Artificial Boundary Method Research Articles

Numerical solution of coupled nonlinear Klein-Gordon equationson unbounded domains.

The numerical solution of coupled nonlinear Klein-Gordon equationson unbounded domains is considered by applying the artificial boundary method. Based on the unified approach to overcome the coupled nonlinearity, local artificial boundary conditions are designed on the introduced artificial boundaries. The original problem is reduced to an initial boundary value problem on a bounded domain, which can be efficiently solved by the finite difference method. Some numerical examples are provided to verify the accuracy and effectiveness of the proposed method.

An unsteady two-dimensional vortex method for the flow separation over a flapping wing

The flow separation over a very flexible or morphing aircraft, especially during flying or oscillation at a high angle of attack, is stronger than that for conventional aircraft. However, a low-order model for reliably predicting the dynamics of the LEV and its influence on the force is still not found. It is valuable to develop such aerodynamic approaches for optimizing the wing structure and kinematics and for designing a flight control strategy considering the flow separation problem. This paper presents a two-dimensional vortex method in which an artificial boundary is constructed for the velocity adjustment of the vortices near the control panels. This artificial boundary is established in accordance with the thickness of the actual wing. A sub-time step is introduced into the solution process of the vortex method to predict the vortex locations and perform velocity adjustments. Crossing vortices retain only their tangential velocity components relative to the artificial boundary in order to avoid entering the range of the airfoil represented by the artificial boundary. Moreover, the excessive induced velocities on the control panels caused by crossing vortices are decreased, and the nonphysical behaviors and discontinuous pressure are avoided. To demonstrate the advantages of the proposed artificial boundary method in modeling an airfoil during high-angle-of-attack flight, the proposed approach is applied to a flat-plate starting problem and a flapping problem. The simulation results are then compared with the results of several experiments and numerical data for both attached and separated test cases, verifying the capabilities of the proposed method.

Nonlinear seismic response and amplification effect of 3D sedimentary basin based on bounding surface constitutive model

Sedimentary basins can have significant amplification effects on seismic ground motion. It is very necessary to use appropriate nonlinear soil constitutive model to quantitatively analyze the seismic response of three-dimensional (3D) sedimentary basins subjected to strong ground motion. This paper employs a bounding surface constitutive model to simulate the soil nonlinearity, which is successfully incorporated into ABAQUS software. A finite element model is then established to investigate the nonlinear seismic response of 3D sedimentary employing the constitutive model along with viscoelastic artificial boundary and equivalent nodal force method. The nonlinear amplification effect of a typical 3D layered sedimentary basin is extensively evaluated. The influence of a soft interlayer in the sedimentary basin on the seismic amplification is further analyzed. The results demonstrate that the maximum surface peak acceleration usually occurs in specific areas; as the input ground motion intensity increases, the focusing and amplification effects become more significant in deeper soil layers. A soft interlayer will significantly increase the short period component of the surface response spectrum, and shorten the predominant period of sedimentary basin. These results can provide a reference basis for seismic risk analysis and determination of ground motion parameters in a sedimentary basin.

Efficient implementation of the exact artificial boundary condition for the exterior problem of the Stokes system in three dimensions

AbstractIn this paper the Stokes system in an unbounded domain is solved by the artificial boundary method. The novelty lies in an operator form of the exact Dirichlet-to-Neumann (DtN) mapping. With the help of the Chebyshev rational approximation of the square root function, we derive a highly accurate approximate DtN mapping, which can be numerically implemented without resorting to the eigen-decomposition in terms of the vectorial spherical harmonics. In addition, we develop an efficient block preconditioner for the augmented truncated saddle point problem. Numerical experiments demonstrate the effectiveness of the proposed method.

Efficient approach to solve time fractional Kardar–Parisi–Zhang equation on unbounded domains

It is an important issue to numerically solve the time fractional Kardar–Parisi–Zhang equation on unbounded domains, which is a universal model to study the dynamics of the interface with memory effects. The artificial boundary method is adopted to circumvent the problems defined on unbounded domain and retain the same dynamics as the original problem on a bounded domain. The artificial boundary conditions are designed by combining the ideas of operator splitting approach and artificial boundary method to overcome the unboundedness of the physical domain, nonlocality and nonlinearity of the time fractional Kardar–Parisi–Zhang equation. The bounded and convergent properties are analyzed rigorously. The accuracy and feasibility of the proposed method are verified by numerical results.

Artificial boundary conditions for nonlinear time fractional Burgers’ equation on unbounded domains

This paper is concerned with the design of artificial boundary conditions (ABCs) for nonlinear time fractional Burgers’ equation on unbounded domains, that describes the propagation of many kinds of waves in physics. It is a major challenge to design appropriate ABCs to obtain the numerical solution and simulate the real physical phenomena, owing to the unboundedness of the physical domain, nonlocality and nonlinearity. The ABCs on the introduced artificial boundaries are constructed by applying the ideas of operator splitting approach and artificial boundary method to obtain a reduced initial boundary value problem on the truncated computational domain, which can be solved by the finite difference method efficiently. The boundedness and convergence are rigorously analyzed. Numerical simulations are presented to validate accuracy and feasibility of our proposed method.

Immersed boundary method for multiphase transport phenomena

Abstract Multiphase flows with momentum, heat, and mass transfer exist widely in a variety of industrial applications. With the rapid development of numerical algorithms and computer capacity, advanced numerical simulation has become a promising tool in investigating multiphase transport problems. Immersed boundary (IB) method has recently emerged as such a popular interface capturing method for efficient simulations of multiphase flows, and significant achievements have been obtained. In this review, we attempt to give an overview of recent progresses on IB method for multiphase transport phenomena. Firstly, the governing equations, the basic ideas, and different boundary conditions for the IB methods are introduced. This is followed by numerical strategies, from which the IB methods are classified into two types, namely the artificial boundary method and the authentic boundary method. Discussions on the implementation of various boundary conditions at the interphase surface with momentum, heat, and mass transfer for different IB methods are then presented, together with a summary. Then, the state-of-the-art applications of IB methods to multiphase flows, including the isothermal flows, the heat transfer flows, and the mass transfer problems are outlined, with particular emphasis on the latter two topics. Finally, the conclusions and future challenges are identified.

Artificial Boundary Method for European Pricing Option Problem

Artificial Boundary Method for European Pricing Option Problem

Efficient semi-implicit compact finite difference scheme for nonlinear Schrödinger equations on unbounded domain

The compact finite difference scheme is designed to numerically solve the nonlinear Schrödinger equation and coupled nonlinear Schrödinger equations on an unbounded domain in this paper. The original problem on an unbounded domain is reduced to an initial boundary value problem defined on a bounded computational domain by applying the artificial boundary method. Then, the reduced problem on the bounded computational domain is solved by an efficient semi-implicit compact finite difference scheme, which is a fourth-order scheme with respect to spatial variable. The scheme efficiently avoids the time-consuming iteration procedure necessary for the nonlinear scheme and thus is relatively time-saving. Finally, the stability of the proposed scheme is rigorously analyzed. Numerical examples are given to illustrate the accuracy and effectiveness of the proposed method.

Numerical solution of coupled nonlinear Schrödinger equations on unbounded domains

In this paper, we investigate the numerical solution of general N-coupled nonlinear Schrödinger equations on unbounded domains, that describe the multiple solitary waves in physics. Due to the unboundedness of the physical domain and strong nonlinearity, how to design appropriate numerical approach to obtain the numerical solution and simulate the real physical phenomena is a major challenge. Based on the ideas of artificial boundary method and operator splitting approach, we develop local artificial boundary conditions for the N-coupled nonlinear Schrödinger equations on the introduced artificial boundaries. Then the original initial value problem was reduced into an initial boundary value problem on the truncated computational domain, which can be solved by the finite difference method efficiently. We rigorously prove the stability of the reduced problem by introducing a series of auxiliary variables, which are vital to overcome the difficulty of the mixed partial derivatives in the local artificial boundary conditions. Numerical simulations are also presented to validate accuracy and the stability of our proposed method.

Numerical Computations of Nonlocal Schrödinger Equations on the Real Line

The numerical computation of nonlocal Schrodinger equations (SEs) on the whole real axis is considered. Based on the artificial boundary method, we first derive the exact artificial nonreflecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458–3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonreflecting boundary condition and leads us to reformulate the original infinite discrete system into an equivalent finite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are finally provided to demonstrate the effectiveness of our approach.

Fast Evaluation of Artificial Boundary Conditions for Advection Diffusion Equations

An artificial boundary method is developed for solving the one-dimensional advection diffusion equation in the real line. In order to construct a fully discrete fast numerical algorithm with rigorous error analysis, we start with the two-step backward difference formula for time discretization of the advection diffusion equation in the whole real line. Then, we use the discrete analogue of the Laplace transform to derive a second-order time-stepping scheme in a bounded domain equipped with a discrete artificial boundary condition (ABC). The Galerkin finite element method is used for spatial discretization. To expedite the evaluation of time convolution involved in the discrete ABC, we propose a fast algorithm based on the best rational approximation of square root function in subdomains of the complex plane. An estimate for this best rational approximation enables us to prove optimal-order convergence of the fully discrete numerical scheme (integrating the fast approximation algorithm). Several numerical examples are provided to illustrate the convergence of numerical solutions and the effectiveness of the proposed fast approximation algorithm.

1D finite element artificial boundary method for transient response of ocean site under obliquely incident earthquake waves

Ocean site consisting of seawater-seabed-bedrock system is modeled as horizontally layered fluid-poroelastic-solid media. Earthquake wave is assumed as a plane P or SV wave of oblique incidence from the solid bedrock into the site. A time-domain numerical method called one-dimensional finite element artificial boundary (1D-FE-AB) method is proposed to effectively and accurately calculate the transient response of the ocean site under the obliquely incident earthquake waves. In this method, a horizontally line-shaped AB is firstly introduced in the solid bedrock to divide the whole site into the fluid-poroelastic-solid layer and the remaining homogenous solid half space. The former is semi-discretized by FE method along depth, and the latter with the incident earthquake wave is simulated by an exact AB condition. The resulting semi-discrete system with AB is then transformed into a spatially 1D dynamic equation along depth by using the Snell's law of plane body wave propagation in multilayer media. The 1D equation can be solved by the time integration algorithm such as Newmark method. Numerical examples are finally given to demonstrate the effectiveness of the proposed 1D-FE-AB method.

Numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains.

The numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains is considered by applying the artificial boundary method in this paper. In order to design the local absorbing boundary conditions for the coupled nonlinear Schrödinger equations, we generalize the unified approach previously proposed [J. Zhang et al., Phys. Rev. E 78, 026709 (2008)PLEEE81539-375510.1103/PhysRevE.78.026709]. Based on the methodology underlying the unified approach, the original problem is split into two parts, linear and nonlinear terms, and we then achieve a one-way operator to approximate the linear term to make the wave out-going, and finally we combine the one-way operator with the nonlinear term to derive the local absorbing boundary conditions. Then we reduce the original problem into an initial boundary value problem on the bounded domain, which can be solved by the finite difference method. The stability of the reduced problem is also analyzed by introducing some auxiliary variables. Ample numerical examples are presented to verify the accuracy and effectiveness of our proposed method.

Fast and stable evaluation of the exact absorbing boundary condition for the semi-discrete linear Schrödinger equation in unbounded domains

This paper is concerned with the numerical solution of the one-dimensional semi-discrete linear Schrödinger equation in unbounded domains. In order to compute the solution on the domain of physical interest, the artificial boundary method is applied to transform the original unbounded domain problem into an initial boundary value problem on a truncated finite domain. We prove the stability of the truncated semi-discrete problem. Then, a fast algorithm is proposed to approximate the nonlocal absorbing boundary condition. The novelty of this fast algorithm is that the stability of the approximate truncated semi-discrete problem is automatically maintained. In the end, numerical examples are presented to demonstrate the performance of the proposed algorithm.

1D finite element artificial boundary method for layered half space site response from obliquely incident earthquake

Site response analysis is an important topic in earthquake engineering. A time-domain numerical method called as one-dimensional (1D) finite element artificial boundary method is proposed to simulate the homogeneous plane elastic wave propagation in a layered half space subjected to the obliquely incident plane body wave. In this method, an exact artificial boundary condition combining the absorbing boundary condition with the inputting boundary condition is developed to model the wave absorption and input effects of the truncated half space under layer system. The spatially two-dimensional (2D) problem consisting of the layer system with the artificial boundary condition is transformed equivalently into a 1D one along the vertical direction according to Snell’s law. The resulting 1D problem is solved by the finite element method with a new explicit time integration algorithm. The 1D finite element artificial boundary method is verified by analyzing two engineering sites in time domain and by comparing with the frequency-domain transfer matrix method with fast Fourier transform.

An Equation Decomposition Based Tailored Finite Point Method for Linearized Incompressible Flow in Two-Dimensional Space

Abstract In this paper, we propose a tailored finite point method for linearized incompressible flow (Oseen equations) in two dimensions based on the equation decomposition technique. Unlike the usual vorticity-stream function formulation, the velocities are decomposed to irrotational and rotational parts. We only need to solve a system of two elliptic equations which are decoupled in the interior domain. They are only coupled in boundary conditions. When the domain is unbounded, we use the artificial boundary method to reduce the original problem to a problem on a bounded computational domain. Our finite point method has been tailored to some particular properties of the problem. Therefore, our scheme satisfies the discrete maximum principle in the interior domain automatically. We also give some remarks on more generally linearized incompressible flow, and it can be considered as the first step to solve the incompressible Navier–Stokes problem. Finally, several numerical examples show the efficiency and feasibility of our method whatever the Reynolds number is small or large.

An Artificial Boundary Method for Burgers’ Equation inthe Unbounded Domain

An Artificial Boundary Method for Burgers’ Equation inthe Unbounded Domain

Book Reviews

This issue's featured review by K. K. Tung is about SIAM's Mathematics & Climate by Kaper and Engler. This undergrad textbook should inspire students to think about the policy choices that could be made, based on sound math and statistical modeling and more specific conceptual models. Also included are reviews of E's Principles of Multiscale Modeling, Han and Wu's Artificial Boundary Method, Narang-Siddarth and Valasek's Nonlinear Time Scale Systems, Noonburg's Ordinary Differential Equations, Paul and Baschnagel's Stochastic Processes, Schuss's Brownian Dynamics, Shafarevich's two-volume Basic Algebraic Geometry, Shiryaev's Problems in Probability, Shtern's Counterflows, Strang's Differential Equations and Linear Algebra, and Syropoulos's Theory of Fuzzy Computation. You can count on quite a variety of opinions by our expert reviewers.

A survey on artificial boundary method

The artificial boundary method is one of the most popular and effective numerical methods for solving partial differential equations on unbounded domains, with more than thirty years development. The artificial boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and the theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. This paper reviews the development and discusses different forms of the artificial boundary method.