Matthew,

When using an implicit integration method (Newmark in this case, but the same was true when using Backward Euler or BDF2 in the previous Matlab assignment), you must perform a Newton-Raphson (N-R) calculation at

each time-step to calculate the states at the new time (t

n+1). Therefore, you must iterate to obtain better and better approximations to q

n+1, qd

n+1, qdd

n+1, and

ln+1.

So what N-R does is compute the following better and better approximations:

q

0n+1 , qd

0n+1 , qdd

0n+1 ,

l0n+1 (initial guess for N-R, that is N-R iteration 0)

q

1n+1 , qd

1n+1 , qdd

1n+1 ,

l1n+1 (N-R iteration 1)

q

2n+1 , qd

2n+1 , qdd

2n+1 ,

l2n+1 (N-R iteration 2)

... (etc. until you decide that N-R has converged)

As you can see, it's all about quantities at time t

n+1. The quantities at time t

n were already calculated and we're done with them by this time. Their only "use" at time t

n+1 is to provide the initial guess for N-R. In other words, we set:

qdd

0n+1 = qdd

nl0n+1 =

ln(then, using the Newmark integration formulas, we can also evaluate q

0n+1 and qd

0n+1 and so we can start the N-R iterations.)

Bottom line: "old" and "new" in the lecture notes refer to two successive N-R approximations to these quantities. In other words, as an example,

when I say qold this really means qoldn+1.

The

Jacobian should be recalculated at

each time-step but you do have two options:

- only evaluate it once per time-step, at the first iteration of the N-R loop. This means, evaluate it using q0n+1 and then reuse it for all subsequent N-R iterations until you're done calculating the quantities at time tn+1. But when you move on to time tn+2 you'll have to re-evaluate the Jacobian again.
- evaluate it multiple times per time-step, namely at each iteration of the N-R loop. This means, evaluate it continuously, using qoldn+1, i.e., the previous N-R approximation to qn+1.

For efficiency reasons, I suggest you use the first option above.

On the other hand, the

residual vector (i.e., the RHS in the linear system used within N-R to calculate the correction)

must use the values at the previous N-R iteration, that is it must be based on q

oldn+1, qd

oldn+1, qdd

oldn+1, and

loldn+1.

Hope that this clarifies things.

Good luck,

-Radu