WWU. Efficient Reduced Order Simulation of Pore-Scale Lithium-Ion Battery Models. living knowledge. Mario Ohlberger, Stephan Rave ECMI 2018

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1 M Ü N S T E R Efficient Reduced Order Simulation of Pore-Scale Lithium-Ion Battery Models Mario Ohlberger, Stephan Rave living knowledge ECMI 2018 Budapest June 20, 2018

2 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 2 /18 The MULTIBAT Project Experimental Data Mathematical Modeling Multiscale Numerics Model Reduction Integration Validation Understand degradation processes in rechargeable Li-Ion Batteries through mathematical modeling and simulation. Focus: Li-Plating.

3 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 3 /18 Challenge Li-plating is initiated at interface between active particles and electrolyte. Need pore-scale models which resolve active particle geometry. Huge nonlinear discretizations. Cannot be solved at cell scale on current hardware. Parameter studies extremely expensive, even on small domains. Figure: Simulation of half-cell geometry with plated lithium (green) on 65.6µm 44µm 44µm domain.

4 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 4 /18 Stochastic Structure Modeling [Feinauer, Schmidt, Westhoff (Ulm)] Visual comparison of 2D and 3D cut-outs of experimental data (left) and simulated (right) shows good agreement.

5 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 5 /18 Microscale Modeling [Hein, Latz (HIU)] Variables: c : Li + concentration φ : electrical potential Electrolyte: c t (De c) = 0 (κ 1 t+ RT 1 c κ φ) = 0 F c Electrodes: c (Ds c) = 0 t (σ φ) = 0 Coupling: Normal fluxes at interfaces given by Butler-Volmer kinetics j inter = 2k ( ) η c ec s sinh 2RT F η = φ s φ e U 0( cs ) c max N inter = 1 F j inter

6 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 6 /18 Microscale Modeling Lithium Plating

7 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 7 /18 Discretization [Iliev, Schmidt, Zausch (Fraunhofer ITWM)] Cell centered Finite Volume discretization on voxel grid + implicit Euler leads to nonlinear equation systems of the form: Full Order Model Find [c µ (n), φ (n) µ ] V h V h =: V such that [ ] ([ ]) 1 t (c(n+1) µ c µ (n) ) c µ (n+1) + A µ 0 φ (n+1) µ = 0, c (0) µ = c µ,0. Numerical fluxes on interfaces = Butler-Volmer fluxes. Newton scheme with algebraic multigrid solver. Implemented by Fraunhofer ITWM in. µ P indicates dependence on model parameters (e.g. temperature T, charge rate).

8 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 8 /18 Reduced Basis Approximation [Ohlberger, R (Münster)] Reduced Order Model Find [ c µ (n) (n), φ µ ] Ṽc Ṽφ =: Ṽ solving projected equation [ ] 1 t ( c(n+1) µ c µ (n) ) + { ]) } (n+1) ([ c µ PṼ A µ 0 φ (n+1) = 0, c µ (0) = PṼc (c µ,0 ). µ Basis generation: POD of a priori selected solution trajectories, separately for c and φ (different scales). POD (=Proper Orthogonal Decomposition): truncated singular value decomposition of snapshot matrix (a.k.a. principal component analysis) More advanced: Use greedy search to select snapshot parameters.

9 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 9 /18 Empirical Operator Interpolation (a.k.a. DEIM, EIM) Problem: Still expensive to evaluate Solution: Use locality of finite volume operators: PṼ A µ : Ṽ V Ṽ. to evaluate M DOFs of A µ([c, φ]) wee need M C M DOFs of [c, φ]. Approximate where A µ I M [A µ] := I M ÃM,µ R M, R M : V R M Ã M,µ : R M R M I M : R M V restriction to M DOFs needed for evaluation A µ restricted to M interpolation DOFs linear combination with interpolation basis

10 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 10 /18 Empirical Operator Interpolation (2) Empirical Operator Interpolation Given M interpolation DOFs (magic points) and corresponding interpolation basis, approximate: A µ I M [A µ] := I M Ã M,µ R M. Basis Generation: Compute operator evaluations on solution snapshots (including Newton stages). Iteratively extend interpolation basis with worst-approximated evaluation. Choose new interpolation DOF where new vector is maximal (EI-GREEDY). Interpolate Butler-Volmer part of A µ and 1/c c separately (φ-part of A µ vanishes for solutions). More advanced: Build RB and interpolation basis simultaneously using error estimator to select snapshots (POD-EI-GREEDY).

11 Full Reduction WWU M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 11 /18 Reduced Order Model with EI Find [ c µ (n) (n), φ µ ] Ṽc Ṽφ = Ṽ solving projected equation [ ] ]) 1 t ( c(n+1) µ c µ (n) { } (n+1) ) ([ c µ + (PṼ I M ) Ã M,µ R M 0 φ (n+1) = 0, c µ (0) = PṼc (c µ,0 ). µ Offline/Online decomposition Precompute the linear operators PṼ I M and R M w.r.t. basis of Ṽ. Effort to evaluate (PṼ I M ) ÃM,µ R M w.r.t. this basis: O(NM) + O(M) + O(CMN), where N := dim Ṽ.

12 Numerical Results Model: Half-cell with plated Li µ = discharge current DOFs Reduction: Snapshots: 3 N = M = Rel. err.: < Timings: Full model: 15.5h Projection: 14h Red. model: 8m Speedup: 120 WWU M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 12 /18 cell voltage / V avg. Li concentr. / kmol/m transferred charge / nah na 79.6 na na na na full model red. model Figure: Validation of reduced order model output for random discharge currents; solid lines: full order model, markers: reduced order model.

13 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 13 /18 Semi-Heuristic Error Estimate [Hain, Ohlberger, Radic, Urban, 2018] Idea: Use extended ROM as surrogate for FOM solution: Saturation Assumtion: [ĉ n µ, ˆϕ(n) µ ] ˆV c ˆV ϕ Ṽ c Ṽ ϕ ĉ (n) µ c (n) µ Θ c (n) µ c (n) µ, ˆϕ (n) µ ϕ (n) µ Θ ϕ (n) µ ϕ (n) µ Error Estimate: c µ (n) c µ (n) 1 1 Θ c(n) µ ĉ µ (n), ϕ µ (n) ϕ (n) µ 1 1 Θ ϕ(n) µ ˆϕ (n) µ ˆN N = Θ := 0 max rel overest: 1.08 / 3.46 relative error min rel underest: / discharge current / na c ϕ

14 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 14 /18 Outlook: Localized RB Approximation 10 0 rel. error c N 0 1,000 2,000 M 10 0 rel. error φ N 0 1,000 2,000 M Figure: First experiments on small battery geometry; left: local RB dimensions, right: error for varying N, M; top: Li-concentration c, bottom: electrical potential φ.

15 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 15 /18 Software Interfaces in MULTIBAT

16 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 15 /18 Software Interfaces in MULTIBAT Interfaces allow us to: easily exchange Dune solver with. independently develop MOR algorithms. easily apply MOR algorithms to updated models in. reuse MOR algorithms for other problems.

17 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 16 /18 pymor Model Order Reduction with Python dealii model() fenics model() dune model() PDE solver deal.ii FEniCS DUNE... pymor-deal.ii dolfin dune-pymor d, prod =... operator, rhs LincombOperator Discretization pymor example.py U = d.solve(mu) d.visualize(u) apply2 Operator apply2 Operator... product Operator VectorArray axpy scal U = d.solve(mu) apply inverse apply2 greedy(d,...) Greedy reductor(d, basis) Reductor Gram-Schmidt User Code gram schmidt basis extension(basis, U, prod) Generic algorithms... Quick prototyping with Python. Seamless integration with high-performance PDE solvers. Out of box MPI support for reduction algs. and PDE solvers. BSD-licensed, fork us on Github!

18 pymor Some More Applications Linear elasticity (deal.ii) Nonlinear diffusion (FEniCS) Free boundary problem (NGSolve) Localized RB Multiscale Method (DUNE) MPI distributed transport problem (DUNE) Your problem here!

19 M Ü N S T E R Reduction of Pore-Scale Lithium-Ion Battery Models 18 /18 Thank you for your attention! MULTIBAT: Unified Workflow for fast electrochemical 3D simulations of lithium-ion cells combining virtual stochastic microstructures, electrochemical degradation models and model order reduction J. Comp. Sci., Localized Reduced Basis Approximation of a Nonlinear Finite Volume Battery Model with Resolved Electrode Geometry. In: Model Reduction of Parametrized Systems, Springer, pymor Generic Algorithms and Interfaces for Model Order Reduction SIAM J. Sci. Comput., 38(5), My homepage