1. Caughey Absorbing Layer Method: an absorbent diaper method#
By considering the effectiveness of methods that use absorbent layers with multi-directional attenuation properties, Semblat et al. [bia1] _ introduced an absorption technique based on a simple description of the attenuation process. Physical models are generally very efficient in describing mitigation phenomena; however, these models are not always easy to implement. Given the fact that Caughey damping is already available in most finite element codes, Semblat et al. [bia1] _ proposed using this formulation to describe wave attenuation in an absorbent layer of finite thickness. The method is called CALM (Caughey Absorbing Layer Method). This formulation is purely numerical but, as mentioned below, an interpretation is possible in some cases. It should be noted that Rayleigh damping is a classical method for easily constructing damping matrix \(\matrix{C}\) for a finite element model with the following form:
- matrix {C} = a_0matrix {M} + a_1matrix {K}
- label:
eq:calm1
where \(\matrix{M}\) and \(\matrix{K}\) are respectively the mass and stiffness matrices of the entire system (more details on this damping modeling are provided in le chapitre correspondant). This is called the Rayleigh damping matrix; \(a_0\) and \(a_1\) are the Rayleigh coefficients. \(\matrix{C}\) is the sum of two terms: one proportional to the mass matrix \(\matrix{M}\), one to the stiffness matrix \(\matrix{K}\). A more general amortization formulation was proposed by Caughey and is expressed as follows:
- matrix {C} =matrix {M}sum_ {j=0} ^ {m-1} a_j (matrix {M} ^ {-1} K) ^j
- label:
eq:calm2
Observing equation eq:calm2, we can notice that the Rayleigh formulation corresponds to the 2nd order Caughey formulation (\(m=2\)), that is to say a linear combination of the mass and stiffness matrices. This way of constructing the depreciation matrix is very effective because the operation is very simple and it is often available in most calculation codes. Moreover, in a modal approach, the Rayleigh damping matrix (or Caughey) is diagonal in the modal base. This damping is therefore called proportional or classical. In the case of non-proportional damping, complex modes must be calculated (in order to decouple the modal equations).
Rayleigh damping (or Caughey) can also be used to analyze the propagation of a damped elastic wave. For this reason, it may be useful to have a rheological interpretation of such a purely numerical formulation. In the field of mechanical wave propagation, an equivalence between the generalized Rayleigh and Maxwell formulation has already been proposed ([bib2] _, [bib3] _), [bib3] _), briefly recalled below.
Consider Rayleigh damping. The overall loss coefficient can be written as follows:
- eta = 2xi =frac {a_0} {omega} + a_1omega
- label:
eq:calm3
where \(\omega\) is the pulsation and \(\xi\) is the damping coefficient. Assuming a relationship between internal dissipation and frequency for Rayleigh damping, it is possible to build a rheological model with attenuation that is also dependent on frequency. For a viscoelastic rheological model of complex modules \(M=M_R+iM_I\), the expression of the quality factor \(Q\) is given, in the field of geophysics and acoustics, as follows:
For low or moderate Rayleigh damping, there is a simple relationship between the inverse of the quality factor and the damping ratio, i.e. \(Q^{-1} = 2 \xi\). Therefore, the overall loss coefficient is zero for a frequency that tends to zero or to infinity. This clearly shows us how the model behaves through instant and long-term responses. As shown in [bib25] _ and [bib26] _, a rheological model that perfectly meets these needs (frequency-dependent attenuation, instantaneous and long-term effects) is a particular case of a generalized Maxwell model.
In v2.03.201-MaxwellGene, we can observe a rheological interpretation of the model: we connect, in parallel, a classical Maxwell diagram with a simple damper. The generalized Maxwell model can be characterized by its complex modules from which one can easily derive the inverse of the quality factor, which takes the same form as the quality factor of the Rayleigh damping model: it is the sum of two terms, one proportional to the frequency and another that is inversely proportional to the frequency.
By considering equation eq:calm3 and the rheological model presented on v2.03.201-MaxwellGene, the Rayleigh coefficients can be very easily linked to the rheological parameters of the generalized Maxwell model:
- left{begin {array} {ll}
a_0 =frac {K (xi_1 +xi_2)} {xi_1^2}\ a_1 =frac {xi_2} {K} end {array}right. :label: eq:calm5
The equivalence between Rayleigh damping and the generalized Maxwell model is noted for wave propagation problems and in general in dynamics [bib2] _. If there is a need to characterize a real material, Rayleigh coefficients can be identified via experimental results. In the case of approach CALM, the theoretical damping will be chosen for a fictional absorbent material (without the need to identify the parameters experimentally).
We assume, therefore, an elastic medium and an absorbent layer identified as the boundary of the elastic medium. The absorbent layer is modeled with appropriate damping properties in order to attenuate wave reflections at the edges of the domain. This method (CALM) can reduce the amplitude of the elastic wave that comes from the elastic medium and, therefore, the reflection at the artificial boundaries of the medium. Different techniques with different damping variations in the absorbent layer are proposed by Semblat et al. [bia1] _. Spatial variations in damping are controlled by variable damping coefficients in finite elements. Considering Rayleigh damping, the elementary damping matrix for a finite element \(\Omega_e\) is written as follows:
- matrix {C} ^e = a_0^ematrix {M} ^e + a_1^ematrix {K} ^e
- label:
eq:calm6
where \(\matrix{M}^e\) and \(\matrix{K}^e\) are respectively the mass and stiffness matrices for the finite element \(\Omega_e\). The Rayleigh coefficients may be different in each element or chosen to be constant piecewise in the absorbent layer.