Salvo Equations Simulator

A comprehensive web application for naval combat modeling using various implementations of the Salvo Combat Model. This application provides interactive simulations of different naval combat scenarios, from basic salvo equations to complex Monte Carlo simulations.

Features

• Basic Salvo Model: Classic implementation of the salvo equations
• Continuous-Time Salvo: Differential equation-based model for continuous combat simulation
• Stochastic Salvo: Probabilistic model with Gaussian noise in damage calculations
• Multiple Forces Salvo: Extension to handle multiple combatant forces
• Monte Carlo Simulation: Advanced model with multi-domain combat (air, naval, submarine)

Installation

1. Clone this repository:

git clone https://github.com/yourusername/salvo-equations-simulator.git
cd salvo-equations-simulator

2. Create a virtual environment (optional but recommended):

python -m venv venv
source venv/bin/activate  # On Windows, use: venv\Scripts\activate

3. Install the required packages:

pip install -r requirements.txt

Usage

To run the application:

streamlit run app.py

The application will open in your default web browser at http://salvo-equations.streamlit.app.

Project Structure

.
├── app.py              # Main Streamlit application
├── assets/            # Directory containing images and other static files
│   ├── CDDGN.png     # CDDGN logo
│   └── naval-battle.jpg # Background image
├── requirements.txt   # Python dependencies
└── README.md         # This file

Models

Basic Salvo Model

Let:

• $$x_n$$: fraction of Blue force at time step n
• $$y_n$$: fraction of Red force at time step n
• $$f_x$$: firepower per unit of Blue
• $$f_y$$: firepower per unit of Red
• $$q_x$$: fraction of Red’s incoming fire intercepted by Blue
• $$q_y$$: fraction of Blue’s incoming fire intercepted by Red
• $$C_x$$: defensive capacity (saturation threshold) of Blue
• $$C_y$$: defensive capacity of Red
• $$D(\cdot, \cdot)$$: nonlinear damage function (models saturation effects)

Then the equations are:

$$\begin{aligned} \text{Attack on Red:} \quad & A_y = f_x \cdot x_n \cdot (1 - q_y) \\\ \text{Attack on Blue:} \quad & A_x = f_y \cdot y_n \cdot (1 - q_x) \\\ \\\ \text{Damage to Red:} \quad & d_y = D(A_y, C_y) \\\ \text{Damage to Blue:} \quad & d_x = D(A_x, C_x) \\\ \\\ \text{Update Blue force:} \quad & x_{n+1} = \max(0, x_n - d_x) \\\ \text{Update Red force:} \quad & y_{n+1} = \max(0, y_n - d_y) \end{aligned}$$

• Simulates combat between two forces
• Parameters include initial forces, firepower, and interception capabilities

Continuous-Time Salvo

$$\frac{dx}{dt} = - f_y \cdot y(t) \cdot (1 - q_x)$$ $$\frac{dy}{dt} = - f_x \cdot x(t) \cdot (1 - q_y)$$

• Uses differential equations for continuous combat simulation
• Provides smooth force evolution over time

Stochastic Salvo

$$\begin{align*} A_y &= f_x \cdot x_n \cdot (1 - q_y) + \varepsilon_y, \quad \varepsilon_y \sim \mathcal{N}(0, \sigma^2) \\\ A_x &= f_y \cdot y_n \cdot (1 - q_x) + \varepsilon_x, \quad \varepsilon_x \sim \mathcal{N}(0, \sigma^2) \\\ x_{n+1} &= \max(0, x_n - A_x) \\\ y_{n+1} &= \max(0, y_n - A_y) \end{align*}$$

• Adds Gaussian noise to damage calculations
• Accounts for uncertainty in combat outcomes

Multiple Forces Salvo

$$\begin{align*} \text{Damage to Red unit } j: \quad D^y_j &= \sum_{i=1}^{3} F^x_{ij} \cdot x_i \cdot (1 - q_y) \\\ \text{Damage to Blue unit } j: \quad D^x_j &= \sum_{i=1}^{3} F^y_{ij} \cdot y_i \cdot (1 - q_x) \\\ x_j^{(n+1)} &= \max(0, x_j^{(n)} - D^x_j) \\\ y_j^{(n+1)} &= \max(0, y_j^{(n)} - D^y_j) \end{align*}$$

• Extends the basic model to handle multiple combatant forces
• Complex interactions between multiple forces

Monte Carlo Simulation

📐 Mathematical Models

The Monte Carlo simulation in this project uses the following probabilistic equations:

1. Firepower Generation (Poisson-distributed shots per unit)

$$ S_i^{(d)} \sim \text{Poisson}(\lambda_i^{(d)}) $$

2. Probability of Hit and Interception

$$ H_{i \rightarrow j}^{(d_k \rightarrow d_l)} \sim \text{Binomial}\left(S_i^{(d_k)},; p_{\text{hit},,i}^{(d_k \rightarrow d_l)} \cdot (1 - p_{\text{int},,j})\right) $$

3. Unit Loss Update

$$ U_j^{(d_l)}(t+1) = \max\left(0,; U_j^{(d_l)}(t) - \sum_{d_k} H_{i \rightarrow j}^{(d_k \rightarrow d_l)} \right) $$

4. Reinforcement Probability (Adaptive)

$$ P_{\text{reinforce},,i}^{(d)} = \min\left(0.02 + 0.1 \cdot \left(1 - \frac{\sum_d U_i^{(d)}(t)}{\sum_d U_i^{(d)}(0)}\right),; 1.0\right) $$

5. Morale Collapse Condition

If

$$ \frac{\sum_d U_i^{(d)}(t)}{\sum_d U_i^{(d)}(0)} < \theta_{\text{morale}}, $$

then side $i$ may collapse with probability $p_{\text{collapse}}$.

6. Monte Carlo Simulation Loop

Repeat for each simulation $s = 1, 2, \dots, N_{\text{sim}}$ and each round $t = 1, 2, \dots, T_{\text{max}}$.

• Multi-domain combat (air, naval, submarine)
• Poisson-distributed firepower
• Binomial-distributed hit probabilities
• Adaptive reinforcement probabilities
• Morale collapse mechanics

Contributing

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

License

This project is licensed under the MIT License - see the LICENSE file for details.

Contact

For questions and feedback, please contact:

• Email: tpires@id.uff.br or tullio.mozart@marinha.mil.br
• GitHub: KanonStarbringer