3.1.9.30. PML Element

This command is used to construct four-node and and eight-node PML elements. Perfectly Matched Layer (PML) elements are a numerical technique used in simulations to manage boundary conditions and minimize reflections at the edges of a computational domain. PMLs are often implemented as layers or regions at the edges of a computational domain, where the properties of the material in the PML are carefully designed to gradually absorb and attenuate outgoing waves. This absorption process is designed to be as efficient as possible, reducing the reflection of waves back into the domain. The choice of PML parameters, such as the profile of the absorbing material and its thickness, depends on the specific simulation and the desired level of accuracy.

Implementation details for these elements can be found in:

23. Zhang, E. Esmaeilzadeh Seylabi, E. Taciroglu,(2019), An ABAQUS toolbox for soil-structure interaction analysis, Computers and Geotechnics, Volume 114, 2019, https://doi.org/10.1016/j.compgeo.2019.103143.

Command

1. PML2D

   element PML $eleTag $node1 $node2 $node3 $node4 $E $nu $rho $EleType $Thickness $m $R $RD_half_width_x $Depth $alpha $beta

2. PML3D

   element PML $eleTag $node1 $node2 $node3 $node4 $node5 $node6 $node7 $node8 $gamma $beta $eta $E $nu $rho $EleType $Thickness $m $R $RD_half_width_x $RD_half_width_y $Depth $alpha $beta

1. PML2D

Argument	Type	Description
eleTag	integer	unique integer tag identifying element object
node1 $node2 node3 node4	4 integer	the four nodes defining the element input in counterclockwise order (-ndm 2 -ndf 5)
E	float	Young’s modulus
nu	float	Poisson’s Ratio
rho	float	Density
EleType	float	Element type based on the region of the PML it can be 1 2 3 4 5 (you can choose any number the code automaticlly )
Thickness	float	Thickness of the PML around regular domain
m	float	PML parameter (m=2 is recommended)
R	float	PML parameter (R=1e-8 is recommended)
RD_half_width_x	float	Distance of the border of the PML and regular domain to the center of the domain at origin
Depth	float	Depth of the PML from the surface of the regular domain in the negative y direction
alpha	float	Rayleigh damping parameter for PML (alpha will be 0 even if you input a value)
beta	float	Rayleigh damping parameter for PML (beta will be 0 even if you input a value)

2. PML3D

Argument	Type	Description
eleTag	integer	unique integer tag identifying element object
node1 node2 node3 node4 node5 node6 node7 $node8	8 integer	the eight nodes defining the element input in counterclockwise order (-ndm 2 -ndf 5)
\({\gamma}\)	float	Newmark integration parameter
\({\beta}\)	float	Newmark integration parameter
\({\eta}\)	float	Newmark integration parameter
E	float	Young’s modulus
nu	float	Poisson’s Ratio
rho	float	Density
EleType	float	Element type based on the region of the PML it can be 1 2 3 4 5 (you can choose any number the code automaticlly )
Thickness	float	Thickness of the PML around regular domain
m	float	PML parameter (m=2 is recommended)
R	float	PML parameter (R=1e-8 is recommended)
RD_half_width_x	float	Distance of the border of the PML and regular domain to the center of the domain at origin
Depth	float	Depth of the PML from the surface of the regular domain in the negative y direction
alpha	float	Rayleigh damping parameter for PML (alpha will be 0 even if you input a value)
beta	float	Rayleigh damping parameter for PML (beta will be 0 even if you input a value)

Note

1. For 2D PML each node has 5 DOFs (2 translations and Sx, Sy, Sxy)

2. For 3D PML each node has 9 DOFs (3 translations and Sx, Sy, Sz, Sxy, Sxz, Syz)

3. There is no recorder for PML elements. The stresses can be obtained from the node recorders.

4. When using PML 3D only newmark integration method can be used and the parameters using for the integration should be passed to the element(Example: \({\gamma} = 1/2, {\beta} = 1/4, {\eta} =1/12\))

5. The center of the regular domian should be at the origin of the global coordinate system (0,0,0)

6. It is highly recommended to use the same mesh for the PML and the regular domain

7. It is highly recommended to use the same material for the PML and the regular domain

8. It is highly recommended to use uniform and square mesh for the PML elements.

Example

The following example constructs a PML3D element with tag 1 between nodes 1, 2, 3, 4, 5, 6, 7, 8 with the properties E=1e6, nu=0.3, rho=1.0, EleType=1, Thickness=1.0, m=2, R=1e-8, RD_half_width_x=10.0, RD_half_width_y=10.0, Depth=5.0, alpha=0.0, beta=0.0.

1. Tcl Code

element PML 1 1 2 3 4 5 6 7 8 1 0.5 0.25 0.0833333333333333 1e6 0.3 1.0 1 1.0 2 1e-8 10.0 10.0 5.0 0.0 0.0

2. Python Code

element('PML', 1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 0.5, 0.25, 0.0833333333333333, 1e6, 0.3, 1.0, 1, 1.0, 2, 1e-8, 10.0, 10.0, 5.0, 0.0, 0.0)

Code Developed by: W. Zhang, E. Taciroglu, A. Pakzad, P. Arduino (UCLA, UW)