generation of coarse scale continuum flow models from detailed fracture characterizations 1 a procedure for developing coarse scale continuum models from detailed fracture descriptions is developed and applied the coarse models are in the form of a generalized dual porosity representation in which matrix rock and fractures exchange fluid locally while large scale flow occurs through the fracture network the methodology developed here introduces local subgrids to resolve dynamics within the matrix and provides appropriate coarse scale parameters describing fracture fracture matrix fracture and matrix matrix flow the geometry of the local subgrids as well as the required parameters for the coarse scale model are determined from local flow solutions using the underlying discrete fracture model the method is applied to two and three dimensional single phase and two phase flow problems and the accuracy of the coarse models is assessed relative to fully resolved discrete fracture simulations for the cases considered it is shown that the technique is capable of generating highly accurate coarse models with many fewer unknowns than the detailed characterizations and speedups of about a factor of 100 are achieved

finite element formulations for transient dynamic analysis in structural systems with viscoelastic treatments containing fractional derivative models this paper presents finite element formulations for transient dynamic analysis in structural systems with damping treatments in which viscoelastic materials are characterized by means of fractional derivative models in contrast to other formulations such as that of padovan employing the principle of virtual work the proposed formulations begin from the local equation of linear momentum and make use of the weighted residual method providing a matrix equation of motion involving fractional operators the numerical approximations of these are developed through the grunwald letnikov definition which allows to formulate explicit and implicit numerical schemes the principal advantage of the proposed formulations is that the history of the displacements and external forces must be stored but not that of the stress which reduces computational time and storage needs numerical applications are presented for a cantilever beam where a viscoelastic treatment has been applied using damping material modelled by a five parameter fractional derivative model the results of the proposed formulation are compared among them and with those of padovan for different damping values and for different load cases the influence of the truncation of grunwald coefficients and of the integration time step is also investigated copyright 2006 john wiley sons ltd

analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method the fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials motion of a large thin plate in a newtonian fluid creep and relaxation functions for viscoelastic materials the pi ld m controller for the control of dynamical systems etc phenomena in electromagnetics acoustics viscoelasticity electrochemistry and material science are also described by differential equations of fractional order the solution of the differential equation containing fractional derivative is much involved instead of application of the existing methods an attempt has been made in the present analysis to obtain the solution of nonlinear dynamic system containing fractional derivative ji zeng wang et al coiflets based method in the solution of nonlinear dynamic system containing fractional derivative in the fourth international conference on nonlinear mechanics icnm iv shanghai august 2002 pp 1304 1308 by the relatively new adomian decomposition method the results obtained by this method are then graphically represented and then compared with the exact solution a good agreement of the results is observed

the virtual element method for discrete fracture network simulations abstract in this work an optimization based approach presented in berrone et al 2013 2014 10 12 for discrete fracture network simulations is coupled with the virtual element method vem for the space discretization of the underlying darcy law the great flexibility of the vem in handling rather general polygonal elements allows in a natural way for an effective description of irregular solutions starting from an arbitrary triangulation which is built independently of the mesh on other fractures only partial conformity is in fact obtained with this approach numerical results performed on several dfn configurations confirm the viability and efficiency of the resulting method

modeling fractures and barriers as interfaces for flow in porous media we consider a fractured porous medium that is studied at a scale such that the fractures can be modeled individually models for flow in which the fractures are interfaces between subdomains are presented these models take into account interactions between the fractures and the surrounding porous medium existence and uniqueness of the solution to the model problem are proved error estimates show convergence as mathcal o max h d in the l2 norm where h is the mesh size and d the fracture width numerical experiments confirm the theoretical results

partitioning strategies for the interaction of a fluid with a poroelastic material based on a nitsche s coupling approach abstract we develop a computational model to study the interaction of a fluid with a poroelastic material the coupling of stokes and biot equations represents a prototype problem for these phenomena which feature multiple facets on one hand it shares common traits with fluid structure interaction on the other hand it resembles the stokes darcy coupling for these reasons the numerical simulation of the stokes biot coupled system is a challenging task the need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners for the equations at hand are typical shortcomings of classical discretization methods applied to this problem such as the finite element method for spatial discretization and finite differences for time stepping the application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others at each time step in this work we develop and thoroughly analyze a loosely coupled scheme for stokes biot equations the scheme is based on nitsche s method for enforcing interface conditions once the interface operators corresponding to the interface conditions have been defined time lagging allows us to build up a loosely coupled scheme with good stability properties the stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem the error of the resulting method is also analyzed showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes in order to restore good approximation properties while maintaining the computational efficiency of the loosely coupled approach we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

transient analysis of viscoelastic helical rods subject to time dependent loads abstract the transient analysis of viscoelastic helical rods subject to time dependent loads are examined in the laplace domain the governing equations for naturally twisted and curved spatial rods obtained using the timoshenko beam theory are rewritten for cylindrical helical rods the curvature of the rod axis effect of rotary inertia and shear and axial deformations are considered in the formulation the material of the rod is assumed to be homogeneous isotropic and linear viscoelastic the viscoelastic constitutive equations are written in the boltzmann volterra form ordinary differential equations in canonical form obtained in the laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem the solutions obtained are transformed to the real space using an appropriate numerical inverse laplace transform method numerical results for quasi static and dynamic response of viscoelastic models are presented in the form of graphics

influence of transverse shear on stochastic instability of viscoelastic beam abstract the stochastic instability problem associated with an axially loaded timochenko beam made of viscoelastic material is formulated the beam is treated as voigt kelvin body compressed by time dependent deterministic and stochastic forces by using the direct liapunov method bounds of the almost sure instability of beams as a function of retardation time variance of the stochastic force mode number section shape factor and intensity of the deterministic component of axial loading are obtained calculations are performed for the gaussian process with a zero mean and variance 2 as well as for harmonic process with an amplitude a

power law and fractional calculus model of viscoelasticity mathematical relationships between the power law model and the fractional calculus model of viscoelastic behavior are developed it is demonstrated that the two models have mathematically similar relaxation spectra this similarity is seen to produce an asymptotic equivalence of the models at long relaxation times and correspondingly low frequencies of motion in the lower transition and rubbery regions the divergent behavior of the models at higher frequencies in the transition and glassy regions is presented porposals to take advantage of the similarities between power law and fractional calculus models are discussed

uncertainty quantification in discrete fracture network models stochastic geometry we consider the problem of uncertainty quantification analysis of the output of underground flow simulations we consider in particular fractured media described via the discrete fracture network model within this framework we address the relevant case of networks in which the geometry of the fractures is described by stochastic parameters in this context due to a possible lack of smoothness in the quantity of interest with respect to the stochastic parameters well assessed techniques such as stochastic collocation may fail in providing reliable estimates of first order moments of the quantity of interest in this paper we overcome this issue by applying the multilevel monte carlo method using as underlying solver an extremely robust method

power series for inverse jacobian elliptic functions the 12 inverse jacobian elliptic functions are expanded in power series by using properties of the symmetric elliptic integral of the first kind suitable notation allows three series to include all 12 cases three of which have been given previously all coefficients are polynomials in the modulus k that are homogeneous variants of legendre polynomials the four series in each of three subsets have the same coefficients in terms of k

an approximate numerical method for the complex eigenproblem in systems characterised by a structural damping matrix this paper presents an efficient numerical method that approximates the complex eigenvalues and eigenvectors in structural systems with viscoelastic damping materials characterised by the complex structural damping matrix this new method begins from the solution of the undamped system and approximates the complex eigenpair by finite increments using the eigenvector derivatives and the rayleigh quotient it is implemented in three different approaches single step incremental and iterative schemes the single step technique is presented for systems with low and medium damping from numerical examples it can be verified that the errors committed by the approximate single step technique with respect to the exact solutions obtained by the iram method are less than 0 2 when the loss factor of the material damping is lower than 1 this is a considerable improvement on other approximate methods for higher damped systems an improvement to the previous approach is proposed by an incremental technique that keeps the accuracy without significantly increasing the computational time the complex eigenproblem for materials whose mechanical properties are dependent on frequency is solved by a fast iterative approach whose validity is proved using a four parameter fractional derivative model

a generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks the simulation of flow in fractured media requires handling both a large number of fractures and a complex interconnecting network of these fractures networks considered in this paper are three dimensional domains made up of two dimensional fractures intersecting each other and randomly generated due to the stochastic generation of fractures intersections can be highly intricate the numerical method must generate a mesh and define a discrete problem for any discrete fracture network dfn a first approach erhel de dreuzy and poirriez siam j sci comput 31 2009 pp 2688 2705 is to generate a conforming mesh and to apply a mixed hybrid finite element method however the resulting linear system becomes very large when the network contains many fractures hence a second approach pichot erhel and de dreuzy appl anal 89 2010 pp 1629 1643 is to generate a nonconforming mesh using an independent mesh generation for each fracture then a mortar technique applied to the mixed hybrid finite element method deals with the nonmatching grids when intersections do not cross or overlap pairwise mortar relations for each intersection are efficient pichot erhel and de dreuzy 2010 but for most random networks discretized intersections involve more than two fractures in this paper we design a new method generalizing the previous one and that is applicable for stochastic networks the main idea is to combine pairwise mortar relations with additional relations for the overlapping part this method still ensures the continuity of fluxes and heads and still yields a symmetric positive definite linear system numerical experiments show the efficiency of the method applied to complex stochastic fracture networks we also study numerical convergence when reducing the mesh step this method makes it easy to perform mesh optimization and appears to be a very promising tool to simulate flow in multiscale fracture networks

cross sectional analysis of composite beams including large initial twist and curvature effects an asymptotically exact methodology based on geometrically nonlinear three dimensional elasticity is presented for cross sectional analysis of initially curved and twisted nonhomogeneous anisotropic beams the analysis is subject only to the restrictions that the strain is small relative to unity and that the maximum dimension of the cross section is small relative to the wavelength of the deformation and to the minimum radius of curvature and or twist the final one dimensional strain energy per unit length exhibits asymptotically correct second order dependence of the initial curvature and twist parameters cross sectional constants of the one dimensional theory are obtained via finite element discretization over the cross sectional plane numerical results obtained for both isotropic and composite beams are compared with published results from special purpose analyses for initially twisted straight beams as well as initially curved untwisted beams the agreement with previously published results is excellent

a cut finite element method for a model of pressure in fractured media we develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks the crack has its own pressure field and can cut through

quasi static and dynamic response of viscoelastic helical rods in this study the dynamic behaviour of cylindrical helical rods made of linear viscoelastic materials are investigated in the laplace domain the governing equations for naturally twisted and curved spatial rods obtained using the timoshenko beam theory are rewritten for cylindrical helical rods the curvature of the rod axis effect of rotary inertia and shear and axial deformations are considered in the formulation the material of the rod is assumed to be homogeneous isotropic and linear viscoelastic in the viscoelastic material case according to the correspondence principle the material constants are replaced with their complex counterparts in the laplace domain ordinary differential equations in scalar form obtained in the laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem in the solutions the kelvin model is employed the solutions obtained are transformed to the real space using the durbin s numerical inverse laplace transform method numerical results for quasi static and dynamic response of viscoelastic models are presented in the form of graphics

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

complex modal analysis of rods with viscous damping devices abstract the complex modal analysis of rods equipped with an arbitrary number of viscous damping devices is addressed the following types of damping devices are considered external grounded spring damper attached mass spring damper and internal spring damper within a standard 1d formulation of the vibration problem the theory of generalized functions is used to model axial stress and displacement discontinuities at the locations of the damping devices by using the separate variable approach a simple solution procedure of the motion equation leads to exact closed form expressions of the characteristic equation and eigenfunctions which inherently fulfill the required matching conditions at the locations of the damping devices based on the characteristic equation a closed form sensitivity analysis of the eigensolution is implemented the displacement eigenfunctions exhibit orthogonality conditions they can be used with the complex mode superposition principle to tackle forced vibration problems and in conjunction with the stress eigenfunctions to build the exact dynamic stiffness matrix of the rod for complex modal analysis of truss structures numerical results are discussed for a variety of parameters

dynamic stiffness formulation free vibration and wave motion of helical springs a coil spring can only be treated as a simple massless force element at low frequencies the effects of internal resonances leading to significant dynamic stiffening for an automotive suspension spring this occurs at frequencies as low as about 40 hz this paper presents an efficient method for calculating the dynamic stiffness of a helical coil spring the partial differential equations of motion are used to derive the relation between wavenumber and frequency along with the associated wave shapes by expressing the response in terms of these waves the dynamic stiffness matrix is assembled natural frequencies are obtained from the reduced stiffness matrix allowing for different boundary conditions making use of the wittrick williams algorithm the results of the dynamic stiffness method are compared with those of the transfer matrix method and the finite element method the nature of the wave propagation is also investigated although at low frequencies four wave types propagate above a particular frequency only two propagating waves remain these are composite waves which are excited by both axial and transverse motion for lower values of helix angle an intermediate frequency range exists where six propagating waves can occur

dynamic analysis of linear viscoelastic cylindrical and conical helicoidal rods using the mixed fem abstract the objective of this study is to investigate the influence of the rotary inertia on dynamic behavior of linear viscoelastic cylindrical and conical helixes by means of the laplace transform mixed finite element formulation and solution the element matrix is based on the timoshenko beam theory the influence of rotary inertias is considered in the dynamic analysis which is original in the literature rectangular sine and step type of impulsive loads are applied on helices having rectangular cross sections with various aspect ratios the kelvin and standard models are used for defining the linear viscoelastic material behavior and by means of the correspondence principle the elastic viscoelastic analogy the material parameters are replaced with their complex counterparts in the laplace domain the analysis is carried out in the laplace domain and the results are transformed back to time space numerically by modified durbin s algorithm first the solution algorithm is verified using the existing open sources in the literature and afterwards some benchmark examples such as conical viscoelastic rods are handled

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

a local in space timestep approach to a finite element discretization of the heat equation with a posteriori estimates a new numerical method is presented for the heat equation with discontinuous coefficients based on a crank nicolson scheme and a conforming finite element space discretization in the proposed method each node of the spatial discretization may have the global timestep split into an arbitrary number of local substeps in order to pursue a local improvement of the time discretization in the regions of the spatial domain where the solution changes rapidly this method can possibly be used together with adaptive strategies for both the space discretization and the choice of timesteps to suitably produce an efficient space time discretizaton of the problem robust a posteriori upper and lower bounds of the error are proposed to attain this target moreover some indications are given on how to modify the mesh the timestep and the number of substeps to improve the discretization

derivation of equivalent pipe network analogues for three dimensional discrete fracture networks by the boundary element method discrete fracture network dfn models generally require solution of flow and transport equations in three dimensional networks of either disc polygonal or pipe elements pipe network elements have significant advantages in computation for both flow and transport however there is a need to develop an efficient procedure for derivation of the properties of these pipes to ensure that they are hydraulically equivalent to the dfn network of polygonal elements in this study a boundary element procedure for derivation of pipe properties is developed and demonstrated the results show that the hydraulic behavior of pipe networks can be equivalent to that of polygonal element dfn models

methods of interconversion between linear viscoelastic material functions part ii an approximate analytical method abstract a new very simple approximate interconversion method is proposed and verified by examples this technique employing the slope of the source function on logarithmic scales is found to substantially enhance the accuracy compared to existing approximate methods the new method is based on the characteristic mathematical properties of the narrow band weight functions involved in the interrelationships between broad band material functions with the material functions represented locally by a power law they are interrelated in terms of adjustment factors expressed through the local log log slope of the given source function a number of existing approximate interconversion methods are also tested and compared with the new method in part i park and schapery 1998 an efficient numerical interconversion method based on a prony exponential series representation of both the source and target functions was presented such a series representation is not needed here the new method when applied to the prediction of broad band time dependent functions from laplace or fourier transforms is an approximate method of transform inversion that is applicable to functions which are not necessarily viscoelastic material functions

on timoshenko like modeling of initially curved and twisted composite beams abstract a generalized finite element based cross sectional analysis for nonhomogenous initially curved and twisted anistropic beams is formulated from geometrically nonlinear three dimensional 3 d elasticity the 3 d strain field is formulated based on the concept of decomposition of the rotation tensor and is given in terms of one dimensional 1 d generalized strains and a 3 d warping displacement that is obtained from the formulation not assumed the warping is found in terms of the 1 d strains via the variational asymptotic method vam in this paper a timoshenko like model is presupposed for a beam with cross sectional characteristic length h wavelength of deformation given by l and the magnitude of the radius of initial curvature and or twist is taken to be of the order r first a solution for the asymptotically correct refinement of classical anisotropic beam theory for initially curved and twisted beams through o h2 r2 is obtained next the o h2 l2 correction is computed it is known that timoshenko like theory is not capable of capturing all the o h2 l2 corrections for generally anisotropic beams however if all the o h2 l2 terms are known then the corresponding timoshenko like theory is uniquely defined numerical results are presented to illustrate the trends of the various classical extension twist bending twist and extension bending and nonclassical couplings extension shear bending shear and shear torsion as the initial twist and curvatures are varied

power law damping parameter identification abstract the parameter estimation of a nonlinear power damping system is studied the parameter identification method used here assumes a priori the equation of motion describing the system dynamics the method which is based on the measured data acceleration was applied to the free and forced vibrations the identification procedure was found to be robust on the guessed value of parameters at the numerical experimentation the parameter values were estimated with a good accuracy for both modes of system operation free and forced if only the measured time history was sampled at a high enough rate for the noise level contained within it was shown that the steady state of the harmonically excited system is not the best region for the parameter identification with this method during the experimentation the method was applied to the free vibrations in different media air and water the results obtained by the parameter identification method were compared to the ones obtained by separate tests and good agreement was found the identification procedure was found to work fine for all models under consideration and the models responses correspond well to the measured acceleration time histories

steady state response of axially moving viscoelastic beams with pulsating speed comparison of two nonlinear models abstract principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed a nonlinear partial differential equation governing the transverse vibration is derived from the dynamical constitutive and geometrical relations under certain assumption the partial differential reduces to an integro partial differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams the method of multiple scales is applied to two equations to calculate the steady state response closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms the stability of straight equilibrium and nontrivial steady state response are analyzed by use of the lyapunov linearized stability theory numerical examples are presented to highlight the effects of speed pulsation viscoelascity and nonlinearity and to compare results obtained from two equations

numerical analysis of a two phase flow discrete fracture matrix model we present a new model for two phase darcy flows in fractured media in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain matrix fractures can act as drains or as barriers since pressure discontinuities at the matrix fracture interfaces are permitted additionally a layer of damaged rock at the matrix fracture interfaces is accounted for the numerical analysis is carried out in the general framework of the gradient discretisation method compactness techniques are used to establish convergence results for a wide range of possible numerical schemes the existence of a solution for the two phase flow model is obtained as a byproduct of the convergence analysis a series of numerical experiments conclude the paper with a study of the influence of the damaged layer on the numerical solution

flow simulation in three dimensional discrete fracture networks in fractured rocks fluid flows mostly within a complex arrangement of fractures both the fracture network structure and its hydraulic properties are determined at first order by the broad range of fracture lengths and densities to handle the observed wide variety of fracture properties and the lack of direct fracture visualization we develop a general and efficient stochastic numerical model for discrete fracture networks dfns in a three dimensional 3d computational domain we present an original conforming mesh generation method addressing the penalizing configurations stemming from close fractures and acute angles between fracture intersections flows are subsequently computed by using a mixed hybrid finite element mhfe method the lack of direct fracture knowledge is treated by monte carlo simulations requiring simulations with a large number of networks with various characteristics we analyze the complexity in size and in time for the computation of flow in 3d dfns meshed with our method and compare with the complexities for 2d rectangular domains meshed with a regular grid we find out that complexity in size is similar whereas complexity in time is slightly larger for dfns than for 2d regular domains

physics based preconditioners for flow in fractured porous media discrete fracture models are an attractive alternative to upscaled models for flow in fractured media as they provide a more accurate representation of the flow characteristics a major challenge in discrete fracture simulation is to overcome the large computational cost associated with resolving the individual fractures in large scale simulations in this work two characteristics of the fractured porous media are utilized to construct efficient preconditioners for the discretized flow equations first the preconditioners are tailored to the fracture geometry and presumed flow properties so that the dominant features are well represented there this assures good scalability of the preconditioners in terms of problem size and permeability contrast for fracture dominated problems numerical examples show that such geometric preconditioners are comparable or preferable when compared to state of the art algebraic multigrid preconditioners the robustness of the physics based preconditioner for less favorable fracture conditions is further demonstrated by a systematic degradation of the fracture hierarchy second the preconditioners are physics preserving in the sense that conservative fluxes can be computed even for an inexact pressure solutions this facilitates a scheme where accuracy in the linear solver can be traded for efficiency by terminating the iterative solvers based on error estimates and without sacrificing basic physical modeling principles with the combination of these two properties a novel preconditioner is obtained which bridges the gap between multiscale approximations and iterative linear solvers

a hybrid high order method for a coupled stokes darcy problem on general meshes abstract in this work a hybrid high order hho method on general meshes is presented to solve a coupled stokes darcy problem with the beavers joseph saffman interface condition constructed on polynomials of arbitrary degree k 0 the numerical method is established in terms of discrete unknowns attached to mesh faces and cells or elements the unified discrete scheme for stokes equation and darcy equation is given by the continuity condition of the interface the unique solvability of the discrete scheme is proved moreover the energy error estimate for the velocity and l 2 error estimate for pressure of order k 1 are derived finally a series of numerical experiments are reported to illustrate the accuracy mass conservation and robustness of our method

uncertainty quantification in discrete fracture network models we consider flows in fractured media described by discrete fracture network dfn models we perform an uncertainty quantification analysis assuming the fractures transmissivity coefficients to be random variables two probability distributions log uniform and log normal are used within different laws that express the coefficients in terms of a family of independent stochastic variables truncated karhunenlove expansions provide instances of such laws the approximate computation of quantities of interest such as mean value and variance for outgoing fluxes is based on a stochastic collocation approach that uses suitable sparse grids in the range of the stochastic variables whose number defines the stochastic dimension of the problem this produces a non intrusive computational method in which the dfn flow solver is applied as a black box a very fast error decay related to the analytical dependence of the observed quantities upon the stochastic variables is obtained in the low dimensional cases using isotropic sparse grids comparisons with monte carlo results show a clear gain in efficiency for the proposed method for increasing dimensions attained via successive truncations of karhunenlove expansions results are still good although the rates of convergence are progressively reduced resorting to suitably tuned anisotropic grids is an effective way to contrast such curse of dimensionality in the explored range of dimensions the resulting convergence histories are nearly independent of the dimension

calculation of the taylor series expansion coefficients of the jacobian elliptic function introduction not much is known about the taylor series expansion coefficients of the jacobian elliptic functions sn x k cn x k and dn x k in handbooks only the first four or five terms are given see for instance abramowitz and stegun 1 p 575 hancock s book on elliptic functions 4 p 252 and p 486 or gradshteyn and ryshik 5 recently however alois schett gave a combinatorial expression of the coefficients of the taylor series expansion of sn x k and evaluated explicitly the first eight nontrivial terms see 6 p 146 his results were extended by dominique dumont 3 who gave a new combinatorial interpretation of the coefficients of sn x k it is often said that no recurrence formulae exist for these coefficients 1 p 575 or 6 p 143 what is meant by that is probably that no simple recurrence formulae exist using differential and other equations one can deduce a number of useful recurrence relations and with the help of the burmann lagrange theorem even a definite although hard to handle expression of the coefficients in question

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

a discontinuous finite volume method for a coupled fracture model abstract in this paper we present a discontinuous finite volume method for a hybrid dimensional coupled fracture model on triangular meshes in matrix and fracture the pressure is approximated by the discontinuous and continuous piecewise linear elements and in order to retain local conservation property the velocity is sought in the discontinuous lowest order raviart thomas element space and piecewise linear element space the proposed numerical scheme is only associated with the pressure so we avoid solving an indefinite saddle point problem in addition the discontinuous method eliminates the continuity of approximate functions across the interelement boundary so it has the advantage of high localizability and easy handling of complicated geometries optimal order error estimates for the pressure and velocity are proved and numerical experiments are carried out to confirm our theoretical analysis

eliminating the modal truncation problem encountered in frequency responses of viscoelastic systems increasing the number of degrees of freedom used in finite element analysis for mechanical and structural systems with viscoelastic damping the need to consider the modal truncation problem of viscoelastic systems is more than ever before the higher modes may be unnecessary to obtain in dynamic analysis for engineering applications for viscoelastic systems the modal truncation problem may be more frequently encountered since the nonviscous modes are difficult or even impossible to be found accurately even if a small scaled problem is considered for some eigensolution methods this study aims at eliminating the influence of the higher modes on the frequency responses of viscoelastically damped systems a method is presented by making the equilibrium equations of motion into a subspace equation spanned in terms of the columns of a projection basis obtained by considering the use of the contribution of the lower modes and the first two terms of the neumann expansion of the contribution of the unavailable modes finally three example studies are provided to illustrate the effectiveness of the derived results it is shown that the proposed method can reduce the modal truncation error significantly

towards effective flow simulations in realistic discrete fracture networks we focus on the simulation of underground flow in fractured media modeled by means of discrete fracture networks focusing on a new recent numerical approach proposed by the authors for tackling the problem avoiding mesh generation problems we further improve the new family of methods making a step further towards effective simulations of large multi scale heterogeneous networks namely we tackle the imposition of dirichlet boundary conditions in weak form in such a way that geometrical complexity of the dfn is not an issue we effectively solve dfn problems with fracture transmissivities spanning many orders of magnitude and approaching zero furthermore we address several numerical issues for improving the numerical solution also in quite challenging networks

traces extensions and co normal derivatives for elliptic systems on lipschitz domains this is the post print version of the article the official published version can be accessed from the link below copyright 2011 elsevier

a combined approach of the laplace transform and pade approximation solving viscoelasticity problems in this paper the laplace transform method is combined with pade approximations to solve linear viscoelastic problems this approach allows to avoid the usual difficulties of original function determination an algorithm is given to find solution with arbitrary precision as an example the solution for problem of viscoelastic orthotropic half plane stress state under concentrated normal force is given

formal upscaling and numerical validation of unsaturated flow models in fractured porous media abstract in this work we consider a mathematical model for describing flow in an unsaturated porous medium containing a fracture both the flow in the fracture as well as in the matrix blocks are governed by richards equation coupled by natural transmission conditions using formal asymptotics we derive upscaled models as the limit of vanishing e the ratio of the width and length of the fracture our results show that the ratio of porosities and permeabilities in the fracture to matrix determine to the leading order of approximation the appropriate effective model in these models the fracture is a lower dimensional object for which different transversally averaged models are derived depending on the ratio of the porosities and permeabilities of the fracture and respective matrix blocks we obtain a catalogue of effective models which are validated by numerical computations

exact steady states of the periodically forced and damped duffing oscillator abstract we present an analytical methodology to compute the exact steady states of the strongly nonlinear duffing oscillator with arbitrary viscous damping and under the action of a time periodic excitation in contrast to current analytical approaches that consider harmonic excitations to produce approximate steady state solutions the excitations considered in this work are multi harmonic and are computed in closed form we consider the exact fundamental resonance plot of this strongly nonlinear system and present an exact depiction of the dynamic balance between the internal and external forces leading to it in terms of rotating vectors in the complex plane with modulated amplitudes and modulated frequencies of rotation moreover an infinity of exact steady state solutions seems to be possible each corresponding to a different time periodic excitation generalization of the presented methodology to systems with other type of nonlinearities or with many degrees of freedom is presumably straightforward

geometric multigrid methods for darcy forchheimer flow in fractured porous media in this paper we present a monolithic multigrid method for the efficient solution of flow problems in fractured porous media specifically we consider a mixed dimensional model which couples darcy flow in the porous matrix with forchheimer flow within the fractures a suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure in order to solve this system we propose a full approximation scheme fas multigrid solver that appropriately deals with the mixed dimensional nature of the problem by using mixed dimensional smoothing and inter grid transfer operators numerical experiments show that the proposed multigrid method is robust with respect to the fracture permeability the forchheimer coefficient and the mesh size the case of several possibly intersecting fractures in a heterogeneous porous medium is also discussed

an efficient multi point flux approximation method for discrete fracture matrix simulations we consider a control volume discretization with a multi point flux approximation to model discrete fracture matrix systems for anisotropic and fractured porous media in two and three spatial dimensions inspired by a recently introduced approach based on a two point flux approximation we explicitly account for the fractures by representing them as hybrid cells between the matrix cells as well as simplifying the grid generation our hybrid approach excludes small cells in the intersection of the fractures and hence avoids severe time step restrictions associated with small cells excluding the small cells also reduces the condition number of the discretization matrix for examples involving realistic anisotropy ratios in the permeability numerical results show significant improvement compared to existing methods based on two point flux approximations we also investigate the hybrid method by studying the convergence rates for different apertures and fracture matrix permeability ratios finally the effect of removing the cells in the intersections of the fractures are studied together these examples demonstrate the efficiency flexibility and robustness of our new approach

gradient discretization of hybrid dimensional darcy flows in fractured porous media this article deals with the discretization of hybrid dimensional darcy flows in fractured porous media these models couple the flow in the fractures represented as surfaces of codimension one with the flow in the surrounding matrix the convergence analysis is carried out in the framework of gradient schemes which accounts for a large family of conforming and nonconforming discretizations the vertex approximate gradient scheme and the hybrid finite volume scheme are extended to such models and are shown to verify the gradient scheme framework our theoretical results are confirmed by numerical experiments performed on tetrahedral cartesian and hexahedral meshes in heterogeneous isotropic and anisotropic porous media

formulation and integration of the standard linear viscoelastic solid with fractional order rate laws abstract a physically sound three dimensional anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a finite set of the pertinent internal variables is presented it is shown that the internal variables can be expressed in terms of the strain as convolution integrals with kernels of mittag leffler function type a time integration scheme based on the generalized midpoint rule together with the grunwald algorithm for numerical fractional differentiation for integration of the constitutive response is developed the predictive capability of the viscoelastic model for describing creep relaxation and damped dynamic responses is investigated both analytically and numerically the algorithm and the present general linear viscoelastic model are implemented into the general purpose finite element code abaqus the algorithm is then used together with an explicit difference scheme for integration of structural responses in numerical examples the quasi static and damped responses of a viscoelastic ballast material that is subjected to loads simulating the overrolling of a train are investigated

parallel numerical modeling of hybrid dimensional compositional non isothermal darcy flows in fractured porous media this paper introduces a new discrete fracture model accounting for non isothermal compositional multiphase darcy flows and complex networks of fractures with intersecting immersed and non immersed fractures the so called hybrid dimensional model using a 2d model in the fractures coupled with a 3d model in the matrix is first derived rigorously starting from the equi dimensional matrix fracture model then it is dis cretized using a fully implicit time integration combined with the vertex approximate gradient vag finite volume scheme which is adapted to polyhedral meshes and anisotropic heterogeneous media the fully coupled systems are assembled and solved in parallel using the single program multiple data spmd paradigm with one layer of ghost cells this strategy allows for a local assembly of the discrete systems an efficient preconditioner is implemented to solve the linear systems at each time step and each newton type iteration of the simulation the numerical efficiency of our approach is assessed on different meshes fracture networks and physical settings in terms of parallel scalability nonlinear convergence and linear convergence

a new two scale computational model for hydromechanical coupling in jointed rocks abstract we develop a new computational model to describe hydro mechanical coupling in fractured rocks composed of a linear poroelastic biot medium and nonlinear elastic joints with constitutive response governed by the barton bandis bb law the model aims at capturing increase in stiffness induced by fracture closure during fluid withdrawal the nonlinear hydro mechanical formulation is constructed within the framework of the discrete fracture model with flow and geomechanical sub systems coupled through a sequential iterative algorithm the internal contact constraint arising from non overlapping between opposite fracture faces is enforced through the weak fulfillment of the bb law such a constraint is captured within the framework of the augmented lagrangian formulation where the non linear mechanical interaction is enforced adopting successive approximations of the lagrange multiplier interpreted as the contact pressure in the joint supplemented by a penalty component associated with the rock stiffness furthermore adopting a traditional flow based upscaling method macroscopic permeabilities are numerically reconstructed with magnitude strongly dependent on the local stress state such a mechanical dependence of the homogenized properties is represented in a discrete manner through pseudo coupling tables with enormous potential to be explored within a preprocessing stage in reservoir simulators to compute multipliers relative to a chosen pre stressed reference state where input data is available numerical simulations are performed for some fracture arrangements illustrating the potential of the formulation proposed herein in bridging hydromechanical coupling at different scales in jointed rocks

the hybrid laplace transform finite element method applied to the quasi static and dynamic analysis of viscoelastic timoshenko beams the quasi static and dynamic responses of a linear viscoelastic beam are solved numerically by using the hybrid laplace transform finite element method in the analysis the timoshenko beam theory which includes the transverse shear and rotatory inertia effect and conventional beam theory are used to solve this problem the temperature field is assumed to be constant and homogeneous and that the relaxation modulus has the form of the prony series in the hybrid method the laplace transform with respect to time is applied to the coupled equations and the finite element model is developed by applying hamilton s variational principle without any integral transformation the numerical results of quasi static and dynamic responses for the models of maxwell fluid and three parameter solid types are presented and discussed

a lagrange multiplier method for a stokes biot fluid poroelastic structure interaction model we study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium the free fluid is governed by the stokes equations while the flow in the poroelastic medium is modeled using the biot poroelasticity system equilibrium and kinematic conditions are imposed on the interface a mixed darcy formulation is employed resulting in continuity of flux condition of essential type a lagrange multiplier method is employed to impose weakly this condition a stability and error analysis is performed for the semi discrete continuous in time and the fully discrete formulations a series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters

dynamic stability in parametric resonance of axially accelerating viscoelastic timoshenko beams abstract this paper investigates dynamic stability of an axially accelerating viscoelastic beam undergoing parametric resonance the effects of shear deformation and rotary inertia are taken into account by the timoshenko thick beam theory the beam material obeys the kelvin model in which the material time derivative is used the axial speed is characterized as a simple harmonic variation about the constant mean speed the governing partial differential equations are derived from newton s second law euler s angular momentum principle and the constitutive relation the method of multiple scales is applied to the equations to establish the solvability conditions in summation and principal parametric resonances the sufficient and necessary condition of the stability is derived from the routh hurvitz criterion some numerical examples are presented to demonstrate the effects of related parameters on the stability boundaries

the green s matrix and the boundary integral equations for analysis of time harmonic dynamics of elastic helical springs helical springs serve as vibration isolators in virtually any suspension system various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions the method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time harmonic dynamics of elastic springs in the framework of bernoulli euler beam theory in this paper the derivations of the green s matrix of the somigliana s identities and of the boundary integral equations are presented the vibrational power transmission in an infinitely long spring is analyzed by means of the green s matrix the eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations in the course of analysis the essential features and advantages of the method of boundary integral equations are highlighted the reported analytical results may be used to study the time harmonic motion i

multicomponent fluid flow by discontinuous galerkin and mixed methods in unfractured and fractured media 1 a discrete fracture model for the flow of compressible multicomponent fluids in homogeneous heterogeneous and fractured media is presented in single phase in the numerical model we combine the mixed finite element mfe and the discontinuous galerkin dg methods we use the cross flow equilibrium concept to approximate the fractured matrix mass transfer the discrete fracture model is numerically superior to the single porosity model and overcomes limitations of the dual porosity models including the use of a shape factor the mfe method provides a direct and accurate approximation for the velocity field which is crucial for the convective terms in the flow equations the dg method associated with a slope limiter is used to approximate the species balance equations this method can capture the sharp moving fronts the calculation of the fracture fracture flux across three and higher intersecting fracture branches is a challenge in this work we provide an accurate approximation of these fluxes by using the mfe formulation numerical examples in unfractured and fractured media illustrate the efficiency and robustness of the proposed numerical model

an optimization approach for large scale simulations of discrete fracture network flows in recent papers 1 2 the authors introduced a new method for simulating subsurface flow in a system of fractures based on a pde constrained optimization reformulation removing all difficulties related to mesh generation and providing an easily parallel approach to the problem in this paper we further improve the method removing the constraint of having on each fracture a non empty portion of the boundary with dirichlet boundary conditions this way dirichlet boundary conditions are prescribed only on a possibly small portion of dfn boundary the proposed generalization of the method in 1 2 relies on a modified definition of control variables ensuring the non singularity of the operator on each fracture a conjugate gradient method is also introduced in order to speed up the minimization process

linear dynamics of elastic helical springs asymptotic analysis of wave propagation helical springs serve as vibration isolators in virtually any suspension system a variety of theories to describe the dynamic behaviour of these structural elements which involves interaction of flexural torsion and longitudinal waves can be found in the literature alongside this various approximate methods are employed to determine the eigenfrequencies of vibrations of springs in this paper the validity ranges of alternative theories are assessed by comparison of the location of the dispersion curves this paper also contains a rigorous asymptotic analysis of the exact dispersion equation with two small parameters being employed it allows for the identification of significant regimes of linear wave motion in a helical spring in each of these regimes simple formulae for wavenumbers are obtained by the dominant balance method and their validity ranges are checked against direct numerical solution mode shapes associated with each wavenumber are also analysed

hybrid dimensional modelling of two phase flow through fractured porous media with enhanced matrix fracture transmission conditions in this work we extend to two phase flow the single phase darcy flow model proposed in 26 12 in which the d 1 dimensional flow in the fractures is coupled with the d dimensional flow in the matrix three types of so called hybrid dimensional two phase darcy flow models are proposed they all account for fractures acting either as drains or as barriers since they allow pressure jumps at the matrix fracture interfaces the models also permit to treat gravity dominated flow as well as discontinuous capillary pressure at the material interfaces the three models differ by their transmission conditions at matrix fracture interfaces while the first model accounts for the nonlinear two phase darcy flux conservations the second and third ones are based on the linear single phase darcy flux conservations combined with different approximations of the mobilities we adapt the vertex approximate gradient vag scheme to this problem in order to account for anisotropy and heterogeneity aspects as well as for applicability on general meshes several test cases are presented to compare our hybrid dimensional models to the generic equi dimensional model in which fractures have the same dimension as the matrix leading to deep insight about the quality of the proposed reduced models

an inverse model and mathematical solution for inferring viscoelastic properties and dynamic deformations of heterogeneous structures abstract we proposed an adjoint based inverse method and mathematical solutions using lagrange multiplier theorem for inferring dynamic viscoelastic properties of heterogeneous structures with displacement observations to improve computation reliability and efficiency existing methods such as the one using finite difference approximated gradient may not be efficient and accurate enough for considering complex situations of the coupled effects of dynamic loading material deformation memory and structural heterogeneity the proposed method derives both gradient and hessian mathematically to satisfy the first order necessary and second order sufficient optimal conditions which has great advantages compared to other approaches we also proposed robust numerical algorithms for accurate and fast computations including a regularization to control reasonable parameter ranges a newton s method with hessian function to determine search direction for fast convenience and a modified armijo rule to find a stable step length we developed a galerkin time domain finite element method for numerical solutions of resulting partial differential equations and validated the model for a layered structure under dynamic loading tests results indicate that the proposed method has greatly improved computation accuracy and speed as compared to existing approaches it may be applied to a broad range of heterogeneous materials and structures at different length and time scales

dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractionalderivatives frame structures with viscoelastic dampers mounted on them are considered in this paper viscoelastic ve dampers are modelled using two three parameter fractional rheological models the structures are treated as elastic linear systems the equation of motion of the whole system structure with dampers is written in terms of state space variables the resulting matrix equation of motion is the fractional differential equation the proposed state space formulation is new and does not require matrices with huge dimensions the paper is devoted to determine the dynamic properties of the considered structures the nonlinear eigenvalue problem is formulated from which the dynamic parameters of the system can be determined the continuation method is used to solve the nonlinear eigenvalue problem moreover results of typical calculations are presented

methods of interconversion between linear viscoelastic material functions part i a numerical method based on prony series abstract an efficient and accurate numerical method of interconversion between linear viscoelastic material functions based on a prony series representation is presented and tested using experimental data from selected polymeric materials the method is straightforward and applicable to interconversion between modulus and compliance functions in time frequency and laplace transform domains good agreement is shown between solutions obtained from the method in different domains a detailed computational procedure and selection of values for parameters involved in the method are presented and illustrated in particular the effects of different choices of relaxation and retardation times on the accuracy of the method are discussed the mathematical efficiency associated with prony series representations of both source and target transient material functions is fully utilized the method is general enough to cover both viscoelastic solids and liquids the tensile relaxation data from polymethyl methacrylate pmma and the shear storage compliance data from polyisobutylene are used in illustrating the method in a companion paper schapery and park 1998 an extended approximate analytical interconversion method is presented

determination of mechanical spectra from experimental responses abstract a recursive computer algorithm was developed which generates line spectra from experimental response functions the method allows storing information on the mechanical properties of polymeric materials in a convenient way the algorithm also interconverts between relaxation and retardation spectra from the spectra any desired response function can then be recovered the algorithm essentially utilizes the fact that the kernel functions resemble step functions slightly different codes are used for each kernel function the appearance of negative relaxation or retardation lines is obviated mathematically such lines would be acceptable and they do not seriously affect reconstruction of responses within relaxation or retardation behavior however they would seriously interfere with interconversion between the two types of behavior and they would also pose problems in the interpretation of the spectra

impact of eliminating fracture intersection nodes in multiphase compositional flow simulation algebraic elimination of nodes at discrete fracture intersections via the star delta technique has proven to be a valuable tool for making multiphase numerical simulations more tractable and efficient this study examines the assumptions of the star delta technique and exposes its effects in a 3 d multiphase context for advective and dispersive diffusive fluxes key issues of relative permeability saturation capillary pressure kr s pc and capillary barriers at fracture fracture intersections are discussed this study uses a multiphase compositional finite difference numerical model in discrete fracture network dfn and discrete fracture matrix dfm modes it verifies that the numerical model replicates analytical solutions and performs adequately in convergence exercises conservative and decaying tracer one and two phase flow dfm and dfn domains the study culminates in simulations of a two phase laboratory experiment in which a fluid invades a simple fracture intersection the experiment and simulations evoke different invading fluid flow paths by varying fracture apertures as oil invades water filled fractures and as water invades air filled fractures results indicate that the node elimination technique as implemented in numerical model correctly reproduces the long term flow path of the invading fluid but that short term temporal effects of the capillary traps and barriers arising from the intersection node are lost

mechanism of complicated volume deformation in polymers and its fractional time based description abstract in tension and compression of polymers some abnormal phenomena of volume changes were often observed and its mechanism is still unclear in this paper based on the thermodynamics principle it is shown that the combined effects of internal energy and entropy are the basic reason of causing the anomaly in polymers moreover to describe the complicated deformation through a simple model and have a correct modelization of mechanical behaviors a volume strain model is presented in which the variation of volume strain resulted from the combined effects is characterized through a fractional integral expression comparison between fitting results and experimental results reveals the proposed model can describe the volume deformation of polymers

nonlinear vibration of viscoelastic beams described using fractional order derivatives abstract the problem of non linear steady state vibration of beams harmonically excited by harmonic forces is investigated in the paper the viscoelastic material of the beams is described using the zener rheological model with fractional derivatives the constitutive equation which contains derivatives of both stress and strain significantly complicates the solution to the problem the von karman theory is applied to take into account geometric nonlinearities amplitude equations are obtained using the finite element method together with the harmonic balance method and solved using the continuation method the tangent matrix of the amplitude equations is determined in an explicit form the stability of the steady state solution is also examined a parametric study is carried out to determine the influence of viscoelastic properties of the material on the beam s responses

analysis of the coupled navier stokes biot problem abstract we analyze a weak formulation of the coupled problem defining the interaction between a free fluid and a poroelastic structure the problem is governed by the time dependent incompressible navier stokes equations and the biot equations under a small data assumption existence and uniqueness results are proved and a priori estimates are provided

on simulations of discrete fracture network flows with an optimization based extended finite element method following the approach introduced in siam j sci comput 35 2013 pp b487 b510 we consider the formulation of the problem of fluid flow in a system of fractures as a pde constrained optimization problem with discretization performed using suitable extended finite elements the method allows independent meshes on each fracture thus completely circumventing meshing problems usually related to the discrete fracture network dfn approach the application of the method to dfns of medium complexity is fully analyzed here accounting for several issues related to viable and reliable implementations of the method in complex problems

analytical modeling of viscoelastic dampers for structural and vibration control different approaches to the mathematical modeling of viscoelastic dampers are addressed and their theoretical basis and performance are compared the standard mechanical model smm comprising linear springs and dashpots is shown to accurately describe the broad band rheological behavior of common viscoelastic dampers and be more efficient than other models such as the fractional derivative model and the modified power law the smm renders a prony series expression for the modulus and compliance functions in the time domain and the remarkable mathematical efficiency associated with the exponential basis functions of a prony series greatly facilitates model calibration and interconversion while cumbersome nonlinear regression is usually required for other models a simple collocation or least squares method can be used to fit the smm to available experimental data the model allows viscoelastic material functions to be readily determined either directly from the experimental data or through interconversion from a function established in another domain numerical examples on two common viscoelastic dampers demonstrate the advantages of the smm over fractional derivative and power law models detailed computational procedures for fitting and interconversion are discussed and illustrated published experimental data from a viscoelastic liquid damper and a viscoelastic solid damper are used in the examples

an unfitted finite element method for the darcy problem in a fracture network abstract the paper develops an unfitted finite element method for solving the darcy system of equations posed in a network of fractures embedded in a porous matrix the approach builds on the hughes masud stabilized formulation of the darcy problem and the trace finite element method the system of fractures is allowed to cut through the background mesh in an arbitrary way moreover the fractures are not triangulated in the common sense and the junctions of fractures are not fitted by the mesh to couple the flow variables at multiple fracture junctions we extend the hughes masud formulation by including penalty terms to handle interface conditions one observation made here is that by over penalizing the pressure continuity interface condition one can avoid including additional jump terms along the fracture junctions this simplifies the formulation while ensuring the optimal convergence order of the method the application of the trace finite element method allows to treat both planar and curvilinear fractures with the same ease the paper presents convergence analysis and assesses the performance of the method in a series of numerical experiments for the background mesh we use an octree grid with cubic cells the flow in the fracture can be easily coupled with the flow in matrix but we do not pursue the topic of discretizing such coupled system here

a pde constrained optimization formulation for discrete fracture network flows we investigate a new numerical approach for the computation of the three dimensional flow in a discrete fracture network that does not require a conforming discretization of partial differential equations on complex three dimensional systems of planar fractures the discretization within each fracture is performed independently of the discretization of the other fractures and of their intersections an independent meshing process within each fracture is a very important issue for practical large scale simulations making mesh generation easier some numerical simulations are given to show the viability of the method the resulting approach can be naturally parallelized for dealing with systems with a huge number of fractures

orbital stability of periodic traveling wave solutions for the kawahara equation in this paper we investigate the orbital stability of periodic traveling waves for the kawahara equation we prove that the periodic traveling wave under certain conditions minimizes a convenient functional by using an adaptation of the method developed by grillakis et al j funct anal 74 160 197 1987 the required spectral properties to ensure the orbital stability are obtained by knowing the positiveness of the fourier transform of the associated periodic wave established by angulo and natali siam j math anal 40 1123 1151 2008