Truncation Boundary Condition Research Articles

Fractional Second-Grade Fluid Flow over a Semi-Infinite Plate by Constructing the Absorbing Boundary Condition

The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed based on the artificial boundary method (ABM), addressing issues related to the semi-infinite boundary. We adopt the finite difference method (FDM) for deriving the numerical solution by employing the L1 scheme to approximate the fractional derivative. To confirm the precision of this method, a source term is added to establish an exact solution for verification purposes. A comparative evaluation of the ABC versus the direct truncated boundary condition (DTBC) is conducted, with their effectiveness and soundness being visually scrutinized and assessed. This study investigates the impact of the motion of plates at different fluid flow velocities, focusing on the effects of dynamic elements influencing flow mechanisms and velocity. This research’s primary conclusion is that a higher fractional parameter correlates with the fluid flow. As relaxation parameters decrease, the delay effect intensifies and the fluid velocity decreases.

Construction of the absorbing boundary condition for the flow of Oldroyd-B fluid over a semi-infinite plate with magnetic effect

As a kind of non-Newtonian fluid, the Oldroyd-B fluid has widespread applications. To study the flow characteristics deeply is of great significance. In this paper, we consider a simple model of the Oldroyd-B fluid flow over a semi-infinite plate in a magnetic field. The governing equation is formulated, and the numerical solutions are obtained using the finite difference method. To deal with the semi-infinite region, the artificial boundary method is applied to construct the absorbing boundary condition (ABC) with the (inverse) z-transform, which converts the semi-infinite region to a finite one. To test the accuracy of the numerical scheme, a numerical example by introducing the source term is presented. Graphs show the rationality of the ABC by comparing the fluid flow velocity between the direct truncated boundary condition and the ABC. The effects of the amplitude, the frequency, the relaxation time parameter, the retardation time parameter, and the magnetic field on the magnitude and the cycle of flow velocity are investigated and discussed. The main findings are that the retardation time parameter promotes the velocity of the fluid flow, while the relaxation time and magnetic field hinder the fluid flow. When the relaxation time is equal to the retardation time, the Oldroyd-B fluid can approximate the Newtonian fluid. In addition, the oscillating cycle becomes shorter for a smaller relaxation time parameter or a larger magnetic field and frequency.

Time harmonic analysis of concrete gravity dam-reservoir systems by mixed High order-Wavenumber approach for general reservoir bottom condition

Recently, mixed HO-WN (i.e., High order-Wavenumber) method was introduced for dynamic analysis of concrete gravity dam-reservoir systems. This is formulated by FE-(FE-TE) approach (i.e., Finite Element-(Finite Element-Truncation Element)). In this technique, dam and reservoir are discretized by plane solid and fluid finite elements. Moreover, the mixed HO-WN (i.e., High order-Wavenumber) condition is imposed at the reservoir truncation boundary. This task is formulated by employing a truncation element at that boundary. The method is derived by combining the High-order and Wavenumber approaches. The present study is an extension of mixed HO-WN approach which allows it to be effectively utilized in the case of absorptive as well as fully reflective reservoir bottom condition. The formulation is initially explained in details and three alternatives are presented for this method. Subsequently, the response of an idealized triangular dam-reservoir system is studied for an absorptive reservoir condition due to horizontal and vertical ground motions. For each alternative, responses are evaluated from accuracy point of view by comparing them against exact solutions for three different values of normalized reservoir length (i.e., L/H ratio). It should be emphasized that a very low value of L/H = 0.2 is also considered for this investigation which makes it a very challenging test for any kind of truncation boundary condition. Moreover, it is worthwhile to stress that although time harmonic analysis is adopted in the present study, the main part of formulation is explained in the context of time domain. Therefore, the approach can easily be extended for transient type of analysis.

Insight into the significance of absorbing boundary condition for the flow mechanism analysis of fractional Maxwell fluid over a semi-infinite plate

Viscoelastic fluids have many applications in engineering, and studying the complex fluidity of viscoelastic fluids can improve their applicability. Based on the flow caused by the pressure or the moving plate with various velocities, the aim of this paper is to deeply study the significance of absorbing boundary condition for the flow mechanism analysis of the fractional Maxwell fluid, of which the constitutive relation is formulated by introducing the relaxation parameter and the fractional parameter with considering the memory characteristics. For treating the model in a semi-infinite boundary domain, the artificial boundary method is applied to transfer it to a problem in a bounded domain with absorbing boundary condition, which is solved numerically by the finite difference method combined with the L1 formula and verified by numerical examples. The difference of the flow characteristics is subject to the direct truncation boundary condition and the absorbing boundary condition is compared and the effectiveness and rationality are analyzed graphically, and the influences of the dynamic parameters on the velocity and the flow mechanism are also discussed. The main findings of this research are that the larger relaxation parameter plays a role in a stronger delay effect, a larger fractional parameter refers to the stronger memory characteristics of the delay effect, and the smaller Reynolds number leads to the larger viscous force, all of which lead to a slower flow process.

Experimental study on the wave-induced dynamic response and hydrodynamic characteristics of a submerged floating tunnel with elastically truncated boundary condition

Hydrodynamic load and motion response are the first considerations in the structural design of a submerged floating tunnel (SFT). Currently, most of the relevant studies have been based on a two-dimensional model test with a fixed or fully free boundary condition, which inhibits a deep investigation of the hydrodynamic characteristics with an elastic constraint. As a result, a series of difficulties exist in the structural design and analysis of an SFT. In this study, an SFT model with a one-degree-of-freedom vertical elastically truncated boundary condition was established to investigate the motion response and hydrodynamic characteristics of the tube under the wave action. The effect of several typical hydrodynamic parameters, such as the buoyancy-weight ratio, γ, the relative frequency, f/fN, the Keulegan–Carpenter (KC) number, the reduced velocity, Ur, the Reynolds number, Re, and the generalized Ursells number, on the motion characteristics of the tube, were selectively analyzed, and the reverse feedback mechanism from the tube's motion response to the hydrodynamic loads was confirmed. Finally, the critical hydrodynamic parameters corresponding to the maximum motion response at different values of γ were obtained, and a formula for calculating the hydrodynamic load parameters of the SFT in the motion state was established. The main conclusions of this study are as follows: (i) Under the wave action, the motion of the SFT shows an apparent nonlinearity, which is mainly caused by the intensive interaction between the tube and its surrounding water particles, as well as the nonlinearity of the wave. (ii) The relative displacement of the tube first increases and then decreases with increasing values of f/fN, Ur, KC number, Re, and the generalized Ursells number. (iii) γ is inversely proportional to the maximum relative displacement of the tube and the wave force on the tube in its motion direction. (iv) Under the motion boundary condition (as opposed to the fixed boundary condition), the peak frequency of the wave force on the SFT in its motion direction decreases and approaches the natural vibration frequency of the tube, whereas the wave force perpendicular to the motion direction increases. When the incident wave frequency is close to the natural vibration frequency of the tube, the tube resonates easily, leading to an increased wave force in the motion direction. (v) If the velocity in the Morison equation is substituted by the water particle velocity measured when the tube is at its equilibrium position, the inertia coefficient in the motion direction of the tube is linearly related to its displacement, whereas that in the direction perpendicular to the motion direction is logarithmically related to its displacement.

Application of Hagstrom-Warburton high-order truncation boundary condition on time harmonic analysis of concrete arch dam-reservoir systems

PurposeSubsequently, the response of idealized Morrow Point arch dam is studied due to stream, vertical and cross-stream ground motions for reservoir bottom/sidewalls conditions of both fully reflective and absorptive. For each combination, different orders of Hagstrom–Warburton (HW) condition are evaluated from accuracy point of view by comparing them against exact solutions. It should be emphasized that normalized length of reservoir near-field region is taken as a very low value of L/H = 0.2 during this process which makes it a very challenging test for any kind of truncation boundary condition.Design/methodology/approachIn present study, dynamic analysis of concrete arch dam-reservoir systems is formulated by FE-(FE-TE) approach [i.e. finite element-(finite element-truncation element)]. In this technique, dam and reservoir are discretized by solid and fluid finite elements. Moreover, the HW high-order condition imposed at the reservoir truncation boundary. This task is formulated by employing a truncation element at that boundary. It is emphasized that reservoir far-field is excluded from the discretized model. The formulation is initially explained in details.FindingsThe trend in gaining accuracy with increase in order of HW condition were basically the same for all three types of excitations under both full reflective and absorptive reservoir bottom/sidewalls conditions. The only exception was for cross-stream excitation response which was showing less accurate results near the first major peak for moderate orders of HW (e.g. O3-2) in comparison to what was observed for responses due to symmetric excitations (stream and vertical). This is mainly attributed to the selection of evanescent-type parameters of HW condition which is based on the first symmetric mode of reservoir. However, it is noted that error diminishes even for cross-stream excitation as order increases. High orders of HW condition, such as O5-5 considered herein, generate highly accurate responses for all three possible excitations under both types of full reflective and absorptive reservoir bottom/sidewalls conditions. It is such that responses are hardly distinguishable from corresponding exact responses. This reveals that proposed FE-(FE-TE) analysis technique based on HW condition is quite successful, and one may fully rely on that for accurate and efficient analysis of concrete arch dam-reservoir systems.Originality/valueDynamic analysis of concrete arch dam-reservoir system is formulated by new method. HW high-order condition is applied for a very low and challenging reservoir length. Different orders are evaluated against exact solution with excellent agreement. Generalized matrices of truncation element are derived for FEM programmers. The method is discussed for all types of excitation and reservoir base conditions.

Seismic performance of arch dams on non-homogeneous and discontinuous foundations (a case study: Karun 4 Dam)

The present study set out to investigate the nonlinear seismic response of the dam–reservoir–rock foundation system, taking into consideration the effects of change in the material properties of discontinuous foundation. To this end, it is important to provide the proper modeling of truncated boundary conditions at the far-end of rock foundation and reservoir fluid domain and to correctly apply the in situ stresses for rock foundation. The nonlinear seismic response of an arch dam mainly depends on the opening and sliding of the dam body’s contraction joints and foundation discontinuities, failure of the jointed rock and concrete materials, etc. In this paper, a time domain dynamic analysis of the 3D dam–reservoir–foundation interaction problem was performed by developing a nonlinear Finite Element program. The results of the analysis of Karun-4 Dam revealed the essential role of modeling discontinuities and boundary conditions of rock foundation under seismic excitation.

Berenger's Split-Field Perfectly Matching Layer (BS-PML) for Numerical Mode Matching (NMM) Solutions in Lossless Media

In this letter, a Berenger's split-field perfectly matching layer (BS-PML) as an absorbing boundary condition for solutions of the two-dimensional numerical mode matching (NMM) method has been proposed to model the field in media involving lossless layers instead of the traditional truncated boundary condition. The proposed method guarantees the proper absorbing boundary conditions for NMM models and then the accurate numerical solutions for the fields. The characteristic of a cylindrical wave incoming to PML has been derived. Furthermore, the strategy in setting parameters of the PML in an NMM model has been presented. Finally, the accuracy of the NMM solutions resulted from the BS-PML boundary condition and the traditional truncated boundary condition has been compared and discussed.

A scaled boundary finite element method applied to electrostatic problems

Abstract The scaled boundary finite element method (SBFEM) is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. In this paper, the SBFEM is firstly extended to solve electrostatic problems. Two new SBFE coordination systems are introduced. Based on Laplace equation of electrostatic field, the derivations (based on a new variational principle formulation) and solutions of SBFEM equations for both bounded domain and unbounded domain problems are expressed in details, the solution for the inclusion of prescribed potential along the side-faces of bounded domain is also presented in details, then the total charges on the side-faces can be semi-analytically solved, and a particular solution for the potential field in unbounded domain satisfying the constant external field is solved. The accuracy and efficiency of the method are illustrated by numerical examples with complicated field domains, potential singularities, inhomogeneous media and open boundaries. In comparison with analytic solution method and other numerical methods, the results show that the present method has strong ability to resolve singularity problems analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom. The method in electromagnetic field calculation can have broad application prospects.

The closed-form solution of vector finite element integral equations for three dimensional electromagnetic analysis

In this paper, a set of closed-form formulas for vector Finite Element Method (FEM) to analyze three dimensional electromagnetic problems is presented on the basis of Gaussian quadrature integration scheme. By analyzing the open region problems, the first-order Absorbing Boundary Condition (ABC) is considered as the truncation boundary condition and the equation is carried out in a closed-form. Based on the formulas, the hybrid Expanded Cholesky Method (ECM) and MultiFrontal algorithm (MF) is applied to solve finite element equations. Using the closed-form solution, the electromagnetic field of three dimensional targets can be studied sententiously and accurately. Simulation results show that the presented formulas are successfully and concise, which can be easily used to analyze three dimensional electromagnetic problems.

Electromagnetic Scattering from Randomly Rough Surfaces with Hybrid FEM/BIE

The hybrid finite element method (FEM) together with the boundary integral equation (BIE) is firstly applied to scattering from a conducting rough surface. The BIE is used as the truncation boundary condition for the special unbounded half space, whereas the FEM is used to solve the governing equation in the region surrounded by a rough surface and artificial boundary. Tapered wave incidence is employed to cancel the so-called “edge effect". A hybrid FEM/BIE formulation for generalized one-dimensional conducting rough surface scattering is presented, as well as examples that evaluate its validity compared to the method of moments. The bistatic scattering coefficients of a Gaussian rough surface are calculated for transverse-magnetic wave incidence. Conclusions are reached after analyzing the scattering patterns of rough surfaces with different rms heights and correlation lengths

The scaled boundary finite element method applied to electromagnetic field problems

Computation electromagnetic is an important research field of electromagnetic fields and microwave technology subjects. In this paper, the scaled boundary finite element method (SBFEM) is extended to solve one type of electromagnetic field problems-electrostatic field problems. Based on Laplace equation of electrostatic field, the derivations and solutions of SBFEM equations for both bounded and unbounded domain problems are expressed in details, and the solution for the inclusion of prescribed potential along the side-faces of bounded domain is also presented in details, then the total charges on the side-faces can be semi-analytically solved. The accuracy and efficiency of the method are illustrated by numerical examples of electromagnetic field problems with complicated field domains, potential singularities, inhomogeneous media and open boundaries. In comparison with analytic solution method and other numerical methods, the results show that the present method has strong ability to resolve potential field singularities analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom. The method in electromagnetic field calculation can have broad application prospects.

A novel procedure for determination of hydrodynamic pressure along upstream face of dams due to earthquakes

The estimation of hydrodynamic pressures along the upstream face of the dam is a critical parameter for the accurate analysis and design of a dam. The accurate estimation of the hydrodynamic pressures necessitates the consideration of interaction between the dam, the reservoir and the foundation. The interaction effects of the unbounded domain of the reservoir and the absorptive materials deposited at the reservoir bottom are frequency dependent which can be incorporated in a frequency domain procedure easily. But in a time domain procedure the frequency dependent interaction effects are lost. In a frequency domain solution, the excitation frequencies are extracted from the earthquake signal using a Fourier transformation, but do not give any information about how it varies with time. To overcome this, a short-time Fourier transform based formulation is presented in this paper to evaluate the hydrodynamic pressures in time domain to account for the frequency dependent interaction effects of the dam–reservoir system. Thus, the adequate accuracy in the determination of hydrodynamic pressure under earthquake excitation is ensured with the proposed truncation boundary condition.

A combined finite and infinite element approach for modeling spherically symmetric transient subsurface flow

This paper presents a finite element-infinite element coupling approach for modeling a spherically symmetric transient flow problem in a porous medium of infinite extent. A finite element model is used to examine the flow potential distribution in a truncated bounded region close to the spherical cavity. In order to give an appropriate artificial boundary condition at the truncated boundary, a transient infinite element, that is developed to describe transient flow in the exterior unbounded domain, is coupled with the finite element model. The coupling procedure of the finite and infinite elements at their interface is described by means of the boundary integro-differential equation rather than through a matrix approach. Consequently, a Neumann boundary condition can be applied at the truncated boundary to ensure the C 1-continuity of the solution at the truncated boundary. Numerical analyses indicate that the proposed finite element-infinite element coupling approach can generate a correct artificial truncated boundary condition to the finite element model for the unbounded flow transport problem.

Radiation admittance of an open‐ended parallel plate metallic waveguide with exterior planar dielectric coating

A methodology is presented for the calculation of the radiation admittance of an open‐ended, homogeneously filled, parallel plate metallic waveguide with exterior planar dielectric coating, operated at the fundamental (transverse electromagnetic wave) mode. The methodology relies on the definition of an auxiliary electromagnetic boundary value problem that is simple to solve using an integral equation approach. In order to provide for broadband calculation of the radiation admittance of the open‐ended waveguide from DC to several tens of gigahertz, a loop‐tree decomposition scheme is used in the method of moments solution of the resulting integral equation. A rational function‐fitting process with passivity enforcement is used to generate a broadband, passive macromodel for the calculated radiation impedance. Such a macromodel is suitable for use as a local truncation boundary condition in the finite difference/finite element modeling of electromagnetic interactions in multilayered signal and power distribution networks encountered in printed circuit boards and packages.

Balancing Boundary and Discretization Errors Using an Iterative Absorbing Boundary Condition

An iterative scheme is described for solving unbounded electromagnetic scattering problems using the finite-element method. The scheme starts with elements of a low polynomial order and progressively increases the order and improves the quality of the truncation boundary condition, so that the discretization error and boundary error are roughly similar at each stage. For the examples in this paper, errors in the radar cross section (RCS) are estimated. The discretization error estimate is obtained by the approximate solution of a problem in which all of the finite elements are one order higher. Results for a metallic cube, two dielectric spheres, a conducting ogive, and a conducting flat plate show that the scheme can obtain accurate bistatic RCS values very efficiently.

An exact truncation boundary condition for incompressible fluid domains in dam–reservoir interaction analysis

In this paper, an exact truncation boundary condition is derived and implemented in a finite element code for the analysis of dam–reservoir interaction for incompressible, inviscid and unbounded fluid domains. The reservoir domain is divided into two regions in the derivation of the truncation boundary condition. These are the near field having a complex geometry and the far field with a uniform cross-section. The proposed boundary condition is obtained using the analytical solution for the far field and used as a truncation boundary condition at the truncation surface for the near field including the dam–reservoir system. This method has the advantage of geometrical flexibility in the near field and of being a semi-analytical approach giving close results to the exact solutions for the cases analyzed. In addition, accurate results are also obtained when the reservoir domain is truncated very close to the dam–reservoir interface.

A non-reflecting boundary condition for the finite element modeling of infinite reservoir with layered sediment

The design of seismic resistant concrete gravity dam necessitates accurate determination of hydrodynamic pressure developed in the adjacent reservoir. The hydrodynamic pressure developed on structure is dependent on the physical characteristics of the boundaries surrounding the reservoir including reservoir bottom. The sedimentary material in the reservoir bottom absorbs energy at the bottom, which will affect the hydrodynamic pressure at the upstream face of the dam. The fundamental parameter characterizing the effect of absorption of hydrodynamic pressure waves at the reservoir bottom due to sediment is the reflection coefficient. The wave reflection coefficient is determined from parameters based on sediment layer thickness, its material properties and excitation frequencies. An analytical or a closed-form solution cannot account for the arbitrary geometry of the dam or reservoir bed profile. This problem can be efficiently tackled with finite element technique. The need for an accurate truncation boundary is felt to reduce the computational domain of the unbounded reservoir system. An efficient truncation boundary condition (TBC) which accounts for the reservoir bottom effect is proposed for the finite element analysis of infinite reservoir. The results show the efficiency of the proposed truncation boundary condition.

An accurate waveguide port boundary condition for the time-domain finite-element method

This paper presents an accurate time-domain waveguide port boundary condition (WPBC) that can be used as a mesh truncation boundary condition to terminate a finite-element mesh. The formulation for deriving the boundary condition is based on multimodal expansion, and thus is related to the modal absorbing boundary conditions developed for the finite-difference time-domain method. The boundary condition is formulated for the general three-dimensional analysis of waveguides with an arbitrary cross section. All waveguide modes, including transverse electromagnetic, transverse electric, and transverse magnetic modes, are accounted for in the boundary condition in a unified formulation. Both propagating and evanescent modes can be properly absorbed. The proposed formulation is validated by numerical examples, and the accuracy and stability of the algorithm are investigated.

A novel far-boundary condition for the finite element analysis of infinite reservoir

The present paper deals with the finite element analysis of the reservoir of infinite extent using a novel far-boundary condition. The equations of motion are expressed in terms of the pressure only assuming water as inviscid and incompressible. The truncation boundary condition is developed numerically from the classical wave equation. Comparative studies show that the proposed far-boundary condition is numerically efficient and accurate over the existing ones, available in the literature. The effect of the geometry of the reservoir bed and the adjacent structure on the development hydrodynamic pressure has been studied. The results show that the geometry of the reservoir bed and as well as the adjacent structure has considerable effect on the development of hydrodynamic pressure at the dam–reservoir interface.