# 2D formulation for Electrostatic Problems

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\int_{\Omega^{(e)}} \mathbf{N^T} \rho_S \partial \Omega^{(e)} | \int_{\Omega^{(e)}} \mathbf{N^T} \rho_S \partial \Omega^{(e)} | ||

− | = 2 A^{(e)} | + | = 2 A^{(e)} \left ( \left [ \frac{1}{2} \quad \frac{1}{2} \quad 0 \right ]^T |

+ \left [ 0 \quad \frac{1}{2} \quad \frac{1}{2} \right ]^T | + \left [ 0 \quad \frac{1}{2} \quad \frac{1}{2} \right ]^T | ||

− | + \left [ \frac{1}{2} \quad 0 \quad \frac{1}{2} \right ]^T | + | + \left [ \frac{1}{2} \quad 0 \quad \frac{1}{2} \right ]^T \right ) |

\frac{1}{6} \rho_S | \frac{1}{6} \rho_S | ||

= A^{(e)} \frac{\rho_S}{3} \left [ 1 \quad 1 \quad 1 \right ]^T | = A^{(e)} \frac{\rho_S}{3} \left [ 1 \quad 1 \quad 1 \right ]^T |

## Latest revision as of 10:47, 27 November 2009

The 2D Electrostatic Poisson's equation given by the governing PDE and its boundary conditions:

can be written as (see the General formulation for Electrostatic Problems):

with (* n* is the number of nodes of the element):

## 2D formulation for Triangular Elements

After applying the numerical integration for triangular elements by using the natural coordinates, we obtain:

### Stiffness Matrix K^{(e)}

### Source Vector f^{(e)}

**Linear case**(**n**=1 integration point):_{p}

**Quadratic case**(**n**=3 integration points):_{p}