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2.7 Assembling

For a problem described by a system of PDEs a local matrix for each element $ T$ of the discretization $ T_h(\Omega)$, nucleus, is constructed. This nucleus is integrated into the general matrix of the system through a process called assembling. In this section the assembling algorithm applied in numerical schemes of problems discussed in Chapters 3 and 4 is derived.

The Jacobi matrix needed for the Newton method in the case of a finite element discretization is,

$\displaystyle \mathbf{G}=\frac{\partial (\ell_{1,1},\dots,\ell_{1,N},\dots,\ell...
...ots,\ell_{M,N})}{\partial (c_{1,1},\dots,c_{1,N},\dots,c_{M,1},\dots,c_{M,N})}.$ (2.40)

In the further text we will call this Jacobi matrix general matrix. The inputs of the general matrix are calculated according to,

$\displaystyle g(p+N(q-1),a+N(b-1))=\frac{\partial\ell_{qp}(c_{1,1},\dots,c_{1,N},\dots,c_{M,1},\dots,c_{M,N})}{\partial c_{ab}},$ (2.41)

where $ p,a\in\{1,\dots,N\}$, $ q,b\in\{1,\dots,M\}$, and the discrete operators $ \ell_{qp}$ are calculated as integrals over the discretization $ T_h(\Omega)$.
The calculation of the general matrix $ \mathbf{G}$ by the inner product $ (\varphi_i,\mathcal{L}(\mathbf{c}_h))_{h}$ over the whole discretisized domain $ T_h(\Omega)$ is rarely utilized in computer programs. A common approach is to construct the matrix $ \mathbf{G}$ by assembling it out of the nucleus matrix $ \Pi_T$ which is calculated for each element $ T\in T_h(\Omega)$ [11],

$\displaystyle \mathbf{\Pi}_T=\frac{\partial (\ell_{1,1}^e,\ell_{1,2}^e,\ell_{1,...
...rtial (c_{1,1},c_{1,2},c_{1,3},c_{1,4},\dots,c_{M,1},c_{M,2},c_{M,3},c_{M,4})},$ (2.42)

with the inputs,

$\displaystyle \mathbf{\pi}_T(s+4(p-1),r+4(l-1))=\frac{\partial\ell_{sp}^e(c_{1,1},\dots,c_{1,N},\dots,c_{M,1},\dots,c_{M,N})}{\partial c_{rl}},$ (2.43)

where $ s,r\in\{1,2,3,4\}$ and $ p,l\in\{1,\dots,M\}$.

The basic nodal function $ \varphi_A$, defined at the arbitrary point $ A\in T_h(\Omega)$ is non-zero only on the patch $ P(A)$. Furthermore $ \varphi_A$ can be represented as,

$\displaystyle \varphi_A = \sum_{T\in P(A)} \varphi_A^T.$ (2.44)

$ \varphi_A^T=1$ at the point $ A$ and $ \varphi_A^T=0$ at three other points of the tethraedal element $ T$.

The discrete operators $ \ell_{ap}$ obtained by testing of $ p$-th equation of the system with the basic nodal function $ \varphi_A$ is,

$\displaystyle \ell_{ap}=(\varphi_A,\mathcal{L}_p(\mathbf{c}_h))_{T_h(\Omega)}=\sum_{T\in P(A)}(\varphi_A^T,\mathcal{L}_p(\mathbf{c}_h))_{T}.$ (2.45)

The partial derivative of (2.45) with respect to $ c_{kl}$ is,

$\displaystyle \frac{\partial \ell_{ap}}{\partial c_{kl}}=\sum_{T\in P(A)}\frac{...
... c_{1,i} \varphi_i,\dots,\sum_{i=1}^4 c_{M,i}\varphi_i))_{T}}{\partial c_{kl}}.$ (2.46)

Obviously in (2.46) $ \varphi_A^T$ is equal to one of the basic nodal functions $ \varphi_1,\varphi_2,\varphi_3,\varphi_4$ at the element $ T$. Furthermore, the partial derivative is non-zero only if the $ c_{kl}$ stays for one nodal value of $ T$.

We now define the operator $ \chi(T,i)$, $ T\in T_h(\Omega)$, $ i=1,2,3,4$, which assigns a single global index $ I$ to every local index $ i$ of vertex belonging to the tethraedal element $ T$ . The inverse function $ \chi^{-1}(T,I)$ is also well-defined.

From (2.41), (2.43), and (2.46) we have,

$\displaystyle g(a+N(p-1),k+N(l-1))=\frac{\partial \ell_{ap}}{\partial c_{kl}}=\sum_{T\in P(A)}\mathbf{\pi}_T(\chi^{-1}(T,a)+4(p-1),\chi^{-1}(T,k)+4(l-1)).$ (2.47)

The assembling process consists of calculating of nucleus matrix $ \mathbf{\Pi}_T$ for each element of the mesh $ T_h(\Omega)$ and building the inputs of the global matrix $ \mathbf{G}$ as the sum on the right side of the equation (2.47). For the implementation of the assembling algorithm into software tools the following algorithm is used,

Image assembling_snapshot_cut

At the end of the assembling process the general matrix $ \mathbf{G}$ contains the values given by (2.40).

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Next: 2.8 General Solving Procedure Up: 2. Finite Element Method Previous: 2.6 Newton Methods

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