Variational principles for hydrodynamic impact problems

We first establish the rigorous field equations of the two continuous stages before and after entering water. Then correspondently, we obtain the specific variational principles, bounded theorems, and boundary integral equations of the second stage problems. The existence of solutions are proved and the scheme of solving the solutions are provided. Finally, as a numerical example, the ship's wave resistence problem is used to demonstrate the specific application of the second stage problems and its accuracy. Then we provide a rigorous and sound theoretical basis of variational finite element method and boundary element method for calculating the accurately fundamental equations.     

A boundary-type finite element model for water surface wave problems

A new combinative method of boundary-type finite elements and boundary solutions is presented to study wave diffraction-refraction and harbour oscillation problems. The numerical model is based on the mild-slope equation. The key feature of this method is that the discretized matrix equation can be formulated only by the calculation of a line integral, since the interpolation equation which satisfies the governing equation in each element is used. The numerical solutions are compared with existing analytical, experimental, observed and other numerical results. The present method is shown to be an effective and accurate method for water surface wave problems.     

Boundary element method for long-time water wave propagation over rapidly varying bottom topography

We study numerically the linear water wave equations for shallow channels with rapidly varying bottom topography. We do not use the shallow water approximation because it is not valid when the bottom is rapidly varying. We use the boundary element method because it allows accurate tracking of the surface waves for long times. We present the results of a range of numerical validation experiments and a comparison between propagation over a periodic and a random rough bottom topography.     

Numerical analysis of a moving boundary problem in coastal hydrodynamics

A finite element model to tackle the moving boundary problem of wave run-up on moderately steep slopes is developed. The special aspects considered in this study are (1) the modification of shallow water equations to accommodate the effect of vertical accelerations and (2) the use of Lagrangian acceleration coupled with an element that adapts itself to the moving boundary closely. The pressure term in the one-dimensional momentum equation is derived using the Eulerian equation in the vertical direction. This takes care of the vertical accelerations which are significant during the motion of a wave on moderately steep slopes. The element near the boundary is allowed to change its dimension so that the fluid boundary is closely followed. Such a flexible element precludes the need for approximation of the variables with regard to the indefinite position of the boundary. This element is split into two when its dimension becomes unduly large compared to the unchanging elements. The need for such a splitting is shown by an examination of the entries in the global matrix. Results of water profile as a wave runs up a structure are given. A brief history of the work on similar problems is outlined.     

流固耦合地震波动问题的显式谱元模拟方法

流固耦合地震波动问题主要研究由流体和固体构成的复杂系统中地震波传播特性及其规律. 传统模拟方法中一般以声波方程、弹性波方程的数值解分别描述理想流体和弹性固体中的波动, 并实时地处理两种不同性质介质之间的相互耦合作用, 数值格式复杂且限制数值模拟精度与计算效率. 本文采用谱元法结合多次透射公式人工边界条件实现了一种流固耦合地震波动问题的高阶显式数值计算方法. 该方法利用了流固耦合问题统一计算框架,可将饱和多孔介质的Biot波动方程分别退化为理想流体的声波方程和弹性固体的弹性波方程. 通过P波垂直入射的水平成层理想流体-饱和多孔介质-弹性固体场地模型、P波斜入射的不规则层状界面以及任意形状界面的理想流体-饱和多孔介质-弹性固体场地模型等三个算例, 与传递函数法解析解以及集中质量有限元法计算结果进行对比分析, 证明了本文方法的正确性与有效性. 数值模拟结果表明, 本文方法相较传统有限元法可以少得多的节点数量获得更高的数值精度, 并且在较宽的频率范围内都能可靠地模拟出流固耦合系统的动力响应, 充分体现出本文方法兼顾高精度、计算效率和复杂场地建模灵活的特点.      

Fracture mechanics analysis of curved stiffened panels using BEM

A dual boundary element method is developed for a analysis of reinforced cracked shallow shells. Boundary integral equations are derived from the Betti’s reciprocal theorem for a cracked shallow shell with transverse frames and longitudinal stiffeners. The effect of frames and stiffeners are treated as a distribution of line body forces. The radial basis function is used to transform domain integrals to boundary integrals. Stress intensity factors are evaluated from crack opening displacements. The effect of curvature on the stress intensity factors is illustrated by numerical examples. Three examples are presented to demonstrate the accuracy of this method compared with solutions obtained using the finite element method.     

含孔曲板弹性波散射与动应力分析

基于敞口浅柱壳弹性波动方程及摄动方法，对无限大含孔曲板弹性波散射及动应力问题进行了分析研究，将经典薄板弯曲波动问题的分析解作为本问题的主项，给出了在稳态波下孔洞附近散射波的零阶渐近解。建立了求解含孔曲板弹性波散射与动应力问题的边界积分方程法，利用积分方程法可获得问题的近似分析解。并给出了无限大曲板圆孔附近动应力集中系数的数值结果，且对计算结果进行了分析与讨论。     

A modified Galerkin/finite element method for the numerical solution of the Serre‐Green‐Naghdi system

A new modified Galerkin/finite element method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low‐order Lagrange finite element spaces, despite the fact that the system contains third order spatial partial derivatives for the depth averaged velocity of the fluid. After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions with laboratory experiments and with available theoretical asymptotic results. Copyright © 2016 John Wiley & Sons, Ltd.     

Shallow Water Flow Analysis with Moving Boundary Technique Using Least-squares Bubble Function

This paper presents a computational simulation method for a river problem. For the actual flow problem, it is necessary to compute flow velocity, water elevation and water region at the same time. For the basic formulation, the unsteady shallow water equations are used. As the numerical approach, implicit FEM is proposed by bubble function. To control numerical stability and accuracy, LSBF (Least-Squares Bubble Function) is used to solve the finite element equations. Also, the fixed boundary technique is combined to deal with wet and dry areas in the moving finite element mesh. Some numerical tests are shown to check this method.     

基于混合变量条形传递函数方法的圆柱壳屈曲分析

提出了一种分析交各向异性圆柱壳和阶梯圆柱壳稳定性问题的混合变量条形传递函数方法。首先基于Fluegge薄壳理论，通过定义广义位移变量和对应的广义力变量，建立了圆柱壳混合变量能量泛函；然后通过引入条形单元，定义混合状态变量和采用传递函数方法对超级壳单元求解，得到具有多种边界条件圆柱壳屈曲问题的半解析解；最后通过位移连续和力平衡条件，可以得到阶梯圆柱壳屈曲问题的解。理论解推导过程表明此方法在引入边界条件和进行阶梯圆柱壳求解时非常方便。算例分析的结果验证了本方法的正确性。     

水下爆炸非均熵二维定常流的三族特征线解法

针对二维定常可压缩超声速非等熵柱状流,提出一种特征线差分解法,通过在沿马赫线的相容方程中添加沿流线的熵变项以描述非等熵效应,得到等熵流和非等熵流均适用的三族特征线方程组。根据水下爆炸近场特点,建立无限长柱状装药的定常模型,将三族特征线方程组用有限差分法离散求解,通过构造合适的网格保证计算格式可以数值上收敛,由此编制程序并计算几种柱状炸药的水下爆炸近场冲击波。对比有限元模拟结果和实验结果发现,特征线差分法可以比较准确地捕捉冲击波形状并计算冲击波后流场,从而验证了所提出的三族特征线差分法的准确性。     

Exact Riemann Solution and Weighted Average Flux Method for Power-law Channel Section

The aim of this work is to extend the Weighted Averaged Flux (WAF) method by E.F. Toro from rectangular to generic regular cross sections, in order to represent the geometric effects of the cross-section on wave propagation along channels. Assuming a power-law variation of the channel width, the 1D conservative shallow water equations, their characteristic form and the shock propagation equations are presented. The exact Riemann solver is derived and is applied to the dam-break problem in valleys with different shape in order to test its efficiency and to check the accuracy order of solutions obtained by approximating the real cross-section with an equivalent rectangle. The WAF method is extended to a power-law channel section and is used to solve the 1D proposed equations taking into account all the source terms that are incorporated into the local Riemann problem. A code based on this method has been developed and results of numerical applications to a Venturi channel and to the attenuation of waves are presented, in order to verify, for well known situations, how accurately the source terms are represented. The code is applied to reproduce one of Brock's experiments (1969; 1970) on roll waves generation in a rectangular channel and results are compared with those obtained with a Godunov-type code developed at CEMAGREF, which is based on a Roe scheme with a later average evaluation of source terms.     

Scattering of Plane SH-Wave by a Cylindrical Hill of Arbitrary Shape

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Surface waves propagation in shallow water: A finite element model

A two-dimensional (in-plane) numerical model for surface waves propagation based on the non-linear dispersive wave approach described by Boussinesq-type equations, which provide an attractive theory for predicting the depth-averaged velocity field resulting from that wave-type propagation in shallow water, is presented. The numerical solution of the corresponding partial differential equations by finite-difference methods has been the subject of several scientific works. In the present work we propose a new approach to the problem: the spatial discretization of the system composed by the Boussinesq equations is made by a finite element method, making use of the weighted residual technique for the solution approach within each element. The model is validated by comparing numerical results with theoretical solutions and with results obtained experimentally.     

STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH LARGE OPENINGS

Based on Donnell's shallow shell equation, a new method is given in this paper to ana-lyze theoretical solutions of stress concentrations about cylindrical shells with large openings. With themethod of complex variable function, a series of conformal mapping functions are obtained from dif-ferent cutouts' boundary curves in the developed plane of a cylindrical shell to the unit circle. And,the general expressions for the equations of a cylindrical shell, including the solutions of stress concen-trations meeting the boundary conditions of the large openings' edges, are given in the mapping plane.Furthermore, by applying the orthogonal function expansion technique, the problem can be summa-rized into the solution of infinite algebraic equation series. Finally, numerical results are obtained forstress concentration factors at the cutout's edge with various opening's ratios and different loadingconditions. This new method, at the same time, gives a possibility to the research of cylindrical shellswith large non-circular openings or with nozzles.     

An improved wave motion input method for application of multi-transmitting boundary

To solve scattering problems with multi-transmitting boundary, we present an improved wave motion input method based on the idea that error caused by the difference between incident wave field used in calculation and waves propagating in finite element grids can be eliminated to suppress drift instability. In this method, a calculation scheme is proposed to obtain the numerical solution of incident wave field, which establishes boundary region models with the multi-transmitting boundary. Numerical experiments demonstrate that this improved wave motion input method not only eliminates drift instability but also effectively improves the calculation accuracy of low-frequency components. Furthermore, the method is easy to implement and, unlike other approaches, does not need artificial parameters. Thus, this method is proposed for use in wave scattering simulation such as seismic response analyses of structures, particularly for long-period structures and those that are sensitive to low frequency.     

主动约束层阻尼部分覆盖圆柱壳耦合振动控制

采用分布参数建模,从Ham ilton变分原理出发推导了主动约束层阻尼(ACLD)覆盖圆柱壳耦合振动的运动微分方程和边界条件,并扩展适合一维连续结构分析的基于解析解的谱传递矩阵法(STMM)用于ACLD部分覆盖圆柱壳。通过数值计算,研究了ACLD的长度和位置对固有频率和模态耗散因子的影响。STMM能有效克服有限元法单元数目多、动力学方程阶数高及ACLD长度和位置变化时须重复建模的缺点,以最少的单元数目建立低阶控制方程。典型算例显示了STMM的有效性和精确性。     

Comparative Analysis of Nonlinear Dispersive Shallow Water Models

The results of comparative analysis of some nonlinear dispersive models of shallow water are presented. The aim is to find their individual properties relevant for the numerical solution of some model problems of long wave transformation over submerged obstacles The study considers basic properties of the listed models and their numerical implementation. Computations are obtained compared with the analytical solution and experimental data. Attention is primarily focused on the models suggested by Peregrine (1967); Zheleznyak and Pelinovsky (1985); Kim, Reid, Whitakcr (1988): Fedotova and Pashkova (1997). Also classical equations of shallow water are considered in both linear and nonlinear approximations.     

The influence of normal flow boundary conditions on spurious modes in finite element solutions to the shallow water equations

Finite element solutions of the primitive equation (PE) form of the shallow water equations are notorious for the severe spurious 2Δx modes which appear. Wave equation (WE) solutions do not exhibit these numerical modes. In this paper we show that the severe spurious modes in PE solutions are strongly influenced by essential normal flow boundary conditions in the coupled continuity-momentum system of equations. This is demonstrated through numerical examples that avoid the use of essential normal flow boundary conditions either by specifying elevation values over the entire boundary or by implementing natural flow boundary conditions in the weak weighted residual form of the continuity equation. Results from a series of convergence tests show that PE solutions are of nearly the same quality as WE solutions when spurious modes are suppressed by alternative specification of the boundary conditions. Network intercomparisons indicate that varying nodal support does not excite spurious modes in a solution, although it does enhance the spurious modes introduced when an essential normal flow boundary condition is used. Dispersion analysis of discrete equations for interior and boundary nodes offers an explanation of the observed solution behaviour. For certain PE algorithms a mixed situation can arise where the boundary nodes exhibit a monotonic (noise-free) dispersion relationship and the interior nodes exhibit a folded (noisy) dispersion relationship. We have found that the mixed situation occurs when all boundary nodes are specified elevation nodes (which are enforced as essential conditions in the continuity equation) or when specified flow boundary nodes are treated as natural boundary conditions in the continuity equation. In either case the effect is to generate a solution that is essentially free of noise. Apparently, the monotonic dispersion behaviour at the boundaries suppresses the otherwise noisy behaviour caused by the folded dispersion relation on the interior.     

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

A numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non-linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non-linearity and the coupling with the thermal boundary layer equation. Because the non-linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.