Prompt: In section 2.3 of the text (pg. 60-64) the strong form and weak form of the equations for linear heat conduction are given. Using a Galerkin based method of weighted residuals, show the construction of the given weak form starting from the strong form, stated as:
Given $f : \Omega \rightarrow \mathbb{R}, q : \Gamma{o} \rightarrow \mathbb{R} \text { and } h : \Gamma{h} \rightarrow \mathbb{R}, \text { find } u : \overline{\Omega} \rightarrow \mathbb{R}$ such that
\begin{align}
q_{i, i}&=f \quad \text { in } \Omega &\quad \text { (Heat Equation) }
\label{eqnHeat}
\\
u
&=
g \quad \text { on } \Gamma_{g}
\quad &\text { (Essential Boundaries) } \\
-q_{i} n_{i}
&=
h \quad \text { on } \Gamma_{h}
\quad &\text { (Natural Boundaries) }
\end{align}
With the functions from candidate spaces: $u \in \delta$ and $w \in \mathcal{V}$.
δV={uu∈H1(Ω),u(Γ)=g}={ww=H1(Ω),w(Γ)=0}
Beginning with the Heat Equation $[\ref{eqnHeat}]$, we cannot solve in the strong form presented. Instead, we minimize a weighting function $w$ in the domain $\Omega$, where the closure of $\Omega$ is $\Gamma$. Distributing the weight function to produce two integral terms. Solving the first term using integration by parts (IBP) produces
\begin{align}
0
&=
\int_{\Omega}
\overbrace{w\left(q_{i, i}-f\right) }^\text{distribute}
d \Omega
=
\overbrace{\int_{\Omega} wq_{i, i} d \Omega }^\text{weaken w/ IBP}
- \int_{\Omega} w f d \Omega \\
&=
-
\overbrace{\int_{\Omega} w_{, i} q_{i} d \Omega
+
\int_{\Gamma_h} w
\underbrace{q_{i} n_{i}}_\text{BC}
d \Gamma
}^\text{from IBP}-
\int_{\Omega} w f d \Omega \label{eqnLambda}\\
&=-\int_{\Omega} w_{, i} q_{i} d \Omega-\int_{\Gamma_{h}} w h d \Gamma-\int_{\Omega} w f d \Omega
\end{align}
Algebraic rearrangement with knowns on the right and the unknown on the left:
\boxed{
-\int_{\Omega} w_{, i} q_{i} d \Omega
=
\int_{\Gamma_{h}} w h d \Gamma
+\int_{\Omega} w f d \Omega
}
\label{eqn2BC}
Problem 2
$\textcolor{red}{\texttt{To Be Graded }}$ Exercise 1 on page 68 of the text book. This exercise is a multidimensional analog of the the one contained in Sec. 1.8. Let
Γint=(e=1⋃nelΓe)−Γ (interior element boundaries)
One side of $\Gamma{int}$ is arbitrarily designated to be the “+ side” and the other is the “- side”. Let $n^+$ and $n^-$ be unit normals to $\Gamma{int}$, with the relationship:
Consider the weak formulation$[\ref{eqn2BC}]$ and assume that $w$ and $u$ are smooth on the element interiors but may experience discontinuities in gradient across element boundaries. Restating $[\ref{eqn2BC}]$, the equation is made homogenous by moving the left hand side (LHS) term to the right hand side (RHS) of the equation.
−∫Ωw,iqidΩ=∫ΓhwhdΓ+∫ΩwfdΩ0=∫Ωw,iqidΩ+∫ΓhwhdΓ+∫ΩwfdΩ0=IBP∫Ωw,iqidΩ+∫ΩwfdΩcollect same domain+∫ΓhwhdΓ
The normal flux relationship of Eqn $[\ref{eqnJump}]$ is applied into the middle term. Decomposing the $\Gamma = \cancelto{0}{\Gamma_g} + \Gamma_h$, produces
${\bf d}^e$ is the element temperature vector. Show that the heat flux vector at point $x \in \Omega^e$ can be calculated from the formulation
q(x)=−D(x)B(x)de=−D(x)a=1∑nenBadae
Not sure how this expression was constructed, to move the $B$ matrix as a summation of element nodes. It looks like collocation because of evaluation at specific locations but I’m not sure how to prove it.
Problem 4
Exercise 2 on page 71 of the textbook, Consider the strong statement where the replacement on $\Gamma_h$ is
λu−qini=h on Γh
where $\lambda\geq0$ is a function of $x \in\Gamma_h$. To generalize, continue from $[\ref{eqnLambda}]$, where the additional term creates:
Assuming that the original weak form expression was also positive definite, then the additional contribution to $k_{ab}^e$ is based purely on $\lambda$. To prove that $\bf K$ is positive definite, then we would need to show that $\bf c^T\cdot K \cdot c \geq 0$ and $0$ only when $\bf c$ is the 0 vector, $\bf 0 $.