A Fourier Transform 3D-Model for Wave Propagation in layered orthotropic Media

Transcription

1 A Fourier Transform 3D-Model for Wave Propagation in layered orthotropic Media F. Kramer*, H. Waubke Abstract In order to assess and prevent noise emmission in the environment of traffic routes, prognosis models for wave propagation are of interest for many applications. Guided by experience in railway traffic simulation, this contribution introduces a new model for wave propagation and simulation in layered orthotropic media. The model uses a component-wise Fourier transform to obtain the general solution within one layer; by a variation of constants, this yields the solution within a loaded layer. Then, balances at layer boundaries are used to derive the solution at the layer boundaries. Thus, a grid of stress respectively displacement vectors is obtained, and back transformation is performed. The model has been tested for physical accuracy concerning the splitting of layers and for numerical accuracy. The results are presented. INTRODUCTION The simulation of vibrations induced by machinery and traffic has become of increasing concern within the last decades, due to frequent construction of high speed trains and mass rapid transport systems. Theoretical studies on the propagation of waves in anisotropic layered media have been an important field of research for many years, and have been of particular use for applications in geophysics and seismology [1, 2]. Therefore, much research has been done on wave propagation in anisotropic layered media [3], layered orthotropic media however, has rarely been focused upon. Waubke [4] has suggested extensions and developments for this model in earlier papers. This paper draws these ideas together, and offers some concrete numerical results. In the first section, a model of wave propagation is derived by a Fourier transform of the classical wave equation. The balance of stresses and displacements at the layer boundary delivers a system of linear equations, yielding the stress and displacement level at the layer boundaries as functions ûk x,ky,z,ω and ˆσk x,k y,z,ω.

2 This model applied to a system of layers gives a transport matrix for the layer boundaries, yielding a system whose solution permits the calculation of displacement and stress levels at arbitrary points within the media. Due to the numerical stabililty of the approach, this is possible for large distances from the point of excitation. At first, the focus lies on one layer and the equations describing the process are derived. The strain vector ǫ is a vector of local derivatives of the displacement vector u, denoted by ǫ := D x, y, z u. The soil is assumed to consist of n orthotropic layers. For each layer, the following parameters are needed: the shear modulus G, the Young s modulus E, the Poisson ratios ν, the density ρ and width d. Additionally, homogenous behavior in the horizontal directions is assumed: E y = E x, G yz = G zx, and ν yz = ν zx hold. So, the elasticity matrix takes block diagonal form with blocks E 11 := 1 E x νxy E x νxy E x 1 E x νzx νzx νzx νzx 1 1, E 22 := diag2g xy, 2G zx, 2G zx 1 As can be found in [1], the wave equation takes the special form Bσ + ρω 2 û = ˆBE ˆDû + ρω 2 û =: Âû = ˆb. 2 for a differential operator B, where b is some external force. We denote the Fourier transform ˆf of a function fα by F α,kα fk α, and the Fourier back transform by F 1 k α,α.the displacement vector u := ux,y,z,t is Fourier transformed in all directions, yielding ûk x,k y,k z,ω. Thus, the differential operator D is substituted by the polynomial operator ˆD. Simple equations are obtained, expressing the link of the transformed displacement vector and the transformed stress vector: ˆσ = Eˆǫ = E ˆDû, where ˆD = ˆDk x,k y,k z is the transformed differential operator D. Note that E ˆD is symmetric. The transformed wave equation then takes the form ˆBE ˆDû + ρω 2 û =: Âû = ˆb. 3 In order to cope with the boundary conditions, it is useful to implement a grid in k x,k y,ω. Therefore k x,k y,ω are treated as constants, and are omitted in some cases. The general form of a solution DERIVING THE EQUATIONS The condition detâ = 0 is necessary for solutions of Âû = 0, û 0 to exist. The roots of the polynomial detâk z called eigenvalues are denoted by κ, and the vectors of the corresponding kernel are denoted as Ψ called eigenvectors. Note that

3 ICSV13, July 2-6, 2006, Vienna, Austria due to the symmetry of Â, it holds that detâk z = detâ k z. The homogeneous solution u h and the corresponding stress function σ h take the form û h k z = c h,i Ψ i δk z κ i, ˆσ h k z = c h,i E ˆDκ i Ψ i δk z κ i 4 for arbitrary coefficients c h,i for 1 i 6. We define Hk x,k y,κ i,ω := E ˆDk x,k y,κ i,ω for better readability, and write Hκ i =: H i for the sake of simplicity. As the model assumes the media to be layered with respect to the z-direction, û is therefore transformed back over the z direction. Equations for ũk x,k y,z,ω are derived by performing the transformation. This yields ũ h = c h,i Ψ i e jκiz, σ h = c h,i Hκ i Ψ i e jκiz, 5 By a variation of constants, the general form of the solution can be expressed as ũ = c h,i + c p,i zψ i e jκiz, σ = c h,i + c p,i zh i Ψ i e jκ iz 6 for z-dependent coefficients c p,i z from a particular solution. Calculation of the stress and displacement vector at layer boundaries In order to derive the equations for the particular solution, a discrete impact ata depth d inside a layer is assumed. Concretely, the impact at a point x 0,y 0,z 0,t 0 with constant direction f is assumed in the form px,y,z,t = δx x 0 δy y 0 δz z 0 δt t 0 f. The Fourier transform is applied on the pz. This yields ˆpk x,k y,k z,ω = expjx 0 k x + y 0 k y + z 0 k z + t 0 ωf. 7 As the model is invariant to translation, the origin of the coordinates x, y, z, ω is set to the point of impact, so that ˆp = f. Note that if the origin is assumed in the middle of the layer, a transformation factor expz p is yielded see Figure 1. A Fourier backward transform gives ˆp as right hand side of Eq. 3. It is to be taken into account that the balance of stresses only holds for the components that act at the interface of the layer boundary. the vector σ is reduced to dimension 3 by dropping the other components; the matrix H r is similarly reduced. A layer with width d is considered, loaded with an impact vector ˆp in depth d p. The layer is split at the depth of the impact d p, and two layers with identical material parameters are derived. For the upper virtual layer let ũ u k x,k y,z = d p,ω and ũ m k x,k y,z = 0,ω be the transformed displacement at the upper and lower boundary, respectively. For the lower virtual layer, let ũ n

4 Figure 1: Treatment of a loaded layer: the layer is split by a virtual layer boundary, at the depth of the impact. and ũ n be the displacement vector at the upper boundary and ũ d the one at the lower boundary. Obviously, ũ m = ũ n holds. For the virtual layer we establish the solution for the upper ũ z and the lower ũ z part: ũ z := a i Ψ i e jκiz, ũ z := b i Ψ i e jκiz, 8 such define ũ u := ũ d p,ũ := ũ 1 0, as well as ũ n := ũ 0,ũ d := ũ d d p, The equations for displacements and stress at the point of impact can be established now. The stress vectors σ r,m and σ r,n at the virtual layer boundary differ by ˆp. The coefficient vector a and b consist of a homogenoeous and particular part, a = a h +a p, b = b h + b p where a h = b h. The particular part b p = 0 as the lower virtual layer is treated as an unloaded layer. The particular part a p has to satisfy ũ p,m = a p,i Ψ i = 0, σ p,m = This gives a p as the solution of a linear equation Ψi Ma p = ˆp 0 with M := H r,i Ψ i a p,i H r,i Ψ i = ˆp. 9 1 i 6, ˆp 0 := 0 ˆp. 10 The coefficient of the displacements and stresses of the unified layers are denoted by c = c h + c p. The particular component of the solution is given by c p = a p = M 1ˆp 0. As the origin is in the middle of the layer, correction factors expjκ i z p for the partial components are derived. For an arbitrary z within the layer boundaries, this yields ũz = c h,i +c p,i e jκ iz p Ψ i e jκ iz d 2, σz = c h,i +c p,i e jκ iz p H r,i Ψ i e jκ iz d 2. 11

5 ICSV13, July 2-6, 2006, Vienna, Austria The factors exp jκ i z d/2 are derived from the backward Fourier transform of the i-th component for the upper layer boundary compare with Eq. 5; as a consequence, exp± 1jκ 2 id arise componentwise for the layer boundaries, and expjκ i z p for the particular solution. The matrices Θ and Ξ, as well as the vector p, are defined as follows, in order to express the displacements and stresses at the layer boundaries: This yields Θ := ũ =: Θc h + Ψ i e 1 2 jκ id Ψ i e 1 2 jκ id 1 i 6, Ξ r := H r,i Ψ i e 1 2 jκ id H r,i Ψ i e 1 2 jκ id 1 i 6 12 p := c p,i e jκ iz p 1 i Θu p 0 = ũ h + ũ p, σ r =: Ξ r c h + Ξr,u p 0 = σ r,h + σ r,p. 14 where Θ u, Ξ r,u consist of the upper three rows of Θ resp Ξ r. Substituting c h = Θ 1 ũ ũ p into the stress equations, and defining K := Ξ r Θ 1, it holds that σ r = Ξ r Θ 1 ũ ũ p + σ r,p = Kũ Kũ p + σ r,p, 15 for a loaded layer. The case for an unloaded layer can easily be deduced from the general case by setting the partial component to zero. BALANCE AT LAYER BOUNDARIES Now, a system of n layers is considered. For the i-th layer, let the matrices c i, Θ i, Ξ i, H i, K i denote the matrices defined in the section before, evaluated for the i-th layer. Let the index X u respectively X d denote the upper respectively lower half of a matrix X. The layers are numbered in the canonical way, such that ũ i is the displacement at the upper boundary of the i-th layer. The stress vectors are sometimes denoted alternatively as σ i,u = σ i. The index r is omitted in σ r, Ξ r etc. For unloaded layers, the balance of stresses yields Boundaries of loaded layers K i,d ũi ũ i+1 = K i+1,u ũi+1 ũ i+2 The layer k is assumed to be loaded. Let the indices h and p denote the homogeneous and the particular part of the solution. Eq.14 yields the homogeneous and the particular component of the stress vector as σ p,k = Ξk,u p k 0, σ h,k = Ξ k c k = Kk,uu K k,ud K k,du K k,dd 16 ũ h,k. 17

6 The stress vectors at the layer boundaries equate in the same way, up to the partial component of the solution. With σ k,u = σ k 1,d it holds that σ k 1,d = σ k,u = σ h,k,u + σ p,k,u, and as the stress vector at the upper boundary takes the form σ k,u = Ξ k,u c k + p k see Eq. 14, the balance of stresses at the upper loaded boundary takes the form Ξ k 1,d c k 1 = Ξ k,u c k + p k 18 and leads to K k 1,d ũk 1 ũ k = K k,u ũk ũ k+1 K k,uu ũ p,k + σ p,k,u. 19 after substitution c i = Θ 1 i ũ i ũ p,i for i = k 1,k. The vector ũ p,k is equivalent to ũ p from Eq. 14, and σ p,k,u is equivalent to σ r,p from Eq.14, for the k-th layer. For the lower boundary, since σ k,d = σ k+1,u, this yields K k,d ũk ũ k+1 + K k,du ũ p = K k+1,u ũk+1 ũ k Top boundary and bottom boundary For the displacement ũ 1, which occurs at the boundary between air and soil, the boundary conditions are assumed to be of von Neumann type. For the bottom halfspace, boundary conditions are derived by assuming that for a monofrequent wave with angular frequency Ω, the wave front κ i z + Ωt = 0 travels downward with time: z z Ω t = Re = Re t κ i > 0 ReΩκ i This eliminates three degrees of freedom. Thus, the matrices Θ n, Ξ n and K n shrink to size 3 3. At the bottom layer, this yields K n 1,d ũn 1 ũ n = K n ũ n. 22 Synthesis Again, a system of n layers is considered, including the k-th one, which is assumed to be loaded. Putting together the results, the equation to be solved is established defining the supervector U := ũ 1,...,ũ n T. So, it delivers a linear equation LU = V, where L is an n n block band matrix with 3 3 blocks.

7 ICSV13, July 2-6, 2006, Vienna, Austria Figure 2: Left: The relative error of U for a four layer system; Right: Difference between a four layer system and the same system split within the first layer. Obtaining the grids With the solution U it is possible to compute the stress vector as well as the displacement vector at arbitrary points. Extending the definition of Θ, Ξ and ˆp yields Θ s z := Ψ s,i e jκ s,iz ds 2, Ξ s z := H s,i Ψ s,i e jκ s,iz ds 2 23, 1 i 6 1 i 6 ˆp s z := { cp,i e jκ id p z 1 i 6 if z d p 0 z > d p. 24 for the layer with index s and for 0 z d s. The formulas for the displacement vector uk x,k y,z,ω and the stress vector σk x,k y,z,ω take the form u s k x,k y,z = Θ s zc h,s + p s z, σ s k x,k y,z = Ξ s zc h,s + p s z. 25 The coefficients c h,s and ĉ s are c h,s = Θ 1 s û h from Eq. 14. By implementing a grid in the z-direction, the four dimensional grid in k x,k y,z,ω can easily be derived by evaluating in z. A Fourier back transform in k x and k y finally yields a grid in the coordinates x,y,z,ω, which was the intended result. TESTING THE MODEL Numerical Accuracy: The numerical stability is checked by running a symbolic computation software MAPLE 9.5 with the data, and comparing the results to the ones of the C++ program. Results are computed on various points of the k x,k y [ 10, 10] [ 10, 10] plane for ω = 10.0 khz, accuracy up to the fifth decimal position is detected. See Figure 2 for the results. Physical consistency: The second test is checking physical consistency. For a system of layers, the solution is computed. By introducing a virtual layer boundary at

8 a random depth, one layer is split in two with identical material parameters. The results must coincide at the layer boundaries. Results are calculated for ω = 10.0 khz, k x,k y [ 10, 10] [ 10, 10]. See Figure 2 for the results. CONCLUSIONS The model is designed for a layer structure of n layers consisting of orthotropic media. A general solution within one layer is derived with a Fourier approach, and back Fourier transformed is performed to handle the boundary conditions. Some additional considerations allow the case of the loaded layer to be handled. The balances at the layer boundaries deliver equations for the displacement vectors at the layer boundaries. Thus, the stress and displacement vectors can be calculated at arbitrary points in the soil. The model presents a more realistic approach, as it incorporates the orthotropic behavior of soils. It showed high accuracy in algebraic as well as in physical tests. The problems the model has to deal with are merely caused by the fact that the soil parameters are in general not determined well enough. Empiric tests as well as future research will focus on that, promising attempts have already been made[5] and can probably be applied. ACKNOWLEDGEMENTS This work is financed by the Austrian Federal Ministry for Transport, Innovation and Technology. Thanks to Wolfgang Kreuzer for technical assistance. References [1] W.M. Ewing, W.S. Jardetzky and F. Press, Elastic Waves in Layered Media, McGraw-Hill, New York, [2] F. Ziegler and P. Borejko, Seismic Waves in Layered Soil: The Generalized Ray Theory, in Structural Dynamics-Recent Advances, edited by G. I. Schueller, Springer, Berlin, 1991, pp [3] I. Tsvankin, Seismic Signatures and Analysis of Reflection data in Anisotropic media, volume 29 of Handbook of Geophysical Exploration: Seismic Exploration, Elsevier science, Amsterdam, [4] H. Waubke, Waves in a Layered Orthotropic Medium with and without Random Properties, in 10th International Congress on Sound and Vibration, Stockholm, [5] H. Waubke, P. Balazs, B. Jilge and W. Kreuzer, Waves in Random Layers with Arbitrary Covariance Functions, in 12th International Congress on Sound and Vibration, Lisbon,2005.