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‎docs/f_approx_init_stress.md‎

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# `f_approx_init_stress.m` Documentation
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## Overview
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`f_approx_init_stress` computes an approximate initial Cauchy stress at the bubble maximum radius using analytical corrections for viscoelastic, compressibility, and compressive effects.
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This provides a lower-fidelity but faster estimate compared to the numerical shooting method.
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## Major Sections
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1. **Constant Definitions**: Compute `trc` (time constant) and coefficient `B`.
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2. **Barotropic Correction** (`fbarbc`): Accounts for vapor and compressibility effects.
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3. **Strain Correction** (`fbarst`): Originating from surface tension scaling.
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4. **Compressibility Correction** (`fbarc`): Based on Grüneisen parameter.
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5. **Viscous Correction** (`fbarv`): Derived from Reynolds number.
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6. **Elastic Correction** (`fbare`): Analytical term for elastic effects.
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7. **Stress Summation** (`fsum`): Aggregates all correction terms, with input validation.
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8. **Time Constant Calculation** (`tg`): Computes relaxation time for stress growth.
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9. **Stress Prediction** (`Smaxpred`): Final predicted maximum stress.
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## Inputs
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| Name | Description | Units |
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|----------|---------------------------------------------------|---------|
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| `Ro` | Non-dimensional bubble radius (`R/R_eq`) | – |
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| `kappa` | Ratio of specific heats of gas | – |
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| `al_nd` | Non-dimensional strain-softening exponent | – |
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| `pwv_nd` | Non-dimensional vapor pressure ratio | – |
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| `We` | Weber number (surface tension effects) | – |
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| `Re` | Reynolds number (viscous effects) | – |
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| `De` | Deborah number (viscoelastic relaxation effects) | – |
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| `Ca` | Cauchy number (elastic to inertial ratio) | – |
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| `alpha` | Quadratic KV model exponent | – |
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## Outputs
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| Name | Description | Units |
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|-------------|-----------------------------------------|---------|
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| `Smaxpred` | Predicted maximum Cauchy stress ratio | – |
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## Notes
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- Throws an error if the aggregated stress correction (`fsum`) exceeds 1.
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- Analytical method is faster but less accurate than `f_init_stress` numerical shooting.

‎docs/f_init_stress.md‎

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# `f_init_stress.m` Documentation
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## Overview
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`f_init_stress` determines the initial stress by numerically solving bubble collapse dynamics with a shooting method, ensuring the stress at maximum radius matches the selected viscoelastic model.
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## Major Sections
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1. **Shooting Setup**: Configure `fminsearch` options (`sopts`).
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2. **Initial Guess via Shooting**: Use `f_maxwell_growth_iter` to find wall velocity that maximizes stress integral.
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3. **Time Integration**: Run `ode23tb` on `f_RP_bub_collapse_comp` over nondimensional time span.
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4. **Stress Extraction**: Identify maximum stress from solution matrix.
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### Internal Functions
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- `f_maxwell_growth_iter`: Computes max radius error for shooting.
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- `f_RP_bub_collapse_comp`: Nondimensional Rayleigh–Plesset ODE for collapse with stress evolution.
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## Inputs
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| Name | Description | Units |
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|-----------|----------------------------------------|---------|
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| `Req` | Equilibrium radius | – |
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| `Re` | Reynolds number | – |
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| `Ca` | Cauchy number | – |
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| `De` | Deborah number | – |
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| `We` | Weber number | – |
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| `CL` | Speed of sound ratio (`C/C_ref`) | – |
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| `Pv_star` | Non-dimensional vapor pressure | – |
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## Outputs
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| Name | Description | Units |
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|------|----------------------------------|---------|
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| `S0` | Initial stress at bubble maximum | – |
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## Notes
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- Uses `ode23tb` for stiff ODE integration.
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- Requires consistency between stress model and dimensional parameters.

‎docs/f_odesolve.md‎

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# `f_odesolve.m` Documentation
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## Overview
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`f_odesolve` is a wrapper that selects and executes MATLAB's ODE solvers (`ode15s`, `ode23tb`, `ode45`) with user-defined time stepping and tolerance options.
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## Major Sections
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1. **Options Configuration**: Sets `RelTol`, `AbsTol`, and `MaxStep` based on `divisions`.
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2. **Solver Selection**: Routes to the correct solver based on `method` code:
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- 15 → `ode15s`
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- 23 → `ode23tb`
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- 45 → `ode45`
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3. **Execution**: Calls the selected solver and returns results.
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## Inputs
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| Name | Description | Units |
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|------------|----------------------------------------------|---------|
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| `bubble` | Function handle for ODE system | – |
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| `init` | Initial condition vector | – |
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| `method` | Solver choice (15, 23, or 45) | – |
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| `divisions`| Maximum number of time steps (0 = auto) | – |
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| `tspan` | Two-element time span `[t0 tf]` | – |
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| `tfin` | Final time (for step size calculation) | – |
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## Outputs
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| Name | Description | Units |
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|------|--------------------------|---------|
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| `t` | Time vector | – |
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| `X` | Solution matrix (states) | – |
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## Notes
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- Throws an error for unsupported `method` codes.
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- `MaxStep` is set to `tfin/divisions` when `divisions > 0`.

‎docs/f_pinfinity.md‎

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# `f_pinfinity.m` Documentation
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## Overview
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`f_pinfinity` calculates the time-dependent external pressure (`p8`) and its derivative (`p8dot`) that drive bubble dynamics, supporting multiple waveform types.
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## Major Sections
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1. **Input Unpacking**: Extracts waveform parameters from `vararg`.
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2. **Custom Waveform Handling**: If `wave_type < 0`, uses polynomial fits (`wave_poly`, `wave_dpoly`).
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3. **Predefined Waveforms**:
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- **Impulse** (`wave_type = 0`)
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- **Gaussian** (`wave_type = 1`)
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- **Histotripsy** (`wave_type = 2`)
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- **Heaviside Impulse** (`wave_type = 3`)
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4. **Inner Functions**: Define `impulse`, `gaussian`, `histo`, and `heaviside_impulse`.
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## Inputs
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| Name | Description | Units |
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|-------------|-------------------------------------------------|---------|
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| `t` | Current time | s |
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| `vararg` | Vector `[om, ee, tw, dt, mn, wave_type]` | – |
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## Outputs
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| Name | Description | Units |
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|----------|---------------------------------------------|---------|
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| `p8` | External pressure at time `t` | Pa |
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| `p8dot` | Time derivative of external pressure | Pa/s |
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## Notes
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- `wave_poly` and `wave_dpoly` must be defined in the caller’s workspace for custom waveforms.
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- Gaussian and histotripsy waveforms depend on `ee` (amplitude), `om` (frequency), `tw` (width), `dt` (delay), and `mn` (exponent).

‎docs/f_pvsat.md‎

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# `f_pvsat.m` Documentation
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## Overview
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`f_pvsat` returns the saturated water vapor pressure as a function of temperature using an empirical exponential fit.
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## Major Sections
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1. **Empirical Formula**: Applies `Pv = 1.17e11 * exp(-5200/T)`.
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## Inputs
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| Name | Description | Units |
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|------|--------------------|-------|
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| `T` | Temperature | K |
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## Outputs
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| Name | Description | Units |
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|------|-------------------------------|-------|
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| `Pv` | Saturated vapor pressure | Pa |
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## Notes
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- Source: A. Preston, 2004 (Thesis).
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- Valid for typical biological temperature ranges (290–310 K).

‎docs/f_radial_eq.md‎

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# `f_radial_eq.m` Documentation
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## Overview
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`f_radial_eq` evaluates the radial acceleration (`Rddot`) of a bubble wall using various models (Rayleigh–Plesset, Keller–Miksis, Gilmore) and equations of state.
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## Major Sections
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1. **Rayleigh–Plesset (Model 1)**
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2. **Keller–Miksis Pressure Form (Model 2)**
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3. **Keller–Miksis Enthalpy Form (Model 3)**
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4. **Gilmore with Tait EoS (Model 4)**
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5. **Keller–Miksis with Mie–Grüneisen EoS (Model 5)**
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6. **Gilmore with Mie–Grüneisen EoS (Model 6)**
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7. **EOS Helper Functions**:
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- `f_mie_gruneisen_eos_scalar`
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- `f_mie_rho_from_p_scalar`
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## Inputs
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| Name | Description | Units |
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|-------------|--------------------------------------------|-------|
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| `radial` | Model selector (1–6) | – |
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| `P` | Internal bubble pressure (non-dim) | – |
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| `Pdot` | Time derivative of `P` | – |
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| `Pf8` | External pressure (non-dim) | – |
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| `Pf8dot` | Time derivative of `Pf8` | – |
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| `iWe` | Inverse Weber number | – |
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| `R` | Bubble radius (non-dim) | – |
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| `Rdot` | Bubble wall velocity (non-dim) | – |
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| `S` | Stress term (non-dim) | – |
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| `Sdot` | Time derivative of `S` | – |
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| `Cstar` | Non-dim sound speed | – |
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| `sam` | Tait EOS parameter | – |
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| `no` | Tait exponent | – |
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| `GAMa` | Mie–Grüneisen parameter | – |
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| `nstate` | EOS exponent | – |
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| `nog` | (nstate–1)/2 for MG EoS | – |
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| `hugoniot_s`| Hugoniot slope | – |
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| `JdotA` | Viscous and thermal coupling parameter | – |
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| `ddintfnu` | Second derivative of non-Newtonian term | – |
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| `iDRe` | Inverse Debye–Stokes number | – |
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## Outputs
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| Name | Description | Units |
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|----------|-----------------------------------------|-------|
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| `Rddot` | Bubble wall acceleration | – |
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## Notes
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- EOS helper functions perform scalar EOS and density calculations.
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- Ensure physical consistency when mixing different models.

‎docs/f_stress_calc.md‎

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# `f_stress_calc.m` Documentation
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## Overview
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`f_stress_calc` computes the stress integral and its time derivative for a range of viscoelastic models (Kelvin–Voigt, Maxwell, Jeffreys, Zener, Oldroyd-B).
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## Major Sections
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1. **No Stress (Model 0)**
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2. **Kelvin–Voigt Neo-Hookean (Model 1)**
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3. **Quadratic KV Neo-Hookean (Model 2)**
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4. **Linear Maxwell/Jeffreys/Zener (Model 3)**
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5. **Quadratic Maxwell Variant (Model 4)**
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6. **Upper-Convected Maxwell / Oldroyd-B (Model 5)**
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7. **Special Placeholder (Model -1)**
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## Inputs
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| Name | Description | Units |
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|--------------|----------------------------------------|-------|
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| `stress` | Model selector (0–5, -1) | – |
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| `X` | State vector containing auxiliary vars | – |
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| `Req`, `R` | Equilibrium and current radius | – |
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| `Ca`, `De` | Cauchy and Deborah numbers | – |
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| `Re8` | Reynolds number | – |
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| `Rdot` | Wall velocity | – |
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| `alphax` | Quadratic KV exponent | – |
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| `ivisco1` | Index of first viscous state variable | – |
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| `ivisco2` | Index of second viscous state variable | – |
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| `LAM` | Dimensionless elastic modulus | – |
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| `zeNO`, `cdd`| Spectral coupling parameters | – |
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| `intfnu`, `dintfnu` | Non-Newtonian integrals | – |
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| `iDRe` | Inverse Debye–Stokes number | – |
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## Outputs
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| Name | Description | Units |
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|---------|------------------------------------|-------|
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| `S` | Stress integral | – |
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| `Sdot` | Time derivative of `S` | – |
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| `Z1dot` | First auxiliary stress derivative | – |
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| `Z2dot` | Second auxiliary stress derivative | – |
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## Notes
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- For spectral solvers, `zeNO` and coupling differ.
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- The placeholder model (-1) is for internal code consistency.

‎docs/f_stress_dissipation.md‎

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# `f_stress_dissipation.m` Documentation
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## Overview
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`f_stress_dissipation` calculates the mechanical dissipation term (`τ:∇u`) contributing to energy loss, accounting for both finite-difference and spectral solvers.
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## Major Sections
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1. **Radial Stretch Calculation**: Compute `Rst`, `x2`, `ix2`, `x4`.
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2. **Finite-Difference Dissipation**: Branch for models 0–5.
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3. **Spectral Dissipation**: Alternative calculation if `spectral == 1`.
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## Inputs
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| Name | Description | Units |
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|----------------|----------------------------------------|-------|
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| `stress` | Stress model selector (0–5) | – |
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| `spectral` | Solver type flag | – |
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| `Req`, `R` | Equilibrium and current radius | – |
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| `Rdot` | Wall velocity | – |
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| `Ca`, `Br` | Cauchy number and Brinkman number | – |
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| `Re8`, `alphax`| Reynolds and quadratic exponent | – |
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| `yT2`, `yT3` | Collocation grid vectors | – |
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| `iyT3`, `iyT4`, `iyT6` | Inverses of grid powers | – |
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| `X`, `ZZT` | State vectors and spectral matrix | – |
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| `ivisco1`, `ivisco2` | Auxiliary stress indices | – |
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| `fnu`, `DRe` | Non-Newtonian friction and number | – |
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## Outputs
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| Name | Description | Units |
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|-------------|---------------------------------------|-------|
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| `taugradu` | Mechanical dissipation (`τ:∇u`) | – |
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## Notes
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- Spectral branch uses `ZZT*(X(ivisco1)-X(ivisco2))` for dissipation.

‎docs/f_viscosity.md‎

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# `f_viscosity.m` Documentation
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## Overview
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`f_viscosity` computes non-Newtonian viscosity integrals for Carreau, Cross, Powell–Eyring, and related models, providing stress and its derivatives for bubble dynamics.
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## Major Sections
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1. **Shear Rate Computation**: `gammadot_R`, `gammadot_num`, `dgammadot`, `ddgammadot`.
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2. **Model Selection**: Branch for `nu_model` (1–7).
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3. **Integral Calculations**: Perform numerical integration with `integral`.
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4. **Internal Functions**: Definitions for `sf_carreau`, `sf_carreau_yasuda`, etc.
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## Inputs
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| Name | Description | Units |
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|-------------|----------------------------------------|---------|
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| `nu_model` | Viscosity model selector (1–7) | – |
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| `Rdot` | Wall velocity | – |
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| `R` | Bubble radius | – |
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| `a`, `nc` | Model parameters (exponents) | – |
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| `lambda` | Time constant for viscosity model | s |
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## Outputs
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| Name | Description | Units |
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|----------|--------------------------------------------------|---------|
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| `f` | Local viscosity factor | – |
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| `intf` | Stress integral for non-Newtonian term | – |
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| `dintf` | First derivative of `intf` | – |
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| `ddintf` | Second derivative of `intf` | – |
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## Notes
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- Use high tolerance (`1e-8`) for integrals.
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- Models:
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- 1: Carreau
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- 2: Carreau–Yasuda
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- 3: Powell–Eyring
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- 4: Modified Powell–Eyring
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- 5: Cross
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- 6: Simplified Cross
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- 7: Modified Cross

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