Correction to: Numerical Simulation and Experimental Validation of Nondendritic Structure Formation in Magnesium Alloy Under Oscillation and Ultrasonic Vibration

Correction to: Metallurgical and Materials Transactions B https://doi.org/10.1007/s11663-019-01654-5

The equation 1 “1 − ϕ” should be “1 − ϕ²”.

The equation 2 last “∂u/∂t” should be “∂ϕ/∂t” .

The corrected equations are

$$ \begin{aligned} &\tau \left( \varvec{n} \right)\left[ {1 + (1 - k)u} \right]\frac{\partial \varphi^{2}}{\partial t} \\ &\quad = \varphi (1 - \varphi ^{2}) - \lambda \left( {1 - \varphi^{2} } \right)^{2} \left( {u + \theta_{\text{sys}} } \right) + \nabla \cdot \left\{ {\left[ {W(\varvec{n})} \right]^{2} \nabla \varphi } \right\} \\ & \qquad - \,\frac{\partial }{\partial x}\left[ {W(\varvec{n})W^{\prime}(\varvec{n})\frac{\partial \varphi }{\partial y}} \right] + \frac{\partial }{\partial y}\left[ {W(\varvec{n})W^{\prime}(\varvec{n})\frac{\partial \varphi }{\partial x}} \right] \\ \end{aligned} $$

(1)

$$ \left( {\frac{1 + k}{2} - \frac{1 - k}{2}\varphi } \right)\left( {\frac{\partial u}{\partial t} + U \cdot \nabla u} \right) = \nabla \left( {D\frac{1 - \varphi }{2}\nabla u + J_{AT} } \right) + \frac{1}{2}\frac{\partial \varphi}{\partial t}\left[ {1 + \left( {1 - k} \right)u} \right] $$

(2)

The caption of Figure 21 should say, “Microstructure of solidified AZ91D at a fixed oscillation frequency (1 Hz), ultrasonic power (1000 W), and oscillation amplitude (4π/3) for inclined angles of (a) 25 deg, (b) 20 deg, and (c) 15 deg.