The ultimate optimization in the development of components such as SAW (Surface Acoustic Wave) and BAW (Bulk Acoustic Wave), is to increase the capability to simulate the real shape of the resonator. In the case of SAW devices, the main effort is currently devoted to account for lateral resonance [
When one solves an equation numerically by volume discretization like in Finite Element Analysis (FEA), one must truncate the computational grid in some way. In such cases, the important thing is how to perform this truncation without introducing significant artifacts into the computation. Two cases may arise. If the computational domain is large enough and the sides are seen by a very attenuated wave so there is no problem. In the other hand, if the wave doesn’t vanish at the mesh boundary we must apply absorbing condition to avoid spurious reflection. The application of boundary conditions as rigid walls (Dirichlet or Neumann) or periodic boundary conditions leads unacceptable artifacts reflections at the edges of the mesh. Therefore, the wave equations require an absorption condition which makes vanish the waves that strike the edges without reflection into the domain of interest. A well known method is the Boundary Element Method [
In 1994, however, the problem of absorbing boundaries for wave equations was transformed in a seminal paper by Berenger [
even if it reflects o the boundary, the returning wave after one round trip through the absorbing layer is exponentially tiny.
The problem with this approach is that, whenever you have a transition from one material to another, waves generally reflect, and the transition from non-absorbing to absorbing material is no exception so, instead of having reflections from the grid boundary, you now have reflections from the absorber boundary. However, Berenger showed that a special absorbing medium could be constructed so that waves do not reflect at the interface: a perfectly matched layer, or PML. Although PML was originally derived for electromagnetism (Maxwell’s equations), the same ideas are immediately applicable to other wave equations. A lot of work has been devoted to transpose this approach to elasticity problems [4-6].
In the following, we first expose the theoretical development for piezoelectricity-based problems, focusing on the simple case of the scalar plane wave. We will see that the establishment of PML formulation may be described by the combination of the following two steps: first, changing in complex variable, next a coordinate transformation allowing the back to real coordinates.
A first numerical application is achieved in 2D to address the simulation of finite dimension of a SAW resonator with a periodic FEA code. Indeed, up to now, the devices modelings were most often considered as 2D systems infinitely periodic in the direction of propagation and infinite in the perpendicular one. We just demonstrate that it is now possible to take into account the effects due to finite dimension along the direction of propagation.
We also show, the interests of PML method is the case of an 3D SAW devices. This is especially to show the influence of the numerical aperture which was considered as infinite. This also shows the effects of lateral edges of such components.
As a conclusion, we propose different tracks to further optimize the PML approach, particularly for 3D piezoelectric problems.
First, we start from the variational formulation of the linear dynamic piezoelasticity problem. Here, the unknowns of the problem are mechanical displacement and electrical potential. This approach has been implemented initially by Tiersten [
The terms 
are successively elastic, piezoelectric and dielectric constants.
The method of PML takes care of not changing this formulation. Indeed, we will see that the calculation process leads to change only material constants.
Let us start with the plane wave 
of some wave equation in infinite space.
In 
and extended until
. In particular, we will focus on the problem in the 
direction (the other directions will follow by the same technique). An absorbing domain (PML domain, the blue one in 
to
.
The PML setting occurs in four conceptual steps, summarized as follows:
• Changing x in complex variable;
• Coordinate transformation allowing back to real x;
• Transformation of equations in real coordinates into complex materials constants;
• Optimization of PML parameter and construction of the PML region.
The first step is to proceed to change of variable complex within the whole domain, which transforms the oscillating waves in exponentially decreasing wave in the PML area without reflections at the interface with the main domain (depicted by the dashed line at 
in
where 
growths from the origin of the absorbing area to its end along a defined rate. In the main domain, this function is zero thus there is no changes on the FEA solution). It is clear that the change of variable has the effect of introducing an evanescence of the wave in the PML part. The wave model becomes:
However, since this transform must be efficient for any frequency (we represent the problem in the spectral domain), it is sense to define this function as follows:
We will see thereafter the absorption function 
that has been chosen and the criteria imposed.
The second step, is to perform a coordinate transformation to express the complex x as a function of a real coordinate. We can easily define the Jacobian transformation linking the considered coordinate systems as following:
The entire process of PML can be conceptually summed up by this previous transformation of the dierential equation: In the PML aeras where
, the oscillating solutions turn into exponentially decaying ones.
In the region of interest 
and
, so the solution remains unchanged even though the change of variable is applied. This is one reason why we do not create reflections at the interface
.
The choice to write the absorbing function with a frequency depedance is motived by looking at what happens for the plane wave
:
The term 
is equal to
, the inverse of the phase velocity 
in the x direction. This allows that the attenuation rate of the PML is independent of the frequency ω: all wavelengths decrease at the same rate.
Conformably to Zheng and Huang [
We now rewrite the elasticity equations in the absorbing region turning x to
, using then (6) to express the result in the initial coordinates. In the new coordinates, we have real coordinates and complex materials constants.
As in [
where 
is characterised by its specific function
. 
and ui respectively represent the dynamic stresses and displacements, and ρ is the mass density.
We introduce a non symmetrical stress tensor, expressed in the transformed axis:
where 
is the transformed elastic constant tensor relative to the absorption area. We multiply (8) by
, thus yielding Newton relation for PMLs in the real coordinates
where 
is the mass density relative to the transformed domain. Given that the obtained form of the equilibrium equation complies with the classical expression for usual solids, it is allowed to exploit the standard FEA formulation. It must however be careful to take into account the frequency dependence of the transformed tensors (stress and electrical displacement) in the PML région. These developments of course can be extended to piezoelectricity by rewriting Poisson’s equation and taking into account the piezoelectric coupling in the stress definition as follows:
Poisson’s equation expressed in the transformed system of axes reads:
with 
the electrical displacement vector. To provide an homogeneous formulation, we proceed as for the stress definition (9), multiplying the electrical displacement by
, yielding:
As for the propagation Equation (8), the Poisson’s condition is written accounting for these changes as:
In the same way as the transformation of the stress tensor, we introduce the modified piezoelectric and dielectric constants defined as follows:
We now are able to establish a FEA formulation exploiting these developments in Equation (16) without fundamental changes of the variational formulation developed in Equation (1):
As some care must be paid to the truncation of the computational region and also to the choice of the absorption function, next paragraphs are dedicated to these questions.
Once we have performed the PML transformation of the FEA formulation, the solutions are unchanged in the region of interest 
and exponentially decaying in the PML regions
. At the end of the PML domain, the boundary conditions are not important. Indeed, even if a hard wall is set on this frontier, the reflected waves on it can be neglected after the propagation into the PML.
In order to optimize the operation of the PML, the absorption function 
must be wisely chosen. We discuss here the absorption function and the importance of its parameters.
In our developments, the absorption function is chosen to avoid introducing any brutal change in the physical constants. That must be applied as well at the interface between the main domain and PML domain as throughout the PML area. This signifies it must exhibits derivatives close to zero at each edge of the PML area, but it must continuously vary more rapidly within this area to avoid meshing a too long PML zone. We also want this function to be even (symmetrical) so it can apply on the both sides of a domain centred around
. As proposed in [
that allows to match all these specifications at once
with xa and xp the limits of the PML area and n an integer controlling the absorption rate. The coefficient 
is an adjustable normalisation parameter. This function is plotted in
With
, the first derivates are not zero. So the matching coditions are not observed. On the other hand, for n too large (for instance, n equal to 4 or 5), the slope in the middle of the PML is high and this can lead to numerical reflections. Indeed, in the FEA formalism, the absorbing function must be numerically integrated on
each elements of the mesh. So, considering Gauss rules, a high order polynomial requires more points to converge. If this condition is not respected, some artifacts may appear.
To highlight, the effects of the pml, Figures 4(a) and (b) depict the case without and with PML. We see in
In 
to obtain coherent behavior of PML. For
, the absorbing function reveals a smoothy behavior providing a constant decreasing of the wave .along the PML. In contrast, for
, we see that the displacement seems to be inconsistent. This is due to the fact that the increase of 
leads to a too brutal change in the rate of absorption and induces numerical reflections.
The PML/FEA formalism is achieved for each frequency point and so is time consuming. Then for practical application, we chose 
as the best trade-off between a strong absorption rate and fast computations. The dmax is selected between 106 and 107.
and
. The period is 10 µm. The depth of PML on both sides is 35 µm. The hight of the mesh is 75 µm. The absorbing parameters are set to 
and
.