MontoThermoPOC

A Monte Carlo thermal simulator for solving heat PDEs using the Walk-on-Stars (WoS) algorithm via the zombie library. The project explores WoS as a mesh-free alternative to classical solvers (FEM) for steady-state and transient heat problems, with applications in additive manufacturing (3D printing).

Each example is verified against either an analytical solution or a deterministic FEM reference.

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Getting Started

Prerequisites

• Python 3.12
• The zombie Walk-on-Stars library (included as a git submodule)

Installation

# Clone the repo with submodules
git clone --recurse-submodules https://github.com/MohammadrezaMoeini/MontoThermoPOC.git
cd MontoThermoPOC

# Create and activate a virtual environment
python3 -m venv ~/.MontoThermoPOC312
source ~/.MontoThermoPOC312/bin/activate

# Install dependencies
pip install -r requirements.txt

Run an example

python experimentDev/example01/example01.py

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Examples

Example 01 — 2D Laplace equation (Walk-on-Stars)

Solves the Laplace equation on the unit square $[0,1]^2$ using the zombie WoS library and compares against the analytical solution.

PDE & boundary conditions:

$$\nabla^2 u = 0 \quad \text{on } [0,1]^2$$

Edge	Condition
Top ($y=1$)	$u = 1$
All others	$u = 0$

Output:

• Example 1: WoS solution vs analytical — 2D temperature field on $[0,1]^2$

Example 02 — 3D Laplace equation (Walk-on-Stars)

Extends Example 01 to a 3D unit cube $[0,1]^3$, visualised at three horizontal slices and compared against the analytical double Fourier-series solution.

PDE & boundary conditions:

$$\nabla^2 u = 0 \quad \text{on } [0,1]^3$$

Face	Condition
Top ($z=1$)	$u = 1$
All others	$u = 0$

Output:

• Example 2: Analytical vs Zombie WoS solution and absolute error at three $z$-slices

• Example 2: 3D view of the solution at horizontal slices coloured by temperature

• Example 2: 1D cross-section along $y=0.5$, $z=0.5$ — Analytical vs Zombie WoS

Example 03 — Transient heat equation (Walk-on-Stars)

Solves the transient heat equation on $[0,1]^2$ using WoS with backward Euler time discretisation and compares against the analytical Fourier-series solution.

PDE & boundary conditions:

$$\frac{\partial u}{\partial t} = \alpha \nabla^2 u \quad \text{on } [0,1]^2$$

Edge	Condition
Top ($y=1$)	$u = 1$
All others	$u = 0$

Initial condition: $u(x, y, 0) = 0$

Time discretisation:

Backward Euler converts the PDE into a screened-Poisson (Yukawa) equation at each step:

$$\nabla^2 u^{n+1} - \sigma, u^{n+1} = -\sigma, u^n \qquad \sigma = \frac{1}{\alpha \Delta t}$$

Zombie solves this directly with $\lambda = \sigma$ and source $f = \sigma \cdot u^n$. At steady state ($u^{n+1} \approx u^n$) the equation collapses back to $\nabla^2 u = 0$, recovering the Example 01 Laplace solution.

Key constraint — $\sigma$ must stay small:

The Dirichlet boundary influence on interior points decays as $\exp(-\sqrt{\sigma} \cdot d)$. For large $\sigma$ this signal falls below the MC noise floor, causing the solution to stagnate. Two diagnostic experiments confirmed this:

• Increasing N_walks 10× (2 000 → 20 000) with $\sigma = 50$: error unchanged — not a variance issue
• Reducing $\sigma$ from 50 to 2 ($\Delta t$: 0.02 → 0.5): mean error dropped 9× — confirmed root cause

Parameter	Value
dt	0.5 s
sigma	2
N_steps	6 (t_final = 3.0 s)
N_walks	2 000
Mean error	0.017

Output:

• Example 3: WoS (top) vs analytical (bottom) temperature field at six time snapshots

• Example 3: 1D profile at $x=0.5$ — analytical (line) vs WoS (dots) at each saved time

Example 04 — Transient heat equation (FEM reference)

Solves the same problem as Example 03 deterministically using the Finite Element Method (P1 linear triangles) as a benchmark for the WoS results.

Time discretisation:

Backward Euler with the FEM system matrix $A = M + \alpha \Delta t, K$ (factored once via LU):

$$A, u^{n+1} = M, u^n$$

where $M$ is the consistent mass matrix and $K$ is the stiffness matrix assembled from P1 shape functions on a uniform $50 \times 50$ triangle mesh.

Output:

• Example 4: FEM (top) vs analytical (bottom) temperature field at six time snapshots

• Example 4: 1D profile at $x=0.5$ — analytical (line) vs FEM (dots) at each saved time

• Example 4: WoS (example03) vs FEM vs Analytical — 1D profile at $x=0.5$, backward Euler $\Delta t=0.5$, $\sigma=2$

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License

This project is licensed under the GNU Affero General Public License v3.0 (AGPL-3.0). See LICENSE for details.

For commercial use, please contact the author.