A-posteriori error analysis for lithium-ion concentrations in batteries utilizing the reduced-basis method

Figures & data

Algorithm 4.1 (POD algorithm)

Require: ${\mathrm{C}}^h(\mu )\in {\mathbb{R}}^{\mathcal{N}\times K}$, $0\ll {\epsilon }^{\mathrm{P}\mathrm{O}\mathrm{D}}<1$. $\mathrm{W}\in {\mathbb{R}}^{\mathcal{N}\times \mathcal{N}}$ with W = W^T and W ≻ 0;

1: d $\leftarrow$ $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\hspace{0.17em}{\mathrm{C}}^h(\mu )$;

2: ${\left\{\left({\psi }_i,{\lambda }_i\right)\right\}}_{i=1}^d\leftarrow$ solve (11);

3: $\mathrm{\ell }\leftarrow 1$

4: $\mathcal{E}(\mathrm{\ell })\leftarrow {\lambda }_1$;

5: While $\mathcal{E}(\mathrm{\ell })<{\epsilon }^{\mathrm{P}\mathrm{O}\mathrm{D}}(\mu )$ and $\mathrm{\ell }<d$ do

6: $\mathcal{E}(\mathrm{\ell }+1)\leftarrow \mathcal{E}(\mathrm{\ell })+{\lambda }_{\mathrm{\ell }+1}$;

7: $\mathrm{\ell }\leftarrow \mathrm{\ell }+1$;

8: end while

9: return POD basis ${\left\{{\psi }_i\right\}}_{i=1}^{\mathrm{\ell }}$, eigenvalues ${\left\{{\lambda }_i\right\}}_{i=1}^{\mathrm{\ell }}$ and error $\mathcal{E}(\mathrm{\ell })$;

Algorithm 4.2 (POD-greedy algorithm)

Require: ${\mathcal{D}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}=\left\{{\mu }_1,\dots ,{\mu }_p\right\}\subset \mathcal{D}$, $0<{\epsilon }^{\mathrm{g}\mathrm{r}}\ll 1$, $0\ll {\epsilon }^{\mathrm{P}\mathrm{O}\mathrm{D}}<1$, and $k_{\mathrm{m}\mathrm{a}\mathrm{x}}\in \mathbb{N}$;

1: Choose ${\mu }^{\left(1\right)}\in {\mathcal{D}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}$;

2: ${\mathrm{C}}^h\left({\mu }^{(1)}\right)$ ← solve (5);

3: ${\psi }_1,\dots ,{\psi }_{{\mathrm{\ell }}_1}$ ← call Algorithm 4.2.1 with inputs ${\mathrm{C}}^h\left({\mu }^{(1)}\right)$ and ε^POD;

4: $N\leftarrow {\mathrm{\ell }}_1$, ${\xi }_i$ ← ${\psi }_i$ ($1\le i\le N$) and k ← 1;

5: while $k<k_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and ${max}_{\mu \in {\mathcal{D}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}}{\mathrm{\Delta }}^N(\mu )>{\epsilon }^{\mathrm{g}\mathrm{r}}$ do

6: ${\mu }^{(k+1)}\leftarrow argma{x}_{\mu \in {\mathcal{D}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}}{\mathrm{\Delta }}^N(\mu )$;

7: ${\mathrm{C}}^h\left({\mu }^{(k+1)}\right)$ ← solve (5);

8: ${\psi }_1,\dots ,{\psi }_{{\mathrm{\ell }}_{k+1}}$ ← call Algorithm 4.2.1 with inputs ${\mathrm{C}}^h\left({\mu }^{(k+1)}\right)-{\mathcal{P}}^{\mathrm{\ell }}{\mathrm{C}}^h\left({\mu }^{(k+1)}\right)$ and ε^POD;

9: ${\xi }_1,\dots ,{\xi }_{N+{\mathrm{\ell }}_{k+1}}$ ← using Gram–Schmidt with ${\xi }_1,\dots ,{\xi }_N,{\psi }_1,\dots ,{\psi }_{{\mathrm{\ell }}_{k+1}}$;

10: end while

11: return reduced-basis $\mathrm{\Xi }=\left[{\xi }_1|\dots |\hspace{0.17em}{\xi }_N\right]$;

Table 1. Parameters for the RB model.

$\mathcal{N}$	300
h	0.03
k	0.1
c0	55
cmax	60
K	20

Figure 1. Decay of the error and its estimates for the four runs: run 1 (up left), run 2 (up right), run 3 (bottom left) and run 4 (bottom right).

Table 2. Results for the four different runs: computational offline time; computational time for the computation of a FV solution; computational time for the computation of a RB solution, speed-up, number of basis vectors, value of the estimator after the last POD-greedy step and value of the error after the last POD-greedy step.

	Run 1	Run 2	Run 3	Run 4
toff [s]	7.74 × 10^1	2.97 × 10^2	2.59 × 10^2	5.76 × 10^1
tFVM [s]	1.24	1.24	1.24	1.24
tRBM [s]	5.58 × 10^−^3	5.94 × 10^−^3	6.89 × 10^−^3	6.73 × 10^−^3
Speed-up	222	208	180	184
N	14	13	13	13
Est.	4.81 × 10^−^7	8.66 × 10^−^5	8.66 × 10^−^5	4.50 × 10^−^7
Err.	4.81 × 10^−^7	8.85 × 10^−^7	8.85 × 10^−^7	8.85 × 10^−^7

Table 3. Parameter estimation – test 1: required parameter μ^d, initial parameter μ^init, estimated parameter μ^comp, stopping criteria, computational time for the optimization process, number of iterations, number of function evaluations, residual.

	FVM	ROM
μ^d	(0.1,0.005)
μ^init	(2.0,0.09)
μ^comp	(2.0000,0.0050)	(2.0000,0.0050)
Stopping criteria	Local minimum found: first-order optimality measure TolFun	Local minimum found: first-order optimality measure TolFun
topt [s]	21.01	2.51
Iterations	10	10
Funct. eval.	33	33
Residual	3.57 × 10^−^17	3.57 × 10^−^17

Table 4. Parameter estimation – test 2: required parameter μ^d, initial parameter μ^init, estimated parameter μ^comp, stopping criteria, computational time for the optimization process, number of iterations, number of function evaluations, residual.

	FVM	ROM
μ^d	(2.0,0.09)
μ^init	(0.1,0.005)
μ^comp	(0.1000,0.0912)	(0.1000,0.0912)
Stopping criteria	Local minimum found: first-order optimality measure TolFun	Local minimum found: first-order optimality measure TolFun
topt [s]	19.42	2.46
Iterations	11	11
Funct. eval.	36	36
Residual	5.46 × 10^−^14	5.46 × 10^−^14

Figure 2. SoC in dependency on μ₁ (left) and μ₂ (right).