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Maximization of Supercapacitor Storage via Topology Optimization of Electrode Structures

Jiajie Li, Xiang Ji, Shenggao Zhou, Shengfeng Zhu

TL;DR

The paper tackles maximizing energy storage in supercapacitors by designing electrode topology through a phase-field topology-optimization framework constrained by a modified steady-state Poisson–Nernst–Planck system. It provides a rigorous existence theory for minimizers via the direct method, derives adjoint-based sensitivities for gradient-based optimization, and implements a stabilized gradient-flow scheme to find local optima. The authors develop a finite-element discretization combined with a Gummel-style solver for the state equations and present an efficient optimization loop that updates the design field while controlling volume. Numerical experiments in 2D and 3D demonstrate porous, high-interface-area electrode structures that substantially increase total charge storage, highlighting the practical potential for improving supercapacitor performance through topology optimization.

Abstract

As widely used electrochemical storage devices, supercapacitors deliver higher power density than batteries, but suffer from significantly lower energy density. In this work, we propose a topology optimization model for electrode structure to maximize energy storage in supercapacitors. The existence of minimizers to the resulting optimal control problem, which is constrained by a modified steady-state Poisson--Nernst--Planck system describing ionic electrodiffusion, has been theoretically established by using the direct method in the calculus of variation. Sensitivity analysis of the topology optimization model is performed to derive variational derivatives and corresponding adjoint equations. A gradient flow formulation discretized by a stabilized semi-implicit scheme is developed to solve the resulting topology optimization problem. Extensive numerical experiments present various porous electrode structures that own large area of electrode-electrolyte interface, demonstrating the effectiveness and robustness of the proposed topology optimization model and corresponding algorithm.

Maximization of Supercapacitor Storage via Topology Optimization of Electrode Structures

TL;DR

The paper tackles maximizing energy storage in supercapacitors by designing electrode topology through a phase-field topology-optimization framework constrained by a modified steady-state Poisson–Nernst–Planck system. It provides a rigorous existence theory for minimizers via the direct method, derives adjoint-based sensitivities for gradient-based optimization, and implements a stabilized gradient-flow scheme to find local optima. The authors develop a finite-element discretization combined with a Gummel-style solver for the state equations and present an efficient optimization loop that updates the design field while controlling volume. Numerical experiments in 2D and 3D demonstrate porous, high-interface-area electrode structures that substantially increase total charge storage, highlighting the practical potential for improving supercapacitor performance through topology optimization.

Abstract

As widely used electrochemical storage devices, supercapacitors deliver higher power density than batteries, but suffer from significantly lower energy density. In this work, we propose a topology optimization model for electrode structure to maximize energy storage in supercapacitors. The existence of minimizers to the resulting optimal control problem, which is constrained by a modified steady-state Poisson--Nernst--Planck system describing ionic electrodiffusion, has been theoretically established by using the direct method in the calculus of variation. Sensitivity analysis of the topology optimization model is performed to derive variational derivatives and corresponding adjoint equations. A gradient flow formulation discretized by a stabilized semi-implicit scheme is developed to solve the resulting topology optimization problem. Extensive numerical experiments present various porous electrode structures that own large area of electrode-electrolyte interface, demonstrating the effectiveness and robustness of the proposed topology optimization model and corresponding algorithm.

Paper Structure

This paper contains 13 sections, 4 theorems, 68 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model setting
  3. Phase field model
  4. Existence results
  5. Optimization method
  6. Sensitivity analysis
  7. Gradient flow minimization
  8. Numerical methods
  9. Finite-element discretization of state and adjoint
  10. Optimization algorithm
  11. Numerical examples
  12. Maximize total charge storage
  13. Conclusion

Key Result

Lemma 2.2

Let Assumption assumptionPNP hold, and suppose the boundary data $g$ is sufficiently small. Given $\phi\in \mathcal{A}$, the problem PNPslot admits a unique solution $(\rho_1,\rho_2,\psi)\in [H^2(\Omega)\cap L^{\infty}(\Omega)]^3$, satisfying the $L^{\infty}$-estimates

Figures (10)

  • Figure 1: Schematic plot of a supercapacitor model on a computational domain $\Omega=\Omega_1 \cup \Omega_2$, where $\Omega_1$ shown in gray represents the electrolyte domain and $\Omega_2$ shown in white is the design electrode domain.
  • Figure 2: A rectangular design domain for Example 1 (left) and an annulus design domain for Example 2 (right).
  • Figure 3: Initial and optimal design for Example 2 with $m = 4, 6, 10$ from left to right.
  • Figure 4: Concentrations of cations (upper row) and anions (lower row) for optimized configurations in Example 1 ($m=4, 6, 10$ from left to right).
  • Figure 5: Convergence histories of energy (left) and volume error (right) for Example 1.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Lemma 2.2
  • proof
  • Theorem 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof