tensor product generalized adi methods for separable elliptic problems we consider solving separable second order linear elliptic partial differential equations if an elliptic problem is separable then for certain discretizations the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices in the most general case the discrete problem can be written in the form a 1 otimes b 2 b 1 otimes a 2 c f we present a new tensor product generalized alternating direction implicit tpgadi iterative method for solving such discrete problems we prove convergence and establish computational efficiency the tpgadi method is applied to the hermite bicubic collocation equations we conclude that the tpgadi method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems

on the hyers ulam rassias stability of approximately additive mappings abstract in this paper using an idea from the paper of p gvruta j math anal appl 184 1994 431 436 the results of th m rassias j math anal appl 158 1991 106 113 shall be generalized

direct minimization of the least squares spectral element functional part i direct solver this paper describes an equivalent but improved least squares formulation for the numerical approximation of the solution of partial differential equations instead of using variational analysis to impose the conditions for minimizing the residual the residuals are minimized directly thus leading to a method we will denote by direct minimization dm dm circumvents setting up the normal equations which consists of matrix matrix multiplications matrix matrix multiplications are expensive may lead to loss of accuracy and destroy the sparsity pattern present in the original system the condition number of the dm formulation is the square root of the condition number which would be obtained if variational analysis was employed an element by element procedure will be presented which allows for parallelization of dm a computational comparison between dm and the conventional least squares formulation based on variational analysis will be presented

a parallel element by element solution scheme a parallel element by element scheme is developed for finite element solution of elliptic boundary value problems it is shown that for a broad class of computational grids the solution algorithm is totally parallelizable moreover the conversion of an existing serial ebe code to a parallel code is seen to be quite simple the method is implemented on the alliant fx 8 and sequent balance parallel computers and speedup performance studies are conducted for a representative elliptic pde in two dimensions the present procedure can be applied quite generally to other finite element applications and the associated programs can be modified in a straightforward way to implement the method

quantum fluctuation relations for the lindblad master equation an open quantum system interacting with its environment can be modeled under suitable assumptions as a markov process described by a lindblad master equation in this work we derive a general set of fluctuation relations for systems governed by a lindblad equation these identities provide quantum versions of jarzynski hatano sasa and crooks relations in the linear response regime these fluctuation relations yield a fluctuation dissipation theorem fdt valid for a stationary state arbitrarily far from equilibrium for a closed system this fdt reduces to the celebrated callen welton kubo formula

on the use of nurbs based discretizations in the scaled boundary finite element method for wave propagation problems abstract we discuss the application of non uniform rational b splines nurbs in the scaled boundary finite element method sbfem for the solution of wave propagation problems at rather high frequencies we focus on the propagation of guided waves along prismatic structures of constant cross section comparisons are made between nurbs based discretizations and high order spectral elements in terms of the achievable convergence rates we find that for the same order of shape functions nurbs can lead to significantly smaller errors compared with lagrange polynomials the difference becomes particularly important at very high frequencies where spectral elements are prone to instabilities furthermore we analyze the behavior of nurbs for the discretization of curved boundaries where the benefit of exact geometry representation becomes crucial even in the low frequency range

quantum entropy production as a measure of irreversibility we consider conservative quantum evolutions possibly interrupted by macroscopic measurements when started in a nonequilibrium state the resulting path space measure is not time reversal invariant and the weight of time reversal breaking equals the exponential of the entropy production the mean entropy production can then be expressed via a relative entropy on the level of histories this gives a partial extension of the result for classical systems that the entropy production is given by the source term of time reversal breaking in the path space measure

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

assessment of rod behaviour theories used in spectral finite element modelling abstract in this work different theories of rods have been discussed and compared the investigated theories are widely used in spectral finite element modelling of rod behaviour associated with propagation of symmetric longitudinal waves these are various single two mode and three mode theories including the elementary classical love and mindlin herrmann approaches as well as new two three and four mode theories proposed by the authors dispersion curves associated with each theory obtained by the use of hamilton s principle have been presented and discussed in the paper the investigation programme carried out by the authors aimed to show major differences and similarities between the rod theories and to discuss certain numerical aspects of their application great attention has been paid on properties limitations as well as difficulties associated with the use of the theories the results obtained from a wide program on numerical tests allowed the authors to draw certain general conclusions that are valid not only in the field of the spectral finite element method but also in the field of dynamics of engineering rod structures

fluctuation theorems for quantum master equations a quantum fluctuation theorem for a driven quantum subsystem interacting with its environment is derived based solely on the assumption that its reduced density matrix obeys a closed evolution equation i e a quantum master equation qme quantum trajectories and their associated entropy heat and work appear naturally by transforming the qme to a time dependent liouville space basis that diagonalizes the instantaneous reduced density matrix of the subsystem a quantum integral fluctuation theorem a steady state fluctuation theorem and the jarzynski relation are derived in a similar way as for classical stochastic dynamics

non conforming curved finite element schemes for time dependent elastic acoustic coupled problems high order numerical methods for solving time dependent acoustic elastic coupled problems are introduced these methods based on finite element techniques allow for a flexible coupling between the fluid and the solid domain by using non conforming meshes and curved elements since characteristic waves travel at different speeds through different media specific levels of granularity for the mesh discretization are required on each domain making impractical a possible conforming coupling in between advantageously physical domains may be independently discretized in our framework due to the non conforming feature consequently an important increase in computational efficiency may be achieved compared to other implementations based on conforming techniques namely by reducing the total number of degrees of freedom differently from other non conforming approaches proposed so far our technique is relatively simpler and requires only a geometrical adjustment at the coupling interface at a preprocessing stage so that no extra computations are necessary during the time evolution of the simulation on the other hand as an advantage of using curvilinear elements the geometry of the coupling interface between the two media of interest is faithfully represented up to the order of the scheme used in other words higher order schemes are in consonance with higher order approximations of the geometry concerning the time discretization we analyze both explicit and implicit schemes these schemes are energy conserving and for the explicit case the stability is guaranteed by a cfl condition in order to illustrate the accuracy and convergence of these methods a set of representative numerical tests are presented

squeeze flow of a piezoviscous fluid we investigate squeeze flow of piezoviscous fluids presence of contact line leads to strong pressure variations strong pressure variations induce strong viscosity variations in piezoviscous fluid squeeze flow characteristics are influenced by piezoviscosity of the fluid we investigate the squeeze flow of a piezoviscous fluid the classical navier stokes fluid model and the no slip boundary condition on the plate fluid interface implies the presence of the pressure singularity at the contact line it can be conjectured that the same behaviour is present even for piezoviscous fluids consequently the pressure variation in the fluid could be large and the pressure dependent viscosity could be important even if the piezoviscous coefficient is apparently small we investigate this conjecture by means of numerical simulation based on a variant of the spectral collocation method the numerical simulations indicate that the effects due to pressure dependent viscosity are for certain range of parameter values significant

the pseudospectral method accurate representation of interfaces in elastic wave calculations when finite difference methods are used to solve the elastic wave equation in a discontinuous medium the error has two dominant components dispersive errors lead to artificial wave trains errors from interfaces lead to circular wavefronts emanating from each location where the interface appears jagged to the rectangular grid the pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy with this method dispersive errors are essentially eliminated the mappings introduced in this paper also eliminate the other dominant error source test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost although the mapping method is described here in connection with the pseudospectral method it can also be used with high order finite difference approximations

modelling of wave propagation in composite plates using the time domain spectral element method abstract this paper presents results of numerical simulation of the propagation of transverse elastic waves corresponding to the a0 mode of lamb waves in a composite plate the problem is solved by the spectral element method spectral plate finite elements with 36 nodes defined at gauss lobatto legendre points are used as a consequence of the selection of lagrange polynomials for element shape functions discrete orthogonality is obtained leading to the diagonal form of the element mass matrix this results in a crucial reduction of numerical operations required for the solution of the equation of motion by time integration numerical calculations have been carried out for various orientations and relative volume fractions of reinforcing fibres within the plate the paper shows how the velocities of transverse elastic waves in composite materials depend on the orientation and the relative volume fraction of the reinforcement

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

quantum work fluctuation theorem exact case study for a perturbed oscillator we analytically explore the fluctuation of the work induced entropy production of an externally perturbed quantum harmonic oscillator interacting with several reservoirs the quantum fluctuation of the work amounts to a non gaussian fluctuation of the entropy production which interpolates the gaussian and the poissonian distributions at high and low temperature regimes also it is shown that the corresponding fluctuation theorem symmetry is rigorously satisfied

the problem of s m ulam for approximately multiplicative mappings in the present paper we study the problem of s m ulam for approximately multiplicative mappings and we consider new open problems

unified treatment of the quantum fluctuation theorem and the jarzynski equality in terms of microscopic reversibility there are two related theorems which hold even in far from equilibrium namely fluctuation theorem and jarzynski equality fluctuation theorem states the existence of symmetry of fluctuation of entropy production while the jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes on the other hand the relationship between these theorems was investigated by crooks phys rev e 60 2721 1999 for the classical stochastic systems in this paper we derive quantum analogues of fluctuation theorem and jarzynski equality in terms of microscopic reversibility in other words the quantum analog of the work by crooks is presented also for the quasiclassical langevin system microscopically reversible condition is confirmed

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

stability of a mixed functional equation in several variables on banach modules abstract in this paper we establish the general solution of the functional equation i 1 n f x i 1 n j 1 n x j i 1 n f x i n f 1 n i 1 n x i n 2 and use a fixed point method to prove its generalized hyers ulam stability in banach modules over a unital banach algebra

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

3 d large scale wave propagation modeling by spectral element method on cray t3e multiprocessor highly efficient algorithms and massively parallel computers are needed for acoustic full wave modeling in large scale 3 d realistic unbounded media in this work a spectral element method in conjunction with a new iterative solution technique is presented the very high spatial accuracy of the algorithm allows for increased computational efficiency when compared to standard finite element method in addition the use of an iterative solver based on a local element by element formulation and a matrix vector product which is factored and converted to a faster matrix matrix product allows for a significant reduction both in storage requirements and in the computational complexity the resulting algorithm can easily be implemented on high performance vector and or parallel processors the numerical scheme has been tested on a cray t3e supercomputer our experimental results show that efficiency is very high and that good load balancing can be obtained at run time because relevant blocks of computations are performed at the element level and are totally independent

entropy production for quantum markov semigroups an invariant state of a quantum markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition in this paper we introduce a notion of entropy production for faithful normal invariant states of a quantum markov semigroup on mathcal b mathsf h as a numerical index measuring how far they are from equilibrium the entropy production rate is defined as the derivative of the relative entropy of the one step forward and backward evolution in analogy with the classical probabilistic concept we prove an explicit trace formula expressing the entropy production rate in terms of the completely positive part of the generator of a norm continuous quantum markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds

generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media it is important to develop fast yet accurate numerical methods for seismic wave propagation to characterize complex geological structures and oil and gas reservoirs however the computational cost of conventional numerical modeling methods such as finite difference method and finite element method becomes prohibitively expensive when applied to very large models we propose a generalized multiscale finite element method gmsfem for elastic wave propagation in heterogeneous anisotropic media where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element the application of multiscale basis functions can capture the fine scale medium property variations and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite element method for wave equation while restricting the error to low values we formulate the continuous galerkin and discontinuous galerkin formulation of the multiscale method both of which have pros and cons applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the elastic wave propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system

generalized additive mapping in banach modules and isomorphisms between c algebras abstract let x y be vector spaces it is shown that if an odd mapping f x y satisfies the functional equation 1 r f j 1 d x j r j 0 1 j 1 d j l r f j 1 d 1 j x j r c l d 1 c l 1 d 1 1 j 1 d f x j then the odd mapping f x y is additive and we prove the hyers ulam stability of the functional equation 1 in banach modules over a unital c algebra as an application we show that every almost linear bijection h a b of a unital c algebra a onto a unital c algebra b is a c algebra isomorphism when h 2 n r n u y h 2 n r n u h y for all unitaries u a all y a and n 0 1 2

on the stability of the linear mapping in banach modules abstract we prove the generalized hyers ulam rassias stability of the linear mapping in banach modules over a unital banach algebra

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

a space time finite element method for structural acoustics in infinite domains part 1 formulation stability and convergence abstract a space time finite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an infinite acoustic fluid is formulated in particular time discontinuous galerkin and galerkin least squares gls variational formulations for coupled structural acoustics in unbounded domains are developed and analyzed for stability and convergence the formulation employs a finite computational fluid domain surrounding the structure and incorporates time dependent non reflecting boundary conditions on the fluid truncation boundary energy estimates are obtained which allow us to prove the unconditional stability of the method for the coupled fluid structure problem with absorbing boundaries the methods developed are especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured space time meshes are used to track waves propagating along space time characteristics an important feature of the space time formulation is the incorporation of temporal jump operators which allow for finite element interpolations that are discontinaous in time for additional stability least squares operators based on local residuals of the structural acoustics equations including the non reflecting boundary conditions are incorporated the energy decay estimates and high order accuracy predicted by our a priori error estimates are demonstrated numerically in a simple canonical example

solving elastodynamics in a fluid solid heterogeneous sphere a parallel spectral element approximation on non conforming grids we present a spectral element approach for modeling elastic wave propagation in a solid fluid sphere where the local effects of gravity are taken into account the equations are discretized in terms of the displacement in the solid and the velocity potential in a neutrally stratified fluid the spatial approximation is based upon a spherical mesh of hexahedra in which local refinement allows for adapting the discretization to the variation of elastic parameters in both the solid and the fluid regions continuity constraints across the non conforming interfaces are introduced through lagrange multipliers which are further discretized by the mortar element method due to the spherical nature of the non conforming interfaces the mortar method turns out to be functionally conforming and allows for an equal order interpolation of the primal variables and the lagrange multipliers the method is shown to provide an accurate solution when compared to analytical calculations obtained for radial models of elastic parameters its parallel implementation is based upon a simple domain decomposition strategy which makes it efficient to solve large problems as those imposed by planetary scales

free boundary conditions of arbitrary polygonal topography in a two dimensional explicit elastic finite difference scheme we present a simple method for simulating 2 d elastic waves in a model with free surface topography of polygonal shape i e a continuous but irregular surface composed of line segments our method requires special treatment for each of the six specific cases involving line segments of various slopes as well as transition points between the sloping segments for brevity only nonnegatively sloping segments are specifically included on an inclined free surface vanishing stress conditions are implemented using a rotated coordinate system parallel to the inclined boundary at transition points on the topography between line segments we use a first order approximate boundary condition in a locally rotated coordinate system aligned with the bisector of the corner as in the classical one sided explicit approximation scheme widely used for the flat free surface case these extrapolation formulas are accurate to first order in spatial increment numerical tests indicate that the present scheme is stable over

a numerical homogenization method for heterogeneous anisotropic elastic media based on multiscale theory the development of reliable methods for upscaling finescale models of elastic media has long been an important topic for rock physics and applied seismology several effective medium theories have been developed to provide elastic parameters for materials such as finely layered media or randomly oriented or aligned fractures in such cases the analytic solutions for upscaled properties can be used for accurate prediction of wave propagation however such theories cannot be applied directly to homogenize elastic media with more complex arbitrary spatial heterogeneity therefore we have proposed a numerical homogenization algorithm based on multiscale finite element methods for simulating elastic wave propagation in heterogeneous anisotropic elastic media specifically our method used multiscale basis functions obtained from a local linear elasticity problem with appropriately defined boundary conditions homogenized effective medium parameters were then computed using these basis functions and the approach applied a numerical discretization that was similar to the rotated staggered grid finite difference scheme comparisons of the results from our method and from conventional analytical approaches for finely layered media showed that the homogenization reliably estimated elastic parameters for this simple geometry additional tests examined anisotropic models with arbitrary spatial heterogeneity in which the average size of the heterogeneities ranged from several centimeters to several meters and the ratio between the dominant wavelength and the average size of the arbitrary heterogeneities ranged from 10 to 100 comparisons to finite difference simulations proved that the numerical homogenization was equally accurate for these complex cases

a generalization of the hyers ulam rassias stability of approximately additive mappings abstract in this paper we prove a generalization of the stability of approximately additive mappings in the spirit of hyers ulam and rassias

least squares legendre spectral element solutions to sound propagation problems this paper presents a novel algorithm and numerical results of sound wave propagation the method is based on a least squares legendre spectral element approach for spatial discretization and the crank nicolson proc cambridge philos soc 43 50 67 1947 and adams bashforth d gottlieb and s a orszag numerical analysis of spectral methods theory and applications cbms nsf monograph siam 1977 schemes for temporal discretization to solve the linearized acoustic field equations for sound propagation two types of nasa computational aeroacoustics caa workshop benchmark problems icase larc workshop on benchmark problems in computational aeroacoustics edited by j c hardin j r ristorcelli and c k w tam nasa conference publication 3300 1995a are considered a narrow gaussian sound wave propagating in a one dimensional space without flows and the reflection of a two dimensional acoustic pulse off a rigid wall in the presence of a uniform flow of mach 0 5 in a semi infinite space the f

solution of the equations of dynamic elasticity by a chebychev spectral method we present a spectral method for solving the two dimensional equations of dynamic elasticity based on a chebychev expansion in the vertical direction and a fourier expansion for the horizontal direction the technique can handle the free surface boundary condition more rigorously than the ordinary fourier method the algorithm is tested against problems with known analytic solutions including lamb s problem of wave propagation in a uniform elastic half space reflection from a solid solid interface and surface wave propagation in a haft space containing a low velocity layer agreement between the solutions is very good a fourth example of wave propagation in a laterally heterogeneous structure is also presented results indicate that the method is very accurate and only about a factor of two slower than the fourier method

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

how to render second order accurate time stepping algorithms fourth order accurate while retaining the stability and conservation properties abstract for a fairly general class of second order accurate algorithms a sub stepping procedure is described which achieves fourth order accuracy while retaining the stability conservation properties and implementation of the underlying second order method two features render the proposed approach especially attractive from a computational perspective over conventional fourth order methods first the improved fourth order accuracy is attained merely by two additional residual evaluations with no changes in the implementation of the underlying second order method in particular no additional storage or extra computations of high order gradients are required second for hamiltonian systems the stability properties and possible conservation properties of the underlying second order method such as energy momentum and symplectic character are preserved the new technique is applied to a number of standard algorithms for elastodynamics structural and rigid body dynamics in particular when applied to the recently proposed second order accurate energy momentum method this approach results in a fourth order accurate energy linear and angular momentum conserving algorithm for general hamiltonian systems

time reversal and entropy there is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time reversal symmetry and the entropy production in such processes we explain on an elementary mathematical level the relations between entropy production phase space contraction and time reversal starting from a deterministic dynamics both closed and open systems in the transient and in the steady regime are considered the main result identifies under general conditions the statistical mechanical entropy production as the source term of time reversal breaking in the path space measure for the evolution of reduced variables this provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful in equalities we also discuss the markov approximation important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior

numerical simulation of interface waves by high order spectral modeling techniques few problems in elastodynamics have a closed form analytical solution the others can be investigated with semianalytical methods but in general one is not sure whether these methods give reliable solutions the same happens with numerical techniques for instance finite difference methods solve in principle any complex problem including those with arbitrary inhomogeneities and boundary conditions however there is no way to verify the quantitative correctness of the solutions the major problems are stability with respect to material properties numerical dispersion and the treatment of boundary conditions in practice these problems may produce inaccurate solutions in this paper the study of complex problems with two different numerical grid techniques in order to cross check the solutions is proposed interface waves in particular are emphasized since they pose the major difficulties due to the need to implement boundary conditions the first method is based on global differential operators where the solution is expanded in terms of the fourier basis and chebyshev polynomials while the second is the spectral element method an extension of the finite element method that uses chebyshev polynomials as interpolating functions both methods have spectral accuracy up to approximately the nyquist wave number of the grid moreover both methods implement the boundary conditions in a natural way particularly the spectral element algorithm we first solve lamb s problem and compare numerical and analytical solutions then the problem of dispersed rayleigh waves and finally the two quarter space problem we show that the modeling algorithms correctly reproduce the analytical solutions and yield a perfect matching when these solutions do not exist the combined modeling techniques provide a powerful tool for solving complex problems in elastodynamics

spectral method solution of the stokes equations on nonstaggered grids abstract the stokes equations are solved using spectral methods with staggered and nonstaggered grids numerous ways to avoid the problem of spurious pressure modes are presented including new techniques using the pseudospectral method and a method solving the weak form of the governing equations a variation on the spectral element method developed by patera the pseudospectral methods using nonstaggered grids are simpler to implement and have comparable or better accuracy than the staggered grid formulations three test cases are presented a formulation with an exact solution a formulation with homogeneous boundary conditions and the driven cavity problem the solution accuracy is shown to be greatly improved for the driven cavity problem when the analytical solution of the singular flow behavior in the upper corners is separated from the computational solution

a spectral collocation method for the navier stokes equations abstract a fourier chebyshev spectral method for the incompressible navier stokes equations is described it is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time in this fully spectral algorithm a preconditioned iterative technique is used for solving the implicit equations arising from semi implicit treatment of pressure mean advection and vertical diffusion terms the algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity

dissipation preserving spectral element method for damped seismic wave equations this article describes the extension of the conformal symplectic method to solve the damped acoustic wave equation and the elastic wave equations in the framework of the spectral element method the conformal symplectic method is a variation of conventional symplectic methods to treat non conservative time evolution problems which has superior behaviors in long time stability and dissipation preservation to reveal the intrinsic dissipative properties of the model equations we first reformulate the original systems in their equivalent conformal multi symplectic structures and derive the corresponding conformal symplectic conservation laws we thereafter separate each system into a conservative hamiltonian system and a purely dissipative ordinary differential equation system based on the splitting methodology we solve the two subsystems respectively the dissipative one is cheaply solved by its analytic solution while for the conservative system we combine a fourth order symplectic nystrm method in time and the spectral element method in space to cover the circumstances in realistic geological structures involving complex free surface topography the strang composition method is adopted thereby to concatenate the corresponding two parts of solutions and generate the completed conformal symplectic method a relative larger courant number than that of the traditional newmark scheme is found in the numerical experiments in conjunction with a spatial sampling of approximately 5 points per wavelength a benchmark test for the damped acoustic wave equation validates the effectiveness of our proposed method in precisely capturing dissipation rate the classical lamb problem is used to demonstrate the ability of modeling rayleigh wave in elastic wave propagation more comprehensive numerical experiments are presented to investigate the long time simulation low dispersion and energy conservation properties of the conformal symplectic methods in both the attenuating homogeneous and heterogeneous media

inverse closedness of approximation algebras abstract we prove the inverse closedness of certain approximation algebras based on a quasi banach algebra x using two general theorems on the inverse closedness of subspaces of quasi banach algebras in the first theorem commutative algebras are considered while the second theorem can be applied to arbitrary x and to subspaces of x which can be obtained by a general k method of interpolation between x and an inversely closed subspace y of x having certain properties as application we present some inversely closed subalgebras of c t and c 1 1 in particular we generalize wiener s theorem i e we show that for many subalgebras s of l 1 z the property c k f s c k f being the fourier coefficients of f implies the same property for 1 f if f c t vanishes nowhere on t

simulation of anisotropic wave propagation based upon a spectral element method we introduce a numerical approach for modeling elastic wave propagation in 2 d and 3 d fully anisotropic media based upon a spectral element method the technique solves a weak formulation of the wave equation which is discretized using a high order polynomial representation on a finite element mesh for isotropic media the spectral element method is known for its high degree of accuracy its ability to handle complex model geometries and its low computational cost we show that the method can be extended to fully anisotropic media the mass matrix obtained is diagonal by construction which leads to a very efficient fully explicit solver we demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2 d transversely isotropic test case and by comparing its predictions against those based upon a finite difference method for a 2 d heterogeneous anisotropic medium we show its generality and its flexibility by modeling wave propagation in a 3 d transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid

microscopic origins of irreversible macroscopic behavior time asymmetric behavior as embodied in the second law of thermodynamics is observed in individual macroscopic systems it can be understood as arising naturally from time symmetric microscopic laws in accord with the ideas of thompson maxwell and boltzmann alternative explanations based on equating irreversible macroscopic behavior with the mixing type of behavior already present in the time evolution of ensembles probability distributions of certain systems having only a few degrees of freedom are in my opinion unnecessary misguided and misleading

local time stepping with the discontinuous galerkin method for wave propagation in 3d heterogeneous media modeling and imaging techniques for geophysics are extremely demanding in terms of computational resources seismic data attempt to resolve smaller scales and deeper targets in increasingly more complex geologic settings finite elements enable accurate simulation of time dependent wave propagation in heterogeneous media they are more costly than finite difference methods but this is compensated by their superior accuracy if the finite element mesh follows the sharp impedance contrasts and by their improved efficiency if the element size scales with wavelength hence with the local wave velocity however 3d complex geologic settings often contain details on a very small scale compared to the dominant wavelength requiring the mesh to contain elements that are smaller than dictated by the wavelength also limitations of the mesh generation software may produce regions where the elements are much smaller than desired in both cases this leads to a reduction of the time step required to solve the wave propagation and significantly increases the computational cost local time stepping lts can improve the computational efficiency and speed up the simulation we evaluated a local formulation of an lts scheme with second order accuracy for the discontinuous galerkin finite element discretization of the wave equation we tested the benefits of the scheme by considering a geologic model for a north sea type example

modeling of the acoustic wave equation with transform methods numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data two separate topics are studied the first deals with the solution of the acoustic wave equation the second topic treats wave phenomena whose direction of propagation is restricted within 90 degrees from a given axis in the numerical methods developed here the wave field is advanced in time by using standard time differencing schemes on the other hand expressions including space derivative terms are computed by fourier transform methods this approach to computing derivatives minimizes truncation errors another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves e g to avoid multiple reflections constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain the error analysis of the algorithms shows that truncation errors are due mainly to time discretization such er

optimal blended spectral element operators for acoustic wave modeling spectral element methods based on high order polynomials are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media however to retain computational efficiency very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided in the present work we devise an approach for reducing such errors by considering modified discrete wave operators we analyze consistent and lumped operators together with blended operators weighted averages of consistent and lumped operators furthermore using the operator blending approach and a novel dispersion analysis method we develop optimal spectral element operators that have increased numerical accuracy without resorting to very high order operators the new operators are faster and computationally more efficient than consistent operators our approach is based on the tensor product decomposition of the element matrices into 1d f

a two scale solution algorithm for the elastic wave equation operator based upscaling is a two scale algorithm that speeds up the solution of the wave equation by producing a coarse grid solution which incorporates much of the local fine scale solution information we present the first implementation of operator upscaling for the elastic wave equation by using the velocity displacement formulation of the three dimensional elastic wave equation basis functions that are linear in all three directions and applying mass lumping the subgrid solve first stage of the two step algorithm reduces to solving explicit difference equations at the second stage of the algorithm we upscale both velocity and displacement by using local subgrid information to formulate the coarse grid problem the coarse grid system matrix is independent of time sparse and banded this paper explores both serial and parallel implementations of the method the main simplifying assumption of the method special zero boundary conditions imposed on coarse blocks in the first stage of the algorithm leads to an easily parallelizable algorithm because very little communication is required between processors in fact for this upscaling implementation calculation of the load vector for the coarse solve dominates the cost of a time step we show that for a homogeneous medium convergence is second order in space and time so long as both the coarse and fine grids are simultaneously refined a series of heterogeneous medium numerical experiments demonstrate that the upscaled solution captures the fine scale fluctuations in the input parameters accurately most notable for use in a seismic inversion algorithm the upscaling algorithm accurately locates the depth of reflectors interface changes

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

domain decomposition method for dynamic faulting under slip dependent friction the anti plane shearing problem on a system of finite faults under a slip dependent friction in a linear elastic domain is considered using a newmark method for the time discretization of the problem we have obtained an elliptic variational inequality at each time step an upper bound for the time step size which is not a cfl condition is deduced from the solution uniqueness criterion using the first eigenvalue of the tangent problem finite element form of the variational inequality is solved by a schwarz method assuming that the inner nodes of the domain lie in one subdomain and the nodes on the fault lie in other subdomains two decompositions of the domain are analyzed one made up of two subdomains and another one with three subdomains numerical experiments are performed to illustrate convergence for a single time step convergence of the schwarz algorithm influence of the mesh size influence of the time step convergence in time instability capturing energy dissipation optimal time step and an application to a relevant physical problem interacting parallel fault segments

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

the spectral element method for elastic wave equations application to 2 d and 3 d seismic problems a spectral element method for the approximate solution of linear elastodynamic equations set in a weak form is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two and three dimensional geometries the computational domain is discretized into quadrangles or hexahedra defined with respect to a reference unit domain by an invertible local mapping inside each reference element the numerical integration is based on the tensor product of a gauss lobatto legendre 1 d quadrature and the solution is expanded onto a discrete polynomial basis using lagrange interpolants as a result the mass matrix is always diagonal which drastically reduces the computational cost and allows an efficient parallel implementation absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains the time discretization is based on an energy momentum conserving scheme that can be put into a classical explicit implicit predictor multicorrector format long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions the accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two dimensional problems obtained by other methods the potentiality of the method is then illustrated by studying a simple three dimensional model very accurate modelling of rayleigh wave propagation and surface diffraction is obtained at a low computational cost the method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three dimensional surface topographies copyright 1999 john wiley sons ltd

stability of an explicit high order spectral element method for acoustics in heterogeneous media based on local element stability criteria this paper considers the stability of an explicit leapfrog time marching scheme for the simulation of acoustic wave propagation in heterogeneous media with high order spectral elements the global stability criterion is taken as a minimum over local element stability criteria obtained through the solution of element borne eigenvalue problems first an explicit stability criterion is obtained for the particular case of a strongly heterogeneous and or rapidly fluctuating medium using asymptotic analysis this criterion is only dependent upon the maximum velocity at the vertices of the mesh elements and not on the velocity at the interior nodes of the high order elements second in a more general setting bounds are derived using statistics of the coefficients of the elemental dispersion matrices different bounds are presented discussed and compared several numerical experiments show the accuracy of the proposed criteria in one dimensional test cases as well as in more realistic large scale 3d problems

a simple multi directional absorbing layer method to simulate elastic wave propagation in unbounded domains the numerical analysis of elastic wave propagation in unbounded media may be difficult due to spurious waves reflected at the model artificial boundaries this point is critical for the analysis of wave propagation in heterogeneous or layered solids various techniques such as absorbing boundary conditions infinite elements or absorbing boundary layers e g perfectly matched layers lead to an important reduction of such spurious reflections in this paper a simple absorbing layer method is proposed it is based on a rayleigh caughey damping formulation which is often already available in existing finite element softwares the principle of the caughey absorbing layer method is first presented including a rheological interpretation the efficiency of the method is then shown through 1d finite element simulations considering homogeneous and heterogeneous damping in the absorbing layer 2d models are considered afterwards to assess the efficiency of the absorbing layer method for various wave types and incidences a comparison with the pml method is first performed for pure p waves and the method is shown to be reliable in a more complex 2d case involving various wave types and incidences it may thus be used for various types of problems involving elastic waves e g machine vibrations seismic waves etc copyright 2010 john wiley sons ltd

on approximation of approximately linear mappings by linear mappings abstract assume a is a normed linear space b is a banach space and f a b is a mapping approximately linear we solve the following ulam problem give conditions in order for a linear mapping near an approximately linear mapping to exist

modeling of wave propagation in plate structures using three dimensional spectral element method for damage detection abstract this paper presents the application of three dimensional 3 d spectral element method sem to wave propagation problems in plate structures for the purpose of damage detection the excellent characteristic of the sem is that the mass matrix is diagonal because of the choice of lagrange interpolation function supported on the gauss lobatto legendre gll points in conjunction with the gll integration rule therefore numerical calculation can be significantly efficient in comparison with the classical finite element method fem by taking advantage of this characteristic a 3 d spectral finite elements sfes based model is developed to simulate the wave propagation in plate structures lamb waves propagating in aluminum plates with and without a crack are investigated responses from 3 d sfes and 2 d sfes based models are compared different lamb wave modes are generated using different excitation approaches and subsequently characteristics of those modes are analyzed the results demonstrate that the proposed model can offer efficient and realistic simulation for lamb wave propagation in plate structures so as to detect damages in those structures

mass lumping techniques in the spectral element method on the equivalence of the row sum nodal quadrature and diagonal scaling methods abstract in the context of wave propagation analysis the spectral element method sem in conjunction with a diagonal mass matrix is often the method of choice therefore it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results to this end we compare the performance of three established methods including the row sum method the nodal quadrature method and the diagonal scaling method the theoretical analysis of these methods reveals a close connection between them under certain conditions that are discussed in detail in this article we are able to show a direct equivalence between these three approaches in this regard the attainable accuracy of the numerical integration of the mass matrix plays an important role by means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix sem in comparison to a formulation based on the consistent mass matrix

hyers ulam rassias stability of homomorphisms in quasi banach algebras associated to the pexiderized cauchy functional equation abstract in this paper we prove the hyers ulam rassias stability of homomorphisms in quasi banach algebras associated to the pexiderized cauchy functional equation this is applied to investigate homomorphisms between quasi banach algebras the concept of hyers ulam rassias stability originated from th m rassias stability theorem that appeared in his paper th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300

spectral domain decomposition methods for the solution of acoustic and elastic wave equations a new spectral domain decomposition method is presented for acoustic and elastodynamic wave propagation in 2 d heterogeneous media starting from a variational formulation of the problem two different approaches are proposed for the spatial discretization a mixed fourier legendre and a full legendre collocation the matching conditions at subdomain interfaces are carefully analyzed and the stability and efficiency of time advancing schemes are investigated the numerical validation with some significant test cases illustrates the accuracy flexibility and robustness of our methods these allow the treatment of complex geometries and heterogeneous media while retaining spectral accuracy

isomorphisms between unital c algebras abstract it is shown that every almost linear bijection h a b of a unital c algebra a onto a unital c algebra b is a c algebra isomorphism when h 2 n u y h 2 n u h y for all unitaries u a all y a and n 0 1 2 and that almost linear continuous bijection h a b of a unital c algebra a of real rank zero onto a unital c algebra b is a c algebra isomorphism when h 2 n u y h 2 n u h y for all u v a v v v 1 v is invertible all y a and n 0 1 2 assume that x and y are left normed modules over a unital c algebra a it is shown that every surjective isometry t x y satisfying t 0 0 and t u x u t x for all x x and all unitaries u a is an a linear isomorphism this is applied to investigate c algebra isomorphisms between unital c algebras

spectral approximation of the periodic nonperiodic navier stokes equations in order to approximate the navier stokes equations with periodic boundary conditions in two directions and a no slip boundary condition in the third direction by spectral methods we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure the compatibility between these two spaces is checked via an infsup condition we analyze a spectral and a collocation pseudo spectral method for the stokes problem and a collocation pseudo spectral method for the navier stokes equations we derive error bounds of spectral type i e which behave likem wherem depends on the number of degrees of freedom of the method and represents the regularity of the data

time harmonic elasticity with controllability and higher order discretization methods the time harmonic solution of the linear elastic wave equation is needed for a variety of applications the typical procedure for solving the time harmonic elastic wave equation leads to difficulties solving large scale indefinite linear systems to avoid these difficulties we consider the original time dependent equation with a method based on an exact controllability formulation the main idea of this approach is to find initial conditions such that after one time period the solution and its time derivative coincide with the initial conditions the wave equation is discretized in the space domain with spectral elements the degrees of freedom associated with the basis functions are situated at the gauss lobatto quadrature points of the elements and the gauss lobatto quadrature rule is used so that the mass matrix becomes diagonal this method is combined with the second order central finite difference or the fourth order runge kutta time discretization as a consequence of these choices only matrix vector products are needed in time dependent simulation this makes the controllability method computationally efficient

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes