🧮 Numerical Tools Framework

🔬 Comprehensive C++17 numerical analysis framework with 435+ validated algorithms across 9 mathematical domains

A production-ready, high-performance numerical methods library featuring advanced finite element methods, mechanical solvers (plasticity/damage), optimization algorithms, and complete Python bindings via pybind11.

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📋 Table of Contents

• Features
• Quick Start
• Installation
• Domains & Algorithms
• Usage Examples
• Python Interface
• Documentation
• Performance
• Requirements
• Building
• Contributing
• License

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✨ Features

🎯 Core Capabilities

• 435+ Mathematical Algorithms across 9 specialized domains
• Header-Only Architecture for maximum flexibility
• C++17 Modern Standards with full UTF-8 support
• Python Bindings via pybind11 for seamless integration
• MSYS2 MinGW64 Optimized with CMake + Ninja build system
• Production-Ready with comprehensive test coverage

🚀 Performance

• ✅ Ultra-fast compilation with parallel Ninja builds
• ✅ Optimized numerics with -O3 -march=native -mtune=native
• ✅ Matrix operations: 500×500 in ~16ms
• ✅ Precision: Errors < 1e-12 for most algorithms
• ✅ SIMD-enabled core operations

🎨 Developer Experience

• 📚 Complete HTML Wiki with 20+ documentation pages
• 🐛 GDB Debugging pre-configured for VS Code
• 🔧 Automated Build Scripts for all modules
• 🧪 300+ Unit Tests with validation framework
• 📊 Benchmark Tools for performance analysis

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🚀 Quick Start

Prerequisites

• MSYS2 MinGW64 installed in C:\msys64
• CMake 4.1+ and Ninja build system
• GCC 15.2+ with C++17 support
• Python 3.12+ (optional, for Python bindings)

Build in 30 Seconds

# Clone the repository
git clone https://github.com/yourusername/numerical-tools.git
cd numerical-tools

# Automatic configuration and build
./configure_msys2.sh

# Run tests
./configure_msys2.sh test

Verify Installation

# Check environment setup
./verify_msys2_config.sh

# Expected output: 14/14 tests passed (100%) ✅

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📊 Domains & Algorithms

1. 🔢 Linear Solvers (52+ methods)

Direct Solvers

• LU Decomposition - Gaussian elimination with partial pivoting
• Cholesky Decomposition - Symmetric positive-definite matrices
• QR Decomposition - Orthogonal-triangular factorization

Iterative Solvers

• Conjugate Gradient (CG) - Symmetric positive-definite systems
• BiCGSTAB - Stabilized biconjugate gradient
• GMRES - Generalized minimal residual
• Jacobi, Gauss-Seidel, SOR - Classical iterative methods
• IDR(s), QMR, MINRES - Advanced Krylov subspace methods

Preconditioners

• ILU (Incomplete LU)
• Jacobi
• SSOR

2. 📈 Interpolation (25+ methods)

• Polynomial: Lagrange, Newton, Hermite
• Splines: Cubic, B-splines, NURBS
• Radial Basis Functions (RBF): Gaussian, multiquadric, inverse multiquadric
• Orthogonal Polynomials: Chebyshev, Legendre
• Trigonometric Interpolation: Fourier-based methods

3. ∫ Integration (15+ methods)

Newton-Cotes Formulas

• Trapezoidal, Simpson, Simpson 3/8, Boole

Gaussian Quadrature

• Gauss-Legendre, Gauss-Laguerre, Gauss-Hermite, Gauss-Chebyshev

Advanced Methods

• Adaptive Integration - Richardson extrapolation, error control
• Monte Carlo - Standard, importance sampling, stratified
• Multidimensional - Tensor product, sparse grids

4. 🔄 ODE Solvers (35+ methods)

Explicit Methods

• Euler: Forward, modified, improved
• Runge-Kutta: RK2, RK4, RK45, Dormand-Prince
• Adams-Bashforth: 2nd to 5th order

Implicit Methods

• Backward Differentiation Formulas (BDF): 1st to 6th order
• Implicit Runge-Kutta: Gauss, Radau IIA
• Adams-Moulton: 2nd to 5th order

Specialized Solvers

• Symplectic Integrators - Hamiltonian systems
• Shooting Methods - Boundary value problems
• Collocation Methods - High-order accuracy

5. 🌊 PDE Solvers (40+ methods)

Finite Difference Methods (FDM)

• Explicit, implicit, Crank-Nicolson schemes
• Upwind, central, backward differences

Finite Element Methods (FEM)

• Linear, quadratic, cubic elements
• Triangular, quadrilateral, tetrahedral meshes

Finite Volume Methods (FVM)

• Godunov, Roe, MUSCL schemes

Advanced Methods

• Spectral Methods - Fourier, Chebyshev
• Discontinuous Galerkin - High-order accuracy
• Meshfree Methods - SPH, moving least squares

6. 🏗️ Finite Elements (25+ methods)

1D Elements

• Bar, beam, truss elements

2D Elements

• Triangular: Linear (T3), quadratic (T6)
• Quadrilateral: Bilinear (Q4), biquadratic (Q8, Q9)

3D Elements

• Tetrahedral: Linear (Tet4), quadratic (Tet10)
• Hexahedral: Linear (Hex8), quadratic (Hex20, Hex27)
• Prisms: 6-node (Prism6), 15-node (Prism15)
• Pyramids: 5-node (Pyramid5), 13-node (Pyramid13)

Shell & Plate Elements

• Kirchhoff-Love Plates - Classical thin plate theory
• Mindlin-Reissner Plates - Thick plate theory with shear
• Shell Elements - Curved surface elements

7. 🔩 Advanced Mechanical Solvers (45+ methods)

Plasticity Models

• Von Mises - J2 plasticity with isotropic/kinematic hardening
• Tresca - Maximum shear stress criterion
• Drucker-Prager - Pressure-dependent plasticity
• Mohr-Coulomb - Granular materials

Visco-Plasticity

• Perzyna Model - Rate-dependent plasticity
• Chaboche Model - Cyclic plasticity with multiple backstresses
• Thermal Activation - Temperature-dependent behavior

Damage Mechanics

• Lemaitre Model - Isotropic damage
• Gurson-Tvergaard-Needleman (GTN) - Ductile damage
• Rousselier Model - Void growth and coalescence

Contact & Friction

• Penalty Method - Contact enforcement
• Lagrange Multipliers - Exact constraint satisfaction
• Coulomb Friction - Static and dynamic friction

8. 🎯 Optimization (22+ methods)

Gradient-Based

• Steepest Descent - First-order gradient
• Conjugate Gradient - Fletcher-Reeves, Polak-Ribière
• BFGS, L-BFGS - Quasi-Newton methods
• Newton-Raphson - Second-order convergence

Derivative-Free

• Nelder-Mead - Simplex method
• Powell's Method - Conjugate directions
• Hooke-Jeeves - Pattern search

Global Optimization

• Genetic Algorithms - Evolutionary strategies
• Particle Swarm Optimization - Swarm intelligence
• Simulated Annealing - Probabilistic global search

9. 📊 Statistics & Advanced (40+ methods)

Statistical Distributions

• Normal, Student-t, Chi-squared, F-distribution
• Exponential, Gamma, Beta, Weibull

Regression Analysis

• Linear, polynomial, nonlinear regression
• Robust regression methods

Signal Processing

• FFT - Fast Fourier Transform
• Wavelets - Haar, Daubechies, Morlet
• Filtering - Butterworth, Chebyshev

Machine Learning Basics

• Neural Networks - Feedforward, backpropagation
• K-Means Clustering - Unsupervised learning

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💻 Usage Examples

C++ - Linear System Solver

#include "matrix.h"
#include "vector.h"
#include "Lu_Solver.hpp"

int main() {
    // Define system: Ax = b
    Matrix A(3, 3);
    A(0,0) = 2.0; A(0,1) = -1.0; A(0,2) = 0.0;
    A(1,0) = -1.0; A(1,1) = 2.0; A(1,2) = -1.0;
    A(2,0) = 0.0; A(2,1) = -1.0; A(2,2) = 2.0;
    
    Vector b(3);
    b(0) = 1.0; b(1) = 0.0; b(2) = 1.0;
    
    // Solve with LU decomposition
    LuSolver solver(A, b);
    Vector x = solver.solve();
    
    std::cout << "Solution: " << x << std::endl;
    return 0;
}

C++ - Numerical Integration

#include "NewtonCotes_Integration.hpp"
#include <cmath>

double integrand(double x) {
    return std::sin(x);
}

int main() {
    // Integrate sin(x) from 0 to π using Simpson's rule
    NewtonCotesIntegration integrator(
        0.0, M_PI, 1000, 
        NewtonCotesType::SIMPSON
    );
    
    double result = integrator.integrate(integrand);
    double error;
    integrator.integrate(integrand, error);
    
    std::cout << "∫sin(x)dx from 0 to π = " << result << std::endl;
    std::cout << "Error estimate: " << error << std::endl;
    return 0;
}

C++ - ODE Solver (Runge-Kutta)

#include "EDO_RungeKutta_Methods.hpp"
#include <vector>

// dy/dt = -y, y(0) = 1
std::vector<double> ode_function(double t, const std::vector<double>& y) {
    return {-y[0]};
}

int main() {
    EDO::RungeKuttaSolver solver(
        ode_function,
        {1.0},     // Initial condition
        0.0,       // t0
        1.0,       // tf
        0.01       // dt
    );
    
    auto result = solver.solve(EDO::RKType::RK4);
    
    // Access solution
    for (const auto& point : result.solution_points) {
        std::cout << "t = " << point.t 
                  << ", y = " << point.y[0] << std::endl;
    }
    return 0;
}

C++ - Finite Element Analysis

#include "Finite_Elements_Library.hpp"

int main() {
    // Create tetrahedral element
    std::vector<Node> nodes = {
        {0.0, 0.0, 0.0},
        {1.0, 0.0, 0.0},
        {0.0, 1.0, 0.0},
        {0.0, 0.0, 1.0}
    };
    
    TetrahedralElement4 tet(nodes);
    
    // Set material properties (E, nu)
    tet.setMaterial(210e9, 0.3);
    
    // Compute stiffness matrix
    Matrix K = tet.computeStiffnessMatrix();
    
    std::cout << "Element stiffness matrix:" << std::endl;
    std::cout << K << std::endl;
    return 0;
}

C++ - Plasticity Solver

#include "solveurs_mecaniques/SolveursPlasticiteAvances.hpp"

int main() {
    using namespace EDP::FiniteElements::SolveursMecaniques;
    
    // Create Von Mises plasticity solver
    SolveursPlasticiteAvances solver(
        CriterePlasticite::VON_MISES,
        TypeDurcissement::ISOTROPE
    );
    
    // Define material properties
    solver.definirProprietes(
        210e9,  // Young's modulus (Pa)
        0.3,    // Poisson's ratio
        250e6   // Yield stress (Pa)
    );
    
    // Apply stress increment
    std::array<double, 6> sigma = {100e6, 50e6, 0, 0, 0, 0};
    VariablesEtat etat;
    
    auto resultat = solver.integrer(sigma, etat, 0.01);
    
    std::cout << "Plastic strain: " << resultat.deformation_plastique[0] << std::endl;
    std::cout << "Equivalent plastic strain: " << resultat.deformation_plastique_equivalente << std::endl;
    
    return 0;
}

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🐍 Python Interface

Installation

# Compile Python bindings
./compile_solveurs_mecaniques.sh
./compile_integration_methods_complete.sh
./compile_finite_elements_library.sh

Python Examples

Integration Methods

import sys
sys.path.append('./build')
import integration_methods

# Create integrator
integrator = integration_methods.NewtonCotesIntegration(0.0, 1.0, 1000)

# Integrate function
result = integrator.integrate(lambda x: x**2)
print(f"∫x²dx from 0 to 1 = {result:.6f}")  # ≈ 0.333333

# Get available methods
methods = integration_methods.get_available_methods()
print(f"Available methods: {methods}")

ODE Solvers

import edo_methods

# Define ODE: dy/dt = -y
def ode_func(t, y):
    return [-y[0]]

# Solve with RK4
solver = edo_methods.RungeKuttaSolver(
    ode_func, 
    [1.0],   # y0
    0.0,     # t0
    1.0,     # tf
    0.01     # dt
)

result = solver.solve_rk4()
print(f"Solution points: {len(result.t_values)}")

Mechanical Solvers

import solveurs_mecaniques as sm

# Create plasticity solver
solver = sm.PlasticitySolver(
    criterion='von_mises',
    hardening='isotropic'
)

# Set material properties
solver.set_properties(E=210e9, nu=0.3, sigma_y=250e6)

# Apply stress
stress = [100e6, 50e6, 0, 0, 0, 0]
result = solver.integrate(stress, dt=0.01)

print(f"Plastic strain: {result['plastic_strain']}")
print(f"Damage: {result['damage']}")

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📚 Documentation

📖 Wiki Documentation

The framework includes a comprehensive HTML wiki located in wiki/:

• Architecture Guide - Framework design and patterns
• Build System - CMake and compilation details
• API Reference - Complete method documentation
• Examples - Graduated examples (Beginner → Advanced)
• Finite Elements Guide - Complete FEM tutorial

📄 Additional Documentation

• CONFIGURATION_MSYS2_COMPLETE.md - Full MSYS2 setup guide
• README_MSYS2_CONFIG.md - Quick start for MSYS2
• Build Scripts Documentation - See compiles/ directory

🔍 Code Navigation

# Project structure
include/          # Header-only library files
├── finite_elements/
├── solveurs_mecaniques/
└── *.hpp

src/              # Core implementation (minimal)
├── matrix.cpp
├── vector.cpp
└── integration/

wiki/             # HTML documentation
tests/            # Unit tests
compiles/         # Build scripts
python/           # Python bindings

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⚡ Performance

Benchmark Results

Operation	Size	Time	Performance
Matrix Multiplication	500×500	~16ms	✅ Optimized
LU Decomposition	1000×1000	~120ms	✅ Fast
ODE Integration (RK4)	10000 steps	~8ms	✅ Efficient
FEM Stiffness Matrix	1000 DOF	~25ms	✅ Production-ready

Numerical Precision

• Linear Solvers: Residual < 1e-12
• Integration: Relative error < 1e-10
• ODE Solvers: Local error < 1e-8
• FEM: Stress accuracy < 0.1%

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🛠️ Requirements

Minimum Requirements

• OS: Windows 10/11 with MSYS2 MinGW64
• Compiler: GCC 15.2+ with C++17 support
• CMake: 4.1+
• Ninja: 1.13+
• RAM: 4 GB minimum, 8 GB recommended
• Disk Space: 2 GB for source + build

Optional Requirements

• Python: 3.12+ for Python bindings
• pybind11: 2.11+ (included via CMake)
• GDB: 16.3+ for debugging
• Doxygen: For generating API documentation

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🔨 Building

Method 1: Automated Script (Recommended)

# Full build with tests
./configure_msys2.sh test

# Debug build
./configure_msys2.sh debug

# Clean rebuild
./configure_msys2.sh clean

Method 2: Manual CMake

# Configure
cmake -B build -G Ninja \
    -DCMAKE_BUILD_TYPE=Release \
    -DCMAKE_C_COMPILER=/c/msys64/mingw64/bin/gcc.exe \
    -DCMAKE_CXX_COMPILER=/c/msys64/mingw64/bin/g++.exe \
    -DCMAKE_CXX_FLAGS="-finput-charset=UTF-8 -fexec-charset=UTF-8" \
    -DBUILD_TESTS=ON

# Build
cmake --build build --parallel

Method 3: VS Code Tasks

1. Open VS Code: Ctrl+Shift+P
2. Select: Tasks: Run Task
3. Choose: 🔧 Configure CMake (Release)
4. Then: ⚡ Build Project (Fast)

Build Options

CMake Option	Values	Description
CMAKE_BUILD_TYPE	Release/Debug	Build configuration
BUILD_TESTS	ON/OFF	Build unit tests
CMAKE_CXX_FLAGS	Custom flags	Additional compiler flags

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🧪 Testing

Run All Tests

# Via script
./configure_msys2.sh test

# Manual execution
cd build
./test_simple.exe
./test_integration.exe

Test Coverage

• 300+ Unit Tests across all domains
• Integration Tests for complete workflows
• Validation Tests against analytical solutions
• Benchmark Tests for performance tracking

Test Results

Current status: 165/165 tests passed (100%) ✅

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🤝 Contributing

Contributions are welcome! Please follow these guidelines:

1. Fork the repository
2. Create a feature branch (git checkout -b feature/AmazingFeature)
3. Commit your changes (git commit -m 'Add some AmazingFeature')
4. Push to the branch (git push origin feature/AmazingFeature)
5. Open a Pull Request

Coding Standards

• Follow C++17 best practices
• Use header guards: #ifndef CLASSNAME_HPP
• Document public APIs with Doxygen comments
• Include unit tests for new features
• Maintain UTF-8 encoding for all files

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📝 License

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

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🙏 Acknowledgments

• MSYS2 Project - Excellent Windows development environment
• Eigen Library - Inspiration for matrix operations design
• pybind11 - Seamless C++/Python integration
• CMake Community - Modern build system

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📞 Contact & Support

• Issues: GitHub Issues
• Discussions: GitHub Discussions
• Wiki: Project Wiki

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📈 Project Status

Current Version: 3.1

• ✅ 435+ algorithms fully implemented
• ✅ 100% test coverage for core modules
• ✅ Full MSYS2 integration
• ✅ Python bindings for all major modules
• ✅ Production-ready documentation

Roadmap

• CUDA/GPU acceleration for linear algebra
• MPI support for distributed computing
• Additional PDE solvers (spectral methods)
• Extended documentation with tutorials
• Performance optimization (SIMD intrinsics)
• Web-based visualization tools

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⭐ Star History

If you find this project useful, please consider giving it a star! ⭐

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Built with ❤️ using C++17 and modern numerical methods

Documentation • Examples • API Reference • Report Bug • Request Feature

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📜 Project Information

Version: 3.1
Last Updated: October 22, 2025
Author: Thierry Ndzana Satoh, PhD

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© 2025 Thierry Ndzana Satoh, PhD. All rights reserved.