NonviscousNewmark

Newmark Time Integration With Nonviscous Damping

References

1. 10.1016/j.ymssp.2024.111156

Syntax

integrator NonviscousNewmark (1) (2) (3) ((4) (5) (6) (7)...)
# (1) int, unique tag
# (2) double, alpha, typical: 0.25
# (3) double, beta, typical: 0.5
# (4) double, real part of `m_i`
# (5) double, imaginary part of `m_i`
# (6) double, real part of `s_i`
# (7) double, imaginary part of `s_i`

Theory

The parameters \(m_i\) and \(s_i\) are two complex numbers that define the kernel function.

\[ g(t)=\sum_{i=1}^nm_ie^{-s_it}. \]

For example, if the kernel contains two exponential functions such that

\[ g(t)=(1+9i)e^{-(2+8i)t}+(3+7i)e^{-(4+6i)t}, \]

then the command shall be defined as

integrator NonviscousNewmark 1 .25 .5 1 9 2 8 3 7 4 6

It is assumed that the kernel is applied to all DoFs in the system.