Chapter 6 Strong and weak forms for multidimensional scalar field problems

Divergence theorem and Green's formula

Green's theorem:

If \(\theta(x,y)\in C^0\) and integrable, then

\(\int\limits_{\Omega}\vec{\nabla}\theta d\Omega=\oint\limits_{\Gamma}\theta \vec{n}d\Gamma\) or \(\int\limits_{\Omega}\pmb{\nabla}\theta d\Omega=\oint\limits_{\Gamma}\theta \vec{n}d\Gamma\)

Divergence theorem:

If \(\vec{q}\) is \(C^0\) and integrable, then

\(\int\limits_{\Omega}\vec{\nabla}\cdot\vec{q}d\Omega=\oint\limits_{\Gamma}\vec{q}\cdot\vec{n}d\Gamma\) or \(\int\limits_{\Omega}\pmb{\nabla}^T\pmb{q}d\Omega=\oint\limits_{\Gamma}\pmb{q}^T\pmb{n}d\Gamma\)

Green's formula / Green's first identity:

\(\int\limits_{\Omega}w\vec{\nabla}\cdot\vec{q}d\Omega=\oint\limits_{\Gamma}w\vec{q}\cdot\vec{n}d\Gamma-\int\limits_{\Omega}\vec{\nabla}w\cdot\vec{q}\)

Strong form and weak form for heat conduction

Strong form

Energy balance: \(\vec{\nabla}\cdot\vec{q}-s=0\) on \(\Omega\)

Fourier's law: \(\vec{q}=-k\vec{\nabla}T\) on \(\Omega\)

natural BC: \(q_n=\vec{q}\cdot\vec{n}=\overline{q}\) on \(\Gamma_q\)

essential BC: \(T=\overline{T}\) on \(\Gamma_T\)

Weak form

Find \(T\in U\) such that

\(\int\limits_{\Omega}\vec{\nabla}w\cdot \vec{q}=\int\limits_{\Gamma_q}w\vec{q}d\Gamma-\int\limits_{\Omega}ws d\Omega\) for \(\forall w\in U_0\)

or in matrix notation

\(\int\limits_{\Omega}(\pmb{\nabla}w)^T\pmb{q}=\int\limits_{\Gamma_q}w\vec{q}d\Gamma-\int\limits_{\Omega}ws d\Omega\) for \(\forall w\in U_0\)

\(\int\limits_{\Omega}(\pmb{\nabla}w)^T\pmb{D}\pmb{\nabla}T=-\int\limits_{\Gamma_q}w\vec{q}d\Gamma+\int\limits_{\Omega}ws d\Omega\) for \(\forall w\in U_0\)

where the generalized Fourier's law \(\pmb{q}=-\pmb{D}\pmb{\nabla}T\) is used

Strong form and weak form for scalar steady-state Advection-Diffusion equation in 2D

General form of Advection-Diffusion equation:

\(\vec{\nabla}\cdot (\theta\vec{v})+\vec{\nabla}\cdot\vec{q}-s=0\)

Introduce the continuity equation for steady-state problems of incompressible materials

\(\vec{\nabla}\cdot\vec{v}=0\)

Then the above equation becomes

\(\vec{v}\cdot\vec{\nabla}\theta+\vec{\nabla}\cdot\vec{q}-s=0\)

Use the matrix notations and the generalized Fourier's law, the strong form can be written as

Strong form

\(\pmb{v}^T\pmb{\nabla}\theta-\pmb{\nabla}^T(\pmb{D}\pmb{\nabla}\theta)-s=0\) on \(\Omega\)

\(\theta=\overline{\theta}\) on \(\Gamma_{\theta}\)

\(q_n=\pmb{q}^T\pmb{n}=\overline{q}\) on \(\Gamma_{q}\)

Weak form

find the trial solutions \(\theta(x,y)\in U\) such that

\(\int\limits_{\Omega}w\pmb{v}^T\pmb{\nabla}\theta d\Omega+\int\limits_{\Omega}(\pmb{\nabla}w)^T\pmb{D}\pmb{\nabla}\theta d\Omega+\int\limits_{\Gamma_q}w\vec{q}d\Gamma-\int\limits_{\Omega}ws d\Omega=0\)