README.md

Staggered Grid Thermo Mechanical Solver (SGTM)

The SGTM solver is a transient thermal solver designed to simulate evolving temperature fields in additive manufacturing (AM) and welding processes, where a moving energy source induces large thermal loads.

Governing Equations

The solver addresses the heat transfer equation:

$$ \rho c_p \frac{\partial T}{\partial t} = - \nabla \cdot (k \nabla T) + Q_{in} $$

where $Q_{in}$ is the volumetric heat source term.

The thermal stress is calculated as:

$$ \boldsymbol{\sigma}_h = E\alpha \left( T_h - T^0 \right)\mathbf{I} $$

where $E$ is the Young's modulus, $\alpha$ is the thermal expansion coefficient, $T_h$ is the element temperature, and $T^0$ is a reference temperature.

Numerical Methods

Thermal Evolution

A finite volume-like approach is used to evolve nodal temperatures, employing finite element basis functions for gradients.

$$ T_p^{n+1} = T_p^{n} + \frac{\Delta t}{m_{p}c_{p}} \left( \sum_{h \in p} \mathbf{N}_{hp} \cdot \mathbf{q}_h^{n+1/2} + \sum_{h \in p} \frac{1}{8} \int \limits_{V_{h}}{Q}^{n+1/2}_{h, , in}, dV - \bigg|\bigg|\sum_{i \in p} \mathbf{N}_{ip}\bigg|\bigg| , q_i^{n+1/2} \right) $$

Heat Flux

The heat flux at the element center is calculated as:

$$ \mathbf{q}_h = -k_h\nabla T_h $$

Boundary Conditions

The solver supports convection and radiation heat transfer on the boundaries:

• Convection: $q_i^{conv} = -h(T_p - T_{\infty})$
• Radiation: $q_i^{rad} = \epsilon \sigma (T_{c}^4 - T_{\infty}^4)$

Time Integration

Nodal temperatures are advanced using a second-order Runge-Kutta integration scheme.

Results

Example: Transient thermal simulation of a moving Gaussian heat source.

Implementation Details

• Element Type: Linear hexahedral elements.
• Material Handling: Formulated as a two-material problem (air and metal) for AM processes.
• Coupling: Can be coupled with static mechanical finite element solvers for thermal distortion analysis.