Algorithm Template for Nonlinear Dynamic Solver using the Newmark & Newton Loops

Nonlinear dynamic analysis plays an important role in seismic performance evaluation of structures. The solution of nonlinear dynamic can be solved by using Newmarks's method. This article provides a 1-DOF cookbook for review purposes and to provide basic knowledge for multi-DOF problems. The algorithm consists of Newmark loop and Newton loop as

Time Integration (Newmark)

└── for each time step

└── Newton Iteration

└── Stress Return Mapping

Fig. 1 Program structure

The algorithm can be expanded below

Newmark time step t+1

1) Linear predictor: Keff * Δu = ΔFeff

2) Update u, v, a

3) Newton correction if nonlinear:

r = fs + c*v + m*a + m*ag

Keff_t = Kt + a0*m + a1*c

Keff_t * δu = -r

return mapping → fs, Kt

4) Commit state and move to t+2

By separating the Newmark and Newton loop, the solution of complex nonlinear dynamic analysis shall become obvious and easy to manage.

For a special and most common case (average acceleration)

beta = 1/4, gamma = 1/2

a0 = 4/dt^2

a1 = 2/dt

a2 = 4/dt

a3 = a4 = 1

a5 =0

For each time step n → n+1


    Known:

        un, vn, an, state_n



    Step A: Linear Newmark predictor

        Use tangent/stiffness from previous step

        Keff = Kt_n + a0*m + a1*c



        ΔFeff = F(n+1)-F(n)

                + m*(a2*vn + a3*an)

                + c*(a4*vn + a5*an)



        Solve:

            Keff * Δu = ΔFeff



        Update:

            u = un + Δu

            a = a0*Δu - a2*vn - a3*an

            v = a1*Δu - a4*vn - a5*an



    Step B: Newton correction

        repeat until convergence:



            Return mapping:

                [fs, Kt, state_trial] = material_update(u,state_n)



            Residual:

                r = fs + c*v + m*a + m*ag(n+1)



            Effective tangent:

                Keff_t = Kt + a0*m + a1*c



            Solve:

                δu = -r / Keff_t



            Update:

                u = u + δu

                a = a + a0*δu

                v = v + a1*δu



    Step C: Commit

        U(n+1)=u

        V(n+1)=v

        A(n+1)=a

        state = state_trial

Closure:

By separating Newmark trial step from the Newton iterative step in an implicit solution scheme, the Newmark step gives a trial responses including the residual force vector r. Inside Newton loop, the iteration would bring r to zero, ie at equilibrium.

It shall be noted that inside Newton loop, time is frozen, the solution is similar to nonlinear static procedure. The effect of M and C just resides in Keff because of linearization of da and dv terms not by the time difference scheme.

For explicit dynamic scheme, Newton loop is not used where all variables are updated after the Newmark step. However, stress correction is applied immediately after the Newmark trial step to bring the stress to the yield surface. The process is much faster and more efficient with the cost of very tiny time step to preserve solution stability (Bathe, 1996)

Reference

KJ Bathe, Finite Element Procedures, Prentice-Hall,1996