Perfectly matched layers are artificial anisotropic materials that absorb all incoming elastic waves without any reflections, except for the gazing wave that parallels the PML interface in the propagation direction. PMLs are constructed to propagate elastic waves in homogeneous isotropic elastic media in harmonic response analyses (or in an infinite static fields in static analyses).
Consider two Cartesian coordinate systems: a global system with respect to an orthogonal basis , and a local system with respect to another orthogonal basis . The displacements of such an elastic PML medium in the local coordinate system are governed by the following equations:
(2–103) |
(2–104) |
(2–105) |
where:
= components of the stress tensor in the local Cartesian coordinate system |
= components of the infinitesimal strain tensor in the local Cartesian coordinate system |
= components of the material stiffness tensor in the local Cartesian coordinate system |
= nonzero complex-valued coordinate stretching functions in the direction of the local Cartesian coordinate system |
= density of the elastic medium |
On multiplying Equation 2–103 with the and noting that is a function of only, Equation 2–103 through Equation 2–105 are rewritten in matrix notation:
(2–106) |
(2–107) |
(2–108) |
where:
The tensors and vectors in the local coordinate system are transformed into tensors and vectors in the global coordinate system using the rotation-of-basis transformation matrix with the component , such that . The governing equations of the displacements of the elastic PML medium in the global coordinate system are written as:
(2–109) |
(2–110) |
(2–111) |
By using the Galerkin method and setting the displacements to zero (input as UX, UY and UZ on the D command) on the backed boundary, the "weak" formulation of an elastic wave in the elastic PML medium is expressed as:
(2–112) |
where:
(2–113) |
The PML material is defined by SOLID185, SOLID186, and SOLID187 elements with KEYOPT(15) = 1. Because the PML material is constructed in Cartesian coordinates, the edges of the 3-D PML region must be aligned with the axes of the global Cartesian coordinate system. More than one 1-D PML region may exist in a finite element model. The PML element coordinate system (PSYS) uniquely identifies each PML region. A parabolic attenuation distribution minimizes numerical reflections in the PML.
ANSYS, Inc. recommends using at least three PML element layers to obtain satisfactory accuracy. Some buffer elements between the PML region and objects should be utilized. Because a PML acts as an infinite open domain, any boundary conditions and material properties must be carried over into the PML region. The displacements must be set to zero (input as UX, UY and UZ on the D command) on the exterior surface of the PML, excluding Neumann boundaries (symmetric planes in the model). Any excitation sources such as body force are prohibited in the PML region.
For more information, see Perfectly Matched Layers (PML) in the Acoustic Analysis Guide.