Absorbing Boundary Research Articles

A finite element solution of earthquake-induced hydrodynamic forces and wave forces on multiple circular cylinders

This paper investigates the interaction of water with multiple circular cylinders subjected to earthquake ground motion and incident linear waves respectively. Accurate and efficient finite element models are firstly presented to calculate the earthquake-induced hydrodynamic forces and wave forces on uniform vertical cylinders with arbitrary cross-section. An elliptical boundary surrounding all the cylinders is introduced so that the whole domain is partitioned into unbounded and bounded domains. The elliptical exact absorbing boundary condition for simulating unbounded domain is derived in the elliptical coordinate system. A coupled equation is obtained by adding the impedance matrix of unbounded domain to that of bounded domain. The proposed models are furtherly used to investigate the earthquake-induced hydrodynamic forces and wave forces on pile group with two, three and four piles, where the coefficients of pile group are given. The results indicate that interaction of water with multiple cylinders is significant when the relative distance between two adjacent piles is less than three. This phenomenon depends on the relative distance between piles, width-depth ratio, wave numbers, and the direction of earthquake or incident wave. In addition, the simplified formulas of the coefficients of pile group of added mass are proposed for engineering application.

A local collocation method to construct Dirichlet-type absorbing boundary conditions for transient scalar wave propagation problems

This paper introduces an effective way to equip the standard finite element method (FEM) for the solution of transient scalar wave propagation problems in unbounded domains. Similar to many other methods, we truncate the unbounded domain at an artificial boundary and convert the problem into a bounded one by prescribing appropriate absorbing boundary conditions (ABCs) at the truncating boundary. In the present method, the ABCs are time-dependent, and they are constructed by a simple collocation approach which is local in space and time. Therefore, the method does not make use of any routine schemes such as Fourier and Laplace transform. We shall show that the method is simple, and it can be easily applied to an explicit time domain FEM approach so that the sparsity of the FEM scheme (as well as its efficiency) can be preserved. The proposed method does not require any auxiliary variables as well as any approximating differential operators. This feature roots from the fact that here the ABCs are Dirichlet-type (or first-type) and thus they can be easily imposed to the corresponding boundaries. Therefore, the method shares some similarities with the conventional 1st and 2nd order ABC methods in terms of the simplicity of implementation. The method employs basis functions that exactly satisfy the governing and dispersion wave equations. The basis function can be easily adjusted to act as outgoing waves transmitting energy from the interior domain (near field) towards exterior domain (far field); i.e, they can cope with satisfaction of radiation boundary conditions. Several numerical examples are presented to evaluate the performance and to demonstrate the effectiveness of the approach. We shall show that the present method is capable of yielding results with a proper level of accuracy, similar to that of the perfectly matched layers method (PMLs), and it performs stably even in the case of long-term computations.

Frequency-domain elastic wavefield simulation with hybrid absorbing boundary conditions

AbstractThe frequency-domain seismic modeling has advantages over the time-domain modeling, including the efficient implementation of multiple sources and straightforward extension for adding attenuation factors. One of the most persistent challenges in the frequency domain as well as in the time domain is how to effectively suppress the unwanted seismic reflections from the truncated boundaries of the model. Here, we propose a 2D frequency-domain finite-difference wavefield simulation in elastic media with hybrid absorbing boundary conditions, which combine the perfectly matched layer (PML) boundary condition with the Clayton absorbing boundary conditions (first and second orders). The PML boundary condition is implemented in the damping zones of the model, while the Clayton absorbing boundary conditions are applied to the outer boundaries of the damping zones. To improve the absorbing performance of the hybrid absorbing boundary conditions in the frequency domain, we apply the complex coordinate stretching method to the spatial partial derivatives in the Clayton absorbing boundary conditions. To testify the validity of our proposed algorithm, we compare the calculated seismograms with an analytical solution. Numerical tests show the hybrid absorbing boundary condition (PML plus the stretched second-order Clayton absorbing condition) has the best absorbing performance over the other absorbing boundary conditions. In the model tests, we also successfully apply the complex coordinate stretching method to the free surface boundary condition when simulating seismic wave propagation in elastic media with a free surface.

Analytical Derivation of Ellipsoid Conformal Perfectly Matched Layers for Electromagnetic Simulations

The 3-D ellipsoid conformal perfectly matched layer (PML) is proposed and realized by the discontinuous Galerkin time-domain (DGTD) method. The interface between free space and the PML region is an ellipsoid in this scheme. The analytical formulas are derived from differential geometry. The numerical results show that compared with planar-PML, the ellipsoid-PML (EPML) avoids the corner problem and provides an efficient absorbing boundary condition for the DGTD simulation. Meanwhile, this new EPML scheme has higher accuracy than that of the old EPML scheme that we proposed.

Solution of a P and S wave propagation model using high performance computation

The propagation of seismic waves is affected by the type of transmission media. Therefore, it is necessary to solve a differential equation system in partial derivatives allowing for identification of waves propagating into an elastic media. This paper summarizes a research using a partial differential equation system representing the wave equation using the finite differences method to obtain the elastic media response, using an staggered grid. To prevent reflections in the computational regions, absorbent boundaries were used with the PML method. The implementation of the numerical scheme was made on two computational architectures (CPU and GPU) that share the same type of memory distribution. Finally, different code versions were created to take advantage of the architecture in the GPU memory, performing a detailed analysis of variables such as usage of bandwidth of the GPU internal memory, added to a version that is not limited by the internal memory in the graphic processing unit, but rather by the memory of the whole computational system.

Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}_{+}^{n} $\end{document}</tex-math></inline-formula>

In this paper, the asymptotic wave behavior of the solution for the nonlinear damped wave equation in \begin{document}$ \mathbb{R}^n_+ $\end{document} is investigated. We describe the double mechanism of the hyperbolic effect and the parabolic effect using the explicit functions. With the absorbing and radiative boundary condition, we show that the Green's function for the half space linear problem can be described in terms of the fundamental solution for the Cauchy problem and the reflected fundamental solution coupled with a boundary operator. Using the Duhamel's principle, we see that due to the fast decay property of the Green's function and the high nonlinearity, the pointwise decaying rate for the nonlinear solution and extra time decaying rate for its first order derivative are obtained.

Acoustic radiation from a cylinder in shallow water by finite element-parabolic equation method

It can be a difficult problem to precisely predict the acoustic field radiated from a finite elastic structure in shallow water channel because of its strong coupling with up-down boundaries and the fluid medium, whose acoustic field cannot be calculated directly by existing methods, such as Ray theory, normal mode theory and other different methods, which are adaptable to sound fields from idealized point sources in waveguide. So, there is no reliable research method of predicting the acoustic radiation of elastic structure in shallow water at present. Based on the finite element method (FEM) coupled with the parabolic equation (PE), the theoretical model for structure acoustic radiation in shallow water at low frequency is established in this paper. This model mainly consists of three sections. First, obtaining the near-field vibro-acoustic characteristics of the elastic structure in shallow water by the multi-physics coupling model established by FEM, whose FEM model includes the up-down boundaries and the completely absorbent sound boundaries in the horizontal direction. Second, getting the acoustic information in the depth, which is set as the acoustic input condition i.e. starting field for the PE. Third, the acoustic information in the far-field quickly calculated by the PE and the finite difference method (FDM). The accuracy, efficiency and fast convergence of FEM-PE method are validated by numerical simulation and theoretical analysis through using a monopole source and structural source in the Pekeris waveguide, respectively. The vibro-acoustic characteristics of elastic cylinder influenced by upper and lower fluid boundaries of the Pekeris waveguide are calculated and analyzed. The cylindrical shell material is steel, and it is 1 m in radius and 10 m in length. The shallow water channel is a Pekeris waveguide with 30 m in depth, at the upper boundary, i.e., the free surface, the lower boundary is the semi-infinite liquid boundary. The analyzed frequencies range from 50 Hz to 200 Hz. The study shows that when the cylindrical shell approaches to the sea surface or bottom, the coupled frequency is higher or lower respectively than that of the shell immersed in the free field. When the diving depth reaches a certain distance range, the coupled frequency tends to be the same as that in free field. The acoustic field radiated from an elastic shell in Pekeris waveguide is similar to that from a point source at low frequency, but there exists a significant difference in high frequency between them, so the structural source can be equivalent to a point source conditionally. The sound radiation attenuation of the structure happens in sequence according to the near-field acoustic shadow zone, the spherical wave attenuation zone, the region between spherical wave and the cylindrical wave attenuation zone, and the cylindrical wave attenuation zone.

The Crank–Nicolson hybrid difference method for the time wave equation on exterior domains with the discrete radial absorbing boundary condition

The hybrid difference method (HDM) combined with the discrete radial absorbing boundary condition (ABC) and the Crank–Nicolson time marching is presented. The HDM method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The discrete radial ABC is a time dependent version of the discrete radial ABC for the elliptic equations, that has been proposed in [Y. Jeon, Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition, J. Sci. Comput. 75, 889–905 (2018)]. Through a change of variables the Bayliss–Turkel ABC is renewed. Moreover, it is partially proved that the discrete radial ABC is a non-standard finite difference approximation of the Bayliss–Turkel ABC. The discrete radial ABC is easy to implement and can be applied to an arbitrary convex fictitious domain including the rectangular box since it is degenerated to a quasi-Dirichlet condition for the time dependent problem. Two-dimensional numerical experiments confirming efficiency of our numerical scheme for various wave numbers and several artificial boundaries are presented.

Capacitance Extraction With Provably Good Absorbing Boundary Conditions

3-D field solvers have become popular tools for parasitic capacitance extraction of full custom circuits, IPs, and packages. Traditional field solvers based on finite difference method/finite element method/floating random walk truncate the field on the outer boundary of the numerical region by applying Dirichlet or Neumann boundary conditions. However, to ensure high accuracy in the vicinity of circuit elements, a substantial gridded region between the circuit and the outer boundary is necessary in order to reduce distortion of the field caused by these truncation conditions. In this paper, we make a fundamental contribution to the application of numerical field solvers by proposing a class of absorbing boundary conditions, which when implemented, significantly reduce the distortion of the field at the numerical boundary, and consequently, throughout the numerical region. The absorbing boundary condition we propose will allow the field throughout the numerical region to behave as though there is no numerical boundary, accurately mimicking the fields in an actual circuit. As a result, the size of the numerical region can be significantly reduced, which in turn reduces the run time without sacrificing accuracy. A mathematical development of the proposed absorbing boundary condition is presented. It is shown that the error of the proposed ${n}$ th order absorbing boundary is ${O(1/r^{n+2})}$ , while the error of the traditional Neumann boundary is ${O(1/r^{2})}$ , where ${r}$ is the size of the numerical region. Experimental results for capacitance extraction with interconnects in multilayer dielectrics and silicon on insulator show the proposed methods improve the run time and accuracy of numerical solutions of Laplace’s equation over previous boundary conditions for uniform or nonuniform meshes.

Absorption probabilities of discrete quantum mechanical systems

Absorbing boundaries introduce non-unitary dynamics in quantum mechanical systems. In particular, it has been shown that absorbing boundaries partially reflect quantum information in a variety of one-dimensional quantum walks. The methods used to compute absorption probabilities for quantum walks can be extended to calculate absorption probabilities for general finite and discrete quantum mechanical systems. These methods can also be used to compute moments of the associated absorption time distributions. We conclude the paper by discussing a modification of the quantum Zeno effect in these absorption systems.

Reciprocal absorbing boundary condition with perfectly matched discrete layers for transient analysis of SV-P waves in a layered half-space

We present an extension of the combination of the Reciprocal Absorbing Boundary Condition (RABC) with Perfectly Matched Discrete Layers (PMDLs) for transient analysis of in-plane wave motion in a layered half-space. Due to the accuracy of RABC for multilayered systems, it is employed for domain truncation beyond vertical boundaries. PMDLs are particularly appropriate for handling the homogenous half-space and, herein, they are applied as excellent absorbers of waves into the half-space underlying the layered domain. Such a combination leads to solutions of problems of plane-strain wave-propagation in a layered half-space, directly in the time domain. Numerical examples are presented illustrating the accuracy, stability and effectiveness of the proposed combination. By combining two absorbing boundaries, the use of corner elements becomes unnecessary. Moreover, the long-term solution by RABCs-PMDLs is observed to be stable in the time-window of numerical examples.

Hybrid absorbing boundary condition for piecewise smooth curved boundary in 2D acoustic finite difference modelling

Flexible computational domains and their corresponding irregular absorbing boundary conditions (ABCs) have previously been shown to improve the efficiency of finite difference (FD) modelling. However, these proposed ABCs for irregular boundaries are based on irregular grid methods. Although they can improve geometric flexibility of FD modelling, irregular grid methods are still complex to implement due to computationally expensive meshing process. To avoid complex mesh generation, FDs in mesh-free discretisation have been developed for nontrivial geometric settings, where scattered nodes can be placed suitably with respect to irregular boundaries and arbitrarily shaped anomalies without coordinate mapping or mesh-element forming. Radial-basis-function-generated FD (RBF-FD) has been proven successful in modelling seismic wave propagation based on mesh-free discretisation. Using RBF-FD, we develop a hybrid ABC for piecewise smooth curved boundary in 2D acoustic wave modelling based on straightforward expanding strategies for sampled boundary and corner nodes. The whole irregular computational domain is combined by an objective zone and a transition zone separated by piecewise smooth curved boundary. Nodal distribution in the objective zone is adaptive to the model structure through special alignment and varying density of nodes, while the transition zone are designed for solving one way wave equation more easily and accurately in the presence of piecewise smooth curved boundary. Modelling examples for homogenous and heterogeneous models with differently shaped computational domains demonstrate the effectiveness of our proposed method.

Dynamic Analysis of Anchored Diaphragm Walls

Summary In this paper, the behavior of diaphragm walls with and without anchor have been studied. In addition to static loads, the walls were subjected to seismic load due to earthquake, using finite element program, PLAXIS 2D. Maximum displacement and bending moment of the wall were compared in different cases. Introduction Deep excavations in urban areas require not only the stability of retained soil, but also special attentions due to the nearby existing buildings. They must be designed in such a way that the requirements for both ultimate and serviceability limit state for supporting system and neighboring structures are satisfied. Anchored diaphragm walls are one of the safest lateral supports, which help in overall stability of the excavated areas. Diaphragm walls represent in general a considerably more complex problem than gravity or cantilever walls. Methodology and Approaches The wall displacement, maximum bending moment and anchor extreme force were calculated using a dynamic PLAXIS finite element program. The soil was considered as elasto-plastic material, using Mohr–Coulomb constitutive model. The wall and the anchor were considered to behave elastically. Considering the dynamic applied load, absorbent boundaries were assumed to prevent dynamic wave reflection. Three different historical strong motions namely, Tabas, Bam and Roodbar were applied to the wall. The following outputs were obtained for different excavation geometry and mechanical properties for soil. Results and Conclusions general results can be drawn as following: 1-Change in the soil stiffness (modulus of elasticity and cohesion) from soft to stiff has caused the maximum wall displacement and bending moment to increase by 76% and 20%, respectively. 2- Under dynamic load, the maximum lateral displacement of the wall is at its maximum value, while the static wall displacement is maximum at about one third the wall height measured from its toe. 3-Peak ground acceleration has direct effect on maximum wall displacement and bending moment, while earthquake frequency has inversely changed these output parameters. Based on the obtained results 30% increase in earthquake acceleration has caused the maximum wall displacement increase by 15%. 4- The maximum displacement and bending moments increased noticeably under earthquake load. Therefore, for diaphragm walls in earthquake prone area, dynamic case should be considered in design. Albeit, building codes, accept higher displacement and bending moments under exceptional loadings.

The Vlasov–Poisson–Fokker–Planck equation in an interval with kinetic absorbing boundary conditions

We study the initial–boundary value problem for the Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions. We first prove the existence of weak solutions of the linearized equation in an interval with absorbing boundary conditions. Moreover, the weak solution converges to zero exponentially in time. Then we extend the above results to the fully nonlinear Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions; the existence and the longtime behavior of weak solutions. Finally, we prove that the weak solution is actually a classical solution by showing the hypoellipticity of the solution away from the grazing set and the Hölder continuity of the solution up to the grazing set.

Acoustic reverse-time migration using GPU card and POSIX thread based on the adaptive optimal finite-difference scheme and the hybrid absorbing boundary condition

Reverse-time migration (RTM) is a powerful tool for imaging geologically complex structures such as steep-dip and subsalt. However, its implementation is quite computationally expensive. Recently, as a low-cost solution, the graphic processing unit (GPU) was introduced to improve the efficiency of RTM. In the paper, we develop three ameliorative strategies to implement RTM on GPU card. First, given the high accuracy and efficiency of the adaptive optimal finite-difference (FD) method based on least squares (LS) on central processing unit (CPU), we study the optimal LS-based FD method on GPU. Second, we develop the CPU-based hybrid absorbing boundary condition (ABC) to the GPU-based one by addressing two issues of the former when introduced to GPU card: time-consuming and chaotic threads. Third, for large-scale data, the combinatorial strategy for optimal checkpointing and efficient boundary storage is introduced for the trade-off between memory and recomputation. To save the time of communication between host and disk, the portable operating system interface (POSIX) thread is utilized to create the other CPU core at the checkpoints. Applications of the three strategies on GPU with the compute unified device architecture (CUDA) programming language in RTM demonstrate their efficiency and validity.

Implementation of the infinite-range exterior complex scaling to the time-dependent complete-active-space self-consistent-field method

We present a numerical implementation of the infinite-range exterior complex scaling (irECS) [Phys. Rev. A 81, 053845 (2010)] as an efficient absorbing boundary to the time-dependent complete-active-space self-consistent field (TD-CASSCF) method [Phys. Rev. A 94, 023405 (2016)] for multielectron atoms subject to an intense laser pulse. We introduce Gauss-Laguerre-Radau quadrature points to construct discrete variable representation basis functions in the last radial finite element extending to infinity. This implementation is applied to strong-field ionization and high-harmonic generation in He, Be, and Ne atoms. It efficiently prevents unphysical reflection of photoelectron wave packets at the simulation boundary, enabling accurate simulations with substantially reduced computational cost, even under significant (~ 50%) double ionization. For the case of a simulation of high-harmonic generation from Ne, for example, 80% cost reduction is achieved, compared to a mask-function absorption boundary.

Boundary Homogenization and Capture Time Distributions of Semipermeable Membranes with Periodic Patterns of Reactive Sites

We consider the capture dynamics of a particle undergoing a random walk in a half-space bounded by a plane with a periodic pattern of absorbing pores. In particular, we numerically measure and asymptotically characterize the distribution of capture times. Numerically we develop a kinetic Monte Carlo (KMC) method that exploits exact solutions to create an efficient particle-based simulation of the capture time that deals with the infinite half-space exactly and has a run time that is independent of how far from the pores one begins. Past researchers have proposed homogenizing the surface boundary conditions, replacing the reflecting (Neumann) and absorbing (Dirichlet) boundary conditions with a mixed (Robin) boundary condition. We extend previous work to asymptotically determine the leakage parameter for the mixed boundary condition for arbitrary periodic pore configurations in the dilute fraction limit. In this asymptotic limit, we pose and solve an optimization problem for the Bravais lattice which maximizes the capture rate of the absorbing pores, finding the hexagonal lattice to be the global maximum.

Sound radiation of cylinder in shallow water investigated by combined wave superposition method

It can be a difficult problem to precisely predict the sound field radiated from a finite elastic structure in shallow water channel because of its strong coupling with up-down boundaries and the fluid medium, whose sound field cannot be calculated directly by current methods, such as Ray theory, normal mode theory and other different methods, which are adaptable to sound fields from idealized point sources in waveguide. So far, there is no reliable prediction method to solve this kind of problem. A combined wave superposition method is proposed for such a problem, which combines the traditional wave superposition method with the transfer function in shallow water channel and the multi-physics field coupling numerical model. This method mainly consists of three sections:1) obtaining the normal velocity on the elastic structure surface in shallow water channel by the finite element method (FEM), whose FEM model includes the up-down boundaries and the completely absorbent sound boundaries in the horizontal direction; 2) getting the equivalent point source strength by traditional wave superposition method; 3) calculating the total sound field by adding up each point source field which is obtained by normal mode method. This method is verified by numerical simulation and theoretical analysis by using an imaginary and elastic spherical sound source respectively, and the results demonstrate that the method is valid and has high precision and calculating efficiency. The acoustic radiation characteristics from elastic cylindrical shells is investigated for different acoustic radiation sources, ocean environments and measurements. The cylindrical shell material is steel, whose radius and length are 3 m and 30 m respectively. The shallow water channel is an ideal waveguide with 50 m in depth, at the upper boundary, i.e., the free surface, the lower boundary is the Neumann boundary, i.e., the normal derivative of the acoustic pressure should be zero. The analysis frequency range is from 30 Hz to 200 Hz. The results show that due to a significant coupling effect of up-down direction boundaries on the sound field, the elastic structure can be equivalent to the point source only in low frequency and far field. The spatial field directivity distribution is more obvious at high frequency. The acoustic power measured by vertical line arrayis greatly influenced by ocean boundary and the depth of target.

A Time-Domain Propagator Numerical Method for Computational Electromagnetics

A new time-domain numerical method for full-wave electromagnetic scattering and propagation is presented. The propagator method can be described as a numerical integral operator technique for moving the electromagnetic field through numerical space at successive increments in time. The numerical equations are obtained by discretizing electric and magnetic field propagator integrals. A primary feature of the propagator method is that all six electromagnetic-field components are calculated at each numerical grid point and all components are in time synchronization. Constant spatial and time increments are maintained throughout an inhomogeneous numerical space by employing an extrapolation procedure that ensures causality. A simple and effective first-order absorbing boundary condition, described as the null boundary condition, is introduced. Examples, provided in one-, two-, and three-dimensions, include plane wave reflection from and transmission through a planar boundary, and scattering and radar cross sections for canonical dielectric objects.

An Efficient Second-Order Finite Difference Method for the One-Dimensional Schrödinger Equation with Absorbing Boundary Conditions

A stable and convergent second-order fully discrete finite difference scheme with efficient approximation of the exact absorbing boundary conditions is proposed to solve the Cauchy problem of the o...