direct simulation of the motion of a settling ellipsoid in newtonian fluid in this paper we discuss the generalization of a lagrange multiplier based fictitious domain method to the simulation of the motion of general shape particles in newtonian fluid preliminary numerical results of a settling ellipsoid in a narrow channel filled with a newtonian fluid are presented as expected the ellipsoid turns its broadside to the stream in the simulations

a domain decomposition method for domains with holes using a complementary decomposition in this paper a new non conforming domain decomposition method based on fictitious domains is considered for linear elasticity problems on domains with holes existence and uniqueness of the solution of this alternative approach are showed in the continuous and discrete case although the established a priori estimate for the discretization error is not optimal the scheme is of interest in case of moving geometries and in shape optimization to solve the arising coupled linear system we use a block sor method the convergence rate does not depend on the mesh size numerical results illustrate the flexibility of the discretization scheme

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

the treatment of lubrication forces in boundary integral equations despite their many advantages over other numerical methods boundary integral formulations still fail to provide accurate predictions of mesoscale motion in dense suspensions of rigid particles because the nearly singular flow between surfaces in close proximity cannot be resolved accurately a procedure for incorporating analytical solutions for the lubrication flow within a large scale boundary integral equation method is shown although the method is applied to the case of spherical particles in conjunction with the completed double layer boundary integral equation it can be developed further to treat more complex geometries and can be adapted to other numerical techniques in contrast to other apparently similar approaches the present method does not resort to effective medium approximations and in principle retains all the advantages typical of boundary integral approaches the framework also allows for forces other than those due to hydrodynamic lubrication between particles provided that they are a linear function of the relative velocity or at least can be linearized for example forces due to sub continuum fluid behaviour or forces resulting from surface chemistry it is shown using several benchmarks that the relative motion between two particles in various flows is captured accurately both statically and dynamically in situations where uncorrected simulations fail moreover the computational effort is reduced substantially by the application of the method

the fixed mesh ale approach for the numerical approximation of flows in moving domains in this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization and therefore the formulation to be presented falls within the category of fixed grid methods even though the imposition of boundary conditions is a key ingredient that is very often used to classify the fixed grid method our approach can be applied together with any technique to impose approximately boundary conditions although we also describe the one we actually favor our main concern is to properly account for the advection of information as the domain boundary evolves to achieve this we use an arbitrary lagrangian eulerian framework the distinctive feature being that at each time step results are projected onto a fixed background mesh that is where the problem is actually solved

a hierarchical nxfem for fictitious domain simulations we suggest a fictitious domain method based on the nitsche xfem method of comput meth appl mech engrg 2002 191 5537 5552 that employs a band of elements adjacent to the boundary in contrast the classical fictitious domain method uses lagrange multipliers on a line surface where the boundary condition is to be enforced the idea can be seen as an extension of the chimera method of esaim math model numer anal 2003 37 495 514 but with a hierarchical representation of the discontinuous solution field the hierarchical formulation is better suited for moving fictitious boundaries since the stiffness matrix of the underlying structured mesh can be retained during the computations our technique allows for optimal convergence properties irrespective of the order of the underlying finite element method

a generalized finite difference gfd ale scheme for incompressible flows around moving solid bodies on hybrid meshfree cartesian grids a scheme using the mesh free generalized finite differencing gfd on flows past moving bodies is proposed the aim is to devise a method to simulate flow past an immersed moving body that avoids the intensive remeshing of the computational domain and minimizes data interpolation associated with the established computational fluid methodologies as such procedures are time consuming and are a significant source of error in flow simulation in the present scheme the moving body is embedded and enveloped by a cloud of mesh free nodes which convects with the motion of the body against a background of cartesian nodes the generalized finite difference gfd method with weighted least squares wls approximation is used to discretize the two dimensional viscous incompressible navier stokes equations at the mesh free nodes while standard finite difference approximations are applied elsewhere the convecting motion of the mesh free nodes is treated by the arbitrary lagrangian eulerian ale formulation of the flow equations which are solved by a second order crank nicolson based projection method the proposed numerical scheme was tested on a number of problems including the decaying vortex flow external flows past moving bodies and body driven flows in enclosures

a fictitious domain method with mixed finite elements for elastodynamics we consider in this paper the wave scattering problem by an object with neumann boundary conditions in an anisotropic elastic body to obtain an efficient numerical method permitting the use of regular grids we follow a fictitious domain approach coupled with a first order velocity stress formulation for elastodynamics we first observe that the method does not always converge when the q 1 rm div q 0 finite element is used in particular the method converges for some scattering object geometries but not for others note that the convergence of the q 1 rm div q 0 finite element method was shown in e becache p joly and c tsogka siam j numer anal 39 2002 pp 2109 2132 for the elastodynamic problem in the absence of a scattering object i e without the coupling of the mixed finite elements with the fictitious domain method therefore we propose here a modification of the q 1 rm div q 0 element following the approach in e becache j rodriguez and c tsogka on the convergence of the fictitious domain method for wave equation problems technical report 5802 inria 2006 where the simpler acoustic case was considered to study the numerical properties of the new element we carry out a dispersion analysis several numerical simulations as well as a numerical convergence analysis show that the proposed method provides a good approximate solution

numerical modelling of electromechanical coupling using fictitious domain and level set methods we present a finite element formulation for simulation of electromechanical coupling using a combination of fictitious domain and level set methods the electric field is treated with a fixed eulerian like mesh whereas the structure taken as a perfect conductor is modelled with a conventional lagrangian approach the compatibility between the potential of the conductor and of the electric domain is obtained by introducing a lagrange multiplier function defined on the boundary of the conductor the electromechanical forces are obtained using a variational formulation for the coupled electromechanical domain we use a heaviside function on the level set to remove the electric energy in the conductor domain results are presented for an radio frequency switch and an element of a comb drive

a general fictitious domain method with immersed jumps and multilevel nested structured meshes this study addresses a new fictitious domain method for elliptic problems in order to handle general and possibly mixed embedded boundary conditions e b c robin neumann and dirichlet conditions on an immersed interface the main interest of this fictitious domain method is to use simple structured meshes possibly uniform cartesian nested grids which do not generally fit the interface but define an approximate one a cell centered finite volume scheme with a non conforming structured mesh is derived to solve the set of equations with additional algebraic transmission conditions linking both flux and solution jumps through the immersed approximate interface hence a local correction is devised to take account of the relative surface ratios in each control volume for the robin or neumann boundary condition then the numerical scheme conserves the first order accuracy with respect to the mesh step this opens the way to combine the e b c method with a multilevel mesh refinement solver to increase the precision in the vicinity of the interface such a fictitious domain method is very efficient the l 2 and l norm errors vary like o h l where h l is the grid step of the finest refinement level around the interface until the residual first order discretization error of the non refined zone is reached the numerical results reported here for convection diffusion problems with dirichlet robin and mixed dirichlet and robin boundary conditions confirm the expected accuracy as well as the performances of the present method

a generalization of the vidav jorgens perturbation theorem for semigroups and its application to transport theory on considere des semigroupes t t engendres par l operateur de transport a k comme une perturbation du semi groupe s t engendre par a seul on montre que pour un n tous les produits de la forme ks td 1k ks tnd k t i 0 sont strictement singuliers alors s t et t t ont le meme rayon spectral essentiel

asymptotic spectrum of the linear boltzmann equation with general boundary conditions in finite bodies abstract the purpose of this paper is the study of the spectral properties of both streaming operator and transport operator with general boundary conditions in multidimensional bounded geometry we discuss the asymptotic spectrum existence and nonexistence results of eigenvalues in the half plane c re s t h where s t h stands for the spectral bound of the streaming operator t h next we discuss the irreducibility of the transport semigroup in particular we establish that the transport semigroup is irreducible if the boundary operator is strictly positive afterwards we discuss the strict monotonicity of the leading eigenvalue when it exists of the transport operator with respect to different parameters of the equation our analysis is based essentially on results from the theory of positive linear operators

a fictitious domain approach to the numerical solution of pdes in stochastic domains we present an efficient method for the numerical realization of elliptic pdes in domains depending on random variables domains are bounded and have finite fluctuations the key feature is the combination of a fictitious domain approach and a polynomial chaos expansion the pde is solved in a larger fixed domain the fictitious domain with the original boundary condition enforced via a lagrange multiplier acting on a random manifold inside the new domain a generalized wiener expansion is invoked to convert such a stochastic problem into a deterministic one depending on an extra set of real variables the stochastic variables discretization is accomplished by standard mixed finite elements in the physical variables and a galerkin projection method with numerical integration which coincides with a collocation scheme in the stochastic variables a stability and convergence analysis of the method as well as numerical results are provided the convergence is spectral in the polynomial chaos order in any subdomain which does not contain the random boundaries

on the leading eigenvalue of neutron transport models abstract we give variational characterizations of the leading eigenvalue of neutron transport like operators the proofs rely on sub and super eigenvalues various bounds of the leading eigenvalue are derived

the finite element immersed boundary method with distributed lagrange multiplier we introduce a new formulation for the finite element immersed boundary method which makes use of a distributed lagrange multiplier we prove that a full discretization of our model based on a semi implicit time advancing scheme is unconditionally stable with respect to the time step size

a cartesian grid finite volume method for the advection diffusion equation in irregular geometries we present a fully conservative high resolution finite volume algorithm for advection diffusion equations in irregular geometries the algorithm uses a cartesian grid in which some cells are cut by the embedded boundary a novel feature is the use of a capacity function to model the fact that some cells are only partially available to the fluid the advection portion then uses the explicit wave propagation methods implemented in clawpack and is stable for courant numbers up to 1 diffusion is modelled with an implicit finite volume algorithm results are shown for several geometries convergence is verified and the 1 norm order of accuracy is found to between 1 2 and 2 depending on the geometry and peclet number software is available on the web

an immersed finite element method with integral equation correction summary we propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions the solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution for computing the finite element solution the physical domain is embedded into a slightly larger cartesian box shaped domain and is discretized using a block structured mesh the defect in the essential boundary conditions which occurs along the physical domain boundaries is subsequently corrected with an integral equation method in order to facilitate the mapping between the finite element and integral equation solutions the physical domain boundary is represented with a signed distance function on the block structured mesh as a result only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated except for the non boundary conforming block structured mesh the overall approach is first presented for the poisson equation and then generalized to incompressible viscous flow equations as an example of fluid structure coupling the settling of a heavy rigid particle in a closed tank is considered copyright 2010 john wiley sons ltd

universal meshes a method for triangulating planar curved domains immersed in nonconforming meshes summary we introduce a new method to triangulate planar curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary in the process no new vertices are introduced and connectivities of triangles are left unaltered the method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection to guarantee its robustness we require that the domain be c2 regular the background mesh be sufficiently refined near the boundary and that specific angles in triangles near the boundary be strictly acute the method can render both straight edged and curvilinear triangulations for the immersed domain the latter includes curved triangles that conform exactly to the immersed boundary and ones constructed with isoparametric mappings to interpolate the boundary at select points high order finite elements constructed over these curved triangles achieve optimal accuracy which has customarily proven difficult in numerical schemes that adopt nonconforming meshes aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains with no conformity requirements the same background mesh can be adopted to triangulate a large family of domains immersed in it including ones realized over several updates during the coarse of simulating problems with moving boundaries we term such a background mesh as a universal mesh for the family of domains it can be used to triangulate universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes furthermore because the evolving geometry can be approximated with any desired order numerical solutions can be computed with high order accuracy we present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with stokesian fluids copyright 2014 john wiley sons ltd

stable imposition of stiff constraints in explicit dynamics for embedded finite element methods summary we investigate various strategies to enforce the kinematics at an embedded interface for transient problems within the extended finite element method in particular we focus on explicit time integration of the semi discrete equations of motion and extend both dual and primal variational frameworks for constraint enforcement to a transient regime we reiterate the incompatibility of the dual formulation with purely explicit time integration and the severe restrictions placed by the courant friedrichs levy condition on primal formulations we propose an alternate consistent formulation for the primal method and derive an estimate for the stabilization parameter which is more amenable in an explicit dynamics framework importantly the use of the new estimate circumvents the need for any tolerances as an interface approaches an element boundary we also show that with interfacial constraints existing mass lumping schemes can lead to prohibitively small critical time steps accordingly we propose a mass lumping procedure which provides a more favorable estimate these techniques are then demonstrated on several benchmark numerical examples where we compare and contrast the accuracy of the primal methods against the dual methods in enforcing the constraints copyright 2012 john wiley sons ltd

approximate imposition of boundary conditions in immersed boundary methods we analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods as basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain and therefore dirichlet boundary conditions need to be prescribed in an approximate way as starting variational approach we consider nitsche s methods and we then move to two options that yield non symmetric problems but that turned out to be robust and efficient the essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition

weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non matching meshes this is the accepted version of the following article codina r and baiges j 2015 weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non matching meshes int j numer meth engng 104 624 654 doi 10 1002 nme 4815 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 4815 abstract

derivation and validation of a novel implicit second order accurate immersed boundary method a novel implicit second order accurate immersed boundary method ibm for simulating the flow around arbitrary stationary bodies is developed implemented and validated in this paper the ibm is used to efficiently take into account the existence of bodies within the fluid domain the flow domain consist of simple cartesian cells whereas the body can be arbitrary at the triangulated interface of the body and the fluid the immersed boundary the coefficients obtained from discretizing the navier stokes equations are closed with a second order accurate interpolation arising from the immersed boundary condition employed at the interface two different conditions are developed in this paper and it is shown that for the mirroring method the resulting coefficients lead to a well posed and diagonally dominant system which can be efficiently solved with a preconditioned krylov sub space solver the immersed boundary condition generates a fictitious reversed velocity field inside the immersed boundary which is excluded from the continuity equation to account for the presence of the ib in the pressure correction equation resulting in no mass flux over the ib the force acting on the object from the fluid is determined by integrating the pressure and the viscous forces over the object the method is validated by simulating the flow around a sphere for a range of re numbers it is shown that the drag is very well in agreement with experimental data accuracy and convergence studies are employed proving the second order accuracy of the method and showing the superiority in convergence rate compared to other ibm finally the drag force of a cluster of non spherical particles is employed to show the generality and potential of the method

weak spectral mapping theorems for c0 groups associated to transport equations in slab geometry abstract we show in this paper that the c 0 groups governing neutron transport equations with reflecting boundary conditions in slab geometry satisfy a weak spectral mapping theorem in any l p spaces 1 p the main part of this paper is dealing with collisionless transport c 0 group while the full transport c 0 group is dealt with by stability arguments

on parameter space of complex polynomial vector fields in c abstract the space d of degree d single variable monic and centered complex polynomial vector fields can be decomposed into loci in which the vector fields have the same topological structure this paper analyzes the geometric structure of these loci and describes some bifurcations in particular it is proved that new homoclinic separatrices can form under small perturbation by an example we show that this decomposition of parameter space by combinatorial data is not a cell decomposition the appendix to this article joint work with tan lei shows that landing separatrices are stable under small perturbation of the vector field if the multiplicities of the equilibrium points are preserved

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

configurations of critical points in complex polynomial differential equations abstract in this work we focus on the configuration location and stability of simple critical points of polynomial differential equations of the form z f z z c the case where all the critical points are of center type is studied in more detail finding several new center configurations one of the main tools in our approach is the 1 dimensional euler jacobi formula

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

positivity and stability of the essential spectrum for singular transport semigroups with non vacuum boundary conditions in this paper we prove some compactness results for a large class of singular transport equations our approach relies on some comparison results for positive operators our results extend and complete several earlier works we also provide simpler proofs for several known results in the literature

a least squares fictitious domain method for incompressible viscous flow around obstacles with navier slip boundary condition abstract in this article we discuss a least squares fictitious domain method for incompressible viscous flow around obstacles with navier slip boundary condition assuming that and b are two bounded sub domains of r d with b in order to solve the incompressible navier stokes equations with a navier slip condition on the boundary of the obstacle b we advocate a fictitious domain method where one solves a simpler variant of the original problem on the whole followed by a well chosen correction over b this method is of the virtual control type and relies on a least squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space a detailed discussion of the finite element implementation of the above methodology is also provided numerical results are given they suggest optimal order of convergence

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

on the classical solutions of the boltzmann equation on etudie le probleme aux valeurs initiales pour l equation de boltzmann on montre que des solutions au sens classique existent pour tout temps si la donnee initiale est assez proche de l equilibre on se restreint a un domaine cubique et on considere seulement la condition aux limites periodique

domain imbedding methods for the stokes equations we study direct and iterative domain imbedding methods for the stokes equations on certain non rectangular domains in two space dimensions we analyze a continuous analog of numerical domain imbedding for bounded smooth domains and give an example of a simple numerical algorithm suggested by the continuous analysis this algorithms is applicable for simply connected domains which can be covered by rectangular grids with uniformly spaced grid lines in at least one coordinate direction we also discuss a related fft based fast solver for stokes problems with physical boundary conditions on rectangles and present some numerical results

a fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems part 2 neumann boundary conditions we present a fictitious domain decomposition method for the fast solution of acoustic scattering problems characterized by a partially axisymmetric sound hard scatterer we apply this method to the solution of a mock up submarine problem and highlight its computational advantages and intrinsic parallelism a key component of our method is an original idea for addressing a neumann boundary condition in the general framework of a fictitious domain method this idea is applicable to many other linear partial differential equations besides the helmholtz equation copyright 2003 john wiley sons ltd

optimal spectral theory of the linear boltzmann equation we give optimal compactness results in lp spaces 1 p related to spectral theory of general neutron transport equations on spatial domains with finite lebesgue measure

singular one dimensional transport equations on lp spaces we prove the well posedness of the cauchy problem governed by a linear mono energetic singular transport equation i e transport equation with unbounded collision frequency and unbounded collision operator with specular reflecting and periodic boundary conditions on lp spaces the large time behaviour of its solution is also considered we discuss the compactness properties of the second order remainder term of the dyson phillips expansion for a large class of singular collision operators this allows us to evaluate the essential type of the transport semigroup from which the asymptotic behaviour of the solution is derived

an implicit boundary finite element method with extension to frictional sliding boundary conditions and elasto plastic analyses abstract implicit boundary methods which enrich the interpolation structure with implicit weight functions are straightforward methods for the enforcement of dirichlet boundary conditions in this article we follow the implicit boundary method that uses approximate step functions the step boundary method developed by kumar et al and provide modifications that have several advantages roller boundary conditions have wide practical applications in engineering however the step boundary method for roller boundary conditions with inclinations has yet to be fully formulated through to the final linear system of equations thus we provide a complete derivation that leads to simplified stiffness matrices compared to the original approach which can be implemented directly in fictitious domain finite element analysis the approach is then extended we believe for the first time to the nonlinear cases of frictional boundary conditions and elasto plastic material behaviour the proposed formulation and procedures are validated on a number of example problems that test different aspects of the method

on l1 exponential trend to equilibrium for conservative linear kinetic equations on the torus we compute the growth bound of weighted advection semigroups on the n dimensional flat torus with general not necessarily bounded measurable nonnegative collision frequencies and give a compactness theorem in l1 spaces of interest for the stability of essential type of kinetic semigroups under perturbation by scattering operators this provides us with a characterisation of the existence of a spectral gap for general conservative neutron transport like equations various related questions are also dealt with

simulation of elastic filaments interacting with a viscous pulsatile flow abstract motivated by the need to understand the motion of ruptured atherosclerosis plaque fragments in a flowing blood we simulated flexible elastic filaments interacting with a viscous pulsatile flow in two dimensions using the immersed boundary method our simulations show that 1 a flexible horizontal filament positioned perpendicularly to the flow is a stable configuration and 2 hydrodynamical forces prevent adhesion of filaments but mutual interaction of vortical fields yields clustering of filaments into groups that may be called aggregation

on the rate of convergence to equilibrium for the linear boltzmann equation with soft potentials abstract in this work we present several quantitative results of convergence to equilibrium for the linear boltzmann operator with soft potentials under grad s angular cut off assumption this is done by an adaptation of the famous entropy method and its variants resulting in explicit algebraic or even stretched exponential rates of convergence to equilibrium under appropriate assumptions the novelty in our approach is that it involves functional inequalities relating the entropy to its production rate which have independent applications to equations with mixed linear and non linear terms we also briefly discuss some properties of the equation in the non cut off case and conjecture what we believe to be the right rate of convergence in that case

finite element analysis on implicitly defined domains an accurate representation based on arbitrary parametric surfaces in this paper we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis the novel contributions of this paper are threefold 1 describe and validate a method to represent arbitrary parametric surfaces implicitly 2 represent arbitrary solids implicitly including sharp features using level sets and boolean operations 3 impose arbitrary dirichlet and neumann boundary conditions on the resulting implicitly defined boundaries the methods proposed do not require local refinement of the finite element mesh in regions of high curvature ensure the independence of the domain s volume on the mesh do not rely on boundary regularization and are well suited to methods based on fixed grids such as the extended finite element method xfem numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries this approach is one step in the desired direction of tying numerical simulations to computer aided design cad similarly to the isogeometric analysis paradigm

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

a fictitious domain distributed lagrange multiplier based fluid structure interaction scheme with hierarchical b spline grids abstract we present a numerical scheme for fluid structure interaction based on hierarchical b spline grids and fictitious domain distributed lagrange multipliers the incompressible navier stokes equations are solved over a cartesian grid discretised with b splines the fluid grid near the immersed solids is refined locally using hierarchical b splines the immersed solid is modelled as geometrically exact beam discretised with standard linear lagrange shape functions the kinematic constraint at the fluid solid interface is enforced with distributed lagrange multipliers the unconditionally stable and second order accurate generalised method is used for integration in time for both the fluid and solid domains a fully implicit and fully coupled solution scheme is developed by using the newton raphson method to solve the non linear system of equations obtained with galerkin weak formulation first the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder then the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme the results obtained with the present scheme are compared with the reference values

distributed lagrange multiplier fictitious domain method in the framework of lattice boltzmann method for fluid structure interactions a new implementation of the lattice boltzmann method lbm for fluid structure interactions is presented the idea of the distributed lagrange multiplier fictitious domain method dlm fd is introduced in the framework of the lattice boltzmann algorithm this implementation employs a fixed mesh for the solution of the fluid problem and a lagrangian formulation for the solid problem the main advantage of the method is that the re meshing procedure normally required in the ale method is circumvented numerical examples are provided to verify the algorithm and illustrate the capacity of the method to deal with the fluid elastic solid interactions

a fictitious domain method using equilibrated basis functions for harmonic and bi harmonic problems in physics abstract in this paper we present a new meshless method to solve well known problems in physics and engineering with either constant or variable material properties in 2d space harmonic and bi harmonic problems are considered in this paper the method constructs a set of basis functions called equilibrated basis functions eqbfs through a weighted residual integration over a fictitious domain embedding the main one the bases can satisfy the governing partial differential equations approximately chebyshev polynomials of the first kind are employed to construct the eqbfs and exponential functions are used as the weights in the integrals the parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1d ones over a normalized intervals this reduces the computational efforts significantly either the eqbfs or the results of the integration process may be stored for further use the validity of the results is examined through an extended patch test a set of physical sample problems with variable non variable material properties as the potential flow over a cylinder steady state heat conduction problems in an anisotropic inhomogeneous functionally graded material potential and stokes flow through a channel of finite width obstructed by periodic cylinders and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method as a preliminary study we show that the method may effectively be used in a domain decomposition approach

a least squares finite element method for the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line in this article we discuss the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall the method we employ combines a finite element space approximation with a time discretization by operator splitting to solve the cahn hilliard part of the problem we use a least squares conjugate gradient method we also show that the scheme has the total energy decaying in time property under certain conditions our numerical experiments indicate that the method discussed here is accurate stable and efficient

an explicit formula for the derivative of a class of cost functionals with respect to domain variations in stokes flow domain optimization problems for the two dimensional stationary stokes equations are studied frechet differentiability of a class of cost functionals with respect to the variation of the shape of the computational domain is established an embedding domain technique provides an equivalent formulation of the problem on a fixed domain and moreover a simply computable formula for the derivative of the cost functional with respect to the domain existence of a solution to the class of domain optimization problems is proved numerical examples show the reliability of the derivative formula

a partitioned fluid structure algorithm for elastic thin valves with contact we present a numerical method to simulate the movements of a thin valve immersed in an incompressible viscous fluid the fluid and structure meshes are not matching the kinematic continuity is imposed using lagrange multipliers the method therefore belongs to the fictitious domain fd family this approach allows very large displacements we propose a partitioned fluid structure algorithm which keeps the fluid and structure solvers independent and which is able to manage contact without assuming that the structure solvers include contact capabilities various numerical tests are proposed in particular when the displacement is moderate we show that the results are in good agreement with arbitrary lagrangian eulerian ale simulations we then present results with contact on a rigid wall finally we show that the proposed approach can be mixed with ale when the wall is elastic

a fictitious domain method for the simulation of thermoelastic deformations in nc milling processes summary this paper presents a higher order finite element approach for the simulation of heat diffusion and thermoelastic deformations in nc milling processes the inherent continuous material removal in the process of the simulation is taken into account via continuous removal dependent refinements of a paraxial hexahedron base mesh covering a given workpiece these refinements rely on isotropic bisections of these hexahedrons along with subdivisions of the latter into tetrahedrons and pyramids in correspondence to a milling surface triangulation obtained from the application of the marching cubes algorithm the resulting mesh is employed for an element wise defined characteristic function for the milling dependent workpiece within that paraxial hexahedron base mesh using this characteristic function a higher order fictitious domain method is used to compute the heat diffusion and thermoelastic deformations where the corresponding ansatz spaces are defined for some hexahedron based refinement of the base mesh numerical experiments compared to real physical experiments exhibit the applicability of the proposed approach to predict deviations of the milled workpiece from its designed shape due to thermoelastic deformations in the process this article is protected by copyright all rights reserved

a note on the stability parameter in nitsche s method for unfitted boundary value problems abstract nitsche s method is a popular approach to implement dirichlet type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible the method is widely applied in the context of unfitted finite element methods of the classical symmetric nitsche s method it is well known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems in this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this we show that in specific situations error bounds can deteriorate and give examples of computations where nitsche s method yields large and even diverging discretization errors

an enhanced immersed structural potential method for fluid structure interaction within the group of immersed boundary methods employed for the numerical simulation of fluid structure interaction problems the immersed structural potential method ispm was recently introduced gil et al 2010 1 in order to overcome some of the shortcomings of existing immersed methodologies in the ispm an incompressible immersed solid is modelled as a deviatoric strain energy functional whose spatial gradient defines a fluid structure interaction force field in the navier stokes equations used to resolve the underlying incompressible newtonian viscous fluid in this paper two enhancements of the methodology are presented first the introduction of a new family of spline based kernel functions for the transfer of information between both physics in contrast to classical ibm kernels these new kernels are shown not to introduce spurious oscillations in the solution second the use of tensorised gaussian quadrature rules that allow for accurate and efficient numerical integration of the immersed structural potential a series of numerical examples will be presented in order to demonstrate the capabilities of the enhanced methodology and to draw some key comparisons against other existing immersed methodologies in terms of accuracy preservation of the incompressibility constraint and computational speed

an operator splitting scheme with a distributed lagrange multiplier based fictitious domain method for wave propagation problems we propose a novel fictitious domain method based on a distributed lagrange multiplier technique for the solution of the time dependent problem of scattering by an obstacle we study discretizations that include a fully conforming approach as well as mixed finite element formulations utilizing the lowest order nedelec edge elements in 2d on rectangular grids we also present a symmetrized operator splitting scheme for the scattering problem which decouples the operator that propagates the wave from the operator that enforces the dirichlet condition on the boundary of an obstacle a new perfectly matched layer pml model is developed to model the unbounded problem of interest this model is based on a formulation of the wave equation as a system of first order equations and uses a change of variables approach that has been developed to construct pml s for maxwell s equations we present an analysis of our fictitious domain approach for a one dimensional wave problem based on calculations of reflection coefficients we demonstrate the advantages of our fictitious domain approach over the staircase approximation of the finite difference method we also demonstrate some important properties of the distributed multiplier approach that are not shared by a boundary multiplier fictitious domain approach for the same problem numerical results for two dimensional wave problems that validate the effectiveness of the different methods are presented

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

periodic orbits in complex abel equations abstract this paper is devoted to prove two unexpected properties of the abel equation d z d t z 3 b t z 2 c t z where b and c are smooth 2 periodic complex valuated functions t r and z c the first one is that there is no upper bound for its number of isolated 2 periodic solutions in contrast recall that if the functions b and c are real valuated then the number of complex 2 periodic solutions is at most three the second property is that there are examples of the above equation with b and c being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of b and c this result is also in contrast with the characterization of the center variety for the examples of abel equations d z d t a t z 3 b t z 2 studied in the literature where the center variety is located in a finite number of connected components

multiscale methods for problems with complex geometry we propose a multiscale method for elliptic problems on complex domains e g domains with cracks or complicated boundary for local singularities this paper also offers a discrete alternative to enrichment techniques such as xfem we construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account the corrections only need to be computed in regions surrounding fine scale geometric features we achieve linear convergence rate in the energy norm for the multiscale solution moreover the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh the analytical findings are verified in a series of numerical experiments

a fictitious domain method for conformal modeling of the perfect electric conductors in the fdtd method we present a fictitious domain method to avoid the staircase approximation in the study of perfect electric conductors pec in the finite difference time domain fdtd method the idea is to extend the electromagnetic field inside the pec and to introduce a new unknown the surface electric current density to ensure the vanishing of the tangential components of the electric field on the boundary of the pec this requires the use of two independent meshes a regular three dimensional 3 d cubic lattice for the electromagnetic field and a triangular surface patching for the surface electric current density the intersection of these two meshes gives a simple coupling law between the electric field and the surface electric current density an interesting property of this method is that it provides the surface electric current density at each time step furthermore this method looks like fdtd with a special model for the pec numerical results for several objects are presented

a fictitious domain approach with spread interface for elliptic problems with general boundary conditions the aim of this article is to solve second order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach the main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain as the spatial discretization is being performed in the fictitious domain this method allows the use of structured meshes the discretization is not boundary fitted to the original physical domain this paper describes several ways to impose dirichlet robin or neumann boundary conditions on a spread immersed interface without locally modifying the numerical scheme and without using lagrange multipliers the numerical applications focus on difiusion and convection problems in the unit disk with dirichlet or robin boundary conditions for such problems analytical solutions can be determined for a correctly chosen source term the numerical resolution is performed using a q1 finite element scheme the spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vinicity of the immersed boundary the l2 norm of the errors is computed in order to evaluate the capability of the method immersed boundaries are found in many industrial applications like two phase flow simulations fluid structure interaction etc this article represents a first step towards the simulation of these kinds of applications

implementation of regularized isogeometric boundary element methods for gradient based shape optimization in two dimensional linear elasticity the present work addresses shape sensitivity analysis and optimization in two dimensional elasticity with a regularized isogeometric boundary element method igabem non uniform rational b splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations with the advantage of tight integration of design and analysis the application of igabem in shape optimization reduces the mesh generation regeneration burden greatly the work is distinct from the previous literatures in igabem shape optimization mainly in two aspects 1 the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives which were the main implementation difficulty in igabem and 2 although based on the same computer aided design cad model the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh thus achieving a balance between less design variables for efficiency and refined mesh for accuracy this technique was initially used in isogeometric finite element method and was incorporated into the present igabem implementation

a boundary multiplier fictitious domain method for the steady incompressible navier stokes equations we analyze the error of a fictitious domain method with boundary lagrange multiplier it is applied to solve a non homogeneous steady incompressible navier stokes problem in a domain with a multiply connected boundary the interior mesh in the fictitious domain and the boundary mesh are independent up to a mesh length ratio

a cartesian grid nonconforming immersed finite element method for planar elasticity interface problems in this paper a new nonconforming immersed finite element ife method on triangular cartesian meshes is developed for solving planar elasticity interface problems the proposed ife method possesses optimal approximation property for both compressible and nearly incompressible problems its degree of freedom is much less than those of existing finite element methods for the same problem moreover the method is robust with respect to the shape of the interface and its location relative to the domain and the underlying mesh both theory and numerical experiments are presented to demonstrate the effectiveness of the new method theoretically the unisolvent property and the consistency of the ife space are proved experimentally extensive numerical examples are given to show that the approximation orders in l2 norm and semi h1 norm are optimal under various lam parameters settings and different interface geometry configurations

parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids a novel immersed object method is developed for simulating two dimensional unsteady incompressible viscous flows around arbitrarily moving rigid bodies it has been implemented in a parallel unstructured finite volume incompressible navier stokes solver based on the artificial compressibility ac approach using a higher order characteristics based upwind scheme and matrix free implicit dual time stepping in the immersed object method an object is immersed in the flow field and it is supposed to contain frozen fluid which moves like a solid body this is realized by introducing source terms in the momentum equations during the ac sub iterations an internal mesh within the object is employed to search and locate all the eulerian nodes within the object in every time step for imposing the source terms unlike many existing methods this method does not require complex searching extrapolation and interpolation to find the intersections of the object boundary with the unstructured background mesh and assign flow condition onto the object boundary if it is necessary to capture the boundary layer accurately then a dense overlapping grid can then be constructed around the object for further refined calculation the immersed object method has been used to simulate steady and unsteady incompressible viscous flows over a stationary circular cylinder rotating square cylinder and moving disk in cavity the results agree well with published numerical solutions and experimental measurements

weak impositions of dirichlet boundary conditions in solid mechanics a critique of current approaches and extension to partially prescribed boundaries abstract in this article we first review various approaches developed to date for the weak imposition of dirichlet boundary conditions in fictitious domain analysis for elasticity problems the hellinger reissner h r principle the linked lagrange multiplier llm method the implicit boundary method and the fat boundary method are discussed along with the well known lagrange multiplier penalty and nitsche s methods we state these approaches in a common form starting with energy functionals and weak forms and discretise using the fictitious domain finite element method previous formulations of these methods were in general developed for full prescription along the dirichlet boundary which generally implies no local effect of boundary inclination however partially prescribed conditions such as the structural roller boundary condition with inclination have wide practical applications in engineering here we provide techniques of imposing such boundary conditions in these methods in detail for those methods that contain algorithmic parameters such as the penalty and nitsche s methods extra computation or empirical estimation is necessary to decide values of the parameters and hence we discuss parametric and convergence behaviours through numerical examples to provide guidance on the choice of parameters

the fixed mesh ale approach applied to solid mechanics and fluid structure interaction problems in this paper we propose a method to solve solid mechanics and fluid structure interaction problems using always a fixed background mesh for the spatial discretization the main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves to achieve this we use an arbitrary lagrangian eulerian ale framework the distinctive characteristic being that at each time step results are projected onto a fixed background mesh for solid mechanics problems subject to large strains the fixed mesh fm ale method avoids the element stretching found in fully lagrangian approaches for fsi problems fm ale allows for the use of a single background mesh to solve both the fluid and the structure copyright 2009 john wiley sons ltd

a variational framework for fluid solid interaction problems based on immersed domains theoretical bases abstract in this work the theoretical bases to handle the fluid solid interaction problem based on the conception of immersed domains are developed starting from a well known governing variational principle from the continuum mechanics the purpose of the present work is twofold firstly a variational principle to model the interaction between a fluid and a thin structure characterized as a shell for which its volume is geometrically neglected is devised by incorporating suitable kinematical assumptions into the original problem this leads to a rather general formulation for the fluid shell interaction problem that embraces methodologies such as the immersed boundary method secondly no longer is the volume of the solid disregarded and the variational principle is manipulated in order to attain a formulation that comprises the problem of an augmented fluid which is extended as a fictitious fluid to the whole domain of analysis interacting with an immersed solid in some parts the formulation resembles the immersed finite element method as well as the recently presented immersed continuous method due to the appearance of a volume force defined over the fictitious fluid domain one of the main differences lies in the fact that in the present approach the volume force is the lagrange multiplier that imposes the continuity of the velocity field along the whole overlapped domain the other noteworthy difference dwells in the handling of the mass conservation for the artificial fluid three aspects of this setting must be pointed out as original contributions of the work i a consistent generalization of the theoretical continuous setting of immersed boundary methods is carried out ii the variational basis for the finite element approximations is clearly posed written and easy to understand and iii when dealing with an immersed solid a more general formulation valid for all classes of combinations fluid solid is developed eliminating the restriction to compressible fluid compressible solid of the immersed continuous method and the incompressible fluid incompressible solid of the immersed finite element method

simultaneous bifurcation of limit cycles from two nests of periodic orbits abstract let z f z be an holomorphic differential equation having a center at p and consider the following perturbation z f z e r z z we give an integral expression similar to an abelian integral whose zeroes control the limit cycles that bifurcate from the periodic orbits of the period annulus of p this expression is given in terms of the linearizing map of z f z at p the result is applied to control the simultaneous bifurcation of limit cycles from the two period annuli of z i z z 2 after a polynomial perturbation

convergence analysis of the q1 finite element method for elliptic problems with non boundary fitted meshes summary the aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain s boundary this problematic of the last century e g the finite difference methodology returns to topical studies with the huge development of domain embedding fictitious domain or cartesiangrid methods these methods use regular structured meshes most often cartesian for non aligned domains although non boundary fitted approaches become more and more applied very few studies are devoted to theoretical error estimates in this paper the convergence of a q1 nonconforming finite element method is analyzed for secondorder elliptic problems with dirichlet robin or neumann boundary conditions the finite element method uses standard q1 rectangular finite elements as the finite element approximate space is not contained in the original solution space this method is referred to as nonconforming a stairstep boundary defined from the cartesian mesh approximates the original domain s boundary the convergence analysis of the finite element method for such a kind of non boundary fitted stairstepped approximation is no treated in the literature the study of dirichlet problems is based on similar techniques as those classically used with boundary fitted linear triangular finite elements the estimates obtained for robin problems are novel and use some more technical arguments the rate of convergence is proved to be in o h 1 2 for the h 1 norm for all general boundary conditions and classical duality arguments allow to obtain an o h error estimate in the l 2 norm for dirichlet problems numerical results obtained with fictitious domain techniques that impose original boundary conditions on a non boundary fitted approximate immersed interface are presented these results confirm the theoretical rates of convergence copyright c 2007 john wiley sons ltd

a compactness result for perturbed semigroups and application to a transport model abstract in this paper we are concerned with the compactness properties of remainder terms of the dyson phillips expansion of perturbed semigroups on general banach spaces more specifically we derive conditions which ensure the compactness of the remainder term r n t for some integer n our result applies directly to discuss the time asymptotic behaviour for large times of the solution of a one dimensional transport equation with reentry boundary conditions on l 1 spaces without regularity conditions on the initial data

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

dynamics of structures coupled with elastic media a review of numerical models and methods abstract this paper reviews issues and developments in the field of structure environment interaction problems in which the environment is an elastic body possibly unbounded it covers in particular the fields of soil structure interaction ground borne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium the general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled and the domain decomposition technique classically used for its description is put up extensions for the cases of nonlinear structure and uncertain environment are then discussed finally the ongoing research in the fields of unbounded interface moving interface and multiple interfaces are reviewed and summarized