Mathematical Physics - Volume II - Numerical Methods

2.1 Finite element application to solving of one-dimensional problems

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hence we obtain the balance equation:

d P ( z ) d z

= f ( z ) .

(2.7)

By replacing the constitutive equation (2.1) into the balance equation (2.6), we obtain an ordinary differential equation for the observed one-dimensional elliptical problem:

d dz

d z

d u ( z )

k ( z )

= f ( z ) .

(2.8)

−

At the points where the material module k is continuous, equation (2.8) can be represented as a second-order linear differential equation in the following way: − k ( z ) d 2 u ( z ) d z 2 − d k ( z ) d z d u ( z ) d z = f ( z ) . (2.9) (4) Let us now consider nodes with discontinuities, starting with z = z 3 . We have previously shown that there are no jumps in the flux: [ | P ( z 3 ) | ] = 0 z = z 3 . (2.10) Mean value theorem cannot be applied in this case, due to integrand discontinuity: ku ′ is continuous, but ( ku ′ ) ′ is not defined. Thus, there is no differential equation for z = z 3 !. At point (node) z = z 1 , where f is continuous, but k is not, the flux zero jump condition, ∥ P ( z 1 ) ∥ = 0, also holds. Equation (2.7) is satisfied, however since k ( z ) is not differentiable at z 1 , equation (2.8) cannot be transformed into (2.9). (5) At point z = z 2 , where a concentrated force f = ˆ f δ ( z − z 2 ) is acting, flux balance equation for the area around z 2 is given by: where ¯ f represents the smooth part of f . Same as in the previous case, we obtain a homogeneous jump condition (integral with respect to ¯ f has a limit of 0): ∥∥ = ˆ f for z = z 2 (2.12) where ˆ f does not depend on a and b . This is the exact reason why we cannot obtain a differential equation at z = z 2 . (6) Finally, let us consider the boundary conditions, i.e. conditions at point z = z 0 and z = z 4 . In reality, al physical systems must have some kind of interaction with their environment, which is usually included in the system equations by defining the flux or state variable values on body’s boundaries. Hence, in a general case, approximate modelling of the body’s is necessary in order to define the boundary conditions. Shown in Figure 2.1c, is a small area of a bar which contains the left boundary, where the flux is defined as P 0 . Flux in this area is conserved if the following holds: P ( b ) − P ( a ) = b Z a f ( z ) d z = b Z a ¯ f ( z ) d z = b Z a ˆ f δ ( z − z 2 ) d z (2.11)

a Z 0

P ( a ) − P ( 0 ) =

f ( z ) d z ,

(2.13)

since in the boundary case, when a → 0, P ( 0 ) = P 0 . When defining the flux in the boundary points, we must take into account that it is necessary to know the direction of displacement change rate in