> ## Documentation Index
> Fetch the complete documentation index at: https://docs.ionworks.com/llms.txt
> Use this file to discover all available pages before exploring further.
# 区分的補間子
> Build smooth, differentiable piecewise interpolants for OCP, diffusivity, and SOC- or temperature-dependent battery parameters.
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.
## 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.
## Mathematical Formulation
### 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 $x \gg \theta$.
### 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:
$$
p_i^{\text{seg}}(x) = p_i + (p_{i+1} - p_i) \frac{x - x_i}{x_{i+1} - x_i}
$$
## Piecewise Parameter Functions
### 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.
### 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.
Use different smoothing parameters for dimensions with different scales (e.g., SOC in \[0, 1] vs Temperature in \[273, 323] K).
パイプラインでこれらの補間子を構成する方法については、[Pipelines → Calculations → Piecewise](/ja/pipelines/calculations/piecewise) を参照してください。
### Formulations: Knots vs Slopes
Two equivalent parameterizations are available:
| Formulation | Parameters | Best for |
| ----------- | ------------------------------------------ | --------------------------------- |
| **Knots** | Values at each breakpoint | Direct measurements |
| **Slopes** | Initial value + slopes between breakpoints | Optimization (smoother landscape) |
Both have the same degrees of freedom. Slopes can be converted to knots using `SlopesToKnots`.
## OCP Interpolants
Open-circuit potential (OCP) interpolants are specialized functions $U(\theta)$ that return equilibrium voltage for a given stoichiometry.
### 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.
### 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.
### 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.
### 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.
Always use MSMR blending if your simulation might access stoichiometries outside your experimental range. The MSMR model provides physically reasonable behavior at the extremes.
## Numerical Considerations
### Smoothing Parameter Selection
| Value | Effect |
| ----------- | ------------------------------------------------------------------ |
| Too small | Approaches discontinuous, may cause solver issues |
| Too large | Excessive smoothing, reduces accuracy |
| Recommended | $\epsilon \approx 10^{-4}$ to $10^{-3} \times$ (range of variable) |
### 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.
## Common Use Cases
Capture phase transitions or concentration effects on transport
Define equilibrium voltage for electrochemical models
Non-Arrhenius temperature dependence
Parameters depending on multiple variables simultaneously
## 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