A discontinuous Galerkin method with plane waves for sound absorbing materials
Abstract: Poro-elastic materials are commonly used for passive control of noise and vibration, and are key to reducing noise emissions in many engineering applications, including the aerospace, automotive and energy industries. More efficient computational models are required to further optimise the use of such materials. In this paper we present a Discontinuous Galerkin method (DGM) with plane waves for poro-elastic materials using the Biot theory solved in the frequency domain. This approach offers significant gains in computational efficiency and is simple to implement (costly numerical quadratures of highly-oscillatory integrals are not needed). It is shown that the Biot equations can be easily cast as a set of conservation equations suitable for the formulation of the wave-based DGM. A key contribution is a general formulation of boundary conditions as well as coupling conditions between different propagation media. This is particularly important when modelling porous materials as they are generally coupled with other media, such as the surround fluid or an elastic structure. The validation of the method is described first for a simple wave propagating through a porous material, and then for the scattering of an acoustic wave by a porous cylinder. The accuracy, conditioning and computational cost of the method are assessed, and comparison with the standard finite element method is included. It is found that the benefits of the wave-based DGM are fully realised for the Biot equations and that the numerical model is able to accurately capture both the oscillations and the rapid attenuation of the waves in the porous material
Summary (4 min read)
- The objective of this work is to develop a Discontinuous Galerkin Method (DGM) with plane waves to predict sound absorption in poro-elastic materials (PEM).
- The dynamic behaviour of porous materials is classically obtained from homogenized models and particularly from the Biot–Allard theory [1–4] which is based on a continuous field mechanics approach.
- The mid-frequencies correspond to situations where the standard FEM struggles with the size of the problem, but the statistical methods commonly used for high-frequency problems are not yet applicable.
- It was also shown that the DGM with numerical flux provides a unified framework to describe several wave-based methods [10, 15, 16], including the ultra-weak variational formulation, and the wave-based leastsquare method .
- In this paper the authors present a DGM using plane waves for poro-elastic materials modelled with Biot’s theory in the frequency domain.
2. GOVERNING EQUATIONS
- Throughout this paper, a harmonic time dependence e+iωt is assumed with the angular frequency ω.
- The numerical method and its applications are presented in two dimensions (x, y).
- The coefficient matrices Ax and Ay are assumed constant.
- For the porous material, these matrices are complex valued and vary with frequency ω.
2.1. First-order model for poro-elastic media
- There are several formulations of the Biot equations available in the literature.
- The right-hand side corresponds to elastic effects.
- Â andN are the elastic coefficients of the solid phase in vacuo which are directly obtained from the Lamé coefficients λ and µ.
- It should be noted that it is also possible to formulate a DGM starting from the non-conservative form (4), provided that the numerical flux is defined in a consistent way to ensure conservation of the field variables.
- One might think that the large number of unknowns introduced in this model implies that the computational cost of solving for all these variables will be high.
3. WAVE-BASED DGM
- The authors will now present the formulation and discretisation of the wave-based discontinuous Galerkin method of the conservative equations (1).
- The authors follow the same principles as in [10, 16], but the significant addition presented here is a general approach to incorporate a large class of boundary conditions (section 3.5), as well as coupling conditions between two different media (section 3.4).
- This approach relies heavily on the concept of characteristics which is introduced in section 3.2.
3.1. Variational formulation
- The variational formulation associated with the conservative form (1) is to find a solution u such that∑ ue = u|Ωe and ve = v|Ωe denote the restrictions of the solution and the test function to each element Ωe.
- The authors follow the usual idea from finite volume and discontinuous Galerkin methods of introducing a numerical flux on the interfaces between elements.
- Ωe and Ωe′ , and on this interface define the unit normal n pointing into Ωe′ .
- This aspect will be discussed in more detail in section 3.4.
- The concept of characteristics plays a central role in the analysis of partial differential equations of the form (1), see , and thus in the construction of numerical fluxes .
- Consider the boundary ∂Ω with unit normal vector n and tangential vector τ .
- For the Biot equations these eigenvalues are complex-valued, but the velocity and direction of propagation of each characteristic are easily obtained from the real part of the eigenvalue (which is consistent with group velocity).
- The general principle will guide the formulation of the boundary conditions and interface conditions in sections 3.4 and 3.5.
3.3. Numerical flux
- In the present work the authors use the upwind flux splitting (or exact Roe solver) which is a standard numerical flux for linear hyperbolic system.
- Λ± only contain the positive or negative eigenvalues.
- The two terms on the right correspond to the distinct contributions from the characteristics propagating in the positive and negative direction, respectively.
- If the authors now consider an interface Γee′ between two elements, the first term in (15) is associated with the characteristics travelling from element Ωe to Ωe′ and should therefore be calculated using ue.
- Conversely, the second term is associated with the characteristics travelling in the opposite direction and it is calculated using ue′ .
3.4. Boundary conditions
- As explained in , these boundary conditions should be used to specify the incoming characteristics in terms of the outgoing characteristics and the source terms, if any.
- Meth. (18) Separating the incoming characteristics from the others the authors get C̃−ũ− = s− C̃0+ũ0+ . (19) As mentioned in the previous section (and discussed in detail in ) a well-posed boundary condition specifies completely the incoming characteristics ũ−.
- The calculation of this matrix can be performed numerically in the implementation of the method, or can be done analytically beforehand.
- The authors now introduced some common boundary conditions used in practice for the Biot equations (2).
- There is also no tangential stresses on the surface since the porous material is allowed to slide along the surface.
3.5. Interface conditions
- The procedure described above for the boundary conditions can also be used for interface conditions between two different media.
- The unit normal on Γee′ points into Ωe. (25) A necessary condition is that the number of interface conditions corresponds to the total number of incoming characteristics on either sides of the interface.
- To fix ideas, two different sets of conditions are presented for the interface between a porous material and air.
3.6. Plane-wave discretization
- Finally the authors introduce the use of plane waves to discretise the variational formulations (12).
- Un of the plane waves are determined by requiring that these are solutions of the governing equations (1).
- In the case of the Biot equations (2–3) the calculation of the eigenvectors was found to be easier when using the non-conservative form (4).
- In addition, with other wave-based methods, in particular the partition-of-unity FEM, the presence of integrands involving polynomials and exponentials requires the use of numerical quadrature methods.
4.1. Plane wave propagating on a square
- This standard test case, already used in , is sufficiently simple to provide a detailed assessment of the performance of the method.
- The authors begin by considering the effect of the incident wave direction relative to other plane waves in the basis (33) as this has a profound impact on the accuracy of the method.
- The results are shown in figure 5 for a compression wave and the shear wave.
- These changes in actual element size induce the oscillations seen in the convergence curves for 1/h < 12.
- As expected, the numerical error decreases rapidly as the frequency is reduced (i.e. as the wavelength increases) until the conditioning deteriorates at levels of error of the order of 10−6 %.
4.2. Comparison with standard finite elements
- Numerical predictions for waves in porous material almost always rely on standard finite elements [29–32].
- The authors now compare the wave-based DGM against the standard finite element method with quadratic shape functions.
- Refining the mesh for the finite element method yields only a slow reduction in the absolute level of error but not a change in the rate of convergence (as expected from h-convergence).
- Similar conclusions can be obtained when considering the convergence with frequency presented in Figure 9.
- Several DGM results are presented corresponding to different numbers.
4.3. Sound absorption by a thin poroelastic layer
- The properties of the layer are given in Table I and correspond to Mat.
- The second resonance will be used to check the validity of the numerical scheme.
- While the mesh refinement is isotropic for the FEM, for the DGM a single element is used across the thickness of the layer and only the number of elements in the lateral direction is varied.
- Figure 11 shows the real part of the solutions of the pressure and the x component of the solid displacement, obtained with the FEM and the wave-based DGM.
4.4. Application to the scattering and absorption of sound by a porous cylinder
- The authors now consider a more complex test case, which involves the coupling of waves in air with the waves in a porous material.
- At the interface between the porous material and the surrounding volume of fluid, the two different conditions introduced in section 3.5 are considered.
- These two conditions can be formulated in the form of equation (25) and the methodology described in section 3.5 can be readily applied to implement these interface conditions in the variational formulation.
- A first remark on this figure is that for each solution is continuous which shows that each type of waves, taken individually, is discretised in a consistent manner by the numerical model.
- This paper described the first application of a plane-wave DGM for poroelastic materials modelled using the Biot equations.
- Using the charateristics of the governing equations, a general and systematic procedure was presented to include a large family of boundary conditions and interface conditions in the formulation of the numerical model.
- Interestingly it was found that the accuracy and the condition number are closely linked.
- This results in complex, three-dimensional geometries that will require a careful choice of plane-wave bases to maintain the improved efficiency of the method.
- Also, it might be worth considering hybrid model combining both wave-based elements and standard finite elements.
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