Numerical Modeling of the Hyperbolic Mild-Slope Equation in Curvilinear Coordinates

The mild-slope equation is familiar to coastal engineers as it can effectively describe wave propagation in nearshore regions. However, its computational method in Cartesian coordinates often renders the model inaccurate in areas with irregular shorelines, such as estuaries and harbors. Based on the hyperbolic mild-slope equation in Cartesian coordinates, the numerical model in orthogonal curvilinear coordinates is developed. The transformed model is discretized by the finite difference method and solved by the ADI method with space-staggered grids. The numerical predictions in curvilinear co- ordinates show good agreemenl with the data obtained in three typical physical expedments, which demonstrates that the present model can be used to simulate wave propagation, for normal incidence and oblique incidence, in domains with complicated topography and boundary conditions.

Numerical Simulation of Wave-Induced Currents Combined with Parabolic Mild-Slope Equation in Curvilinear Coordinates

Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.

High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory(WENO)scheme.The exact C-property is investigated,and comparison with the standard finite difference WENO scheme is made.Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems.The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems,indicating smaller errors compared with the Lax-Friedrichs solver.In addition,we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.

Performance of Geometric Multigrid Method for Two-Dimensional Burgers’Equations with Non-Orthogonal,Structured Curvilinear Grids

This paper seeks to develop an efficient multigrid algorithm for solving the Burgers problem with the use of non-orthogonal structured curvilinear grids in L-shaped geometry.For this,the differential equations were discretized by Finite Volume Method(FVM)with second-order approximation scheme and deferred correction.Moreover,the algebraic method and the differential method were used to generate the non-orthogonal structured curvilinear grids.Furthermore,the influence of some parameters of geometric multigrid method,as well as lexicographical Gauss–Seidel(Lex-GS),η-line Gauss–Seidel(η-line-GS),Modified Strongly Implicit(MSI)and modified incomplete LU decomposition(MILU)solvers on the Central Processing Unit(CPU)time was investigated.Therefore,several parameters of multigrid method and solvers were tested for the problem,with the use of nonorthogonal structured curvilinear grids and multigrid method,resulting in an algorithm with the combination that achieved the best results and CPU time.The geometric multigrid method with Full Approximation Scheme(FAS),V-cycle and standard coarsening ratio for this problem were utilized.This article shows how to calculate the coordinates transformation metrics in the coarser grids.Results show that the MSI and MILU solvers are the most efficient.Moreover,theMSI solver is faster thanMILU for both grids generators;and the solutions are more accurate for the Burgers problem with grids generated using elliptic equations.

Electrodynamics in Curvilinear Coordinates and the Equation of a Geodesic Line

The creation of the theory of relativity, which discovered the equivalence of mass and energy, showed that the concept of a point charge, used in the formulation of Coulomb’s law, one of the basic laws of classical electrodynamics, contradicts the famous formula establishing the equivalence of mass and energy. But the discovery of quarks makes it possible to present classical electrodynamics in a form free from the indicated contradiction. In the article, having considered the electromagnetic field in a curvilinear coordinate system, a theory has been created that expands our understanding of the electromagnetic field, the nature of quarks, the nature of strong interaction, and the connection between strong interaction and electromagnetic interaction. This theory is based on the principle of equivalence of an electromagnetic field to a free material particle formulated in the article and the law of formation of elementary particles from an electromagnetic field that follows from it.

A Numerical Model of Nonlinear Wave Propagation in Curvilinear Coordinates

For the simulation of the nonlinear wave propagation in coastal areas with complex boundaries, a numerical model is developed in curvilinear coordinates. In the model, the Boussinesq-type equations including the dissipation terms are em- ployed as the governing equations. In the present model, the dependent variables of the transformed equations are the free surface elevation and the utility velocity variables, instead of the usual primitive velocity variables. The introduction of utility velocity variables which are the products of the contravariant components of the velocity vector and the Jacobi ma- trix can make the transformed equations relatively concise, the treatment of lateral boundary conditions easier and the de- velopment of the program simpler. The predictor-corrector method and five-point finite-difference scheme are employed to discretize the time derivatives and the spatial ones, respectively. The numerical model is tested for three cases. It is found that the numerical results are in good agreement with the analytical results and experimental data.

NEW BOUNDARY ELEMENT METHOD FOR TORSION PROBLEMS OF CYLINDER WITH CURVILINEAR CRACKS

The Saint-Venant torsion problems of a cylinder with curvilinear cracks were considered and reduced to solving the boundary integral equations only on cracks. Using the interpolation models for both singular crack tip elements and other crack linear elements, the boundary element formulas of the torsion rigidity and stress intensity factors were given. Some typical torsion problems of a cylinder involving a straight, kinked or curvilinear crack were calculated. The obtained results for the case of straight crack agree well with those given by using the Gauss-Chebyshev integration formulas, which demonstrates the validity and applicability of the present boundary element method.

Fracture analysis for torsion problems of a composite cylinder with curvilinear cracks

The Salnt-Venant torsion problems of a composite cylinder with curvilinear cracks were investigated. By considering the bimaterial interface as a boundary of the outer bar or inner one, the problem was reduced to the solution of boundary integral equations on the crack, external boundary and interface. Using the new boundary element method, some typical torsion problems of a composite cylinder involving a straight or kinked crack were calculated. The obtained results were compared with data in the literature to show validity and applicability of the present method.

NUMERICAL SOLUTION OF THE UNSTEADYINCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON THECURVILINEAR HALF-STAGGERED MESHNUMERICAL SOLUTION OF THE UNSTEADYINCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON THECURVILINEAR HALF-STAGGERED MESH

In this paper, the Crank-Nicholson + component-consistent pressure correction method for the numerical solution of the unsteady incompressible Navier-Stokes equation of [1] on the rectangular half-Staggered mesh has been extended to the curvilinear half-Staggered mesh. The discrete projection, both for the projection step in the solution procedure and for the related differential-algebraic equations, has been carefully studied and verified. It is proved that the proposed method is also unconditionally (in t) nonlinearly stable on the curvilinear mesh, provided the mesh is not too skewed. It is seen that for problems with an outflow boundary, the half-Staggered mesh is especially advantageous. Results of preliminary numerical experiments support these claims.

WATER WAVE SIMULATION IN CURVILINEAR COORDINATES USING A TIME-DEPENDENT MILD SLOPE EQUATION

The purpose of this article is to model the detailed progress of wave propagation in curvilinear coordinates with an effective time-dependent mild slope equation. This was achieved in the following approach, firstly deriving the numerical model of the equation, i.e., Copeland＇s hyperbolic mild-slope equation, in orthogonal curvilinear coordinates based on principal of coordinate transformation, and then finding the numerical solution of the transformed model by use of the Alternative Directions Implicit （ADI） method with a space-staggered grid. To test the curvilinear model, two cases of a channel with varying cross section and a semi-circular channel were studied with corresponding analytical solutions. The model was further investigated through a numerical simulation in Ponce de Leon Inlet, USA. Good agreement is reached and therefore, the use of the present model is valid to calculate the progress of wave propagation in areas with curved shorelines, nearshore breakwaters and other complicated geometries.

A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations

Based on the high order essentially non-oscillatory(ENO)Lagrangian type scheme on quadrilateral meshes presented in our earlier work[3],in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics.The main purpose of this work is to demonstrate our claim in[3]that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges,which restricts the accuracy of the resulting scheme to at most second order.The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes.Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties.

傅里叶变换红外测定氧化淀粉的羧基含量初探

采用傅里叶变换红外测试氧化淀粉中的羧基含量，以国标法测得的羧基含量为纵坐标，谱图中羰基吸光度与C- H键吸光度比值为横坐标绘制得到标准曲线。对未知氧化淀粉产物进行红外测试，得到吸光度比值通过标准曲线方程，计算其羧基含量，得到的羧基含量与采用国标法测试的羧基含量进行对比，误差在2％～4％之间。为了提高实验的准确度，选取标样绘制标准曲线时，必须保证待测氧化淀粉的羧基含量在标准曲线计算的极限值范围之内，且偏离标准曲线的极限值。与国标法测试羧基含量相比，采用红外测试，方法简单易操作，效率高，准确度好。因此采用红外光谱测试方法对氧化淀粉的羧基含量的预测具有一定重要意义。

三峡工程大江截流流场的数值模拟

通过数值求解二维浅水方程组，模拟三峡工程大江截流过程中的流场，可快速获得任一口门宽度下的全域流场信息，计算结果合理，可为截流方案和施工决策提供依据．

定边界点二维正交曲线网格的数值生成方法

使用一种可以在边界上任意确定网格点的方法生成二维正交曲线网格.结果表明:该法以求解拉普拉斯方程组为基础,物理概念明确,且无需构造"合并"或"聚集"控制函数,使得方程离散简单,经验性因素降低;该法网格线与边界的正交性良好,可随意控制网格的疏密度,而且利用这一特性可将分汊区域或多连域分割成多个单连域分别进行求解,使几何图形复杂的计算区域网格的生成得到简化.

采用Poisson方程生成曲线网格时源项P、Q选择的研究

采用 Poisson方程进行曲线网格生成时 ,如何确定合适的调节因子 P、Q函数是网格生成技术中的一个重要的研究内容。本文提出一种新的构造 P、Q函数的方法 ,该方法直接利用边界网格节点的分布信息来控制区域内部网格节点的分布 ,生成的正交曲线网格令人满意。该方法可应用于河道、湖泊等一类复杂边界的二维流速场的数值模拟中。

边界拟合坐标网格生成方法研究

指出了Thompson及Thomas曲线网格生成方法中控制网格分布的调节函数的问题所在,克服了Thomas曲线网格生成法中边界局部平直近似假定的缺陷,经过严格推导得出一组新的调节函数,并给出了曲线网格生成实例。实例检验表明,该调节函数能够对复杂边界的单连通域或多连通域生成理想的曲线网格,即边界处网格正交,内部网格分布能够适应物理量场变化。

正交曲线坐标系二维浅水方程ELADI有限差分方法

N-S方程数值模拟的精度和效率一直是计算流体力学的重要研究课题。结合欧拉-拉格朗日方法(ELM)和交替方向隐式方法(AD I)建立正交曲线坐标系二维浅水方程的ELAD I(Eu lerian-Lagrangian alternating d irection imp lic itm ethod)有限差分方法,详细阐述了基本原理和离散方法,分析了ELM方法的数值扩散特性,并通过室内水槽和天然河道资料与传统AD I方法进行了验证比较。模拟结果表明,ELAD I方法在达到一定计算精度的同时,计算效率显著提高,对于某些算例,比传统AD I方法提高近90%,Courant数可以达到40以上。

曲线坐标系下哈密顿体系的建立

在任意曲线坐标系下建立了曲六面体区域的哈密顿体系，扩大了哈密顿体系的适用范围，并使系统方程具备了处理任意复杂边界条件的能力．

利用二、三维嵌套技术数值模拟复杂边界下的流场

通过数值模拟复杂几何区域和边界条件下二、三维嵌套紊流模型，获得了计算区域内任意一点的流场信息。这一方法克服了常规的单纯模拟二维或三维紊流控制方程组而无法获得准确的自由表面和三维紊流场的缺点，通过两者有机的结合，并采用动边界处理等先进的计算技术，得出了精度更高、更为合理的紊流场。该方法曾被用于三峡工程大江截流完成之后导流明渠通航的流场模拟中，计算结果合理可信，部分成果的精度也已得到了实测资料的验证。

用于起伏地表条件下地球物理场数值模拟的正交曲网格生成技术

起伏地表条件下地球物理场数值模拟所面临的主要困难在于如何求解自由边界(地—空界面)上的边界条件。如今虽有多种解决途径,如映射法、插值法和不规则网格法,但均存在一些问题,本文针对上述方法的问题提出了一种新的解决途径。首先引入了在计算流体力学中发展起来的贴体正交网格技术,通过对弹性波场和直流电场在地—空界面上所应满足的微分方程(边界条件)的分析,得到了正交曲网格优于非正交曲网格的结论;然后利用基于泊松方程的网格生成方法,采用Hilgenstock法控制源项,并结合代数辅助方法生成了正交贴体曲网格。结果表明,本文所用的网格生成方法适用于解决起伏地表条件下地球物理场的数值模拟问题。