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Optimization of the cut configuration for skin grafts

Helmut Harbrecht, Viacheslav Karnaev

TL;DR

The paper addresses how to optimally orient predefined skin-cut patterns to improve stretch outcomes by formulating a 2D linear-elastic model with holes representing cuts. It derives Hadamard shape gradients for three objectives—$\mathcal{C}$ (compliance), $\mathcal{M}$ (Lp von Mises stress norm), and $\mathcal{A}$ (deformed area)—and solves the resulting shape-optimization problems using gradient descent, augmented by a genetic algorithm to overcome multiple local minima. Existence of solutions is established on the compact configuration set, and the authors provide detailed adjoint-based sensitivity formulas and a concrete numerical implementation (FreeFem++), including a structured cell-block discretization. Numerical experiments show intuitive and interpretable patterns: horizontal cuts minimize compliance in uniaxial tension, zigzag cuts reduce stress, and mixed patterns emerge under biaxial loading, with auxetic configurations arising in some optima. The work lays a rigorous, computationally feasible framework for designing skin-graft cut layouts and suggests natural extensions to nonlinear hyperelastic models and more general geometries.

Abstract

The subject of this work is the problem of optimizing the configuration of cuts for skin grafting in order to improve the efficiency of the procedure. We consider the optimization problem in the framework of a linear elasticity model. We choose three mechanical measures that define optimality via related objective functionals: the compliance, the \(L^p\)-norm of the von Mises stress, and the area of the stretched skin. We provide a proof of the existence of solutions for each problem, but we cannot claim uniqueness. We compute the gradient of the objectives with respect to the cut configuration using shape calculus concepts. To solve the problem numerically, we use the gradient descent method, which performs well under uniaxial stretching. However, in more complex cases, such as multidirectional stretching, its effectiveness is limited due to low sensitivity of the functionals. To avoid this difficulty, we use a combination of the genetic algorithm and the gradient descent method, which leads to a significant improvement in the results.

Optimization of the cut configuration for skin grafts

TL;DR

The paper addresses how to optimally orient predefined skin-cut patterns to improve stretch outcomes by formulating a 2D linear-elastic model with holes representing cuts. It derives Hadamard shape gradients for three objectives— (compliance), (Lp von Mises stress norm), and (deformed area)—and solves the resulting shape-optimization problems using gradient descent, augmented by a genetic algorithm to overcome multiple local minima. Existence of solutions is established on the compact configuration set, and the authors provide detailed adjoint-based sensitivity formulas and a concrete numerical implementation (FreeFem++), including a structured cell-block discretization. Numerical experiments show intuitive and interpretable patterns: horizontal cuts minimize compliance in uniaxial tension, zigzag cuts reduce stress, and mixed patterns emerge under biaxial loading, with auxetic configurations arising in some optima. The work lays a rigorous, computationally feasible framework for designing skin-graft cut layouts and suggests natural extensions to nonlinear hyperelastic models and more general geometries.

Abstract

The subject of this work is the problem of optimizing the configuration of cuts for skin grafting in order to improve the efficiency of the procedure. We consider the optimization problem in the framework of a linear elasticity model. We choose three mechanical measures that define optimality via related objective functionals: the compliance, the -norm of the von Mises stress, and the area of the stretched skin. We provide a proof of the existence of solutions for each problem, but we cannot claim uniqueness. We compute the gradient of the objectives with respect to the cut configuration using shape calculus concepts. To solve the problem numerically, we use the gradient descent method, which performs well under uniaxial stretching. However, in more complex cases, such as multidirectional stretching, its effectiveness is limited due to low sensitivity of the functionals. To avoid this difficulty, we use a combination of the genetic algorithm and the gradient descent method, which leads to a significant improvement in the results.
Paper Structure (14 sections, 6 theorems, 52 equations, 8 figures)

This paper contains 14 sections, 6 theorems, 52 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Notation
  4. Governing equations
  5. Optimization problem
  6. Sensitivity analysis
  7. Fundamentals of shape calculus
  8. Shape derivatives
  9. Numerical realization
  10. Optimization algorithm
  11. Computational setup
  12. Numerical results: Stretching in one axial direction
  13. Numerical results: Stretching in two axial directions
  14. Conclusion

Key Result

Lemma 2.1

Assume $\Omega_{\boldsymbol\alpha^*}\in\boldsymbol O_{2\pi}$ and consider the sequence $\{\Omega_{\boldsymbol\alpha_n}\}_{n\in\mathbb N}\subset\boldsymbol O_{2\pi}$. Let $\boldsymbol u_n := \boldsymbol u_{\boldsymbol\alpha_n}$ for all $n\in\mathbb{N}$ and $\boldsymbol u^* := \boldsymbol u_{\boldsym

Figures (8)

  • Figure 2.1: The model for the elastic body with cuts which are represented by thin elliptical holes.
  • Figure 3.1: Variation $\Omega_{\boldsymbol\alpha_{\epsilon_i}}$of a shape $\Omega_{\boldsymbol\alpha}$ according to a deformation field $\boldsymbol\theta_1$.
  • Figure 4.1: Skin grafting model with a random configuration of cuts. The finite element mesh consists of roughly 165,000 finite elements.
  • Figure 4.2: Stretched skin for the random configuration of cuts: (a) single axis stretching, (b) bi-axial stretching.
  • Figure 4.3: Stretching in one axial direction. Final cut configuration (left), deformed skin (middle), and convergence history (right): first row -- compliance $\mathop{\mathrm{\mathcal{C}}}\nolimits(\Omega_{\boldsymbol\alpha})$, second row -- $L^5$-norm of the von Mises stress $\mathop{\mathrm{\mathcal{M}}}\nolimits(\Omega_{\boldsymbol\alpha})$, third row -- area of the deformed body $\mathop{\mathrm{\mathcal{A}}}\nolimits(\Omega_{\boldsymbol\alpha})$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • Proposition 3.5
  • ...and 3 more