A GPU‐accelerated nodal discontinuous Galerkin method with high‐order absorbing boundary conditions and corner/edge compatibility

Discontinuous Galerkin finite element schemes exhibit attractive features for accurate large‐scale wave‐propagation simulations on modern parallel architectures. For many applications, these schemes must be coupled with nonreflective boundary treatments to limit the size of the computational domain without losing accuracy or computational efficiency, which remains a challenging task. In this paper, we present a combination of a nodal discontinuous Galerkin method with high‐order absorbing boundary conditions for cuboidal computational domains. Compatibility conditions are derived for high‐order absorbing boundary conditions intersecting at the edges and the corners of a cuboidal domain. We propose a GPU implementation of the computational procedure, which results in a multidimensional solver with equations to be solved on 0D, 1D, 2D, and 3D spatial regions. Numerical results demonstrate both the accuracy and the computational efficiency of our approach.     

三维波流耦合水槽模拟模型

水中结构物通常会受到波浪和水流的联合作用,为了得到作用在结构上的波流荷载,该文基于Flow-3D软件建立了三维波流水槽的模拟模型。为了准确模拟波浪和水流的相互作用,避免将造波边界和水流边界同时叠加在入口边界处,采用质量源造波法来实现内造波。利用Flow-3D的二次开发功能在水槽两端设置海绵层消波区,并结合Sommerfeld辐射边界条件来实现水槽的通流消波。对波浪的传播、波流相互作用试验的流场以及波流和柱体结构相互作用试验的流场进行数值模拟,结果表明:所建立的三维波流水槽模型可以准确地模拟波浪和水流的相互作用以及波流对结构的联合作用,并有效地进行消波,避免了二次反射,可用于波浪和水流联合作用下水中结构物波流荷载的研究。     

Role of 0D peripheral vasculature model in fluid–structure interaction modeling of aneurysms

Patient-specific simulations based on medical images such as CT and MRI offer information on the hemodynamic and wall tissue stress in patient-specific aneurysm configurations. These are considered important in predicting the rupture risk for individual aneurysms but are not possible to measure directly. In this paper, fluid–structure interaction (FSI) analyses of a cerebral aneurysm at the middle cerebral artery (MCA) bifurcation are presented. A 0D structural recursive tree model of the peripheral vasculature is incorporated and its impedance is coupled with the 3D FSI model to compute the outflow through the two branches accurately. The results are compared with FSI simulation with prescribed pressure variation at the outlets. The comparison shows that the pressure at the two outlets are nearly identical even with the peripheral vasculature model and the flow division to the two branches is nearly the same as what we see in the simulation without the peripheral vasculature model. This suggests that the role of the peripheral vasculature in FSI modeling of the MCA aneurysm is not significant.     

Transmitting boundary conditions for 1D peridynamics

The peridynamic theory reformulates the equations of continuum mechanics in terms of integro‐differential equations instead of partial differential equations. It is not straightforward to apply the available artificial boundary conditions for continua to peridynamic modeling. We therefore develop peridynamic transmitting boundary conditions (PTBCs) for 1D wave propagation. Differently from the previous method where the matching boundary condition is constructed for only one boundary material point, the PTBCs are established by considering the interaction and exchange of information between a group of boundary material points and another group of inner material points. The motion of the boundary material points is recursively constructed in terms of their locations and is determined through matching the peridynamic dispersion relation. The effectiveness of the PTBCs is examined by reflection analyses, numerical tests, and numerical convergent conditions. Furthermore, two‐way interfacial conditions are proposed. The PTBCs are then applied to simulations of wave propagation in a bar with a defect, a composite bar with interfaces, and a domain with a seismic source. All the analyses and applications demonstrate that the PTBCs can effectively remove undesired numerical reflections at artificial boundaries. The methodology may be applied to modeling of wave propagation by other nonlocal theories. Copyright © 2016 John Wiley & Sons, Ltd.     

Development of a 3D numerical wave tank for modeling tsunami generation by underwater landslides

A three-dimensional (3D) numerical wave tank (NWT) solving fully nonlinear potential flow theory, with a higher-order boundary element method (BEM), is modified to simulate tsunami generation by underwater landslides. New features are added to the NWT to model underwater landslide geometry and motion and specify corresponding boundary conditions in the BEM model. In particular, a new snake absorbing piston boundary condition is implemented to remove reflection from the onshore and offshore boundaries of the NWT. Model results are favorably compared to recent laboratory experiments. Sensitivity analyses of numerical results to the width and length of the discretization are conducted, to determine optimal numerical parameters. The effect of landslide width on tsunami generated is estimated. Results show that the two-dimensional approximation is applicable when the ratio of landslide width over landslide length is greater than 2. Numerical accuracy is examined and found to be excellent in all cases.     

A stable Galerkin reduced order model for coupled fluid–structure interaction problems

A stable reduced order model (ROM) of a linear fluid–structure interaction (FSI) problem involving linearized compressible inviscid flow over a flat linear von Kármán plate is developed. Separate stable ROMs for each of the fluid and the structure equations are derived. Both ROMs are built using the ‘continuous’ Galerkin projection approach, in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. The mode shapes for the structure ROM are the eigenmodes of the governing (linear) plate equation. The fluid ROM basis is constructed via the proper orthogonal decomposition. For the linearized compressible Euler fluid equations, a symmetry transformation is required to obtain a stable formulation of the Galerkin projection step in the model reduction procedure. Stability of the Galerkin projection of the structure model in the standard L² inner product is shown. The fluid and structure ROMs are coupled through solid wall boundary conditions at the interface (plate) boundary. An a priori energy linear stability analysis of the coupled fluid/structure system is performed. It is shown that, under some physical assumptions about the flow field, the FSI ROM is linearly stable a priori if a stabilization term is added to the fluid pressure loading on the plate. The stability of the coupled ROM is studied in the context of a test problem of inviscid, supersonic flow past a thin, square, elastic rectangular panel that will undergo flutter once the non‐dimensional pressure parameter exceeds a certain threshold. This a posteriori stability analysis reveals that the FSI ROM can be numerically stable even without the addition of the aforementioned stabilization term. Moreover, the ROM constructed for this problem properly predicts the maintenance of stability below the flutter boundary and gives a reasonable prediction for the instability growth rate above the flutter boundary. Copyright © 2013 John Wiley & Sons, Ltd.     

弹性介质波传问题的吸收边界条件

本文通过合成一阶偏微分算子，给出了弹性介质中波传问题的吸收边界条件；为检验其吸收能力，利用弹性波的势函数理论导出了Ｐ波和Ｓ波在吸收边界处的反射系数公式。文中给出的吸收条件形式简单，且算例表明吸收效果和稳定性都很好     

Blast response simulation of an elastic structure: Evaluation of the fluid–structure interaction effect

Blast pressure wave interaction with an elastic structure is investigated using a numerical analysis approach, which considers fluid–structure interaction (FSI) within an Arbitrary Lagrange Euler (ALE) framework. Approximate numerical procedures for solving the Riemann problem associated with the shock are implemented within the Godunov finite volume scheme for the fluid domain. The structural displacement predicted by ignoring FSI is larger than the corresponding displacement considering FSI. The influence of the structural and blast pressure wave parameters on the importance of FSI is studied using an analysis of variables. Two non-dimensional parameters corresponding to the ratios of blast duration to the time period of the structure and the velocity of the structure to the particle velocity of the incident blast pressure wave are identified. It is shown that for a given blast pressure wave, the error in the maximum displacement predicted by ignoring FSI effect during structural motion is directly proportional to the ratio of the structure velocity to the particle velocity of the incident blast pressure wave. There is a continuous exchange of energy between the structure and air during the structural motion, which is significant when the structural velocity is significant compared to the particle velocity of incident blast pressure wave. FSI effect become insignificant when the ratio of velocities starts approaching zero.     

Formulation and validation of Berenger's PML absorbing boundary forthe FDTD simulation of acoustic scattering

Berenger's perfectly matched layer (PML) absorbing boundary condition for electromagnetic (EM) waves is derived to absorb 2-D and 3-D acoustic waves in finite difference time domain (FDTD) simulation of acoustic wave propagation and scattering. A PML medium suitable for acoustic waves is constructed. Plane wave propagation in the PML medium is solved for both 2-D and 3-D cases and explicit FDTD boundary conditions are derived. The equations show that a matched PML medium is a perfect simulation of free space in that a plane wave does not change its direction of propagation or its speed when it propagates from free space into a matched PML medium. FDTD simulation of a pulsed point source propagating in two dimensions is carried out to test the performance of the PML boundary for acoustic waves. Results show that an eight layer PML boundary condition reduces the reflected error 40 dB over Mur's second order boundary condition     

Comparison of highly efficient absorbing boundary conditions for the beam propagation method

Numerical experiments using the paraxial finite-difference beam propagation method have been performed with the following boundary conditions: perfectly matched layer, Higdon absorbing boundary conditions, complementary operators method, and extended complementary operators method. We have shown that Higdon operators must be modified for the paraxial wave equation to take into account the spectrum of incident rays on the boundaries of the computational domain. Reflection coefficients, accuracy, numerical dissipation/ gain, memory requirements, and time computation are compared and discussed for these absorbing techniques.     

Immersed smoothed finite element method for fluid–structure interaction simulation of aortic valves

This paper presents a novel numerical method for simulating the fluid?Cstructure interaction (FSI) problems when blood flows over aortic valves. The method uses the immersed boundary/element method and the smoothed finite element method and hence it is termed as IS-FEM. The IS-FEM is a partitioned approach and does not need a body-fitted mesh for FSI simulations. It consists of three main modules: the fluid solver, the solid solver and the FSI force solver. In this work, the blood is modeled as incompressible viscous flow and solved using the characteristic-based-split scheme with FEM for spacial discretization. The leaflets of the aortic valve are modeled as Mooney-Rivlin hyperelastic materials and solved using smoothed finite element method (or S-FEM). The FSI force is calculated on the Lagrangian fictitious fluid mesh that is identical to the moving solid mesh. The octree search and neighbor-to-neighbor schemes are used to detect efficiently the FSI pairs of fluid and solid cells. As an example, a 3D idealized model of aortic valve is modeled, and the opening process of the valve is simulated using the proposed IS-FEM. Numerical results indicate that the IS-FEM can serve as an efficient tool in the study of aortic valve dynamics to reveal the details of stresses in the aortic valves, the flow velocities in the blood, and the shear forces on the interfaces. This tool can also be applied to animal models studying disease processes and may ultimately translate to a new adaptive methods working with magnetic resonance images, leading to improvements on diagnostic and prognostic paradigms, as well as surgical planning, in the care of patients.     

Modeling of uncertainty for flood wave propagation induced by variations in initial and boundary conditions using expectation operator on explicit numerical solutions

This study proposes a framework for uncertainty analysis by incorporating explicit numerical solutions of governing equations for flood wave propagation with the expectation operator. It aims at effectively evaluating the effect of variations in initial and boundary conditions on the estimation of flood waves. Spatiotemporal semivariogram models are employed to quantify the correlation of the variables in time and space. The 1D nonlinear kinematic wave equation for the overland flow (named EVO_NS_KWE) is applied in the model development. Model validation is made by comparison with the Monte Carlo simulation model in the calculation of statistical properties of model outputs (ie, flow depths), that is, the mean, standard deviation, and coefficient of variation. The results from the model validation show that the EVO_NS_KWE model can produce excellent approximations of the mean and less satisfactory approximations of the standard deviation and coefficient of variation compared with those obtained by using the Monte Carlo simulation model. It concludes that the uncertainties of flow depths in the domain are significantly affected by variations in the boundary condition. Future application of the EVO_NS_KWE model enables the evaluation of uncertainty in model outputs induced by the initial and boundary condition subject to uncertainty and will also provide corresponding probabilistic information for risk quantification method.     

2.5D coupled BEM–FEM used to model fluid and solid scattering wave

We describe a hybrid method to study fluid and solid interaction problems in the frequency domain. The numerical method is based on subdomain decomposition. The BEM is used to model unbounded solid mediums, whereas the confined subdomains, both fluid and solid, are represented by the FEM. The analysis is carried out by superposing two‐and‐a‐half dimension (2.5D) problems for different longitudinal wave numbers. A novel 2.5D FEM formulation for inviscid fluids is proposed, which include the energy lost at the fluid boundary enclosure. The fluid and solid subdomains are coupled, and appropriate boundary conditions are imposed at the interfaces. The proposed technique is verified from analytical solutions. A cylindrical cavity located in an unbounded solid medium excited by a dilatational point source is studied. Computed results are in good agreement with the analytical solution. Later, noise and vibration in a concrete tunnel due to an internal pressure load is analysed with the proposed methodology. Results show that tunnel and soil displacements increase with the load speed, as did the air pressure inside the tunnel, according with the travelling ranges defined by the wave propagation velocities in each medium. Copyright © 2014 John Wiley & Sons, Ltd.     

FSI modeling with the DSD/SST method for the fluid and finite difference method for the structure

We present a fluid–structure interaction (FSI) modeling method based on using the deforming-spatial-domain/stabilized space–time (DSD/SST) method for the fluid mechanics part and a finite difference (FD) method for the structural mechanics part. As the structural mechanics model, we focus on the thin-shell model. The fluid mechanics equations with moving boundaries are solved with the DSD/SST method and the thin-shell structural mechanics equation is solved with a FD method, with partitioned coupling between the two parts. The coupling of the DSD/SST and FD solvers makes sure that the boundary conditions on the fluid-structure interface at the end of each time step are matched between the fluid and the structure. A hanging plate in vacuum under gravitational force is performed to validate the structure solver. In addition, a pitching plate in a uniform flow is simulated to validate the FSI solver. The present results are in reasonable agreement with data predicted by other methods.     

Harmonic waves in a prestressed thin elastic tube filled with a viscous fluid

In this work a theoretical analysis is presented for wave propagation ina thin-walled prestressed elastic tube filled with a viscous fluid. Thefluid is assumed to be incompressible and Newtonian, whereas the tubematerial is considered to be incompressible, isotropic and elastic.Considering the physiological conditions that the arteries experience, sucha tube is initially subjected to a mean pressure P_i and anaxial stretch _z. If it is assumed that in the course ofblood flow small incremental disturbances are superimposed on this initialfield, then the governing equations of this incremental motion are obtainedfor the fluid and the elastic tube. A harmonic-wave type of solution issought for these field equations and the dispersion relation is obtained.Some special cases, as well as the general case, are discussed and thepresent formulation is compared with some previous works on the samesubject.     

A stable 3D energetic Galerkin BEM approach for wave propagation interior problems

We consider 3D interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potentials. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the energetic Galerkin boundary element method. Numerical results are presented and discussed in order to show the stability and accuracy of the proposed technique.     

The spectral element method for elastic wave equations—application to 2‐D and 3‐D seismic problems

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two‐ and three‐dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor‐product of a Gauss–Lobatto–Legendre 1‐D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy‐momentum conserving scheme that can be put into a classical explicit‐implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two‐dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three‐dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three‐dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.     

A dynamic infinite element for three-dimensional infinite-domain wave problems

A three-dimensional dynamic infinite element which satisfies the following requirements: (1) displacement compatibility on the interface between finite and infinite elements; (2) definition of the wave propagation and amplitude attenuation behaviours in the infinite element using wave propagation functions; (3) convergence of the generalized integrals related to mass and stiffness matrices of the infinite element: and (4) displacement continuity along the common boundary of neighbouring infinite elements in the case of simulating multiple material layers or multiple wave numbers within the foundation, is presented in this paper. Since P-waves, S-waves and R-waves in the foundation can be simulated Simultaneously in the present infinite element, the seismic response of an arch-dam-foundation system, especially a thin double-curvature arch-dam-foundation system where the boundary element loses its competitive capacity with the finite element, can be economically calculated by coupling this infinite element with conventional finite elements. The good accuracy obtained using the present infinite element and finite element coupling model to simulate foundation wave problems has been proven by comparing the current numerical results with previous analytical results.     

A simple multi‐directional absorbing layer method to simulate elastic wave propagation in unbounded domains

The numerical analysis of elastic wave propagation in unbounded media may be difficult due to spurious waves reflected at the model artificial boundaries. This point is critical for the analysis of wave propagation in heterogeneous or layered solids. Various techniques such as Absorbing Boundary Conditions, infinite elements or Absorbing Boundary Layers (e.g. Perfectly Matched Layers) lead to an important reduction of such spurious reflections. In this paper, a simple absorbing layer method is proposed: it is based on a Rayleigh/Caughey damping formulation which is often already available in existing Finite Element softwares. The principle of the Caughey Absorbing Layer Method is first presented (including a rheological interpretation). The efficiency of the method is then shown through 1D Finite Element simulations considering homogeneous and heterogeneous damping in the absorbing layer. 2D models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types and incidences. A comparison with the PML method is first performed for pure P‐waves and the method is shown to be reliable in a more complex 2D case involving various wave types and incidences. It may thus be used for various types of problems involving elastic waves (e.g. machine vibrations, seismic waves, etc.). Copyright © 2010 John Wiley & Sons, Ltd.     

Coupled BEM–FEM solutions for direct time domain soil–structure interaction analysis

A coupled BEM–FEM methodology is presented for 3D wave propagation and soil–structure interaction analysis in the direct time domain. The employed boundary element method (BEM) uses a new generation of the Stokes fundamental solutions that utilize the B-Spline family of polynomials. A standard finite element methodology for dynamic analysis along with direct integration in time is coupled to the BEM through a staggered solution approach. Each method provides initial conditions to the other at the beginning of each time step. Formulation and computational aspects of the proposed coupling scheme are discussed. A number of numerical examples are presented for the validation and demonstration of the general nature of the proposed methodology.