convergence and superconvergence of staggered discontinuous galerkin methods for the three dimensional maxwell s equations on cartesian grids in this paper a new type of staggered discontinuous galerkin methods for the three dimensional maxwell s equations is developed and analyzed the spatial discretization is based on staggered cartesian grids so that many good properties are obtained first of all our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the gauss law moreover the mass matrices are diagonal thus time marching is explicit and is very efficient our method is high order accurate and the optimal order of convergence is rigorously proved it is also very easy to implement due to its cartesian structure and can be regarded as a generalization of the classical yee s scheme as well as the quadrilateral edge finite elements furthermore a superconvergence result that is the convergence rate is one order higher at interpolation nodes is proved numerical results are shown to confirm our theoretical statements and applications to problems in unbounded domains with the use of pml are presented a comparison of our staggered method and non staggered method is carried out and shows that our method has better accuracy and efficiency

simulating the dynamics of inextensible vesicles by the penalty immersed boundary method in this paper we develop an immersed boundary ib method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid in order to take into account the inextensibility constraint of the vesicle the penalty immersed boundary pib method is used to virtually decouple the fluid and vesicle dynamics as numerical tests of our current pib method the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored and compared with the previous literature the method is also validated by a series of convergence study which confirms its consistent first order accuracy on the velocity field the vesicle configuration the vesicle area and the perimeter errors in addition the method is also applied to study a binary component vesicle problem

a staggered discontinuous galerkin method for wave propagation in media with dielectrics and meta materials some electromagnetic materials exhibit in a given frequency range effective dielectric permittivity and or magnetic permeability which are negative in the literature they are called negative index materials left handed materials or meta materials we propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta material the method we investigate can be considered as an alternative method compared to the method presented by the second author and co workers in particular we shall use the abstract framework they developed to prove well posedness of the exact problem we recast this problem to fit later discretization by the staggered discontinuous galerkin method developed by the first author and co worker a method which relies on introducing an auxiliary unknown convergence of the numerical method is proven with the help of explicit inf sup operators and numerical examples are provided to show the efficiency of the method

a fast numerical method for solving the three dimensional stokes equations in the presence of suspended particles abstract a new fast numerical method for solving the three dimensional stokes equations in the presence of suspended particles is presented the fluid dynamics equations are solved on a lattice a particle is represented by a set of points each of which moves at the local fluid velocity and is not constrained to lie on the lattice these points are coupled by forces which resist deformation of the particle these forces contribute to the force density in the stokes equations as a result a single set of fluid dynamics equations holds at all points of the domain and there are no internal boundaries particle size shape and deformability may be prescribed computational work increases only linearly with the number of particles so large numbers 500 1000 of particles may be studied efficiently the numerical method involves implicit calculation of the particle forces by minimizing an energy function and solution of a finite difference approximation to the stokes equations using the fourier toeplitz method the numerical method has been implemented to run on all cray computers the implementation exploits the cray s vectorized arithmetic and on machines with insufficient central memory it performs efficient disk 1 o while storing most of the data on disk applications of the method to sedimentation of one two and many particle systems are described trajectories and settling speeds for two particle sedimentation and settling speed for multiparticle sedimentation from initial distributions on a cubic lattice or at random give good quantitative agreement with existing theories

modeling a no slip flow boundary with an external force field abstract a novel technique related to peskin s immersed boundary approach is used to introduce solid surfaces into a simulated flow field the navier stokes equations permit the presence of an externally imposed body force that may vary in space and time forces are chosen to lie along a desired surface and to have a magnitude and direction opposing the local flow such that the flow is brought to rest on an element of the surface for unsteady viscous flow the direct calculation of the needed force is facilitated by a feedback scheme in which the velocity is used to iteratively determine the desired value in particular we determine the surface body force from the relation f x 5 t t 0 u x 5 t dt u x 5 t for surface points x 5 velocity u time t and negative constants and examples are presented which include 2d flow around cylinders 3d turbulent channel flow where one boundary is simulated with a force field and turbulent channel flow over a riblet covered surface while the new method may be applied to complex geometries on a non cartesian mesh we have chosen to use a simple cartesian grid all simulations are done with a spectral code in a single computational domain without any mapping of the mesh

a staggered discontinuous galerkin method for the stokes system discontinuous galerkin dg methods are a class of efficient tools for solving fluid flow problems there are in the literature many greatly successful dg methods in this paper a new staggered dg

force evaluation in the lattice boltzmann method involving curved geometry the present work investigates two approaches for force evaluation in the lattice boltzmann equation the momentum exchange method and the stress integration method on the surface of a body the boundary condition for the particle distribution functions on curved geometries is handled with second order accuracy based on our recent works mei et al j comput phys 155 307 1999 ibid 161 680 2000 the stress integration method is computationally laborious for two dimensional flows and in general difficult to implement for three dimensional flows while the momentum exchange method is reliable accurate and easy to implement for both two dimensional and three dimensional flows several test cases are selected to evaluate the present methods including i two dimensional pressure driven channel flow ii two dimensional uniform flow past a column of cylinders iii two dimensional flow past a cylinder asymmetrically placed in a channel with vortex shedding iv three dimensional pressure driven flow in a circular pipe and v three dimensional flow past a sphere the drag evaluated by using the momentum exchange method agrees well with the exact or other published results

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

higher order triangular finite elements with mass lumping for the wave equation in this article we construct new higher order finite element spaces for the approximation of the two dimensional 2d wave equation these elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping these formulas are constructed explicitly error estimates are provided for the corresponding semidiscrete problem finally higher order finite difference time discretizations are proposed and various numerical results are shown

discontinuous galerkin finite element method for the wave equation the symmetric interior penalty discontinuous galerkin finite element method is presented for the numerical discretization of the second order wave equation the resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal hence the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time stepping scheme optimal a priori error bounds are derived in the energy norm and the l 2 norm for the semidiscrete formulation in particular the error in the energy norm is shown to converge with the optimal order cal o h min s ell with respect to the mesh size h the polynomial degree ell and the regularity exponent s of the continuous solution under additional regularity assumptions the l 2 error is shown to converge with the optimal order cal o h min s ell numerical results confirm the expected convergence rates and illustrate the versatility of the method

large eddy simulation in complex geometric configurations using boundary body forces anumericalmethodispresentedthatallowslargeeddysimulation les ofturbulente owsincomplexgeometric cone gurations with moving boundaries and that retains the advantages of solving the navier stokes equations on e xed orthogonal grids the boundary conditions are applied independently of the grid by assigning body forces over surfaces that need not coincide with coordinate lines the use of orthogonal nondeforming grids simplie es grid generation facilitates theimplementation of high order nondissipativediscretization schemes andminimizes the spatial and temporal variations in e lter width that complicate unstructured deforming grid les dynamic subgrid scaleturbulence models areparticularly appealing in combination with the body forceprocedure because the dynamic model accounts automatically for the presence of solid walls without requiring damping functions the method is validated by simulations of the turbulent e ow in a motored axisymmetric piston cylinder assembly for which detailed experimental measurements are available computed mean and rms velocity proe les show very good agreement with measured ensemble averages thepresent numerical code runs on small personal computerlike workstations for a comparable level of accuracy computational requirements memory and cpu time are at least a factor of 10 lower compared to published simulations for the same cone guration obtained using an unstructured boundary e tted deforming grid approach

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

two level overlapping schwarz algorithms for a staggered discontinuous galerkin method two overlapping schwarz algorithms are developed for a discontinuous galerkin finite element approximation of second order scalar elliptic problems in both two and three dimensions the discontinuous galerkin formulation is based on a staggered discretization introduced by chung and engquist siam j numer anal 47 2009 pp 3820 3848 for the acoustic wave equation two types of coarse problems are introduced for the two level schwarz algorithms the first is built on a nonoverlapping subdomain partition which allows quite general subdomain partitions and the second on introducing an additional coarse triangulation that can also be quite independent of the fine triangulation condition number bounds are established and numerical results are presented

convergence analysis of an energy based discontinuous galerkin method for the wave equation in second order form hp version abstract in this paper we study the error estimates of the h p version of a spatial discontinuous galerkin method proposed in appelo and hagstrom 2015 for the wave equation for different degrees of the approximation spaces we give the error estimates depending on the flux parameters in the energy norm and prove that optimal convergence rates can be obtained for some special flux parameters which is an improvement of the existing work of appelo and hagstrom 2015 finally we provide numerical experiments to verify our theoretical findings

application of a ghost fluid approach for a thermal lattice boltzmann method in this paper a ghost fluid gf method is utilized to propose a numerical approach to enhance the capability of thermal lattice boltzmann method tlbm in dealing with complex geometries a ghost fluid approach is imposed on a double population thermal lattice boltzmann method a cartesian grid handles the flow and the boundaries are imposed by a ghost fluid approach the essence of this method is to decompose the unknown distribution functions into equilibrium and non equilibrium parts at each ghost points the major quantities are extrapolated from the image points to the corresponding ghost points to form the equilibrium parts the non equilibrium parts are then determined by using the bounce back scheme the method is relatively easy to apply and second order accurate there is no need to modify the original governing equations and both dirichlet and neumann boundary conditions can be handled the method is applied to couette flow between two concentric circular cylinders natural convection in a square cavity natural convection in an annulus and a forced convection in a lid driven semi circular cavity the results obtained are generally in good agreement with that predicted by other theoretical and numerical efforts

a bddc algorithm for a class of staggered discontinuous galerkin methods a bddc balancing domain decomposition by constraints algorithm is developed and analyzed for a staggered discontinuous galerkin dg finite element approximation of second order scalar elliptic problems on a quite irregular subdomain partition an optimal condition number bound is proved for two dimensional problems in addition a sub optimal but scalable condition number bound is obtained for three dimensional problems these bounds are shown to be independent of coefficient jumps in the subdomain partition numerical results are also included to show the performance of the algorithm

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

regular and slow fast codimension 4 saddle node bifurcations abstract using geometric singular perturbation theory including the family blow up as one of the main techniques we prove that the cyclicity i e maximum number of limit cycles in both regular and slow fast unfoldings of nilpotent saddle node singularity of codimension 4 is 2 the blow up technique enables us to use the well known results for slow fast codimension 1 and 2 hopf bifurcations slow fast bogdanov takens bifurcations and slow fast codimension 3 saddle and elliptic bifurcations

canard cycles for predator prey systems with holling types of functional response abstract by using the singular perturbation theory developed by dumortier and roussarie and recent work of de maesschalck and dumortier we study the canard phenomenon for predator prey systems with response functions of holling types we first develop a formula for computing the slow divergence integrals by using the formula we prove that for the systems with the response function of holling types iii and iv the cyclicity of any limit periodic set is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations we also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two

limit cycles near hyperbolas in quadratic systems abstract in this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane we study a strip of hyperbolas occurring in some quadratic systems we deal with the cyclicity of the degenerate graphics d i 2 a from the programme set up in f dumortier r roussarie c rousseau hilbert s 16th problem for quadratic vector fields j differential equations 110 1994 86 133 to solve the finiteness part of hilbert s 16th problem for quadratic systems techniques from geometric singular perturbation theory are combined with the use of the bautin ideal we also rely on the theory of darboux integrability

multi layer canard cycles and translated power functions abstract the paper deals with two dimensional slow fast systems and more specifically with multi layer canard cycles these are canard cycles passing through n layers of fast orbits with n 2 the canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n we prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions

slow fast bogdanov takens bifurcations abstract in this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of bogdanov takens bt bifurcations we introduce among other precise definitions the notion of slow fast bt bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits based on geometric singular perturbation theory including blow up we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

an analysis of new mixed finite elements for the approximation of wave propagation problems we construct and analyze a new family of rectangular two dimensional or cubic three dimensional mixed finite elements for the approximation of the acoustic wave equations the main advantage of this element is that it permits us to obtain through mass lumping an explicit scheme even in an anisotropic medium nonclassical error estimates are given for this new element

multiple canard cycles in generalized lienard equations the paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields the results deal with any number of parameters proofs are based on the techniques introduced in canard cycles and center manifolds f dumortier and r roussarie 1996 mem amer math soc 121 the presentation is limited to generalized lienard equations ex x c x x c 0

a new modification of the immersed boundaries method for fluid solid flows moderate reynolds numbers in this study we consider a combination of an interpolation technique with the d alambert principle that allows the direct numerical simulation of fluid solid flows on a regular rectangular grid the method models a no slip boundary with the second order accuracy interpolation scheme without explicit calculation of the forces that the particles exert at the fluid the method is applicable for a wide range of reynolds numbers from creeping flows up to re 100

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

cyclicity of a fake saddle inside the quadratic vector fields abstract this paper concerns the study of small amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity this degenerate singularity is also known as an impassable grain the canonical form of the unperturbed vector field is like a degenerate flow box near the singularity the phase portrait consists of parallel fibers all but one of which have no singular points and at the singular fiber there is one node we demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic

a posteriori error analysis for a staggered discontinuous galerkin discretization of the time harmonic maxwell s equations in this paper we present the first a posteriori error analysis for the staggered discontinuous galerkin sdg method specifically we consider the approximation of the time harmonic maxwell s equations by a sdg method and prove that our residual based a posteriori error indicator is both reliable and efficient we validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems

periodic orbits and chaos in fast slow systems with bogdanov takens type fold points abstract the existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast slow type having bogdanov takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow up method in particular the blow up method is effectively used for analyzing the flow near the bogdanov takens type fold point in order to show that a slow manifold near the fold point is extended along the boutroux s tritronquee solution of the first painleve equation in the blow up space

limit cycles in slow fast codimension 3 saddle and elliptic bifurcations abstract this paper deals with local bifurcations occurring near singular points of planar slow fast systems in particular it is concerned with the study of the slow fast variant of the unfolding of a codimension 3 nilpotent singularity the slow fast variant of a codimension 1 hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the van der pol system similarly codimension 2 slow fast bogdanov takens bifurcations have been characterized here the singularity is of codimension 3 and we distinguish slow fast elliptic and slow fast saddle bifurcations we focus our study on the appearance on small amplitude limit cycles and rely on techniques from geometric singular perturbation theory and blow up

a bddc algorithm with adaptive primal constraints for staggered discontinuous galerkin approximation of elliptic problems with highly oscillating coefficients a bddc balancing domain decomposition by constraints algorithm for a staggered discontinuous galerkin approximation is considered after applying domain decomposition method a global linear system on the subdomain interface unknowns is obtained and solved by the conjugate gradient method combined with a preconditioner to construct a preconditioner that is robust to the coefficient variations a generalized eigenvalue problem on each subdomain interface is solved and primal unknowns are selected from the eigenvectors using a predetermined tolerance by the construction of the staggered discontinuous galerkin methods the degrees of freedom on subdomain interfaces are shared by only two subdomains and hence the construction of primal unknowns are simplified the resulting bddc algorithm is shown to have the condition number bounded by the predetermined tolerance a modified algorithm for parameter dependent problems is also introduced where the primal unknowns are only computed in an offline stage numerical results are included to show the performance of the proposed method and to verify the theoretical estimate

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

numerical simulation of a cylinder in uniform flow in this study a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow this approach readily allows the imposition of a no slip boundary within the flow field by a feedback forcing term added to the momentum equations in the present work this technique is used with a high order finite difference method effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers very good agreement is found between the present calculations and previous computational and experimental results for steady and time dependent flow at low reynolds numbers

a deluxe feti dp algorithm for a hybrid staggered discontinuous galerkin method for h curl elliptic problems summary convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered dg finite element approximation of h curl elliptic problems in two dimensions in addition to the advantages of staggered dg methods the basis functions of the new hybrid staggered dg method are all locally supported in the triangular elements and a lagrange multiplier approach is applied to enforce the global connections of these basis functions the interface problem on the lagrange multipliers is further reduced to the resulting problem on the subdomain interfaces and a dual and primal finite element tearing and interconnecting algorithm with an enriched weight factor is then applied to the resulting problem our algorithm is shown to give a condition number bound ofc 1c log h h 2 independent of the two parameters whereh his the number of triangles across each subdomain numerical results are included to confirm our theoretical bounds copyright 2013 john wiley sons ltd received 12 september 2013 accepted 11 november 2013 in this paper a dual and primal finite element tearing and interconnecting feti dp algorithm is developed for a fast and stable solution of a new hybrid staggered dg hsdg method applied to h curl problems staggered dg sdg methods were first introduced in 1 3 for wave propagation problems the idea was subsequently applied to other problems such as convection diffusion equations 4 electromagnetic problems 5 9 stokes equations 10 and multiscale wave simulations 11 12 similar to 5 the advantages of using sdg for h curl problems are the preservation of the structures of the differential operators the local conservation property and the optimal convergence in particular the discretizations of the two curl operators by the sdg method are adjoint to each other and the null space of the discrete curl operator is exactly the gradients moreover the divergence condition is automatically satisfied in an appropriate weak sense despite the aforementioned advantages the implementation of sdg methods requires careful numbering of the supports of the basis functions to allow an easier implementation of the sdg method the hsdg method is introduced in this paper one distinctive advantage of the hsdg method is that the basis functions are all locally supported in the triangular elements hence the numbering can be easily performed the fact that the basis functions are totally discontinuous requires some global couplings in particular we will use the lagrange multiplier approach to enforce the global connection of the basis functions this idea is also related to the hybridized dg methods 13 16 in fact it is shown that the sdg method is the limit of a hybridized dg method

singular perturbations and vanishing passage through a turning point abstract the paper deals with planar slow fast cycles containing a unique generic turning point we address the question on how to study canard cycles when the slow dynamics can be singular at the turning point we more precisely accept a generic saddle node bifurcation to pass through the turning point it reveals that in this case the slow divergence integral is no longer the good tool to use but its derivative with respect to the layer variable still is we provide general results as well as a number of applications we show how to treat the open problems presented in artes et al 2009 1 and dumortier and rousseau 2009 13 dealing respectively with the graphics di 2 a and df 1 a from dumortier et al 1994 14

two dimensional simulation of rayleigh waves with staggered sine cosine transforms and variable grid spacing simulation of rayleigh waves requires high accuracy and an adequate spatial sampling at the surface discrete cosine and sine transforms are used to compute spatial derivatives along the direction perpendicular to the surface of the earth unlike the standard fourier method these transforms allow nonperiodicboundaryconditionstobesatisfied inparticular the stress free conditions at the surface because simulation of surface waves requires more points per minimum wavelengthatthesurfacethansimulationofbodywaves theequispaced grid is not efficient to overcome this problem a grid compression is performed at the surface to obtain a denser spatial sampling grid size is minimal at the surface and increases with depth until reaching at a relatively shallow depth the grid points per wavelength required by the body waves the stress free boundary conditions are naturally handled by expanding the appropriate stress components in terms of the discrete sine transform the wave equation is solved in the particle velocity and stress formulation using a runge kutta time integration and the convolutional pml cpmlmethodtopreventreflectionsfromthemeshboundaries the simulations are very accurate for shallow sources andreceiversandlargeoffsets

canard phenomenon for an sis epidemic model with nonlinear incidence abstract by using the singular perturbation theory especially on canard cycles the canard phenomenon for an sis epidemic model with nonlinear incidence is investigated the phenomenon suggests the existence of disease outbreak in the infection transmission it is proved that the cyclicity of any possible slow fast cycle is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the slow fast cycle by small perturbations we also indicate the regions in parameter space where the corresponding slow fast cycle has cyclicity at most one or at most two

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

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

an immersed boundary method based on the gas kinetic bgk scheme for incompressible viscous flow in this paper an immersed boundary ib method based on the gas kinetic bhatnagar gross krook bgk scheme is proposed to simulate the incompressible viscous flow around stationary and moving rigid bodies in the presented ib method a set of lagrangian points represent the solid boundary and will exert force on the surrounding eulerian points which are the cell centers within the finite volume framework this force is calculated by a special iterative procedure and has an effect on the flux at the interface of cell which guarantees the fulfillment of the no slip boundary condition and entirely avoids the flow penetration the flow field is obtained from the bgk scheme with local grid refinement without complex mesh generation and transformation the present technique can be conveniently applied to simulations with complex and moving solid boundaries the second order temporal accuracy and better than first order spatial accuracy in l 2 norm are testified by the simulation of stokes first problem other three different test cases including flows around a stationary circular cylinder two circular cylinders in tandem and an oscillating circular cylinder demonstrate the good capability of the present method expanding the immersed boundary ib method to the finite volume gas kinetic bhatnagar gross krook bgk scheme the flux across the cell s interface will be influenced by the immersed boundary force the flow penetration to the immersed boundary is completely eliminated the present method can be conveniently applied to problems with moving objects

boundary conditions for lattice boltzmann simulations with complex geometry flows boundary conditions of lattice boltzmann method to simulate flows embedded with a solid object is proposed the closest nodes adjacent to the boundary in the fluid domain are used as boundary nodes of the flow domain the fluid velocity of the boundary node is obtained by linear interpolation between the velocities of the solid object and the second fluid node further away then distribution functions originating from the solid domain at the boundary nodes are modified using known distribution functions and correctors to satisfy the momentum this boundary condition is an extended form of a method proposed by hou et al c f hou c chang c a lin consistent boundary conditions for 2d and 3d lattice boltzmann simulations submitted for publication for plane wall and regular geometry the technique is examined by simulating decaying vortex transient flow induced by an abruptly rotating ring and flow over an asymmetrically placed cylinder numerical simulations indicate that this method is second order accurate and all the numerical results are compatible with the benchmark solutions

an improved rotated staggered grid finite difference method with fourth order temporal accuracy for elastic wave modeling in anisotropic media abstract the rotated staggered grid rsg finite difference method is a powerful tool for elastic wave modeling in 2d anisotropic media where the symmetry axes of anisotropy are not aligned with the coordinate axes we develop an improved rsg scheme with fourth order temporal accuracy to reduce the numerical dispersion associated with prolonged wave propagation or a large temporal step size the high order temporal accuracy is achieved by including high order temporal derivatives which can be converted to high order spatial derivatives to reduce computational cost dispersion analysis and numerical tests show that our method exhibits very low temporal dispersion even with a large temporal step size for elastic wave modeling in complex anisotropic media using the same temporal step size our method is more accurate than the conventional rsg scheme our improved rsg scheme is therefore suitable for prolonged modeling of elastic wave propagation in 2d anisotropic media

canards heteroclinic and homoclinic orbits for a slow fast predator prey model of generalized holling type iii abstract for a classical ratio dependent predator prey model with the generalized holling type iii functional response it was previously investigated in 20 by hsu and huang for global stability of an equilibrium and in 21 by huang ruan and song for subcritical hopf and bogdanov takens bifurcations here in this model when prey reproduces much faster than predator by using geometric singular perturbation theory we achieve much richer new dynamical phenomena than the existing ones such as the existence of canard cycles canard explosion and relaxation oscillations heteroclinic and homoclinic orbits cyclicity of slow fast cycles and the coexistence of the hopf cycle and the relaxation oscillation on global stability of the equilibrium we also provide less restricted conditions than the existing ones

immersed boundary methods for general finite difference and finite volume navier stokes solvers we present an immersed boundary algorithm for incompressible flows with complex boundaries suitable for cartesian or curvilinear grid system the key stages of any immersed boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary a line in 2d a surface in 3d and the spreading of a force field from the immersed boundary to the neighboring mesh points to enforce the desired boundary conditions on the immersed boundary points we propose a technique that uses the reproducing kernel particle method w k liu s jun y f zhang reproducing kernel particle methods int j numer methods fluids 20 8 1995 1081 1106 for the interpolation and spreading unlike other methods presented in the literature the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved independent of the grid topology uniform or non uniform cartesian or curvilinear the technique is easy to implement and is able to maintain the order of the original underlying spatial discretization applications to two and three dimensional flows in cartesian and non cartesian grid system with uniform and non uniform meshes are presented

rotational invariance in the three dimensional lattice boltzmann method is dependent on the choice of lattice the lattice boltzmann method has recently gained popularity as a tool for simulating complex fluid flows it uses discrete sets of velocity vectors or lattices to create a reduced model of the molecular dynamics of a continuum fluid while several lattices are believed to behave isotropically there are reports of qualitatively incorrect results however thus far the reason as to why a lack of isotropy occurs is not known based on the hypothesis that lower order lattices may not display rotational invariance this study tests the isotropy of the d3q15 d3q19 and d3q27 lattices by performing simulations at intermediate reynolds numbers 50 500 and low knudsen number 250 the d3q15 and d3q19 lattices produced different results depending on the plane of the lattice with which the flow was aligned lattice planes with fewer than six velocity vectors consistently produced results which were qualitatively different from the planes with six or more velocity vectors these errors were not observed at re 50 or when a d3q27 lattice was used they appeared to be independent of grid density collision operator and ma this suggests that the lattices which contain these planes are not fully isotropic and therefore do not properly replicate the behavior of a real fluid in this particular situation notably downstream from the expansion predictions made using these models in more complex geometries may therefore be affected by the orientation of the lattice when using lbm in cfd simulation including validation this study highlights the need for caution to ensure that the solution obtained is independent of the lattice orientation throughout the domain

an improved vacuum formulation for 2d finite difference modeling of rayleigh waves including surface topography and internal discontinuities rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography modeling of rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics for simulation of rayleigh waves the traction free boundary condition needs to be satisfied on the free surface avacuum formulation naturally incorporates surface topographyinfinite difference fd modeling by treating the surface grid nodes as the internal grid nodes however theconventionalvacuumformulationdoesnotcompletely fulfill the free surface boundary condition and becomes unstableformodelingusinghigh orderfdoperators wedeveloped a stable vacuum formulation that fully satisfies the free surface boundary condition by choosing an appropriate combination of the staggered grid form and a parameter averaging scheme the elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes the improved vacuum formulation can accurately and stably simulate rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high poisson s ratios 0 4 this method requires fewer grid points per wavelength than the stress image based methods internal discontinuities in a model can be handled without modification of the algorithm onlyminorchangesarerequiredtoimplementtheimproved vacuum formulation in existing 2d fd modeling codes

one point boundary condition for the lattice boltzmann method we propose a correction to the bounce back boundary condition for lattice boltzmann algorithms which improves the accuracy of pressure from zero to first order and the accuracy of velocity from first to second order compared to interpolation based corrections our method has the advantage of being completely local in fact methods using interpolation face difficulties at boundary points where not enough neighboring nodes are available we show that a combination with our method offers a natural solution to this problem

slow divergence integral and its application to classical lienard equations of degree 5 abstract the slow divergence integral is a crucial tool to study the cyclicity of a slow fast cycle for singularly perturbed planar vector fields in this paper we deduce a useful form for this integral in order to apply it to various problems as an example we use it to prove that the slow divergence integral along any non degenerate slow fast cycle for singular perturbations of classical lienard equations of degree 5 has at most one zero and the zero is simple if it exists hence the cyclicity of any non degenerate slow fast cycle in this class of equations is at most 2 up to now there were many interesting results about lienard equations of degree 3 4 and 6 but almost nothing is known about degree 5 the result in this paper can be seen as a first stage to study the uniform property for classical lienard equations of degree 5

discontinuous galerkin methods with staggered hybridization for linear elastodynamics abstract in this paper we will develop a new staggered hybridization technique for discontinuous galerkin methods to discretize linear elastodynamic equations the idea of hybridization is used extensively in many discontinuous galerkin methods but the idea of staggered hybridization is new our new approach offers several advantages namely energy conservation high order optimal convergence preservation of symmetry for the stress tensor block diagonal mass matrices as well as low dispersion error the key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh we prove the stability and the convergence of the proposed scheme in both the semi discrete and the fully discrete settings numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion furthermore an application of this method to rayleigh wave propagation is presented

simulation of flexible filaments in a uniform flow by the immersed boundary method an improved version of the immersed boundary ib method is developed for simulating flexible filaments in a uniform flow the proposed ib method is based on an efficient navier stokes solver adopting the fractional step method and a staggered cartesian grid system the fluid motion defined on an eulerian grid and the filament motion defined on a lagrangian grid are independently solved and their interaction force is explicitly calculated using a feedback law a direct numerical method is developed to calculate the filament motion under the constraint of inextensibility when applied to the case of a swinging filament analogous to a rope pendulum the proposed method gave results very similar to those of the analytical solution derived using the perturbation method for a flexible filament flapping in a uniform flow the mechanism by which small vortex processions are produced was investigated the bistable property of the system was observed by altering the filament length and the effects of the boundary condition at the fixed end simply supported or clamped were studied for two side by side filaments in a uniform flow both in phase flapping and out of phase flapping were reproduced in the present simulations a repulsive force was included in the formulation to handle collisions between the free ends of side by side filaments undergoing out of phase flapping

an improved direct forcing immersed boundary method for finite difference applications a modified immersed boundary method is developed using the direct forcing concept an improved bilinear interpolation extrapolation algorithm is implemented for more accurate boundary forcing expressions and easier implementation detailed discussions of the method are presented on the stability velocity interpolation on the immersed boundary direct forcing extrapolation to the grid points resolution of the immersed boundary points and internal treatment the method can achieve second order accurate solutions the method is then applied to a finite difference scheme to compute flow over a stationary cylinder an oscillating cylinder and a stationary sphere the accuracy of the computational results is verified using numerous computational and experimental results in the literature

numerical simulation of dendritic solidification with convection two dimensional geometry a front tracking method is presented for simulations of dendritic growth of pure substances in the presence of flow the liquid solid interface is explicitly tracked and the latent heat released during solidification is calculated using the normal temperature gradient near the interface a projection method is used to solve the navier stokes equations the no slip condition on the interface is enforced by setting the velocities in the solid phase to zero the method is validated through a comparison with an exact solution for a stefan problem a grid refinement test and a comparison with a solution obtained by a boundary integral method three sets of two dimensional simulations are presented a comparison with the simulations of beckermann et al j comput phys 154 468 1999 a study of the effect of different flow velocities and a study of the effect of the prandtl number on the growth of a group of dendrites growing together the simulations show that on the upstream side the dendrite tip velocity is increased due to the increase in the temperature gradient and the formation of side branches is promoted the flow has the opposite effect on the downstream side the results are in good qualitative agreement with published experimental results even though only the two dimensional aspects are examined here

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

higher order conformal decomposition fem cdfem abstract a higher order accurate finite element method is proposed which uses automatically generated meshes based on implicit level set data for the description of boundaries and interfaces in two and three dimensions the method is an alternative for fictitious domain and extended finite element methods the domain of interest is immersed in a background mesh composed by higher order elements the zero level sets are identified and meshed followed by a decomposition of the cut background elements into conforming sub elements adaptivity is a crucial ingredient of the method to guarantee the success of the mesh generation it ensures the successful decomposition of cut elements and enables improved geometry descriptions and approximations it is confirmed that higher order accurate results with optimal convergence rates are achieved with the proposed conformal decomposition finite element method cdfem

a staggered discontinuous galerkin method for the simulation of seismic waves with surface topography accurate simulation of seismic waves is of critical importance in a variety of geophysical applications based on recent works on staggered discontinuous galerkin methods we have developed a new method for the simulations of seismic waves which has energy conservation and extremely low grid dispersion so that it naturally provided accurate numerical simulations of wave propagation useful for geophysical applications and was a generalization of classical staggered grid finite difference methods moreover it could handle with ease irregular surface topography and discontinuities in the subsurface models our new method discretized the velocity and the stress tensor on this staggered grid with continuity imposed on different parts of the mesh the symmetry of the stress tensor was enforced by the lagrange multiplier technique the resulting method was an explicit scheme requiring the solutions of a block diagonal system and a local saddle point system in each time step and it was therefore very efficient to tailor our scheme to rayleigh waves we developed a mortar formulation of our method specifically a fine mesh was used near the free surface and a coarse mesh was used in the rest of the domain the two meshes were in general not matching and the continuity of the velocity at the interface was enforced by a lagrange multiplier the resulting method was also efficient in time marching we also developed a stability analysis of the scheme and an explicit bound for the time step size in addition we evaluated some numerical results and found that our method was able to preserve the wave energy and accurately computed the rayleigh waves moreover the mortar formulation gave a significant speed up compared with the use of a uniform fine mesh and provided an efficient tool for the simulation of rayleigh waves

slow divergence integral on a mobius band abstract the slow divergence integral has proved to be an important tool in the study of slow fast cycles defined on an orientable two dimensional manifold e g r 2 the goal of our paper is to study 1 canard cycle and 2 canard cycle bifurcations on a non orientable two dimensional manifold e g the mobius band by using similar techniques our focus is on smooth slow fast models with a hopf breaking mechanism the same results can be proved for a jump breaking mechanism and non generic turning points the slow fast bifurcation problems on the mobius band require the study of the 2 return map attached to such 1 and 2 canard cycles we give a simple sufficient condition expressed in terms of the slow divergence integral for the existence of a period doubling bifurcation near the 1 canard cycle we also prove the finite cyclicity property of singular 1 and 2 homoclinic loops regular 1 homoclinic loops of finite codimension have been studied by guimond

regular article an accurate curved boundary treatment in the lattice boltzmann method the lattice boltzmann equation lbe is an alternative kinetic method capable of solving hydrodynamics for various systems major advantages of the method are due to the fact that the solution for the particle distribution functions is explicit easy to implement and natural to parallelize because the method often uses uniform regular cartesian lattices in space curved boundaries are often approximated by a series of stairs that leads to reduction in computational accuracy in this work a second order accurate treatment of the boundary condition in the lbe method is developed for a curved boundary the proposed treatment of the curved boundaries is an improvement of a scheme due to o filippova and d hanel 1998 j comput phys 147 219 the proposed treatment for curved boundaries is tested against several flow problems 2 d channel flows with constant and oscillating pressure gradients for which analytic solutions are known flow due to an impulsively started wall lid driven square cavity flow and uniform flow over a column of circular cylinders the second order accuracy is observed with a solid boundary arbitrarily placed between lattice nodes the proposed boundary condition has well behaved stability characteristics when the relaxation time is close to 1 2 the zero limit of viscosity the improvement can make a substantial contribution toward simulating practical fluid flow problems using the lattice boltzmann method

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

an immersed boundary method for fluid flexible structure interaction an immersed boundary method for simulation of fluid flexible structure interaction has been proposed in the method an efficient navier stokes solver adopting the fractional step method and a staggered cartesian grid system is used to solve the incompressible fluid motion in an eulerian domain on the other hand a moving lagrangian grid is used to discretize the structure domain and the equation of structure motion is derived by the energy method and is solved by an iterative method the fluid structure interaction is formulated by an additional momentum forcing which is obtained on the lagrangian grid using the structure motion equation and is spread to the nearby eulerian girds by applying a smoothed approximation of the dirac delta function in this way the massive boundary is handled easily without any special treatment while for the neutrally buoyant case our formulation becomes equivalent with previous ones in the simulations we mainly use small additional boundary mass toward biofluid applications three numerical examples are simulated a lid driven cavity with a deformable volume at the bottom a swimming jellyfish and a deformable circular ring moving through a channel with contraction the results are compared with those of previous studies and the theoretical solutions the temporal and spatial convergence rates of the present method are specified the tendency of the numerical solution as the additional boundary mass approaches zero is investigated

an immersed boundary method based on discrete stream function formulation for two and three dimensional incompressible flows an immersed boundary method is proposed in the framework of discrete stream function formulation for incompressible flows in order to impose the non slip boundary condition the forcing term is determined implicitly by solving a linear system the number of unknowns of the linear system is the same as that of the lagrangian points representing the body surface thus the extra cost in force calculation is negligible if compared with that in the basic flow solver in order to handle three dimensional flows at moderate reynolds numbers a parallelized flow solver based on the present method is developed using the domain decomposition strategy to verify the accuracy of the immersed boundary method proposed in this work flow problems of different complexity decaying vortices flows over stationary and oscillating cylinders and a stationary sphere and flow over low aspect ratio flat plate are simulated and the results are in good agreement with the experimental or computational data in previously published literatures

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

singular perturbations of the holling i predator prey system with a focus abstract we study the occurrence of limit cyles in a model for two species predator prey interaction referred to as holling i this model is based on the phenomenological fitting of the so called functional response function to observed data in nature by a continuous piecewise differentiable function of the original models presented by holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet we extend the work of liu who showed that for certain values of parameters of the holling i system at least two nested limit cycles will occur surrounding a focus his case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities i e a singular perturbation problem with slow fast solutions we prove that exactly two hyperbolic limit cycles occur after perturbation

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

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 pre conditioned implicit direct forcing based immersed boundary method for incompressible viscous flows a novel immersed boundary ib method based on an implicit direct forcing idf scheme is developed for incompressible viscous flows the key idea for the present idf method is to use a block lu decomposition technique in momentum equations with taylor series expansion to construct the implicit ib forcing in a recurrence form which imposes more accurate no slip boundary conditions on the ib surface to accelerate the ib forcing convergence during the iterative procedure a pre conditioner matrix is introduced in the recurrence formulation of the ib forcing a jacobi type parameter is determined in the pre conditioner matrix by minimizing the frobenius norm of the matrix function representing the difference between the ib forcing solution matrix and the pre conditioner matrix in addition the pre conditioning parameter is restricted due to the numerical stability in the recurrence formulation consequently the present pre conditioned idf pidf enables accurate calculation of the ib forcing within a few iterations we perform numerical simulations of two dimensional flows around a circular cylinder and three dimensional flows around a sphere for low and moderate reynolds numbers the result shows that pidf yields a better imposition of no slip boundary conditions on the ib surfaces for low reynolds number with a fairly larger time step than ib methods with different direct forcing schemes due to the implicit treatment of the diffusion term for determining the ib forcing finally we demonstrate the robustness of the present pidf scheme by numerical simulations of flow around a circular array of cylinders flows around a falling sphere and two sedimenting spheres in gravity

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed

the staggered dg method is the limit of a hybridizable dg method we show in the framework of steady state diffusion boundary value problems that the staggered discontinuous galerkin sdg method siam j numer anal 47 2009 pp 3820 3848 can be obtained from a hybridizable discontinuous galerkin hdg method siam j numer anal 47 2009 pp 1319 1365 by setting its stabilization function to zero at some suitably chosen element faces and by letting it go to infinity at all the remaining others we then show that this point of view allows the sdg method to immediately acquire new properties all inherited from the hdg methods namely their efficient implementation by hybridization their postprocessings and their superconvergence properties

combined immersed boundary finite difference methods for three dimensional complex flow simulations a second order accurate highly efficient method is developed for simulating unsteady three dimensional incompressible flows in complex geometries this is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid the governing equations therefore can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures two different forcings are tested showing that while the quality of the results is essentially the same in both cases the efficiency of the calculation strongly depends on the particular expression a major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme this ranges from zeroth order for the most commonly used interpolations up to second order for an ad hoc velocity interpolation the present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature finally in the last example we show the flow inside an ic piston cylinder assembly at high reynolds number to our knowledge this is the first example in which the immersed boundary technique is applied to a full three dimensional complex flow with moving boundaries and with a reynolds number high enough to require a subgrid scale turbulence model

viscous flow and colloid transport near air water interface in a microchannel in order to understand the transport behavior of colloids near an air water interface awi two computational methods are applied to simulate the local water flow field near a moving awi in a 2d microfluidic channel the first method is a mesoscopic multicomponent and multiphase lattice boltzmann lbm model and the second is the macroscopic navier stokes based volume of fluid interface tracking method in the lbm it is possible to predict the dynamic contact angles after the static contact angle is correctly set and the predicted dynamic contact angles are in good agreement with previous observations it is demonstrated that the two methods can yield a similar flow velocity field if they are applied properly the flow field relative to awi depends on the direction of the flow and exhibits curved streamlines that transport fluid between the center of the channel and the wall region using the obtained flow the motion of sub micron colloids in a de ionized water solution is then studied by a lagrangian approach the observed colloid trajectories are in qualitative agreement with our visualizations using a confocal microscope