Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of s...Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.展开更多
In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional...In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.展开更多
A 3D rankine panel method was developed for calculating the linear wave-making resistance of a tri-maran with Wigley hulls. In order to calculate the normal vector and derivative of the body surface accurately, non-un...A 3D rankine panel method was developed for calculating the linear wave-making resistance of a tri-maran with Wigley hulls. In order to calculate the normal vector and derivative of the body surface accurately, non-uniform rational B-spline (NURBS) was adopted to represent body surface and rankine source density. The radiation condition is satisfied using the numerical technology of staggered grids. Numerical results show that the linear wave-making resistance of the trimaran can be calculated effectively using this method.展开更多
The hull form optimization concerns one of the most important applications of wave-making resistance theories.In recent years,scholars can determine the hull form by using the optimization method based on the computat...The hull form optimization concerns one of the most important applications of wave-making resistance theories.In recent years,scholars can determine the hull form by using the optimization method based on the computational fluid dynamics(CFD)and other mathematical techniques.In this paper,the hull form optimization method based on the Rankine source method and nonlinear programming(NLP)is discussed;in the optimization process,a hull form modification function is introduced to represent an improved hull surface and to generate a new smooth hull surface by changing its frame lines and bow stem profiles under the prescribed design constraints. Numerical example is given for a practical container hull form.Finally,shape optimization of bow bulls is shown for non-protruding and protruding bow bulls.This study presents a simplified and practical design method to the select frame lines of bow bulls.展开更多
Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms.In our earlier work[31–33],we designed high order well-balanced schemes to a cl...Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms.In our earlier work[31–33],we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms.In this paper,we present a different approach to the same purpose:designing high order well-balanced finite volume weighted essentially non-oscillatory(WENO)schemes and RungeKutta discontinuous Galerkin(RKDG)finite element methods.We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly,if a small modification on either the initial condition or the flux is provided.The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method.The same idea can be applied to the finite volume WENO schemes.We will first describe the algorithms and prove the well balanced property for the shallow water equations,and then show that the result can be generalized to a class of other balance laws.We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions,the non-oscillatory property for general solutions with discontinuities,and the genuine high order accuracy in smooth regions.展开更多
In previous work, we have analyzed the feasibility of the estimation for a source term S（x, y, z） in a transversal section, The present study is concerned with a twodimensional inverse phase change problem. The goal...In previous work, we have analyzed the feasibility of the estimation for a source term S（x, y, z） in a transversal section, The present study is concerned with a twodimensional inverse phase change problem. The goal is the estimation of the dissipated heat flux in the liquid zone （reconstruction of a source term in the energy equation） from experimentally measured temperatures in the solid zone. This work has an application in the electron beam welding of steels of thickness about 8cm. The direct thermometallurgical problem is treated in a quasi steady two-dimensional longitudinal section （x, y）. The beam displacement is normally in the y direction. But in the quasi steady simulation, the beam is steady in the study section. The sample is divided in the axial direction z in few sections. At each section, a source term is defined with a part of the beam and creates a vaporized zone and a fused zone. The goal of this work is the rebuilding of the complete source term with the estimations at each section. In this paper, we analyze the feasibility of the estimation. For this work, we use only the simulated measurements without noise.展开更多
基金the National Science Council of Taiwan for funding this research (NSC 96-2221-E-019-061).
文摘Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.
基金supported by the National Natural Science Foundation of China(11961044)the Doctor Fund of Lan Zhou University of Technologythe Natural Science Foundation of Gansu Provice(21JR7RA214)。
文摘In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.
基金the National Natural Science Founda-tion of China (No. 10572094)the Special ResearchFund for the Doctoral Program of Higher Education(No. 20050248037)
文摘A 3D rankine panel method was developed for calculating the linear wave-making resistance of a tri-maran with Wigley hulls. In order to calculate the normal vector and derivative of the body surface accurately, non-uniform rational B-spline (NURBS) was adopted to represent body surface and rankine source density. The radiation condition is satisfied using the numerical technology of staggered grids. Numerical results show that the linear wave-making resistance of the trimaran can be calculated effectively using this method.
基金the National Natural Science Foundation of China(No.51009087)
文摘The hull form optimization concerns one of the most important applications of wave-making resistance theories.In recent years,scholars can determine the hull form by using the optimization method based on the computational fluid dynamics(CFD)and other mathematical techniques.In this paper,the hull form optimization method based on the Rankine source method and nonlinear programming(NLP)is discussed;in the optimization process,a hull form modification function is introduced to represent an improved hull surface and to generate a new smooth hull surface by changing its frame lines and bow stem profiles under the prescribed design constraints. Numerical example is given for a practical container hull form.Finally,shape optimization of bow bulls is shown for non-protruding and protruding bow bulls.This study presents a simplified and practical design method to the select frame lines of bow bulls.
基金supported by ARO grant W911NF-04-1-0291,NSF grant DMS-0510345 and AFOSR grant FA9550-05-1-0123.
文摘Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms.In our earlier work[31–33],we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms.In this paper,we present a different approach to the same purpose:designing high order well-balanced finite volume weighted essentially non-oscillatory(WENO)schemes and RungeKutta discontinuous Galerkin(RKDG)finite element methods.We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly,if a small modification on either the initial condition or the flux is provided.The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method.The same idea can be applied to the finite volume WENO schemes.We will first describe the algorithms and prove the well balanced property for the shallow water equations,and then show that the result can be generalized to a class of other balance laws.We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions,the non-oscillatory property for general solutions with discontinuities,and the genuine high order accuracy in smooth regions.
文摘In previous work, we have analyzed the feasibility of the estimation for a source term S（x, y, z） in a transversal section, The present study is concerned with a twodimensional inverse phase change problem. The goal is the estimation of the dissipated heat flux in the liquid zone （reconstruction of a source term in the energy equation） from experimentally measured temperatures in the solid zone. This work has an application in the electron beam welding of steels of thickness about 8cm. The direct thermometallurgical problem is treated in a quasi steady two-dimensional longitudinal section （x, y）. The beam displacement is normally in the y direction. But in the quasi steady simulation, the beam is steady in the study section. The sample is divided in the axial direction z in few sections. At each section, a source term is defined with a part of the beam and creates a vaporized zone and a fused zone. The goal of this work is the rebuilding of the complete source term with the estimations at each section. In this paper, we analyze the feasibility of the estimation. For this work, we use only the simulated measurements without noise.
