immersed interface finite element methods for elliptic interface problems with nonhomogeneous jump conditions in this work a class of new finite element methods called immersed interface finite element methods is developed to solve elliptic interface problems with nonhomogeneous jump conditions simple non body fitted meshes are used a single function that satisfies the same nonhomogeneous jump conditions is constructed using a level set representation of the interface with such a function the discontinuities across the interface in the solution and flux are removed and an equivalent elliptic interface problem with homogeneous jump conditions is formulated special finite element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate these methods are designed as an efficient component of the finite element level set methodology for fast simulation of interface dynamics that does not require remeshing

vibration of beams with general boundary conditions due to a moving harmonic load the transverse vibrations of elastic homogeneous isotropic beams with general boundary conditions due to a moving random force with constant mean value are analyzed the boundary conditions considered are pinned pinned fixed fixed pinned fixed and fixed free based on the bernoulli beam theory the problem is described by means of a partial differential equation closed form solutions for the variance and the coefficient of variation of the beam deflection are obtained and compared for three types of force motion accelerated decelerated and uniform the effects of beam damping and speed of the moving force on the dynamic response of beams are studied in detail

a reconstruction based chebyshev collocation method for the poisson equation an accurate treatment of the gibbs wilbraham phenomenon on irregular interfaces abstract an accurate numerical solution of the laplace equation and poisson equation is desirable in physical problems related to fluid flows and spectral methods are well suited towards this goal however in the presence of an interface the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the gibbs wilbraham phenomenon leading to a reduced solution accuracy in order to resolve the discontinuity at the interface we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely differentiable smooth function and a modified heaviside function the smooth function is composed of chebyshev polynomials while the modified heaviside function is expressed in a weak form using the jump conditions across the interface the reconstruction framework allows us to impose the interfacial conditions exactly to motivate further application of the technique we first present the underlying rationale by revisiting the work of wilbraham 1848 13 followed by the numerical implementation we present numerical solution of four equations to illustrate the application of the proposed technique i an ordinary differential equation ode ii the euler bernoulli equation iii the laplace equation and iv the poisson equation the laplace and poisson equation are solved on a square domain embedded with complex interfacial geometries in two dimensions and on a cube with a spherical interface in three dimensions the method achieves error in the maximum norm o 10 14 using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems we also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub grid scale perturbations to interface location an alternate derivation for the modified heaviside function is proposed that is distinct from the taylor series expansion used to treat interface conditions in difference methods the pointwise error plots are subsequently presented to demonstrate that the gibbs wilbraham phenomenon is accurately resolved finally we present concluding thoughts on the potential applications of the reconstruction technique

numerical method for solving matrix coefficient elliptic equation with sharp edged interfaces solving elliptic equations with sharp edged interfaces is a challenging problem for most existing methods especially when the solution is highly oscillatory nonetheless it has wide applications in engineering and science an accurate and efficient method is desired we propose an efficient non traditional finite element method with non body fitting grids to solve the matrix coefficient elliptic equations with sharp edged interfaces extensive numerical experiments show that this method is second order accurate in the l norm and that it can handle both sharp edged interface and oscillatory solutions

a fourth order finite difference method for solving elliptic interface problems with the fft acceleration abstract in this paper a new cartesian grid finite difference method is introduced based on the fourth order accurate matched interface and boundary mib method and fast fourier transform fft the proposed augmented mib method consists of two major parts for solving elliptic interface problems in two dimensions for the interior part in treating a smoothly curved interface zeroth and first order jump conditions are enforced repeatedly by the mib scheme to generate fictitious values near the interface for the exterior part two layers of zero padding solutions are introduced beyond the original rectangular domain so that the fft inversion becomes feasible different types of boundary conditions including dirichlet neumann robin and their mix combinations can be imposed via the mib scheme to generate fictitious values near boundaries based on fictitious values at both interfaces and boundaries the augmented mib method reconstructs cartesian derivative jumps as auxiliary variables then by treating such variables as unknowns an enlarged linear system is obtained in the schur complement solution of such system the fft algorithm will not sense the solution discontinuities so that the discrete laplacian can be efficiently inverted therefore the fft based augmented mib not only achieves a fourth order of accuracy in dealing with interfaces and boundaries but also produces an overall complexity of o n 2 log n for a n n uniform grid moreover the augmented mib scheme can provide fourth order accurate approximations to solution gradients and fluxes

a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces we present a finite volume method for the solution of the two dimensional elliptic equation s x u x f x with variable discontinuous coefficients and solution discontinuities on irregular domains the method uses bilinear ansatz functions on cartesian grids for the solution u x resulting in a compact nine point stencil the resulting linear problem has been solved with a standard multigrid solver singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two step asymptotic approach the method achieves second order of accuracy in the l and l2 norm

vibration analysis of non linear beams subjected to a moving load using the finite element method this paper is concerned with the non linear vibration of immovably supported variable beams in which the geometric non linearity due to the axial force generated by stretching the middle surface is taken into account when the beam is subjected to a moving load the dynamic deflections of the beam are computed by using a galerkin finite element formulation and the time differential terms are integrated by using the implicit direct integration method three types of approximations for solving the non linear problem are considered in the first of which both the longitudinal deflections and inertia are considered with only the longitudinal deflections being considered and in the second neither the longitudinal deflections nor inertia in the third the results obtained by using these non linear models are numerically compared with that obtained by the linear model

a galerkin formulation of the mib method for three dimensional elliptic interface problems we develop a three dimensional 3d galerkin formulation of the matched interface and boundary mib method for solving elliptic partial differential equations pdes with discontinuous coefficients i e the elliptic interface problem the present approach builds up two sets of elements respectively on two extended subdomains which both include the interface as a result two sets of elements overlap each other near the interface fictitious solutions are defined on the overlapping part of the elements so that the differentiation operations of the original pdes can be discretized as if there was no interface the extra coefficients of polynomial basis functions which furnish the overlapping elements and solve the fictitious solutions are determined by interface jump conditions consequently the interface jump conditions are rigorously enforced on the interface the present method utilizes cartesian meshes to avoid the mesh generation in conventional finite element methods fems we implement the proposed mib galerkin method with three different elements namely rectangular prism element five tetrahedron element and six tetrahedron element which tile the cartesian mesh without introducing any new node the accuracy stability and robustness of the proposed 3d mib galerkin are extensively validated over three types of elliptic interface problems in the first type interfaces are analytically defined by level set functions these interfaces are relatively simple but admit geometric singularities in the second type interfaces are defined by protein surfaces which are truly arbitrarily complex the last type of interfaces originates from multiprotein complexes such as molecular motors near second order accuracy has been confirmed for all of these problems to our knowledge it is the first time for an fem to show a near second order convergence in solving the poisson equation with realistic protein surfaces additionally the present work offers the first known near second order accurate method for c 1 continuous or h 2 continuous solutions associated with a lipschitz continuous interface in a 3d setting

a kernel free boundary integral method for elliptic boundary value problems this paper presents a class of kernel free boundary integral kfbi methods for general elliptic boundary value problems bvps the boundary integral equations reformulated from the bvps are solved iteratively with the gmres method during the iteration the boundary and volume integrals involving green s functions are approximated by structured grid based numerical solutions which avoids the need to know the analytical expressions of green s functions the kfbi method assumes that the larger regular domain which embeds the original complex domain can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast fourier transform fft based poisson helmholtz solvers or those based on geometric multigrid iterations are applicable the structured grid based solutions are obtained with standard finite difference method fdm or finite element method fem where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme numerical results demonstrating the efficiency and accuracy of the kfbi methods are presented it is observed that the number of gmres iterations used by the method for solving isotropic and moderately anisotropic bvps is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals with the standard second order fems and fdms the kfbi method shows a second order convergence rate in accuracy for all of the tested dirichlet neumann bvps when the anisotropy of the diffusion tensor is not too strong

a numerical method for solving elasticity equations with interface involving multi domains and triple junction points solving elasticity equations with interfaces on multiple domains has wide applications in engineering and science corner singularities make it difficult for most existing solvers to deal with a triple junction in the case of nonelastic problems therefore constructing an efficient and accurate solver for an elasticity problem with multiple domains is a challenge in this paper an efficient non traditional finite element method with non body fitted grids is proposed to solve elliptical elasticity equations with multi domains and triple junction points numerical experiments show that this method is approximately second order accurate in the l norm for piecewise smooth solutions

nonlinear dynamic analysis of an eccentrically prestressed damped beam under a concentrated moving harmonic load this paper focuses on the geometrically nonlinear dynamic analysis of an eccentrically prestressed simply supported damped beam subjected to a concentrated moving harmonic load the nonlinear dynamic deflections of the beam are obtained by polynomial functions the kelvin voigt model for the material of the beam is used two coupled nonlinear systems of equations of motion are derived using lagrange s equations under the assumptions of the euler bernoulli beam theory with the von karman s nonlinear strain displacement relationships the rotary inertia axial displacement and axial inertia are included in the formulation the nonlinear equations of motion are solved by using the implicit time integration method of newmark in conjunction with the newton raphson iteration method in this study the effects of large deflections the internal damping of the beam the velocity of the moving harmonic load the prestress load the eccentricity of the prestress load and the excitation frequency on the dynamic response of the beam are discussed the obtained results are compared with the results based on the linear beam theory convergence studies are performed numerical results show that the above mentioned effects play a very important role in the deflections of the beam

vibration analysis of a non linear beam subjected to moving loads by using the galerkin method abstract this paper presents the analysis of dynamic deflections of a beam including the effects of geometric non linearity subjected to moving vehicle loads the beam is assumed to be elastic and simply supported with immovable ends and the vehicles are assumed to be single degree of freedom spring mass damper with the vehicles moving on the beam from one end to the other the dynamic reflections of the beam and vehicles are computed by using the galerkin method thus the dynamic deflections are assumed to be a set of time functions multiplied by approximate functions respectively and the time functions are numerically computed by solving the non linear differential equations by the newmark method

vibration of multi span non uniform beams under moving loads by using modified beam vibration functions based on hamilton s principle the vibration of a multi span non uniform beam subjected to a moving load is analysed by using modified beam vibration functions as the assumed modes the modified beam vibration functions satisfy the zero deflection conditions at all the intermediate point supports as well as the boundary conditions at the two ends of the beam numerical results are presented for both uniform and non uniform beams under moving loads of various velocities examples show that this method converges very quickly and good results are obtained

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

a new weak galerkin finite element method for elliptic interface problems a new weak galerkin wg finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces comparing with the existing wg algorithm for solving the same type problems the present wg method has a simpler variational formulation and fewer unknowns moreover the new wg algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders optimal order error estimates in both h 1 and l 2 norms are established for the present wg finite element solutions extensive numerical experiments have been conducted to examine the accuracy flexibility and robustness of the proposed wg interface approach in solving regular elliptic interface problems high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees moreover the wg method is shown to be able to accommodate very complicated interfaces due to its flexibility in choosing finite element partitions finally in dealing with challenging problems with low regularities the piecewise linear wg method is capable of delivering a second order of accuracy in l norm for both c 1 and h 2 continuous solutions

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

an improved non traditional finite element formulation for solving three dimensional elliptic interface problems solving elliptic equations with interfaces has wide applications in engineering and science the real world problems are mostly in three dimensions while an efficient and accurate solver is a challenge some existing methods that work well in two dimensions are too complicated to be generalized to three dimensions although traditional finite element method using body fitted grid is well established the expensive cost of mesh generation is an issue in this paper an efficient non traditional finite element method with non body fitted grids is proposed to solve elliptic interface problems the special cases when the interface cuts though grid points are handled carefully rather than perturbing the cutting point away to apply the method for general case both dirichlet and neumann boundary conditions are considered numerical experiments show that this method is approximately second order accurate in the l norm and l2 norm for piecewise smooth solutions the large sparse matrix for our linear system also has nice structure and properties

interface solutions of partial differential equations with point singularity abstract partial differential equations pdes such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips existing numerical methods for such pdes suffer from the constraint of this low regularity and fail to deliver an optimal high order rate of convergence we approach this problem by i choosing an artificial interface to enclose the center of low regularity and ii representing the solution in the interior of artificial interface as unknown linear combination of known low regularity modes of the solution this results in an interface formulation of the original pde and the linear combination is represented in the interface conditions by enforcing the smooth component of the numerical solution in the interior domain to be approximately zero a least squares problem is obtained for the unknown coefficients the solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the pde in the exterior domain the potential of our interface formulation is favorably demonstrated by numerical experiments on one and two dimensional poisson equations with low regularity solutions high order numerical solutions of unknown coefficients and pdes are obtained this proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems we address the relations between our interface formulation and domain decomposition methods as well as a regularization strategy for the poisson boltzmann equation with singular charge density

error estimates in weighted sobolev norms for finite element immersed interface methods abstract when solving elliptic partial differential equations in a region containing immersed interfaces possibly evolving in time it is often desirable to approximate the problem using an independent background discretisation not aligned with the interface itself optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface at the cost of a higher complexity in the implementation a much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions as done in immersed boundary methods these methods are known to have inferior convergence properties depending on the global regularity of the solution across the interface when compared to enriched methods in this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon restricted to a small neighbourhood of the interface in particular we show that optimal approximations can be constructed in a natural and inexpensive way simply by reformulating the problem in a distributionally consistent way and by resorting to weighted norms when computing the global error of the approximation

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

the extended finite element method for rigid particles in stokes flow a new method for the simulation of particulate flows based on the extended finite element method x fem is described in this method the particle surfaces need not conform to the finite element boundaries so that moving particles can be simulated without remeshing the near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh we present a weak form of the equations of motion useful for the simulation of freely moving particles and solve example problems for particles with prescribed and unknown velocities copyright 2001 john wiley sons ltd

dynamic finite element analysis of a nonlinear beam subjected to a moving load abstract in this paper the deterministic and random vibration analysis of a nonlinear beam on an elastic foundation subjected to a moving load which may simulate railway track runway etc has been performed the effects of longitudinal deflection and inertia have been considered so that the coupled equations of longitudinal and transverse deflections can be derived based on bernoulli euler hypothesis the randomness of the beam profile has been considered in such a way that the mean line of the beam is variable with respect to position in the vertical plane and is superimposed by stochastic uncertainty and the moving load travels along the beam with constant velocity or acceleration the deterministic and statistical dynamic responses of the beam have been calculated by using the galerkin s method in conjunction with the finite element method and the derived nonlinear system differential equation has been solved by using the implicit direct integration method in particular the standard deviation of the transverse deflection of the nonlinear beam has been calculated and presented by using the monte carlo simulation technique also the distribution of the midpoint deflection of the beam has been investigated by using the probability paper plot

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

matched interface and boundary mib method for elliptic problems with sharp edged interfaces elliptic problems with sharp edged interfaces thin layered interfaces and interfaces that intersect with geometric boundary are notoriously challenging to existing numerical methods particularly when the solution is highly oscillatory this work generalizes the matched interface and boundary mib method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems we classify these problems into five distinct topological relations involving the interfaces and the cartesian mesh lines flexible strategies are developed to systematically extends the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point the concepts of primary and secondary fictitious values are introduced to deal with sharp edged interfaces for corner singularity or tip singularity an appropriate polynomial is multiplied to the solution to remove the singularity extensive numerical experiments confirm the designed second order convergence of the proposed method

convergence of the ghost fluid method for elliptic equations with interfaces this paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps a weak formulation of the problem is first presented which then yields the existence and uniqueness of a solution to the problem by classical methods it is shown that the application of the ghost fluid method by fedkiw kang and liu to this problem can be obtained in a natural way through discretization of the weak formulation an abstract framework is given for proving the convergence of finite difference methods derived from a weak problem and as a consequence the ghost fluid method is proved to be convergent

vibrations of non uniform continuous beams under moving loads abstract the dynamic behavior of multi span non uniform beams transversed by a moving load at a constant and variable velocity is investigated the continuous beam is modelled using bernoulli euler beam theory the solution is obtained by using both the modal analysis method and the direct integration method the natural frequencies and mode shapes used in the solution of this problem are obtained exactly by deriving the exact dynamic stiffness matrices for any polynomial variation of the cross section along the beam using the exact element method the mode shapes are expressed as infinite polynomial series using the exact mode shapes yields the exact solution for general variation of the beam section in case of constant and variable velocity numerical examples are presented in order to demonstrate the accuracy and the effectiveness of the present study and the results are compared to previously published results

dynamic response of a beam with intermediate point constraints subject to a moving load abstract the transverse vibration of a beam with intermediate point constraints subject to a moving load is analyzed by using the euler beam theory and the assumed mode method the point constraints in the form of supports are assumed to be linear springs of large stiffness results of numerical simulations are presented for various combinations of constant axial velocity of the moving load and number of point constraints

a simple weak formulation for solving two dimensional diffusion equation with local reaction on the interface abstract in this paper we propose a numerical method for solving two dimensional diffusion equation with nonhomogeneous jump condition and nonlinear flux jump condition located at the interface we use finite element method coupled with newton s method to deal with the jump conditions and to linearize the system it is easy to implement the grid used here is body fitting grids based on the idea of semi cartesian grid numerical experiments show that this method is nearly second order accurate in the l norm

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

a weak formulation for solving elliptic interface problems without body fitted grid a typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations pde in different regions which are coupled together with interface conditions such as jumps in solution and flux across the interface in many situations such as the interface is moving the challenge is how to solve such a problem accurately robustly and efficiently without generating a body fitted mesh the key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the pde is not valid across the interface in this work we present a systematic formulation and further study of a second order accurate numerical method proposed in hou and liu 2005 16 for elliptic interface problem the key idea is to decompose the solution into two parts a singular part and a regular part the singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain which can be solved by a standard finite element formulation in a general setup the two parts are coupled together we give an explicit study of the construction of the singular part and the discretized system for the regular part one key advantage of using weak formulation is that one can avoid assuming using more regularity than necessary of the solution and the interface we present the numerical algorithm and numerical tests in 3d to demonstrate the accuracy and other properties of our method

vibration of multi span timoshenko beams to a moving force abstract a method of modal analysis is proposed in this paper to investigate the forced vibration of multi span timoshenko beams the ratio of the radius of gyration of the cross section to one span length is defined as a parameterr the effect ofron the first modal frequency of a beam is studied a concentrated force traversing on the beam is used as an example the effects of span number rotary inertia and shear deformation on the maximum moment the maximum deflection and the critical velocity of a beam are examined the results are compared with those of a multi span bernoulli euler beam

an unfitted finite element method using discontinuous galerkin in this paper we present a new approach to simulations on complex shaped domains the method is based on a discontinuous galerkin dg method using trial and test functions defined on a structured grid essential boundary conditions are imposed weakly via the dg formulation this method offers a discretization where the number of unknowns is independent of the complexity of the domain we will show numerical computations for an elliptic scalar model problem in 2 and 3 convergence rates for different polynomial degrees are studied

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a sharp interface finite volume method for elliptic equations on cartesian grids we present a second order sharp interface finite volume method for the solution of the three dimensional elliptic equation x u x f x with variable coefficients on cartesian grids in particular we focus on interface problems with discontinuities in the coefficient the source term the solution and the fluxes across the interface the method uses standard piecewise trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27 point stencil singularities associated with vanishing partial volumes of intersected grid cells are removed by a two term asymptotic approach in contrast to the 2d method presented by two of the authors in m oevermann r klein a cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces journal of computational physics 219 2006 749 769 we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz this simplifies the treatment of the different cut cell types and avoids additional special operations for degenerated interface topologies the resulting set of linear equations has been solved with a bicgstab solver preconditioned with an algebraic multigrid in various testcases including large ratios and non smooth interfaces the method achieves second order of accuracy in the l and l 2 norm

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

a hybridizable discontinuous galerkin method for elliptic interface problems in the formulation of boundary integral equations abstract a novel numerical method is presented to solve the general elliptic interface problem with piecewise constant diffusion coefficients to overcome the difficulties caused by piecewise constant diffusion coefficients a combination of the kernel free boundary integral kfbi method and the unfitted mesh based hybridizable discontinuous galerkin hdg method is considered to this end we first reformulate the elliptic interface problem into a fredholm boundary integral equation of the second kind and further solve the well conditioned discrete integral equation iteratively with the gmres method in each iteration evaluation of boundary and volume integrals is made by interpolating numerical solutions to equivalent but simple interface problems which is efficiently calculated with a hdg method on unfitted meshes the advantage of using the unfitted hdg method to treat the equivalent but simple interface problem is that the interpolation along boundary integrals on the interface can be easily evaluated via a linear lagrange polynomial element by element which is an important improvement from ying and henriquez 2007 numerical experiments are given to verify the effectiveness of the novel method which has second order accuracy even in the case of high contrast diffusion coefficients

gradient recovery for elliptic interface problem iii nitsche s method abstract this is the third paper on the study of gradient recovery for elliptic interface problems in our previous works guo and yang 2017 23 we developed gradient recovery methods for elliptic interface problems based on body fitted meshes and immersed finite element methods despite the efficiency and accuracy that these methods bring to recover the gradient there are still some cases in unfitted meshes where skinny triangles appear in the generated local body fitted triangulations that destroy the accuracy of recovered gradient near the interface in this paper we propose a gradient recovery technique based on nitsche s method for elliptic interface problems which avoids the loss of accuracy of gradient near the interface caused by skinny triangles we analyze the supercloseness between the gradient of the numerical solution by the nitsche s method and the gradient of the interpolation of the exact solution which leads to the superconvergence of the proposed gradient recovery method we also present several numerical examples to validate the theoretical results

precise time step integration for the dynamic response of a continuous beam under moving loads abstract the dynamic response of a non uniform continuous euler bernoulli beam is analyzed with hamilton s principle and the eigenpairs are obtained by the ritz method a high precision integration method is used to calculate the dynamic responses of this beam numerical results show that the method is more accurate in the prediction of the vibration responses under the moving loads than the newmark method

a simple method for matrix valued coefficient elliptic equations with sharp edged interfaces abstract the traditional finite element method has a number of nice properties and thus it is highly desired for matrix valued coefficient elliptic equations with sharp edged interfaces however its efficient implementation with body fitting grids for such problems is highly nontrivial in this paper we propose a simple finite element method with body fitting grids based on semi cartesian grid which makes its implementation fairly straightforward even for complicated geometry all possible situations that the interface cuts the grid are considered further the symmetry and positive definiteness of the resulting matrix are ensured for positive definite coefficients numerical experiments show that it is second order accurate in the l norm and numerically very stable

non linear vibration of timoshenko beam due to a moving force and the weight of beam the large deflection theory is employed to derive the equations of motion of the timoshenko beam due to the coupling effect of an external force with the weight of the beam galerkin s method is employed to obtain the dynamic responses of the beam a set of two discrete moving forces is taken as an example to investigate the characteristics of these responses results show that the effect of weight of the beam decreases the fundamental period of the structure further both the dynamic deflection and the dynamic moment of the beam predicated by the theory including the effect of weight of the beam are less than those of the beam predicated either by the small deflection theory or by the large deflection theory without including the effect of weight of the beam

a second order virtual node method for elliptic problems with interfaces and irregular domains we present a second order accurate geometrically flexible and easy to implement method for solving the variable coefficient poisson equation with interfacial discontinuities or on irregular domains handling both cases with the same approach we discretize the equations using an embedded approach on a uniform cartesian grid employing virtual nodes at interfaces and boundaries a variational method is used to define numerical stencils near these special virtual nodes and a lagrange multiplier approach is used to enforce jump conditions and dirichlet boundary conditions our combination of these two aspects yields a symmetric positive definite discretization in the general case we obtain the standard 5 point stencil away from the interface for the specific case of interface problems with continuous coefficients we present a discontinuity removal technique that admits use of the standard 5 point finite difference stencil everywhere in the domain numerical experiments indicate second order accuracy in l

symmetric interior penalty galerkin approaches for two dimensional parabolic interface problems with low regularity solutions abstract this work presents novel finite element approaches for solving a parabolic partial differential equation with discontinuous coefficients and low regularity solutions in a bounded convex polyhedral domain a spatial semi discretization based on symmetric interior penalty galerkin sipg approximations is constructed and analyzed by using discontinuous piecewise linear functions for smooth initial data spatial errors in the broken l 2 h 1 and l 2 h 1 norms are proven to be optimal with respect to low regularity solutions which are only piecewise h 1 s smooth with 0 s 1 furthermore we present the full sipg discretizations based on an euler backward finite difference time discretization the proposed approximations are shown to be unconditionally stable and have nearly the optimal l 2 l 2 and l 2 h 1 error estimates even when the regularities of the solutions are low on the whole domain the error estimates are optimal with respect to time sharply depending on the indexes in the global and local regularity numerical experiments for two dimensional parabolic interface problems verify the theoretical convergence rates

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

on the structure of the k additive fuzzy measures in this paper we present some results concerning the vertices of the set of fuzzy measures being at most k additive we provide an algorithm to compute them we give some examples of the results obtained with this algorithm and give lower bounds on the number of vertices for the n 1 additive case proving that it grows much faster than the number of vertices of the general fuzzy measures the results in the paper suggest that the structure of k additive measures might be more complex than expected from their definition and in particular that they are more complex than general fuzzy measures

the adaptive immersed interface finite element method for elliptic and maxwell interface problems we propose self adaptive finite element methods with error control for solving elliptic and electromagnetic problems with discontinuous coefficients the meshes in the methods do not need to fit the interfaces new error indicators are introduced to control the error due to non body fitted meshes flexible h adaptive strategies are developed which can be systematically extended to a large class of interface problems extensive numerical experiments are performed to support the theoretical results and to show the competitive behavior of the adaptive algorithm even for interfaces involving corner or tip singularities

moving forces identification on a multi span continuous bridge abstract a continuous bridge is modelled as a multi span continuous timoshenko beam with non uniform cross section the vibration behaviour of this beam subjected to moving loads is analyzed by hamilton s principle with the intermediate point constraints represented by very stiff linear springs a method based on the modal superposition and optimization technique is developed to identify the moving forces in the time domain the damped least squares method is used in the identification to provide bounds to the results the errors in the moving force identification are discussed in the paper both computation simulations and laboratory tests show that the method is effective for identifying the moving loads using different combinations of measured responses

gradient recovery for elliptic interface problem ii immersed finite element methods abstract this is the second paper on the study of gradient recovery for elliptic interface problem in our previous work guo and yang 2016 17 we developed a novel gradient recovery technique for finite element method based on the body fitted mesh in this paper we propose new gradient recovery methods for two immersed interface finite element methods symmetric and consistent immersed finite method ji et al 2014 23 and petrov galerkin immersed finite element method hou et al 2004 22 and hou and liu 2005 20 compared to the body fitted mesh based gradient recovery method the new methods provide a uniform way of recovering gradient on regular meshes numerical examples are presented to confirm the superconvergence of both gradient recovery methods moreover they provide asymptotically exact a posteriori error estimators for both immersed finite element methods

vibration of viscoelastic beams subjected to an eccentric compressive force and a concentrated moving harmonic force abstract the problem of lateral vibration of a beam subjected to an eccentric compressive force and a harmonically varying transverse concentrated moving force is analyzed within the framework of the bernoulli euler beam theory the lagrange equations are used to examine the free vibration characteristics of an axially loaded beam and the dynamic response of a beam subjected to an eccentric compressive force and a moving harmonic concentrated force the constraint conditions of supports are taken into account by using the lagrange multipliers in the study trial function denoting the deflection of the beam is expressed in a polynomial form by using the lagrange equations the problem is reduced to the solution of a system of algebraic equations results of numerical simulations are presented for various combinations of the value of the eccentricity the eccentric compressive force excitation frequency and the constant velocity of the transverse moving harmonic force convergence studies are made the validity of the obtained results is demonstrated by comparing them with exact solutions based on the bernoulli euler beam theory obtained for the special cases of the investigated problem

a boundary condition capturing multigrid approach to irregular boundary problems we propose a geometric multigrid method for solving linear systems arising from irregular boundary problems involving multiple interfaces in two and three dimensions in this method we adopt a matrix free approach i e we do not form the fine grid matrix explicitly and we never form nor store the coarse grid matrices as many other robust multigrid methods do the main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function numerical results are given to compare our multigrid method with black box and algebraic multigrid methods

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim

an augmented matched interface and boundary mib method for solving elliptic interface problem abstract in this paper a second order accurate augmented matched interface and boundary mib is introduced for solving two dimensional 2d elliptic interface problems with piecewise constant coefficients the augmented mib seamlessly combines several key ingredients of the standard mib augmented immersed interface method iim and explicit jump iim to produce a new fast interface algorithm based on the mib zeroth and first order jump conditions are enforced across an arbitrarily curved interface which yields fictitious values on cartesian nodes near the interface by using such fictitious values a simple procedure is proposed to reconstruct cartesian derivative jumps as auxiliary variables and couple them with the jump corrected taylor series expansions which allow us to restore the order of the central difference across the interface to two moreover by using the schur complement to disassociate the algebraic computation of auxiliary variables and function values the discrete laplacian can be efficiently inverted by using the fast fourier transform fft it is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size as a consequence the total computational cost of the augmented mib is about o n 2 log n for a cartesian grid with dimension n n in 2d therefore the augmented mib outperforms the classical mib in all cases by significantly reducing the cpu time while keeping the same second order of accuracy in dealing with complicated interfaces

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19