Mathematical Physics - Volume II - Numerical Methods

3.5 Finite element approximations

71

Based on conservation laws, [ | σ n | ] equals zero through the inter-surface where there are no point or line sources, hence for smooth f we have that: S ( 0 ) 1 = 0 . (3.105) On the other hand, if the source function f contains sources within a point or a line, then the jump [ | σ | ] n represents a source equation and is not zero. If the source is a point, then, strictly speaking, variation formulation is not applicable. Thus, it is assumed that f has the following form: f ( x , y ) = ¯ f ( x , y )+ ˆ f δ ( x − x i , y − y i ) (3.106) where ¯ f is the smooth (integrable) part of f , and ˆ f δ ( x − x i , y − y i ) denotes the source in point ( x i , y i ) ∈ Ω h , with a magnitude f . As was the case with the one-dimensional problem, mesh nodes Ω h are placed in the source point, since then integrals (3.91) and (3.96) only contain the smooth part of f : S ( 0 ) 1 = 4 ∑ e = 1 Z ∂ Ω e σ n φ 1 d s = 4 ∑ m = 1 Z Γ m [ | σ n | ] φ 1 d s . (3.107) The presence of the base function φ 1 1 indicates that S ( 0 ) 1 represents the weight mean of these jumps in the internal node 1. If we (figuratively) balance these flux jumps which differ from zero, with a source in point ˆ f , we obtain: S ( 0 ) 1 = ˆ f (3.108) when (3.106). holds. It should also be noted that the point source in the case of a two-dimensional problem provides a singular solution. According to essential boundary conditions, values u h are given in nodes at ∂ Ω 1 h . Since σ n is unknown at ∂ Ω 1 h , S ( 1 ) i cannot be defined here. However, if all nodal displacements, u 1 , u 2 , · · · , u N , are known, S ( 1 ) i can be approximated directly from (3.103). On the boundary ∂ Ω 2 h , natural boundary conditions are defined, thus the following holds: σ n ( s ) = ˆ σ ( s ) , (3.109)

thus, we have the following approximation:

N ∑ i = 1 Z ∂ Ω

S ( 2 )

i ∼ =

ˆ σφ i d s .

(3.110)

2 h

Symbol ∂ Ω e

2 h refers to the part of ∂ Ω e which intersects with ∂ Ω 2 h . Now we have reached a linear

algebraic system of equations as follows:

( 2 ) i , i , j = 1 , 2 , . . . , N ,

K i j u j = F i − S

(3.111)

where

E ∑ e = 1 E ∑ e = 1

K e

K i j =

i j

(3.112)

F e

F i =

i .