Non-reflecting Boundary Conditions Research Articles

Wave diffraction by floating bodies in water of finite depth using an exact DtN boundary condition

For the exact Non-Reflecting Boundary Conditions (NRBCs) or so-called Dirichlet-to-Neumann (DtN) boundary conditions, the artificial boundary is usually chosen as a spherical or ellipsoidal surface by which the computational domain is enclosed fully with finite volume. For the wave diffraction problem in finite-depth water unbounded horizontally, the free water boundary condition and wall boundary condition should be satisfied on still water level (SWL) and seabed, respectively. In this paper, the artificial boundary for DtN condition is chosen as only a lateral cylindrical surface by which it is impossible to envelope the objects and fluid domain fully, but possible by adding still water surface and seabed. A novel DtN condition on the artificial boundary is suggested and then Boundary Element Method (BEM) implementation is described to solve the three-dimensional wave diffraction problems by arbitrary-shaped floating bodies in water of finite depth. Upon successful verification for the case of a bottom-truncated cylinder, the scheme is applied to a single chamfer box and an array of chamfer boxes.

Effect of Soil Properties and Input Motion on Site Amplification Using Validated Nonlinear Soil Model

Three-dimensional (3-D) nonlinear site response analyses are conducted using finite element models of actual soil profiles from ten nuclear power plant (NPP) sites in the United States to investigate the effects of soil properties and input motions on site amplification. The modeling approach developed in this study combines several novel elements, such as 3-D analysis (including vertical motions), nonlinear inelastic behavior of soil (strain-dependent shear modulus reduction and hysteretic damping), formulation of nonreflecting boundary conditions at the base, and generation of realistic outcrop ground motions for specific sites. All these elements of the modeling approach are first validated using actual data from five earthquakes at three downhole array stations recorded in the Kiban-Kyoshin network (KiK-net), Japan. The same approach is then used to develop site models of ten NPP sites in the United States and corresponding ground motions that are spectrally matched to the site hazard spectra. Eight sets of three-component input motions are used in the study and are categorized on the basis of presence or absence of a near-field pulse in the seed ground motions used for spectral matching. It is found that all sites retain a definite site amplification function regardless of the input motion, provided that the seed motion is spectrally matched to the site hazard spectra. The magnitude of site amplification and frequencies at which they occur depend upon soil properties, particularly the shear wave velocity profile and the constitutive relationship (strain-dependent shear modulus reduction and hysteretic damping) of soil. Amplification of spectral acceleration in the vertical direction (up-down motion) is found to be just as much as, if not more than, the amplification in the horizontal direction. Peak shear strain is found to be about 20% larger for near-field motions compared to far-field motions whereas maximum horizontal site amplification for far-field motions is found to be consistently larger than that of near-field motions, even though the differences between the two remain within the scatter resulting from individual ground motions.

An approach for permeable boundary conditions in SPH

Smoothed Particle Hydrodynamics (SPH) has intrinsic advantages over the Finite Volume (FV) method for predicting multiphase flows, since the need for fluid phase interface reconstructions is avoided due to the natural advection of those interfaces based on the Lagrangian description of the control volumes. However, permeable boundary conditions are less straightforward to handle with SPH, even though domain inlets and outlets are needed in most engineering problems. In this work we propose a new framework for permeable boundary conditions for the Weakly Compressible SPH based on ghost particles. The major characteristics of our approach are the application of Navier Stokes Characteristic Boundary Conditions (NSCBC) and a particle mass generation and removal procedure at the boundary. This approach enables the application of non-reflective boundary conditions and redundantizes the need for advecting ghost particles from an external buffer zone to the internal domain. Additionally, the particle mass variation can exactly be balanced with the mass flux defined at the boundary. Furthermore, we are proposing a novel approach to set the ghost particle state based on an extrapolation of the state of particles located in the internal domain and on the domain boundary. We validate the proposed framework for permeable boundaries with one- and two-dimensional test cases. The major findings are as follows. Due to the particle mass generation and removal at the domain boundary, spurious oscillations are introduced due to a non-optimal particle distribution. Those oscillations are reduced in our work to an acceptable level (1×10−5 to 1×10−4 relative to the values of the fluid dynamics variables) using existing stabilization schemes as well as the aforementioned novel methods for setting ghost particle states and particle masses at the permeable boundaries. We observe a very good agreement between the solutions obtained with the presented approach and analytical as well as numerical solutions obtained with the FV method. The proposed approach for setting the ghost particle states is indispensable for achieving a perfect balance between particle mass variation and mass flux across the permeable interface as well as the steady state and the temporal accuracy of the solution. Additionally, the pressure reflectivity at the permeable boundary is found to be in a very good agreement with the theory of the NSCBC method. Furthermore, for a SPH-FV coupled simulation we could demonstrate an excellent agreement compared to the FV method as well as the capability of our approach to handle significant flow unsteadiness and gradients at the permeable coupling interfaces without the introduction of spurious flow behavior.

A numerically exact nonreflecting boundary condition applied to the acoustic Helmholtz equation

When modeling wave propagation, truncation of the computational domain to a manageable size requires nonreflecting boundaries. To construct such a boundary condition on one side of a rectangular domain for a finite-difference discretization of the acoustic wave equation in the frequency domain, the domain is extended on that one side to infinity. Constant extrapolation in the direction perpendicular to the boundary provides the material properties in the exterior. Domain decomposition can split the enlarged domain into the original one and its exterior. Because the boundary-value problem for the latter is translation-invariant, the boundary Green functions obey a quadratic matrix equation. Selection of the solvent that corresponds to the outgoing waves provides the input for the remaining problem in the interior. The result is a numerically exact nonreflecting boundary condition on one side of the domain. When two nonreflecting sides have a common corner, the translation invariance is lost. Treating each side independently in combination with a classic absorbing condition in the other direction restores the translation invariance and enables application of the method at the expense of numerical exactness. Solving the quadratic matrix equation with Newton’s method turns out to be more costly than solving the Helmholtz equation and may select unwanted incoming waves. A proposed direct method has a much lower cost and selects the correct branch. A test on a 2D acoustic marine seismic problem with a free surface at the top, a classic second-order Higdon condition at the bottom, and numerically exact boundaries at the two lateral sides demonstrates the capability of the method. Numerically exact boundaries on each side, each computed independently with a free-surface or Higdon condition, provide even better results.

Traction open boundary condition for incompressible, turbulent, single- or multi-phase flows, and surface wave simulations

In simulations, artificial boundaries need to be introduced to limit the size of computational domains and thereby lower computational cost. At these boundaries, flow variables need to be calculated in a way that will not induce any perturbation of the interior solution, which poses a great challenge in incompressible flows. In this paper, we demonstrate the potential of a new traction open boundary condition to address the classical problems encountered in simulations with open boundary conditions: backflow instability, wave reflections, and confinement caused by the proximity of the outlet. This novel boundary treatment, based on a Lagrangian estimation of the traction in the outlet section coupled to a stabilization term, is shown to provide accuracy and stability for turbulent, single- or multi-phase flows, test cases. Using a simulation of surface gravity waves, we show that if special care is given to the computation of the estimated traction, it is possible to get a fully non-reflective open boundary condition.

Accurate Full-Vectorial Finite Element Method Combined with Exact Non-Reflecting Boundary Condition for Computing Guided Waves in Optical Fibers

Abstract The vector electromagnetic problem for eigenwaves of optical fibers, originally formulated on the whole plane, is equivalently reduced to a linear parametric eigenvalue problem posed in a circle, convenient for numerical solution. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Asymptotic properties of the dispersion curves and their smoothness are investigated for the new formulation of the problem. A numerical method based on finite element approximations combined with an exact non-reflecting boundary condition is developed. Error estimates for approximating eigenvalues and eigenfunctions are derived.

Numerical strategy to perform direct numerical simulations of hypersonic shock/boundary-layer interaction in chemical nonequilibrium

Literature regarding hypersonic shock/boundary-layer interaction is mostly restricted to calorically perfect gases, even though this condition is far from reality when temperature rises. High-temperature effects alter physical and transport properties of the fluid, due to vibrational excitation and gas dissociation, and chemical reactions must be considered in order to compute the flow field. In this work, a code for hypersonic aerodynamics with reactions using parallel machines (CHARLIE) is described and a numerical methodology is developed to perform direct numerical simulations of shock/boundary-layer interaction in chemical nonequilibrium. The numerical scheme and the characterization of non-reflecting boundary conditions are addressed. Results show that the flow properties differ considerably if chemical reactions are taken into account. A direct numerical simulation of a shock interacting with a turbulent boundary layer in the hypersonic regime with high-temperature effects is also presented for the first time.

Smoothed Particle Hydrodynamics Simulations of Turbulent Flow in Curved Pipes With Different Geometries: A Comparison With Experiments

Abstract The swirling secondary flow in curved pipes is studied in three-space dimensions using a weakly compressible smoothed particle hydrodynamics (WCSPH) formulation coupled to new nonreflecting outflow boundary conditions. A large eddy simulation (LES) model for turbulence is benchmarked with existing experimental data. After validation of the present model against experimental results for a 90 deg pipe bend, a detailed numerical study aimed at reproducing experimental flow measurements for a wide range of Reynolds numbers has been performed for different pipe geometries, including U pipe bends, S-shaped pipes, and helically coiled pipes. In all cases, the SPH calculated behavior shows reasonably good agreement with the measurements across and downstream the bend in terms of streamwise velocity profiles and cross-sectional contours. Maximum mean-root-square deviations from the experimentally obtained profiles are always less than ∼1.8%. This combined with the very good matching between the SPH and the experimental cross-sectional contours shows the uprising capabilities of the present scheme for handling engineering applications with streamline curvature, such as flows in bends and manifolds.

A Low Dissipative and Stable Cell-Centered Finite Volume Method with the Simultaneous Approximation Term for Compressible Turbulent Flows

A simultaneous-approximation term is a non-reflecting boundary condition that is usually accompanied by summation-by-parts schemes for provable time stability. While a high-order convective flux based on reconstruction is often employed in a finite-volume method for compressible turbulent flow, finite-volume methods with the summation-by-parts property involve either equally weighted averaging or the second-order central flux for convective fluxes. In the present study, a cell-centered finite-volume method for compressible Naiver–Stokes equations was developed by combining a simultaneous-approximation term based on extrapolation and a low-dissipative discretization method without the summation-by-parts property. Direct numerical simulations and a large eddy simulation show that the resultant combination leads to comparable non-reflecting performance to that of the summation-by-parts scheme combined with the simultaneous-approximation term reported in the literature. Furthermore, a characteristic boundary condition was implemented for the present method, and its performance was compared with that of the simultaneous-approximation term for a direct numerical simulation and a large eddy simulation to show that the simultaneous-approximation term better maintained the average target pressure at the compressible flow outlet, which is useful for turbomachinery and aerodynamic applications, while the characteristic boundary condition better preserved the flow field near the outlet.

Efficiency of using adaptive artificial boundary conditions at computer simulation of contrast spatio‐temporal laser‐induced structures in a semiconductor

Many problems of modern laser physics are governed by equations or sets of equations in an unbounded domain. For solving these problems using the computer simulation, it is necessary to introduce the bounded domain, which size should be extended significantly to avoid the spurious wave reflection from the domain boundaries. Alternatively, the artificial (non-reflective or transparent) boundary conditions should be stated. This approach is also effective for enhancing computation performance at the numerical solution of the nonlinear partial differential equations (PDEs). In the current paper, we investigate the laser pulse propagation in a semiconductor, governed by the Schrödinger equation, under the appearance of spatio-temporal contrast structures of semiconductor characteristics. Their evolution is described by a set of PDEs. The optical pulse is partly reflected from the boundaries of these structures. Consequently, even a little reflection of the optical pulse from the artificial boundaries can essentially distort the numerical solution. Thus, these artificial boundary conditions must possess a high quality to minimize their reflection coefficients. With this aim, we propose the method for constructing adaptive artificial boundary conditions and discuss their advantages.

Amplitudes of characteristic waves and non-reflecting outflow conditions for high-fidelity numerical simulations

Non-reflecting boundary conditions are crucial for aeroacoustic simulations to prevent non-physical reflections at boundaries from contaminating the interior acoustic solutions. Various non-reflecting outflow conditions, mainly based on the characteristic theory, have been developed. By using a locally one-dimensional inviscid approximation, the characteristic wave amplitudes are identified as the projection of flow solutions into the eigenspace of the flux Jacobian matrix. When the weighted transverse terms are included, the definitions are essentially changed and produce different levels of non-physical reflections. This study aims to resolve the ambiguity in defining the amplitudes of characteristic waves. A new analysis in the frequency and wave number domain is proposed to identify the characteristic wave amplitudes in multidimensional space, and a second order in terms of the lateral wave number non-reflecting outflow condition that is more physically reasonable is derived. Numerical experiments are carried out to evaluate the performances of these non-reflecting conditions.

Study on the influence of structure factors of diesel engine exhaust purification muffler on transmission loss in different frequency bands

The main purpose of exhaust mufflers design is to improve the TL (transmission loss) of the target frequency band. Therefore, this paper first proposed an exhaust muffler model with foam ceramics as sound-absorbing material for a diesel engine. The finite element AML (Automatic Matched Layer) method is used to simulate the non-reflective boundary conditions and calculate the TL. In addition, the validity and reliability of the solution model are verified by comparing the experimental value and simulation value of the TL. Compared with the prototype muffler, the TL of the exhaust purification muffler has increased by an average of 6.96 dB in the frequency range of 20–4500 Hz. Finally, the influence law of structure factors m (expansion ratio), n (cavity-length ratio) and p (length-diameter ratio) on TL in different frequency bands shows that: when changing the structure factors to increase the TL value of the exhaust purification muffler, the expansion ratio should be prioritized in the low-medium frequency bands, the cavity-length ratio should be prioritized in the medium–high frequency bands, the length-diameter ratio should be prioritized in the high frequency bands.

An analytical model of full-field displacement and strain induced by amplitude-modulated focused ultrasound in harmonic motion imaging

The majority of disease processes involves changes in the micro-structure of the affected tissue, which can translate to changes in the mechanical properties of the corresponding tissue. Harmonic motion imaging (HMI) is an elasticity imaging technique that allows the study of the mechanical parameters of tissue by detecting the tissue response by a harmonic motion field, which is generated by oscillatory acoustic radiation force. HMI has been demonstrated in tumor detection and characterization as well as monitoring of ablation procedures. In this study, an analytical HMI model is demonstrated and compared with a finite element model (FEM), allowing rapid and accurate computation of the displacement, strain, and shear wave velocity (SWV) at any location in a homogenous linear elastic material. Average absolute differences between the analytical model and the FEM were respectively 1.2% for the displacements and 0.5% for the strains for 41 940 force voxels at 0.22 s per displacement evaluation. A convergence study showed that the average difference could be further decreased to 1.0% and 0.15% for the displacements and strains, respectively, if force resolution is increased. SWV fields, as calculated with the FEM and the analytical model, have regional differences in velocities up to 0.57 m s−1 with an average absolute difference of 0.11 ± 0.07 m s−1, primarily due to imperfections in the non-reflecting FEM boundary conditions. The apparent SWV differed from the commonly used plane-wave approximation by up to 1.2 m s−1 due to near and intermediate field effects. Maximum displacement amplitudes for a model with an inclusion stabilize within 10% of the homogenous model at an inclusion radius of 10 mm while the maximum strain reacts faster, stabilizing at an inclusion radius of 3 mm. In conclusion, an analytical model for HMI stiffness estimation is presented in this paper. The analytical model has advantages over FEM as the full-field displacements do not need to be calculated to evaluate the model at a single measurement point. This advantage, together with the computational speed, makes the analytical model useful for real-time imaging applications. However, the analytical model was found to have restrictive assumptions on tissue homogeneity and infinite dimensions, while the FEM approaches were shown adaptable to variable geometry and non-homogenous properties.

Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

AbstractWe develop method of characteristics schemes based on explicit Runge–Kutta and pseudo‐Runge–Kutta third‐ and fourth‐order solvers along the characteristics. Schemes based on Runge–Kutta solvers are found to be strongly unstable for certain physics‐motivated models. In contrast, schemes based on pseudo‐Runge–Kutta solvers are shown to be only weakly unstable for periodic boundary conditions and essentially stable for the more physically relevant nonreflecting boundary conditions. Our implementation of nonreflecting boundary conditions does not rely on interpolation.

Stable Perfectly Matched Layers with Lorentz transformation for the convected Helmholtz equation

Perfectly Matched Layers (PMLs) appear as a popular alternative to non-reflecting boundary conditions for wave-type problems. The core idea is to extend the computational domain by a fictitious layer with specific absorption properties such that the wave amplitude decays significantly and does not produce back reflections. In the context of convected acoustics, it is well-known that PMLs are exposed to stability issues in the frequency and time domain. It is caused by a mismatch between the phase velocity on which the PML acts, and the group velocity which carries the energy of the wave. The objective of this study is to take advantage of the Lorentz transformation in order to design stable perfectly matched layers for generally shaped convex domains in a uniform mean flow of arbitrary orientation. We aim at presenting a pedagogical approach to tackle the stability issue. The robustness of the approach is also demonstrated through several two-dimensional high-order finite element simulations of increasing complexity.

EFFECT OF GEOMETRY OF STENOSIS ON THE COMPUTATION OF FLOW CHARACTERISTICS OF UNSTEADY BLOOD FLOW IN A COLLAPSIBLE VESSEL

This paper discusses the effect of different geometric representations of stenosis on the numerical solution of one-dimensional unsteady blood flow in stenotic blood vessel (or stenosis) taking into account fluid-structure interaction. In the formulation, a collapsible pressure-area constitutive relation is added to the coupled mass and momentum equations to allow for the interaction between the cross sectional area, volumetric flow rate and pressure of the flow and hence the prevalence of the one-dimensional fluid-structure interaction. The formulation is stabilized by employing Streamline-Upwind Petrov-Galerkin scheme. Non-reflecting boundary conditions are imposed based on the method of characteristics. Flow characteristics and the geometrical effects of the stenosis are then discussed. Numerical results show that stenosis with irregular shape is more prone to collapse as compared to the smooth one for a given baseline conditions. This study, thus, highlights the importance of representing the shape of the stenosis as close as possible as it might give information otherwise missing in the simplistic smooth representation of the stenosis.

On Time-Domain NRBC for Maxwell’s Equations and Its Application in Accurate Simulation of Electromagnetic Invisibility Cloaks

In this paper, we present analytic formulas of the temporal convolution kernel functions involved in the time-domain non-reflecting boundary condition (NRBC) for the electromagnetic scattering problems. Such exact formulas themselves lead to accurate and efficient algorithms for computing the NRBC for domain reduction of the time-domain Maxwell’s system in $${\mathbb {R}}^3$$ . A second purpose of this paper is to derive a new time-domain model for the electromagnetic invisibility cloak. Different from the existing models, it contains only one unknown field and the seemingly complicated convolutions can be computed as efficiently as the temporal convolutions in the NRBC. The governing equation in the cloaking layer is valid for general geometry, e.g., a spherical or polygonal layer. Here, we aim at simulating the spherical invisibility cloak. We take the advantage of radially stratified dispersive media and special geometry, and develop an efficient vector spherical harmonic-spectral-element method for its accurate simulation. Compared with limited results on FDTD simulation, the proposed method is optimal in both accuracy and computational cost. Indeed, the saving in computational time is significant.

ANALYSIS OF THE CAPABILITIES OF THE LOGOS SOFTWARE PACKAGE FOR CALCULATING THE SEISMIC IMPACT ON A STRUCTURE

The possibilities of the LOGOS software package for calculating seismic vibrations of buried structures, considering the contact interaction with the ground and the gravity field, are investigated and expanded. To reduce computational costs, the LOGOS software package includes a method for modeling non-reflective boundary conditions that was developed earlier and implemented in the “Dynamics-2” software package, which allows reducing the size of the computational domain. The results of numerical simulation in the LOGOS of shear wave propagation in an elastic bounded subdomain of a continuous medium are presented, demonstrating the effectiveness of non-reflecting boundary conditions for a three-dimensional formulation. The dynamic relaxation technique used in the dynamic strength module of the LOGOS software package for calculating the initial static stresses from the action of the gravity field with the subsequent solution of the non-stationary problem is investigated. Numerical estimates of the expected accuracy of the dynamic solution are obtained, depending on the specified accuracy of the static calculation. Thus, the relative error in the change in kinetic energy when using the dynamic relaxation algorithm 10–4 gives a relative error of 10–3 in velocities and 10–2 in stresses, reducing the specified calculation error by 2 orders of magnitude reduces the relative error in calculating velocities by 3 times, stresses – by 5 times. The calculation of the initial stress-strain state of the building-ground system from the action of the gravity field is compared using the dynamic relaxation procedure and using the stationing procedure implemented in the “Dynamics-2” software package. In all problems in the three-dimensional formulation, 8-node hexahedra with one-point integration are used, in the two-dimensional formulation – equivalent 4-node finite elements, for the integration of the defining system of equations, an explicit “cross” scheme is used. Between the subdomains, variants of contact with gluing and contact with friction are implemented. The capabilities of LOGOS for conducting multiprocessor calculations allowed us to make estimates of the convergence of the problem under consideration based on a series of computational experiments.

Three dimensional seismic deformation-shear strain-swelling performance of America-California Oroville Earth-Fill Dam

Structural design of the vertical displacements and shear strains in the earth fill (EF) dams has great importance in the structural engineering problems. Moreover, far fault earthquakes have significant seismic effects on seismic damage performance of EF dams like the near fault earthquakes. For this reason, three dimensional (3D) earthquake damage performance of Oroville dam is assessed considering different far-fault ground motions in this study. Oroville Dam was built in United States of America-California and its height is 234.7 m (770 ft.). 3D model of Oroville dam is modelled using FLAC3D software based on finite difference approach. In order to represent interaction condition between discrete surfaces, special interface elements are used between dam body and foundation. Non-reflecting seismic boundary conditions (free field and quiet) are defined to the main surfaces of the dam for the nonlinear seismic analyses. 6 different far-fault ground motions are taken into account for the full reservoir condition of Oroville dam. According to nonlinear seismic analysis results, the effects of far-fault ground motions on the nonlinear seismic settlement and shear strain behaviour of Oroville EF dam are determined and evaluated in detail. It is clearly seen that far-fault earthquakes have very significant seismic effects on the settlement-shear strain behaviour of EF dams and these earthquakes create vital important seismic damages on the swelling behaviour of dam body surface. Moreover, it is proposed that far-fault ground motions should not be ignored while modelling EF dams.

High-Order WENO Schemes with an Immersed Boundary Method for Shallow Water Equations on the Tsunami Mitigation with Configurations of Cylinder Array

In this paper, we investigated the flow around vertical circular cylinders with the numerical simulation of shallow water equations. Since tsunami tidal waves have large potential energy due to height difference, the man-made structures near coast are heavily damaged during the propagation of waves. For the damage mitigation of natural disaster, a careful configuration of such obstructive structures is worthy of attention, and it is of great practical importance to provide numerical simulation based on the ideal models of natural or artificial structures in a coastal area. The simplified numerical models are listed as the single large cylinder and four small cylinders of double inline, double staggered, and single four arrangement with fixing the same blockage ratio. Among various up-to-date schemes, the fifth-order WENO scheme is selected with a ghost-cell immersed boundary method, which enabled the easy detection of the arbitrary body shapes emerged in background Cartesian grids. In addition, we proposed the non-reflecting characteristic boundary condition based on the characteristic compatibility equations derived in shallow water equations to suppress the unexpected reflection waves at the open boundary. The analysis on a benchmark problem on solitary wave reflection led to an expected validation with the present numerical model. Finally, reflection and diffraction of a tsunami wave are analyzed against a single and four cylinders of the same blockage density, and it is believed to reduce 20.80% points for the best case of single four cylinders in the view of dissipation ratio defined by the kinematic power.