The aim of this paper is to introduce a new semi-analytical method named precise integration method of lines(PIMOL),which is developed and used to solve the ordinary differential equation(ODE)systems based on the fini...The aim of this paper is to introduce a new semi-analytical method named precise integration method of lines(PIMOL),which is developed and used to solve the ordinary differential equation(ODE)systems based on the finite difference method of lines and the precise integration method.The irregular domain problem is mainly discussed in this paper.Three classical examples of Poisson^equation problems are given,including one regular and two irregular domain examples.The PIMOL reduces a semi-discrete ODE problem to a linear algebraic matrix equation and does not require domain mapping for treating the irregular domain problem.Numerical results show that the PIMOL is a powerful method.展开更多
In this work,we adapt and compare implicity linear collocation method and iterated implicity linear collocation method for solving nonlinear two dimensional Fredholm integral equations of Hammerstein type using IMQ-RB...In this work,we adapt and compare implicity linear collocation method and iterated implicity linear collocation method for solving nonlinear two dimensional Fredholm integral equations of Hammerstein type using IMQ-RBFs on a non-rectangular domain.IMQs show to be the most promising RBFs for this kind of equations.The proposed methods are mesh-free and they are independent of the geometry of domain.Convergence analysis of the proposed methods together with some benchmark examples are provided which support their reliability and numerical stability.展开更多
An indirect-forcing immersed boundary method for solving the incompressible Navier-Stokes equations involving the interfaces and irregular domains is developed.The rigid boundaries and interfaces are represented by a ...An indirect-forcing immersed boundary method for solving the incompressible Navier-Stokes equations involving the interfaces and irregular domains is developed.The rigid boundaries and interfaces are represented by a number of Lagrangian control points.Stationary rigid boundaries are embedded in the Cartesian grid and singular forces at the rigid boundaries are applied to impose the prescribed velocity conditions.The singular forces at the interfaces and the rigid boundaries are then distributed to the nearby Cartesian grid points using the immersed boundary method.In the present work,the singular forces at the rigid boundaries are computed implicitly by solving a small system of equations at each time step to ensure that the prescribed velocity condition at the rigid boundary is satisfied exactly.For deformable interfaces,the forces that the interface exerts on the fluid are computed from the configuration of the elastic interface and are applied to the fluid.The Navier-Stokes equations are discretized using finite difference method on a staggered uniform Cartesian grid by a second order accurate projection method.The ability of the method to simulate viscous flows with interfaces on irregular domains is demonstrated by applying to the rotational flow problem,the relaxation of an elastic membrane and flow in a constriction with an immersed elastic membrane.展开更多
The spatial domain of Molecular Dynamics simulations is usually a regular box that can be easily divided in subdomains for parallel processing.Recent efforts aimed at simulating complex biological systems,like the blo...The spatial domain of Molecular Dynamics simulations is usually a regular box that can be easily divided in subdomains for parallel processing.Recent efforts aimed at simulating complex biological systems,like the blood flow inside arteries,require the execution of Parallel Molecular Dynamics(PMD)in vessels that have,by nature,an irregular shape.In those cases,the geometry of the domain becomes an additional input parameter that directly influences the outcome of the simulation.In this paper we discuss the problems due to the parallelization of MD in complex geometries and show an efficient and general method to perform MD in irregular domains.展开更多
Tension Leg Platform(TLP) is a hybrid structure used as oil drilling and production facility within water depths of 1200 m. The extension of this TLP concept to deeper waters is a challenge and warrants for some inn...Tension Leg Platform(TLP) is a hybrid structure used as oil drilling and production facility within water depths of 1200 m. The extension of this TLP concept to deeper waters is a challenge and warrants for some innovative design concepts. In this paper, a relatively new concept of TLP which is christened as Tension-Based Tension Leg Platform(TBTLP) and patented by Srinivasan(1998) has been chosen for study. Response analysis of TLP with one tension base under irregular waves for three different sea states has been performed using hydrodynamic tool ANSYS? AQWA?. Results are reported in terms of RAOs, response spectrums for surge, heave and pitch degrees of freedom from which spectral statistics have been obtained. The statistics of TBTLP have been compared with TLPs(without tension base) for two different water depths to highlight the features of the new concept. The effect of viscous damping and loading effects on the RAOs are also investigated.展开更多
A local refinement hybrid scheme(LRCSPH-FDM)is proposed to solve the two-dimensional(2D)time fractional nonlinear Schrodinger equation(TF-NLSE)in regularly or irregularly shaped domains,and extends the scheme to predi...A local refinement hybrid scheme(LRCSPH-FDM)is proposed to solve the two-dimensional(2D)time fractional nonlinear Schrodinger equation(TF-NLSE)in regularly or irregularly shaped domains,and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross-Pitaevskii equation(TF-GPE)with the rotating Bose-Einstein condensate.It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge.The proposed LRCSPH-FDM(which is based on a local refinement corrected SPH method combined with FDM)is derived by using the finite difference scheme(FDM)to discretize the Caputo TF term,followed by using a corrected smoothed particle hydrodynamics(CSPH)scheme continuously without using the kernel derivative to approximate the spatial derivatives.Meanwhile,the local refinement technique is adopted to reduce the numerical error.In numerical simulations,the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method.The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D(where 1D stands for one-dimensional)analytical TF-NLSEs in a rectangular region(with regular or irregular particle distribution)or in a region with irregular geometry.The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain,and the results from the posed method are compared with those from the FDM.All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.展开更多
A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth...A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains.In the solution of differential equations,various derivatives of the unknown function are denoted as new functions.Then,the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals.Therefore,the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative.During the application of the proposed method,boundary conditions can be automatically included in the integration operations,and relevant matrices can be assured to exhibit perfect sparse patterns.As examples,we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes.By comparing the solutions obtained by the proposed method with the exact solutions,the new multiresolution method is found to have a convergence rate of fifth order.The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method(FEM)with tens of thousands of elements.In addition,because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order,we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.展开更多
无网格局部彼得洛夫-伽辽金(meshless local Petrov-Galerkin,MLPG)法是一种具有代表性的无网格方法,在计算力学领域得到广泛应用.然而,这种方法在边界上需执行积分运算,通常很难处理不规则求解域问题.为了克服MLPG法的这种局限性,提出...无网格局部彼得洛夫-伽辽金(meshless local Petrov-Galerkin,MLPG)法是一种具有代表性的无网格方法,在计算力学领域得到广泛应用.然而,这种方法在边界上需执行积分运算,通常很难处理不规则求解域问题.为了克服MLPG法的这种局限性,提出了无网格局部强弱(meshless local strong-weak,MLSW)法.MLSW法采用MLPG法离散内部求解域,采用无网格介点(meshless intervention-point,MIP)法施加自然边界条件,并采用配点法施加本质边界条件,避免执行边界积分运算,可适用于求解各类复杂的不规则域问题.从理论上讲,这种结合式方法,既保持了MLPG法稳定而精确计算的优势,同时兼备配点型方法在处理复杂结构问题时简洁而灵活的优势,实现了弱式法和强式法的优势互补.此外,MLSW法采用移动最小二乘核(moving least squares core,MLSc)近似法来构造形函数,是对传统移动最小二乘(moving least squares,MLS)近似法的一种改进.MLSc使用核基函数代替通常的基函数,有利于数值求解的精确性和稳定性,而且其导数近似计算变得更为简单.数值算例结果初步表明:这种新方法实施简单,求解稳定、精确,表现出适合工程运用的潜力.展开更多
文摘The aim of this paper is to introduce a new semi-analytical method named precise integration method of lines(PIMOL),which is developed and used to solve the ordinary differential equation(ODE)systems based on the finite difference method of lines and the precise integration method.The irregular domain problem is mainly discussed in this paper.Three classical examples of Poisson^equation problems are given,including one regular and two irregular domain examples.The PIMOL reduces a semi-discrete ODE problem to a linear algebraic matrix equation and does not require domain mapping for treating the irregular domain problem.Numerical results show that the PIMOL is a powerful method.
文摘In this work,we adapt and compare implicity linear collocation method and iterated implicity linear collocation method for solving nonlinear two dimensional Fredholm integral equations of Hammerstein type using IMQ-RBFs on a non-rectangular domain.IMQs show to be the most promising RBFs for this kind of equations.The proposed methods are mesh-free and they are independent of the geometry of domain.Convergence analysis of the proposed methods together with some benchmark examples are provided which support their reliability and numerical stability.
基金The last author’s research is supported by the grant AcRF RG59/08 M52110092.
文摘An indirect-forcing immersed boundary method for solving the incompressible Navier-Stokes equations involving the interfaces and irregular domains is developed.The rigid boundaries and interfaces are represented by a number of Lagrangian control points.Stationary rigid boundaries are embedded in the Cartesian grid and singular forces at the rigid boundaries are applied to impose the prescribed velocity conditions.The singular forces at the interfaces and the rigid boundaries are then distributed to the nearby Cartesian grid points using the immersed boundary method.In the present work,the singular forces at the rigid boundaries are computed implicitly by solving a small system of equations at each time step to ensure that the prescribed velocity condition at the rigid boundary is satisfied exactly.For deformable interfaces,the forces that the interface exerts on the fluid are computed from the configuration of the elastic interface and are applied to the fluid.The Navier-Stokes equations are discretized using finite difference method on a staggered uniform Cartesian grid by a second order accurate projection method.The ability of the method to simulate viscous flows with interfaces on irregular domains is demonstrated by applying to the rotational flow problem,the relaxation of an elastic membrane and flow in a constriction with an immersed elastic membrane.
文摘The spatial domain of Molecular Dynamics simulations is usually a regular box that can be easily divided in subdomains for parallel processing.Recent efforts aimed at simulating complex biological systems,like the blood flow inside arteries,require the execution of Parallel Molecular Dynamics(PMD)in vessels that have,by nature,an irregular shape.In those cases,the geometry of the domain becomes an additional input parameter that directly influences the outcome of the simulation.In this paper we discuss the problems due to the parallelization of MD in complex geometries and show an efficient and general method to perform MD in irregular domains.
文摘Tension Leg Platform(TLP) is a hybrid structure used as oil drilling and production facility within water depths of 1200 m. The extension of this TLP concept to deeper waters is a challenge and warrants for some innovative design concepts. In this paper, a relatively new concept of TLP which is christened as Tension-Based Tension Leg Platform(TBTLP) and patented by Srinivasan(1998) has been chosen for study. Response analysis of TLP with one tension base under irregular waves for three different sea states has been performed using hydrodynamic tool ANSYS? AQWA?. Results are reported in terms of RAOs, response spectrums for surge, heave and pitch degrees of freedom from which spectral statistics have been obtained. The statistics of TBTLP have been compared with TLPs(without tension base) for two different water depths to highlight the features of the new concept. The effect of viscous damping and loading effects on the RAOs are also investigated.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11501495,51779215,and 11672259)the Postdoctoral Science Foundation of China(Grant Nos.2015M581869 and 2015T80589)the Jiangsu Government Scholarship for Overseas Studies,China(Grant No.JS-2017-227)。
文摘A local refinement hybrid scheme(LRCSPH-FDM)is proposed to solve the two-dimensional(2D)time fractional nonlinear Schrodinger equation(TF-NLSE)in regularly or irregularly shaped domains,and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross-Pitaevskii equation(TF-GPE)with the rotating Bose-Einstein condensate.It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge.The proposed LRCSPH-FDM(which is based on a local refinement corrected SPH method combined with FDM)is derived by using the finite difference scheme(FDM)to discretize the Caputo TF term,followed by using a corrected smoothed particle hydrodynamics(CSPH)scheme continuously without using the kernel derivative to approximate the spatial derivatives.Meanwhile,the local refinement technique is adopted to reduce the numerical error.In numerical simulations,the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method.The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D(where 1D stands for one-dimensional)analytical TF-NLSEs in a rectangular region(with regular or irregular particle distribution)or in a region with irregular geometry.The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain,and the results from the posed method are compared with those from the FDM.All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.
基金Project supported by the National Natural Science Foundation of China(No.11925204)the 111 Project(No.B14044)。
文摘A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains.In the solution of differential equations,various derivatives of the unknown function are denoted as new functions.Then,the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals.Therefore,the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative.During the application of the proposed method,boundary conditions can be automatically included in the integration operations,and relevant matrices can be assured to exhibit perfect sparse patterns.As examples,we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes.By comparing the solutions obtained by the proposed method with the exact solutions,the new multiresolution method is found to have a convergence rate of fifth order.The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method(FEM)with tens of thousands of elements.In addition,because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order,we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.
文摘无网格局部彼得洛夫-伽辽金(meshless local Petrov-Galerkin,MLPG)法是一种具有代表性的无网格方法,在计算力学领域得到广泛应用.然而,这种方法在边界上需执行积分运算,通常很难处理不规则求解域问题.为了克服MLPG法的这种局限性,提出了无网格局部强弱(meshless local strong-weak,MLSW)法.MLSW法采用MLPG法离散内部求解域,采用无网格介点(meshless intervention-point,MIP)法施加自然边界条件,并采用配点法施加本质边界条件,避免执行边界积分运算,可适用于求解各类复杂的不规则域问题.从理论上讲,这种结合式方法,既保持了MLPG法稳定而精确计算的优势,同时兼备配点型方法在处理复杂结构问题时简洁而灵活的优势,实现了弱式法和强式法的优势互补.此外,MLSW法采用移动最小二乘核(moving least squares core,MLSc)近似法来构造形函数,是对传统移动最小二乘(moving least squares,MLS)近似法的一种改进.MLSc使用核基函数代替通常的基函数,有利于数值求解的精确性和稳定性,而且其导数近似计算变得更为简单.数值算例结果初步表明:这种新方法实施简单,求解稳定、精确,表现出适合工程运用的潜力.