A general numerical method for solution of gravity wave problems. Part 2: Steady nonlinear gravity waves

A type of numerical scheme for 2D and 3D steady non-linear water wave problems is described. It is based on the finite process method and is insensitive to initial solutions. The relationship between the finite process method and iterative techniques is discussed. As a numerical example the flow past a submerged vortex is solved and the results are compared with those of other authors.     

A simple method for simulating viscoelastic fluid flows in slit channel

A low-cost semi-analysis finite element technique, named the finite piece method (FPM) is presented in this article. It aims to solve three-dimensional (3D) viscoelastic slit flows. The viscoelastic stress of the fluid is modelled using an K-BKZ integral constitutive equation of the Wagner type. Picard iteration is used to solve non-linear equations. The FPM is tested on flow problems in both planar and contraction channels. The accuracy of the method is assessed by comparing flow distributions and pressure with results obtained by 3D finite element method (FEM). It shows that the solution accuracy is excellent and a substantial amount of computing time and memory requirement can be saved.     

Analysis of efficient preconditioned defect correction methods for nonlinear water waves

Robust computational procedures for the solution of non‐hydrostatic, free surface, irrotational and inviscid free‐surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable prediction of free‐surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow equations. We present new detailed fundamental analysis using finite‐amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high‐order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow models. Our study is particularly relevant for fast and efficient simulation of non‐breaking fully nonlinear water waves over varying bottom topography that may be limited by computational resources or requirements. To gain insight into algorithmic properties and proper choices of discretization parameters for different PDC strategies, we study systematically limits of accuracy, convergence rate, algorithmic and numerical efficiency and scalability of the most efficient known PDC methods. These strategies are of interest, because they enable generalization of geometric multigrid methods to high‐order accurate discretizations and enable significant improvement in numerical efficiency while incuring minimal storage requirements. We demonstrate robustness using such PDC methods for practical ranges of interest for coastal and maritime engineering, that is, from shallow to deep water, and report details of numerical experiments that can be used for benchmarking purposes. Copyright © 2014 John Wiley & Sons, Ltd.     

Traveling gravity water waves in two and three dimensions

This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity.     

The interaction of equatorial waves with a barotropic shear: a potential test case for climate model dynamical cores

We compare the performances of two different numerical methods to solve the equatorial shallow water equations in a background meridional shear: A coarse-resolution Galerkin-truncation-based method and a finite-volume method, for the case of an equatorial Rossby wave. In the presence of the barotropic shear, the Rossby wave quickly loses its energy through shear interaction and excites other equatorially trapped waves despite the fact that the PDE system is linear. The two methods handle this sudden energy exchange across scales very differently. While the finite volume converges statistically to a coherent large-scale solution, the Galerkin truncation method results in large-scale and slow oscillations where energy is bumped back and forth within the resolved-scale spectrum, involving higher-order equatorial wave modes heavily depending on numerical resolution, that is, the number of Galerkin basis functions. The addition of an artificial viscosity for the coarse-resolution Galerkin method heavily damps the energy oscillations without significantly changing its transient dynamics. This provides an example of interaction across scales where the resolution of scales that are apparently not representative of the statistical solution are important for the transient dynamics. Therefore, non-linear interactions of equatorially trapped waves may provide an interesting test bed for the validation of climate model dynamical cores.     

Nonlinear dynamics of hydrostatic internal gravity waves

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.      

Minimum fuel trajectory transfer in a general gravity field using a finite number of engine firings to cause velocity discontinuities

A method is presented to compute the firing logic for a rocket to transfer from one arbitrary trajectory to another such trajectory in a fixed maximum number of firings of the engine in a manner to minimize the amount of fuel consumed while satisfying various orbital constraints.

The iterative procedure presented may be used to solve other constrained non-linear parameter minimization problems. The method involves iteratively minimizing the first differential of the “cost” function subject to improving the first differentials of the constraints by using linear programming. Both equality and inequality constraints are amenable to solution by this method.     

Three-dimensional hydrodynamics on finite elements. Part II: Non-linear time-stepping model

The development and application of a non-linear 3D hydrodynamic model are described. The model is based on the wave equation rearrangement of the primitive 3D shallow water equations with a general eddy viscosity formulation for the vertical shear. A Galerkin procedure is used to discretize these on simple sixnode elements: linear triangles in the horizontal with linear variations in the vertical. Resolution of surface, bottom and interfacial boundary layers is facilitated and total flexibility is preserved for specifying spatial and temporal variations in the vertical viscosity and density fields. A semi-implicit time-stepping algorithm allows the solutions for elevation and velocity to be uncoupled during each time step. The elevation solution is essentially a 2D wave equation calculation with a stationary sparse matrix representing the gravity waves. With nodal quadrature the subsequent velocity calculation is achieved by factoring only a tridiagonal diffusion matrix representing the vertical viscous terms. As a result the overall calculation scales computationally as only a 2D problem but provides the full 3D solution. Application to field-scale problems is illustrated for the English Channel/Southern Bight system and the Lake Maracaibo system.     

The numerical study of roll‐waves in inclined open channels and solitary wave run‐up

In this paper, we introduce a finite‐volume kinetic BGK scheme and its applications to the study of roll and solitary waves. The current scheme is based on the numerical solution of the gas‐kinetic Bhatnagar–Gross–Krook model in the flux evaluation across each cell interface. An intrinsic connection between the BGK model and time‐dependent, non‐linear, non‐homogeneous shallow‐water equations enables us to solve shallow‐water equations automatically with our kinetic scheme. The analytical solution, experimental measurements, and numerical calculations for problems associated with roll‐waves down an inclined open channel and solitary waves incident on a sloped beach are also presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Representation of Normal Cone Inclusion Problems in Dynamics Via Non-linear Equations

Analyzing non-smooth mechanical systems requires often the solution of inclusion problems of normal cone type. These problems arise for example in the event-driven or time-stepping simulation approaches. Such inclusion problems can be written as non-linear equations, which can be solved iteratively. In this paper we discuss three different methods to derive the non-linear equations representing the inclusions arising in the event-driven simulation approach. First, we formulate inclusions describing the individual non-smooth constraints and solve them successively. Secondly, we interpret the non-linear equations as the conditions for the saddle point of the augmented Lagrangian function. As a third possibility we discuss the exact regularization of set-valued force laws. All three methods lead to the same numerical scheme, but give different insight into the problem. Especially the factor r occurring in the non-linear equations is discussed. Two iterative methods for solving the non-linear equations are presented together with some remarks on convergence.     

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.     

Global convergence of inexact Newton methods for transonic flow

In computational fluid dynamics, non-linear differential equations are essential to represent important effects such as shock waves in transonic flow. Discretized versions of these non-linear equations are solved using iterative methods. In this paper an inexact Newton method using the GMRES algorithm of Saad and Schultz is examined in the context of the full potential equation of aerodynamics. In this setting, reliable and efficient convergence of Newton methods is difficult to achieve. A poor initial solution guess often leads to divergence or very slow convergence. This paper examines several possible solutions to these problems, including a standard local damping strategy for Newton's method and two continuation methods, one of which utilizes interpolation from a coarse grid solution to obtain the initial guess on a finer grid. It is shown that the continuation methods can be used to augment the local damping strategy to achieve convergence for difficult transonic flow problems. These include simple wings with shock waves as well as problems involving engine power effects. These latter cases are modelled using the assumption that each exhaust plume is isentropic but has a different total pressure and/or temperature than the freestream.     

A meshless numerical wave tank for simulation of nonlinear irregular waves in shallow water

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed. Copyright © 2008 John Wiley & Sons, Ltd.     

Rotationally symmetric internal gravity waves

Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.     

基于碎点法的动态断裂分析

工程中的冲击防护结构在撞击、爆炸等冲击载荷下可能发生动态断裂并最终破坏, 抑制结构的动态断裂是提升结构防护能力的重要手段, 为此需要准确预测结构在动态载荷下的断裂行为. 数值仿真是预测动态断裂的重要手段, 然而当前工程中常用的有限元法在模拟断裂方面仍存在网格畸变和难以显式引入裂纹等问题. 碎点法是近年来提出的一种不连续型伽辽金弱形式无网格方法, 适合模拟断裂问题, 本文提出一种显式动力学格式的碎点法并将该方法应用于动态断裂分析. 一方面, 碎点法参考弱形式无网格类方法, 将求解域离散为空间中的节点和子域, 并基于支持域内的节点群构造子域的位移试函数, 因此该方法的子域具有抵抗畸变的能力. 另一方面, 碎点法参考间断伽辽金有限元法, 使用分片连续的位移试函数, 并引入内部界面数值通量修正保证方法的一致性和稳定性, 因此该方法易于在结构中显式引入裂纹. 本文首先介绍碎点法的核心思想和离散形式, 接着推导了动力学碎点法弱形式动量方程, 然后建立了碎点法的显式动力学求解格式, 最后通过算例验证动力学碎点法预测应力波传播和动态断裂行为的能力.      

求解浅水波方程的并行物理信息神经网络算法

双曲守恒律方程是一类比较特殊的偏微分方程,其数值求解方法的研究一直是一个热点问题,一个显著特性是即使初始条件是光滑的,其解也可能会发展成间断。浅水波方程作为非线性双曲守恒律方程,由于间断解的存在,其精确求解存在很大困难。针对浅水波方程数值求解问题,本文基于PINN(Physics informed neural networks)反问题网络结构构造新的网络,构造的网络结构包括两个并行的神经网络,其中一个网络与已知状态数据(熵稳定格式加密求出)相关,另一个网络与方程本身相关。利用已知速度数据结合浅水波方程本身求解未知水深,最终通过一些数值算例验证网络的可行性。结果表明,新的网络结构可用于浅水波方程求解,利用速度数据可以较为精确地推算出水深。     

Accurate simple approximation for the solitary waveApproximation simple et précise de l'onde solitaire

This Note investigates the effect of a renormalization technique on high-order shallow water approximations of gravity waves. The method is illustrated for the solitary surface wave. Applied to the solution of a generalized KdV equation, it is shown that the renormalization significantly increases the accuracy. To cite this article: D. Clamond, D. Fructus, C. R. Mecanique 331 (2003).     

Flexural gravity wave motion over poroelastic bed

Using Biot’s consolidation theory, effect of poroelastic bed on flexural gravity wave motion is analyzed in both the cases of single-layer and two-layer fluids. The model for the flexural gravity waves is developed using linear water wave theory and small amplitude structural response in finite water depth. The effects of permeability and shear modulus of poroelastic bed and time period on flexural gravity wave motion are studied by analyzing the dispersion relation, phase speed, plate deflection, interface elevation and pressure distribution along water depth. Various results for surface gravity waves are analyzed as special cases. The study reveals that bed permeability retards the hydrodynamic pressure distribution along the water depth significantly compared to shear modulus whilst, floating plate deflection decreases significantly with change in shear modulus compared to permeability of the poroelastic bed. The present study can be generalized to analyze various wave–structure interaction problems over poroelastic bed.     

A finite point method for adaptive three‐dimensional compressible flow calculations

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A high‐order PIC method for advection‐dominated flow with application to shallow water waves

A hybrid Eulerian‐Lagrangian particle‐in‐cell–type numerical method is developed for the solution of advection‐dominated flow problems. Particular attention is given over to the high‐order transfer of flow properties from the particles to the grid. For smooth flows, the method presented is of formal high‐order accuracy in space. The method is applied to solve the nonlinear shallow water equations resulting in a new, and novel, shock capturing shallow water solver. The approach is able to simulate complex shallow water flows, which can contain an arbitrary number of discontinuities. Both trivial and nontrivial bottom topography is considered, and it is shown that the new scheme is inherently well balanced, exactly satisfying the ‐property. The scheme is verified against several one‐dimensional benchmark shallow water problems. These include cases that involve transcritical flow regimes, shock waves, and nontrivial bathymetry. In all the test cases presented, very good results are obtained.