Mathematical Physics - Volume II - Numerical Methods

3. Comparison of finite element method and finite difference method

As an example for comparing finite element method and finite difference method, we will consider an equation describing the free oscillations of a beam:

∂ 4 Ω ∂ x 4

∂ 2 Ω ∂ t 2

EI

+ m

= 0 ,

(3.1)

which will be transformed by replacing Ω ( x , t ) = W ( x ) e i ω t , for the purpose of determining its exact solution. We now have: d 4 W d x 4 − λ W = 0 (3.2) where m is unit mass per unit length, ω is the eigenfrequency, E is the elasticity module and I is the moment of inertia, λ = m ω 2 EI = β 4 . (3.3) The general solution of (3.1) is: W ( x ) = C 1 sin ( β x )+ C 2 cos ( β x )+ C 3 sinh ( β x )+ C 4 cosh ( β x ) . (3.4) Constants C 1 − C 4 are determined from the initial conditions: W ( 0 ) = W ( L ) = 0 dW dx ( 0 ) = dW dx ( L ) = 0 (3.5) which results in four homogeneous equations with four unknowns. In order for a non-trivial solution to exist, the system matrix needs to be equal to zero, from which we obtain the frequency equation