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
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
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
To achieve high parallel efficiency for the global MASNUM surface wave model, the algorithm of an irregular quasirectangular domain decomposition and related serializing of calculating points and data exchanging schem...To achieve high parallel efficiency for the global MASNUM surface wave model, the algorithm of an irregular quasirectangular domain decomposition and related serializing of calculating points and data exchanging schemes are developed and conducted, based on the environment of Message Passing Interface(MPI). The new parallel version of the surface wave model is tested for parallel computing on the platform of the Sunway BlueLight supercomputer in the National Supercomputing Center in Jinan. The testing involves four horizontal resolutions, which are 1°×1°,(1/2)°×(1/2)°,(1/4)°×(1/4)°, and(1/8)°×(1/8)°. These tests are performed without data Input/Output(IO) and the maximum amount of processors used in these tests reaches to 131072. The testing results show that the computing speeds of the model with different resolutions are all increased with the increasing of numbers of processors. When the number of processors is four times that of the base processor number, the parallel efficiencies of all resolutions are greater than 80%. When the number of processors is eight times that of the base processor number, the parallel efficiency of tests with resolutions of 1°×1°,(1/2)°×(1/2)° and(1/4)°×(1/4)° is greater than 80%, and it is 62% for the test with a resolution of(1/8)°×(1/8)° using 131072 processors, which is the nearly all processors of Sunway BlueLight. When the processor's number is 24 times that of the base processor number, the parallel efficiencies for tests with resolutions of 1°×1°,(1/2)°×(1/2)°, and(1/4)°×(1/4)° are 72%, 62%, and 38%, respectively. The speedup and parallel efficiency indicate that the irregular quasi-rectangular domain decomposition and serialization schemes lead to high parallel efficiency and good scalability for a global numerical wave model.展开更多
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
Compact higher-order(HO)schemes for a new finite difference method,referred to as the Cartesian cut-stencil FD method,for the numerical solution of the convection-diffusion equation in complex shaped domains have been...Compact higher-order(HO)schemes for a new finite difference method,referred to as the Cartesian cut-stencil FD method,for the numerical solution of the convection-diffusion equation in complex shaped domains have been addressed in this paper.The Cartesian cut-stencil FD method,which employs 1-D quadratic transformation functions to map a non-uniform(uncut or cut)physical stencil to a uniform computational stencil,can be combined with compact HO Pad´e-Hermitian formulations to produce HO cut-stencil schemes.The modified partial differential equation technique is used to develop formulas for the local truncation error for the cut-stencil HO formulations.The effect of various HO approximations for Neumann boundary conditions on the solution accuracy and global order of convergence are discussed.The numerical results for second-order and compact HO formulations of the Cartesian cut-stencil FD method have been compared for test problems using the method of manufactured solutions.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
In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19...In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimen- sional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function inte- gral is approximated by applying standard one dimensional high order numerical quadra- tures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the in this paper achieve or exceed the expected second to fourth-order methods implemented accuracy.展开更多
In this paper,numerical sensitivity analysis with respect to the Reynolds number for the flow past obstacle problem is presented.To carry out such analysis,at each time step,we need to solve the incompressible Navier-...In this paper,numerical sensitivity analysis with respect to the Reynolds number for the flow past obstacle problem is presented.To carry out such analysis,at each time step,we need to solve the incompressible Navier-Stokes equations on irregular domains twice,one for the primary variables;the other is for the sensitivity variables with homogeneous boundary conditions.The Navier-Stokes solver is the augmented immersed interface method for Navier-Stokes equations on irregular domains.One of the most important contribution of this paper is that our analysis can predict the critical Reynolds number at which the vortex shading begins to develop in the wake of the obstacle.Some interesting experiments are shown to illustrate how the critical Reynolds number varies with different geometric settings.展开更多
We present a solver for the Poisson-Boltzmann equation and demonstrate its applicability for biomolecular electrostatics computation.The solver uses a level set framework to represent sharp,complex interfaces in a sim...We present a solver for the Poisson-Boltzmann equation and demonstrate its applicability for biomolecular electrostatics computation.The solver uses a level set framework to represent sharp,complex interfaces in a simple and robust manner.It also uses non-graded,adaptive octree grids which,in comparison to uniform grids,drastically decrease memory usage and runtime without sacrificing accuracy.The basic solver was introduced in earlier works[16,27],and here is extended to address biomolecular systems.First,a novel approach of calculating the solvent excluded and the solvent accessible surfaces is explained;this allows to accurately represent the location of the molecule’s surface.Next,a hybrid finite difference/finite volume approach is presented for discretizing the nonlinear Poisson-Boltzmann equation and enforcing the jump boundary conditions at the interface.Since the interface is implicitly represented by a level set function,imposing the jump boundary conditions is straightforward and efficient.展开更多
The parallel implementation of MUPHY,a concurrent multiscale code for large-scale hemodynamic simulations in anatomically realistic geometries,for multi-GPU platforms is presented.Performance tests show excellent resu...The parallel implementation of MUPHY,a concurrent multiscale code for large-scale hemodynamic simulations in anatomically realistic geometries,for multi-GPU platforms is presented.Performance tests show excellent results,with a nearly linear parallel speed-up on up to 32GPUs and a more than tenfold GPU/CPU acceleration,all across the range of GPUs.The basic MUPHY scheme combines a hydrokinetic(Lattice Boltzmann)representation of the blood plasma,with a Particle Dynamics treatment of suspended biological bodies,such as red blood cells.To the best of our knowledge,this represents the first effort in the direction of laying down general design principles for multiscale/physics parallel Particle Dynamics applications in non-ideal geometries.This configures the present multi-GPU version of MUPHY as one of the first examples of a high-performance parallel code for multiscale/physics biofluidic applications in realistically complex geometries.展开更多
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.
基金supported by National Basic Research Program of China (Grant Nos. 2010CB950300, 2010CB950500)Public Science and Technology Research Funds Projects of Ocean (Grant No. 201105019)+1 种基金Key Supercomputing Science-Technology Project of Shandong Province of China (Grant No. 2011YD01107)Scientific Research Foundation of the First Institute of Oceanography, State Oceanic Administration (Grant No. GY02-2010G22)
文摘To achieve high parallel efficiency for the global MASNUM surface wave model, the algorithm of an irregular quasirectangular domain decomposition and related serializing of calculating points and data exchanging schemes are developed and conducted, based on the environment of Message Passing Interface(MPI). The new parallel version of the surface wave model is tested for parallel computing on the platform of the Sunway BlueLight supercomputer in the National Supercomputing Center in Jinan. The testing involves four horizontal resolutions, which are 1°×1°,(1/2)°×(1/2)°,(1/4)°×(1/4)°, and(1/8)°×(1/8)°. These tests are performed without data Input/Output(IO) and the maximum amount of processors used in these tests reaches to 131072. The testing results show that the computing speeds of the model with different resolutions are all increased with the increasing of numbers of processors. When the number of processors is four times that of the base processor number, the parallel efficiencies of all resolutions are greater than 80%. When the number of processors is eight times that of the base processor number, the parallel efficiency of tests with resolutions of 1°×1°,(1/2)°×(1/2)° and(1/4)°×(1/4)° is greater than 80%, and it is 62% for the test with a resolution of(1/8)°×(1/8)° using 131072 processors, which is the nearly all processors of Sunway BlueLight. When the processor's number is 24 times that of the base processor number, the parallel efficiencies for tests with resolutions of 1°×1°,(1/2)°×(1/2)°, and(1/4)°×(1/4)° are 72%, 62%, and 38%, respectively. The speedup and parallel efficiency indicate that the irregular quasi-rectangular domain decomposition and serialization schemes lead to high parallel efficiency and good scalability for a global numerical wave model.
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.
基金support from the Natural Sciences and Engineering Research Council of Canada through the Discovery Grants program.
文摘Compact higher-order(HO)schemes for a new finite difference method,referred to as the Cartesian cut-stencil FD method,for the numerical solution of the convection-diffusion equation in complex shaped domains have been addressed in this paper.The Cartesian cut-stencil FD method,which employs 1-D quadratic transformation functions to map a non-uniform(uncut or cut)physical stencil to a uniform computational stencil,can be combined with compact HO Pad´e-Hermitian formulations to produce HO cut-stencil schemes.The modified partial differential equation technique is used to develop formulas for the local truncation error for the cut-stencil HO formulations.The effect of various HO approximations for Neumann boundary conditions on the solution accuracy and global order of convergence are discussed.The numerical results for second-order and compact HO formulations of the Cartesian cut-stencil FD method have been compared for test problems using the method of manufactured solutions.
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In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimen- sional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function inte- gral is approximated by applying standard one dimensional high order numerical quadra- tures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the in this paper achieve or exceed the expected second to fourth-order methods implemented accuracy.
基金The first and second authors are partially supported by US-ARO grant 49308-MAUS-AFSOR grant FA9550-06-1-0241+2 种基金The second author is also partially supported by US-NSF grant DMS-0911434the US-NIH grant 096195-01,and CNSF 11071123The third author is partially supported by the Hong Kong RGC Grant HKBU201710。
文摘In this paper,numerical sensitivity analysis with respect to the Reynolds number for the flow past obstacle problem is presented.To carry out such analysis,at each time step,we need to solve the incompressible Navier-Stokes equations on irregular domains twice,one for the primary variables;the other is for the sensitivity variables with homogeneous boundary conditions.The Navier-Stokes solver is the augmented immersed interface method for Navier-Stokes equations on irregular domains.One of the most important contribution of this paper is that our analysis can predict the critical Reynolds number at which the vortex shading begins to develop in the wake of the obstacle.Some interesting experiments are shown to illustrate how the critical Reynolds number varies with different geometric settings.
基金supported in part by the W.M.Keck Foundation,by the Institute for Collaborative Biotechnologies through contract no.W911NF-09-D-0001 from the U.S.Army Research Officeby ONR under grant agreement N00014-11-1-0027+1 种基金by the National Science Foundation under grant agreement CHE 1027817by the Department of Energy under grant agreement DE-FG02-08ER15991.
文摘We present a solver for the Poisson-Boltzmann equation and demonstrate its applicability for biomolecular electrostatics computation.The solver uses a level set framework to represent sharp,complex interfaces in a simple and robust manner.It also uses non-graded,adaptive octree grids which,in comparison to uniform grids,drastically decrease memory usage and runtime without sacrificing accuracy.The basic solver was introduced in earlier works[16,27],and here is extended to address biomolecular systems.First,a novel approach of calculating the solvent excluded and the solvent accessible surfaces is explained;this allows to accurately represent the location of the molecule’s surface.Next,a hybrid finite difference/finite volume approach is presented for discretizing the nonlinear Poisson-Boltzmann equation and enforcing the jump boundary conditions at the interface.Since the interface is implicitly represented by a level set function,imposing the jump boundary conditions is straightforward and efficient.
文摘The parallel implementation of MUPHY,a concurrent multiscale code for large-scale hemodynamic simulations in anatomically realistic geometries,for multi-GPU platforms is presented.Performance tests show excellent results,with a nearly linear parallel speed-up on up to 32GPUs and a more than tenfold GPU/CPU acceleration,all across the range of GPUs.The basic MUPHY scheme combines a hydrokinetic(Lattice Boltzmann)representation of the blood plasma,with a Particle Dynamics treatment of suspended biological bodies,such as red blood cells.To the best of our knowledge,this represents the first effort in the direction of laying down general design principles for multiscale/physics parallel Particle Dynamics applications in non-ideal geometries.This configures the present multi-GPU version of MUPHY as one of the first examples of a high-performance parallel code for multiscale/physics biofluidic applications in realistically complex geometries.