Mathematical Physics - Volume II - Numerical Methods

2.4 Determining of finite element matrices and finite element system matrices 43

Consider a typical finite element Ω e with nodes S e

e 2 . Variation formulation of equation

1 and S

(2.79) for finite elements Ω e , regardless of boundary conditions in z = 0 and z = l is: Z S e 2 S e 1 k ( u e h ) ′ ( v e h ) ′ d z = Z S e 2 S e 1 ¯ f v e h d z + P ( S e 1 ) v e h ( S e 1 ) − P ( S e 2 ) v e h ( S e 2 ) , 2 ) are exact, rather than approximate, and represent the natural boundary conditions in their corresponding nodes. It should be noted that equation (2.80) is defined for global coordinate z, and not the local ξ as usual, in order to make monitoring of the global stiffness matrix and load vector forming easier. Stiffness matrix and load vectors are typically determined in a local coordinate system, and then are translated into the global coordinate system using coordinate transformations, for all finite elements. Let us observe the form of u e h : (2.80) where u e h and v e h are the limits for u h and v h in Ω e . Values of fluxes P ( S e 1 ) and P ( S e

N e ∑ j = 1

u e

u e

e j ( z ) ,

j ψ

h ( z ) =

(2.81)

where N e is the number of nodes in the finite element Ω e , ψ e

j is the shape function, and u e

j is the

value of u e

h and z = z e j .

u e

e h ( z

e j ) j = 1 , . . . , N e .

j = u

(2.82)

By replacing (2.81) into (2.80), and assuming that v e h = ψ e

i we obtain a system of linear algebraic

equations in the following form: N e ∑ j = 1 k e i j u e

e i + P ( S

e 1 ) ψ

e i ( S

e 1 ) − P ( S

e 2 ) ψ

e 2 ( S

e 2 ) ,

j = f

(2.83)

where k e

i j is the stiffness matrix, and f e

i is the load vector for finite element Ω e :

i j = Z i = Z f e

S e 2

k e

k ( ψ e

e j ) ′ d z

i ) ′ ( ψ

(2.84)

S e 1

S e 2

¯ f ψ e

i d z .

(2.85)

S e 1

In practice, integrals in (2.84-2.85) are not determined analytically, but numerically, with sufficient accuracy. The load vector f e i is usually determined using interpolants and not f itself; e.g. if ¯ f is the continuous part of f (no concentrated forces) and if the following holds:

N e ∑ i = 1

¯ f ( z e

f e

e i ( z ) .

i ) ψ

h ( z ) =

(2.86)

then instead (2.85), we use:

i = Z

S e 1

f e

f e

e i d z .

h ψ

(2.87)

S e 1

In this way, the load vector can be determined based on nodal values. Since we have determined the necessary matrices and finite element equations, it is now necessary to form global equation systems, which will be used to represent the body being considered here. Let us adopt linear shape functions, i.e. two-node finite elements, defined by the following equations: k e 11 u e 1 + k e 12 u e 2 = f e 1 + P ( S e 1 ) k e 21 u e 1 + k e 22 u e 2 = f e 2 − P ( S e 2 ) , (2.88)