Artificial Boundary Conditions Research Articles

A novel investigation of Casson fluid flow over an infinite plate with constant temperature by constructing the absorbing boundary conditions

Casson fluid has been widely used in blood, honey, chocolate, and various polymer fluids, for which the investigation of its underlying flow and heat transfer mechanisms in complex environments is of vital significance. In this paper, the Casson fluid flow over an infinite moving plate with constant temperature is studied, which leads to a coupled system of the velocity and temperature fields. To treat the infinite coupled system more accurately, the artificial boundary conditions in terms of fractional derivatives are developed using the Laplace transform, which transfers the problem in the infinite domain to a model in the bounded domain. The finite difference method is applied to solve the new bounded coupled system. Furthermore, three numerical experiments are presented to demonstrate the validity and accuracy of the transferred model and the numerical scheme. The influence of some important physical parameters on the velocity and temperature fields is also explored. The main findings are that the viscosity of the Casson fluid decreases with an increasing Grashof number leading to an accelerated flow, the thermal conductivity diminishes as the Prandtl number increases resulting in a weakened thermal effect within the flow, and the velocity at the fixed position decreases as the Casson parameter increases.

An Efficient Local Artificial Boundary Condition for Infinite Long Finite Element Euler–Bernoulli Beam

To solve the wave propagation problems of the Euler–Bernoulli beam in an unbounded domain effectively and efficiently, a new local artificial boundary condition technology is proposed. It replaces the residual right-hand side of the truncated discrete equation with an equivalent linear algebraic system. First, the equivalent Schrodinger equation is discussed. Its artificial boundary condition is obtained by first rationalizing the Dirichlet-to-Neumann condition in the frequency domain with a Pade approximation and then inverse transforming each Pade term back into the time domain by introducing auxiliary degrees of freedom. Frequency shifting is employed such that it performs better near a prescribed frequency. Then, the artificial boundary condition of the finite element Euler–Bernoulli beam is obtained by simple algebraic manipulations on that of the corresponding Schrodinger equation. This method only makes local changes to the original truncated discrete dynamic system and thus is very efficient and easy to use. The accuracy of the proposed method can be improved by using more Pade terms and a proper shift frequency. The numerical example shows, with only a few additional degrees of freedom, the proposed artificial boundary condition effectively eliminates the spurious reflection. The idea of the proposed method can also be used in other dispersive wave systems.

Artificial Boundary Condition Processing Technique for Phased-Resolved Wave Surface Reconstruction

Artificial Boundary Condition Processing Technique for Phased-Resolved Wave Surface Reconstruction

Time-domain instability mechanism for artificial boundary condition of semi-infinite medium described by discrete rational approximation

The use of discrete rational approximation functions to represent foundation dynamic impedance plays a crucial role in the time-domain analysis of soil-structure interaction regarding artificial boundary. In modeling, the rational approximation functions need to be transformed into time-domain recursive models for incorporation into time-history analysis. However, this method also suffers from uncertainties in parameter identification and instability in time-domain analysis when the time-domain recursive models of soil are coupled with upper structure. The stability issue in artificial boundary coupled time-domain computation has not been effectively addressed. This paper establishes the closed-loop transfer function of the coupled system and analyzes the causes of instability in the coupled system based on gain margin analysis. Different numerical simulations demonstrate that the major cause of artificial boundary system instability is the description errors of discrete rational approximation functions beyond the fitting frequency range.

A stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions for a 2D peridynamic diffusion model

In this work, a stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions (ABCs) is designed for solving a peridynamic diffusion model on the whole two dimensional domain. To do so, we first use a high-order quadrature-based finite difference scheme to approximate the spatially nonlocal operator and use the BDF2 scheme to approximate the temporal direction. For the resulting fully discrete system, we apply the nonlocal potential theory to obtain the discrete Dirichlet-to-Dirichlet (DtD)-type ABCs. After that, we further derive the Dirichlet-to-Neumann (DtN)-type ABCs based on the nonlocal Neumann data defined from the discrete nonlocal Green's first identity. For the stability and convergence analysis, we reformulate the BDF2 operator into a discrete convolution sum form, then use the technique of the discrete orthogonal convolution kernels to obtain the optimal convergence order. Finally, numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

Free Vibrations of a New Three-Phase Composite Cylindrical Shell

The novel concept of a functionally graded three-phase composite structure is derived from the urgent need to improve the mechanical properties of traditional two-phase composite structures in aviation. In this paper, we study the free vibrations of a new functionally graded three-phase composite cylindrical shell reinforced synergistically with graphene platelets and carbon fibers. We calculate the equivalent elastic properties of the new three-phase composite cylindrical shell using the Halpin-Tsai and Mori-Tanaka models. The governing equations of this three-phase composite cylindrical shell are derived by using first-order shear deformation theory and Hamilton’s principle. We obtain the natural frequencies and mode shapes of the new functionally graded three-phase composite cylindrical shell under artificial boundary conditions. By comparing the results of this paper with the numerical results of finite element software, the calculation method is verified. The effects of the boundary spring stiffness, GPL mass fraction, GPL functionally graded distributions, carbon fiber content, and the carbon fiber layup angle on the free vibrations of the functionally graded three-phase composite cylindrical shell are analyzed in depth. The conclusions provide a certain guiding significance for the future application of this new three-phase composite structure in the aerospace and engineering fields.

Constructing artificial boundary condition of dispersive wave systems by deep learning neural network

To solve one dimensional dispersive wave systems in an unbounded domain, a uniform way to establish localized artificial boundary conditions is proposed. The idea is replacing the half-infinite interval outside the region of interest with a super element which exhibits the same dynamics response. Instead of designing the detailed mechanical structures of the super element, we directly reconstruct its stiffness, mass, and damping matrices by matching its frequency-domain reaction force with the expected one. An artificial neural network architecture is thus specifically tailored for this purpose. It comprises a deep learning part to predict the response of generalized degrees of freedom under different excitation frequencies, along with a simple linear part for computing the external force vectors. The trainable weight matrices of the linear layers correspond to the stiffness, mass, and damping matrices we need for the artificial boundary condition. The training data consists of input frequencies and the corresponding expected frequency domain external force vectors, which can be readily obtained through theoretical means. In order to achieve a good result, the neural network is initialized based on an optimized spring-damper-mass system. The adaptive moment estimation algorithm is then employed to train the parameters of the network. Different kinds of equations are solved as numerical examples. The results show that deep learning neural networks can find some unexpected optimal stiffness/damper/mass matrices of the super element. By just introducing a few additional degrees of freedom to the original truncated system, the localized artificial boundary condition works surprisingly well.

Numerical solution of nonlinear Schrödinger equation with damping term on unbounded domain

The artificial boundary method is applied to numerically solve the nonlinear Schrödinger equation with damping term on unbounded domain. A novel transformation is developed to eliminate the troublesome damping term, which yields the difficulty to design the artificial boundary conditions. The artificial boundary conditions are investigated to reduce the original problem into an initial boundary value problem, based on the idea of operator splitting approach. The stability of the reduced problem is analyzed by the energy estimation. The accuracy and effectiveness of the proposed method are verified by numerical examples.

Scaled boundary perfectly matched layer for wave propagation in a three-dimensional poroelastic medium

In this paper, we introduce a novel direct time-domain artificial boundary condition, called the scaled boundary perfectly matched layer, for wave problems in 3D unbounded fluid-saturated poroelastic medium. The poroelastic medium is conceptualized as a two-phase medium composed of a solid skeleton and an interstitial fluid based on the u-p formulation of Biot's theory. The scaled boundary perfectly matched layer proposed in this work originates from an extension of a recently developed perfectly matched layer for wave problems in a single-phase solid medium. The dependence of the conventional perfectly matched layer on the global coordinate system is overcome in the present method. As a result, this method permits the utilization of an artificial boundary with general geometry (not necessarily convex) and can consider planar physical surfaces and interfaces extending to infinity. The proposed method is formulated by a mixed unsplit-field form, which includes both the mixed displacement-stress field and pore pressure-relative displacement field. The resulting scaled boundary perfectly matched layer formulation can be directly described by a system of third-order ordinary differential equations in time, which can be easily integrated into the standard finite-element framework of poroelastic theory. Finally, numerical benchmark examples are presented to demonstrate the accuracy and robustness of the proposed scaled boundary perfectly matched layer, as well as the versatility in practical engineering problems.

A second-order absorbing boundary condition for two-dimensional peridynamics

The aim of this paper is to develop numerical analysis for the two-dimensional peridynamics which depicts nonlocal phenomena with artificial boundary conditions (ABCs). To this end, the artificial boundary conditions for the fully discretized peridynamics are proposed. Then, the numerical analysis of the fully discretized scheme is developed such that the ABCs solve the corner reflection problem with second-order accuracy. Finally numerical examples are given to verify theoretical results.

Gauge–Uzawa-based, highly efficient decoupled schemes for the diffuse interface model of two-phase magnetohydrodynamic

In this paper, we design two totally decoupled, linear and unconditionally energy stable schemes with first-order time accuracy for the multi-physics diffuse interface model of two-phase magnetohydrodynamic (MHD) problem. These schemes combine the semi-implicit stabilization method/invariant energy quadratization (IEQ) method for the phase-field equations, Gauge–Uzawa method for the MHD equations, with some subtle implicit–explicit treatments for nonlinear coupled terms. Then this strong nonlinear and coupled complex system is split into a series of small linear elliptic equations, which makes the calculation more efficient. Additionally, compared to the standard projection method, the given schemes can overcome the selection of the initial value about pressure p and the treatment of artificial boundary condition on p which can lead to boundary layers and lose accuracy. Finally, rigorous proof of unconditional energy stability and several simulations of numerical experiment are conducted to demonstrate the validity of the decoupled schemes.

Hybrid Artificial Boundary Conditions for the Application of Blunt-Body Aerodynamic Noise Prediction

A hybrid artificial boundary condition (HABC) that combines the volume-based acoustic damping layer (ADL) and the local face-based characteristic boundary condition (CBC) is presented to enhance the absorption of acoustic waves near the computational boundaries. This method is applied to the prediction of aerodynamic noise from a circular cylinder immersed in uniform compressible viscous flow. Different ADLs are designed to assess their effectiveness whereby the effect of the mesh-stretch direction on wave absorption in the ADL is analysed. Large eddy simulation (LES) and FW-H acoustic analogy method are implemented to predict the far-field noise, and the sensitivities of each approach to the HABC are compared. In the LES computed propagation field of the fluctuation pressure and the frequency-domain results, the spurious reflections at edges are found to be significantly eliminated by the HABC through the effective dissipation of incident waves along the wave-front direction in the ADL. Thereby, the LES results are found to be in a good agreement with the acoustic pressure predicted using FW-H method, which is observed to be just affected slightly by reflected waves.

Evaluation of the effects of the seismic wave incidence angle on soil-structure interactions via advanced finite beam elements

This work investigates the interactions between the soil and a superstructure subjected to various seismic waves using variable-kinematics finite beam elements. Artificial boundary conditions based on springs and dashpots with appropriate elastic and viscous coefficients were adopted to ensure the absorption of the incident waves and simulate the infinite domain condition. The one-dimensional finite element formulation is obtained with a unified formalism that enables arbitrary structural models to be adopted. Lagrange-type expansions have been used in this work to approximate the beam kinematics and facilitate the application of artificial boundary conditions. Both pressure and shear waves with different incidence angles have been considered, and the dynamic responses calculated with the current approach have been compared with analytical solutions when available. The results have demonstrated the accuracy and potentialities of the methodology and provided reasonable confidence for future applications.

Artificial boundary condition for Klein-Gordon equation by constructing mechanics structure

An innovative local artificial boundary condition is proposed to numerically solve the Cauchy problem of the Klein-Gordon equation in an unbounded domain. Initially, the equation is considered as the axial wave propagation in a bar supported on a spring foundation. The numerical model is then truncated by replacing the half-infinitely long bar with an equivalent mechanical structure. The effective frequency-dependent stiffness of the half-infinitely long bar is expressed as the sum of rational terms using Pade approximation. For each term, a corresponding substructure composed of dampers and masses is constructed. Finally, the equivalent mechanical structure is obtained by parallelly connecting these substructures. The proposed approach can be easily implemented within a standard finite element framework by incorporating additional mass points and damper elements. Numerical examples show that with just a few extra degrees of freedom, the proposed approach effectively suppresses artificial reflections at the truncation boundary and exhibits first-order convergence.

A scaled boundary finite element method for soil dynamic impedance of pile groups using hybrid quadtree mesh considering horizontal vibration

The numerical modeling of pile-soil dynamic interaction has been a challenging problem in engineering for many years because of the unbounded domain, which typically involves significant computational cost. This paper develops a novel scaled boundary finite element method (SBFEM) that can accurately calculate the soil dynamic impedance of pile groups with end bearing subjected to horizontal vibration. The three-dimensional (3D) problem is first transformed to a two-dimensional (2D) Helmholtz equation in the horizontal plane, exploiting the analytical solution for the vertical modes using separation of variables. The SBFEM is then applied to solve the 2D computational domain, which is divided into bounded and unbounded domains. Based on the SBFEM, an accurate circular artificial boundary condition (ABC) is formulated to simulate the unbounded domain of the soil. A special hybrid quadtree mesh is devised for this problem, where a high-quality structured quadrilateral mesh is generated surrounding the piles. The accuracy of the proposed method is verified using a single pile problem with known analytical solution. The effects of dimensionless frequency, pile numbers, soil depth and relative distance between piles on the soil dynamic impedance are considered.

Existence of a steady flow through a rotating radial turbine with an arbitrarily large inflow and an artificial boundary condition on the outflow

AbstractWe prove the existence of a steady strong solution to the Navier–Stokes‐type problem in a 2D multiply connected domain, modeling a flow of an incompressible viscous fluid through a rotating radial turbine. We consider the inhomogeneous Dirichlet boundary condition on the inflow Γin and an artificial boundary condition of the “do nothing” type on the outflow Γout. The conditions admit an arbitrarily large flux through the turbine. This is, however, compensated by the requirement that the distance between Γout and the profile family is “sufficiently small”. (Theorem 1.1.) The solution is steady in the rotating frame, that is, in the frame, attached to the rotating turbine. As an auxiliary result, we prove that a function, that takes just two different values in a 1D interval on two complementary measurable sets M and of positive 1D measure, is not in . (Lemma 3.4.)

An unconditionally stable artificial compression method for the time‐dependent groundwater‐surface water flows

AbstractIn this article, we propose a second order, unconditionally stable artificial compression method for the fully evolutionary Stokes/Darcy and Navier‐Stokes/Darcy equations that model the coupling surface and groundwater flows. It uncouples the surface from the groundwater flow by the Crank‐Nicolson Leapfrog scheme for the discretization in time, and through the artificial compression method without artificial pressure boundary conditions to decouple the velocity and pressure of the incompressible flow. Finally, we have verified the stability and second‐order convergence of the algorithm from theoretical analysis and numerical experiments.

Numerical modeling of connected piled raft foundation under seismic loading in layered soils

AbstractUntil recently, the behavior of connected piled raft foundation was not fully understood in the seismically active region due to the complex dynamic soil–pile–foundation structure interaction. This concern arises when the soil deposit-supported foundations are stratified or heterogynous and subjected to high ground motion intensity. In the current study, a series of numerical analyses using ABAQUS software have been conducted on a pile group of (3 × 3) arranged into a square pattern to investigate the seismic response of piled foundations embedded in dry sandy soil (homogenous and layered), and how the amplification of propagated waves affects the bending moment along piles. For mesh generation, an artificial boundary condition using the tied-nodes approach was adopted to simulate the free-field motion of soil under earthquake excitation. The structure used a single degree of freedom with a lumped mass. Moreover, Mohr–Coulomb and linear elastic models have been chosen for soil and pile–raft, respectively. The results demonstrate that the foundation rocking increases in stratified soil compared to homogenous soil, irrespective of the seismic intensity. The maximum bending moment was observed at the pile head in homogenous soil and shallow depths in layered soil because of the kinematic interaction at the soil interface. The results also indicated that the amplification factor (acceleration at a certain depth to the acceleration at bedrock) was found to be 203 and 189% in homogenous soil for PGA values of 0.1 and 0.33 g, respectively. Almost there were no effects of seismic intensity in layered soil on the amplified waves transmitted into the soil surface.

Mechanical behavior of high-pressure pipeline installed through horizontal directional drilling under seismic loads

The present study investigates the mechanical behavior of high-pressure pipelines installed through horizontal directional drilling (HDD), a crucial trenchless installation method, under seismic loads. Utilizing the finite element method and artificial viscoelastic boundary conditions, this paper analyzes the stress response of pipelines subjected to seismic loads in two directions. The seismic loads are applied using the response spectrum method, and the pipeline stress is analyzed in time history. A comparison of the stress in pipelines installed through the HDD method and the traditional open-cut method is conducted, and the impact of earthquake magnitude on pipeline stress is evaluated. The results indicate that the pipeline stress is greater when the seismic acceleration acts transversely, with bending deformation being the dominant mode of response. A 9.29% increase in maximum stress is observed for each increment in earthquake magnitude. Furthermore, the stress in pipelines installed through the HDD method is 48% lower than that of pipelines installed using the open-cut method when subjected to transverse seismic loads.

Wave diffraction and radiation by a fully-submerged body in front of a vertical wall by using an exact DtN artificial boundary condition

This paper is concerned with wave diffraction and radiation by an arbitrarily-shaped submerged body in front of a vertical wall. An artificial boundary is chosen as a virtual cylindrical surface so that it can enclose two bodies (one is the original body and another is its imaginary body by the method of mirror image), and the entire water domain is divided into an interior subdomain and an exterior one by the artificial boundary. An exact DtN artificial boundary condition on the artificial boundary is derived from an analytical solution in the exterior subdomain, and it is used as a boundary condition to solve hydrodynamic interaction of the bodies with the interior subdomain numerically using boundary integral equation. The present numerical model is validated by comparison with precedent analytical results in case of a submerged circular cylinder. The effects of heading angle, body submergence and distance from the wall on wave elevation and exciting forces are considered in case of a chamfer box or a capsule.