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  1. Integrator
  2. Implicit
  3. Newmark

LeeNewmarkFull

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Last updated 2 years ago

Newmark Algorithm With Lee Damping Model (Full Modes)

Please check the references for theory.

For the moment, MPC cannot be considered in all global damping models.

Syntax

integrator LeeNewmarkFull (1) (2) (3) ((4) (5) (6) [7...]...)
# (1) int, unique integrator tag
# (2) double, alpha in Newmark method
# (3) double, beta in Newmark method
# (4) string, type identifier
# (5) double, \zeta_p
# (6) double, \omega_p
# (7...) double/int, parameters associated with the mode

By default, the converged stiffness from the last substep is used in assembling damping matrix, this leads to quadratic convergence rate. For special purposes, one can also use the following commands to explicitly use the certain stiffness matrix.

integrator LeeNewmarkFullInitial (1) (2) (3) ((4) (5) (6) [7...]...)
integrator LeeNewmarkFullCurrent (1) (2) (3) ((4) (5) (6) [7...]...)
integrator LeeNewmarkFullTrial (1) (2) (3) ((4) (5) (6) [7...]...)

Remarks

  1. User shall ensure all active DoFs in the system are displacement DoFs.

  2. The LeeNewmarkFull integrator uses a standard Newmark algorithm and the damping model proposed by Lee (2020). Several types of basic functions are defined to control the bandwidth.

  3. The static condensation procedure is reversed so that the damping matrix C\mathbf{C}C is a sparse matrix of a different size.

  4. Currently, the modified equation of motion (of larger size) is stored as a sparse matrix and the SuperLU solver is used to solve sparse systems. To switch to other solver, one can use the following command.

    set system_solver MUMPS

    If MKL is enabled, it is possible to use PARDISO solver by setting the following command.

    set system_solver PARDISO

    If CUDA is enabled, it is possible to use CUDA solver by setting the following command.

    set system_solver CUDA
  5. The final global matrix is expected to be considerable huge compared to the original matrix. Thus, the sparse storage scheme is also used for assembling original stiffness and mass matrices by default. Please always add the following command in the corresponding step.

    set sparse_mat true
  6. The following type identifiers are available:

    1. -type0 --- zeroth order model, need no additional parameter, viz., [7] can be empty

    2. -type1 --- type 1 model, need one integer parameter, viz., [7] (npn_pnp​) is a non-negative integer

    3. -type2 --- type 2 model, need two integer parameters, viz., [7] (nprn_{pr}npr​) and [8] (npln_{pl}npl​) are non-negative integers (zero allowed)

    4. -type3 --- type 3 model, need a double parameter, viz., [7] (γ\gammaγ) is a double number that is greater than γ>−1\gamma>-1γ>−1

    5. -type4 --- type 4 model, need five parameters, [7...10] are four non-negative integers (nprn_{pr}npr​, npln_{pl}npl​, npkn_{pk}npk​, npmn_{pm}npm​), [11] is a double parameter greater than γ>−1\gamma>-1γ>−1

  7. Multiple types can be mixed in any preferred orders. See examples below.

Examples

Each of the following commands defines 5%5\%5% peak damping on ω=1\omega=1ω=1 via different types with different parameters.

integrator LeeNewmarkFull 1 .25 .5 -type0 .05 1
integrator LeeNewmarkFull 1 .25 .5 -type1 .05 1 2
integrator LeeNewmarkFull 1 .25 .5 -type2 .05 1 2 1
integrator LeeNewmarkFull 1 .25 .5 -type2 .05 1 3 1
integrator LeeNewmarkFull 1 .25 .5 -type2 .05 1 2 3
integrator LeeNewmarkFull 1 .25 .5 -type3 .05 1 -.25
integrator LeeNewmarkFull 1 .25 .5 -type3 .05 1 .5
integrator LeeNewmarkFull 1 .25 .5 -type3 .05 1 1 1 1 1 .5
integrator LeeNewmarkFull 1 .25 .5 -type3 .01 1 .5 -type2 .01 1 2 3 -type2 .01 1 2 1 -type0 .01 1 -type1 .01 1 2

With the above commands, we use α=0.25\alpha=0.25α=0.25 and β=0.5\beta=0.5β=0.5 in Newmark method. Users need to manually compute ζp\zeta_pζp​, ωp\omega_pωp​ and other parameters to obtain desired curve.

There is no validation of input parameters. It is users' responsibility to ensure parameters are valid, and the resulting curve is desired. Please refer to the references for customization of damping curves.

One can also use to generate parameter sets.

10.1016/j.jsv.2020.115312
10.1016/j.engstruct.2020.110178
10.1016/j.compstruc.2020.106423
10.1016/j.compstruc.2021.106663
Damping Dolphin