2 DIMENTIONAL FLUID MECHANICS
The problem of linear elastostatics described in detail in can be extended to include the effects of inertia. The resulting equations of motion take the form
where u = u(x1 , x2, x3 , t) is the unknown displacement field, ρ is the mass density, and I = (0, T ) with T being a given ti me. Also, u0 and v0 are the prescribed initi al displacement and velocity fields. Clearly, two sets of boundary conditions are set on Γu and Γq , respectively, and are assumed to hold throughout the time interval I . Likew ise, two sets of initial conditions are set for the whole domain Ω at time t = 0. The stron g form of the resulting initial/boundary- value problem is stated as follows: given functions f , t, u¯ , u0 and v0, as well as a constitutive equation for σ, find u in Ω × I , such that the equations are satisfied.
A Galerkin-based weak form of the linear elastostatics problem has been derived in Sec-tion In the elastodynamics case, the only substantial difference involves the inclusion of the term RΩ w • ρu¨ dΩ, as long as one adopts the semi-discrete approach. As a result, the weak form at a fixed time can be expressed as
Following a standard proced ure, the contribution of the forcing vector Fi nt,e due to interele- ment tractions is neglected upon assembly of the global equations . As a result, the equations is give rise to their assembled counterparts in the form
Mu + Kuˆ = F ,
where uˆ is the global unknown displacement vector1 . The preceding equat ions are, of course, subject to initial conditions t hat can be written in vectorial form as uˆ(0) = uˆ0 and vˆ(0) = vˆ0
The most commonly emp loyed method for the numerical solution of t he system of coupled linear second-order ordi nary differential equations is the Newmark m ethod. This
method is based on a time series expansion of ˆu and ˆ u˙ := v.ˆ Concentrating on the time interval (tn ,tn+1], the New mark method is defined by the equations
It is clear that the Newmark equations define a whole family of time inte grators.
It is important to distinguish this family into two categories, namely implicit and explicit integrators, corresponding to β > 0 and β = 0, respectively.
The overhead “hat” symbol is used to distinguish between the vector field u and the solution vector uˆemanating fr om the finite element approximation of the vector field u.
The general implicit Newmar k integration method may be implemented as follows: first, solve (9.18)1 for aˆn+1 , namely write
Then, substitute (9.19) into the semi-discrete form (9.17) evaluated at tn+1 to find that
After solving for uˆn+1, one ma y compute the acceleration aˆn+1 from and the velocity vˆn+1 from.
Finally, the general explicit N ewmark integration method may be implemented as follows: starting from the semi-discrete e quations evaluated at tn+1, one may substitute uˆn+1from to find that
If M is rendered diagonal (see discussion in our pages ), then aˆn+1 can be determined without solving any coupled linear algebraic equations. Then, ˆ are the velocities bvn+1 immediately computed from (9. 18)2. Also, the displacements uˆ n+1 are computed from indepen-dently of the acceleratio ns aˆn+1 .