The elastic equations can then be rewritten in much simpler a form:
$$
-
- \nabla (\lambda+\mu) (\div \vec u)
+ \nabla \lambda (\div \vec u)
-
(\nabla \cdot \mu \nabla) \vec u
+ -
+ \div \mu (\nabla \vec u)^T
=
\vec f,
$$
$$
a(\vec u, \vec v) =
\left(
- (\lambda+\mu) \div \vec u, \div \vec v
+ \lambda \div \vec u, \div \vec v
\right)_\Omega
+
- \sum_k
+ \sum_{i,j}
\left(
- \mu \nabla u_k, \nabla v_k
+ \mu \partial_i u_j, \partial_i v_j
+ \right)_\Omega,
+ +
+ \sum_{i,j}
+ \left(
+ \mu \partial_i u_j, \partial_j v_i
\right)_\Omega,
$$
-or as a sum over components:
+or also writing the first term a sum over components:
$$
a(\vec u, \vec v) =
- \sum_{k,l}
+ \sum_{i,j}
\left(
- (\lambda+\mu) \partial_l u_l, \partial_k v_k
+ \lambda \partial_l u_l, \partial_k v_k
\right)_\Omega
+
\sum_{k,l}
\left(
- \mu \partial_l u_k, \partial_l v_k
+ \mu \partial_i u_j, \partial_i v_j
+ \right)_\Omega,
+ +
+ \sum_{i,j}
+ \left(
+ \mu \partial_i u_j, \partial_j v_i
\right)_\Omega.
$$
\sum_{k,l}
\left\{
\left(
- (\lambda+\mu) \partial_l (\Phi_i)_l, \partial_k (\Phi_j)_k
+ \lambda \partial_l (\Phi_i)_l, \partial_k (\Phi_j)_k
\right)_\Omega
+
\left(
\mu \partial_l (\Phi_i)_k, \partial_l (\Phi_j)_k
\right)_\Omega
+ +
+ \left(
+ \mu \partial_l (\Phi_i)_k, \partial_k (\Phi_j)_l
+ \right)_\Omega
\right\}
\\
=
\end{multline*}
\end{center}
We note that here and in the following, the indices $k,l$ run over spatial
-directions, i.e. $0\le k,l \le d-1$, and that indices $i,j$ run over degrees
+directions, i.e. $0\le k,l < d$, and that indices $i,j$ run over degrees
of freedoms.
The local stiffness matrix on cell $K$ therefore has the following entries:
\sum_{k,l}
\left\{
\left(
- (\lambda+\mu) \partial_l (\Phi_i)_l, \partial_k (\Phi_j)_k
+ \lambda \partial_l (\Phi_i)_l, \partial_k (\Phi_j)_k
\right)_K
+
\left(
\mu \partial_l (\Phi_i)_k, \partial_l (\Phi_j)_k
\right)_K
+ +
+ \left(
+ \mu \partial_l (\Phi_i)_k, \partial_k (\Phi_j)_l
+ \right)_K
\right\},
$$
-where $i,j$ now are local degrees of freedom and therefore $0\le i,j \le N-1$.
+where $i,j$ now are local degrees of freedom and therefore $0\le i,j < N$.
In these formulas, we always take some component of the vector shape functions
$\Phi_i$, which are of course given as follows (see their definition):
$$
\sum_{k,l}
\Bigl\{
\left(
- (\lambda+\mu) \partial_l \phi_i\ \delta_{l,comp(i)},
- \partial_k \phi_j\ \delta_{k,comp(j)}
+ \lambda \partial_l \phi_i\ \delta_{l,comp(i)},
+ \partial_k \phi_j\ \delta_{k,comp(j)}
\right)_K
\\
&\qquad\qquad +
\mu \partial_l \phi_i\ \delta_{k,comp(i)},
\partial_l \phi_j\ \delta_{k,comp(j)}
\right)_K
+ +
+ \left(
+ \mu \partial_l \phi_i\ \delta_{k,comp(i)},
+ \partial_k \phi_j\ \delta_{l,comp(j)}
+ \right)_K
\Bigr\}
\\
&=
\left(
- (\lambda+\mu) \partial_{comp(i)} \phi_i,
- \partial_{comp(j)} \phi_j
+ \lambda \partial_{comp(i)} \phi_i,
+ \partial_{comp(j)} \phi_j
\right)_K
+
\sum_l
\partial_l \phi_j
\right)_K
\ \delta_{comp(i),comp(j)}
+ +
+ \left(
+ \mu \partial_{comp(j)} \phi_i,
+ \partial_{comp(i)} \phi_j
+ \right)_K
\\
&=
\left(
- (\lambda+\mu) \partial_{comp(i)} \phi_i,
- \partial_{comp(j)} \phi_j
+ \lambda \partial_{comp(i)} \phi_i,
+ \partial_{comp(j)} \phi_j
\right)_K
+
\left(
\mu \nabla \phi_i,
\nabla \phi_j
\right)_K
+ +
+ \left(
+ \mu \partial_{comp(j)} \phi_i,
+ \partial_{comp(i)} \phi_j
+ \right)_K
\ \delta_{comp(i),comp(j)}.
\end{align*}
\end{center}