3.1.10.32. 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 $matTag $PMLThickness meshType meshParams… [-Cp $Cp] [-m $m] [-R $R] [-alphabeta $alpha $beta] [-Newmark $gamma $beta $eta $keisi]

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 3 -ndf 9)
matTag	integer	material tag (must be an ElasticIsotropicMaterial)
PMLThickness	float	Thickness of the PML around regular domain
meshType	string	Mesh type defining the PML region (General, Box, Sphere, Cylinder)
meshParams…	float	Additional parameters depending on meshType (see below)
-Cp	float	PML parameter defining wave speed (default calculated from material properties)
-m	float	PML parameter (default m=2)
-R	float	PML parameter (default R=1e-8)
-alphabeta	|float float|	User-defined alpha and beta parameters
-Newmark	|float float float float|	Newmark integration parameters (gamma, beta, eta, keisi)

Mesh Type Parameters

Mesh Type	Required Parameters	Description
General	Xref, Yref, Zref, N1, N2, N3	Reference point and normal vector components
Box	XC, YC, ZC, L, W, H	Center coordinates at the surface and dimensions of the box (e.g., 20, 50, 0.0, widthX, widthY, depthZ)

Note

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

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

3. For PML 2D, the center of the regular domian should be at the origin of the global coordinate system (0,0,0)

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

5. When using PML 3D, only the Newmark integration method can be used. If the parameters for the integration are not passed to the element, the default parameters will be used (Example: \(\gamma = 0.5, \beta = 0.25, \eta = 0.0833333333333333, \xi = 0.0208333333333333\)).

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

9. If the user provides the alpha and beta parameters explicitly using the -alphabeta flag, the following formulas for calculating alpha_0 and beta_0 are skipped, and the user-defined parameters are used instead:

   \[\begin{split}\begin{aligned} C_p &= \sqrt{\frac{E \cdot (1 - \nu)}{\rho \cdot (1 + \nu) \cdot (1 - 2 \cdot \nu)}} \\ PML_b &= \frac{\text{PMLThickness}}{1.0} \\ \alpha_0 &= \frac{(m_{\text{coeff}} + 1) \cdot PML_b}{2.0 \cdot PML_b} \cdot \log_{10}\left(\frac{1}{R}\right) \\ \beta_0 &= \frac{(m_{\text{coeff}} + 1) \cdot C_p}{2.0 \cdot PML_b} \cdot \log_{10}\left(\frac{1}{R}\right) \end{aligned}\end{split}\]

Example

The following example constructs a PML 3D element with tag 1 between nodes 1, 2, 3, 4, 5, 6, 7, 8 using material tag 1, PML thickness 1.0, and a box-type mesh with parameters 10.0, 10.0, 5.0 20 20 20.

1. Tcl Code

element PML 1 1 2 3 4 5 6 7 8 1 1.0 Box 10.0 10.0 5.0 20.0 20.0 20.0 -m 2 -R 1e-8 -Newmark 0.5 0.25 0.0833333333333333 0.0208333333333333

2. Python Code

element('PML', 1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1.0, 'Box', 10.0, 10.0, 5.0, 20.0, 20.0, 20.0, '-m', 2, '-R', 1e-8, '-Newmark', 0.5, 0.25, 0.0833333333333333, 0.0208333333333333)

Example

The following example constructs a PML 2D element with tag 2 between nodes 1, 2, 3, 4 using Young’s modulus 3000, Poisson’s ratio 0.3, density 2500, element type 1, thickness 1.0, and PML parameters 2, 1e-8, 5.0, 10.0, 0.0, 0.0.

1. Tcl Code

element PML 2 1 2 3 4 3000 0.3 2500 1 1.0 2 1e-8 5.0 10.0 0.0 0.0

2. Python Code

element('PML', 2, 1, 2, 3, 4, 3000, 0.3, 2500, 1, 1.0, 2, 1e-8, 5.0, 10.0, 0.0, 0.0)