A fast solver for elastic scattering from axisymmetric objects by boundary integral equations

Abstract

Fast and high-order accurate algorithms for three-dimensional elastic scattering are of great importance when modeling physical phenomena in mechanics, seismic imaging, and many other fields of applied science. In this paper, we develop a novel boundary integral formulation for the three-dimensional elastic scattering based on the Helmholtz decomposition of elastic fields, which converts the Navier equation to a coupled system consisted of Helmholtz and Maxwell equations. An FFT-accelerated separation of variables solver is proposed to efficiently invert boundary integral formulations of the coupled system for elastic scattering from axisymmetric rigid bodies. In particular, by combining the regularization properties of the singular boundary integral operators and the FFT-based fast evaluation of modal Green’s functions, our numerical solver can rapidly solve the resulting integral equations with a high-order accuracy. Several numerical examples are provided to demonstrate the efficiency and accuracy of the proposed algorithm, including geometries with corners at different wavenumbers.

Appendix.: Fourier expansion for the surface gradient and divergence

Let Γ be the boundary of a three-dimensional axisymmetric object with parametrization given by

$$\left( r(s)\cos\theta,r(s)\sin\theta,z(s)\right), $$

and (τ₁,τ₂) be the unit orthogonal tangential vectors on Γ. Here we do not require the parameter s to be the arclength of the generating curve of Γ.

Lemma 1

The surface gradient of a function σ ∈ C^1,α(Γ) with Fourier expansion (??) is given by

$$ \begin{array}{@{}rcl@{}} \text{Grad} \sigma ={\sum}_{m} \left( \frac{1}{\sqrt{r^{\prime 2}+z^{\prime 2}}}\frac{\partial\sigma_{m}}{\partial s} \boldsymbol{\tau}_1 + \frac{\mathrm{i} m }{r} \sigma_{m} \boldsymbol{\tau}_2 \right)e^{\mathrm{i} m\theta }. \end{array} $$

(A.1)

The surface divergence of a tangential vector $\boldsymbol {J}=J^{1}\boldsymbol {\tau }_1+J^{2}\boldsymbol {\tau }_2\in T_{d}^{0,\alpha }({\varGamma })$ with Fourier expansion (??) is given by

$$ \begin{array}{@{}rcl@{}} \text{Div} \boldsymbol{J}={\sum}_{m} \left( \frac{1}{r\sqrt{r^{\prime 2}+z^{\prime 2}}}\left( r^{\prime}(s){J_{m}^{1}}+r ({J_{m}^{1}})^{\prime}\right) + \frac{\mathrm{i} m }{r} {J_{m}^{2}}\right)e^{\mathrm{i} m\theta }. \end{array} $$

(A.2)

We first show the expression of Grad_y(n(x) ⋅n(y)) and Div_y(n(y) ×τ_i(x) ×n(y)), i = 1, 2 in the parametrization form. Since

$$ \begin{array}{@{}rcl@{}} \boldsymbol{n}(\boldsymbol{x})&=& C_{t}\left( z^{\prime}(t)(\cos \theta_{t}, \sin\theta_{t},0)-r^{\prime}(t)(0,0,1)\right),\\ \boldsymbol{n}(\boldsymbol{y})&=& C_{s}\left( z^{\prime}(s)(\cos \theta, \sin\theta,0)-r^{\prime}(s)(0,0,1)\right), \end{array} $$

where $C_{s} = 1/\sqrt {r^{\prime }(s)^{2}+z^{\prime }(s)^{2}}$, $C_{t} = 1/\sqrt {r^{\prime }(t)^{2}+z^{\prime }(t)^{2}}$, it holds

$$ \begin{array}{@{}rcl@{}} &&\text{Grad}_{\boldsymbol{y}}(\boldsymbol{n}(\boldsymbol{x})\cdot \boldsymbol{n}(\boldsymbol{y})) \\ &=& C_{s} \left( C_{t}C^{\prime}_{s}(z^{\prime}(t)z^{\prime}(s)\cos(\theta_{t}-\theta)+r^{\prime}(t)r^{\prime}(s)) \right.\\&&\left.+C_{t}C_{s}(z^{\prime}(t)z^{\prime\prime}(s)\cos(\theta_{t}-\theta)+r^{\prime}(t)r^{\prime\prime}(s))\right)\boldsymbol{\tau}_1 \\ &+&1/r(s)C_{t}C_{s} z^{\prime}(t)z^{\prime}(s)\sin(\theta_{t}-\theta)\boldsymbol{\tau}_2. \end{array} $$

Similarly, since

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\tau}_1(\boldsymbol{x})\!&=&\! C_{t}\left( r^{\prime}(t)(\cos \theta_{t}, \sin\theta_{t},0)+z^{\prime}(t)(0,0,1)\right), \quad \boldsymbol{\tau}_2(\boldsymbol{x})= (-\sin \theta_{t}, \cos\theta_{t},0), \\ \boldsymbol{\tau}_1(\boldsymbol{y})\!&=&\! C_{s}\left( r^{\prime}(s)(\cos \theta, \sin\theta,0)+z^{\prime}(s)(0,0,1)\right), \quad \boldsymbol{\tau}_2(\boldsymbol{y})= (-\sin \theta, \cos\theta,0), \end{array} $$

by Lemma 7, it holds

$$ \begin{array}{@{}rcl@{}} &&\text{Div}_{\boldsymbol{y}} (\boldsymbol{n}(\boldsymbol{y})\times \boldsymbol{\tau}_1(\boldsymbol{x}) \times \boldsymbol{n}(\boldsymbol{y})) \\ &=&C_{s}/r(s) \left[ r^{\prime}(s)C_{t}C_{s}\left( r^{\prime}(t)r^{\prime}(s)\cos(\theta_{t}-\theta)+z^{\prime}(t)z^{\prime}(s)\right)\right. \\ &+&r(s)C_{t}C^{\prime}_{s}\left( r^{\prime}(t)r^{\prime}(s)\cos(\theta_{t}-\theta)+z^{\prime}(t)z^{\prime}(s)\right) \\ &+&\left. r(s)C_{t}C_{s}\left( r^{\prime}(t)r^{\prime\prime}(s)\cos(\theta_{t}-\theta)+z^{\prime}(t)z^{\prime\prime}(s)\right)\right] \\ &-&1/r(s)C_{t}r^{\prime}(t)\cos(\theta_{t}-\theta), \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} &&\text{Div}_{\boldsymbol{y}} (\boldsymbol{n}(\boldsymbol{y})\times \boldsymbol{\tau}_2(x) \times \boldsymbol{n}(\boldsymbol{y})) \\ &=&C_{s}/ r(s) \left[-r^{\prime}(s)C_{s} r^{\prime}(s)\sin(\theta_{t}-\theta)-r(s)(C^{\prime}_{s}r^{\prime}(s)+C_{s}r^{\prime\prime}(s))\sin(\theta_{t}-\theta) \right]\\&+&1/r(s)\sin(\theta_{t}-\theta). \end{array} $$

To numerically construct the surface gradient operator for an unknown function σ, we make use of the fact that σ is discretized on p scaled Gauss-Legendre nodes on each panel as discussed in Section ??. Following the notation in Section ??, σ on the i th panel γ_i is approximated by

$$\sigma(\boldsymbol{y}) \approx \sum\limits_{j=1}^{p}c_{ij}{P^{i}_{j}}(\boldsymbol{y})= \sum\limits_{j=1}^{p}\sum\limits_{n=1}^{p} U^{i}_{jn} \sigma(\boldsymbol{y}_{in}) {P^{i}_{j}}(\boldsymbol{y}). $$

Therefore the surface divergence can be approximated as

$$\text{Grad}_{\boldsymbol{y}}\sigma(\boldsymbol{y}) \approx \sum\limits_{j=1}^{p} \sum\limits_{n=1}^{p} U^{i}_{jn} \sigma(\boldsymbol{y}_{in}) \text{Grad}_{\boldsymbol{y}} {P^{i}_{j}}(\boldsymbol{y})=\sum\limits_{n=1}^{p} \sum\limits_{j=1}^{p} U^{i}_{jn}\text{Grad}_{\boldsymbol{y}} {P^{i}_{j}}(\boldsymbol{y}) \sigma(\boldsymbol{y}_{in}) .$$

By taking y to be the p scaled Gauss-Legendre nodes on the i th panel, we obtain the discretized surface gradient operator for σ. The discretization of surface divergence operator for a tangential vector function J can be constructed similarly.

Combining the results above with Lemmas 3 and 4, we are able to obtain the azimuthal Fourier decomposition of boundary operators ${\mathscr{H}}$ and $\mathcal {N}$.