Class Of Analytic Solutions Research Articles

A polar diatomic molecule under a high-frequency laser field: classical analytical solution

AbstractDynamics of a polar diatomic molecule (represented as the oscillating rotator model) under external fields is the most fundamental problem and the testbed in molecular physics—analogously to the dynamics of hydrogenic atom (represented as the model of an electron in the Coulomb field) under external fields being the most fundamental problem and the testbed in atomic physics. We present the classical analytical study of the dynamical Stark effect in the oscillating rotator under a high-frequency laser field. We obtain the analytical results by using the method of effective potentials. We demonstrate that under the laser field, the rotation frequency increases, while the oscillation frequency decreases, and the amplitude of the oscillations increases. More significantly, the laser field causes the generation of the second harmonic in the oscillations of this system. This is an important, counterintuitive result in studying this fundamental physical system. We also drew attention to some flaws in the literature on this subject.

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The C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^3$$\\end{document} approach is an invariant formalism that utilizes the eigenvalues of the Riemann curvature tensor to match spacetimes across a specific matching surface. We apply this approach to match an anisotropic fluid with an exterior vacuum solution, including the case in which discontinuities appear on the matching surface. As a particular example, a class of analytic solutions, which describe the gravitational field of realistic neutron stars, is matched to the exterior Schwarzschild spacetime.

Response of semicircular canyons and movable cylindrical cavities to SH waves in anisotropic half-space geology

The development of tunnels or the laying of underground pipelines are essential engineering projects in modern society, and in canyon tunnels and underground pipeline projects, the surface motion and cavity edge motion have been topics of concern in ground vibration problems. We investigate the wave scattering problem in an elastic half-space anisotropic medium containing a semicircular canyon and a subsurface movable cylindrical cavity by using the wave function expansion method, the complex function method, and the mirror method. By deriving the governing equation and transforming it into the standard form of the Helmholtz equation satisfying the zero-stress boundary condition, we solve the corresponding displacement functions. Introducing a position correction coefficient, the scattered wavefield in a half-space anisotropic medium is constructed by the mirror method, which improves the problem of scattered wave source singularity in anisotropic half-space media. Then, combining the free boundary conditions with a Fourier series expansion method, we solve the unknown coefficients in the equations. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, using frequency- and time-domain analysis, we investigate the effects of the relevant parameters on the surface motion [Formula: see text], the dynamic stress concentration factor, and the displacement amplitude [Formula: see text]. The results indicate that rock anisotropy and the presence of semicircular canyons have a significant effect on the dynamic response of subsurface structures. This not only provides a theoretical basis for practical unlined tunnels or pipeline projects but also can provide a basis for seismic design of underground structures.

Scattering of SH waves by elliptical cavity and type-III crack in deep anisotropic geology

Most of the underground rock formations are anisotropic, and the study of the dynamic response of anisotropic geology containing internal defects is necessary for the development of underground tunnels or caverns. In this paper, the wave function expansion method, the complex function method, and the Green's function method are used to solve the problem of scattering of SH waves in anisotropic rock in the presence of elliptical cavity and type-III crack. Boundary conditions for solving the problem are given using the conformal transformation method, and crack in the medium is constructed using the "crack cutting" technique. The scattered wave field generated by the crack is constructed based on the Green function method, and the displacement and stress fields in the model are obtained by the wave field superposition principle in the case of SH wave incidence. The dynamic stress concentration factor (DSCF) on the boundary of the elliptical cavity and the dynamic stress intensity factor (DSIF) at the crack tip are derived. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, the effects of the anisotropic strength of the rock mass, the dimensions of the cavern, and the incidence angle and wave number of the SH wave on the DSCF and DSIF are analyzed by numerical calculations in both the frequency and time domains. The results show that the anisotropy of the rock mass, the shape of the cavern and the defects of the surrounding medium have a great influence on the dynamic response of the underground structures, which should be given enough attention in the development and design of deep underground caverns or tunnels.

Analytical solution model of heat transfer for energy soldier piles during excavation to backfilling

This paper presents a two-dimensional heat transfer model for energy soldier piles during excavation and after backfilling. Based on Green's function and separation of variables method, this model considers the heat transfer process from various angles during excavation and after backfilling, deriving a series of analytical solutions for the heat transfer processes. It is more suitable for solving the heat transfer problem of energy soldier piles. The accuracy of the analytical solution was verified by numerical simulation and classical analytical solutions. Then, the effects of air convection coefficient and backfill materials' thermal diffusivity on heat transfer are analyzed. The results show that the temperature rise of both the soil and the pile in the non-excavated case (ordinary energy pile) is lower than that of the various convective coefficient cases, and the convective heat transfer contributes to the heat transfer capacity of energy soldier piles. After backfilling, materials with higher thermal diffusivity have smaller temperature rise. Conversely, materials with lower thermal diffusivity have higher temperature rise. Under the long-term operation strategy, the heat flux within the soil and backfill is higher and favorable for heat transfer by selecting the backfill with lower thermal diffusivity.

Fractional-order numerical modeling to study chloride ion transport in concrete with fly ash or slag additions.

To better study the chloride ion migration in concrete with fly ash or ground granulated blast furnace slag under low fatigue load, a Caputo time fractional-order chloride diffusion model is developed in this paper. The model, grounded in Fick's second law with a fractional-order derivative, employs an implicit numerical method for discretization, resulting in a fractional-order numerical scheme. The stability and convergence of the scheme are rigorously proven within the paper. The model's unknown parameters are estimated using genetic algorithm with a grid method. To validate the model's effectiveness, its numerical solution is juxtaposed with experimental results from chloride erosion studies. Furthermore, the fitting efficacy of the Caputo time fractional-order numerical scheme is compared with that of the classical Fick's second law numerical scheme and analytical solution. The research findings demonstrate that the fractional-order numerical scheme can more accurately simulate the chloride concentration in concrete containing fly ash or slag. Additionally, the model shows promise in predicting the service life of fly ash or slag concrete.

Accuracy evaluation of heat pulse method to determine ice thermal conductivity

The heat pulse (HP) method is among the most widely used methods to measure thermal properties of frozen soil, snow or glaciers. The phase change or ice melting caused by the released heat of the HP method considerably affects the measurement accuracy. Accurate estimation of thermal properties at high subfreezing soil temperatures (e.g., −5–0 °C) remains as a challenge for decades, given that ice melting effects are too complicated to be accounted in the analytical solutions to heat conduction equation. Ice is the fundamental component that determines hydrothermal properties of frozen soil and snow/ice. Therefore, this study observed the different thermal responses of the ice to 6 heating strategies for the heat pulse (combination of 2 heating durations and 3 heating strengths). Meanwhile, the results of finite element method based on COMSOL numerical simulations and two classical analytical solutions were compared. The results showed that accuracy of the results was improved by optimizing the phase change parameters of latent heat term that is included in the governing equation in COMSOL. The main findings are: (1) The numerical simulations and SLS (short-time linear heat source) method well fit the observed temperature change as a function of time at a distance from the heater, ΔT(r, t) curves, with differences in phase change temperatures; (2) Modification of phase change parameters in COMSOL considerably optimized the ΔT(r, t) curve for heating duration over 15 s; (3) The numerical method effectively improved the calculation accuracy of ice thermal conductivity by 9.66% at −5 °C to −1 °C and 17.53% at −3 °C to −1 °C, respectively.

Reflected wave energy by two collinear submerged wavy breakwaters

Under the assumption of a potential flow, we obtain an asymptotic solution up to the second order in a regular expansion, for the problem of the interaction between linear long waves and two submerged floating breakwaters of wavy surfaces, placed in a collinear manner. The hydrodynamics that results from the interaction between the waves and the structures is modeled with the aid of the well-known shallow water wave equations together with the Laplace equation. For this purpose, the domain perturbation method is used to obtain the solution of the governing equations and assume, as is common, small amplitudes of the breakwater's wavy surfaces. This solution is compared with classical analytical solutions reported in the specialized literature, and they adjust properly. Several geometrical configurations of the breakwaters are analyzed. As the breakwaters are near the free surface elevation, just at one-third of the total water depth, larger values of the reflection coefficients are obtained. The maximum wave reflection occurs for four undulations of the breakwater surfaces. In addition, the wave reflection increases as the amplitude of the surfaces of the breakwaters increases. The results of this study are expected to be used by coastal engineers for preliminary feasibility and desk design of submerged breakwaters with wavy surfaces.

Frequency‐domain fundamental solutions and boundary element method for consistent couple stress elastodynamic problems

AbstractIn this article, the elastodynamic problems are investigated within the framework of a consistent couple‐stress theory (CCST). The basic equations for the CCST elastodynamic problems are derived at first. These dynamic formulas are then analyzed in the frequency domain by the Fourier transform. The frequency‐domain fundamental solutions to the transformed elastodynamic problems are subsequently derived for the two‐dimensional (2D) plane‐strain state. Based on the fundamental solutions and the elastodynamic reciprocal theorem, the frequency‐domain boundary integral equations (BIEs) are established. Then, the BIEs are numerically solved by a collocation method in conjunction with analytical and numerical treatments of the arising singular integrals. To obtain the time‐domain dynamic responses, an exponential window method (EWM) is adopted to transform the frequency‐domain results to the time‐domain solutions. By using the developed boundary element method (BEM), several typical couple‐stress elastodynamic problems are numerically investigated and the size‐effects are studied by using different values of the length‐scale parameter in the CCST. Good agreements between the results obtained by the developed BEM for a tiny length‐scale parameter and the corresponding classical BEM results or analytical solutions are achieved, which validates the high accuracy of the present BEM.

A new discrete model to better consider tracer distribution along boreholes during the Finite Volume Point Dilution Method

The Finite Volume Point Dilution Method (FVPDM) is a single-well tracer experiment which has been successfully used in many hydrogeological contexts to quantify groundwater fluxes. During continuous injection of tracer into a well, the tracer concentration evolution measured within the tested well directly depends on the groundwater flow crossing the well screens. Up to now, the FVPDM mathematical formulation used to simulate the tracer concentration evolution measured in the tested well assumed perfect homogenization of the tracer along the tested interval, which is a reasonable assumption in many cases. However, when FVPDM are performed in long-screened boreholes or in very permeable aquifer materials, the recirculation flow rate imposed to ensure mixing is suspected to be too low to perfectly homogenize the tracer. In order to assess the effect of non-perfect mixing on FVPDM results, we introduce here a new discrete model that explicitly considers the recirculation flow rate. The mathematical developments are validated using field measurements, and a sensitivity analysis is proposed to assess the effect of the mixing flow rate on tracer concentration homogenization within the well. Results confirm that, when the recirculation flow rate applied is not high enough compared to the groundwater flow rate, the tracer distribution is not uniform in the tested interval. In this case, the use of the classical analytical solution, commonly used to interpret the concentration evolution leads to highly overestimated groundwater fluxes. The discrete model introduced here can be used instead to properly estimate groundwater fluxes and assess the tracer distribution within the tested interval. The discrete model offers the possibility of interpreting field measurements conducted under non-perfect mixing conditions and increases the range of fluxes that can be investigated through FVPDM.

New black hole solutions in three-dimensional f(R) gravity

We construct two new classes of analytical solutions in three-dimensional spacetime in the framework of f(R) gravity whose field equations did not have cosmological constant. These solutions cannot coincide with Einstein’s general relativity due to the non-allowable vanishing of the dimensional parameter that characterizes the higher-order curvature. The first class of this solutions represents a non-rotating black hole (BH) while the second class corresponds to a rotating BH solution. The Ricci scalar of these BH solutions have non-trivial values and are described by the gravitational mass M, two angular momenta J and J1, and an effective cosmological constant Λeff that mainly comes from the contribution of the higher-order curvature terms. Interestingly enough, we observe that in contrast to Bañados–Teitelboim–Zanelli (BTZ) solution, which has only causal singularities and its scalar invariants are constant everywhere, the scalar invariants of our solutions indicate strong singularities for spacetime. This implies that the higher order derivative terms that come from f(R) gravity changes the casual structure of the spacetime. Indeed, the scalar invariants of the solutions behave as O1rn where n>0. The behavior of the invariants as r→0 become strong as compared with the behavior of black holes that reproduce from GR. Furthermore, we construct the forms, rotating/non-rotating, of the f(R) function showing that they behave as polynomial functions. Finally, we show that the obtained solutions are stable because of the positivity of their heat capacity, and also from the condition of Ostrogradski which states that the second derivative of f(R) should have a positive value.

Fully Coupled Model for One-Dimensional Large-Strain Consolidation and Heat Conduction in Saturated Clay

The consolidation characteristics of soils are affected by both mechanical loading and ambient temperature. However, research on this coupled theory is lacking. In this study, a fully coupled model for one-dimensional large-strain consolidation and heat conduction is established, where the influences of temperature on the physical-mechanical properties of saturated clay are considered. Based on the finite difference method, the numerical solutions for the coupled model are developed. Moreover, the correctness is validated by comparing the calculational results of the proposed model with those of the COMSOL simulation (a finite element software simulation) and the classical analytical solutions, respectively. Finally, the effects of different factors on consolidation behaviors are discussed. It is found that the increase in temperature increment ΔT generally accelerates the dissipation rate of excess pore-water pressure (EPWP) and increases the final settlement. The settlement is gradually reduced with an increasing effective yield stress σcR. A larger permeability coefficient kvr,R leads to an increasing EPWP dissipation rate. Furthermore, it is observed that the influence of σcR on the settlement is slightly enhanced with an increasing ΔT, while the effect of kvr,R on the dissipation rate of EPWP becomes less remarkable under a higher ΔT. In conclusion, the proposed coupled model can properly describe the large-strain consolidation behaviors of saturated clay when the effect of heat conduction is incorporated.

Geometry‐Based Prediction of Solute Transport Process in Single 3D Rock Fractures Under Laminar Flow Regime

AbstractSubsurface substance migration in the fractured rock aquifer is mainly controlled by fractures, and prediction of solute transport in fractures is important to many geophysical processes and engineering activities. This study explores the possibility of predicting transport process in single rock fractures from measurable physical properties. For this purpose, we conducted a large number of pore‐scale simulations of solute transport in single 3D rock fractures with different apertures and various degrees of roughness. The numerically‐obtained breakthrough curves under laminar flow regime were reproduced with high fidelity by the classic analytical solution within the framework of macroscopic advection‐dispersion theory. The fitted transport coefficients (hydrodynamic dispersion coefficient and transport velocity) were normalized by the parallel‐plate model's values, and were further parameterized by aperture b and roughness parameter σ/b in form of ωeb/ψ + λ(mσ/b − 1)/bn and 1 − α(σ/b)/βb, respectively. By incorporating these parameterized transport coefficients and the modified cubic law into the classic analytical solution, we proposed a new transport predictive model. This model was adaptively degenerated into the classic analytical solution of parallel‐plate model, and was proved to be capable of predicting macroscopic solute transport only relying on arithmetic‐mean mechanical aperture and its standard deviation, without solving the velocity and concentration fields. By the established model, we quantitatively revealed the impacts of aperture and roughness on the breakthrough curve under laminar flow regime. This study puts a step toward predicting solute transport process in fractured rock aquifers with measurable geometric properties.

Effects of Wave-Induced Doppler Velocity on the Sea Surface Current Measurements by Ka-Band Real-Aperture Radar with Small Incidence Angle

The Doppler shift of microwave radar sea surface echoes serves as the foundation for sea surface current field retrieval; it includes the shift caused by satellite platform motion, ocean waves, and sea surface currents. The Doppler shift caused by ocean waves is known as the wave-induced Doppler velocity (UWD), and its removal is critical for the accurate retrieval of sea surface current fields. The low-incidence Ka-band real-aperture radar rotary scan regime has the capability of directly observing wide-swath two-dimensional current fields, but as a new regime to be further explored and validated, simulation and analysis of UWD in this regime have a significant influence on the hardware design and currently observed applications of this satellite system in its conceptual stage. In this study, we simulated and investigated the impacts of radar parameters and sea-state conditions on the UWD obtained from small-incidence-angle Ka-band rotational scanning radar data and verified the simulation results with the classical analytical solution of average specular scattering point velocity. Simulation results indicate that the change in the azimuth direction of platform observation affects UWD accuracy. Accuracy is the lowest when the antenna is in a vertical side-view. The UWD increases slowly with the incidence angle. Ocean waves are insensitive to polarization in the case of small-incidence-angle specular scattering. The increase in wind speed and the development of wind waves result in a substantial increase in UWD. We classified swell by wavelength and wave height and found that UWD increases with swell size, especially the contribution of swell height to UWD, which is more significant. The contribution of the swell to UWD is smaller than that of wind waves to UWD. Furthermore, the existence of sea surface currents changes the contribution of ocean waves to UWD, and the contribution weakens with increasing wind speed and increases with wind wave development.

Effects of Soil Type and Thermal Boundary on Predicting Temperature Profiles and Groundwater Fluxes.

In the last few years, several articles have studied heat as a groundwater tracer and developed analytical geothermal solutions to predict the subsurface temperature and groundwater fluxes. These solutions can be sorted into steady-state and transient solutions. The steady-state solutions cannot describe the time-varying subsurface temperature, while the transient solutions ignore subsurface thermal boundary effects. Moreover, soil type may be another crucial factor significantly affecting the prediction results. This study compares six existing classical analytical solutions to examine the effects of soil types and subsurface thermal boundaries on simulating temperature-depth profiles and estimating groundwater fluxes. Several synthetic cases are built by considering the common soil types, sand and clay, to demonstrate their effects on predicting the profiles. A field case is used to show the effect of subsurface thermal boundaries on the groundwater flux estimated by an inverse approach. The study results indicate that the soil types have significant influences on simulating the profiles, and the influences grow with time. Some existing solutions may give inaccurate estimations of the field groundwater flux since they merely consider the heat source from the temperature variations on the ground surface but ignore possible thermal boundary effects in the subsurface. These findings will be valuable to those applying heat as a tracer to investigate infiltration.

A Fully Coupled Hydro-Mechanical Approach for Multi-Fracture Propagation Simulations

Hydraulic fracturing is a complex nonlinear hydro-mechanical coupled process. Accurate numerical simulation is of great significance for reducing fracturing costs and improving reservoir development benefits. The aim of this paper is to propose an efficient numerical simulation method for the fracturing-to-production problem under a unified framework that has good convergence and accuracy. A hydro-mechanical coupled fracturing model (HMFM) is established for poroelastic media saturated with a compressible fluid, and the local characteristics of the physical field are fully considered. Each fracture is explicitly characterized using the discrete fracture model (DFM), which can better reflect the physical characteristics near fractures. Based on the extended finite element method (XFEM) and the Newton–Raphson method, a fully coupled approach named Unified Extended Finite Element (UXFEM) is developed, which can solve the nonlinear system of equations that describe the solution under a unified framework. UXFEM can accurately capture the local physical characteristics of different physical fields on the orthogonal structured grids. It realizes the grid-fracture decoupling, and fractures can propagate in any direction, which shows greater flexibility in simulating fracture propagation. The fully coupled approach can better reflect the essential relationship between pressure, stress, and fracture, which is beneficial to studying hydro-mechanical coupled problems. To validate the UXFEM, UXFEM is compared with the classical KGD model, analytic solution, and COMSOL solution. Finally, based on UXFEM, the interference phenomenon and fracturing-to-production study are carried out to prove the broad practical application prospect of this new fully coupled approach.

Simulation of fully coupled hydro-mechanical behavior based on an analogy between hydraulic fracturing and heat conduction

The hydraulic fracturing process involves complex coupling problems of the deformation and fracturing of solid and fluid flow in fractures. In this paper, a simple numerical tool for simulating the coupled hydro-mechanical behavior is developed by using an analogy between hydraulic fracturing and heat conduction. Using this analogy, the numerical implementation can be simply realized in the framework of a commercial finite element package ABAQUS with two user material subroutines to fully characterize the complex coupling problems of hydraulic fracturing. It is thus considerably simplified without any new development of user-defined element and fluid–solid solver but directly with the commercial finite element package. The proposed numerical tool along with the virtual crack method for elastic–brittle material is firstly verified by the classical 2D plain strain KGD analytical solution and then applied to several representative hydraulic fracturing problems and laboratory tests to illustrate its advantageous features. The present method is aiming to develop a useful numerical tool to study the coupled hydro-mechanical behavior of hydraulic fracturing. However, it is not limited to any specific materials and can be extended to other constitutive models that featured with the fully coupling problems of fluid and solid due to its simplicity.

Phreatic seepage around a rectilinear cutoff wall: The Zhukovsky-Vedernikov-Polubarinova-Kochina dispute revisited

The problem of steady 2-D potential Darcian flow from a flat-bottomed reservoir of a coffer or other type of dam, separated from an empty tailwater by an impermeable vertical sheet piling, is one of the pillars in groundwater hydrology and soil mechanics. Analytical solutions to this problem are presented in the books by Forchheimer, Zhukovsky, Vedernikov, Polubarinova-Kochina, Harr, among others. For the case of seepage with a phreatic line, descending from the downstream face of the cutoff, the formulae presented in classical textbooks - despite the simplicity of conformal mappings of the complex potential quarter-plane onto the Zhukovsky half-strip - have lacunae and flaws. We discuss and clean the mismatch in the abovementioned formulae. The classical analytical solutions in the textbooks deal with the regime of no backwater, for which the magnitude of the specific discharge vector at infinity coincides with hydraulic conductivity of the soil. We obtain a general solution for most general flows with backwater, for which the specific discharge at infinity is zero and the free surface has an inflection point in the physical plane. The capillary fringe in the hodograph domain is imaged by a circular cut. The pressure head infinitely deep under the cutoff-wall is infinite. A circular tetragon in the hodograph domain is conformally mapped onto the quadrant in the complex potential plane via a reference plane. In the case with backwater, one extra condition is needed to fix a mathematical solution. For this purpose, the location of a point on the free surface is fixed. Mathematically, two affixes in the reference plane are found by solving a system of two nonlinear equations (in the no backwater case, only one affix is found and one equation is solved). We also examine flow with a phreatic line in a low-permeable layer (e.g. concrete apron of a dam) with an Λ-shaped topology of streamlines, bounded from the left by a cutoff. Seepage from a dam reservoir through a subjacent highly-permeable alluvium induces the pore water motion in the upper layer. A Hilbert boundary-value problem is solved for the complex potential in this cell. In the Kirkham-Brock type boundary condition along the interface between the two layers, the horizontal component of the specific discharge is constant. Phreatic surfaces rising on the downstream face of the cutoff caps Λ−shaped streamlines, which turn around a hinge point on the interface between two layers of contrasting permeability.

Analytical GNI Analysis for Lateral-torsional Behavior of Thin-walled Beams with Doubly-symmetric I-Sections

In this paper elastic lateral-torsional behavior of simple beams is discussed. The motivation of the presented research is the observation that classic analytical prediction and finite element prediction are, typically, considerably different, when the second-order nonlinear behavior of beams with initial imperfections is analyzed. In order to understand and explain the observed differences, a novel analytical solution is presented for the geometrically nonlinear analysis of beams with initial geometric imperfection. The presented analytical solution is derived for doubly-symmetric cross-sections, but with the novelty that it takes into consideration the changing geometry as the load is increasing. The most important steps of the derivations are summarized, and the resulted formulae are briefly discussed. Numerical studies are performed, too: the results of the new analytical formulae are compared to those from shell finite element analysis. The results suggest that the new formulae are able to capture the most important elements of the behavior. By the analytical and numerical results, it is proved that classic analytical solutions for the geometrically nonlinear analysis of beams with geometric imperfections are necessarily different from the numerical results obtained by incremental-iterative procedures.

Low Earth orbit constellation design using a multi-objective genetic algorithm for GNSS reflectometry missions

Spaceborne global navigation satellite system reflectometry (GNSS-R) is an innovative bistatic radar remote sensing technique utilizing low Earth orbit (LEO) based GNSS-R instruments to acquire GNSS L-band opportunistic signals for measuring geophysical parameters. A GNSS-R LEO constellation with an optimization design for its specialized missions is very significant and necessary. However, the constellation design involves multi-parameter and multi-objective optimization, and the classical analytic solution is not capable of such a complicated issue. This study proposes a multi-objective LEO constellation design method with a genetic algorithm (GA) and presents a framework for designing two GNSS-R LEO constellations, termed “lower-latitude constellation” for typhoons and hurricanes observation in the tropics and “global constellation” for global geophysical parameter measurements. Then, the observation capability of both designed constellations is evaluated in terms of the number of reflection points, spatial coverage density, and revisit time to verify the GA efficiency in LEO constellation design. Results show that the two designed LEO constellations with high fitness function values possess optimal orbit parameter set configuration and outperform the existing CyGNSS constellations in observation performance. Compared with CyGNSS, the number of reflection points observed by the lower-latitude constellation and the global constellation increases by 38% and 45%, as well as the spatial coverage density increases by 28% and 36%. The revisit time for the lower-latitude constellation is reduced by 0.29 h, whereas the revisit time for the global constellation increases by one hour.