pseudo time stepping methods for space time discontinuous galerkin discretizations of the compressible navier stokes equations the space time discontinuous galerkin discretization of the compressible navier stokes equations results in a non linear system of algebraic equations which we solve with pseudo time stepping methods we show that explicit runge kutta methods developed for the euler equations suffer from a severe stability constraint linked to the viscous part of the equations and propose an alternative to relieve this constraint while preserving locality to evaluate its effectiveness we compare with an implicit explicit runge kutta method which does not suffer from the viscous stability constraint we analyze the stability of the methods and illustrate their performance by computing the flow around a 2d airfoil and a 3d delta wing at low and moderate reynolds numbers

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

chaotic scattering in a near integrable system the melnikov method is used to study the transition from a completely integrable classical potential scattering system to a non integrable one the authors investigate the heteroclinic bifurcations and prove the existence of arbitrarily many elliptic and hyperbolic periodic orbits in a bounded scattering region for selected parameter values some of the analytical predictions are checked by numerical scattering experiments

b spline level set method for shape reconstruction in electrical impedance tomography a b spline level set bls based method is proposed for shape reconstruction in electrical impedance tomography eit we assume that the conductivity distribution to be reconstructed is piecewise constant transforming the image reconstruction problem into a shape reconstruction problem the shape interface of inclusions is implicitly represented by a level set function lsf which is modeled as a continuous parametric function expressed using b spline functions starting from modeling the conductivity distribution with the b spline based lsf we show that the shape modeling allows us to compute the solution by restricting the minimization problem to the space spanned by the b splines as a consequence the solution to the minimization problem is obtained in terms of the b spline coefficients we illustrate the behavior of this method using simulated as well as water tank data in addition robustness studies considering varying initial guesses differing numbers of control points and modeling errors caused by inhomogeneity are performed both simulation and experimental results show that the bls based approach offers clear improvements in preserving the sharp features of the inclusions in comparison to the recently published parametric level set method

utilizing prior information in the estimation of volume fraction distribution for many industrial processes the bulk volume fraction of one phase such as gas or solid can be of considerable importance for this reason it is often controlled within certain limits and therefore is known at least vaguely tomographic imaging of volume fraction can provide very valuable information not only on the bulk volume fraction but also on the spatial distribution of the phase of interest prior information on the bulk volume fraction can assist in tomographic inversion even if this information is not precise in this paper we propose an electrical impedance tomography approach for imaging volume fraction distribution a vague bulk volume fraction prior is utilized in the estimation two dimensional numerical examples are used to demonstrate the proposed method copyright 2005 john wiley sons ltd

the factorization method applied to the complete electrode model of impedance tomography the factorization method is a tool for recovering inclusions inside a body when the neumann to dirichlet operator which maps applied currents to measured voltages is known in practice this information is never at hand due to the discreteness and physical properties of the measurement devices the complete electrode model of impedance tomography includes these physical characteristics but leads to a finite dimensional data set called the resistivity matrix the main result of this work is an approximation link relating the resistivity matrix to the neumann to dirichlet operator in the l 2 operator norm this result allows us to extend the factorization method to the framework of real life electrode measurements using a regularized series criterion which is easy to implement in practice the truncation index of the sequence criterion which represents the stopping index of the regularization scheme can be computed solely from the measured perturbed and finite dimensional data the functionality of

a weak formulation of roe s approximate riemann solver abstract this paper presents a weak formulation of roe s approximate riemann solver based on a notion of weak solution to nonlinear hyperbolic systems in nonconservative form this weak formulation is applied to the euler equations for real gases and the homogeneous equilibrium two phase flow model numerical results for a two phase flow heat addition problem are given

generalized linearization techniques in electrical impedance tomography electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages as the current voltage pairs depend nonlinearly on the conductivity impedance tomography leads to a nonlinear inverse problem often the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such in this paper we compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity it is numerically demonstrated that the conductivity linearization often results in compromised accuracy in both forward and inverse computations inspired by these observations we present and analyze a new linearization technique which is based on the logarithm of the neumann to dirichlet operator the method is directly applicable to discrete settings including the complete electrode model we also consider frechet derivatives of the logarithmic operators numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography

finite element implementation of maxwell s equations for image reconstruction in electrical impedance tomography traditionally image reconstruction in electrical impedance tomography eit has been based on laplace s equation however at high frequencies the coupling between electric and magnetic fields requires solution of the full maxwell equations in this paper a formulation is presented in terms of the maxwell equations expressed in scalar and vector potentials the approach leads to boundary conditions that naturally align with the quantities measured by eit instrumentation a two dimensional implementation for image reconstruction from eit data is realized the effect of frequency on the field distribution is illustrated using the high frequency model and is compared with laplace solutions numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation the results show that scalar vector potential reconstruction produces images which are essentially indistinguishable from a laplace algorithm for frequencies below 1 mhz but superior at frequencies reaching 10 mhz

tracer dynamics in open hydrodynamical flows as chaotic scattering abstract methods coming from the theory of chaotic scattering are applied to the advection of passive particles in an open hydrodynamical flow in a region of parameters where a von karman vortex street is present with a time periodic velocity field behind a cylinder in a channel particles can temporarily be trapped in the wake they exhibit chaotic motion there due to the presence of a nonattracting chaotic set the experimentally well known concept of streaklines is interpreted as a structure visualising asymptotically the unstable manifold of the full chaotic set the evaluation of streaklines can also provide characteristic numbers of this invariant set e g topological entropy lyapunov exponent escape rate the time delay distributions are also evaluated we demonstrate these ideas with the aid of both computer simulations of the navier stokes equations and analytical model computations properties that could be measured in a laboratory experiment are discussed

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

hllc type riemann solver for the baer nunziato equations of compressible two phase flow we first construct an approximate riemann solver of the hllc type for the baer nunziato equations of compressible two phase flow for the subsonic wave configuration the solver is fully nonlinear it is also complete that is it contains all the characteristic fields present in the exact solution of the riemann problem in particular stationary contact waves are resolved exactly we then implement and test a new upwind variant of the path conservative approach such schemes are suitable for solving numerically nonconservative systems finally we use locally the new hllc solver for the baer nunziato equations in the framework of finite volume discontinuous galerkin finite element and path conservative schemes we systematically assess the solver on a series of carefully chosen test problems

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

devil staircase behavior of dynamical invariants in chaotic scattering a crisis in chaotic scattering is characterized by the merging of two or more nonattracting chaotic saddles the fractal dimension of the resulting chaotic saddle increases through the crisis we present a rigorous analysis for the behavior of dynamical invariants associated with chaotic scattering by utilizing a representative model system that captures the essential dynamical features of crisis our analysis indicates that the fractal dimension and other dynamical invariants are a devil staircase type of function of the system parameter our results can also provide insight for similar devil staircase behaviors observed in the parametric evolution of chaotic saddles of general dissipative dynamical systems and in communicating with chaos 2000 elsevier science b v all rights reserved

a discontinuous galerkin method for the simulation of compressible gas gas and gas water two medium flows abstract in this paper we develop a new discontinuous galerkin method for the simulation of shocks and interfaces in compressible gas gas and gas water two medium flows by solving the based model the spatial discretization is carefully designed to possess the following features discrete conservation in terms of the total mass total momentum and total energy high order accuracy and consistency for smooth flows free of oscillations at an isolated material interface in order to handle potential discontinuities arising in the simulation a nonlinear limiter based on the weighted essentially non oscillatory weno strategy is employed to suppress numerical oscillations and to preserve high order of accuracy in regions of smooth flows the weno reconstruction is imposed on suitably selected quantities rather than the conserved ones in case for discontinuities with large pressure ratio low density and dramatic change of material property where unphysical variables may be encountered a posteriori solution correction on the subcell level is locally adopted to enhance the robustness a series of typical test cases for both one and two dimensional problems are provided to demonstrate the performance of the proposed method

a devil s staircase into chaotic scattering abstract we discuss analytically the transition from a regular to a chaotic classical dynamics of a sample scattering model introduced earlier the transition arises as a homoclinic bifurcation of an atypical type in phase space as a discontinuity for the grammatical complexity and as an onset of a devil s staircase for the topological entropy

application of stochastic galerkin fem to the complete electrode model of electrical impedance tomography abstract the aim of electrical impedance tomography is to determine the internal conductivity distribution of some physical body from boundary measurements of current and voltage the most accurate forward model for impedance tomography is the complete electrode model which consists of the conductivity equation coupled with boundary conditions that take into account the electrode shapes and the contact resistances at the corresponding interfaces if the reconstruction task of impedance tomography is recast as a bayesian inference problem it is essential to be able to solve the complete electrode model forward problem with the conductivity and the contact resistances treated as a random field and random variables respectively in this work we apply a stochastic galerkin finite element method to the ensuing elliptic stochastic boundary value problem and compare the results with monte carlo simulations

period doubling cascades of attractors a prerequisite for horseshoes this paper shows that if a horseshoe is created in a natural manner as a parameter is varied then the process of creation involves the appearance of attracting periodic orbits of all periods furthermore each of these orbits will period double repeatedly with those periods going to infinity

a degenerate bifurcation to chaotic scattering in a multicentre potential many scattering systems can be described as scattering of a point particle off a multicentre potential in this paper we present a two centre system which shows either regular or chaotic scattering depending on the kinetic energy i e the velocity of the incoming particle the transition points to chaotic scattering can be derived analytically by linearization of the poincare map at one of these transition velocities there is a degenerate bifurcation where the invariant set contains a parabolic surface and where the time delay statistics is algebraic with power sigma 2

electrical conductivity reconstruction using nonlocal boundary conditions in this paper we study existence uniqueness and solution estimates to the mixed problem nabla cdot sigma nabla u 0 with dirichlet to neumann map boundary conditions and neumann boundary conditions we then show how this can be used in the reconstruction of sigma given the relationship between u and its normal derivative on the boundary portion where we do not apply the dirichlet to neumann map a numerical reconstruction scheme is also derived

cantor set structures in the singularities of classical potential scattering a classical mechanical system is analysed which exhibits complicated scattering behaviour in the set of all incoming asymptotes there is a fractal subset on which the scattering angle is singular though in the complement of this cantor set the deflection function is regular one can choose impact parameter intervals leading to arbitrarily complicated trajectories the authors show how the complicated scattering behaviour is caused by unstable periodic orbits having homoclinic and heteroclinic connections thereby a hyperbolic invariant set is created leading to horseshoe chaos in the flow this invariant set contains infinitely many unstable localised orbits periodic and aperiodic ones the stable manifolds of these orbits reach out into the asymptotic region and create the singularities of the scattering function

fractal properties of scattering singularities we discuss properties of the set of scattering singularities in regions of irregular scattering we show how a symbolic organisation of the set can be used to determine the fractal dimension and the scaling function this yields information on the distribution of lyapunov exponents of bounded orbits the specific model studied is the motion of a particle in a plane elastically reflected by three circular discs centred on the corners of an equilateral triangle

resolution and stability analysis of an inverse problem in electrical impedance tomography dependence on the input current patterns electrical impedance tomography is a procedure by which one finds the conductivity distribution inside a domain from measurements of voltages and currents at the boundary this work addresses the i

globally hyperbolic moment model of arbitrary order for one dimensional special relativistic boltzmann equation this paper extends the model reduction method by the operator projection to the one dimensional special relativistic boltzmann equation the derivation of arbitrary order globally hyperbolic moment system is built on our careful study of two families of the complicate grad type orthogonal polynomials depending on a parameter we derive their recurrence relations calculate their derivatives with respect to the independent variable and parameter respectively and study their zeros and coefficient matrices in the recurrence formulas some properties of the moment system are also proved they include the eigenvalues and their bound as well as eigenvectors hyperbolicity characteristic fields linear stability and lorentz covariance a semi implicit numerical scheme is presented to solve a cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method the results show that the solutions of our hyperbolic moment system converge to the solution of the special relativistic boltzmann equation as the order of the hyperbolic moment system increases

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

chaotic scattering off the magnetic dipole classical scattering with singularities of cantor set type can be observed if unstable localised orbits exist whose homoclinic structures are transported to infinity by the hamiltonian flow an electron moving in the field of a magnetic dipole is a simple example of physical relevance to demonstrate this transport mechanism in the asymptotic plane spanned by impact parameter and incoming direction the deflection function is singular for initial conditions leading to captured orbits using the method of poincare sections the authors find a correspondence between this set of singularities and the stable manifolds of localised orbits the scattering data which are measured in the asymptotic region of free motion provide information about chaotic motion in a finite part of the position space

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

emergence of continual directed flow in hamiltonian systems we propose a minimal model for the emergence of a directed flow in autonomous hamiltonian systems it is shown that internal breaking of the spatio temporal symmetries via localised initial conditions that are unbiased with respect to the transporting degree of freedom and transient chaos conspire to form the physical mechanism for the occurrence of a current most importantly after passage through the transient chaos trajectories perform solely regular transporting motion so that the resulting current is of continual ballistic nature this has to be distinguished from the features of transport reported previously for driven hamiltonian systems with mixed phase space where transport is determined by intermittent behaviour exhibiting power law decay statistics of the duration of regular ballistic periods

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

improving the forward solver for the complete electrode model in eit using algebraic multigrid image reconstruction in electrical impedance tomography is an ill posed nonlinear inverse problem linearization techniques are widely used and require the repeated solution of a linear forward problem to account correctly for the presence of electrodes and contact impedances the so called complete electrode model is applied implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method however preconditioners are essential for efficient convergence preconditioners based on incomplete factorization methods are commonly used but their performance depends on user tuned parameters to avoid this deficiency we apply black box algebraic multigrid using standard commercial and freely available software the suggested solution scheme dramatically reduces the time cost of solving the forward problem numerical results are presented using an anatomically detailed model of the human head

space time discontinuous galerkin discretization of rotating shallow water equations a space time discontinuous galerkin dg discretization is presented for the rotating shallow water equations over varying topography we formulate the space time dg finite element discretization in an efficient and conservative discretization the hllc flux is used as numerical flux through the finite element boundaries when discontinuities are present we locally apply dissipation around these discontinuities with the help of krivodonova s discontinuity indicator such that spurious oscillations are suppressed the non linear algebraic system resulting from the discretization is solved using a pseudo time integration with a second order five stage runge kutta method a thorough verification of the space time dg finite element method is undertaken by comparing numerical and exact solutions we also carry out a discrete fourier analysis of the one dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error the numerical scheme is validated in a novel way by considering various simulations of bore vortex interactions in combination with a qualitative analysis of pv generation by non uniform bores finally the space time dg method is particularly suited for problems where dynamic grid motion is required to demonstrate this we simulate waves generated by a wave maker and verify these for low amplitude waves where linear theory is approximately valid

on the hp adaptive solution of complete electrode model forward problems of electrical impedance tomography electrical impedance tomography is a noninvasive imaging technique based on measurements of currents and voltages on the boundary of the object of interest the most accurate forward model for impedance tomography is the complete electrode model that takes into account the electrode shapes and the contact impedances at the corresponding interfaces many practical reconstruction algorithms of electrical impedance tomography require repetitive computation of accurate forward solutions for the complete model in this work we introduce an hp adaptive finite element method for solving the complete electrode forward problem and demonstrate its functionality via two dimensional numerical studies

electrical impedance tomography problem with inaccurately known boundary and contact impedances in electrical impedance tomography eit electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body the resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data all the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode skin contact impedances are known a priori however in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances and therefore approximate model domain and contact impedances have to be used in the image reconstruction however it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images leading to loss of diagnostically relevant information in a recent paper kolehmainen 2006 we showed how in the 2 d case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of teichmuller mappings in this paper the theory and reconstruction method are extended to include the estimation of unknown contact impedances the method is implemented numerically and tested with experimental eit data the results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method

in vivo impedance imaging with total variation regularization we show that electrical impedance tomography eit image reconstruction algorithms with regularization based on the total variation tv functional are suitable for in vivo imaging of physiological data this reconstruction approach helps to preserve discontinuities in reconstructed profiles such as step changes in electrical properties at interorgan boundaries which are typically smoothed by traditional reconstruction algorithms the use of the tv functional for regularization leads to the minimization of a nondifferentiable objective function in the inverse formulation this cannot be efficiently solved with traditional optimization techniques such as the newton method we explore two implementations methods for regularization with the tv functional the lagged diffusivity method and the primal dual interior point method pd ipm first we clarify the implementation details of these algorithms for eit reconstruction next we analyze the performance of these algorithms on noisy simulated data finally we show reconstructed eit images of in vivo data for ventilation and gastric emptying studies in comparison to traditional quadratic regularization tv regulariza tion shows improved ability to reconstruct sharp contrasts

weakly noisy chaotic scattering received 30 april 2013 published 23 september 2013 the effect of a weak source of noise on the chaotic scattering is relevant to situations of physical interest we investigate how a weak source of additive uncorrelated gaussian noise affects both the dynamics and thetopology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regimeof the h enon heiles hamiltonian system we have found long transients for the time escape distributions forcritical values of the noise intensity for which the particles escape slower as compared with the noiseless case an analysis of the survival probability of the scattering function versus the gaussian noise intensity shows asmooth curve with one local maximum and with one local minimum which are related to those long transientsand with the basin structure in phase space on the other hand the computation of the exit basins in phase spaceshows a quadratic curve for which the basin boundaries lose their fractal like structure as noise turned on doi 10 1103 physreve 88 032914 pacs number s 05

a staggered semi implicit spectral discontinuous galerkin scheme for the shallow water equations a spatially arbitrary high order semi implicit spectral discontinuous galerkin dg scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed the free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis and as test functions each unknown is however defined on a different set of control volumes that are spatially staggered with respect to each other a semi implicit time integration yields a stable and efficient mass conservative algorithm the use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block penta diagonal system must be solved for the new free surface location which is a scalar quantity subsequently the new momentum components are directly obtained the staggered semi implicit approach makes the present dg scheme different from other published dg schemes the sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix free gmres algorithm furthermore for the shallow water equations the proposed scheme can be written as a quadrature free method where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors for the special case n 0 the proposed method reduces to a classical semi implicit finite difference scheme the proposed semi implicit scheme is particularly well suited for low froude number flows the method is validated on some typical academic benchmark problems using polynomial degrees of up to n 20

reconstruction of permittivity profile from boundary capacitance data we present numerical methods of reconstructing permittivity profile of material flow from data collected in electrical capacitance tomography ect based on a formula for the sensitivity maps from a partial differential equation model we propose a unified framework of iterative algorithms using updated sensitivity maps to properly incorporate the non linear nature of the problem and a regularization mechanism to address the ill posedness for two and three phase flows the total variation tv regularization method is shown to be effective in reconstructing the permittivity profiles accurately in a few iterations without any a priori knowledge of the unknown permittivity values

a space time hybridizable discontinuous galerkin method for incompressible flows on deforming domains we present the first space time hybridizable discontinuous galerkin hdg finite element method for the incompressible navier stokes and oseen equations major advantages of a space time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space time elements our formulation is related to the hdg formulation for incompressible flows introduced recently in e g n c nguyen j peraire b cockburn a hybridizable discontinuous galerkin method for stokes flow comput methods appl mech eng 199 2010 582 597 however ours is inspired in typical dg formulations for compressible flows which allow for a more straightforward implementation another difference is the use of polynomials of fixed total degree with space time hexahedral and quadrilateral elements instead of simplicial elements we present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time

a laplace method for under determined bayesian optimal experimental designs abstract in long et al 2013 a new method based on the laplace approximation was developed to accelerate the estimation of the post experimental expected information gains kullback leibler divergence in model parameters and predictive quantities of interest in the bayesian framework a closed form asymptotic approximation of the inner integral and the order of the corresponding dominant error term were obtained in the cases where the parameters are determined by the experiment in this work we extend that method to the general case where the model parameters cannot be determined completely by the data from the proposed experiments we carry out the laplace approximations in the directions orthogonal to the null space of the jacobian matrix of the data model with respect to the parameters so that the information gain can be reduced to an integration against the marginal density of the transformed parameters that are not determined by the experiments furthermore the expected information gain can be approximated by an integration over the prior where the integrand is a function of the posterior covariance matrix projected over the aforementioned orthogonal directions to deal with the issue of dimensionality in a complex problem we use either monte carlo sampling or sparse quadratures for the integration over the prior probability density function depending on the regularity of the integrand function we demonstrate the accuracy efficiency and robustness of the proposed method via several nonlinear under determined test cases they include the designs of the scalar parameter in a one dimensional cubic polynomial function with two unidentifiable parameters forming a linear manifold and the boundary source locations for impedance tomography in a square domain where the unknown parameter is the conductivity which is represented as a random field

high order ader schemes for a unified first order hyperbolic formulation of newtonian continuum mechanics coupled with electro dynamics abstract in this paper we propose a new unified first order hyperbolic model of newtonian continuum mechanics coupled with electro dynamics the model is able to describe the behavior of moving elasto plastic dielectric solids as well as viscous and inviscid fluids in the presence of electro magnetic fields it is actually a very peculiar feature of the proposed pde system that viscous fluids are treated just as a special case of elasto plastic solids this is achieved by introducing a strain relaxation mechanism in the evolution equations of the distortion matrix a which in the case of purely elastic solids maps the current configuration to the reference configuration the model also contains a hyperbolic formulation of heat conduction as well as a dissipative source term in the evolution equations for the electric field given by ohm s law via formal asymptotic analysis we show that in the stiff limit the governing first order hyperbolic pde system with relaxation source terms tends asymptotically to the well known viscous and resistive magnetohydrodynamics mhd equations furthermore a rigorous derivation of the model from variational principles is presented together with the transformation of the euler lagrange differential equations associated with the underlying variational problem from lagrangian coordinates to eulerian coordinates in a fixed laboratory frame the present paper hence extends the unified first order hyperbolic model of newtonian continuum mechanics recently proposed in 110 42 to the more general case where the continuum is coupled with electro magnetic fields the governing pde system is symmetric hyperbolic and satisfies the first and second principle of thermodynamics hence it belongs to the so called class of symmetric hyperbolic thermodynamically compatible systems shtc which have been studied for the first time by godunov in 1961 61 and later in a series of papers by godunov and romenski 67 69 119 an important feature of the proposed model is that the propagation speeds of all physical processes including dissipative processes are finite the model is discretized using high order accurate ader discontinuous galerkin dg finite element schemes with a posteriori subcell finite volume limiter and using high order ader weno finite volume schemes we show numerical test problems that explore a rather large parameter space of the model ranging from ideal mhd viscous and resistive mhd over pure electro dynamics to moving dielectric elastic solids in a magnetic field

intermittency in connected hamiltonian systems abstract intermittency as extensively examined in dissipative systems is studied in hamiltonian systems consisting of integrable or chaotic billiards connected through a hole near the transition to intermittency we obtain the scaling law p pc 1 where is the mean residence time in a billiard p is d or a 1 d hole length a area of the billiard dc a 1c 0 in cases with particular geometrical distortions in the neighborhood of the hole this law holds if d is replaced by a conveniently defined effective hole size this work is a first step for studying chaotic scattering systems whose exits are connected to confining potentials

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

on variational and symplectic time integrators for hamiltonian systems various systems in nature have a hamiltonian structure and therefore accurate time integrators for those systems are of great practical use in this paper a finite element method will be explored to derive symplectic time stepping schemes for non autonomous systems in a systematic way the technique used is a variational discontinuous galerkin finite element method in time this approach provides a unified framework to derive known and new symplectic time integrators an extended analysis for the new time integrators will be provided the analysis shows that a novel third order time integrator presented in this paper has excellent dispersion properties these new time stepping schemes are necessary to get accurate and stable simulations of forced water waves and other non autonomous variational systems which we illustrate in our numerical results

a weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging we consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude of one current density field generated by applying current on a set of electrodes the required interior data can be obtained by means of mri measurements on the boundary we only require knowledge of the electrodes their impedances and the corresponding average input currents from the mathematical point of view this practical question leads us to consider a new weighted minimum gradient problem for functions satisfying the boundary conditions coming from the complete electrode model cem of somersalo cheney and isaacson siam j appl math 52 1992 pp 1023 1040 we show that this variational problem has nonunique solutions the surprising discovery is that the physical data is still sufficient to determine the geometry of the connected components of the level sets of the minimizers we thus obtain an interesting phase retrieval result knowledge of the input current at the boundary allows determination of the full current vector field from its magnitude we characterize locally the nonuniqueness in the variational problem in two and three dimensions we also show that additional measurements of the voltage potential along a curve joining the electrodes yield unique determination of the conductivity the proofs involve a maximum principle and a new regularity up to the boundary result for the cem boundary conditions a nonlinear algorithm is proposed and implemented to illustrate the theoretical results

numerical solution of navier stokes korteweg systems by local discontinuous galerkin methods in multiple space dimensions compressible liquid vapor flow with phase transitions can be described by systems of navier stokes korteweg type they extend the navier stokes equations by nonlinear higher grade terms which take the form of either differential or nonlocal integral operators a numerical approximation method on the basis of the local discontinuous galerkin method in multiple space dimensions is suggested for isothermal flows it relies on a specific discretization of a non conservative formulation to enhance the performance of the overall scheme two techniques are used i local spatial adaptivity based on gradient indicators for the density and ii parallelism based on domain decomposition the paper concludes with numerical experiments in two and three space dimensions they show the reliability and efficiency of the proposed approach as well as they demonstrate the applicability of the models for several important phase transition phenomena

variational space time dis continuous galerkin method for nonlinear free surface water waves a new variational finite element method is developed for nonlinear free surface gravity water waves using the potential flow approximation this method also handles waves generated by a wave maker its formulation stems from miles variational principle for water waves together with a finite element discretization that is continuous in space and discontinuous in time one novel feature of this variational finite element approach is that the free surface evolution is variationally dependent on the mesh deformation vis a vis the mesh deformation being geometrically dependent on free surface evolution another key feature is the use of a variational dis continuous galerkin finite element discretization in time moreover in the absence of a wave maker it is shown to be equivalent to the second order symplectic stormer verlet time stepping scheme for the free surface degrees of freedom these key features add to the stability of the numerical method finally the resulting numerical scheme is verified against nonlinear analytical solutions with long time simulations and validated against experimental measurements of driven wave solutions in a wave basin of the maritime research institute netherlands

chaotic scattering by steep repelling potentials consider a classical two dimensional scattering problem a ray is scattered by a potential composed of several tall repelling steep mountains of arbitrary shape we study when the traditional approximation of this nonlinear far from integrable problem by the corresponding simpler billiard problem of scattering by hard wall obstacles of similar shape is justified for one class of chaotic scatterers named here regular sinai scatterers the scattering properties of the smooth system indeed limit to those of the billiards for another class the singular sinai scatterers these two scattering problems have essential differences though the invariant set of such singular scatterers is hyperbolic possibly with singularities that of the smooth flow may have stable periodic orbits even when the potential is arbitrarily steep it follows that the fractal dimension of the scattering function of the smooth flow may be significantly altered by changing the ratio between the steepness parameter and a parameter which measures the billiards deviation from a singular scatterer thus even in this singular case the billiard scattering problem is utilized as a skeleton for studying the properties of the smooth flow finally we see that corners have nontrivial and significant impact on the scattering functions

space time discontinuous galerkin finite element method for two fluid flows a novel numerical method for two fluid flow computations is presented which combines the space time discontinuous galerkin finite element discretization with the level set method and cut cell based interface tracking the space time discontinuous galerkin stdg finite element method offers high accuracy an inherent ability to handle discontinuities and a very local stencil making it relatively easy to combine with local hp refinement the front tracking is incorporated via cut cell mesh refinement to ensure a sharp interface between the fluids to compute the interface dynamics the level set method lsm is used because of its ability to deal with merging and breakup also the lsm is easy to extend to higher dimensions small cells arising from the cut cell refinement are merged to improve the stability and performance the interface conditions are incorporated in the numerical flux at the interface and the stdg discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative the numerical method is applied to one and two dimensional two fluid test problems using the euler equations

space time discontinuous galerkin method for the compressible navier stokes equations a space time discontinuous galerkin finite element method for the compressible navier stokes equations is presented we explain the space time setting derive the weak formulation and discuss our choices for the numerical fluxes the resulting numerical method allows local grid adaptation as well as moving and deforming boundaries which we illustrate by computing the flow around a 3d delta wing on an adapted mesh and by simulating the dynamic stall phenomenon of a 2d airfoil in rapid pitch up maneuver

a posteriori error estimate and convergence analysis for conductivity image reconstruction in mreit magnetic resonance electrical impedance tomography mreit takes advantage of internal information to solve its nonlinear inverse problem of recovering a conductivity distribution inside an imaging object when we inject current into the imaging object there occurs a distribution of internal magnetic flux density mathbf b b x b y b z in mreit we utilize a magnetic resonance imaging scanner with its main magnetic field in the z direction to acquire b z data the harmonic b z algorithm was invented in 2001 to reconstruct cross sectional conductivity images from b z data sets subject to multiple injection currents utilizing internal b z data it overcomes the inherent ill posedness in electrical impedance tomography we can set up the inverse problem in mreit as a coefficient identification problem of finding sigma appearing in nabla cdot sigma nabla u 0 from acquired data of the z component of nabla times sigma nabla u the harmonic b z algorithm has shown an excellent perfor

an efficient hyperbolic relaxation system for dispersive non hydrostatic water waves and its solution with high order discontinuous galerkin schemes abstract in this paper we propose a novel set of first order hyperbolic equations that can model dispersive non hydrostatic free surface flows the governing pde system is obtained via a hyperbolic approximation of the family of non hydrostatic free surface flow models recently derived by sainte marie et al in 1 our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth averaged non hydrostatic pressure similar to the hyperbolic divergence cleaning applied in generalized lagrangian multiplier methods glm for magnetohydrodynamics mhd we suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing pde we then use arbitrary high order accurate ader discontinuous galerkin dg finite element schemes with an a posteriori subcell finite volume limiter in order to solve the proposed pde system numerically the final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet dry fronts it is well balanced making use of a path conservative formulation of hll type riemann solvers based on the straight line segment path furthermore the proposed ader dg scheme with a posteriori subcell finite volume limiter adapts very well to modern gpu architectures resulting in a very accurate robust and computationally efficient computational method for non hydrostatic free surface flows the new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions in all cases the achieved agreement with analytical solutions or experimental data is very good thus showing the validity of both the proposed mathematical model and the numerical solution algorithm

fractal properties of escape from a two dimensional potential abstract this paper summarizes a numerical investigation of the escape of particles from the two dimensional potential v x y 1 2 x 2 1 2 y 2 x 2 y 2 for variable but fixed energy h 0 12 for 1 1 4h 2 08 escape is impossible but for 1 particles will escape for such larger the final outcome as characterized by the time and direction in which particles escape is a smooth function of initial conditions in certain phase space regions however in other regions the evolution evidences an extremely sensitive dependence on initial conditions indicative of a complex microscopic evolution for sufficiently large values of 2 4 90 0 01 these sensitive regions appear to exhibit a fractal structure with simple scaling properties implying that at a coarse grained level the late time behavior is quite regular specifically the evolution is asymptotically markovian in the sense that for particles that have not yet escaped at late times the coarse grained escape probabilities per unit time tend toward time independent values p 0 independent of the initial conditions the asymptotic values p and the rate of convergence towards these values were examined as a function of and the scale of the coarse graining and were observed to exhibit simple scaling behaviour at least for values of 5 7 p 2 with a critical exponent 0 49 0 05 for a coarse graining corresponding to a linear scale r 0 05 the time t required for convergence towards this value scales as t 2 with 0 39 0 14 0 06 the difference 0 09 0 04 0 02 the value p appears independent of r but the convergence time scales as t r r with 0 08 0 03 seemingly independent of to within statistical uncertainties 0

statistical estimation of resistance conductance by electrical impedance tomography measurements this paper is built upon the assumption that in electrical impedance tomography vectors of voltages and currents are linearly dependent through a resistance matrix this linear relationship was confirmed experimentally and may be derived analytically under certain assumptions regarding electrodes isaacson 1991 given measurement data consisting of voltages and currents we treat this relationship as a linear statistical model thus our goal is not to reconstruct the image but directly estimate its electromagnetic properties reflected in the resistance and or conductance matrix using electrical impedance tomography eit measurements of voltages and currents on the periphery of the body since no inverse problem is involved the algorithm for estimation merely reduces to one matrix inversion we estimate the impedance resistance matrix using well established statistical inference techniques for linear regression models we provide a comprehensive treatment for a two dimensional homogeneous body of a circular shape by which many concepts of electrical impedance tomography such as width of electrodes the difference between voltage current and current voltage systems are illustrated our theory may be applied to various tests including eit hardware calibration and whether the medium is homogeneous these tests are illustrated on phantom agar data

runge kutta discontinuous galerkin method for the approximation of baer and nunziato type multiphase models a high order numerical method is developed for the computation of compressible multiphase flows the model we use is based on the baer and nunziato type systems 4 among all the other available models in the literature these systems present the advantage to be able to simulate either interface or mixture problems nevertheless they still raise some issues mainly based on their non conservative feature the numerical method we propose is a discontinuous galerkin type in this work the interior side integrals are computed thanks to 2 robustness and high order of accuracy of the method are proved on classical interface problems but also on suitably derived analytical solutions

discontinuous galerkin finite element methods for hyperbolic nonconservative partial differential equations we present space and space time discontinuous galerkin finite element dgfem formulations for systems containing nonconservative products such as occur in dispersed multiphase flow equations the main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form then the formulation must reduce to that for conservative systems standard dgfem formulations cannot be applied to nonconservative systems of partial differential equations we therefore introduce the theory of weak solutions for nonconservative products into the dgfem formulation leading to the new question how to define the path connecting left and right states across a discontinuity the effect of different paths on the numerical solution is investigated and found to be small we also introduce a new numerical flux that is able to deal with nonconservative products our scheme is applied to two different systems of partial differential equations first we consider the shallow water equations where topography leads to nonconservative products in which the known possibly discontinuous topography is formally taken as an unknown in the system second we consider a simplification of a depth averaged two phase flow model which contains more intrinsic nonconservative products

multifrequency trans admittance scanner mathematical framework and feasibility a trans admittance scanner tas is a device for breast cancer diagnosis based on numerous experimental findings that complex conductivities of breast tumors significantly differ from those of surrounding normal tissues in tas we apply a sinusoidal voltage between a hand held electrode and a scanning probe placed on the breast skin to make current travel through the breast the scanning probe has an array of electrodes at zero voltage we measure exit currents neumann data through the electrodes that provide a map of trans admittance data over the breast surface the inverse problem of tas is to detect a suspicious abnormality underneath the breast skin from the measured neumann data previous anomaly detection methods used the difference between the measured neumann data and a reference neumann data obtained beforehand in the absence of anomaly however in practice the reference data is not available and its computation is not possible since the inhomogeneous complex conductivity of the normal breast is unknown to deal with this problem we propose a frequency difference tas fdtas in which a weighted frequency difference of the trans admittance data measured at a certain moment is used for anomaly detection this paper provides a mathematical framework and the feasibility of fdtas by showing the relationship between the anomaly information and the weighted frequency difference of the neumann data

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

tikhonov regularization and prior information in electrical impedance tomography the solution of impedance distribution in electrical impedance tomography is a nonlinear inverse problem that requires the use of a regularization method the generalized tikhonov regularization methods have been popular in the solution of many inverse problems the regularization matrices that are usually used with the tikhonov method are more or less ad hoc and the implicit prior assumptions are thus in many cases inappropriate in this paper the authors propose an approach to the construction of the regularization matrix that conforms to the prior assumptions on the impedance distribution the approach is based on the construction of an approximating subspace for the expected impedance distributions it is shown by simulations that the reconstructions obtained with the proposed method are better than with two other schemes of the same type when the prior is compatible with the true object on the other hand when the prior is incompatible with the true object the method will still give reasonable estimates

projective electrical impedance reconstruction with two measurements this paper presents a projective reconstruction method of electrical impedance distribution using two boundary measurements of current voltage data the proposed method has a major advantage over existing electrical impedance tomography systems relatively high resolution impedance image in a near surface region underneath the voltage sensing probe that is placed on a skin surface we attach two pairs of electrodes on the border of the probe in such a way that each pair of electrodes generates electrical current flowing underneath the voltage sensing probe and two currents flow across each other a careful analysis of the measured data with respect to local perturbation of admittivity leads to the projective reconstruction method and its rationale numerical simulations show that the proposed method successfully reconstructs high resolution image of local perturbation of admittivity due to object position changes or admittivity value changes or the both in a near surface region underneath the probe

convergence and regularization for monotonicity based shape reconstruction in electrical impedance tomography the inverse problem of electrical impedance tomography is severely ill posed meaning that only limited information about the conductivity can in practice be recovered from boundary measurements of electric current and voltage recently it was shown that a simple monotonicity property of the related neumann to dirichlet map can be used to characterize shapes of inhomogeneities in a known background conductivity in this paper we formulate a monotonicity based shape reconstruction scheme that applies to approximative measurement models and regularizes against noise and modelling error we demonstrate that for admissible choices of regularization parameters the inhomogeneities are detected and under reasonable assumptions asymptotically exactly characterized moreover we rigorously associate this result with the complete electrode model and describe how a computationally cheap monotonicity based reconstruction algorithm can be implemented numerical reconstructions from both simulated and real life measurement data are presented

an approximate linearized riemann solver for a two fluid model an approximate linearized riemann solver is presented for the numerical simulation of two phase flows this new solver is based on a linearization of nonconservative products and uses an extension of roe s approximate riemann solver the scheme is applied to shock tube problem and to a standard test for two fluid codes

discontinuous galerkin finite element method for shallow two phase flows we present a discontinuous galerkin finite element method for two depth averaged two phase flow models one of these models contains nonconservative products for which we developed a discontinuous galerkin finite element formulation in rhebergen et al 2008 j comput phys 227 1887 1922 the other model is a new depth averaged two phase flow model we introduce for shallow two phase flows that does not contain nonconservative products we will compare numerical results of both models and qualitatively validate the models against a laboratory experiment furthermore because of spurious oscillations that may occur near discontinuities a weno slope limiter is applied in conjunction with a discontinuity detector to detect regions where spurious oscillations appear

multipeaked probability distributions of recurrence times we determine probabilities of recurrence time into finite sized physically meaningful subsets of phase space we consider three different autonomous chaotic systems i scattering in a three peaked potential ii connected billiards and iii lorenz equations we find multipeaked probability distributions similar to the distributions found in driven stochastically resonant systems in nondriven systems such as ours only monotonic decaying distributions exponentials stretched exponentials power laws and slight variations or combinations of these have hitherto been reported discrete peaks in autonomous systems have as yet escaped attention in autonomous systems and correspond to specific trajectory subsets involving an integer number of loops

bifurcation to chaotic scattering abstract we investigate a novel type of bifurcation to chaos which occurs in the context of chaotic scattering in chaotic scattering the deflection angle versus impact parameter is singular on a set of impact parameters which is fractal this behavior is caused by the presence of a chaotic invariant set of unstable bounded orbits in the bifurcation considered here the chaotic set arises abruptly in a sense to be discussed as the particle energy e decreases from above a critical value em to below we call this transition an abrupt bifurcation to fully developed chaotic scattering numerical computation of the dimension d of the chaotic set shows that in agreement with theoretical prediction d 1 in em e 1 near the abrupt bifurcatio

repellers semi attractors and long lived chaotic transients abstract we study the chaotic transients observed in many deterministic systems in general they are related to strange repellers or semi attractors if they are repelling in some and attracting in other directions we propose formulas relating the average life time of the transient to dimensions of the repeller and to lyapunov exponents of the flow on it the formulas are tested numerically in a number of cases

on scattering trajectories of dynamical systems dynamical systems on rn presenting geometric chaos i e open domains where bounded and unbounded orbits are intermingled have been constructed the opposite situation open scattering has been studied for integrable hamiltonian and non hamiltonian vector fields and for equations of type x v x when the potential has the form v a r b r f in spherical coordinates

existence and uniqueness for electrode models for electric current computed tomography the following experiment is considered to a body of given conductivity a number of electrodes are attached through which current is sent on the same electrodes the resulting voltages are measured this experiment can be described by a number of mathematical models k s cheng et al ieee transactions on biomedical engineering 36 1989 pp 918 924 these models are discussed and their predictions compared with experiment in particular a model is exhibited that is capable of predicting the experimentally measured voltages to within 0 1 percent for this model existence and uniqueness of the associated electrical potential is proved