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Shape Optimization of Supercapacitor Electrode to Maximize Charge Storage

Jiajie Li, Shenggao Zhou, Shengfeng Zhu

TL;DR

This work presents a shape optimization framework to maximize charge storage in a supercapacitor by modifying the electrode–electrolyte interface within a steady-state Poisson–Nernst–Planck model. It derives an Eulerian shape derivative via the velocity method, couples state and adjoint systems solved with a Gummel fixed-point solver, and advances domain evolution through $H^1$ and CT-H(sym) gradient flows with volume and perimeter regularization. The approach is validated through comprehensive 2D and 3D numerical experiments across square, irregular, and porous geometries, demonstrating substantial increases in stored charge and smoother electrode morphologies. While the method maintains fixed topology (no hole creation/destruction), it provides a robust computational tool for electrode design and lays groundwork for future topology-optimization extensions to further boost surface area and charge storage efficiency.

Abstract

We build a new mathematical model of shape optimization for maximizing ionic concentration governed by the multi-physical coupling steady-state Poisson-Nernst-Planck system. Shape sensitivity analysis is performed to obtain the Eulerian derivative of the cost functional. The Gummel fixed-point method with inverse harmonic averaging technique on exponential coefficient is used to solve efficiently the steady-state Poisson-Nernst-Planck system. Various numerical results using a shape gradient algorithm in 2d and 3d are presented.

Shape Optimization of Supercapacitor Electrode to Maximize Charge Storage

TL;DR

This work presents a shape optimization framework to maximize charge storage in a supercapacitor by modifying the electrode–electrolyte interface within a steady-state Poisson–Nernst–Planck model. It derives an Eulerian shape derivative via the velocity method, couples state and adjoint systems solved with a Gummel fixed-point solver, and advances domain evolution through and CT-H(sym) gradient flows with volume and perimeter regularization. The approach is validated through comprehensive 2D and 3D numerical experiments across square, irregular, and porous geometries, demonstrating substantial increases in stored charge and smoother electrode morphologies. While the method maintains fixed topology (no hole creation/destruction), it provides a robust computational tool for electrode design and lays groundwork for future topology-optimization extensions to further boost surface area and charge storage efficiency.

Abstract

We build a new mathematical model of shape optimization for maximizing ionic concentration governed by the multi-physical coupling steady-state Poisson-Nernst-Planck system. Shape sensitivity analysis is performed to obtain the Eulerian derivative of the cost functional. The Gummel fixed-point method with inverse harmonic averaging technique on exponential coefficient is used to solve efficiently the steady-state Poisson-Nernst-Planck system. Various numerical results using a shape gradient algorithm in 2d and 3d are presented.
Paper Structure (13 sections, 5 theorems, 82 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 5 theorems, 82 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Shape optimization model
  3. Shape sensitivity analysis
  4. Optimization algorithm
  5. Numerical schemes
  6. Finite-element discretization of state and adjoint
  7. Discrete shape gradient flow
  8. Numerical results
  9. Experiment 1 (Square domain)
  10. Experiment 2 (Irregular domain)
  11. Experiment 3 (Porous domain)
  12. Experiment 4 (Irregular domain in 3d)
  13. Conclusions

Key Result

Proposition 3.3

HaslingerMakinen2003 For a deformation field $\hbox{\boldmath$\theta$}=[\theta_1, \theta_2,\cdots,\theta_d]^{\rm T}\in {W}^{1,\infty}$$(\Omega)^d$, the following results hold where ${\rm Id}$ is an identity matrix and $J_t=\det{({\rm D}T_t)}$ denotes the determinant of the Jacobian.

Figures (9)

  • Figure 1: Schematic illustration of a supercapacitor model with binary electrolytes. The bulk electrolyte domain $\Omega_1$ with boundaries $\Gamma_1$ and $\Gamma_{\text{in}}$ interconnects the domain $\Omega_2$ with a design boundary $\Gamma_2$ that is the electrolyte-electrode interface.
  • Figure 2: Illustration of computational domains: a square domain (left), an irregular domain (middle), and a porous domain (right).
  • Figure 3: Initial domain (upper left) and optimized domain with mesh (upper right), counterion concentration $c_1$ (lower left) and electric potential $\phi$ (lower right) by $H^1$ shape gradient flow for Case 1.1.
  • Figure 4: Optimal mesh computed by CT-H(sym) with boundary type of shape gradient (upper left) and distributed shape gradient (upper right), counterion concentration $c_1$ (middle left) and electric potential $\phi$ (middle right) computed with the boundary type of shape gradient and CT-H(sym) gradient flow for Case 1.2, and convergence histories of objective (lower left) and volume error (lower right) for all Cases 1.1 - 1.3.
  • Figure 5: Initial mesh(upper left), convergence histories of the objective (upper right), counterion concentration $c_1$ (lower left), and electric potential $\phi$ (lower right) at the optimized shape computed by CT-H(sym) gradient flow for Experiment 2.
  • ...and 4 more figures

Theorems & Definitions (13)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Lemma 3.6
  • proof
  • Theorem 3.7
  • ...and 3 more