B2. Analysis based on a direct temporal integration

B2. Analysis based on a direct temporal integration 

Schemes of integration

Main Principles:

The principle of the different direct integration methods is to divide the studied interval into n smaller intervals of length ∆t = T/n and to verify the equilibrium at the discrete times Ti = i ∆t = i T/n. The differences between the methods (centered differences, Wilson, Newmark…) rely on the hypothesis that is considered about the variation of the kinematic values on the interval ∆t.

Explicit scheme:

If the displacement value can be calculated directly from the value (or values) of the previous step, the method is said explicit. In this case, the equilibrium is considered at the beginning of the increment.

Implicit scheme: 

The implicit methods must consider the equilibrium at the end of the interval, which requires the resolution of a linear system. 

As stated earlier, there exist many schemes of integration. The objective here is not to make an exhaustive description of all of them (see the specialized publications for that). Only two schemes will be presented in this guideline, the centered differences method: an explicit and conditionally stable method, and the Newmark method: unconditionally stable (for linear problems).

A scheme is unconditionally stable if, for any initial condition and time step ∆t, the solution is bounded, particularly when ∆t/T is large. On the contrary, a scheme is conditionally stable if the obtained solution is bounded only if ∆t is smaller than a critical value ∆t_crit.

Precision is a concept that is different from stability and it is extremely important for unconditionally stable schemes. Besides the inevitable rounding errors, the precision has an impact on two sources of approximations: an (artificial) increase of the period and a decrease of the amplitude. The influence of these two phenomena gains importance with the increase of ∆t, but independently. 

Centered Differences Method:
It is a scheme resembling finite differences. It relies on the approximation of the acceleration (Taylor series of the second order):

To obtain the same order of error for the velocity, one uses:

The displacements at t+∆t are obtained considering the equilibrium at t:

Thus, introducing the approximations of the acceleration and the velocity:

Of the form: 

This method requires a starting procedure to calculate q(-∆t) (from the equilibrium at t=0). The damping introduced by the scheme is equal to zero (no decrease in amplitude). 

Newmark method:

The scheme is based on the following approximations of the velocity and the displacement at the end of the intervals:

From the values of both parameters α and δ (ranging between 0 and 1) depends on the precision and the stability of the method: the values (δ = 1/2 and α = 1/4) lead to an unconditionally stable scheme named the Newmark method: it corresponds to considering the mean constant acceleration. Considering the equilibrium at the end of the studied interval (at t+Δt), one obtains: 

with 

and 

the coefficients being defined as follows:

The scheme also requires a starting procedure: the value of q¨(0) is obtained considering the equilibrium at t=0. Alike the centered differences method, the Newmark scheme with basis (δ = 1/2 and α = 1/4) does not introduce numerical damping. The values (α = 0, δ = ½) enable to get back to the centered differences method. 

Choice of the spatial and temporal discretization

Criteria about the element's sizes that satisfy the wavelengths

Different criteria can intervene. They are related to the precision of the expected results and the type of calculations used. 

For the transitory analysis, the general recommendation is to have a range of 8 to 10 elements per wavelength. The stationary waves being composed of a sum of propagation waves, the stationary dynamics analysis (in the case of earthquakes typically) will be using the same type of criteria. 

As a reminder, the general equation relating wavelength λ, frequency f, wave celerity c, and natural frequency ω is:

According to the type of elements used and the type of waves of interest, the following formulations are to be remembered: 

Volumetric elements in an isotropic environment (typical case of the modeling of an elastic soil od with Young’s modulus E, Poisson’s ratio , and density ρ):

Shell isotropic elements:

These formulations can be written in a more general form for anisotropic media with 2 principal directions 1 and 2:

One can find the size required to target a cutoff frequency of 40Hz for shells, plates, and volumetric elements. The size required is given by the numbers below / λ for 10 elements. 

This criterion is not necessarily sufficient. In the cases where a temporal analysis by transformation to the modal bases, or a spectral response is sought, a sensitivity analysis must be conducted to verify if the refining of the mesh leads to significant variations in the modal participation factors (or the modal mass if this criterion is examined). These elements are described later in the chapter, and their relations are expressed.

The criterion for the time steps

The conditionally stable temporal schemes, like the scheme of centered differences, must satisfy a condition concerning the time step chosen. This condition is often called CFL (Courant-Friedrichs-Levy). 

---

Example of application for a model of steel truss whose smallest element has a length L:

Let us consider a bar element with two nodes 1 and 2.

The bars have a length L, a section A, a density ρ, and a Young modulus E.

The mass matrix is considered as the total mass uniformly distributed at both ends. 

The matrices K and M are expressed as follows:

The characteristic polynomial is expressed such as:

Which leads to:

Since the propagation velocity of compression waves in a continuous medium is expressed as follows:

One can find that:

In this model whose mesh is irregular, it is the smallest time step that controls the global time step.