three dimensional volume conserving immersed boundary model for two phase fluid flows abstract we present a volume preserving scheme for two phase immiscible incompressible flows using an immersed boundary method ibm in a three dimensional space the two phase ibm employs a mixture of eulerian and lagrangian variables where the fluid interface is represented by discrete lagrangian markers exerting surface tension forces to the eulerian fluid domain and the markers are advected by the fluid velocity the interactions between the lagrangian markers and the fluid variables are linked by the discretized dirac delta function the present study extends the previous two dimensional research li et al volume preserving immersed boundary methods for two phase fluid flows int j numer meth fluids 69 2012 842 858 to the three dimensional space the key idea of the proposed method is relocating surface points along the normal directions to conserve the total volume we perform a number of numerical experiments to show the efficiency and accuracy of the proposed method

the overlapped radial basis function finite difference rbf fd method a generalization of rbf fd abstract we present a generalization of the rbf fd method that computes rbf fd weights in finite sized neighborhoods around the centers of rbf fd stencils by introducing an overlap parameter 0 1 such that 1 recovers the standard rbf fd method and 0 results in a full decoupling of stencils we provide experimental evidence to support this generalization and develop an automatic stabilization procedure based on local lebesgue functions for the stable selection of stencil weights over a wide range of values we provide an a priori estimate for the speedup of our method over rbf fd that serves as a good predictor for the true speedup we apply our method to parabolic partial differential equations with time dependent inhomogeneous boundary conditions neumann in 2d and dirichlet in 3d our results show that our method can achieve as high as a 60 speedup in 3d over existing rbf fd methods in the task of forming differentiation matrices

fast and accurate computation of hypersingular integrals in galerkin surface integral equation formulations via the direct evaluation method hypersingular 4 d integrals arising in the galerkin discretization of surface integral equation formulations are computed by means of the direct evaluation method the proposed scheme extends the basic idea of the singularity cancellation methods usually employed for the regularization of the singular integral kernel by utilizing a series of coordinate transformations combined with a reordering of the integrations the overall algebraic manipulation results in smooth 2 d integrals that can be easily evaluated via standard quadrature rules finally the reduction of the dimensionality of the original integrals together with the smooth behavior of the associated integrands lead up to unmatched accuracy and efficiency

stable and accurate pressure approximation for unsteady incompressible viscous flow how to properly specify boundary conditions for pressure is a longstanding problem for the incompressible navier stokes equations with no slip boundary conditions an analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the laplacian and leray projection operators here we make use of this formula to a improve the accuracy of computing pressure in two kinds of existing time discrete projection methods implicit in viscosity only and b devise new higher order accurate time discrete projection methods that extend a slip correction idea behind the well known finite difference scheme of kim and moin we test these schemes for stability and accuracy using various combinations of c 0 finite elements for all three kinds of time discretization one can obtain third order accuracy for both pressure and velocity without a time step stability restriction of diffusive type furthermore two kinds of projection methods are found stable using piecewise linear elements for both velocity and pressure

numerical evaluation of surface integrals in three dimensions in this paper we consider the evaluation of surface integrals over piecewise smooth surfaces in three dimensions the method consists in first replacing a parametrization for the surface and the integrand function by piecewise polynomial interpolants of them and second using a numerical integration scheme for the resulting integral the order of convergence is higher than would be expected based on the underlying interpolation theory

convergence analysis of finite element methods for h curl ω elliptic interface problems in this article we investigate the analysis of a finite element method for solving h curl elliptic interface problems in general three dimensional polyhedral domains with smooth interfaces the continuous problems are discretized by means of the first family of lowest order nedelec h curl conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter that quantifies the mismatch between the smooth interface and the triangulation optimal error estimates in the h curl norm are obtained for the first time the analysis is based on a strip argument a new extension theorem for h 1 curl functions across smooth interfaces a novel non standard interface aware interpolation operator and a perturbation argument for degrees of freedom for h curl conforming finite elements numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution

meshfree methods on manifolds for hydrodynamic flows on curved surfaces a generalized moving least squares gmls approach abstract we utilize generalized moving least squares gmls to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds we use exterior calculus to formulate incompressible hydrodynamic equations in the stokesian regime and handle the divergence free constraints via a generalized vector potential this provides less coordinate centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth order governing equations in terms of a system of second order elliptic operators using a hodge decomposition we develop methods for manifolds having spherical topology we show the methods exhibit high order convergence rates for solving hydrodynamic flows on curved surfaces the methods also provide general high order approximations for the metric curvature and other geometric quantities of the manifold and associated exterior calculus operators the approaches also can be utilized to develop high order solvers for other scalar valued and vector valued problems on manifolds

level set based topology optimization for electromagnetic dipole antenna design this paper presents a level set framework for a typical electromagnetic design problem of dipole antenna in this study the geometrical configuration of an antenna is represented by the zero level contour of a higher dimensional level set function the governing equation for the induced current flow on a metal surface is the electric field integral equation efie which takes into account the electric component of the incident wave the design objective is formulated in terms of the surface current and incident electric field the normal velocity of the level set model which reflects the sensitivity of the objective function is derived from the adjoint variable method and shape derivative by optimizing the objective function the area with the highest current density to which the voltage feeding should be applied can be reshaped the advantages of adopting the level set technique for electromagnetic design lie in its capacity for capturing sophisticated topological changes and facilitation in mathematical representation of the design configuration the demonstrative examples of dipole antenna design show that the level set method results in a fairly smooth optimization process where the vacuum metal interface gradually attains its optimal configuration a series of design cases with self adjoint and non self adjoint sensitivity analyses are studied and compared to the benchmarking problems in dipole antenna

a one domain approach for modeling and simulation of free fluid over a porous medium we propose a one domain approach based on the brinkman model for the modeling and simulation of the transport phenomenon between free fluid and a porous medium a thin transition layer is introduced between the free fluid region and the porous media region across which the porosity and permeability undergo a rapid but continuous change we study the behavior of the solution to the one domain model analytically and numerically using the method of matched asymptotic expansion we recover the beavers joseph saffman bjs interface condition as the thickness of the transition layer goes to zero we also calculate the error estimates between the leading order solution of the one domain model and the standard darcy stokes model of two domain model with bjs condition numerical methods are developed for both the one domain model and the two domain model numerical results are presented to support the analytical results thereby justifying the one domain model as a good approximation to the two domain stokes darcy model

time evolution of coherent structures in networks of hindmarch rose neurons abstract in the regime of partial synchronization networks of diffusively coupled hindmarch rose neurons show coherent structures developing in a region of the phase space which is wider than in the correspondent single neuron such structures are kept without important changes during several bursting periods in this work we study the time evolution of these structures and their dynamical stability under damage this system may model the behavior of ensembles of neurons coupled through a bidirectional gap junction or in a broader sense it could also account for the molecular cascades present in the formation of flash and short time memory

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3

convergence analysis of an extended krylov subspace method for the approximation of operator functions in exponential integrators we analyze the convergence of an extended krylov subspace method for the approximation of operator functions that appear in exponential integrators for operators the size of the polynomial part of the extended krylov subspace is restricted according to the smoothness of the initial data this restriction for the continuous operator has a significant influence on the approximation of matrix functions evaluated for matrices stemming from space discretizations of the continuous operator we prove convergence of the method for the continuous operator and in the discrete case this leads to a convergence independent of the norm of the discretized operator uniformly over all possible grids the analysis is illustrated by numerical experiments

entropy stable shock capturing space time discontinuous galerkin schemes for systems of conservation laws we present a streamline diffusion shock capturing spacetime discontinuous galerkin dg method to approximate nonlinear systems of conservation laws in several space dimensions the degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes we show entropy stability of the formally arbitrarily high order accurate method for a general system of conservation laws furthermore we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown numerical experiments are presented to illustrate the robustness of the proposed schemes

solutions of nonlinear delay and advanced partial difference equations in the space ln n1 abstract a functional analytic method is used to prove a general theorem which establishes the existence and the uniqueness of a solution of a class of nonlinear delay and advanced partial difference equations in the banach space l n n 1 the proof of the theorem has a constructive character which enables us to obtain a bound of the solution and a region depending on the initial conditions and the parameters of the equation under consideration where this solution holds some known nonlinear partial difference equations which appear in applications are studied as particular cases of the theorem

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

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

a least squares method for linear elasticity using a patch reconstructed space abstract we propose a discontinuous least squares finite element method for solving the linear elasticity the approximation space is obtained by patch reconstruction with only one unknown per element we apply the l 2 norm least squares principle with the weak imposition of the continuity across the interior faces to the stress displacement formulation the optimal convergence order under the energy norm is attained numerical results of the linear elasticity are presented to verify the error estimates our method enjoys the advantages of discontinuous galerkin methods and in addition we illustrate the great simplicity in implementation the robustness and the improved efficiency of our method

adaptive mesh refinement and superconvergence for two dimensional interface problems adaptive mesh refinement and the borgers algorithm are combined to generate a body fitted mesh which can resolve the interface with fine geometric details standard linear finite element method bas

fully symmetric integration formulas for the surface of the sere in s dimensions methods of polynomial degree d to approximate integrals over the surface of the s dimensional sphere are discussed such methods are constructed by choosing the abscissas and weights so that the formulas integrate a certain subset of the polynomials of maximum degree d exactly a simple set of monomials is described the exact integration of which will ensure that the method has the required polynomial degree results are obtained which enable the derivation of consistency conditions on the rule structures an example of a new s dimensional degree 11 rule is given

a quadratically convergent unstructured remeshing strategy for shape optimization a novel unstructured remeshing environment for gradient based shape optimization using triangular finite elements is presented the remeshing algorithm is based on a truss structure analogy in solving for the equilibrium position of the truss system the quadratically convergent newton s method is used exact analytical sensitivity information is therefore made available to the shape optimization algorithm the overall computational efficiency in gradient based shape optimization is very high in solving the truss structure analogy we compare our quadratically convergent newton solver with a previously proposed forward euler solver this includes notes regarding mesh uniformity element quality convergence rates and efficiency we present three numerical examples it is then shown that remeshing may introduce discontinuities and local minima we demonstrate that the effects of these on gradient based algorithms are alleviated to some extent through mesh refinement and may largely be overcome with a simple multi start strategy copyright 2005 john wiley sons ltd

a level set based topology optimization method targeting metallic waveguide design problems in this paper we propose a level set based topology optimization method targeting metallic waveguide design problems where the skin effect must be taken into account since the metallic waveguides are generally used in the high frequency range where this effect critically affects performance one of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal to implement this approach we develop a boundary tracking scheme for the arbitrary lagrangian eulerian ale mesh pertaining to the zero iso contour of the level set function that is given in an eulerian mesh and impose dirichlet boundary conditions at the nodes on the zero iso contour in the ale mesh to compute the electric field since the ale mesh accurately tracks the zero iso contour at every optimization iteration the electric field is always appropriately computed during optimization for the sensitivity analysis we compute the nodal coordinate sensitivities in the ale mesh and smooth them by solving a helmholtz type partial differential equation the obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the eulerian mesh and the level set function is updated based on the computed normal velocity finally the utility of the proposed method is discussed through several numerical examples copyright 2011 john wiley sons ltd

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

periodic states with functional phase relation in weakly coupled chaotic hindmarsh rose neurons abstract periodic states with functional phase relation are found for extremely weak coupling in the coupled hindmarsh rose hr neurons model and the coupled pikovsky circuit model where all oscillators take a same orbit while the phases of different oscillators have functional relation by analyzing a periodically forced chaotic neuron the phase distribution of various oscillators of the coupled hr neurons system is obtained the response curves of phase distribution correspond to different spike lock for the pikovsky circuit model similar phase distribution is also obtained

explicit cost bounds of stochastic galerkin approximations for parameterized pdes with random coefficients this work analyzes the overall computational complexity of the stochastic galerkin finite element method sgfem for approximating the solution of parameterized elliptic partial differential equations with both affine and non affine random coefficients to compute the fully discrete solution such approaches employ a galerkin projection in both the deterministic and stochastic domains produced here by a combination of finite elements and a global orthogonal basis defined on an isotopic total degree index set respectively to account for the sparsity of the resulting system we present a rigorous cost analysis that considers the total number of coupled finite element systems that must be simultaneously solved in the sgfem however to maintain sparsity as the coefficient becomes increasingly nonlinear in the parameterization it is necessary to also approximate the coefficient by an additional orthogonal expansion in this case we prove a rigorous complexity estimate for the number of floating point operations flops required per matrix vector multiplication of the coupled system based on such complexity estimates we also develop explicit cost bounds in terms of flops to solve the stochastic galerkin sg systems to a prescribed tolerance which are used to compare with the minimal complexity estimates of a stochastic collocation finite element method scfem shown in our previous work galindo et al 2015 finally computational evidence complements the theoretical estimates and supports our conclusion that in the case that the coefficient is affine the coupled sg system can be solved more efficiently than the decoupled sc systems however as the coefficient becomes more nonlinear it becomes prohibitively expensive to obtain an approximation with the sgfem

a consistent grayscale free topology optimization method using the level set method and zero level boundary tracking mesh summary this paper proposes a level set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming because the boundary tracking mesh is always conformed to the structural boundary good approximation to the boundary is maintained during optimization therefore structural design problems are solved completely without grayscale material previously we introduced the boundary tracking mesh generating method into level set based topology optimization and updated the design variables by solving the level set equation in order to adapt our previous method to general structural optimization frameworks the incorporation of the method with nonlinear programming is investigated in this paper to successfully incorporate nonlinear programming the optimization problem is regularized using a double well potential furthermore the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming we expect the investigation to open up a new class of grayscale free topology optimization the usefulness of the proposed method is demonstrated using several numerical examples targeting two dimensional compliant mechanism and metallic waveguide design problems copyright 2014 john wiley sons ltd

the newhouse set has a positive hausdorff dimension the newhouse phenomenon of infinitely many coexisting periodic attractors is studied in its simplest form one shows that the corresponding parameter set the newhouse set jn has a strictly positive hausdorff dimension this result is stronger than that of tedeschini lalli and yorke commun math phys 106 635 1986 concerning the lebesgue measure of the newhouse set and is complementary to our knowledge on the topological properties ofjn namely it is a residual set hence uncountable and everywhere dense in a parameter interval

a geometry projection method for continuum based topology optimization with discrete elements abstract this article describes a method for the continuum based topology optimization of structures made of discrete elements in particular we examine the optimization of linearly elastic planar structures made of bars of fixed width and semicircular ends the design space for the optimization consists of the endpoint locations of the bar s medial axes and their out of plane thicknesses to circumvent re meshing upon design changes we project the design onto a fixed analysis grid using a differentiable geometry projection that results in a density field indicating the fraction of solid material anywhere in the design space as in density based topology optimization methods the out of plane thickness is penalized so that the optimizer is capable of removing bars during the optimization the differentiability of the projection allows for the computation via the chain rule of design sensitivities of responses of interest and therefore it allows for the use of robust and efficient gradient based optimization methods notably this approach makes it easier to fabricate optimal designs by using off the shelf stock material furthermore the method considers the case where bars overlap at their joints i e their thicknesses are added at the joint and when they do not finally our proposed method naturally accommodates the imposition of several fixed length scales we demonstrate the proposed approach on classical problems of compliance based topology optimization and identify its benefits as well as research directions to be addressed in the future

near field imaging of infinite rough surfaces this paper is concerned with an inverse rough surface scattering problem in near field optical imaging which is to reconstruct the scattering surface with a resolution beyond the diffraction limit the surface is assumed to be a small and smooth deformation of a plane surface based on a transformed field expansion the boundary value problem with complex scattering surface is converted into a successive sequence of two point boundary value problems in the frequency domain where an analytic solution for the direct scattering problem is derived from the method of integrated solution by neglecting the high order terms in the power series expansion the nonlinear inverse problem is linearized and an explicit inversion formula is obtained a spectral cut off regularization is adopted to suppress the exponential growth of the noise in the evanescent wave components which carry high spatial frequency of the scattering surface and contribute to the superresolution in the near field regime the method works f

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

rotationally invariant quadratures for the sphere we construct near optimal quadratures for the sphere that are invariant under the icosahedral rotation group these quadratures integrate all n 1 2 linearly independent functions in a rotationally invariant subspace of maximal order and degree n the nodes of these quadratures are nearly uniformly distributed and the number of nodes is only marginally more than the optimal n 1 2 3 nodes using these quadratures we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of lagrange interpolation on the sphere this representation uses function values at the quadrature nodes in addition the representation yields an expansion that uses a single function centred and mostly concentrated at nodes of the quadrature thus providing a much better localization than spherical harmonic expansions we show that this representation may be localized even further we also describe two algorithms of complexity for using these grids and representations finally we note that our approach is also applicable to other discrete rotation groups

simulation of the phase field cahn hilliard and tumor growth models via a numerical scheme element free galerkin method abstract the main aim of this research work is to find the numerical solution based on a meshless technique for both the time dependent cahn hilliard and tumor growth partial differential equations the temporal variable is discretized using a second order method based on semi implicit backward differential formula and the stabilized term is added to the considered time discretization also an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization besides to approximate the spatial variables the element free galerkin method is considered for both mathematical models the result of full discrete schemes in studied models is solved via biconjugate gradient stabilized algorithm some numerical simulations are reported to show the capability of the numerical scheme presented here

finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement in this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces in particular we adopt a stabilizat

the mini mixed finite element for the stokes problem an experimental investigation abstract o h 3 2 superconvergence in pressure and velocity has been experimentally investigated for the two dimensional stokes problem discretized with the mini mixed finite element even though the classic mixed finite element theory for the mini element guarantees linear convergence for the total error recent theoretical results indicate that superconvergence of order o h 3 2 in pressure and of the linear part of the computed velocity to the piecewise linear nodal interpolation of the exact velocity is in fact possible with structured three directional triangular meshes the numerical experiments presented here suggest a more general validity of o h 3 2 superconvergence possibly to automatically generated and unstructured triangulations in addition the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise linear part of the computed velocity finding that the former is generally closer to the exact velocity whereas the latter conserves mass better

a computational method for the coupled solution of reaction diffusion equations on evolving domains and manifolds in this paper we devise a moving mesh finite element method for the approximate solution of coupled bulk surface reaction diffusion equations on an evolving two dimensional domain fundamental to the success of the method is the robust generation of bulk and surface meshes for this purpose we use a novel moving mesh partial differential equation mmpde approach the developed method is applied to model problems with known analytical solutions these experiments indicate second order spatial and temporal accuracy coupled bulk surface problems occur frequently in many areas in particular in the modelling of eukaryotic cell migration and chemotaxis we apply the method to a model of the two way interaction of a migrating cell in a chemotactic field where the bulk region corresponds to the extracellular region and the surface to the cell membrane

partial difference equations arising in numerical schemes and game theory a functional analytic method is used to prove that some linear and non linear partial difference equations of two three and four variables which arise in numerical schemes and game theory have a unique solution in the spaces l 2 n 2 or l 1 n 2 l 1 n 3 and l 1 n 4 respectively in the case of non linear difference equations a region which depends on the initial conditions and the parameters of the equation where the solution holds is given for both linear and non linear difference equations a bound of the solution is determined

generalized finite differences using fundamental solutions summary it is well known that solutions for linear partial differential equations may be given in terms of fundamental solutions the fundamental solutions solve the homogeneous equation exactly and are obtained from the solution of the inhomogeneous equation where the inhomogeneous term is described by a dirac delta distribution fundamental solutions are the building blocks of the boundary element method and of the method of fundamental solutions and are traditionally used to build boundary only global approximations in the domain of interest in this work the same characteristic of the fundamental solutions that of solving the homogeneous equation exactly is used but not to build a global approximation on the contrary local approximations are built in such a manner that it is possible to construct finite difference operators that are free from any form of structured grid copyright q 2009 john wiley sons ltd

qualitative measures for initial meshes this paper reviews geometric measures used to assess the shape of finite elements in two and three dimensional meshes measures have been normalized and made scale invariant whenever possible this paper also introduces a universal similarity region that enhances comparisons of triangles and their measures as a byproduct the usr provides a dynamic way to compare improved triangular meshes copyright 2000 john wiley sons ltd

a hybrid variational front tracking level set mesh generator for problems exhibiting large deformations and topological changes we present a method for generating 2 d unstructured triangular meshes that undergo large deformations and topological changes in an automatic way we employ a method for detecting when topological changes are imminent via distance functions and shape skeletons when a change occurs we use a level set method to guide the change of topology of the domain mesh this is followed by an optimization procedure using a variational formulation of active contours that seeks to improve boundary mesh conformity to the zero level contour of the level set function our method is advantageous for arbitrary lagrangian eulerian ale type methods and directly allows for using a variational formulation of the physics being modeled and simulated including the ability to account for important geometric information in the model such as for surface tension driven flow furthermore the meshing procedure is not required at every time step and the level set update is only needed during a topological change hence our method does not significantly affect computational cost

online planning of multiaxial loading path for elastic material identification while anisotropic materials provide more capabilities than regular isotropic materials the directional behavior of the anisotropic materials complicates their elastic and inelastic behavior such nonlinear behavior can be effectively observed and characterized by multiaxial testing but how to plan the loading path of a multiaxial test given a specimen and sensor readings while identifying material properties remains as an untouched issue this paper presents a methodology that updates the loading path at every sensor reading to identify reliable elastic moduli of anisotropic materials the multiaxial test must be quantified in order to find an effective loading path the proposed methodology uses distinguishability and uniqueness as such quantities by firstly analyzing the specimen on a continuum basis and then applying singular value decomposition svd associating the distinguishability and uniqueness to the reliability of elastic moduli the path planning problem is formulated such that an effective loading path can be found efficiently by a standard optimization method numerical examples first investigate the validity of the distinguishability and the uniqueness as performance measures to evaluate loading paths and the reliability of elastic moduli identified the efficacy of the proposed methodology has been then confirmed by analyzing it with and applying it to identification of elastic moduli

rbf loi augmenting radial basis functions rbfs with least orthogonal interpolation loi for solving pdes on surfaces abstract we present a new method for the solution of pdes on manifolds m r d of co dimension one using stable scale free radial basis function rbf interpolation our method involves augmenting polyharmonic spline phs rbfs with polynomials to generate rbf finite difference rbf fd formulas these polynomial basis elements are obtained using the recently developed least orthogonal interpolation technique loi on each rbf fd stencil to obtain local restrictions of polynomials in r 3 to stencils on m the resulting rbf loi method uses cartesian coordinates does not require any intrinsic coordinate systems or projections of points onto tangent planes and our tests illustrate robustness to stagnation errors we show that our method produces high orders of convergence for pdes on the sphere and torus and present some applications to reaction diffusion pdes motivated by biology

finite element approximation of convection diffusion problems using an exponentially graded mesh we present the analysis of an h version finite element method for the approximation of the solution to convection diffusion problems the method uses piece wise polynomials of degree p 1 defined on an exponentially graded mesh optimally constructed for the approximation of exponential layers we consider a model convection diffusion problem posed on the unit square and establish robust optimal convergence rates in the energy and in the maximum norm we also present the results of some numerical computations that illustrate our theoretical findings and compare the proposed method with others found in the literature

analysis of chaotic saddles in low dimensional dynamical systems the derivative nonlinear schrodinger equation abstract in this paper we present a computational study of nonattracting chaotic sets known as chaotic saddles in a low dimensional dynamical system describing stationary solutions of the derivative nonlinear schrodinger equation a driven dissipative model for alfven waves these chaotic saddles have gaps which are filled at chaotic transitions such as a saddle node bifurcation and an interior crisis we give a detailed explanation of how to numerically determine the chaotic saddles and describe how a chaotic attractor after an interior crisis point can be decomposed into two chaotic saddles dynamically connected by a set of coupling unstable periodic orbits created by a gap filling explosion after the crisis this coupling between two chaotic saddles is responsible for the intermittent dynamics displayed by the chaotic system after the interior crisis

a recovery based a posteriori error estimator for h curl interface problems abstract this paper introduces a new recovery based a posteriori error estimator for the lowest order nedelec finite element approximation to the h curl interface problem the error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results under certain assumptions it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients

the orthogonal gradients method a radial basis functions method for solving partial differential equations on arbitrary surfaces much work has been done on reconstructing arbitrary surfaces using the radial basis function rbf method but one can hardly find any work done on the use of rbfs to solve partial differential equations pdes on arbitrary surfaces in this paper we investigate methods to solve pdes on arbitrary stationary surfaces embedded in r 3 using the rbf method we present three rbf based methods that easily discretize surface differential operators we take advantage of the meshfree character of rbfs which give us a high accuracy and the flexibility to represent the most complex geometries in any dimension two out of the three methods which we call the orthogonal gradients ogr methods are the result of our work and are hereby presented for the first time

incomplete approach to homoclinicity in a model with bent slow manifold geometry the dynamics of a model originally proposed for a type of instability in plastic flow has been investigated in detail the bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical behavior including period bubbling and period adding or farey sequences the complex bifurcation sequences characterized by mixed mode oscillations exhibit partial features of shilnikov and gavrilov shilnikov scenario utilizing the fact that the model has disparate time scales of dynamics we explain the origin of the relaxation oscillations using the geometrical structure of the bent slow manifold based on a local analysis we calculate the maximum number of small amplitude oscillations s in the periodic orbit of l s type for a given value of the control parameter this further leads to a scaling relation for the small amplitude oscillations the incomplete approach to homoclinicity is shown to be a result of the finite rate of softening of the eigenvalues of the saddle focus fixed point the latter is a consequence of the physically relevant constraint of the system which translates into the occurrence of back to back hopf bifurcation

a poroelastic immersed boundary method with applications to cell biology the immersed boundary method is a widely used mixed eulerian lagrangian framework for simulating the motion of elastic structures immersed in viscous fluids in the traditional immersed boundary method the fluid and structure move with the same velocity field in this work a model based on the immersed boundary method is presented for simulating poroelastic media in which the fluid permeates a porous elastic structure of small volume fraction that moves with its own velocity field two distinct methods for calculating elastic stresses are presented and compared the methods are validated on a radially symmetric test problem by comparing with a finite difference solution of the classical equations of poroelasticity finally two applications of the modeling framework to cell biology are provided cellular blebbing and cell crawling it is shown that in both examples poroelastic effects are necessary to explain the relevant mechanics

genesis of bursting oscillations in the hindmarsh rose model and homoclinicity to a chaotic saddle abstract we present two hypotheses on the mathematical mechanism underlying bursting dynamics in a class of differential systems 1 that the transition from continuous firing of spikes to bursting is caused by a crisis which destabilizes a chaotic state of continuous spiking and 2 that the bursting corresponds to a homoclinicity to this unstable chaotic state these propositions are supported by a numerical test on the hindmarsh rose model a prototype of its kind we conclude by a unified view for three types of complex multi modal oscillations homoclinic systems bursting and the pomeau manneville intermittency

perturbation based stochastic multi scale computational homogenization method for the determination of the effective properties of composite materials with random properties quantifying uncertainty in the overall elastic properties of composite materials arising from randomness in the material properties and geometry of composites at microscopic level is crucial in the stochastic analysis of composites in this paper a stochastic multi scale finite element method which couples the multi scale computational homogenization method with the second order perturbation technique is proposed to calculate the statistics of the overall elasticity properties of composite materials in terms of the mean value and standard deviation the uncertainties associated with the material properties of the constituents are considered performance of the proposed method is evaluated by comparing mean values and coefficients of variation for components of the effective elastic tensor against corresponding values calculated using monte carlo simulation for three numerical examples results demonstrate that the proposed method has sufficient accuracy to capture the variability in effective elastic properties of the composite induced by randomness in the constituent material properties

superconvergence property of an over penalized discontinuous galerkin finite element gradient recovery method a polynomial preserving recovery method is introduced for over penalized symmetric interior penalty discontinuous galerkin solutions to a quasi linear elliptic problem as a post processing method the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns as well as general meshes satisfying condition by means of the averaging technique we prove the polynomial preserving recovery method for averaged solutions is superconvergent satisfying similar estimates as those for conforming finite element methods we deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator consequently the a posteriori error estimator based on the recovered gradient is asymptotically exact extensive numerical results consistent with our analysis are presented

a local radial basis function method for advection diffusion reaction equations on complexly shaped domains abstract time dependent advection diffusion reaction and diffusion reaction equations are used as models in biology chemistry physics and engineering as representative examples we focus on a chemotaxis model and a turing system from biology and apply a local radial basis function method to numerically approximate the solutions the numerical method can efficiently approximate large scale problems in complexly shaped domains

planar morphometry shear and optimal quasi conformal mappings to characterize the diversity of planar shapes in such instances as insect wings and plant leaves we present a method for the generation of a smooth morphometric mapping between two planar domains which matches a number of homologous points our approach tries to balance the competing requirements of a descriptive theory which may not reflect mechanism and a multi parameter predictive theory that may not be well constrained by experimental data specifically we focus on aspects of shape as characterized by local rotation and shear quantified using quasi conformal maps that are defined precisely in terms of these fields to make our choice optimal we impose the condition that the maps vary as slowly as possible across the domain minimizing their integrated squared gradient we implement this algorithm numerically using a variational principle that optimizes the coefficients of the quasi conformal map between the two regions and show results for the recreation of a sample historical grid deformation mapping of d arcy thompson we also deploy our method to compare a variety of drosophila wing shapes and show that our approach allows us to recover aspects of phylogeny as marked by morphology

a finite element method for the numerical solution of the coupled cahn hilliard and navier stokes system for moving contact line problems in this paper a semi implicit finite element method is presented for the coupled cahn hilliard and navier stokes equations with the generalized navier boundary condition for the moving contact line problems in our method the system is solved in a decoupled way for the cahn hilliard equations a convex splitting scheme is used along with a p1 p1 finite element discretization the scheme is unconditionally stable a linearized semi implicit p2 p0 mixed finite element method is employed to solve the navier stokes equations with our method the generalized navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way the efficiency and capacity of the present method are well demonstrated with several numerical examples

the laplacian eigenvalues of a polygon the difficulties are almost always at the boundary that statement applies to the solution of partial differential equations with a given boundary and also to shape optimization with an unknown boundary these problems require two decisions closely related but not identical o1 how to discretize the boundary conditions 2 how to discretize the boundary itself that second problem is the one we discuss here the region w is frequently replaced by a polygon or polyhedron the approximate boundary email protected n may be only a linear interpolation of the true boundary email protected a perturbation theory that applies to smooth changes of domain is often less successful for a polygon this paper concentrates on a model problem the simplest we could find and we look at eigenvalues of the laplacian the boundary email protected will be the unit circle the approximate boundary email protected n is the regular inscribed polygon with n equal sides it seems impossible that the eigenvalues of regular polygons have not been intensively studied but we have not yet located an authoritative reference the problem will be approached numerically from three directions without attempting a general theory those directions are o1 finite element discretizations of the polygons w n 2 a taylor series based on piecewise smooth perturbations of the circle 3 a series expansion of the eigenvalues in powers of 1 n the second author particularly wishes that we could have consulted george fix about this problem his harvard thesis demonstrated the tremendous improvement that singular elements can bring to the finite element method particularly when w has a reentrant corner or even a crack his numerical experiments in 1 came at the beginning of a long and successful career in applied mathematics we only wish it had been longer

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

an immersed boundary element method for level set based topology optimization summary in this paper we propose a new bem for level set based topology optimization in the proposed bem the nodal coordinates of the boundary element are replaced with the nodal level set function and the nodal coordinates of the eulerian mesh that maintains the level set function because this replacement causes the nodal coordinates of the boundary element to disappear the boundary element mesh appears to be immersed in the eulerian mesh therefore we call the proposed bem an immersed bem the relationship between the nodal coordinates of the boundary element and the nodal level set function of the eulerian mesh is clearly represented and therefore the sensitivities with respect to the nodal level set function are strictly derived in the immersed bem furthermore the immersed bem completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization by using the immersed bem we construct a concrete topology optimization method for solving the minimum compliance problem we provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results copyright 2012 john wiley sons ltd

a numerical study of a semi algebraic multilevel preconditioner for the local discontinuous galerkin method in this work we consider the local discontinuous galerkin ldg method applied to second order elliptic problems arising in the modeling of single phase flows in porous media it has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of o h 2 on structured and unstructured meshes where h is the mesh size thus efficient preconditioners are mandatory we present a semi algebraic multilevel preconditioner for the ldg method using local lagrange type interpolatory basis functions we show numerically that its performance does not degrade or at least the number of iterations increases very slowly as the number of unknowns augments the preconditioner is tested on problems with high jumps in the coefficients which is the typical scenario of problems arising in porous media

high order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles a high order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle by converting the implicitly defined geometry into the graph of an implicitly defined height function the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one dimensional root finding and one dimensional gaussian quadrature the computed quadrature scheme yields strictly positive quadrature weights and inherits the high order accuracy of gaussian quadrature a range of different convergence tests demonstrate orders of accuracy up to 20th order also presented is an application of the quadrature algorithm to a high order embedded boundary discontinuous galerkin method for solving partial differential equations on curved domains

the method of fundamental solutions applied to boundary eigenvalue problems we develop methods based on fundamental solutions to compute the steklov wentzell and laplace beltrami eigenvalues in the context of shape optimization in the class of smooth simply connected two dimensional domains the numerical method is accurate and fast a theoretical error bound is given along with comparisons with mesh based methods we illustrate the use of this method in the study of a wide class of shape optimization problems in two dimensions we extend the method to the computation of the laplace beltrami eigenvalues on surfaces and we investigate some spectral optimal partitioning problems

on the role of polynomials in rbf fd approximations ii numerical solution of elliptic pdes rbf generated finite differences rbf fd have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of pdes we find in the present study that combining polyharmonic splines phs with multivariate polynomials offers an outstanding combination of simplicity accuracy and geometric flexibility when solving elliptic equations in irregular or regular regions in particular the drawbacks on accuracy and stability due to runge s phenomenon are overcome once the rbf stencils exceed a certain size due to an underlying minimization property test problems include the classical 2 d driven cavity and also a 3 d global electric circuit problem with the earth s irregular topography as its bottom boundary the results we find are fully consistent with previous results for data interpolation

effect of domain shape modeling and measurement errors on the 2 d d bar method for eit the d bar algorithm based on nachman s 2 d global uniqueness proof for the inverse conductivity problem nachman 1996 is implemented on a chest shaped domain the scattering transform is computed on this chest shaped domain using trigonometric and adjacent current patterns and the complete electrode model for the forward problem is computed with the finite element method in order to obtain simulated voltage measurements the robustness and effectiveness of the method is demonstrated on a simulated chest with errors in input currents output voltages electrode placement and domain modeling

stencil reduction algorithms for the local discontinuous galerkin method the problem of reducing the stencil of the local discontinuous galerkin method applied to second order differential operator is discussed heuristic algorithms to minimize the total number of non zero blocks of the reduced stiffness matrix are presented and tested on a wide variety of unstructured and structured grids in 2d and 3d copyright 2009 john wiley sons ltd

cooperative behavior of a chain of synaptically coupled chaotic neurons abstract a coupled linear chain of hindmarsh rose model neurons with reciprocal inhibition between neighboring neurons exhibited synchronous oscillations in which neighboring neurons burst out of phase and next nearest neighbor neurons burst in phase the bifurcations observed inside this out of phase regime were qualitatively the same for all chains with an even number of neurons and were similar to those observed in a single isolated cell although the organization of the behavior of a chain of coupled neurons was more regular than that of an isolated cell when noise was added to the synaptic coupling strengths there was less hysteresis in the system and many of the bifurcations with smaller basins of attraction were eliminated making the system even more regular these results suggest that in populations of bursting neurons with reciprocal inhibition the chaotic behavior found in single cells is suppressed

on the radius edge condition in the control volume method in this paper we show that the control volume algorithm for the solution of poisson s equation in three dimensions will converge even if the mesh contains a class of very flat tetrahedra slivers these tetrahedra are characterized by the fact that they have modest ratios of diameter to shortest edge but large circumscribing to inscribed sphere radius ratios and therefore may have poor interpolation properties elimination of slivers is a notoriously difficult problem for automatic mesh generation algorithms we also show that a discrete poincare inequality will continue to hold in the presence of slivers

computational cost of the fekete problem i the forces method on the 2 sphere here we study the computational complexity of the fekete point problem namely we give an exhaustive description of the main properties of an algorithm for the minimization of the logarithmic potential energy on the 2 sphere and we characterize the probability distribution of the cost of the different minima in particular we show that a local minimum can be found with an average cost of about o n 2 8

numerical quadrature over smooth closed surfaces the numerical approximation of definite integrals or quadrature often involves the construction of an interpolant of the integrand and its subsequent integration in the case of one dimension it is natural to rely on polynomial interpolants however their extension to two or more dimensions can be costly and unstable an efficient method for computing surface integrals on the sphere is detailed in the literature reeger fornberg 2016 stud appl math 137 174 188 doi 10 1111 sapm 12106 the method uses local radial basis function interpolation to reduce computational complexity when generating quadrature weights for any given node set this article generalizes this method to arbitrary smooth closed surfaces

a meshfree generalized finite difference method for surface pdes abstract in this paper we propose a novel meshfree generalized finite difference method gfdm approach to discretize pdes defined on manifolds derivative approximations for the same are done directly on the tangent space in a manner that mimics the procedure followed in volume based meshfree gfdms as a result the proposed method not only does not require a mesh it also does not require an explicit reconstruction of the manifold in contrast to some existing methods it avoids the complexities of dealing with a manifold metric while also avoiding the need to solve a pde in the embedding space a major advantage of this method is that all developments in usual volume based numerical methods can be directly ported over to surfaces using this framework we propose discretizations of the surface gradient operator the surface laplacian and surface diffusion operators possibilities to deal with anisotropic and discontinuous surface properties with large jumps are also introduced and a few practical applications are presented

fast high dimensional node generation with variable density abstract we present an algorithm for producing discrete distributions with a prescribed nearest neighbor distance function our approach is a combination of quasi monte carlo q mc methods and weighted riesz energy minimization the initial distribution is a stratified q mc sequence with some modifications a suitable energy functional on the configuration space is then minimized to ensure local regularity the resulting node sets are good candidates for building meshless solvers and interpolants as well as for other purposes where a point cloud with a controlled separation covering ratio is required applications of a three dimensional implementation of the algorithm in particular to atmospheric modeling are also given

the discrete origin of fetd newmark late time instability and a correction scheme the finite element time domain fetd method is commonly used for transient simulation of electromagnetic wave phenomena most practitioners consider fetd when time integrated using the newmark beta method to be unconditionally stable when s 0 25 unlike the finite difference time domain fdtd courant criterion fetd newmark has no limiting timestep above which the method exhibits exponential growth however herein the stability properties of fetd newmark will be rigorously investigated by deducing the jordan canonical form of the fetd newmark amplification matrix and it will be demonstrated that the method does exhibit linear growth for certain field configurations these modes are none other than the pure gradient fields associated with late time instability though many practical simulations are of short duration and will never observe a linearly growing gradient solution it can be problematic for simulations which require long time periods to be integrated a correction scheme for eliminating this late time instability shall be suggested and numerical results will verify its performance