Piecewise Interpolants

Many battery parameters depend on state of charge, temperature, or both. Piecewise interpolation enables the creation of smooth parameter functions suitable for physics-based models. This guide covers both general piecewise interpolants and specialized OCP interpolants.

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Overview

Piecewise interpolation addresses several needs:

• SOC-dependent parameters: Diffusivity, exchange current density, and other transport properties often vary significantly with lithiation state
• Temperature dependence: Non-Arrhenius behavior requires more flexible functional forms
• OCP functions: Open-circuit potential must be defined across the full stoichiometry range for simulation

The implementation uses smooth heaviside functions to create continuous, differentiable interpolants that work well with differential equation solvers.

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Mathematical Formulation

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Smooth Heaviside Function

Traditional piecewise functions use discontinuous step functions, which cause convergence issues in ODE/DAE solvers. Instead, we use a smooth approximation: $$H_{\text{smooth}}(x; \theta, \epsilon) = \frac{1}{2}\left(1 + \tanh\left(\frac{x - \theta}{2\epsilon}\right)\right)$$ where:

• $x$ is the input variable (e.g., SOC)
• $\theta$ is the threshold value
• $\epsilon$ is the smoothing parameter (larger = smoother transition)

The smooth heaviside is infinitely differentiable ($C^\infty$) and transitions from ~0 for $x \ll \theta$ to ~1 for $x \gg \theta$.

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1D Piecewise Linear Interpolation

For a parameter $p$ varying with $x$, given breakpoints $\{x_0, x_1, \ldots, x_{N-1}\}$ with values $\{p_0, p_1, \ldots, p_{N-1}\}$: $$p(x) = p_0 H_0^-(x) + \sum_{i=0}^{N-2} p_i^{\text{seg}}(x) H_i^{\text{seg}}(x) + p_{N-1} H_{N-1}^+(x)$$ Each linear segment interpolates between adjacent knots: $$p_i^{\text{seg}}(x) = p_i + (p_{i+1} - p_i) \frac{x - x_i}{x_{i+1} - x_i}$$

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Piecewise Parameter Functions

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1D Interpolation

For parameters that vary with a single variable (typically SOC), a PiecewiseInterpolation1D direct entry takes a base parameter name, a list of breakpoint values along the independent variable, and a smoothing parameter. The interpolant reads one parameter per breakpoint (e.g. "Negative particle diffusivity at SOC 0.3 [m2.s-1]") and returns a smooth function of the independent variable.

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2D Interpolation

For parameters varying with two variables (e.g., SOC and temperature), a PiecewiseInterpolation2D direct entry takes breakpoints along both axes and separate smoothing parameters for each.

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Formulations: Knots vs Slopes

Two equivalent parameterizations are available: Both have the same degrees of freedom. Slopes can be converted to knots using SlopesToKnots.

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OCP Interpolants

Open-circuit potential (OCP) interpolants are specialized functions $U(\theta)$ that return equilibrium voltage for a given stoichiometry.

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From Experimental Data

An OCPDataInterpolant calculation builds a smooth interpolant from half-cell OCP measurements. Data should be collected at low C-rates (C/20 or slower) to approximate equilibrium.

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From MSMR Parameters

An OCPMSMRInterpolant calculation evaluates the MSMR model over a voltage range to create an interpolant. This provides thermodynamic consistency and reliable extrapolation.

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Blended MSMR / Experimental OCP

The recommended approach combines experimental data with MSMR extrapolation via OCPDataInterpolantMSMRExtrapolation. The blending function smoothly transitions between data and MSMR: $$U(x) = w(x) \cdot U_{\text{data}}(x) + (1 - w(x)) \cdot U_{\text{MSMR}}(x)$$ where $w(x)$ is a bump function that equals ~1 inside the data range and ~0 outside.

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The Inaccessible Lithium Problem

Experimental OCP data never covers the full 0-1 stoichiometry range due to:

• Electrolyte decomposition at low voltages
• Structural instability at high voltages

Without physics-based extrapolation, simulations that access stoichiometries outside the data range produce unreliable results.

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Numerical Considerations

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Smoothing Parameter Selection

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Extrapolation Behavior

Both piecewise and OCP interpolants use constant extrapolation outside their defined ranges. This prevents unbounded values and is physically reasonable for many battery parameters.

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Common Use Cases

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Best Practices

1. Breakpoint placement: Place more breakpoints where the parameter changes rapidly
2. Smoothing selection: Start with defaults; increase if solver has trouble
3. Visualization: Always plot interpolants across the full range to check for artifacts
4. MSMR for OCP: Use blended interpolants for robust extrapolation