Unconditionally Stable System Incorporated Factorization-Splitting Algorithm for Blackout Re-Entry Vehicle

Abstract

A high-temperature plasma sheath is generated on the surface of the re-entry vehicle through the conversion of kinetic energy to thermal and chemical energy across a strong shock wave at the hypersonic speed. Such a condition results in the forming of a blackout which significantly affects the communication components. To analyze the re-entry vehicle at the hypersonic speed, an unconditionally stable system incorporated factorization-splitting (SIFS) algorithm is proposed in finite-difference time-domain (FDTD) grids. The proposed algorithm shows advantages in the entire performance by simplifying the update implementation in multi-scale problems. The plasma sheath is analyzed based on the modified auxiliary difference equation (ADE) method according to the integer time step scheme in the unconditionally stable algorithm. Higher order perfectly matched layer (PML) formulation is modified to simulate open region problems. Numerical examples are carried out to demonstrate the performance of the algorithm from the aspects of target characteristics and antenna model. From resultants, it can be concluded that the proposed algorithm shows considerable accuracy, efficiency, and absorption during the simulation. Meanwhile, plasma sheath significantly affects the communication and detection of the re-entry vehicle.

1. Introduction

A plasma sheath is formed on the surface of the re-entry vehicle due to the conversion of kinetic energy to thermal and chemical energy across a strong shock wave at the hypersonic speed [1]. The occurrence of the blackout phenomenon significantly affects the detection and communication of the re-entry vehicle [2,3]. The accurate simulation of plasma sheaths has become a frontier science during the investigation of blackout phenomena. Several methods have been developed including the physics optic method, shooting bouncing ray method, and so on. However, most of them focus on the rapid calculation of electrically large objects [4,5,6,7,8]. To analyze the time variation behavior, the finite-difference time-domain (FDTD) algorithm shows potential in the simulation of the blackout phenomenon [9]. The conventional FDTD algorithm shows limitations in a multi-scale problem that is composed of antenna and vehicle models. By applying the conventional algorithm to the multi-scale problem directly, the mesh size must be chosen as fine enough to ensure the Courant–Friedrichs–Levy (CFL) condition [10]. Such a condition results in an extremely large computational domain and extensively long simulation duration. To overcome the CFL condition and improve the calculation efficiency, unconditionally stable algorithms are proposed [11,12,13,14]. Most of the unconditionally stable algorithms are the sub-step procedure. The sub-step procedure solves the single iteration with several sub-equations. The numerical dispersion increases with the increment of sub-step equations [15]. Meanwhile, memory consumption also increases in the sub-step procedure [16]. Such a condition leads to a decrement in efficiency and accuracy. To overcome such conditions, one-step schemes are introduced including leapfrog scheme, fundamental scheme, and so on [17,18]. However, there are still limitations in the arithmetical manipulations [19,20]. The one-step Crank-Nicolson (CN) scheme can update Maxwell’s equations within a single iteration which is merely efficient in one-dimensions. By applying the CN scheme to multi-dimension problems, a large sparse matrix must be calculated at each time step which results in a more expensive calculation. To alleviate such conditions, approximate CN schemes are introduced which include approximate decoupling and Douglas-Gunn schemes [21,22]. However, two-dimensional algorithms cannot be directly employed in three-dimensional problems [23]. Thus, approximate-factorization-splitting (AFS) and directly-splitting (DS) schemes are introduced into three-dimensional problems [24,25]. However, massive components, auxiliary variables, and coefficients must be calculated at each time step. In order to alleviate such conditions, a system incorporated directly-splitting (SIDS) algorithm is introduced based on the digital signal processing (DSP) technique [26]. According to the system-incorporated scheme, the entire algorithm can be regarded as the implementation of a signal processing technique. Thus, the algorithm can be simplified by avoiding the calculation of repeated coefficients, field components, and auxiliary variables. However, it has been testified that the DS algorithm shows accuracy degeneration due to the direct approximation of field components [27,28].

A plasma sheath is formed on the surface of the re-entry vehicle at the hypersonic speed. For the investigation of micro-behavior, plasma sheaths can be expressed by the anisotropic magnetized plasma [29]. Several techniques are carried out which are mainly based on the conventional explicit FDTD algorithm [30,31]. Most of them hold field components at a half-integer time step. By applying them directly to the one-step unconditionally stable algorithms, the algorithm will become inaccurate or unstable. To alleviate such conditions, algorithms are modified at the integer time step. Among the existing algorithms, the auxiliary difference equation (ADE) method shows considerable efficiency and accuracy [32,33].

For the open region problems, adequate absorbing boundary conditions must be introduced to efficiently simulate the unbounded grids. The perfectly matched layer (PML) is regarded as one of the most powerful absorbing boundary conditions [34]. The original PML formulation is based on the split-field scheme with four auxiliary variables which results in the decrement of absorption and efficiency. In order to alleviate such conditions, unsplit-field schemes are introduced including stretched coordinate (SC), complex-frequency-shifted (CFS), and convolutional PML (CPML) schemes [35,36,37]. Compared with the SC-PML scheme, CFS-PML formulation shows advantages in late-time reflections and low-frequency evanescent waves. However, both cannot efficiently absorb low-frequency propagation waves. A higher-order scheme is proposed which is implemented by multiplying the stretched coordinate variables together into a single term [38]. The original higher-order PML formulation is the split-field scheme which holds six auxiliary variables. The modified higher-order schemes are carried out with four auxiliary variables to improve the entire performance [38,39,40]. Recently, an unconditionally stable higher-order CPML scheme is proposed to further simplify the implementation [40,41].

It can be concluded that several limitations still should be addressed in the simulation of blackout re-entry vehicles. The problems mainly include the solution of multi-scale problems, calculation of magnetized plasma, and absorbing boundary conditions for complex media.

Here, an unconditionally stable system incorporated factorization-splitting (SIFS) algorithm is proposed based on the DSP technique to efficiently simulate the multi-scale problem of the blackout re-entry vehicle. The plasma sheath can be expressed by the anisotropic magnetized plasma and solved by the modified ADE method. Higher order CPML formulation is proposed for the complex anisotropic media. Numerical examples including the antenna performance and target characteristics are carried out to illustrate the entire behavior of the algorithm. The results indicates that the proposed algorithm shows considerable accuracy, efficiency, and absorption. The employment of the SI method can simplify the implementation by avoiding the calculation of repeated coefficients, field components, and auxiliary variables. The existence of a plasma sheath significantly affects the communication and detection of the re-entry vehicle.

2. Formulation

Inside the anisotropic magnetized plasma regions, Maxwell’s equations in the higher-order PML regions can be written as [42]

$$j\omega D={\nabla }_s\times H$$

(1)

$$j\omega {\mu }_{0}H=-{\nabla }_s\times E$$

(2)

where $D={\epsilon }_0{\epsilon }_{r}E=E+J$. The polarization current density of the anisotropic plasma can be expressed as

$$j\omega J={\epsilon }_0{\omega }_p^{2}E-\nu J+{\omega }_{b}J$$

(3)

where $\nu$, ${\omega }_p$ and ${\omega }_b$ are the damping constant, plasma frequency, and electron gyro-frequency, respectively [43]. The operator ${\nabla }_s$ can be expressed as

$${\nabla }_s=\widehat{x}\frac{1}{S_x}\frac{\partial }{\partial x}+\widehat{y}\frac{1}{S_y}\frac{\partial }{\partial y}+\widehat{z}\frac{1}{S_z}\frac{\partial }{\partial z}$$

(4)

where $S_{\eta }\left(\eta =x,y,z\right)$ is the stretched coordinate variables of the higher order formulation with CFS factor, defined as

$$S_{\eta }=\left({\kappa }_{\eta 1}+\frac{{\sigma }_{\eta 1}}{{\alpha }_{\eta 1}+j\omega {\epsilon }_0}\right)\left({\kappa }_{\eta 2}+\frac{{\sigma }_{\eta 2}}{{\alpha }_{\eta 2}+j\omega {\epsilon }_0}\right)$$

(5)

where ${\kappa }_{\eta n}\ge 1,n=1,2$ is employed, ${\sigma }_{\eta n}$ and ${\alpha }_{\eta n}$ are assumed to be positive real [44]. Maxwell’s equations can be divided into six individual components which almost hold the same forms [42]. Thus, the x-projection of electric components is employed as an example for demonstration. The other components can be obtained according to a similar approach. It can be observed that the algorithm can be solved from the aspects of plasma analyzation and unconditionally stable scheme implementation.

2.1. Modified ADE Algorithm for Unconditionally Stable SI Scheme

Supposing that the plasma is magnetized along the z-direction, the polarization current density can be expressed as

$$j\omega J_x+\nu J_x={\epsilon }_0{\omega }_p^2{E}_x-{\omega }_b{J}_y$$

(6)

$$j\omega J_y+\nu J_y={\epsilon }_0{\omega }_p^2{E}_y+{\omega }_b{J}_x$$

(7)

$$j\omega J_z={\epsilon }_0{\omega }_p^2{E}_z$$

(8)

According to the ADE method [45], components are discretized at the integer time step which can be given as

$$\frac{J_x^{n+1}-J_x^n}{\Delta t}+\frac{\nu \left(J_x^{n+1}+J_x^n\right)}{2}=\frac{{\epsilon }_0{\omega }_p^2\left(E_x^{n+1}+E_x^n\right)}{2}-\frac{{\omega }_b\left(J_y^{n+1}+J_y^n\right)}{2}$$

(9)

$$\frac{J_y^{n+1}-J_y^n}{\Delta t}+\frac{\nu \left(J_y^{n+1}+J_y^n\right)}{2}=\frac{{\epsilon }_0{\omega }_p^2\left(E_y^{n+1}+E_y^n\right)}{2}+\frac{{\omega }_b\left(J_x^{n+1}+J_x^n\right)}{2}$$

(10)

$$\frac{J_z^{n+1}-J_z^n}{\Delta t}=\frac{{\epsilon }_0{\omega }_p^2\left(E_z^{n+1}+E_z^n\right)}{2}$$

(11)

It can be observed that the components are coupled which cannot be updated directly. To implement the algorithm, field components must be decoupled. By substituting Equation (9) into Equation (10) and Equation (10) into Equation (11), results can be given as

$$J_x^{n+1}=p_{1a}{J}_x^n-p_{2a}{J}_y^n+p_{3a}{E}_x^{n+1}+p_{3a}{E}_x^n-p_{4a}{E}_y^{n+1}-p_{4a}{E}_y^n$$

(12)

$$J_y^{n+1}=p_{1a}{J}_y^n+p_{2a}{J}_x^n+p_{3a}{E}_y^{n+1}+p_{3a}{E}_y^n+p_{4a}{E}_x^{n+1}+p_{4a}{E}_x^n$$

(13)

$$J_z^{n+1}=J_z^n+p_{5a}{E}_z^{n+1}+p_{5a}{E}_z^n$$

(14)

where

$$p_{1a}=[1-{(\nu \Delta t/4)}^2-{({\omega }_b\Delta t/4)}^2]/p_{6a},p_{2a}={\omega }_b\Delta t/p_{6a},p_{3a}=[{\epsilon }_0\Delta t{\omega }_p^2(1+\nu \Delta t/4)]/p_{6a},$$

$$p_{4a}=({\epsilon }_0\Delta t^2{\omega }_p^2{\omega }_b)/(2p_{6a}),p_{5a}=\left[{(1+\nu \Delta t/4)}^2+{({\omega }_b\Delta t/4)}^2\right]/2\mathrm{and}{p}_{6a}={\epsilon }_0{\omega }_p^2\Delta t/4.$$

Through the employment of Equations (12)–(14), the plasma sheath can be analyzed. Meanwhile, it can be observed that the components at the time step of n + 1 still exists in these equations which leads to the decoupling of components in an unconditionally stable algorithm. Thus, they should be further decoupled during the implementation of the SIFS algorithm.

2.2. consumption, simulation duration, and time CPML Formulation

With the employment of higher-order CPML formulation, Maxwell’s equations can be given as [46]

$$\frac{\partial D_x}{\partial t}=\frac{1}{S_y^{}}\frac{\partial H_z}{\partial y}-\frac{1}{S_z^{}}\frac{\partial H_y}{\partial z}$$

(15)

Equation (15) can be reorganized in the time domain by employing the inverse Fourier transform (IFT) method, given as

$$\frac{\partial D_x}{\partial t}={\overline{S}}_y*\frac{\partial H_z}{\partial y}-{\overline{S}}_z^{}*\frac{\partial H_y}{\partial z}$$

(16)

where $*$ represents the convolution operation. The stretched coordinate variable ${\overline{S}}_{\eta }$ can be written as

$${\overline{S}}_{\eta }\left(t\right)=\left({\alpha }_{\eta 1}\left(t\right)+\frac{\delta \left(t\right)}{{\kappa }_{\eta 1}}\right)\left({\alpha }_{\eta 2}\left(t\right)+\frac{\delta \left(t\right)}{{\kappa }_{\eta 2}}\right)$$

(17)

where $\delta \left(t\right)$ represents the impulse function in the signal processing technique. ${\alpha }_{\eta }\left(t\right)$ can be expressed as

$${\alpha }_{\eta n}\left(t\right)={\beta }_{\eta n}{e}^{-{\gamma }_{\eta n}t}u\left(t\right)$$

(18)

where the coefficients can be given as

$${\beta }_{\eta n}=-\frac{{\sigma }_{\eta n}}{{\kappa }_{\eta n}^2{\epsilon }_0},{\gamma }_{\eta n}=\frac{{\alpha }_{\eta n}}{{\epsilon }_0}+\frac{{\sigma }_{\eta n}}{{\kappa }_{\eta n}{\epsilon }_0}.$$

$u\left(t\right)$ represents the unit function in the signal processing technique. By substituting Equation (18) into Equation (16) and employing the relationship of polarization current density, results can be given as

$$\begin{array}{ll}{\epsilon }_0\frac{\partial E_x}{\partial t}& -{\epsilon }_0\frac{\partial J_x}{\partial t}={\alpha }_{y1}\left(t\right){\alpha }_{y2}\left(t\right)*\frac{\partial H_z}{\partial y}-{\alpha }_{z1}\left(t\right){\alpha }_{z2}\left(t\right)*\frac{\partial H_y}{\partial z}\\ & -\frac{1}{{\kappa }_{z1}{\kappa }_{z2}}\frac{\partial H_y}{\partial z}-\frac{1}{{\kappa }_{z1}}{\alpha }_{z2}\left(t\right)*\frac{\partial H_y}{\partial z}-\frac{1}{{\kappa }_{z2}}{\alpha }_{z1}\left(t\right)*\frac{\partial H_y}{\partial z}\\ & +\frac{1}{{\kappa }_{y1}{\kappa }_{y2}}\frac{\partial H_z}{\partial y}+\frac{1}{{\kappa }_{y1}}{\alpha }_{y2}\left(t\right)*\frac{\partial H_z}{\partial y}+\frac{1}{{\kappa }_{y2}}{\alpha }_{y1}\left(t\right)*\frac{\partial H_z}{\partial y}\end{array}$$

(19)

According to the FDTD discretized form and recursive convolution method, results can be rewritten as

$$\begin{array}{c}{E}_x^{n+1}=E_x^n+\frac{\Delta t}{{\epsilon }_0}(\frac{{\delta }_y\left(H_z^n\right)}{{\kappa }_{y1}{\kappa }_{y2}}-\frac{{\delta }_z\left(H_y^n\right)}{{\kappa }_{z1}{\kappa }_{z2}}+\frac{F_{xy1}^n}{{\kappa }_{y2}}+\frac{F_{xy2}^n}{{\kappa }_{y1}}-\frac{F_{xz1}^n}{{\kappa }_{z2}}-\frac{F_{xz2}^n}{{\kappa }_{z1}}\\ +{\chi }_{y1}{\chi }_{y2}{e}^{-\left({\gamma }_{y1}+{\gamma }_{y2}\right)\Delta t}{\delta }_y\left(H_z^{n+1}+H_z^n\right)-{\chi }_{z1}{\chi }_{z2}{e}^{-\left({\gamma }_{y1}+{\gamma }_{y2}\right)\Delta t}{\delta }_z\left(H_y^{n+1}+H_y^n\right)+J_x^{n+1}-J_x^n)\end{array}$$

(20)

where ${\delta }_{\eta }$ represents the first-order discretized form in space

$${\delta }_{\eta }\left\{H_y\right\}=\frac{{H_y|}_{\eta +1/2}^{n+1/2}-{H_y|}_{\eta -1/2}^{n+1/2}}{\Delta \eta }$$

(21)

where ${\chi }_{\eta n}=\frac{{\sigma }_{\eta n}}{{\sigma }_{\eta n}{\kappa }_{\eta n}+{\alpha }_{\eta n}{\kappa }_{\eta n}^2}\left(e^{-{\gamma }_{\eta n}\Delta t}-1\right)e^{-{\gamma }_{\eta n}m\Delta t}$. By introducing the coefficients, results can be given as

$$\begin{array}{ll}{E}_x^{n+1}=E_x^n& +p_{1ey}{F}_{xy2}^n+p_{2ey}{F}_{xy1}^n-p_{1ez}{F}_{xz2}^n-p_{2ez}{F}_{xz1}^n+p_{3ey}{\delta }_y\left(H_z^n\right)-p_{3ez}{\delta }_z\left(H_y^n\right)\\ & +p_{4ey}{\delta }_y\left(H_z^{n+1}+H_z^n\right)-p_{4ez}{\delta }_z\left(H_y^{n+1}+H_y^n\right)+J_x^{n+1}-J_x^n)\end{array}$$

(22)

where the coefficients can be given as $p_{1e\eta }=\Delta t/{\epsilon }_0{\kappa }_{\eta 1}$, $p_{2e\eta }=\Delta t/{\epsilon }_0{\kappa }_{\eta 2}$, $p_{3\eta }=\Delta t/\left({\epsilon }_0{\kappa }_{\eta 1}{\kappa }_{\eta 2}\right)$, and $p_{4e\eta }=\left({\chi }_{\eta 1}{\chi }_{\eta 2}{e}^{-\left({\gamma }_{\eta 1}+{\gamma }_{\eta 2}\right)\Delta t}\right)/2{\epsilon }_0$. Magnetic components can be obtained according to a similar procedure. It can be observed from Equation (22) that the components at the time step of n + 1 exists on both sides of the equations, including the magnetic components and polarization current density. In order to alleviate such condition, components at the time step n + 1 are substituted into Equation (22) as

$$\begin{array}{l}{E}_x^{n+1}=E_x^n+p_{1y}{F}_{xy2}^n+p_{2y}{F}_{xy1}^n-p_{1z}{F}_{xz2}^n-p_{2z}{F}_{xz1}^n+p_{3y}{\delta }_y\left(H_z^n\right)-p_{3z}{\delta }_z\left(H_y^n\right)\\ \left(1-D_{2y}-D_{2z}\right)E_x^{n+1}=\left(1+D_{2y}+D_{2z}\right)E_x^n\\ +p_{1ey}{F}_{xy2}^n+p_{2ey}{F}_{xy1}^n-p_{1ez}{F}_{xz2}^n-p_{2ez}{F}_{xz1}^n+p_{3ey}{\delta }_y\left(H_z^n\right)-p_{3ez}{\delta }_z\left(H_y^n\right)\\ +2p_{4ey}{p}_{4hx}{\delta }_y{\delta }_x{E}_y^n-2p_{4ez}{p}_{4hy}{\delta }_z{\delta }_x{E}_z^n+2p_{4ey}{\delta }_{2y}{H}_z^n-2p_{4ez}{\delta }_z{H}_y^n\\ -p_{1hx}{p}_{4ez}{\delta }_z{G}_{yx2}^n-p_{2hx}{p}_{4ez}{\delta }_z{G}_{yx1}^n+p_{1hz}{p}_{4ez}{\delta }_z{G}_{yz2}^n+p_{2hz}{p}_{4ez}{\delta }_z{G}_{yz1}^n\\ +p_{1hy}{p}_{4ey}{\delta }_y{G}_{zy2}^n+p_{2hy}{p}_{4ey}{\delta }_y{G}_{zy1}^n-p_{1hx}{p}_{4ey}{\delta }_y{G}_{zx2}^n-p_{2hx}{p}_{4ey}{\delta }_y{G}_{zx1}^n\\ -p_{3hx}{p}_{4ez}{\delta }_z{\delta }_x\left(E_z^n\right)+p_{3hz}{p}_{4ez}{\delta }_z{\delta }_z\left(E_z^n\right)+p_{3hy}{p}_{4ey}{\delta }_y{\delta }_y\left(E_x^n\right)-p_{3hx}{p}_{4ey}{\delta }_y{\delta }_x\left(E_y^n\right)\\ +\left(p_{1a}-1\right)J_x^n-p_{2a}{J}_y^n+2p_{3a}{E}_x^n-2p_{4a}{E}_y^n\end{array}$$

(23)

where the operator can be defined as $D_{2\eta }=p_{4\eta }^2{\delta }_{\eta }$. By applying Equation (23) directly as an algorithm, large sparse matrices are formed at both sides of the equations which makes the computation much more expensive. To overcome such conditions, disturbance terms are added at both sides of Equation (23), $(-D_{2x}+D_{2x}{D}_{2z}+D_{2x}{D}_{2y}+D_{2y}{D}_{2z}-D_{2x}{D}_{2y}{D}_{2z})E_x^{n+1}$ and $(-D_{2x}+D_{2x}{D}_{2z}+D_{2x}{D}_{2y}+D_{2y}{D}_{2z}-D_{2x}{D}_{2y}{D}_{2z})E_x^n$. The results can be re-organized as

$$\left(1-D_{2x}\right)\left(1-D_{2y}\right)\left(1-D_{2z}\right)E_x^{n+1}=\left(1+D_{2x}+D_{2y}-D_{2z}\right)E_x^n+A^n$$

(24)

where $A^n$ represents the other components at the time step of n. Although equation (24) can be solved according to the CNAFS algorithm in [25]. A large number of auxiliary variables, field components, and coefficients must be solved at each time step which results in the increment of memory consumption and simulation duration. To alleviate such conditions, the SI method is introduced in the z-domain based on the DSP technique. Such an operation avoids the calculation of repeated variables and multi-equation solutions. At the first step of the SI method, auxiliary variables are introduced as

$$\begin{array}{l}{f}_x^n=p_{1ey}{F}_{xy2}^n+p_{2ey}{F}_{xy1}^n-p_{1ez}{F}_{xz2}^n-p_{2ez}{F}_{xz1}^n+p_{3ey}{\delta }_y\left(H_z^n\right)-p_{3ez}{\delta }_z\left(H_y^n\right)\\ g_x^n=p_{1hy}{p}_{4ey}{\delta }_y{G}_{zy2}^n+p_{2hy}{p}_{4ey}{\delta }_y{G}_{zy1}^n-p_{1hx}{p}_{4ey}{\delta }_y{G}_{zx2}^n-p_{2hx}{p}_{4ey}{\delta }_y{G}_{zx1}^n\end{array}$$

(25)

$$-p_{1hx}{p}_{4ez}{\delta }_z{G}_{yx2}^n-p_{2hx}{p}_{4ez}{\delta }_z{G}_{yx1}^n+p_{1hz}{p}_{4ez}{\delta }_z{G}_{yz2}^n+p_{2hz}{p}_{4ez}{\delta }_z{G}_{yz1}^n$$

(26)

$$e_{\eta }=p_{3\eta }^{-1}{\delta }_y{E}_{\eta }^{},e_{\eta }^1=E_{\eta }^1-E_{\eta }^n,e_{\eta }^2=E_{\eta }^2-E_{\eta }^n,e_{\eta }^{n+1}=E_{\eta }^{n+1}-E_{\eta }^n$$

(27)

$$h_z^n=2p_{4ey}{\delta }_{2y}{H}_z^n\mathrm{and}{h}_y^n=2p_{4ez}{\delta }_z{H}_y^n$$

(28)

$$j_x^n=\left(p_{1a}-1\right)J_x^n-p_{2a}{J}_y^n+2p_{3a}{E}_x^n-2p_{4a}{E}_y^n$$

(29)

By employing auxiliary variables and field components, the original equation can be given as

$$\left(1-D_{2y}-D_{2z}\right)E_x^{n+1}=\left(1+D_{2y}+D_{2z}\right)E_x^n+f_x^n+g_x^n+j_x^n+h_z^n-h_y^n$$

(30)

In such circumstances, the FS scheme is employed to further solve the equation, given as

$$\left(1-D_{2x}\right)e_x^1=E_x^n+f_x^n+g_x^n+j_x^n+h_z^n-h_y^n$$

(31)

$$\left(1-D_{2y}\right)e_x^2=e_x^1$$

(32)

$$\left(1-D_{2z}\right)e_x^{n+1}=e_x^2$$

(33)

As can be observed that tri-diagonal matrices are formed on the left side of Equations (31)–(33). The matrices can be solved by the Thomas algorithm implicitly [47]. Through Equations (31)–(33), the entire algorithm can be given according to the Fourier method as a matrix whose features are smaller than 1 which ensures unconditional stability [48]. Meanwhile, the algorithm cannot only be implemented but also simplified according to the SIFS algorithm. The update procedure of the entire algorithm is shown in Figure 1. The blank and red colors represent the original CNAFS algorithm and SIFS algorithm, respectively.

It can be concluded that all of the components in the proposed SIFS algorithm can be solved individually which results in avoiding the calculation of repeated field components, auxiliary variables, and coefficients. Meanwhile, by employing proper auxiliary variables, the algorithm can be further simplified compared with the original CNAFS algorithm which also decreases the memory consumption and simulation duration.

3. Numerical Results and Discussion

Numerical examples are carried out to further demonstrate the effectiveness of the algorithm. The occurrence of blackout phenomenon results in significant influence on the target characteristic and antenna radiation which leads to the detecting failure and communication interruption.

3.1. Target Characteristic Problem: Radar Echo Wave of the Blackout Re-Entry Vehicle

The existence of the plasma sheath shows a significant influence on the radar echo wave resulting in the variation of the target characteristic. Figure 2 shows the sketch picture of the re-entry vehicle model and the location of the antenna. The entire model can be regarded as a combination of half-sphere and half-pyramid. The radius of the sphere and upper radius of the half-pyramid hold the same dimensions as 0.2 mm. The bottom radius of the half-pyramid and the height are both 0.8 mm.

The plasma sheath can be expressed by the anisotropic plasma. The parameters of the plasma sheath can be simulated by the RAM-C based on the Computational Fluid Dynamics method as [49]. According to the same approach, the parameters under the condition of 80 km height in the atmosphere with a speed of 25 Ma. The temperature and the pressure of the atmosphere are 247.0 K and 30.0 Pa. A high temperature of 1700 K is formed due to the conversion of kinetic energy to thermal and chemical energy across a strong shock wave at the hypersonic speed. The plasma sheath is with the parameters and thickness of ${\omega }_p=2\pi \times 28.7$ GHz, ${\omega }_b=1\times {10}^{11}$ GHz, $\nu =20$ GHz, and 0.1 m. On the surface of the re-entry vehicle, the monopole antenna is located along the normal vertical direction. The vehicle is made up of metal which can be expressed by the perfect electronic conductor (PEC) during the simulation.

Figure 3 shows the sketch picture of the re-entry vehicle computational domain. As is demonstrated that the vehicle model is located in the middle of the computational domain. The entire structure is a cube with a length of 1.5 m. The Gaussian pulse plane wave with the maximum frequency of 0.8 GHz incidents from the left of the structure. The rest part of the computational domain is filled with a vacuum. At the boundaries of the domain, they are terminated by 10-cell-PML regions to absorb the outgoing waves and reduce the wave reflections. The parameters inside the PML regions are chosen to obtain the best performance both in time and frequency domains. For comparison, the explicit CFS-PML in [50] and higher order formulation in [51], the original CNAFS algorithm with CFS-PML in [52], and the higher order formulation in [49] are chosen as examples.

According to the unconditionally stable algorithm, mesh size can be chosen to satisfy the demand of calculation rather than the limitation of the CFL condition. In order to satisfy the calculation accuracy, mesh sizes must be chosen as fine enough to conformal to the curves. Thus, the optimal mesh size which shows a compromise between accuracy and efficiency is chosen as $\Delta x=\Delta y=\Delta z=\Delta =3$ mm. According to such circumstances, the entire computational domain can be discretized as $500\Delta x\times 500\Delta y\times 500\Delta z$ in FDTD grids. The maximum time step obtained by the CFL condition in the explicit algorithms can be chosen as $\Delta t_{\max }^{FDTD}=5.78$ ps. CFL number (CFLN) which represents the enlargement of the time step can be obtained as $CFLN=\Delta t/\Delta t_{\max }^{FDTD}$, where $\Delta t$ is the time step employed in the unconditionally stable algorithms. The target characteristic can be reflected by the radar cross-section (RCS) [53]. The RCS with the variation of the frequency obtained by different algorithms and CFLNs is shown in Figure 4. Meanwhile, the RCS parameter of the re-entry vehicle without the plasma sheath can be obtained by changing the plasma sheath to vacuum which is shown in Figure 5.

As can be concluded from Figure 4a and Figure 5a that the curves are almost overlapped with CFLN = 1. Such a condition indicates that the algorithms hold the same calculation accuracy with a small time step. In addition, such a condition also indicates that the calculation of anisotropic magnetized plasma is efficient. From Figure 4b and Figure 5b, curves show some shifting compared with the explicit FDTD-PML and FDTD-HPML schemes. Such a condition is caused by the enlargement of numerical dispersion with the enlargement of the time step. However, the calculation accuracy still shows a considerable level. As can be compared and observed between Figure 4 and Figure 5, the existence of the plasma sheath significantly affects the target characteristic. Nowadays, the communication system between the vehicle and the ground depends on the VHF and UHF band. The variation of scattering waves shows an influence on the detection. The effectiveness of the algorithm can be reflected by the memory consumption, simulation duration, and time reduction obtained by different algorithms and CFLN, as shown in

Table 1. The memory consumption, simulation duration, and time reduction obtained by different algorithms and CFLN of the RCS problem.

Algorithms	CFLN	Steps	Memory (G)	Time (min)	Reduction (%)
FDTD-PML	1	32768	27.5	43.4	-
FDTD-HPML	1	32768	35.9	91.7	−111.3
CNAFS-PML	1	32768	40.4	372.3	−757.8
CNAFS-HPML	1	32768	53.1	440.1	−1006.2
SIFS-HPML	1	32768	47.3	389.5	−785.9
CNAFS-PML	10	3277	40.4	21.9	49.5
CNAFS-HPML	10	3277	53.1	26.4	39.2
SIFS-HPML	10	3277	47.3	22.1	49.1

.

As shown in Table 1, the explicit FDTD-PML scheme shows the least memory consumption and simulation duration due to the non-calculation of matrices with the explicit calculation procedure. The implicit schemes calculate nine matrices during each iteration which results in such a condition. In order to improve the efficiency, the time step can be enlarged according to the unconditionally stable algorithm which can also decrease the total iteration steps. Compared with the original scheme, the proposed algorithm shows improvement in efficiency due to the non-calculation of repeated auxiliary variables, coefficients, and field components according to the SI method. Meanwhile, the memory consumption and simulation duration of the higher order formulation also shows significant improvement due to the multiplying of stretched coordinate variables. The proposed algorithm still shows considerable behavior compared with the original scheme.

3.2. Wave Radiation Problem: Antenna Communication in the Plasma Sheath on the Surface of the Blackout Re-Entry Vehicle

As is demonstrated that the existence of plasma sheath shows a significant influence on the communication system. Among the communication system, the antenna is one of the most important elements. As shown in Figure 2 that the antenna model is located on the side surface of the vehicle. In order to demonstrate the influence of the plasma sheath, Figure 6 shows the computational domain of the antenna. The entire computational domain holds the dimensions of $20\times 20\times 10$ mm³ in each direction. It can be observed that the antenna is situated on the surface of the vehicle which can be expressed by the PEC metal plate with the size of $10\times 10\times 1$ mm³ in each direction. As shown in Figure 6, the monopole antenna with a length of 1.75 mm is located at the center of the domain. The modulated Gaussian source excites from the antenna with the center frequency and bandwidth of 4 GHz and 1 GHz, respectively. The rest part of the computational domain is filled with anisotropic magnetized plasma with the parameters of ${\omega }_p=2\pi \times 28.7$ GHz, ${\omega }_b=1\times {10}^{11}$ GHz, and $\nu =20$ GHz. The boundaries of the computational domain are terminated by the 10-cell-PML regions which hold the best absorption both in time and frequency domains. The observation point is located at the left top corner with 1 cell from three sides of the PML regions.

In the unconditionally stable algorithms, mesh size can be chosen according to the calculation accuracy rather than the CFL condition. Thus, in order to satisfy the accuracy requirement of the extremely thin metal plate along the vertical z-direction, a uniform mesh size can be obtained as $\Delta x=\Delta y=\Delta z=\Delta =0.3$ mm. According to the CFL condition, a maximum time step of the explicit algorithms can be chosen as $\Delta t_{\max }^{FDTD}=0.58$ ps. In order to demonstrate the absorption of absorbing boundary condition, relative reflection error in the time domain is employed which can be defined as

$$R_{dB}\left(t\right)=20{\log }_{10}\left[\frac{\left|E_z^t\left(t\right)-E_z^r\left(t\right)\right|}{\left|\max \left\{E_z^r\left(t\right)\right\}\right|}\right]$$

(34)

where $E_z^t\left(t\right)$ is the test solution which can be obtained directly at the observation point. $E_z^r\left(t\right)$ is the reference solution that can be obtained by 20 times enlarged computational domain and 128-cell-PML regions. Due to the employment of a larger computational domain and thicker PML regions, reflection waves can be minimized without changing the relative position between the observation point and source whose solution is close to the reference solution. Figure 7 shows the relative reflection error obtained by different algorithms.

As shown in Figure 7a that the conventional explicit FDTD-HPML scheme holds the best performance due to the employment of higher-order formulation and explicit calculation procedure. With the employment of an implicit scheme, absorption decreases with the introduction of matrices at each time step. Among the implicit schemes, the proposed scheme almost holds the same performance compared with the original CNAFS-HPML scheme. Compared with the original CNAFS-PML and FDTD-PML schemes, absorption can be increased significantly. It can be concluded that the employment of higher-order formulation improves absorption, especially in late-time reflections. As shown in Figure 7b that the absorption decreases with the enlargement of time steps. The reason is that numerical dispersion increases in such circumstances resulting in a decrement in calculation accuracy and absorption. Meanwhile, it can be concluded that the proposed scheme still shows a similar performance compared with the original CNAFS-HPML scheme. Compared with the FDTD-PML, absorption still can be improved with larger time steps. The effectiveness of the calculation can also be reflected by the memory consumption, simulation duration, and time reduction obtained by different algorithms and CFLN which are shown in

Table 2. The memory consumption, simulation duration, and time reduction obtained by different algorithms and CFLN in the antenna radiation problem.

Algorithms	CFLN	Steps	Memory (G)	Time (min)	Reduction (%)
FDTD-PML	1	32768	1.1	4.7	-
FDTD-HPML	1	32768	1.9	22.3	−377.5
CNAFS-PML	1	32768	2.5	75.1	−1497.9
CNAFS-HPML	1	32768	3.0	90.4	−1823.4
SIFS-HPML	1	32768	2.6	81.6	−1636.2
CNAFS-PML	10	3277	2.5	2.6	44.7
CNAFS-HPML	10	3277	3.0	4.1	12.8
SIFS-HPML	10	3277	2.3	3.3	29.8

. As shown in Table 2, the explicit FDTD-PML and FDTD-HPML schemes hold the best calculation efficiency due to the free matrix form during the iteration. The implicit schemes including CNAFS and SIFS algorithms solve matrices at each time step resulting in the calculation being much more expensive from the aspects of memory consumption and simulation duration. Meanwhile, according to the higher-order formulation, memory consumption, and simulation duration also increase by multiplying stretched coordinate variables together into a single form. The simulation duration can be decreased by employing a larger CFLN to enlarge the simulation time step. It can be observed that the simulation duration can be further decreased with CFLN = 10. Meanwhile, compared with the CNAFS algorithm, the proposed SIFS algorithm can avoid repeated calculation of auxiliary variables, field components, and coefficients resulting in a decrement in simulation duration and memory storage.

The calculation accuracy and absorption can also be evaluated by the scattering parameters in the frequency domain. Figure 8 and Figure 9 shows the return loss (S11) with and without plasma sheath obtained by a different algorithm. As demonstrated in Figure 8a and Figure 9a that the curves are almost overlapped which indicates the same calculation accuracy with a small time step. From Figure 8b and Figure 9b, curves show shifting compared with that obtained by the explicit FDTD-HPML algorithm due to the increment of numerical dispersion with larger CFLNs. It can be observed that the proposed algorithm also holds the most considerable performance. As can be compared between Figure 8 and Figure 9 that the return loss varies with the existence of plasma sheath. Thus, the plasma sheath significantly affects the return loss parameters which results in the failure of the communication system.

4. Conclusions

Here, an unconditionally stable SIFS algorithm is proposed with higher order CPML formulation to efficiently analyze the re-entry vehicle at the hypersonic speed. The ADE method is modified according to the integer time step scheme to analyze the magnetized plasma. Multi-scale numerical examples including target characteristic problem and antenna radiation problem are carried out to demonstrate the effectiveness of the algorithm. The results indicates that the proposed SIFS algorithm can simplify the formulation which shows advantages in efficiency and memory consumption. The proposed CPML algorithm shows considerable absorption and accuracy both in time and frequency domains. Plasma sheath shows significant influence on the target characteristics and antenna performance which affects the communication and detection of the blackout re-entry vehicle.