Heat Transfer Equation: Weak Form

Heat Transfer Equation

• Describes how heat is transferred through a material over time
• The differential equation is given by: $\rho c \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + Q$
• $\rho$ represents density
• $c$ represents specific heat
• $T$ represents temperature
• $t$ represents time
• $k$ represents thermal conductivity
• $Q$ represents the heat source

Weak Form

• Obtained by multiplying the heat transfer equation by a test function $v$ and integrating over the domain $\Omega$
• $\int_{\Omega} \rho c \frac{\partial T}{\partial t} v , d\Omega = \int_{\Omega} \nabla \cdot (k \nabla T) v , d\Omega + \int_{\Omega} Q v , d\Omega$
• Integration by parts (Green's theorem) is applied: $\int_{\Omega} \nabla \cdot (k \nabla T) v , d\Omega = \int_{\Gamma} v k \nabla T \cdot \hat{n} , d\Gamma - \int_{\Omega} k \nabla T \cdot \nabla v , d\Omega$
• $\Gamma$ represents the boundary of $\Omega$
• $\hat{n}$ represents the outward normal vector to $\Gamma$

Boundary Conditions

• Dirichlet boundary condition: $T = T_D$ on $\Gamma_D$, where $T_D$ is the specified temperature
• Neumann boundary condition: $-k \nabla T \cdot \hat{n} = q$ on $\Gamma_N$, where $q$ is the heat flux
• Robin boundary condition: $-k \nabla T \cdot \hat{n} = h(T - T_{\infty})$ on $\Gamma_R$, where $h$ is the heat transfer coefficient and $T_{\infty}$ is the ambient temperature

FEM Formulation

• Approximates the temperature $T(x, t)$ as a sum of nodal temperatures $T_i(t)$ multiplied by basis functions $\phi_i(x)$: $T(x, t) \approx \sum_{i=1}^{N} T_i(t) \phi_i(x)$
• Substituting this into the weak form yields: $\int_{\Omega} \rho c \sum_{i=1}^{N} \frac{\partial T_i}{\partial t} \phi_i v , d\Omega = -\int_{\Omega} k \nabla (\sum_{i=1}^{N} T_i \phi_i) \cdot \nabla v , d\Omega + \int_{\Gamma} v k \nabla T \cdot \hat{n} , d\Gamma + \int_{\Omega} Q v , d\Omega$

Matrix Form

• Choosing the test function $v = \phi_j$ leads to the matrix form: $C \dot{T} + K T = F$
• $C_{ji} = \int_{\Omega} \rho c \phi_i \phi_j , d\Omega$ is the heat capacity matrix
• $K_{ji} = \int_{\Omega} k \nabla \phi_i \cdot \nabla \phi_j , d\Omega + \int_{\Gamma_R} h \phi_i \phi_j , d\Gamma$ is the stiffness matrix
• $F_j = \int_{\Omega} Q \phi_j , d\Omega + \int_{\Gamma_N} q \phi_j , d\Gamma + \int_{\Gamma_R} h T_{\infty} \phi_j , d\Gamma$ is the force vector

Time Integration

• Backward Euler: $(C + \Delta t K) T^{n+1} = C T^n + \Delta t F^{n+1}$
• Forward Euler: $C T^{n+1} = (C - \Delta t K) T^n + \Delta t F^n$

Element-wise Calculation

• The matrices $C$, $K$, and $F$ are assembled from element-wise contributions ($C^e, K^e, F^e$)
• The global matrices are then assembled: $C = \sum_{e=1}^{N_{elem}} C^e$, $K = \sum_{e=1}^{N_{elem}} K^e$, and $F = \sum_{e=1}^{N_{elem}} F^e$