Mathematical Physics - Volume II - Numerical Methods

Chapter 2. Finite element method

38

we obtain the following expressions for constants C 1 and C 0 : C 1 = − k h ( u i + 1 − u i )

(2.45)

k 2

( u i + u i + 1 ) ,

(2.46)

C 0 = −

hence the exact solution is:

1 2

h

1 2

h

2 z

2 z

u =

1 +

u i + 1 +

1 −

u i

(2.47)

If we now introduce the non-dimensional constant ξ = 2 z h

(for z = − h / 2, ξ = − 1; z = h / 2,

ξ = 1), it follows that:

u i u i + 1

u = 1

1 2 ( 1 + ξ )

2 ( 1 − ξ )

(2.48)

.

If we introduce the so-called interpolation functions, defined as ψ 1 = 1 1 2 ( 1 + ξ ) , we can observe that they define a linear interpolation of displacement u (state variable) between nodes i and i + 1 in a system, i.e. elements 1 and 2, Fig. 2.2. It is easily noticeable that these two functions have values of 1 on one end, and 0 on the other. 2 ( 1 − ξ ) and ψ 2 =

1 ψ (ξ)

2 ψ (ξ)

1

1

1

2

Figure 2.2: Linear interpolation function.

It is also evident that these two functions are linearly independent and continuous at the element boundaries. The second conclusion follows from the fact that two base functions, constructed from two adjacent segments of an interpolation function in a node, have the ordinate value of 1 in their own corresponding nodes. Let us now observe the non-homogeneous differential equation with a constant term: ( ku ′ ) ′ = C 2 . (2.49) This equation physically corresponds to the elastic stress state problem with a constant volumet ric force. The exact solution of equation (2.49-53) is: (2.50) Formally, we can consider C 2 as the unknown, as long as the displacement value u is defined in one of the points within the interval, e.g. in its middle, in addition to the previously defined boundary conditions. Boundary conditions in this case can be written as: u = u i − 1 for z = − h / 2 (2.51) u = u i for z = 0 (2.52) u = u i + 1 for z = h / 2 . (2.53) ku + 1 2 C 2 z 2 + C 1 z + C 0 = 0 .