an explicit description of the global attractor of the damped and driven sine gordon equation we prove that the size of the finite dimensional attractor of the damped and driven sine gordon equation stays small as the damping and driving amplitude become small a decomposition of finite dimensional attractors in banach space is found into a part that attracts all of phase space except sets whose finitedimensional projections have lebesgue measure zero and a partc that only attracts sets whose finite dimensional projections have lebesgue measure zero we describe the components of the attractor andc which is called the hyperbolic structure for the damped and driven sine gordon equation is low dimensional but the dimension ofc which is associated with transients is much larger we verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the attractor in this small parameter region

an immersed boundary method with direct forcing for the simulation of particulate flows we present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid the main idea is to incorporate peskin s regularized delta function approach acta numerica 11 2002 1 into a direct formulation of the fluid solid interaction force in order to allow for a smooth transfer between eulerian and lagrangian representations while at the same time avoiding strong restrictions of the time step this technique was implemented in a finite difference and fractional step context a variety of two and three dimensional simulations are presented ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles the accuracy and efficiency of the current method are clearly demonstrated

the discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of poisson equation by using the direct method of lines the numerical simulation for the exterior problem of poisson equation is considered we introduced a polygonal artificial boundary ce and designed a discrete artificial boundary condition on it by using the direct method of lines then the original problem is reduced to a boundary value problem defined in a bounded computational domain with a polygonal boundary the finite element approximation of this reduced boundary value problem is considered and it is proved that the finite element approximate problem is well posed furthermore numerical examples show that the discrete artificial boundary condition is very eaective and more accurate than the neumann boundary condition which is often used in engineering literatures 1999 elsevier science s a all rights reserved

artificial boundary conditions for two dimensional incompressible viscous flows around an obstacle abstract in this paper we consider numerical simulation of incompressible viscous flow around an obstacle in velocity pressure formulation two horizontal straight line artificial boundaries are introduced and the original flow is approximated by a flow in an infinite channel with slip boundary condition on the wall then two vertical segment artificial boundaries are introduced to limit the channel to a bounded computational domain in the region of the channel between the vertical boundaries and infinity the velocity of the flow is almost a constant vector in which the navier stokes equations can be linearised by oseen equations and thus a general solution can be derived by using separation of variables artificial boundary conditions on the vertical segments are then designed by imposing the continuity of velocity and normal stress therefore the original problem is reduced to a boundary value problem on a bounded computational domain numerical example shows that our artificial boundary conditions are very effective

artificial boundary conditions for incompressible navier stokes equations a well posed result abstract numerical simulation of two dimensional incompressible viscous flows around an obstacle is considered two horizontal straight line artificial boundaries are introduced and the original flow is approximated by a flow in an infinite channel with slip boundary condition on the wall then two vertical segment artificial boundaries are introduced and a series of approximate artificial boundary conditions on them are derived by imposing the continuity of velocity and the normal stress thus the original problem is reduced to a problem defined in a bounded computational domain the well posedness of the reduced problem is proved the finite element approximation of this problem is given and error estimates are obtained furthermore numerical examples show the accuracy and efficiency of our artificial boundary conditions

asymptotic boundary conditions for dissipative waves general theory an outstanding issue in the computational analysis of time dependent problems is the imposition of appropriate radiation boundary conditions at artificial boundaries accurate conditions are developed which are based on the asymptotic analysis of wave propagation over long ranges employing the method of steepest descents dominant wave groups are identified and simple approximations to the dispersion relation are considered in order to derive local boundary operators the existence of a small number of dominant wave groups may be expected for systems with dissipation estimates of the error as a function of domain size are derived under general hypotheses leading to convergence results some practical aspects of the numerical construction of the asymptotic boundary operators are also discussed

the direct method of lines for the problem of infinite elastic foundation we consider the numerical simulation for the problem of infinite elastic foundation a polygonal artificial boundary is introduced and a discrete artificial boundary condition on it is presented by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain which is solved by the finite element method in addition we prove an optimal a priori error bound for the displacement in the bounded computational domain finally numerical example shows that the discrete artificial boundary condition given in this paper is very effective and more accurate than neumann boundary condition which is often used in engineering literatures

pairwise interaction extended point particle model for particle laden flows abstract in this work we consider the pairwise interaction extended point particle piep model for euler lagrange simulations of particle laden flows by accounting for the precise location of neighbors the piep model goes beyond local particle volume fraction and distinguishes the influence of upstream downstream and laterally located neighbors the two main ingredients of the piep model are i the undisturbed flow at any particle is evaluated as a superposition of the macroscale flow and a microscale flow that is approximated as a pairwise superposition of perturbation fields induced by each of the neighboring particles and ii the forces and torque on the particle are then calculated from the undisturbed flow using the faxen form of the force relation the computational efficiency of the standard euler lagrange approach is retained since the microscale perturbation fields induced by a neighbor are pre computed and stored as piep maps here we extend the piep force model of akiki et al 3 with a corresponding torque model to systematically include the effect of perturbation fields induced by the neighbors in evaluating the net torque also we use dns results from a uniform flow over two stationary spheres to further improve the piep force and torque models we then test the piep model in three different sedimentation problems and compare the results against corresponding dns to assess the accuracy of the piep model and improvement over the standard point particle approach in the case of two sedimenting spheres in a quiescent ambient the piep model is shown to capture the drafting kissing tumbling process in cases of 5 and 80 sedimenting spheres a good agreement is obtained between the piep simulation and the dns for all three simulations the dem piep was able to recreate to a good extent the results from the dns while requiring only a negligible fraction of the numerical resources required by the fully resolved dns

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

a fast computation technique for the direct numerical simulation of rigid particulate flows in this paper we present a computation technique for the direct numerical simulation of freely moving rigid bodies in fluids we solve three dimensional laminar flow problems using a control volume approach the key feature of this approach is that the computational overhead relative to a pure fluid solver to solve for the motion of rigid particle is very small the formulation is convenient for handling irregular geometries we present results for the sedimentation of particles of different shapes convergence tests are presented to assess the order of accuracy of the numerical scheme various test cases are considered and the numerical results are compared with experimental values to validate the code due to the ability to perform fast computations this method has been used for animations and its application to the direct numerical simulation of turbulent particulate flows merits investigation the technique is not restricted to any constitutive model of the suspending fluid hence it may potentially be used in large eddy simulations les or reynolds averaged navier stokes rans type simulations

numerical simulations of fracture problems by coupling the fem and the direct method of lines in this paper we propose a method for calculating stress intensity factors of plane fracture problems with cracks by coupling the finite element method fem and the direct method of lines we introduce polygonal artificial boundaries and divide the physical domain into two parts a large domain without any crack tips and small polygonal domains surrounding the crack tips we solve the problem defined in the small domains with crack tips by using the direct method of lines and design discrete nonlocal artificial boundary conditions on the polygonal artificial boundaries by imposing the continuity of displacement and normal stress then the original problem is reduced to a boundary value problem defined in a domain without any crack tips the finite element approximation of the reduced problem is considered and we can prove that the finite element approximation is well posed numerical examples and results of a fracture problem with exact solution and two typical fracture problems demonstrate the efficiency and accuracy of the present method

an open boundary condition for the computation of the steady incompressible navier stokes equations abstract we design a new boundary condition on the stream function at outflow for the steady incompressible navier stokes equations written in the stream function vorticity formulation by using a fourier transform in the direction perpendicular to the flow our open boundary condition is compared to the dirichlet and neumann boundary conditions by computing the flow past an ellipse of excentricity 50 our boundary condition proves to be the most accurate and thus the most time saving

boundary conditions at outflow for a problem with transport and diffusion abstract we present a numerical test of a class of approximate boundary conditions at an artificial boundary for the solution of time dependent problems in semi infinite spatial domains the conditions are constructed by use of a general method developed by the author in another work a problem modeling the transport and diffusion of soluble matter by a parallel flow is taken as the test case we demonstrate the effectiveness of the method in obtaining accurate results on small computational domains

error bounds for the finite element approximation of an incompressible material in an unbounded domain in this paper we design high order local artificial boundary conditions and present error bounds for the finite element approximation of an incompressible elastic material in an unbounded domain the finite element approximation is formulated in a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one in fact there are a family of nonlocal approximate artificial boundary conditions with increasing accuracy and computational cost and a family of local ones for a given artificial boundary our error bounds indicate how the errors of the finite element approximations depend on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary numerical examples of an incompressible elastic material outside a circle in the plane is presented numerical results demonstrate the performance of our error bounds

artificial boundary conditions for the linear advection diffusion equation a family of artificial boundary conditions for the linear advection diffusion equation with small viscosity is developed well posedness for the associated initial boundary value problem is analyzed the error produced by truncating the domain is estimated numerical results are presented 1 introduction when computing the solution of a partial differential equation in an unbounded domain one often introduces artificial boundaries in order to limit the computational cost these boundaries must be chosen not too far from the domain of interest therefore the boundary conditions must be good approxima tions to the so called transparent i boundary condition i e such ithat the solution of the problem in the bounded domain is equal to the solution in the original domain the transparent boundary condition is usually an integral relation in time and space between u and its normal derivative on the boundary which makes it impractical from a numerical point of view one must approximate this relation to get local boundary conditions they are often called absorbing or artificial boundary conditions this question is of crucial interest in such different areas as geophysics plasma physics fluid dynamics 1 2 3 and the use of such conditions is now classical in geophysics our interest for the linear advection diffusion equation comes from the navier stokes equation but it arises also in other fields as for example meteorology 6 the incompressible navier stokes equation can be written as

numerical simulation for the problem of infinite elastic foundation in this paper we consider the numerical simulation for the problem of infinite elastic foundation a half circle artificial boundary is introduced and discrete artificial boundary condition on it is designed by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain furthermore finite element approximation of this boundary value problem defined in the bounded domain is considered the numerical results show that the artificial boundary condition given in this paper is very effective and more accurate than the neumann boundary condition which is often used in engineering literatures

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

on the domain of analyticity for solutions of second order analytic nonlinear differential equations abstract the radius of analyticity of periodic analytic functions can be characterized by the decay of their fourier coefficients this observation has led to the use of so called gevrey norms as a simple way of estimating the time evolution of the spatial radius of analyticity of solutions to parabolic as well as non parabolic partial differential equations in this paper we demonstrate using a simple explicitly solvable model equation that estimates on the radius of analyticity obtained by the usual gevrey class approach do not scale optimally across a family of solutions nor do they scale optimally as a function of the physical parameters of the equation we attribute the observed lack of sharpness to a specific embedding inequality and give a modified definition of the gevrey norms which is shown to finally yield a sharp estimate on the radius of analyticity

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

infinite domain elements a parametric element is formulated which enables the economic modelling of infinite domain type problmes a typical problem is an opening in a stress field in an infinite medium either in two or three dimensions the strategy is to model around the opening with two or three layers of conventional isoparametric finite elements and surround these with a single layer of infinite domain elements several sample problems has been analysed for circular square and spherical openings in infinite media and the results compared with either theoretical or boundary element solutions which include the infinite boundary in their solution technique copyright

exact artificial boundary conditions for continuum and discrete elasticity for the continuum and discrete elastic equations we derive exact artificial boundary conditions abcs often referred to as transparent boundary conditions that can be applied at a planar interface below which there are no forces solution of the elasticity equations can then be performed using this interface as an artificial boundary often with greatly reduced computational effort but without loss of accuracy a general solvability requirement is presented for the existence of an artificial boundary operator for discrete systems such as discrete elasticity on an unbounded semi infinite domain the solvability requirement is validated by introducing a sum of exponentials ansatz for the solution below the artificial boundary we also derive a new expression for the total energy for the system involving only the region above the artificial boundary numerical examples are provided to confirm and illustrate the accuracy and effectiveness of the results

nonlocal artificial boundary conditions for the incompressible viscous flow in a channel using spectral techniques in this paper the numerical simulation of the steady incompressible viscous flow in a no slip channel is considered a sequence of approximate nonlocal artificial boundary conditions on a given segment artificial boundary is derived by a system of linearized navier stokes equations and spectral techniques then the original problem is reduced to a boundary value problem in a bounded computational domain the numerical examples show that these artificial boundary conditions are very effective and are also more accurate than dirichlet and neumann boundary conditions which are often used in the engineering literature

a novel boundary infinite element a new improved infinite element based on mapping is proposed for modelling unbounded domain problems mapping has previously been used in such problems not only for the entire domain but also for the infinite element itself the novelty of the present approach is the simplicity of the mapping and the retention of standard integration abscissae and weights the implementation of these elements is very straightforward and they are ideally suited for the analysis of infinite domain static problems in two and three dimensions

error estimates for the finite element approximation of linear elastic equations in an unbounded domain in this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain the finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one in fact there are a family of nonlocal approximate boundary conditions with increasing accuracy and computational cost and a family of local ones for a given artificial boundary our error estimates show how the errors of the finite element approximations depend on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary a numerical example for navier equations outside a circle in the plane is presented numerical results demonstrate the performance of our error estimates

data assimilation algorithm for 3d benard convection in porous media employing only temperature measurements abstract in this paper we propose a continuous data assimilation downscaling algorithm for the benard convection in porous media using only discrete spatial mesh measurements of the temperature in this algorithm we incorporate the observables as a feedback nudging term in the evolution equation of the temperature we show that under an appropriate choice of the nudging parameter and the size of the mesh and under the assumption that the observed data is error free the solution of the proposed algorithm converges at an exponential rate asymptotically in time to the unique exact unknown reference solution of the original system associated with the observed finite dimensional projection of temperature data moreover we note that in the case where the observational measurements are not error free one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements

the artificial boundary conditions for incompressible materials on an unbounded domain in this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in frak r 2 a series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given then the original problem is reduced to a problem on a bounded domain which be solved numerically by a mixed finite element method the numerical example shows that our artificial boundary conditions are very effective

remark on the rate of decay of higher order derivatives for solutions to the navier stokes equations in rn we present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced navier stokes equations in rn the method based on so called gevrey estimates also yields explicit bounds on the growth of the radius of analyticity of the solution in time moreover under the assumption that the navier stokes solution stays sufficiently close to a solution of the heat equation in the l2 norm a result known to be true for a large class of initial data lower bounds on the decay of higher order derivatives can be obtained

gevrey regularity for the attractor of a partially dissipative model of benard convection in a porous medium abstract convective flow though a porous medium can be modeled by darcy s law a linear weakly damped momentum equation coupled with an advection diffusion equation for the energy the solution semigroup for this system is not smoothing and the solution of the momentum equation does not gain regularity with respect to its initial value in finite time however it is known that the semigroup is asymptotically smoothing so that the system possesses a finite dimensional global attractor as well as exponential attractors we show that the global attractor is contained in a special gevrey class of regularity and in particular is real analytic the key idea is the use of a fourier splitting method to approximate every orbit asymptotically in time by a gevrey regular function

one point second order curved boundary condition for lattice boltzmann simulation of suspended particles abstract the lattice boltzmann method lbm has been widely considered as a distinctive and reliable approach for simulating the complex particulate flows as an intrinsic kinetic scheme it is quite convenient for lbm to apply the bounce back bb type methods to handle the moving boundaries with complicated geometry which is a tricky task for general interface resolved methods however the two major schemes in lbm i e the simple bb rule and the curved boundary condition cbc are presently encountered by the problems of low precision and loss of local computation respectively to overcome those two deficiencies in the boundary treatment simultaneously a one point second order cbc is proposed in this paper information of only a single node is required in the present scheme and the second order accuracy is validated in the channel flow and cylindrical couette flow applications to the particulate flows are further implemented to verify the present scheme numerical results are in good agreement with those in the literature

immersed boundary method with non uniform distribution of lagrangian markers for a non uniform eulerian mesh this study presents a technique to incorporate spheres in a channel flow that uses a non uniform eulerian grid using immersed boundary methods with direct forcing an efficient algorithm is presented which distributes the lagrangian markers non uniformly to match the fluid grid and keep the number of markers optimized also a novel method to calculate the area weights of the lagrangian markers is given it is observed that even the best available algorithms for uniform distribution of markers on a sphere result in a finite error using vector spherical harmonics this error is quantified and reduced to machine precision a series of simulations of a stationary and moving sphere in a periodic channel at reynolds number range of 1 100 are presented results for a sphere in an ambient shear flow in close proximity of a wall are also shown where the present non uniform distribution offers an order of magnitude reduction over uniform distribution of lagrangian markers simulations of a random cluster of 640 monodisperse spherical particles show a 77 reduction in lagrangian markers with an error of 0 135 in computing the total drag

finite element method with nonlocal boundary condition for solving the nondestructive testing problem of wood moisture content abstract in this paper we use the finite element method fem with nonlocal boundary condition for solving the nondestructive testing problem of wood moisture content based on a planar capacitance sensor model i e the dtn fem for solving the mathematical model which is described by the exterior problems of a class of 3d laplace equation with complicated boundary conditions the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a dirichlet to neumann dtn mapping represented in terms of the fourier expansion series for numerical computation a series of approximate problems with higher accuracy can be derived if one truncates the series term in the variational formulation which is equivalent to the reduced problem the error estimate is presented to show how the error depends on the finite element discretization and the accuracy of the approximate problem based on the numerical results the relationship between the dielectric constant dc of tested wood and the capacitance value of the sensor is discussed finally we applied a least squares fitting method lsfm to reconstruct the wood moisture content wmc from the data measured with a planar capacitance sensor compared with popular statistical methods the hybrid experimental computational method is more convenient and faster and a large number of experiments are avoided the costs of testing are reduced

high order artificial boundary conditions for ideal axisymmetric irrotational flow around 3d obstacles in this paper the numerical solution for ideal irrotational incompressible flow around axisymmetric 3d obstacles is discussed with the artificial boundary method by introducing an artificial boundary we divide the exterior unbounded domain into a bounded part and an unbounded part after setting up a proper artificial boundary condition we get an approximate reduced problem defined on the bounded part both non local and local artificial boundary conditions are designed numerical experiment is also presented and its result demonstrates the effectiveness of these artificial boundary conditions copyright 2002 john wiley sons ltd

analyticity of solutions for a generalized euler equation abstract we consider the so called lake and great lake equations which are shallow water equations that describe the long time motion of an inviscid incompressible fluid contained in a shallow basin with a slowly spatially varying bottom a free upper surface and vertical side walls under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude if these equations are posed on a space periodic domain and the initial data are real analytic the solution remains real analytic for all times the proof is based on a characterization of gevrey classes in terms of decay of fourier coefficients in particular our result recovers known results for the euler equations in two and three spatial dimensions we believe the proof is new

high order local artificial boundary conditions for problems in unbounded domains in this paper we present error estimates for the finite element approximation of poisson and modified helmholtz equations outside an obstacle or in a semi infinite strip in the plane the finite element approximation is formulated on a bounded domain using a local approximate artificial boundary condition in fact there is a sequence of local approximate boundary conditions for a given artificial boundary our error estimates are based on the mesh size and the location of the artificial boundary the numerical stability and robustness of the method are discussed numerical experiments are presented to demonstrate the performance of the method and our error estimates

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

mixed finite element method and high order local artificial boundary conditions for exterior problems of elliptic equation the mixed finite element method is used to solve the exterior elliptic problem with high order local artificial boundary conditions new unknowns are introduced to reduce the order of the derivatives to two this leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary error estimates are obtained for some local artificial boundary conditions with prescribed order numerical examples are presented and the results demonstrate the effectiveness of this method

error estimates for the finite element approximation of problems in unbounded domains in this paper we present error estimates for the finite element approximation of linear elliptic problems in unbounded domains that are outside an obstacle and a semi infinite strip in the plane the finite element approximation is formulated on a bounded domain using a nonlocal approximate artificial boundary condition in fact there is a family of approximate boundary conditions with increasing accuracy and computational cost for a given artificial boundary our error estimates are based on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary numerical examples for poisson s problem outside a circle and in a semi infinite strip are presented numerical results demonstrate the performance of our error estimates