Mathematical Physics - Volume II - Numerical Methods

Chapter 2. Finite element method

34

these points, in order to know whether the flux is entering or exiting the body. In other words, we define the following: − k ( 0 ) − d u ( 0 ) d z = P 0 , (2.14) − k ( ℓ ) d u ( ℓ ) d z = P ℓ , (2.15) where d u ( 0 ) d z and d u ( ℓ ) d z is obviously derived from the same side. In many cases, the boundary flux is known, thus equations (2.14) and (2.15) become boundary conditions themselves. For second-order equation (e.g. equation (2.9)), these boundary conditions, which contain the first derivative of the unknown u are called natural boundary condition . In some other cases, it is assumed that the boundary flux is proportional to the difference in values of state variables at the boundaries and it values at a certain distance in its vicinity, which requires a simplified constitutive equation for the environment. For example, for z = z 0 = 0, this conditions is given in the following form: P 0 = p 0 [ u ( 0 ) − u 0 ] , (2.16) where p 0 is a known constant dependent on the material module of the environment, and u0 is the known value of environment state variable. By replacing equation (2.16) into (2.14), we obtain: (2.17) Since u ′ ( 0 ) once again appears here, the above equation is also a natural boundary condition for equation (2.8). The selection of boundary conditions in a boundary problem is, of course, influenced by its physical nature. For elastic deformation, this selection involves: • Essential boundary conditions , which define the boundary displacements. • Natural boundary conditions , which define: (a) boundary stresses, equations (2.14) and (2.15) or (b) linear combination of stresses and displacement at the boundary, Eq. (2.17). It should be noted that when the problem requires a stress-related boundary condition (flux) at both boundaries, the defined value of boundary stress must fulfil the global conservation condition: k ( 0 ) d u ( 0 ) d z = p 0 [ u ( 0 ) − u 0 ] ..

l Z 0

P l + P 0 =

f ( z ) d z .

(2.18)

We can now formulate the general physical problem:

d d z

d z

d u ( z )

= f ( x ) , z ∈ Ω i , i = 1 , 2 , 3 , 4 ,

k ( z )

(2.19)

−

with the following jump conditions in discontinuity points:

∥ P ( z 1 ) ∥ = 0 , ∥ P ( z 2 ) ∥ = ˆ f , ∥ P ( z 3 ) ∥ = 0

(2.20)

and the following boundary conditions:

u ( z ) = u 0 or u l

for z = 0 or z = l

P ( z ) = − P 0 P ( z ) = P l

for z = 0 for z = l

(2.21)

P ( z ) − p 0 u ( z ) = − p 0 u 0 for z = 0 P ( z ) − p l u ( z ) = − p l u l for z = l .