Mathematical Physics - Volume II - Numerical Methods

2.4 Determining of finite element matrices and finite element system matrices 45

2.4.1 Application of finite element method to parabolic and hyperbolic partial differential equations Let us now consider time-dependent problems, i.e. parabolic and hyperbolic partial differential equation. For the sake of simplicity, we will once again observe only a single space variable. By introducing time-dependence for the state variable, we obtain the law of balance in the following form: ∂ P ( z , t ) ∂ z − f ( z , t )+ ∂ G ( z , t ) ∂ t = 0 0 < z < l t > 0 , (2.95) where P is the flux, f is the density of distributed internal sources, and G is a quantity which is being "conserved" during the process. For example, if energy conservation (balance) is in question, ∂ G ∂ t , it can represent the rate of entropy change per unit length and unit temperature, in which case it is related to the state variable (temperature) via the state equation: ∂ G ( z , t ) ∂ t = C ( z , t ) ∂ u ( z , t ) ∂ t , (2.96) where C ( z , t ) is a material property – specific heat. If, for example, (2.95) represent the law of motion quantity balance in a deformable body, and u is the displacement field, then the motion quantity G in z at a time t is: G ( z , t ) = ρ ( z ) ∂ u ( z , t ) ∂ t ; ∂ G ( z , t ) ∂ t = ρ ( z ) ∂ 2 u ( z , t ) ∂ t 2 , (2.97) where ρ ( z ) is the mass density in point z . The following state equation applies to flux P :

∂ u ( z , t ) ∂ z

P ( z , t ) = − k ( z )

(2.98)

where | k ( z ) | ≥ k 0 = const > 0 for ∀ z ∈ [ 0 , l ] , wherein a convention is adopted according to which k is considered positive in the case of heat transfer, and negative for elasticity. By using equations (2.96)-(2.98) in order to eliminate G from (2.95), we obtain two type of partial differential equations, one of which is parabolic:

∂ ∂ z

∂ z −

∂ u ( z , t )

∂ u ( z , t )

C ( z , t )

k ( z )

f ( z , t ) = 0 0 < z < l t > 0 ,

(2.99)

∂ t −

which holds for state equation (2.96-98) and is related to heat conduction problems, and the second, hyperbolic equation:

∂ ∂ z

∂ z −

∂ 2 u ( z , t )

∂ u ( z , t )

ρ ( z )

k ( z )

f ( z , t ) = 0 0 < z < l t > 0 .

(2.100)

∂ t 2 −

We will first analyse finite element method application to parabolic partial differential equation, (2.99). For the sake of simplicity, we will assume that C ( z ) is independent from t , and we then have that C ( z ) ≥ c 0 > 0 for ∀ z , where c 0 is a positive constant. In addition to boundary conditions, defined in the following simple form: u ( 0 , t ) = u ( l , t ) = 0 t ≥ 0 , We also need to define the initial conditions:

u ( z , 0 ) = ˜ u ( z ) 0 < z < l ,