A symmetric formula of transformed elasticity tensor in PML domain for elastic wave problem
AA symmetric formula of transformed elasticity tensor in PML domain for elastic wave problem
Yingshi Chen Institute of Electromagnetics and Acoustics, and Department of Electronic Science, Xiamen University, Xiamen 361005, China E-mail: [email protected]
Abstract
PML(Perfectly matched layer) is very important for the elastic wave problem in the frequency domain. Generally, the formulas of elasticity tensor in PML region are derived from the transformed momentum equation. In this note, we proved that the transformed elasticity tensor derived in this way lost its symmetry. Therefore, these formulas are inconsistency in theory and itโs hard to explain its numerical performance. We present a new symmetrical formula of elasticity tensor from the weak form. So the theory of elasticity is still applicable in PML domain.
Keywords: Perfectly matched layer, symmetric elasticity tensor, elastic wave
PML(Perfectly matched layer) is very important for the elastic wave problem in the frequency domain. The formulas of many papers are derived from the transformed momentum equation. However, in this way, the transformed elasticity tensor lost its symmetry. Therefore, in theory, these formulas are inconsistency and itโs hard to explain its numerical performance [5]. In this note, we get a new symmetrical formula of elasticity tensor from the weak form. We use Cartesian tensors such as ๐ ๐๐ , where the indices ๐, ๐ =1,2,3. We also use Einstein Summation Convention: if subscripted variables appearing twice in any term, the subscripted variables are assumed to be summed over. Along each coordinate axis, we define the unit orthonormal base vector ๐ ๐ . From Eulerโs momentum equation [1, 2], we get elastic wave equation in the frequency domain. โ๐ ๐๐ข ๐ โ โ๐ ๐๐ โ๐ฅ ๐ = ๐ ๐ (1) r โ๐ ๐๐ข ๐ โ ๐(๐ถ ๐๐๐๐ : ๐๐ข๐๐๐ฅ๐ )๐๐ฅ ๐ = ๐ ๐ (1.1) (Note: ๐กโ๐ ๐๐ฅ๐๐๐๐ ๐๐ ๐ ๐๐ ๐๐ 1.1 ๐๐๐๐ ๐กโ๐ ๐ ๐ฆ๐๐๐๐ก๐๐ฆ ๐๐ ๐ถ ๐๐๐๐ ) To absorb wave in the PML region, we use the famous complex coordinate stretching technique [6,7,8].
Unsymmetrical formula from transformed momentum equation
Many papers [9,3] try to get transformed momentum equation, which includes some new mathematic objects ๐ฬ , ๐ฬ ๐๐ , ๐ถฬ ๐๐๐๐ . โ๐ ๐ฬ๐ข ๐ โ โ๐ฬ ๐๐ โ๐ฅ ๐ = ๐ฬ ๐ (2) where ๐ฬ = ๐๐ ๐ ๐ ๐ฬ ๐๐ = ๐ถฬ ๐๐๐๐ ๐๐ข ๐ ๐๐ฅ ๐ (3) ๐ถฬ ๐๐๐๐ = ๐ถ ๐๐๐๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ (4) Itโs easy to verify ๐ถฬ ๐๐๐๐ โ ๐ถฬ ๐๐๐๐ (e.g. ๐ถฬ = ๐ถ ๐ ๐ ๐ , ๐ถฬ = ๐ถ ๐ ๐ ๐ ), then ๐ฬ ๐๐ โ ๐ฬ ๐๐ (5) The asymmetry of ๐ฬ ๐๐ means that theory of elasticity breaks down in the PML domain! Then any formula based on (2) are suspicious. As [5] showed: โif the stretched stresses fail equilibrium, then none of the classical theorems of elasticity which underpin the finite element method โfor example, the principle of virtual workโ can be assumed without much ado to remain strictly valid for PMLs.โ So this method is unconvincing and hard to explain its numerical performance Symmetrical formula from weak form
Apply the standard Galerkin procedure, we get weak form โ๐ โซ โ ๐๐ข ๐ ๐ ๐๐ฃ โ โซ โ โ๐ ๐๐ โ๐ฅ ๐ ๐ ๐๐ฃ = โซ โ ๐ ๐ ๐ ๐๐ฃ (6) where โ is any scalar testing functionใ After integration by parts โ๐ โซ โ ๐๐ข ๐ ๐ ๐๐ฃ + โซ ๐โ ๐๐ฅ ๐ ๐ ๐๐ ๐ ๐๐ฃ โ โซ โ(โ ๐ ๐๐ )โ๐ฅ ๐ ๐ ๐๐ฃ = โซ โ ๐ ๐ ๐ ๐๐ฃ (7) Letโs check the third term, which includes second order of the displacement vector ๐ฎ . By gauss (divergence) theorem: โซ โ(โ ๐ ๐๐ )โ๐ฅ ๐ ๐ ๐๐ฃ = โซ ( โ(โ ๐ )โ๐ฅ + โ(โ ๐ )โ๐ฅ + โ(โ ๐ )โ๐ฅ ) ๐ ๐๐ฃ = โซ โ (๐ ๐ + ๐ ๐ + ๐ ๐ ) ๐ ๐๐ (8) n the linear elastic theory, at free surface , ๐ ๐๐ = ๐ ๐๐ =0, So this term is zero. Or use zero dirichlet boundary condition, โ = 0 in the surface, this term is also 0. Letโ check the second term โซ ๐โ ๐๐ฅ ๐ ๐ ๐๐ ๐ ๐๐ฃ . For example, let i=1 โซ ๐โ ๐๐ฅ ๐ ๐ ๐1 ๐๐ฃ ๐ = โซ ๐โ ๐๐ฅ ๐ (๐ถ ๐1๐๐ : ๐๐ข ๐ ๐๐ฅ ๐ ) ๐๐ฃ ๐ธ = โ ๐โ ๐๐ฅ ๐ (๐ถ ๐1๐๐ : ๐๐ข ๐ ๐๐ฅ ๐ ) = โ ๐โ ๐๐ฅ ๐ (๐ถ โ๐ข + ๐ถ โ๐ข +๐ถ โ๐ข ) = โโ โ (๐ถ โ๐ข ) + โโ โ (๐ถ โ๐ข ) + โโ โ (๐ถ โ๐ข ) (9) or โ ๐โ ๐๐ฅ ๐ (๐ถ ๐1๐๐ : ๐๐ข ๐ ๐๐ฅ ๐ ) = โโ โ [๐ถ โ๐ข + ๐ถ โ๐ข + ๐ถ โ๐ข ] (10) where ๐ถ = (๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ) ๐ถ = (๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ) ๐ถ = (๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ) let i=2,3 We can get similar matrix ๐ถ ๐๐ ๐, ๐ =1,2,3. The elements of ๐ถ ๐๐ are same as the formula in [3]. In the PML domain, apply the complex stretch operator to (10): โ ๐โ ฬ๐๐ฅฬ ๐ (๐ถ ๐1๐๐ : ๐๐ขฬ ๐ ๐๐ฅฬ ๐ ) = โ ๐โ ฬ๐ ๐ ๐๐ฅ ๐ (๐ถ ๐1๐๐ : ๐๐ขฬ ๐ ๐ ๐ ๐๐ฅ ๐ ) = (ฮโโ ฬ) โ [๐ถ (ฮโ๐ขฬ ) + ๐ถ (ฮโ๐ขฬ ) + ๐ถ (ฮโ๐ขฬ )] (11) where ฮ = [1/๐ ] (12) Since ฮ is Diagonal, for each vector ๐โ ๐๐๐ ๐โโ : (ฮ๐โ) โ ๐โโ = ๐ ๐ ๐ + ๐ ๐ ๐ + ๐ ๐ ๐ = ๐โ โ (ฮ๐โโ) โ ๐โ ฬ๐๐ฅฬ ๐ (๐ถ ๐1๐๐ : ๐๐ขฬ ๐ ๐๐ฅฬ ๐ ) = ( โโ ฬ) ฮ โ [ ๐ถ ( ฮโ๐ข ฬ ) + ๐ถ ( ฮโ๐ข ฬ ) + ๐ถ ( ฮโ๐ข ฬ )] (13) apply the associate law of matrix product ฮC(ฮ๐โ) = (ฮCฮ)๐โ we get โ ๐โ ฬ๐๐ฅฬ ๐ (๐ถ ๐1๐๐ : ๐๐ขฬ ๐ ๐๐ฅฬ ๐ ) = โโ ฬ โ [(ฮ๐ถ ฮ)(โ๐ขฬ ) + (ฮ๐ถ ฮ)(โ๐ขฬ ) + (ฮ๐ถ ฮ)(โ๐ขฬ )] (14) Compare to (10), we get transformed ๐ถฬ ๐๐ in the PML domain: ๐ถฬ ๐๐ = ฮ๐ถ ๐๐ ฮ (14) or ๐ถฬ ๐๐๐๐ = ๐ถ ๐๐๐๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ (14.1) Itโs easy to verify that ๐ถฬ ๐๐๐๐ = ๐ถฬ ๐๐๐๐ and ๐ถฬ ๐๐๐๐ = ๐ถฬ ๐๐๐๐ . So ๐ฬ ๐๐ = ๐ฬ ๐๐ . Reference
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