Finite difference simulation of nonlinear waves generated by ships of arbitrary three-dimensional configuration

A modified marker-and-cell method is developed in order to simulate nonlinear wave making in the near-field of ships of arbitrary three-dimensional (3D) configuration advancing steadily in deep water. The 3D Navier-Stokes equations are solved by a finite difference scheme under proper boundary conditions. Efforts are particularly focused on the treatment of the boundary conditions on the body surface and free surface which have complicated 3D configurations. An orthogonal cell system with more than 70,000 cells is used for the computation of the waves and flow field of ships. The agreement of computational results with experiment is good, and it promises effectiveness for engineering purposes.     

Scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems

This paper presents a solution for the displacement of a uniform elastic thin plate with an arbitrary cavity, modelled using the biharmonic plate equation. The problem is formulated as a system of boundary integral equations by factorizing the biharmonic equation, with the unknown boundary values expanded in terms of a Fourier series. At the edge of the cavity we consider free-edge, simply-supported and clamped boundary conditions. Methods to suppress ill-conditioning which occurs at certain frequencies are discussed, and the combined boundary integral equation method is implemented to control this problem. A connection is made between the problem of an infinite plate with an arbitrary cavity and the vibration problem of an arbitrarily shaped finite plate, using the jump discontinuity present in single-layer distributions at the boundary. The first few frequencies and modes of displacement are computed for circular and elliptic cavities, which provide a check on our numerics, and results for the displacement of an infinite plate are given for four specific cavity geometries and various boundary conditions.     

A novel mesh regeneration algorithm for 2D FEM simulations of flows with moving boundary

A novel mesh regeneration algorithm is proposed to maintain the mesh structure during a finite element simulation of flows with moving solid boundary. With the current algorithm, a new body-fitted mesh can be efficiently constructed by solving a set of Laplace equations developed to specify the displacements of individual mesh elements. These equations are subjected to specific boundary conditions determined by the instantaneous body motion and other flow boundary conditions. The proposed mesh regeneration algorithm has been implemented on an arbitrary Lagrangian–Eulerian (ALE) framework that employs an operator-splitting technique to solve the Navier–Stokes equations. The integrated numerical scheme was validated by the numerical results of four existing problems: a flow over a backward-facing step, a uniform flow over a fixed cylinder, the vortex-induced vibration of an elastic cylinder in uniformly incident flow, and a complementary problem that compares the transient drag coefficient for a cylinder impulsively set into motion to that measured on a fixed cylinder in a starting flow. Good agreement with the numerical or experimental data in the literature was obtained and new transient flow dynamics was revealed. The scheme performance is further examined with respect to the parameter employed in the mesh regeneration algorithm.     

The lattice Boltzmann equation on irregular lattices

A general framework to extend the lattice Boltzmann equation to arbitrary lattice geometries is presented and numerically demonstrated for the case of a two-dimensional Poiseuille flow. The new scheme considerably extends the range of applicability of the Boltzmann method to problems requiring the use of nonuniform grids.     

振动NACA0012翼型的移动网格数值模拟

利用流体力学方程的积分形式给出非结构移动网格上离散格式,利用自适应移动网格方法移动网格,进而得到网格速度.对振动Naca0012翼型问题,分三种类型确定网格速度,再结合Riemann问题的解法器构造数值通量,得到移动网格单元上新的物理量.数值实验表明这种格式同时具有高效、高分辨率的特点.     

Solutions of Laplace's equation with simple boundary conditions,and their applications for capacitors with multiple symmetries

We find solutions of Laplace's equation with special boundary conditions, using a general curvilinear system of coordinates. We call this purely geometrical solutions Basic Harmonic Functions (BHF's). From them we obtain more general solutions with arbitrary constant values on the boundaries. Further, the BHF's are used to obtain the capacitance of many electrostatic configurations of conductors. Applications in complex geometries are given. Finally, expressions for electric fields between two conductors and surface charge densities are obtained in terms of generalized curvilinear coordinates. The present method can be extrapolated to other linear homogeneous differential equations.     

An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems

The aim of the present paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics flow problems for regular and irregular geometries subject to Dirichlet, Neumann and Robin boundary conditions. Toward this, the meshless point collocation method (MPCM) is used for MHD flow problems in channels with fully insulating or partially insulating and partially conducting walls, having rectangular, circular, elliptical or even arbitrary cross sections. MPC is a truly meshless and computationally efficient method. The maximum principle for the discrete harmonic operator in the meshfree point collocation method has been proven very recently, and the convergence proof for the numerical solution of the Poisson problem with Dirichlet boundary conditions have been attained also. Additionally, in the present work convergence is attained for Neumann and Robin boundary conditions, accordingly. The shape functions are constructed using the Moving Least Squares (MLS) approximation. The refinement procedure with meshless methods is obtained with an easily handled and fully automated manner. We present results for Hartmann number up to 10⁵${10}^5$. The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the exact solution and the conventional FEM and BEM, for the Dirichlet, Neumann and Robin boundary conditions of interior problems with simple or complex boundaries.     

A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles

An efficient numerical scheme to compute flows past rigid solid bodies moving through viscous incompressible fluid is presented. Solid obstacles of arbitrary shape are taken into account using the volume penalization method to impose no-slip boundary condition. The 2D Navier–Stokes equations, written in the vorticity-streamfunction formulation, are discretized using a Fourier pseudo-spectral scheme. Four different time discretization schemes of the penalization term are proposed and compared. The originality of the present work lies in the implementation of time-dependent penalization, which makes the above method capable of solving problems where the obstacle follows an arbitrary motion. Fluid–solid coupling for freely falling bodies is also implemented. The numerical method is validated for different test cases: the flow past a cylinder, Couette flow between rotating cylinders, sedimentation of a cylinder and a falling leaf with elliptical shape.     

基于非结构网格间断有限元方法在可压缩流体中的应用

本文开发了一套基于非结构网格的间断有限元方法(DG)程序,并对与单元形状无关的斜率限制器进行了研究。此程序支持多种网格类型,能够方便应用于具有混合单元的非结构网格,具有处理复杂几何结构的能力,为研究叶轮机械内部复杂流动现象提供了有效的研究工具。本文利用该程序对若干典型无黏和黏性问题进行数值模拟,结果表明,该程序具有较高的可信度,能够处理具有混合单元的非结构网格,并给出良好的数值模拟结果。     

Quantum string cosmologies

Different models of quantum string minisuperspaces for both isotropic and anisotropic geometries are presented. As for the isotropic case the general model of axion‐dilaton cosmology of arbitrary curvature is studied and solved explicitly. In the anisotropic case Bianchi type I and IX and Kantowski‐Sachs geometries are investigated. A short review of string cosmology is given with the emphasis on its “graceful‐exit” problem. The problems of boundary conditions, wave packets in minisuperspace and possible implications for the arrow of time in cosmology are briefly discussed.     

Implementation Aspects of 3D Lattice-BGK: Boundaries, Accuracy, and a New Fast Relaxation Method

In many realistic fluid-dynamical simulations the specification of the boundary conditions, the error sources, and the number of time steps to reach a steady state are important practical considerations. In this paper we study these issues in the case of the lattice-BGK model. The objective is to present a comprehensive overview of some pitfalls and shortcomings of the lattice-BGK method and to introduce some new ideas useful in practical simulations. We begin with an evaluation of the widely used bounce-back boundary condition in staircase geometries by simulating flow in an inclined tube. It is shown that the bounce-back scheme is first-order accurate in space when the location of the non-slip wall is assumed to be at the boundary nodes. Moreover, for a specific inclination angle of 45 degrees, the scheme is found to be second-order accurate when the location of the non-slip velocity is fitted halfway between the last fluid nodes and the first solid nodes. The error as a function of the relaxation parameter is in that case qualitatively similar to that of flat walls. Next, a comparison of simulations of fluid flow by means of pressure boundaries and by means of body force is presented. A good agreement between these two boundary conditions has been found in the creeping-flow regime. For higher Reynolds numbers differences have been found that are probably caused by problems associated with the pressure boundaries. Furthermore, two widely used 3D models, namelyD₃Q₁₅andD₃Q₁₉, are analysed. It is shown that theD₃Q₁₅model may induce artificial checkerboard invariants due to the connectivity of the lattice. Finally, a new iterative method, which significantly reduces the saturation time, is presented and validated on different benchmark problems.     

Optimization of One-Way Quantum Computation Measurement Patterns

In one-way quantum computation (1WQC), an initial highly entangled state, called a graph state, is used to perform universal quantum computations by a sequence of adaptive single-qubit measurements and post-measurement Pauli-X and Pauli-Z corrections. 1WQC computation can be represented by a measurement pattern (or simply a pattern). The entanglement operations in a pattern can be shown by a graph which together with the identified set of its input and output qubits is called the geometry of the pattern. Since a pattern is based on quantum measurements, which are fundamentally nondeterministic evolutions, there must be conditions over geometries to guarantee determinism. These conditions are formalized by the notions of flow and generalized flow (gflow). Previously, three optimization methods have been proposed to optimize 1WQC patterns which can be performed using the measurement calculus formalism by rewriting rules. However, the serial implementation of these rules is time consuming due to executing many ineffective commutation rules. To overcome this problem, in this paper, a new scheme is proposed to perform the optimization techniques simultaneously on patterns with flow and only gflow based on their geometries. Furthermore, the proposed scheme obtains the maximally delayed gflow order for geometries with flow. It is shown that the time complexity of the proposed approach is improved over the previous ones.     

General Considerations on the Finite-Size Corrections for Coulomb Systems in the Debye--Hückel Regime

We study the statistical mechanics of classical Coulomb systems in a low coupling regime (Debye--Hückel regime) in a confined geometry with Dirichlet boundary conditions for the electric potential. We use a method recently developed by the authors which relates the grand partition function of a Coulomb system in a confined geometry with a certain regularization of the determinant of the Laplacian on that geometry with Dirichlet boundary conditions. We study several examples of fully confining geometry in two and three dimensions and semi-confined geometries where the system is confined only in one or two directions of the space. We also generalize the method to study systems confined in arbitrary geometries with smooth boundary. We find a relation between the expansion for small argument of the heat kernel of the Laplacian and the large-size expansion of the grand potential of the Coulomb system. This allow us to find the finite-size expansion of the grand potential of the system in general. We recover known results for the bulk grand potential (in two and three dimensions) and the surface tension (for two-dimensional systems). We find the surface tension for three-dimensional systems. For two-dimensional systems our general calculation of the finite-size expansion gives a proof of the existence a universal logarithmic finite-size correction predicted some time ago, at least in the low coupling regime. For three-dimensional systems we obtain a prediction for the curvature correction to the grand potential of a confined system.     

Theory of Spin Echo in Restricted Geometries under a Step-wise Gradient Pulse Sequence

A closed matrix form solution of the Bloch-Torrey equation is presented for the magnetization density of spins diffusing in a bounded region under a steady gradient field and for the Stejskal-Tanner gradient pulse sequence, assuming straightforward generalization to any step-wise gradient profile. The solution is expressed in terms of the eigenmodes of the diffusion propagator in a given geometry with appropriate boundary conditions (perfectly reflecting or relaxing walls). Applications to rectangular, cylindrical, and spherical geometries are discussed. The relationship with the multiple propagator approach is established and an alternative step-wise gradient discretization procedure is suggested to handle arbitrary gradient waveforms.     

Finite difference scheme for planar waveguides with arbitrary index profiles and its implementation for anisotropic waveguides with a diagonal permitivity tensor

A finite difference scheme with a uniform mesh for planar waveguides with arbitrary refractive index profiles that takes full account of any smooth index variation and index discontinuity is derived for TE and TM-polarized waves. Discretizations that lead to a second-order error in the effective indices are given for TE and TM polarizations. At the computational boundaries, transparent boundary conditions are used. The scheme was implemented for anisotropic waveguides with a diagonal permitivity tensor and examined by using samples with various refractive index profiles, ranging from simple step- and graded-index up to complicated refractive index profile structures composed of either isotropic or anisotropic materials. For simple cases where the results of other methods are available in the literature, the proposed scheme shows very good agreement.     

A primitive-variable Riemann method for solution of the shallow water equations with wetting and drying

A Riemann flux that uses primitive variables rather than conserved variables is developed for the shallow water equations with nonuniform bathymetry. This primitive-variable flux is both conservative and well behaved at zero depth. The unstructured finite-volume discretization used is suitable for highly nonuniform grids that provide resolution of complex geometries and localized flow structures. A source-term discretization is derived for nonuniform bottom that balances the discrete flux integral both for still water and in dry regions. This primitive-variable formulation is uniformly valid in wet and dry regions with embedded wetting and drying fronts. A fully nonlinear implicit scheme and both nonlinear and time-linearized explicit schemes are developed for the time integration. The implicit scheme is solved by a parallel Newton-iterative algorithm with numerically computed flux Jacobians. A concise treatment of characteristic-variable boundary conditions with source terms is also given. Computed results obtained for the one-dimensional dam break on wet and dry beds and for normal-mode oscillations in a circular parabolic basin are in very close agreement with the analytical solutions. Other results for a forced breaking wave with friction interacting with a sloped bottom demonstrate a complex wave motion with wetting, drying and multiple interacting wave fronts. Finally, a highly nonuniform, coastline-conforming unstructured grid is used to demonstrate an unsteady simulation that models an artificial coastal flooding due to a forced wave entering the Gulf of Mexico.     

Lattice Boltzmann computational fluid dynamics in three dimensions

The recent development of the lattice gas method and its extension to the lattice Boltzmann method have provided new computational schemes for fluid dynamics. Both methods are fully paralleled and can easily model many different physical problems, including flows with complicated boundary conditions. In this paper, basic principles of a lattice Boltzmann computational method are described and applied to several three-dimensional benchmark problems. In most previous lattice gas and lattice Boltzmann methods, a face-centered-hyper-cubic lattice in four-dimensional space was used to obtain an isotropic stress tensor. To conserve computer memory, we develop a model which requires 14 moving directions instead of the usual 24 directions. Lattice Boltzmann models, describing two-phase fluid flows and magnetohydrodynamics, can be developed based on this simpler 14-directional lattice. Comparisons between three-dimensional spectral code results and results using our method are given for simple periodic geometries. An important property of the lattice Boltzmann method is that simulations for flow in simple and complex geometries have the same speed and efficiency, while all other methods, including the spectral method, are unable to model complicated geometries efficiently.     

A mass,energy, vorticity,and potential enstrophy conserving lateral boundary scheme for the shallow water equations using piecewise linear boundary approximations

A numerical scheme for treating fluid–land boundaries in inviscid shallow water flows is derived that approximates boundary profiles with piecewise linear segments (shaved cells) while conserving the domain-summed mass, energy, vorticity, and potential enstrophy. The new scheme is a generalization of a previous scheme that also conserves these quantities but uses stairsteps to approximate boundary profiles. Numerical simulations are carried out demonstrating the conservation properties and accuracy of the piecewise linear boundary scheme (the PLS) for inviscid flows and comparing its performance with that of the stairstep scheme (the STS). It is found that while both schemes conserve all four domain-summed quantities, the PLS generates depth and velocity fields that are one-half to one order more accurate than those generated by the STS, and it generates vorticity and potential vorticity fields that are at least as accurate as those generated by the STS and often more accurate. The higher accuracy of the PLS is due to its ability to generate smoother flow fields near boundaries of arbitrary shape.     

On the outflow conditions for spectral solution of the viscous blunt-body problem

The purpose of this paper is to study and identify suitable outflow boundary conditions for the numerical simulation of viscous supersonic/hypersonic flow over blunt bodies, governed by the compressible Navier–Stokes equations, with an emphasis motivated primarily by the use of spectral methods without any filtering. The subsonic/supersonic composition of the outflow boundary requires a dual boundary treatment for well-posedness. All compatibility relations, modified to undertake the hyperbolic/parabolic behaviour of the governing equations, are used for the supersonic part of the outflow. Regarding the unknown downstream information in the subsonic region, different subsonic outflow conditions in the sense of the viscous blunt-body problem are examined. A verification procedure is conducted to make out the distinctive effect of each outflow condition on the solution. Detailed comparisons are performed to examine the accuracy and performance of the outflow conditions considered for two model geometries of different surface curvature variations. Numerical simulations indicate a noticeable influence of pressure from subsonic portion to supersonic portion of the boundary layer. It is demonstrated that two approaches for imposing subsonic outflow conditions namely (1) extrapolating all flow variables and (2) extrapolation of pressure along with using proper compatibility relations are more suitable than the others for accurate numerical simulation of viscous high-speed flows over blunt bodies using spectral collocation methods.