In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body...In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.展开更多
This paper presents the extension of the global description approach of a discontinuous function, which is proposed in the previous paper, to a spectral domain decomposition method. This multi-domain spectral immersed...This paper presents the extension of the global description approach of a discontinuous function, which is proposed in the previous paper, to a spectral domain decomposition method. This multi-domain spectral immersed interlace method（IIM） divides the whole computation domain into the smooth and discontinuous parts. Fewer points on the smooth domains are used via taking advantage of the high accuracy property of the spectral method, but more points on the discontinuous domains are employed to enhance the resolution of the calculation. Two that the domain decomposition technique can placed around the discontinuity. The present reached, in spite of the enlarged computational discontinuous problems are tested to verify the present method. The results show reduce the error of the spectral IIM, especially when more collocation points are method is t：avorable for the reason that the same level of the accuracy can be domain.展开更多
An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in ...An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.展开更多
A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immerse...A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initial problem. These auxiliary unknowns allow imposing various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.展开更多
Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interface...Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed.Note that,while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature,there is a gap in dealing with Navier-Stokes equations,where the non-linear term includes an additional component involving the coordinateφ,even if the geometry and force term are axis-symmetric.Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity,in addition to the pole singularity at r=0.In the newly developed algorithm,we have derived the jump conditions under the cylindrical coordinates.The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates.The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface.As expected,the method is shown to be second-order accurate for the velocity.The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.展开更多
We study the L^l-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into ...We study the L^l-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L^l-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order L^l-error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].展开更多
In this paper we give proof of three binomial coefficient inequalities. These inequalities are key ingredients in [Wen and Jin, J. Comput. Math. 26, （2008）, 1-22] to establish the L^1-error estimates for the upwind ...In this paper we give proof of three binomial coefficient inequalities. These inequalities are key ingredients in [Wen and Jin, J. Comput. Math. 26, （2008）, 1-22] to establish the L^1-error estimates for the upwind difference scheme to the linear advection equations with a piecewise constant wave speed and a general interface condition, which were further used to establish the L^1-error estimates for a Hamiltonian-preserving scheme developed in [Jin and Wen, Commun. Math. Sci. 3, （2005）, 285-315] to the Liouville equation with piecewise constant potentials [Wen and Jin, SIAM J. Numer. Anal. 46, （2008）, 2688-2714].展开更多
In this paper,a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces...In this paper,a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface.The method is to apply the Taylor’s expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes.A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy.The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method.The major advantages of the present method are the special simplicity,the ability in handling the Dirichlet boundary conditions,and no need of the pressure boundary condition.The method can also preserve the volume conservation and the discrete divergence free condition very well.The numerical results show that the proposed method is second order accurate and efficient.展开更多
In this paper,an immersed interface method is presented to simulate the dynamics of inextensible interfaces in an incompressible flow.The tension is introduced as an augmented variable to satisfy the constraint of int...In this paper,an immersed interface method is presented to simulate the dynamics of inextensible interfaces in an incompressible flow.The tension is introduced as an augmented variable to satisfy the constraint of interface inextensibility,and the resulting augmented system is solved by the GMRES method.In this work,the arclength of the interface is locally and globally conserved as the enclosed region undergoes deformation.The forces at the interface are calculated from the configuration of the interface and the computed augmented variable,and then applied to the fluid through the related jump conditions.The governing equations are discretized on a MAC grid via a second-order finite difference scheme which incorporates jump contributions and solved by the conjugate gradient Uzawa-type method.The proposed method is applied to several examples including the deformation of a liquid capsule with inextensible interfaces in a shear flow.Numerical results reveal that both the area enclosed by interface and arclength of interface are conserved well simultaneously.These provide further evidence on the capability of the present method to simulate incompressible flows involving inextensible interfaces.展开更多
An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm.The iteration is constructed by intr...An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm.The iteration is constructed by introducing an unsteady term in the pressure Poisson equation.In each iteration step,a Helmholtz equation is solved on the Cartesian grid using FFT.The combination of the iteration and the immersed interface method enables the solver to handle various jump conditions across twofluid interfaces.This solver can also be used to solve Poisson equations on irregular domains.展开更多
The immersed interface technique is incorporated into CIP method to solve one-dimensional hyperbolic equations with piecewise constant coefficients.The proposed method achieves the third order of accuracy in time and ...The immersed interface technique is incorporated into CIP method to solve one-dimensional hyperbolic equations with piecewise constant coefficients.The proposed method achieves the third order of accuracy in time and space in the vicinity of the interface where the coefficients have jump discontinuities,which is the same order of accuracy of the standard CIP scheme.Some numerical tests are given to verify the accuracy of the proposed method.展开更多
A second order isoparametric finite element method （IPFEM） is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the...A second order isoparametric finite element method （IPFEM） is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.展开更多
An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources.The interface is repre...An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources.The interface is represented by the zero level set of a Lipschitz functionϕ(x,y).Our adaptive mesh refinement is done within a small tube of|ϕ(x,y)|δwith finer Cartesian meshes.The discrete linear system of equations is solved by a multigrid solver.The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically,therefore,reduce the size of the linear system of the equations.Numerical examples presented show the efficiency of the grid refinement strategy.展开更多
The presence of Dirac delta function in differential equation can lead to a discontinuity,which may degrade the accuracy of related numerical methods.To improve the accuracy,a secondorder numerical method for elliptic...The presence of Dirac delta function in differential equation can lead to a discontinuity,which may degrade the accuracy of related numerical methods.To improve the accuracy,a secondorder numerical method for elliptic equations with singular sources is introduced by employing a local kernel flter.In this method,the discontinuous equation is convoluted with the kernel function to obtain a more regular one.Then the original equation is replaced by this fltered equation around the singular points,to obtain discrete numerical form.The unchanged equations at the other points are discretized by using a central difference scheme.1D and 2D examples are carried out to validate the correctness and accuracy of the present method.The results show that a second-order of accuracy can be obtained in the fltering framework with an appropriate integration rule.Furthermore,the present method does not need any jump condition,and also has extremely simple form that can be easily extended to high dimensional cases and complex geometry.展开更多
This paper is to present a finite volume element(FVE)method based on the bilinear immersed finite element(IFE)for solving the boundary value problems of the diffusion equation with a discontinuous coefficient(interfac...This paper is to present a finite volume element(FVE)method based on the bilinear immersed finite element(IFE)for solving the boundary value problems of the diffusion equation with a discontinuous coefficient(interface problem).This method possesses the usual FVE method’s local conservation property and can use a structured mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient has discontinuity along piecewise smooth nontrivial curves.Numerical examples are provided to demonstrate features of this method.In particular,this method can produce a numerical solution to an interface problem with the usual O(h2)(in L2 norm)and O(h)(in H1 norm)convergence rates.展开更多
In this paper,we present a discontinuity and cusp capturing physicsinformed neural network(PINN)to solve Stokes equations with a piecewiseconstant viscosity and singular force along an interface.We first reformulate t...In this paper,we present a discontinuity and cusp capturing physicsinformed neural network(PINN)to solve Stokes equations with a piecewiseconstant viscosity and singular force along an interface.We first reformulate the governing equations in each fluid domain separately and replace the singular force effect with the traction balance equation between solutions in two sides along the interface.Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface,we hereby use a network consisting of two fully-connected sub-networks that approximate the pressure and velocity,respectively.The two sub-networks share the same primary coordinate input arguments but with different augmented feature inputs.These two augmented inputs provide the interface information,so we assume that a level set function is given and its zero level set indicates the position of the interface.The pressure sub-network uses an indicator function as an augmented input to capture the function discontinuity,while the velocity sub-network uses a cusp-enforced level set function to capture the derivative discontinuities via the traction balance equation.We perform a series of numerical experiments to solve two-and three-dimensional Stokes interface problems and perform an accuracy comparison with the augmented immersed interface methods in literature.Our results indicate that even a shallow network with a moderate number of neurons and sufficient training data points can achieve prediction accuracy comparable to that of immersed interface methods.展开更多
This paper applies bilinear immersed finite elements （IFEs） in the interior penalty discontinuous Galerkin （DG） methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous ...This paper applies bilinear immersed finite elements （IFEs） in the interior penalty discontinuous Galerkin （DG） methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous bihnear IFE space is constructed and applied to both the symmetric and nonsymmetric interior penalty DG formulations. The new methods can solve an interface problem on a Cartesian mesh independent of the interface with local refinement at any locations needed even if the interface has a nontrivial geometry. Numerical examples are provided to show features of these methods.展开更多
Abstract.In this paper,a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces.The velocity components ...Abstract.In this paper,a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces.The velocity components along the interface are introduced as two augmented variables and the resulting augmented equation is then solved by the GMRES method.The augmented variables and/or the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity,and are interpolated using cubic splines and are then applied to the fluid through the jump conditions.The Stokes equations are discretized on a staggered Cartesian grid via a second order finite difference method and solved by the conjugate gradient Uzawa-typemethod.The numerical results show that the overall scheme is second order accurate.The major advantages of the present IIM-Stokes solver are the efficiency and flexibility in terms of types of fluid flow and different boundary conditions.The proposed method avoids solution of the pressure Poisson equation,and comparisons are made to show the advantages of time savings by the present method.The generalized two-phase Stokes solver with correction terms has also been applied to incompressible two-phase Navier-Stokes flow.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
In this paper,we numerically investigate the effects of surfactant on dropdrop interactions in a 2D shear flow using a coupled level-set and immersed interface approach proposed in(Xu et al.,J.Comput.Phys.,212(2006),...In this paper,we numerically investigate the effects of surfactant on dropdrop interactions in a 2D shear flow using a coupled level-set and immersed interface approach proposed in(Xu et al.,J.Comput.Phys.,212(2006),590–616).We find that surfactant plays a critical and nontrivial role in drop-drop interactions.In particular,we find that the minimum distance between the drops is a non-monotone function of the surfactant coverage and Capillary number.This non-monotonic behavior,which does not occur for clean drops,is found to be due to the presence of Marangoni forces along the drop interfaces.This suggests that there are non-monotonic conditions for coalescence of surfactant-laden drops,as observed in recent experiments of Leal and co-workers.Although our study is two-dimensional,we believe that drop-drop interactions in three-dimensional flows should be qualitatively similar as the Maragoni forces in the near contact region in 3D should have a similar effect.展开更多
基金supported by the US ARO grants 49308-MA and 56349-MAthe US AFSOR grant FA9550-06-1-024+1 种基金he US NSF grant DMS-0911434the State Key Laboratory of Scientific and Engineering Computing of Chinese Academy of Sciences during a visit by Z.Li between July-August,2008.
文摘In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.
基金National Natural Science Foundation of China(51076006)
文摘This paper presents the extension of the global description approach of a discontinuous function, which is proposed in the previous paper, to a spectral domain decomposition method. This multi-domain spectral immersed interlace method（IIM） divides the whole computation domain into the smooth and discontinuous parts. Fewer points on the smooth domains are used via taking advantage of the high accuracy property of the spectral method, but more points on the discontinuous domains are employed to enhance the resolution of the calculation. Two that the domain decomposition technique can placed around the discontinuity. The present reached, in spite of the enlarged computational discontinuous problems are tested to verify the present method. The results show reduce the error of the spectral IIM, especially when more collocation points are method is t：avorable for the reason that the same level of the accuracy can be domain.
基金supported by the National Natural Science Foundation of China(No.51174236)the National Basic Research Program of China(973 Program)(No.2011CB606306)the Opening Project of State Key Laboratory of Porous Metal Materials(No.PMM-SKL-4-2012)
文摘An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.
文摘A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initial problem. These auxiliary unknowns allow imposing various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.
基金supported by Fundación Sáneca Grant 21728/EE/22(Este trabajo es resultado de la estancia financiada por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad,Colaboración Internacional e Intercambio de Conocimiento“Jiménez de la Espada”)B.Dong is partially supported by the National Natural Science Foundation of China Grant No.12261070+1 种基金Ningxia Key Research and Development Project of China Grant No.2022BSB03048Z.Li is partially supported by Simons Grant 633724 and by Fundación Séneca Grant 21760/IV/22(Este trabajo es resultado de la estancia financiada por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad,Colaboración Internacional Intercambio de Conocimiento“Jiménez de la Espada”).
文摘Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates.In this paper,the immersed interface method(IIM)tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed.Note that,while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature,there is a gap in dealing with Navier-Stokes equations,where the non-linear term includes an additional component involving the coordinateφ,even if the geometry and force term are axis-symmetric.Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity,in addition to the pole singularity at r=0.In the newly developed algorithm,we have derived the jump conditions under the cylindrical coordinates.The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates.The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface.As expected,the method is shown to be second-order accurate for the velocity.The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.
基金supported in part by the Knowledge Innovation Project of the Chinese Academy of Sciences Nos. K5501312S1 and K5502212F1, and NSFC grant No. 10601062supported in part by NSF grant Nos. DMS-0305081 and DMS-0608720, NSFC grant No. 10228101 and NSAF grant No. 10676017
文摘We study the L^l-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L^l-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order L^l-error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].
基金supported in part by the Knowledge Innovation Project of the Chinese Academy of Sciences grants K5501312S1,K5502212F1,K7290312G7 and K7502712F7NSFC grant 10601062
文摘In this paper we give proof of three binomial coefficient inequalities. These inequalities are key ingredients in [Wen and Jin, J. Comput. Math. 26, （2008）, 1-22] to establish the L^1-error estimates for the upwind difference scheme to the linear advection equations with a piecewise constant wave speed and a general interface condition, which were further used to establish the L^1-error estimates for a Hamiltonian-preserving scheme developed in [Jin and Wen, Commun. Math. Sci. 3, （2005）, 285-315] to the Liouville equation with piecewise constant potentials [Wen and Jin, SIAM J. Numer. Anal. 46, （2008）, 2688-2714].
基金supported by the Special Project on High-performance Computing under the National Key R&D Program(No.2016YFB0200604)National Natural Science Foundation of China(11971502,11571385)Guangdong Natural Science Foundation(2017A030313017).
文摘In this paper,a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface.The method is to apply the Taylor’s expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes.A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy.The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method.The major advantages of the present method are the special simplicity,the ability in handling the Dirichlet boundary conditions,and no need of the pressure boundary condition.The method can also preserve the volume conservation and the discrete divergence free condition very well.The numerical results show that the proposed method is second order accurate and efficient.
基金The authors would like to thank the referees for the valuable suggestions on the revision of the manuscript.The research of the first author was partially supported by Guangdong Provincial Government of China through the“Computational Science Innovative Research Team”program,the Sun Yat-sen University“Hundred Talents Program”(34000-3181201)the National Natural Science Foundation of China(No.11101446).
文摘In this paper,an immersed interface method is presented to simulate the dynamics of inextensible interfaces in an incompressible flow.The tension is introduced as an augmented variable to satisfy the constraint of interface inextensibility,and the resulting augmented system is solved by the GMRES method.In this work,the arclength of the interface is locally and globally conserved as the enclosed region undergoes deformation.The forces at the interface are calculated from the configuration of the interface and the computed augmented variable,and then applied to the fluid through the related jump conditions.The governing equations are discretized on a MAC grid via a second-order finite difference scheme which incorporates jump contributions and solved by the conjugate gradient Uzawa-type method.The proposed method is applied to several examples including the deformation of a liquid capsule with inextensible interfaces in a shear flow.Numerical results reveal that both the area enclosed by interface and arclength of interface are conserved well simultaneously.These provide further evidence on the capability of the present method to simulate incompressible flows involving inextensible interfaces.
基金the support of this work by the NSF grant DMS 0915237.
文摘An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm.The iteration is constructed by introducing an unsteady term in the pressure Poisson equation.In each iteration step,a Helmholtz equation is solved on the Cartesian grid using FFT.The combination of the iteration and the immersed interface method enables the solver to handle various jump conditions across twofluid interfaces.This solver can also be used to solve Poisson equations on irregular domains.
基金the Army Research Office under DAAD19-02-1-0394,US-ARO grant 49308-MA,and US-AFSOR grant FA9550-06-1-0241.
文摘The immersed interface technique is incorporated into CIP method to solve one-dimensional hyperbolic equations with piecewise constant coefficients.The proposed method achieves the third order of accuracy in time and space in the vicinity of the interface where the coefficients have jump discontinuities,which is the same order of accuracy of the standard CIP scheme.Some numerical tests are given to verify the accuracy of the proposed method.
基金Supported by the National Natural Science Foundation of China (11071216 and 11101361)
文摘A second order isoparametric finite element method （IPFEM） is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.
文摘An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources.The interface is represented by the zero level set of a Lipschitz functionϕ(x,y).Our adaptive mesh refinement is done within a small tube of|ϕ(x,y)|δwith finer Cartesian meshes.The discrete linear system of equations is solved by a multigrid solver.The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically,therefore,reduce the size of the linear system of the equations.Numerical examples presented show the efficiency of the grid refinement strategy.
基金supported by the National Natural Science Foundation in China(Grant Nos.51076006,11202013)BUAA SJP ‘‘111’’ Program(Grant No.B08009)+1 种基金the National Basic Research Program of China(2012CB720200)the Open Research Fund of MOE Key Lab-oratory of High-speed Railway Engineering,Southwest Jiao-tong University and the European Community’s Seventh Framework Program(FP7/2007-2013)under Grant agreement 225967‘‘NextMuSE’’
文摘The presence of Dirac delta function in differential equation can lead to a discontinuity,which may degrade the accuracy of related numerical methods.To improve the accuracy,a secondorder numerical method for elliptic equations with singular sources is introduced by employing a local kernel flter.In this method,the discontinuous equation is convoluted with the kernel function to obtain a more regular one.Then the original equation is replaced by this fltered equation around the singular points,to obtain discrete numerical form.The unchanged equations at the other points are discretized by using a central difference scheme.1D and 2D examples are carried out to validate the correctness and accuracy of the present method.The results show that a second-order of accuracy can be obtained in the fltering framework with an appropriate integration rule.Furthermore,the present method does not need any jump condition,and also has extremely simple form that can be easily extended to high dimensional cases and complex geometry.
基金This work is partially supported by NSF grant DMS-0713763 and NSERC(Canada).
文摘This paper is to present a finite volume element(FVE)method based on the bilinear immersed finite element(IFE)for solving the boundary value problems of the diffusion equation with a discontinuous coefficient(interface problem).This method possesses the usual FVE method’s local conservation property and can use a structured mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient has discontinuity along piecewise smooth nontrivial curves.Numerical examples are provided to demonstrate features of this method.In particular,this method can produce a numerical solution to an interface problem with the usual O(h2)(in L2 norm)and O(h)(in H1 norm)convergence rates.
基金supports by National Science and Technology Council,Taiwan,under research grants 111-2115-M-390-002 and 110-2115-M-A49-011-MY3,respectively.
文摘In this paper,we present a discontinuity and cusp capturing physicsinformed neural network(PINN)to solve Stokes equations with a piecewiseconstant viscosity and singular force along an interface.We first reformulate the governing equations in each fluid domain separately and replace the singular force effect with the traction balance equation between solutions in two sides along the interface.Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface,we hereby use a network consisting of two fully-connected sub-networks that approximate the pressure and velocity,respectively.The two sub-networks share the same primary coordinate input arguments but with different augmented feature inputs.These two augmented inputs provide the interface information,so we assume that a level set function is given and its zero level set indicates the position of the interface.The pressure sub-network uses an indicator function as an augmented input to capture the function discontinuity,while the velocity sub-network uses a cusp-enforced level set function to capture the derivative discontinuities via the traction balance equation.We perform a series of numerical experiments to solve two-and three-dimensional Stokes interface problems and perform an accuracy comparison with the augmented immersed interface methods in literature.Our results indicate that even a shallow network with a moderate number of neurons and sufficient training data points can achieve prediction accuracy comparable to that of immersed interface methods.
基金supported by NSF grant DMS-0713763the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 501709)the AMSS-PolyU Joint Research Institute for Engineering and Management Mathematics, and NSERC (Canada)
文摘This paper applies bilinear immersed finite elements （IFEs） in the interior penalty discontinuous Galerkin （DG） methods for solving a second order elliptic equation with discontinuous coefficient. A discontinuous bihnear IFE space is constructed and applied to both the symmetric and nonsymmetric interior penalty DG formulations. The new methods can solve an interface problem on a Cartesian mesh independent of the interface with local refinement at any locations needed even if the interface has a nontrivial geometry. Numerical examples are provided to show features of these methods.
基金supported by Guangdong Provincial Government of China through the“Computational Science Innovative Research Team”program and the Sun Yat-sen University“Hundred Talents Program”(34000-3181201)and the National Natural Science Foundation of China(No.11101446).
文摘Abstract.In this paper,a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces.The velocity components along the interface are introduced as two augmented variables and the resulting augmented equation is then solved by the GMRES method.The augmented variables and/or the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity,and are interpolated using cubic splines and are then applied to the fluid through the jump conditions.The Stokes equations are discretized on a staggered Cartesian grid via a second order finite difference method and solved by the conjugate gradient Uzawa-typemethod.The numerical results show that the overall scheme is second order accurate.The major advantages of the present IIM-Stokes solver are the efficiency and flexibility in terms of types of fluid flow and different boundary conditions.The proposed method avoids solution of the pressure Poisson equation,and comparisons are made to show the advantages of time savings by the present method.The generalized two-phase Stokes solver with correction terms has also been applied to incompressible two-phase Navier-Stokes flow.
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support by a research fund from Xiangtan University.Z.Li was partially supported by US ARO grants 56349-MA,AFSOR grant FA9550-09-1-0520NSF grant DMS-0911434support of the National Science Foundation,Division of Mathematical Sciences.H.Zhao is partially supported by NSF grant DMS0811254.
文摘In this paper,we numerically investigate the effects of surfactant on dropdrop interactions in a 2D shear flow using a coupled level-set and immersed interface approach proposed in(Xu et al.,J.Comput.Phys.,212(2006),590–616).We find that surfactant plays a critical and nontrivial role in drop-drop interactions.In particular,we find that the minimum distance between the drops is a non-monotone function of the surfactant coverage and Capillary number.This non-monotonic behavior,which does not occur for clean drops,is found to be due to the presence of Marangoni forces along the drop interfaces.This suggests that there are non-monotonic conditions for coalescence of surfactant-laden drops,as observed in recent experiments of Leal and co-workers.Although our study is two-dimensional,we believe that drop-drop interactions in three-dimensional flows should be qualitatively similar as the Maragoni forces in the near contact region in 3D should have a similar effect.
