Project Overview This project investigates heat-resistant tiles on next-generation reusable spacecraft during atmospheric re-entry. Using advanced numerical methods, we modeled temperature distribution through ceramic tiles to ensure spacecraft safety during the extreme conditions of re-entry (temperatures up to 1477°C). Key Achievements
Optimal Method Selection: Crank-Nicolson method identified as most stable and accurate Parameter Optimization: Timestep of 67.8s and spatial step of 8.33mm for ±0.5K tolerance Safety Engineering: Calculated minimum tile thickness of 139mm with 1.4 safety factor Enhanced Modeling: 2D representation with radiative heat transfer implementation
Repository Contents This repository contains the three most critical functions from our numerical modeling suite.
1. timestepInv() - Optimization Algorithm Primary Application: Algorithmic Trading Time Interval Optimization matlabfunction [bestMethod, maxdt] = timestepInv(tmax, ntMin, ntMax, ntIncr, thick, nx, tol, timeData, tempData) What it does:
Finds optimal time steps for numerical stability within specified tolerance Compares multiple numerical methods (Forward, Backward, DuFort-Frankel, Crank-Nicolson) Identifies maximum stable timestep for each method
2. plot_error_surface() - Risk Surface Analysis Primary Application: Financial Risk Surface Modeling matlabfunction [first_i,first_j,errors,smooth_colors,custom_colormap] = plot_error_surface(target_error, same_value, method, imageName, timeData, tempData, unit) What it does:
Creates 3D error surfaces across parameter combinations Visualizes stability regions and error boundaries Provides color-coded risk mapping
3. last_temp() - Iterative Target Solver Primary Application: Options Pricing and Calibration matlabfunction [x, t, u, thick, temp1] = last_temp(tmax, nt, nx, ~, timeData, tempData, tmin, max_iterations)
What it does: Uses secant method to find parameters that achieve target values Iteratively converges to optimal solutions Handles convergence criteria and maximum iterations
Technical Specifications Numerical Methods Implemented: Forward Differencing: O(Δt, Δx²) accuracy, conditionally stable DuFort-Frankel: O(Δt², Δx², (Δt/Δx)²) accuracy, unconditionally stable Backward Differencing: O(Δt, Δx²) accuracy, more stable than forward Crank-Nicolson: O(Δt², Δx²) accuracy, unconditionally stable (OPTIMAL)
Key Parameters Material Properties: Silica-based ceramic tiles Temperature Range: Up to 1092K surface temperature Safety Factor: 1.4 (NASA recommended) Thermal Diffusivity: 5.35 × 10⁻⁷ m²s⁻¹
Installation and Usage Prerequisites MATLAB R2019b or later Signal Processing Toolbox (recommended)
Research Results Our analysis demonstrated that the Crank-Nicolson method provides:
Superior Stability: Maintains accuracy up to 67.8s timesteps Optimal Accuracy: O(Δt², Δx²) error order Unconditional Stability: No restrictive stability conditions Engineering Safety: 1.4 safety factor ensures 139mm minimum tile thickness
Citation If you use this work in academic research or commercial applications, please cite: Numerical Modelling of a Spacecraft Heat Shield Tile ME20021: Modelling Techniques 2 – Group 99 Authors: Diego Hackel, Martin Gutierrez, Javier Escribano