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UDNewmark

Newmark Time Integration With Uniform/Universal Damping

This damping formulation is an generalisation of 10.1016/j.compstruc.2018.10.016 and the nonviscous damping used in NonviscousNewmark. Both can be expressed in the convolution form.

While NonviscousNewmark convolutes velocity vector, UDDNewmark convolutes hysteresis and inertial forces, respectively.

Pros

  1. Allows a scalar definition of the specific damping factor.
  2. Allows customization of damping curve in the frequency space.
  3. Very cheap to compute, almost trivial, on par with Rayleigh damping.

Cons

  1. Induces dynamic hardening of the system, this hardening is affected by the damping factor over the whole frequency spectrum.

References

  1. 10.1016/j.compstruc.2018.10.016
  2. todo: to be added

Syntax

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integrator UDDNewmark (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 \(m_j\) and \(s_j\) are two complex numbers that define the kernel function.

\[ g(t)=\sum_{j=1}^nm_je^{-s_jt}. \]

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

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

then the command shall be defined as

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integrator UDDNewmark 1 .25 .5 -4 3 2 1 -4 -3 2 -1

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

These parameters contribute to the energy dissipation action in the system. Assuming free vibration and no velocity related forces, the equation of motion can be expressed as follows.

\[ \left\{\begin{array}{l} \mathbf{M}\ddot{\mathbf{u}}+\mathbf{K}\mathbf{u}+\sum\mathbf{f}_j+\cdots=\mathbf{0},\\ \dot{\mathbf{f}}_j=-s_j\mathbf{f}_j+m_j\mathbf{Ku}. \end{array}\right. \]

The real part of \(m_j\) shall be negative. The real part of \(s_j\) shall be positive.

Using the paremeter set for UDDNewmark, the specific damping factor as defined in 10.1016/j.compstruc.2018.10.016 can be expressed as

\[ \zeta(\omega)=-\sum\dfrac{m_j\omega}{s_j^2+\omega^2}. \]

One can customize the specific damping curve using various numerical methods.

Remarks

Two forms UDDNewmark and UDANewmark are equivalent if only hysteretic and inertial actions exist. The following function converts from one to another.

Python
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import numpy as np


def map(m, s):
    """
    Transform damping coefficients between UDD and UDA.
    """
    m, s = np.asarray(m, np.complex128), np.asarray(s, np.complex128)

    scalar = 1 - np.sum(ms := m / s)
    poles, ev = np.linalg.eig(np.diag(s) * scalar + np.outer(ms, s))

    ev = ev[:, idx := np.argsort(poles)]
    poles = poles[idx]

    ms = s @ ev * np.linalg.solve(ev, ms) / (-scalar * poles)
    s = poles / scalar

    return ms * s, s


print(map([-2], [10])) # UDD -> UDA