Mechanics shape function of quadrilateral element composed of subdivision triangles for recursive transfer method

Abstract

The recursive transfer method (RTM) provides a unique numerical means of extracting a localised wave in a scattering problem. However, various shapes of scatterers were hitherto difficult to adapt RTM, because mesh elements were limited to rectangles. In this study, we propose what we call mechanics shape function using subdivision triangles of a quadrilateral element, and we extend RTM to be valid for various shapes of scatterers. The effect of the mechanics shape function is evaluated using a uniform waveguide with a distorted mesh.

Appendices

Appendix A: Vanising of the flux divergence

Expressing the time derivative of the field as $\dot{u}$, the energy flux $\mathbf{J}$ was defined as $\mathbf{J} = \sum _{r=0}^2\big \{ \dot{u} \varDelta _r ( D_r \varDelta _r u) ) - \nabla \dot{u} \cdot \hat{\gamma }_r D_r (\varDelta _r u) \big \}$ [8, 21] Because the deformation $u(\mathbf r)$ satisfies (5), we can express the divergence of the flux as $\nabla \cdot \mathbf{J} = -q(\dot{u},u)$, where $q(\dot{u},u)$ is the spurious energy generation defined by (3). Assuming that the field is in a steady state and expressed in a complex representation, the integral of the flux about the subdivision region $S^{(n,e)}$ can be transformed as follows:

$$\begin{aligned} \int _{S^{(n,e)}}\nabla {}\cdot {}{} \mathbf{J} dS = {\omega \over 2}\mathrm{Im}\big [ F^{(n,e)}[u^*,u] \big ], \end{aligned}$$

(40)

where ‘*’ denotes the complex conjugate. Using the interpolation expression (31), the expression on the right-hand side in (40) can be discretised to derive the following expression:

$$\begin{aligned} \int _{S^{(n,e)}}\nabla {}\cdot {}{} \mathbf{J} dS \approx {\omega \over 2}\mathrm{Im}\big [{\mathbf{U}^{(n,e)*} \tilde{f}_\mathrm{mch}^{(n,e)} \mathbf{U}^{(n,e)}} \big ] . \end{aligned}$$

(41)

According to (41), the divergence definitely vanishes in the discretised system for any complex fields $\mathbf{U}^{(n,e)}$, because the matrix $\tilde{f}_\mathrm{mch}^{(n,e)}(\mathbf{r})$ is symmetric.

Appendix B: Cubic Hermite element for a rhombus

A conventional shape function, that is valid in a rhombus element and compatible with the weak-form discretisation for RTM, can consist of the direct product of the one-dimensional cubic polynomials of a Hermite element. Two fundamental shape functions in a one-dimensional space have the form as $\ell (\xi ) = 1-3\xi ^2 + 2\xi ^3$ and $n(\xi ) = \xi (1-\xi )^2$ [17] for $0 \le \xi \le 1$. Three elementary functions for the first vertex at $(\xi , \eta ) = (0, 0)$ can be created by direct products as follows:

$$\begin{aligned} L_\mathrm{H}^{(1)}(\xi ,\eta )= & {} \ell (\xi )\ell (\eta ), \end{aligned}$$

(42)

$$\begin{aligned} L_{\mathrm{H},\xi }^{(1)}(\xi ,\eta )= & {} n(\xi )\ell (\eta ), \end{aligned}$$

(43)

$$\begin{aligned} L_{\mathrm{H},\eta }^{(1)}(\xi ,\eta )= & {} \ell (\xi )n(\eta ). \end{aligned}$$

(44)

Using a column vector $\mathbf{h}_\mathrm{lcl}^{(1)}(\xi ,\eta )$ = $[{\ } L_\mathrm{H}^{(1)}(\xi ,\eta ){\ }$ $L_{\mathrm{H},\xi }(\xi ,\eta ){\ }$ $L_{\mathrm{H},\eta }(\xi ,\eta ){\ } ]^T$ a set of shape functions for the other vertices can be defined as follows: $\mathbf{h}_\mathrm{lcl}^{(2)}(\xi ,\eta ) = \mathbf{h}_\mathrm{lcl}^{(1)}(1-\xi ,\eta )$, $\mathbf{h}_\mathrm{lcl}^{(3)}(\xi ,\eta ) = \mathbf{h}_\mathrm{lcl}^{(1)}(1-\xi ,1-\eta )$ and $\mathbf{h}_\mathrm{lcl}^{(4)}(\xi ,\eta ) = \mathbf{h}_\mathrm{lcl}^{(1)}(\xi ,1-\eta )$. For transforming the local coordinate $(\xi ,\eta )$ to the global position vector $\mathbf{r}$, the relation (12) was used provided that p = 1. Using the transformation matrix $m_q^{(n,e)}$ that is defined by (17), the directional derivatives in $\mathbf{h}_\mathrm{lcl}^{(q)}(\xi ,\eta )$ can be transformed into the derivatives with respect to x and y from $\xi$ and $\eta$ as follows:

$$\begin{aligned} \mathbf{h}^{(q)}(\mathbf{r}(\xi ,\eta )) = \mathbf{h}_\mathrm{lcl}^{(q)}(\xi ,\eta ) m_q^{(n,e)}, \end{aligned}$$

(45)

where q = 1, 2,..., $P^{(n,e)}$.

A $3P^{(n,e)}$-dimensional row vector, $\mathbf{L}(\mathbf{r})$, that is composed of the shape functions can be defined as follows:

$$\begin{aligned} \mathbf{L}_\mathrm{H}(\mathbf{r}) = [{\ } \mathbf{h}^{(1)}(\mathbf{r}) {\ } \mathbf{h}^{(2)}(\mathbf{r}) {\ } \mathbf{h}^{(3)}(\mathbf{r}) {\ } \mathbf{h}^{(4)}(\mathbf{r}) {\ }] . \end{aligned}$$

(46)

Then, the field $u(\mathbf{r})$ can be interpolated with the $3P^{(n,e)}$-dimensional column vector $\mathbf{U}^{(n,e)}$ of the vertex field, that was defined in Sect. 3.2, as follows:

$$\begin{aligned} u(\mathbf{r}) \approx \mathbf{L}_\mathrm{H}(\mathbf{r}) \mathbf{U}^{(n,e)}, \end{aligned}$$

(47)

which is valid in rhombus elements.

The Hermite shape function can be extended to a scalene rhombus or quadrilateral. In this case, the mapping from the $\xi \eta$-plane to the xy-plane is no longer linear, and the Jacobian depends on $\xi$ and $\eta$; therefore, the integrands in the expressions (24) and (25) become rational functions of $\xi$ and $\eta$, and the analytic expressions of the integrals cannot be derived by symbolic formula manipulation. The integrals can be obtained using numerical methods, but the calculation efficiency and accuracy are significantly degraded. In contrast, the mechanics shape function has no such problems because the Jacobian is constant as discussed in Sect. 3.1.1.