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4.1.2 Assembly

Applying the finite element method to solve a given PDE leads to an algebraic system of equations. In order to solve this system of equations, the global stiffness matrix, $ \mathbf{A}$, and the load vector, $ \mathbf{b}$, have to be determined. However, instead of computing them using directly (4.11) and (4.12), in practice they are computed by summing the contributions from the different elements [152,153,154] according to

$\displaystyle a_{ij} = \sum_{T\in T_h(\symDomain)} \left(L[N_i], N_j\right)_T =...
...\symDomain)} \int_{T} L[N_i(\vec r)] N_j(\vec r) d\symDomain,\qquad i,j=1,...,N$ (4.13)

$\displaystyle b_{j} = \sum_{T\in T_h(\symDomain)} \left(f, N_j\right)_T = \sum_...
... T_h(\symDomain)} \int_{T} f(\vec r) N_j(\vec r) d\symDomain, \qquad j=1,...,N.$ (4.14)

Note that $ \left(L[N_i], N_j\right)_T=0$ unless both $ N_i$ and $ N_j$ belong to the same element $ \T$. Thus, the calculations (4.13) and (4.14) can be limited to the nodes of the element $ \T$, so that $ i,j = 1,..., N_V$, where $ N_V$ is the number of vertices of the element. In this way, for each element $ T\in T_h(\symDomain)$, a $ N_V\times N_V$ matrix is obtained, which is called element stiffness or nucleus matrix. Thus, the general system matrix, $ \mathbf{A}$, can be computed by first computing the nucleus matrices for each $ T\in T_h(\symDomain)$ and then summing the contributions from each element according to (4.13) [152]. The right-hand side vector, $ \mathbf{b}$, is computed in the same way. This process of constructing the general system matrix is called assembly [152]. The main advantage of this assembly process is that it greatly simplifies the computation of the system matrix and right-hand side vector, since (4.11) and (4.12) can be easily calculated for each element of the domain discretization.


next up previous contents
Next: 4.1.3 Shape Function Up: 4.1 The Finite Element Previous: 4.1.1 Galerkin's Method

R. L. de Orio: Electromigration Modeling and Simulation