Finite Element Method

1 Weak Form of the Heat Equation

Strong form:

[ c_p = (kT) ]

Multiply by a test function (v), integrate over (), and integrate by parts:

[ c_p v,d + kTv,d = _{q}q’’v,d - {h}h(T-T)v,d ]

For a Robin boundary:

[ -kTn=h(T-T_) ]

the weak contribution is:

[ _{h}hTv,d = {h}hTv,d ]

2 Backward Euler Time Discretization

[ c_p v,d + kT^{n+1}v,d + _{_h} hT^{n+1}v,d = ]

[ c_p v,d + {h} hTv,d + _{_q} q’’v,d ]

3 Gridap.jl Skeleton

using Gridap

domain = (0.0, 1.0, 0.0, 1.0)
partition = (40, 40)
model = CartesianDiscreteModel(domain, partition)

order = 1
reffe = ReferenceFE(lagrangian, Float64, order)

V = TestFESpace(model, reffe; conformity=:H1)
U = TrialFESpace(V)

Ω = Triangulation(model)
dΩ = Measure(Ω, 2)

k = 1.0

a(T,v) = ∫(k*∇(T)⋅∇(v))dΩ
l(v) = 0

op = AffineFEOperator(a, l, U, V)
T = solve(op)

4 When to Use FEM

FEM is especially useful for:

complex geometries,
multi-material heat conduction,
weak-form boundary conditions,
structural thermal analysis,
coupled thermomechanical problems.

---
title: "Finite Element Method"
---

## Weak Form of the Heat Equation

Strong form:

\[
\rho c_p\frac{\partial T}{\partial t}
=
\nabla\cdot(k\nabla T)
\]

Multiply by a test function \(v\), integrate over \(\Omega\), and integrate by parts:

\[
\int_\Omega
\rho c_p
\frac{\partial T}{\partial t}
v\,d\Omega
+
\int_\Omega
k\nabla T\cdot\nabla v\,d\Omega
=
\int_{\Gamma_q}q''v\,d\Gamma
-
\int_{\Gamma_h}h(T-T_\infty)v\,d\Gamma
\]

For a Robin boundary:

\[
-k\nabla T\cdot n=h(T-T_\infty)
\]

the weak contribution is:

\[
\int_{\Gamma_h}hTv\,d\Gamma
=
\int_{\Gamma_h}hT_\infty v\,d\Gamma
\]

## Backward Euler Time Discretization

\[
\int_\Omega
\rho c_p
\frac{T^{n+1}}{\Delta t}
v\,d\Omega
+
\int_\Omega
k\nabla T^{n+1}\cdot\nabla v\,d\Omega
+
\int_{\Gamma_h}
hT^{n+1}v\,d\Gamma
=
\]

\[
\int_\Omega
\rho c_p
\frac{T^n}{\Delta t}
v\,d\Omega
+
\int_{\Gamma_h}
hT_\infty v\,d\Gamma
+
\int_{\Gamma_q}
q''v\,d\Gamma
\]

## Gridap.jl Skeleton

```julia
using Gridap

domain = (0.0, 1.0, 0.0, 1.0)
partition = (40, 40)
model = CartesianDiscreteModel(domain, partition)

order = 1
reffe = ReferenceFE(lagrangian, Float64, order)

V = TestFESpace(model, reffe; conformity=:H1)
U = TrialFESpace(V)

Ω = Triangulation(model)
dΩ = Measure(Ω, 2)

k = 1.0

a(T,v) = ∫(k*∇(T)⋅∇(v))dΩ
l(v) = 0

op = AffineFEOperator(a, l, U, V)
T = solve(op)
```

## When to Use FEM

FEM is especially useful for:

- complex geometries,
- multi-material heat conduction,
- weak-form boundary conditions,
- structural thermal analysis,
- coupled thermomechanical problems.