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Formulation and analysis of a DPG discretization for a simplified electrochemical model

Jaime Mora-Paz

TL;DR

The paper analyzes a simplified electrochemical model for lithium-ion batteries by reducing the multiphysics system to two linear elliptic boundary-value problems for concentration $c$ and electric potential $\phi$, coupled via Butler-Volmer kinetics and a Robin boundary condition. It develops both classical weak and mixed variational formulations and then introduces broken variational formulations on a mesh, establishing well-posedness under mild assumptions. The Discontinuous Petrov-Galerkin (DPG) discretization is formulated with optimal test functions and Fortin-enriched test spaces to guarantee discrete stability and convergence, achieving an order of convergence $\mathcal{O}(h^p)$. The results provide a rigorous foundation for applying DPG to battery multiphysics problems and point to benefits in adaptive mesh refinement and efficient solver design for more complex, time-dependent, and nonlinear configurations.

Abstract

We present a simplified model consisting on two linear elliptic boundary-value problems that represent a single step and single fixed-point iteration in an electrochemical battery model. The main variables are the concentration and the electric potential, whose equation is assigned a Robin BC with a very important physical interpretation. The solvability of both equations is studied in different funcional settings, to finally prove the well-posedness of a broken mixed variational formulation. The latter formulation opens the opportunity of performing discretization and numerical solution via the Discontinuous Petrov-Galerkin (DPG) method, which guarantees discrete stability thanks to optimal test functions. With only the usual assumptions on the data and the discretization, we show that the method herein proposed is convergent. This analytical effort complements other recent works on batttery multiphysics, that have been more modeling and computing oriented, and establishes the DPG approach as a valuable tool to contribute toward an efficient analysis and design of batteries.

Formulation and analysis of a DPG discretization for a simplified electrochemical model

TL;DR

The paper analyzes a simplified electrochemical model for lithium-ion batteries by reducing the multiphysics system to two linear elliptic boundary-value problems for concentration and electric potential , coupled via Butler-Volmer kinetics and a Robin boundary condition. It develops both classical weak and mixed variational formulations and then introduces broken variational formulations on a mesh, establishing well-posedness under mild assumptions. The Discontinuous Petrov-Galerkin (DPG) discretization is formulated with optimal test functions and Fortin-enriched test spaces to guarantee discrete stability and convergence, achieving an order of convergence . The results provide a rigorous foundation for applying DPG to battery multiphysics problems and point to benefits in adaptive mesh refinement and efficient solver design for more complex, time-dependent, and nonlinear configurations.

Abstract

We present a simplified model consisting on two linear elliptic boundary-value problems that represent a single step and single fixed-point iteration in an electrochemical battery model. The main variables are the concentration and the electric potential, whose equation is assigned a Robin BC with a very important physical interpretation. The solvability of both equations is studied in different funcional settings, to finally prove the well-posedness of a broken mixed variational formulation. The latter formulation opens the opportunity of performing discretization and numerical solution via the Discontinuous Petrov-Galerkin (DPG) method, which guarantees discrete stability thanks to optimal test functions. With only the usual assumptions on the data and the discretization, we show that the method herein proposed is convergent. This analytical effort complements other recent works on batttery multiphysics, that have been more modeling and computing oriented, and establishes the DPG approach as a valuable tool to contribute toward an efficient analysis and design of batteries.
Paper Structure (30 sections, 7 theorems, 59 equations, 1 figure)

This paper contains 30 sections, 7 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Butler-Volmer Reaction Kinetics
  4. Boundary and interface conditions in the original electrochemical model
  5. Boundary, interface and initial conditions for concentration in the anode
  6. Boundary, interface and initial conditions for concentration in the cathode
  7. Boundary, interface and initial conditions for concentration in the electrolyte
  8. Boundary and interface conditions for potential in the anode
  9. Boundary and interface conditions for potential in the cathode
  10. Boundary and interface conditions for potential in the electrolyte
  11. Connection with the model potential problem
  12. General assumptions and notation
  13. Goal, scope and outline of this paper
  14. Well-posedness of the continuous problem
  15. Classical weak formulation
  16. ...and 15 more sections

Key Result

Lemma 1

Let Assumption 1 hold. Then, problem eq:conce_classical_weak is well-posed and for its unique solution $c\in H^1(\Omega)$ there holds

Figures (1)

  • Figure 1: Battery diagram showing boundary portions that are referenced in the specification of boundary conditions.

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Corollary 1
  • proof
  • ...and 5 more