Multi-Objective Optimization with Pymoo / PHV_optimizer /

This repository contains a multi-objective optimization script for prosthetic heart valve leaflet design improvement using the pymoo library. The custom problem is described by parameters, multiple objectives, and constraints. The algorithm used is NSGA-II.

Table of Contents

• Key Components
• Problem Definition
• Optimization Process
• Visualization Functions
• Dependencies
• TODO

Key Components

• optimizer.py: Contains an optimization algorithm based on the class Problem. Custom class Problem (derived from pymoo's ElementwiseProblem) defines objectives, parameters, and constraints, along with the _evaluate method to calculate them.
• test_problem.py: Contains optimization_problem_test, which evaluates a specific problem ("welded_beam") based on given parameters and returns objective and constraint results.
• visualization.py: Contains functions to create various plots, such as Pareto fronts, objective convergence, hypervolume, scatter plots, and parallel coordinates.

Problem Definition

The custom problem class Problem is defined with:

• Parameters: Dictionary of parameter names and their lower and upper bounds (e.g., 'param1': (0.01, 10.0)).
• Objectives: List of objective names to optimize (e.g., ['objective1', 'objective2']).
• Constraints: List of constraint names representing conditions to meet (e.g., ['constraint1', 'constraint2', 'constraint3', 'constraint4']):
  • Constraints are defined numerically, where each constraint function outputs a value representing the degree of violation.
  • A constraint value less than or equal to zero (<= 0) indicates that the constraint is satisfied (non-violated).
  • A positive constraint value (> 0) indicates a violation.

Optimization Process

The optimization is using the following components:

• Algorithm Initialization: Defines population size, crossover, mutation, and other parameters.
• Termination Criteria: Conditions for ending the optimization, such as maximum generations and evaluations.
• Results Extraction: A function extract_optimization_results that extracts optimization results into DataFrames for analysis and storage.

Visualization Functions

The repository contains several visualization functions to analyze the optimization results:

• Pareto Front: Shows the Pareto-optimal solutions.
• Objective Convergence: Displays how objectives improve over generations.
• Hypervolume: Shows convergence by hypervolume.
• Scatter Plots: Plots objectives against parameters and constraints.
• Parallel Coordinates: Plots multiple variables on parallel axes to understand relationships.

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Mathematical Basis of the Unfolding Algorithm

1. Developability and Gaussian Curvature

For a smooth surface $S \subset \mathbb{R}^3$ the Gaussian curvature is the product of the principal curvatures:

$$ K(p)=\kappa_1(p),\kappa_2(p). $$

• Gauss’ Theorema Egregium. Gaussian curvature is invariant under isometric deformations. Hence a surface is developable (can be unfolded to the plane without stretching) iff $K\equiv0$.

2. “Almost‑Developable” Engineering Approximation

In practice we admit small curvatures $|K|\le |K|_{\max}$. Let $a$ be the radius of a small circular patch. For sufficiently small $a$

$$ \frac{\delta A}{A};\approx;\frac{K,a^{2}}{6}, $$

where $\delta A/A$ is the relative area change when the patch is flattened.

Practical design formula

Using equation above and setting the characteristic linear size $L=2a$ (e.g. diameter of leaflet apparatus),

$$ |K|_\max = \frac{24(\delta{A}/A) _ \max}{L^2} $$

This equation underlies gaussian_tolerance_from_area_strain, converting an admissible area strain percentage into a tolerance for $|K|$.

3. Engineering Interpretation of Tolerances

$(\delta{A}/A) _ \max$	abs(K_max) at L=12 mm, mm^-2	Equivalent radius R=sqrt(1/abs(K))	Typical linear strain
1%(0.01)	$$1.4\times10^{-3}$$	31.6mm	≈0.5%
3%(0.03)	$$4.2\times10^{-3}$$	17.3mm	≈1.5%
6%(0.06)	$$8.3\times10^{-3}$$	11.0mm	≈3%
10%(0.10)	$$1.4\times10^{-2}$$	8.4mm	≈5%

*Linear strain is approximated by $\tfrac12,\delta A/A$ for small deformations.

• Polymeric leaflets (e.g. Formlabs Elastic50A, ShoreA50) sustain ≥100% elastic elongation; thus $|K|\le10^{-2},\text{mm}^{-2}$ (≤6% area strain) is mechanically safe and geometrically accurate.
• A tolerance of $10^{-1},\text{mm}^{-2}$ corresponds to 60% area change and is acceptable only when large material draw‑in is permissible.

4. Tolerance Selection Algorithm in Code

# Pseudocode (see adaptive_tolerance in /gaussian_curvature_v2.py)

tol = gaussian_tolerance_from_area_strain(
        diameter_mm=L,
        max_area_strain=desired_area_strain)

result = evaluate_developability(mesh, tol)

adaptive_tolerance iterates over a set of allowable $\delta A/A$ values (1%,3%,6%,10%) and returns the smallest $|K|_{\max}$ that passes evaluate_developability. This yields the least‑distorting geometry consistent with manufacturability.

Dependencies

Necessary dependencies are listed in requirements.txt.

Configuration file

Configure optimization with .yaml files located in ./configuration folder. You can run script with input keys -c config_name or --config config_name

List of parameters:

• parameters: - mandatory

  • param 1: [min, max]
  • param 2: [min, max]
  • .....
  • param N: [min, max]

• objectives: - mandatory

  • - objective 1
  • - objective 2
  • .....
  • - objective N

• constrains: - optional

  • - constr 1
  • - constr 2
  • .....
  • - constr N

• problem_definition:

  • name: XXXX - XXXX - name of your problem. Optional.
  • position: XXXX - mandatory! XXXX may be in [ao, mitr]. This affect to boundary conditions
  • problem_name: XXXX - mandatory! XXXX may be [leaflet_contact, leaflet_single, test]
  • DIA: XX - XX is diameter of lealet apparatus. Measured in mm.
  • Lift: XX - how far leaflet would be lifted to simulate frame. Fully optional, by default assumed by 0 mm
  • SEC: XX - sector of circle occupied by one leaflet
  • mesh_step: XX - size of the mesh. Used in leaflet points generation. Default value - 0.35. More value - coarser mesh
  • material: - mandatory! Following part of yaml defining material properties
    • material_definition_type: XX - mandatory! Define type of used material model: [linear, polynomial, ortho
    • material_name: XX - just name of used material, small QoL
    • poisson_coeff: XX - Poisson coefficient. Used with linear or polynomial model
    • Dens: XX - mandatory! Density of material. By default - 1e-9 tonn/mm
    • s_lim: XX - UTS for material
    • material_csv_path: XX - name of the csv-file located in ./configuration folder. Used with polynomial material model. Format - stress, strain
    • ortho_coeffs_E: - this is array of Young's modulus for ortho material model. Using cylindrical coodrinate system in this point
      • - E1 - Young's modulus in radial direction
      • - E1 - Young's modulus in circumferential direction
      • - E1 - Young's modulus in Z direction
    • ortho_coeffs_poisson: - this is array of Poisson coefficients for ortho material model. Using cylindrical coodrinate system in this point
      • - p1 - Poisson coefficients in radial direction
      • - p2 - Poisson coefficients in circumferential direction
      • - p3 - Poisson coefficients in Z direction
  • Abaqus: - FEA related part of configuration file
    • abq_cpus: XX - how much cpus were used for FEA
    • tangent_behavior: XX - tangential stiffness used in contact problem leaflet_contact
    • normal_behavior: XX - normal stiffness used in contact problem leaflet_contact
  • optimizer: - optimizer-related parameters. everything here is mandatory!. Read PyMoo manuals
    • pop_size: XX
    • offsprings: XX
    • crossover_chance: XX
    • mutation_chance: XX
    • crossover_eta: XX
    • mutation_eta: XX
    • termination_parameters:
      • xtol: XX
      • cvtol: XX
      • ftol: XX
      • period: XX
      • n_max_gen: XX
      • n_max_evals: XX

  You can add additional Hydra-related parameters below.

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TODO

• Try R-NSGA-II.
• Try NSGA-III.
• Try U-NSGA-III.
• Try R-NSGA-III.
• Try MOEA/D.

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For more information about pymoo and its multi-objective optimization features, refer to the [pymoo documentation] (https://pymoo.org/documentation.html).