of local update rules derived from optimality conditions.

Key characteristics:

• Simple and efficient to implement.
• Iteratively updates densities using sensitivity information (e.g., compliance derivatives).
• Often includes move limits to stabilize convergence.

Update rule (simplified):

$$\rho_i^{(new)} = \text{clip}\left(\rho_i \cdot \left(-\frac{\partial C}{\partial \rho_i} / \lambda \right)^{\eta}, \rho_{min}, \rho_{max} \right)$$

where:

• ρ_i: density of element i
• dC/dρ_i: compliance sensitivity
• λ: Lagrange multiplier (to satisfy volume constraint)
• η: damping factor

---

Modified OC (MOC) Method

The Modified OC method (MOC) extends the classic OC method by introducing enhancements such as:

• Log-domain updates to improve numerical stability,
• Dynamic lambda adjustment to better handle volume constraints,
• Stress constraints or connectivity penalties (optional).

Advantages of MOC:

• Improved convergence in difficult optimization problems.
• Better control over numerical instability (e.g., checkerboard patterns).
• More flexibility to incorporate complex constraints.

---

Both methods are particularly suited for density-based approaches (e.g., SIMP), and can be combined with filters (e.g., sensitivity or density filters) to control minimum feature size and avoid numerical issues.

---

Techinical Components

Material Interpolation: SIMP and RAMP

In density-based topology optimization, the material stiffness is interpolated as a function of the element density.

SIMP (Solid Isotropic Material with Penalization)

SIMP is the most commonly used interpolation scheme:

$$E(ρ) = ρ^p * E₀$$

• ρ: element density (range: 0 to 1)
• p: penalization factor (typically p ≥ 3)
• E0: Young’s modulus of solid material

This method penalizes intermediate densities and encourages a 0–1 (black-and-white) design.

RAMP (Rational Approximation of Material Properties)

RAMP is another interpolation scheme used to reduce numerical instabilities like checkerboarding:

$$E(ρ) = E₀ * ρ / (1 + q * (1 - ρ))$$

• q: penalization parameter (higher q gives stronger 0–1 behavior)

RAMP can sometimes provide smoother convergence than SIMP.

---

Heaviside Projection

Heaviside projection is used to sharpen the boundaries between solid and void regions after filtering:

$$ρ̃ = (tanh(β * η) + tanh(β * (ρ - η))) / (tanh(β * η) + tanh(β * (1 - η)))$$

• ρ: filtered density
• ρ̃: projected density
• β: steepness parameter (higher = sharper transitions)
• η: threshold level (usually 0.5)

As beta → ∞, the projection approaches a binary function.

---

Helmholtz Filter (Density Smoothing)

The Helmholtz filter smooths the density field to prevent checkerboard patterns and enforce a minimum feature size.

It solves the PDE:

$$(-r² ∇² + 1) ρ̃ = ρ$$

• ρ: raw density field
• ρ̃: filtered density
• r: filter radius (controls the minimum length scale)

This is often implemented via solving a sparse linear system using finite elements.

Benefits:

• Enforces minimum feature size
• Suppresses numerical instabilities
• Improves manufacturability of the design

---

Acknowledgements

Standing on the shoulders of proverbial giants

This software does not exist in a vacuum. Scikit-Topt is standing on the shoulders of proverbial giants. In particular, I want to thank the following projects for constituting the technical backbone of the project:

• Scipy
• Scikit-fem
• PyAMG
• Numba
• MeshIO
• Matplotlib
• PyVista
• Topology Optimization Community

📖 Citation

If you use this repository in your work, please cite it as follows:

If you use Scikit Topt in your research or software, please cite it as:

@misc{scikit-topt2025,
  author       = {Kohei Watanabe},
  title        = {Scikit Topt: A Python library for topology optimization using scikit-fem},
  year         = {2025},
  publisher    = {GitHub},
  howpublished = {\url{https://github.com/kevin-tofu/scikit-topt}},
  note         = {Accessed: 2025-04-24}
}