Free Surface Boundary Conditions Research Articles

A Three-dimensional Immersed Boundary Method for Accurate Simulation of Acoustic Wavefields with Complex Surface Topography

Abstract Irregular topography of the free surface significantly affects seismic wavefield modelling, especially when employing finite-difference methods on rectangular grids. These methods represent the free surface as discrete points, resulting in a boundary that resembles a “staircase”. This approximation inaccurately represents surface topography, introducing errors in surface reflection traveltimes and generating artificial diffractions in wavefield simulation. We introduce a stable three-dimensional immersed boundary method (3DIBM) employing Cartesian coordinates to address these challenges. The 3DIBM enables the simulation of acoustic waves in media with complex topography through standard finite difference, extending the two-dimensional immersed boundary approach to compute spatial coordinates for ghost and mirror points in a three-dimensional space. Wavefield values at these points are obtained by three-dimensional spatial iterative symmetric interpolation, specifically through the Kaiser windowed sinc method. By implicitly implementing the free surface boundary condition in three dimensions, this method effectively reduces artificial diffractions and enhances the accuracy of reflection traveltime. The effectiveness and accuracy of 3DIBM are validated through numerical tests and pre-stack depth migration (PSDM) imaging with simulated data, demonstrating its superiority as a modelling engine for migration imaging and waveform inversion in three-dimensional land seismic analysis.

Enhanced schemes for resolution of the continuity equation in projection-based SPH

To the best knowledge of the authors, the numerical resolution of the continuity equation, i.e., the simultaneous achievement of accurate and physically consistent velocity divergence and density fields, has not been investigated in the context of projection-based particle methods. In this study, based on the Velocity-divergence Error Mitigating (VEM)/Volume Conservation Shifting (VCS) schemes developed for the WCSPH method, which improve the numerical resolution of the continuity equation, VEM/VCS schemes are proposed for projection-based particle methods. Furthermore, two other novel schemes are introduced. First, a Dynamic VEM (DVEM) scheme is proposed, where VEM is dynamically activated only when the velocity divergence instantaneously diverge from the reference solution at individual particles. The second scheme is referred to as Enhanced VCS by reference density (EVCS), where precise imposition of dynamic free surface boundary condition is considered. Several benchmark tests with different features have been conducted, and it is shown that the proposed schemes enhance the numerical resolution of the continuity equation for incompressible flows, i.e., the divergence free velocity and the constant density conditions. Application of DVEM and EVCS has provided enhanced resolution of the continuity equation along with clear reductions in the numerical energy dissipation associated with the approximated additional numerical terms.

Unsteady hydrodynamics of a surface piercing and fully submerged body when entering a lock

Ship hydrodynamics in a constant waterway have been well studied. With a coordinate system fixed on the moving ship, the boundary value problem (BVP) is usually treated as a steady one. However, the hydrodynamics of a ship moving in confined waterways with abrupt changes in width or depth are complex due to their unsteady nature. The studies on such unsteady problems are insufficient. In the engineering practice, the hydrodynamic unsteadiness can be fully reflected by a scenario when a ship enters a lock. Prior studies have predominantly focused on predicting the ships' hydrodynamic forces without considering the unsteady terms on the free-surface boundary conditions. Obviously, such steady or quasi-steady methods overlooked the crucial unsteady phenomena on free water surface. To address this gap, the present study introduces a novel three-level difference scheme to discretize the free surface condition, preserving unsteady terms while maintaining temporal continuity of cells on the free surface. With the implementation of such fully unsteady BVP, we observed some interesting unsteady free surface motions, which were not well documented in the existing literature. To verify our new observations, as well as to validate the numerical method proposed in this study, two physical model tests were designed and conducted in a towing tank: a submerged ellipsoid enters a deep lock at relatively high speeds, and a box enters a shallow and narrow lock at very low speeds. The discussion is highlighted on the unsteady waves in front of the moving bodies, as well as the unsteady resistance induced by such waves.

Wave drag and wave patterns by ships moving in a single-file formation

To minimize energy expenditure for each individual, animals adopt distinctive formations, such as fish schooling, “V” formation by flying birds, and single-file formation by waterfowls. The phenomenon of ducklings following their mothers in a single-file configuration has been revealed by the mechanisms of wave-riding and wave-passing. Drawing inspiration from this phenomenon, an investigation is undertaken on ships moving in a single-file formation. The objective is to quantify how much energy can be saved in different configurations. In this study, a three-dimensional boundary element method incorporating linear free-surface boundary conditions is used to obtain the wave drag and wave patterns. It is found that when constructive wave interference occurs in a two-ship formation, the wave resistance of the trailing ship increases and the leading ship experiences a decrease in its wave drags, especially when the two ships are in close proximity. Mutual benefit arises when destructive wave interference occurs between two ships. In addition, increasing the size of the trailing vessel facilitates the effect of wave-riding by the leading ship, but this effect becomes less pronounced as the speed increases. In a multi-ship formation configuration, changing the size of the leading ship will have a localized effect on the wave-passing, but the fleet will eventually tend to a dynamic equilibrium. When the position of the first trailing vessel is changed, there is similarly a localized effect on the wave-passing. Adjusting the first trailing ship to the position of the constructive wave interference is not favorable to reducing its own drag but enhances the wave-riding effect of its close follower. Finally, to achieve wave-passing, the trailing ship does not necessarily have to occupy an optimum position. This can still be accomplished if the trailing ship moves backward by an integer multiple of wavelength.

Stress localization investigation of additively manufactured GRCop-42 thin-wall structure

A full-field crystal plasticity (CP) framework is presented for the GRCop-42 alloy to study microscopic mechanical behavior and local stress heterogeneities. The microstructures of additively manufactured (AM) materials are often unique relative to conventionally processed materials, and the local thermal histories drive these differences during the build process. These thermal histories depend on the process parameters (laser power, scan speed, and scan strategy) and the part geometry. Prior research has shown that the mechanical properties of thin-walled structures can vary significantly with wall thickness due to changes in the thermal boundary conditions during manufacturing. It is, therefore, desirable to perform CP simulations based on the phenomenological constitutive model to predict the local mechanical responses induced by microstructural heterogeneities. This work generates representative microstructures based on experimentally collected grain information (i.e., texture) for grain scale stress analysis, and the material constitutive parameters are calibrated using the experimental mechanical testing data. We specifically investigated the effect of crystallographic texture and grain morphologies on the size-dependent mechanical properties of AM GRCop-42. The selection of appropriate material properties for implementing an effective free surface boundary condition and the influence of adjacent buffer layers are also discussed. Analysis of local field results reveals a strong correlation between stress localization and the initial grain orientation. However, no significant relationship between the misorientation of the individual adjacent grains and the average misorientation is observed.

Droplet asymmetry bouncing on structured surfaces: A simulation based on SPH method

The impact of a liquid droplet onto a macro-structured super-hydrophobic surface, results in an asymmetric spreading-retraction process and a significant decrease in contact time. The process involves large deformation of droplets and evolution of surface morphology, so it is difficult to efficiently simulate using traditional grid-based methods. In this study, a numerical model is established using the mesh-free smoothed particle hydrodynamics (SPH) method to simulate the interaction of liquid droplets with solid surfaces exhibiting complex macro-textures. Leveraging the Lagrange nature of SPH, liquid droplets can be model using free surface boundary condition. The weakly compressible SPH formulation is employed to discretize the governing equations of liquid. Surface tension is captured by introducing an additional negative pressure term, inducing attractive forces among fluid particles. Kernel functions are carefully chosen to mitigate stress instability resulting from droplet spreading and retraction. The model is then applied to simulate the dynamic processes of droplets impact on flat and non-flat super-hydrophobic surfaces. The bouncing dynamics of droplets are investigated under varying conditions, including Weber number, size, and morphology of macro-scale surface structures. Our simulation results for contact time, maximum diameter, and droplet morphology align closely with experimental observations. The contact time of the droplet exhibits a dependency on its size and surface geometry, with a transition towards planar geometry diminishing asymmetric spreading effects. The findings highlight the significance of Weber number and cylinder diameter in determining contact time, providing insights for the design of optimized surface structures for specific droplet impact parameters. In addition, the use of a free-surface condition for modeling proved computationally efficient for 3D simulations. The model effectively reproduces nonlinear behaviors such as droplet spreading, splashing, and bouncing, highlighting the significant advantage of SPH in simulating such problems.

Numerical simulation of 3-D seismic wave based on alternative flux finite-difference WENO scheme

SUMMARY High-frequency non-physical oscillations may occur due to shock waves in seismic wavefield and dynamic rupture simulation. In this study, we introduced the alternative flux finite-difference weighted essentially non-oscillatory scheme to address potential shock wave issues in computational seismology effectively. The wavefield of the body-fitted curvilinear domain was accurately computed through conservative grid mapping, ensuring accurate implementation of free surface boundary conditions on irregular surfaces using characteristic boundary conditions and minimizing artificial boundary reflections with exponential decay absorbing layers. Finally, we compared our scheme with the GRTM for flat surfaces and the CGFDM3D-EQR for irregular surfaces to demonstrate its correctness and accuracy, and validated its non-oscillatory characteristics. The aforementioned scheme is anticipated to assume a significant function in simulating more intricate seismic wavefields or dynamic ruptures.

Linear Spectral Method for Simulating the Generation of Regular Waves by a Moving Bottom in a 3-dimensional Space

In this study, we introduce a linear spectral method capable of simulating wave generation and transformation caused by a moving bottom in a 3-dimensional space. The governing equations are linear dynamic free-surface boundary conditions and linear kinematic free-surface boundary conditions, which are solved in Fourier space. Solved velocity potential and free-surface displacement should satisfy continuity equation and kinematic bottom boundary condition. For numerical analysis, a 4th order Runge-Kutta method was utilized to analyze the time integral. The results obtained in Fourier space can be converted into velocity potential and free-surface displacement in a real space using inverse Fourier transform. Regular waves generated by various types of moving bottoms were simulated with the linear spectral method. Additionally, obliquely generated regular waves using specified bottom movements were simulated. The results obtained from the spectral method were compared to analytical solutions, showing good agreement between the two.

An efficient localized Trefftz method for the simulation of two-dimensional sloshing behaviors

This study proposes a new method for effectively and accurately simulating sloshing in a two-dimensional numerical tank. This technique uses the localized Trefftz method (LTM), a meshless numerical approach. Sloshing in a numerical tank is mathematically described as a space-time boundary-value problem rooted in potential flow theory. This occurrence is governed by a second-order partial differential equation and two nonlinear free-surface boundary conditions. In this study, the moving boundary problem undergoes discretization in time and space by using the LTM and the explicit Euler method, respectively. After discretization with the explicit Euler method, the elevation of the free surface is updated, and a boundary-value problem is generated at each time step. This boundary-value problem can be effectively examined using the recently developed LTM, which eliminates the need for complex meshing. Contrary to conventional methods, such as the method of fundamental solutions and the boundary element method, the LTM generates sparse matrices, allowing large-scale numerical simulations. Four numerical examples are presented to demonstrate the clarity and accuracy of the proposed meshless approach.

Numerical and experimental study of ultrasonic seismic waves propagation and attenuation on high‐quality factor samples

AbstractWe propose an approach for measuring seismic attenuation at ultrasonic frequencies on laboratory‐scale samples. We use a Gaussian filter to select a bandwidth of frequencies to identify the attenuation in a small window and, by moving the window across the frequency content of the data, we determine the frequency‐dependent attenuation function. We assess the validity of the method with three‐dimensional numerical simulations of seismic wave propagation across different sample geometries, using free surface boundary conditions. We perform the simulations using viscoelastic media under various seismic attenuation models. Our numerical results indicate that we can successfully recover the representative viscoelastic attenuation parameters of the media, regardless of the sample geometry, by processing the seismic signal recorded either within the volume or at the boundaries. Due to the equipartition phenomenon, the energy of S‐waves is consistently higher in seismic records than that of P‐waves. Therefore, we systematically recover the attenuating properties of S‐waves in the medium. We also conduct experiments of seismic wave propagation on samples of aluminum and Fontainebleau sandstone to validate our approach with real data. The quality factor of the S‐wave in the aluminum medium increases from 300 to 7000 between 60 kHz and 1.2 MHz. The Fontainebleau sandstone, which is more attenuating, exhibits a that increases from 200 at 60 kHz to 1000 at 1.2 MHz. With our approach, we are not only able to recover the attenuation property but also identify the frequency‐dependent attenuation model of the samples. Our method allows for seismic attenuation recovery at ultrasonic frequencies in low‐attenuating media.

A size-dependent 3D solution of functionally graded shallow nanoshells

Abstract An unavailable semi-analytical non-local 3D solution for functionally graded nanoshells with constant radii of curvature is presented. The small length scale effect is included in Eringen’s nonlocal elasticity theory. The constitutive and equilibrium equations are written in terms of curvilinear orthogonal coordinates systems which are only valid for spherical and cylindrical shells, and rectangular plates. The stresses and displacements are assumed in terms of the Navier method which is applicable for simply supported structures. The derivatives in terms of thickness are approximated by the differential quadrature method (DQM). The thickness domain is discretized by the Chebyshev–Gauss–Lobatto grid distribution. Lagrange interpolation polynomials are considered as the basis function for DQM. The correct free surface boundary condition for out-of-plane stresses is considered. Several problems of isotropic and functionally graded shells subjected to different types of loads are analyzed. The results are compared with other three-dimensional solutions and higher-order theories. It is important to emphasize that the radii of curvature are crucial at nanoscale, so it should be considered in the design of nanodevices.

A dispersion analysis of uniformly high order, interior and boundaries, mimetic finite difference solutions of wave propagation problems

A preliminary stability and dispersion study for wave propagation problems is developed for mimetic finite difference discretizations. The discretization framework corresponds to the fourth-order staggered-grid Castillo-Grone operators that offer a sextuple of free parameters. The parameter-dependent mimetic stencils allow problem discretization at domain boundaries and at the neighbor grid cells. For arbitrary parameter sets, these boundary and near-boundary mimetic stencils are lateral, and we here draw first steps on the parametric dependency of the stability and dispersion properties of such discretizations. As a reference, our analyses also present results based on Castillo-Grone parameters leading to mimetic operators of minimum bandwidth that have been previously applied in similar physical problems. The most interior parameter-dependent mimetic stencils exhibit a specific Toeplitz-like structure, which reduces to the standard central finite difference formula for staggered differentiation at grid interior. Thus, our results apply to the whole discretization grid. The study done for the 1-D problem could be applied to the discretization of a free surface boundary condition along an orthogonal gridline to this boundary.

Time-domain numerical simulation for multi-ships moving in waves with forward speed

With the development of the shipping industry, the impact of hydrodynamic interaction between two or more ships on navigational safety has received widespread attention. However, limited research has been conducted on multi-ships moving in waves with forward speed due to the increased complexity of hydrodynamic interactions caused by speed. Therefore, this paper establishes a numerical model of multi-ship hydrodynamics under the wave impact based on the time-domain Rankine source panel method. The hydrodynamic performance and motion characteristics of ships advancing in waves are investigated. An artificial damping layer is applied to free surface boundary conditions to absorb reflected waves. The fourth-order Runge-Kutta method is used for time integration to solve the motion of the ship. The hydrodynamic and motion response of the classical Wigley III ship model are calculated at each wavelength, and the simulation results are compared with other calculations to verify the reasonable reliability of the numerical method employed in this study. On this basis, sensitivity analysis of different parameters is conducted to reveal the influence of various parameters such as speed, wave direction, transverse distance, and water depth on the motion response of the Wigley III ship model in waves.

Strongly Imposing the Free Surface Boundary Condition for Wave Equations with Finite Difference Operators

Strongly Imposing the Free Surface Boundary Condition for Wave Equations with Finite Difference Operators

Equatorially trapped Rossby waves in radiative stars

AbstractObservations by recent space missions reported the detection of Rossby waves (r‐modes) in light curves of many stars (mostly A, B, and F spectral types) with outer radiative envelope. This article aims to study the theoretical dynamics of Rossby‐type waves in such stars. Hydrodynamic equations in a rotating frame were split into horizontal and vertical parts connected by a separation constant (or an equivalent depth). Vertical equations were solved analytically for a linear temperature profile and the equivalent depth was derived through free surface boundary condition. It is found that the vertical modes are concentrated in the near‐surface layer with a thickness of several tens of surface density scale height. Then with the equivalent width, horizontal structure equations were solved, and the corresponding dispersion relation for Rossby, Rossby‐gravity, and inertia‐gravity waves was obtained. The solutions were found to be confined around the equator, leading to the equatorially trapped waves. It was shown that the wave frequency depends on the vertical temperature gradient as well as on stellar rotation. Therefore, observations of wave frequency in light curves of stars with known parameters (radius, surface gravity, rotation period) could be used to estimate the temperature gradient in stellar outer layers. Consequently, the Rossby mode may be considered as an additional tool in asteroseismology apart from acoustic and gravity modes.

A regularized high-order diffusive smoothed particle hydrodynamics scheme without tensile instability

In the present work, we derive a novel high-order weakly compressible smoothed particle hydrodynamics scheme based on an accurate approximation of the pressure gradient and on the use of numerical Riemann fluxes. Specifically, a switch between non-conservative and conservative formulations of the pressure gradient is adopted close to the free surface, in order to fulfill the dynamic free-surface boundary condition and, at the same time, prevent the onset of the tensile instability in inner regions of the fluid domain. The numerical diffusion is obtained using Riemann solvers, with reconstruction/limitation of the left and right states derived from the Monotonic Upstream-centered Scheme for Conservation Laws technique. These allow for a high-order convergence rate of the diffusive terms that, for increasing spatial resolutions, results in a low numerical dissipation without tuning parameters. Regular particle distributions, which are crucial for the model accuracy, are obtained thanks to recent improvements in Particle Shifting Techniques. These are taken into account within the constitutive equations through a quasi-Lagrangian formalism. The energy balance of such a non-conservative formulation is derived, and an in-depth analysis of the term contributing to numerical dissipation is performed. The numerical investigation is carried out on several problems, illustrating the advantages of the present scheme with respect to conservative formulations. Since the proposed formulation does not intrinsically guarantee momenta conservation, the latter are monitored showing that the overall errors are generally small.

An improved acoustic/elastic interface approach for 2D staggered grid finite-difference modeling of Rayleigh waves in the presence of surface topography

Rayleigh waves are generated along the free-surface and their propagation can be strongly influenced by surface topography. Therefore, a critical aspect of Rayleigh-wave study is to understand the wave-propagation behavior in the presence of surface topography. The acoustic/elastic boundary approach was incorporated into a ‘stair-case’ mesh for Rayleigh-wave modeling in the presence of surface topography. The conventional acoustic/elastic boundary approach, however, fails to completely fulfill the free-surface boundary condition at some grid nodes, leading to unphysical simulation results. To solve this problem, we develop a stable acoustic/elastic boundary approach that fully satisfies the free-surface boundary condition by adjusting the free boundary conditions at problematic nodes in the mesh. The accuracy is demonstrated by modeling tests in two-dimensional isotropic models in the presence of surface topography. And compared to the improved vacuum formulation, the proposed method takes less time. Our approach optimizes the free-surface boundary condition and enables the high-precision numerical simulation of Rayleigh waves.

MULTI-LINEAR-ELEMENT DEPTH-INTEGRATED MODELS FOR FLOWS WITH A FREE SURFACE

In this study, we derive a new set of depth-integrated models for solving unsteady flows that satisfy the Euler equations and nonlinear free surface boundary conditions in the sigma -coordinates. One of the obvious differences between the present approach and the ones directly solving the 3D Euler equations using Finite Element Method (FEM) is that in the present model the vertical velocity and the pressure field are eliminated by integrating the continuity equation and vertical momentum equation, respectively. Therefore, the present models only solve the horizontal velocity components and the free surface displacement in the two-dimensional horizontal (2DH) space. The new models are also more advantageous in using fewer elements in the vertical direction while achieving better performance. At this stage, the capability and application of the new models in dealing with free surface wave propagation problems are addressed.

Finite-difference method for modeling the surface wave propagation with surface topography in anisotropic-viscoelastic media

The growth of urbanization makes it increasingly important to accurately investigate underground spaces. Surface-wave-based inversion methods are becoming popular due to their cost-effectiveness and efficiency in reconstructing underground space. However, these methods may produce inaccurate results in the presence of media anisotropy or/and viscoelasticity. To investigate the influence of media anisotropy or viscoelasticity on surface waves, it is crucial to develop an efficient and accurate forward modeling method in anisotropic-viscoelastic (AV) media. The finite-difference (FD) method is a widely used forward modeling method in surface-wave-based inversion methods. However, implementing the free-surface boundary condition and representing non-flat topography can be challenging. Considering the challenges described above, this study proposes a simple and efficient FD method for modeling surface-wave propagation. The proposed method employs a parameter-modified strategy to implement the stress-free condition by modifying the model parameters near the (non-) flat free-surface boundary. A staircase discretization strategy is then used to represent the non-flat free surface. To suppress the staircase diffractions caused by the staircase discretization strategy, an independent wavefield superposition is adopted with modeling results of different parameter configurations. Compared with conventional FD methods, the proposed method can reduce computational costs while accurately simulating surface waves. Numerical experiments have demonstrated the accuracy and feasibility of the proposed method. The proposed method provides a theoretical basis for surface-wave-based inversion methods and contributes to the development of accurate underground space investigation.

Effect of free surface boundary condition on reflected waves in a half-space piezoelectric semiconductor medium

Effect of free surface boundary condition on reflected waves in a half-space piezoelectric semiconductor medium