on the numerical solution of two point discrete boundary value problems we propose several different methods to construct the solutions of linear two point discrete boundary value problems these methods are applied to solve some continuous boundary value problems which are known to be numerically unstable

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

solving quasi static equations with the material point method summary the material point method models continua by following a set of unconnected material points throughout the deformation of a body this set of points provides a lagrangian description of the material and geometry information from the material points is projected onto a background grid where equations of motion are solved the grid solution is then used to update the material points this paper describes how to use this method to solve quasi static problems the resulting discrete equations are a coupled set of nonlinear equations that are then solved with a jacobian free newton krylov algorithm the technique is illustrated by examining two problems the first problem simulates a compact tension test and includes a model of material failure the second problem computes effective macroscopic properties of a polycrystalline thin film copyright 2015 john wiley sons ltd

computational methods for discrete boundary value problems on considere des methodes constructives pour les problemes aux valeurs limites discrets y t 1 g t y t t i n f y 0 ou y et g t y sont des vecteurs a n dimensions i n est l ensemble des points discrets 0 1 n f y est un operatuer de b i n dans r n et b i n est l espace de toutes les fonctions n vectorielles reelles definies sur i n

initial value methods for discrete boundary value problems motivated by the work of angel and kalaba s and golberg 8 on twopoint boundary value problems for difference equations in this paper we shall study several initial value methods for multipoint boundary value problems for linear and nonlinear difference systems we generalize method of adjoints goodman and lance 9 method of complementary functions tifford 17 d an method of particular solutions miele 111 proposed for two point continuous problems to linear multipoint problems for difference equations next we show that method of adjoint and method of complementary functions can be used in an iterative way to solve nonlinear problems to justify the results obtained here several examples are also given the methods discussed here are similar to our earlier work l 3 on continuous problems we will take up the initial adjusting method of ojika kasue and welsh 12 15 18 for the discrete case later throughout the paper we shall assume the existence uniqueness and stability properties of solutions of the problems discussed and refer to 4 for some of these questions

enhanced local maximum entropy approximation for stable meshfree simulations abstract we introduce an improved meshfree approximation scheme which is based on the local maximum entropy strategy as a compromise between shape function locality and entropy in an information theoretical sense the improved version is specifically designed for severe finite deformation and offers significantly enhanced stability as opposed to the original formulation this is achieved by i formulating the quasistatic mechanical boundary value problem in a suitable updated lagrangian setting ii introducing anisotropy in the shape function support to accommodate directional variations in nodal spacing with increasing deformation and eliminate tensile instability iii spatially bounding and evolving shape function support to restrict the domain of influence and increase efficiency iv truncating shape functions at interfaces in order to stably represent multi component systems like composites or polycrystals the new scheme is applied to benchmark problems of severe elastic and elastoplastic deformation that demonstrate its performance both in terms of accuracy as compared to exact solutions and where applicable finite element simulations and efficiency importantly the presented formulation overcomes the classical tensile instability found in most meshfree interpolation schemes as shown for stable simulations of e g the inhomogeneous extension of a hyperelastic block up to 100 or the torsion of a hyperelastic cube by 200 both in an updated lagrangian setting and without the need for remeshing

a fixed mesh method for incompressible flow structure systems with finite solid deformations a fixed mesh algorithm is proposed for simulating flow structure interactions such as those occurring in biological systems in which both the fluid and solid are incompressible and the solid deformations are large several of the well known difficulties in simulating such flow structure interactions are avoided by formulating a single set of equations of motion on a fixed eulerian mesh the solid s deformation is tracked to compute elastic stresses by an overlapping lagrangian mesh in this way the flow structure interaction is formulated as a distributed body force and singular surface force acting on an otherwise purely fluid system these forces which depend on the solid elastic stress distribution are computed on the lagrangian mesh by a standard finite element method and then transferred to the fixed eulerian mesh where the joint momentum and continuity equations are solved by a finite difference method the constitutive model for the solid can be quite general for the force transfer standard immersed boundary and immersed interface methods can be used and are demonstrated we have also developed and demonstrated a new projection method that unifies the transfer of the surface and body forces in a way that exactly conserves momentum the interface is still effectively sharp for this approach the spatial convergence of the method is observed to be between first and second order as in most immersed boundary methods for membrane flows the algorithm is demonstrated by the simulations of an advected elastic disk a flexible leaflet in an oscillating flow and a model of a swimming jellyfish

orbital hp clouds for solving schrodinger equation in quantum mechanics solving schrodinger equation in quantum mechanics presents a challenging task in numerical methods due to the high order behavior and high dimension characteristics in the wave functions in addition to the highly coupled nature between wave functions this work introduces orbital and polynomial enrichment functions to the partition of unity for solution of schrodinger equation under the framework of hp clouds an intrinsic enrichment of orbital function and extrinsic enrichment of monomial functions are proposed due to the employment of higher order basis functions a higher order stabilized conforming nodal integration is developed the proposed methods are implemented using the density functional theory for solution of schrodinger equation analysis of several single and multi electron nucleus structures demonstrates the effectiveness of the proposed method

computational methods for discrete boundary value problems ii abstract for a given nonlinear discrete system together with nonlinear boundary conditions we use newton s iterative scheme and provide sufficient conditions for its convergence to the unique solution then the computational aspects of this iterative method on a floating point system are discussed this includes necessary and sufficient conditions for the convergence of the approximate newton s method an oscillatory state and the stopping criterion to make the analysis widely applicable all the results are proved component wise

mass matrix formulation of the flip particle in cell method a refinement of flip brackbill and ruppel j comput phys 65 314 1986 is described which uses a mass matrix formulation to achieve greater accuracy and less numerical diffusion over the previous version without the refinement there is a significant dissipation of energy in modeling elastic vibrations of a solid moreover in modeling an initial flow discontinuity there are sub grid scale oscillations in the particle velocity field in the neighborhood of the discontinuity these difficulties are eliminated using the mass matrix in addition the mass matrix formulation conserves kinetic energy linear and angular momentum and is galilean invariant

efficient spectral sparse grid methods and applications to high dimensional elliptic equations ii unbounded domains this is the second part in a series of papers on using spectral sparse grid methods for solving higher dimensional pdes we extend the basic idea in the first part j shen and h yu siam j sci comp 32 2010 pp 3228 3250 for solving pdes in bounded higher dimensional domains to unbounded higher dimensional domains and apply the new method to solve the electronic schrodinger equation by using modified mapped chebyshev functions as basis functions we construct mapped chebyshev sparse grid methods which enjoy the following properties i the mapped chebyshev approach enables us to build sparse grids with smolyak s algorithms based on nested spectrally accurate quadratures and allows us to build fast transforms between the values at the sparse grid points and the corresponding expansion coefficients ii the mapped chebyshev basis functions lead to identity mass matrices and very sparse stiffness matrices for problems with constant coefficients and allow us to construct a matrix vector product algorithm with quasi optimal computational cost even for problems with variable coefficients and iii the resultant linear systems for elliptic equations with constant or variable coefficients can be solved efficiently by using a suitable iterative scheme ample numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithms

boundary value problems for general discrete systems on infinite intervals this paper continues the study of boundary value problems for a discrete system which in particular includes the prototype equation x k 1 f k x k equations with finite as well as infinite delays equations of neutral type and the discrete integral equations of volterra type while the finite discrete interval case regular as well as at resonance has been discussed in 1 2 in the present paper the infinite interval case will be addressed

an implicit particle in cell method for granular materials an implicit in time method for granular materials is described the method combines the material point method a first order contact algorithm and a newton krylov equation solver to give improved energy conservation stabilization of the finite grid instability and the correct description of collisions between grains

material point methods applied to one dimensional shock waves and dual domain material point method with sub points using a simple one dimensional shock problem as an example the present paper investigates numerical properties of the original material point method mpm the generalized interpolation material point gimp method the convected particle domain interpolation cpdi method and the dual domain material point ddmp method for a weak isothermal shock of ideal gas the mpm cannot be used with accuracy with a small number of particles per cell gimp and cpdi produce reasonable results however as the number of particles increases the methods fail to converge and produce pressure spikes the ddmp method behaves in an opposite way with a small number of particles per cell ddmp results are unsatisfactory as the number of particles increases the ddmp results converge to correct solutions but the large number of particles needed for convergence makes the method very expensive to use in these types of shock wave problems in two or three dimensional cases the cause for producing the unsatisfactory ddmp results is identified a simple improvement to the method is introduced by using sub points with this improvement the ddmp method produces high quality numerical solutions with a very small number of particles although in the present paper the numerical examples are one dimensional all derivations are for multidimensional problems with the technique of approximately tracking particle domains of cpdi the extension of this sub point method to multidimensional problems is straightforward this new method preserves the conservation properties of the ddmp method which conserves mass and momentum exactly and conserves energy to the second order in both spatial and temporal discretizations

a mesh grading material point method and its parallelization for problems with localized extreme deformation abstract as a kind of meshless method material point method mpm applies an eulerian background grid served as a finite element mesh in each time step and therefore its accuracy and efficiency are mainly dependent on the cell size setting of background grid however the conventional mpm commonly uses a regular background grid with uniform cells which is not apposite for localized extreme deformation problems from the view point of computation efficiency where in fact a local refined background grid is preferable hence a mesh grading material point method mgmpm is proposed here for such problems to supply mpm with the ability for local refinement simulation the edge displacement continuity associated with mesh grading is embedded in the nodal shape functions besides the truss element is incorporated into mgmpm to model the steel reinforcement bars in reinforced concrete impacting problems based on our previous work furthermore the proposed method is parallelized using openmp open multi processing to take advantage of pc power with multi core and hyper threading technologies for large scale engineering problems where both loop level parallelism and code block parallelism are used several numerical examples including stress wave propagation taylor bar impact and penetration problems are studied which show that the efficiency of mgmpm is much higher than that of conventional mpm and with lower memory requirement

a particle method for history dependent materials abstract a broad class of engineering problems including penetration impact and large rotations of solid bodies causes severe numerical problems for these problems the constitutive equations are history dependent so material points must be followed this is difficult to implement in a eulerian scheme on the other hand purely lagrangian methods typically result in severe mesh distortion and the consequence is ill conditioning of the element stiffness matrix leading to mesh lockup or entanglement remeshing prevents the lockup and tangling but then interpolation must be performed for history dependent variables a process which can introduce errors proposed here is an extension of the particle in cell method in which particles are interpreted to be material points that are followed through the complete loading process a fixed eulerian grid provides the means for determining a spatial gradient because the grid can also be interpreted as an updated lagrangian frame the usual convection term in the acceleration associated with eulerian formulations does not appear with the use of maps between material points and the grid the advantages of both eulerian and lagrangian schemes are utilized so that mesh tangling is avoided while material variables are tracked through the complete deformation history example solutions in two dimensions are given to illustrate the robustness of the proposed convection algorithm and to show that typical elastic behavior can be reproduced also it is shown that impact with no slip is handled without any special algorithm for bodies governed by elasticity and strain hardening plasticity

monotone methods for a boundary value problem of second order discrete equation abstract the comparison and existence results of the upper and lower solutions are established for a boundary value problem of second order discrete equation a monotone iterative scheme is proposed the local as well as global existence and uniqueness of the solution are obtained the global convergence of the iteration is investigated the convergence rate of the iteration is analyzed and compared some examples are given

an adaptive finite element material point method and its application in extreme deformation problems abstract taking advantages of both lagrangian and eulerian methods material point method mpm is suitable for modeling problems with extreme deformation however mpm is less accurate and less efficient than finite element method fem for small deformation problems due to particle quadrature and mappings between particles and background grid applied in mpm to take advantages of both fem and mpm an adaptive finite element material point method is developed for modeling the dynamic behavior of material under extreme loading bodies are initially discretized by finite elements and then the elements with large strain are adaptively converted into mpm particles based on their degree of distortion or plastic strain during the solution process the interaction between the remaining finite elements and mpm particles is implemented based on the background grid in mpm framework several numerical examples are presented to validate the efficiency and accuracy of the proposed method and the numerical results are in good agreement with experiments while the efficiency of the method is higher than that of both mpm and fem

imposition of essential boundary conditions in the material point method summary there is increasing interest in the material point method mpm as a means of modelling solid mechanics problems in which very large deformations occur e g in the study of landslides and metal forming however some aspects vital to wider use of the method have to date been ignored in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges in this paper we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions to the mpm expanding its capabilities to boundaries of any inclination to the authors knowledge this is the first time that a method has been proposed that allows arbitrary dirichlet boundary conditions zero and non zero values at any inclination to be imposed in the mpm the method presented in this paper is different from other mpm boundary approximation approaches in that i the boundaries are independent of the background mesh ii artificially stiff regions of material points are avoided and iii the method does not rely on spurious mirroring of the problem domain to imposed symmetry the main contribution of this work is equally applicable to standard finite elements and the mpm this article is protected by copyright all rights reserved

an analytical and numerical study of failure waves based on recent observations in shock experiments on glasses a new failure process has been suggested for a certain type of brittle solids in which a failure wave propagates through a solid at some distance behind the compressive stress wave near but below the hugoniot elastic limit since the failure wave phenomenon is different from the usual inelastic shock waves a combined analytical and numerical effort is made in this paper to explore the impact failure mechanisms associated with the failure wave based on the experimental data available it appears that the physical picture of failure wave is related to local dilatation due to shear induced microcracking a mathematical argument then leads to the conclusion that the failure wave should be described by a diffusion equation instead of a wave equation which is in line with the bifurcation analysis for localization problems however the occurrence of different governing equations in a single computational domain imposes both an analytical and a numerical challenge on the design of an efficient solution scheme with the use of a partitioned modeling approach a simple solution procedure is proposed for failure wave problems which is verified by the comparison with data

geometrically exact time integration mesh free schemes for advection diffusion problems derived from optimal transportation theory and their connection with particle methods we develop an optimal transportation mesh free particle method for advection diffusion in which the concentration or density of the diffusive species is approximated by dirac measures we resort to an incremental variational principle for purposes of time discretization of the diffusive step this principle characterizes the evolution of the density as a competition between the wasserstein distance between two consecutive densities and entropy exploiting the structure of the euler lagrange equations we approximate the density as a collection of diracs the interpolation of the incremental transport map is effected through mesh free max ent interpolation remarkably the resulting update is geometrically exact with respect to advection and volume we present three dimensional examples of application that illustrate the scope and robustness of the method

the material point method for simulation of thin membranes the material point method mpm is extended to handle membranes which are discretized by a collection of unconnected material points placed along each membrane surface these points provide a lagrangian description of the membrane to solve for the membrane motion data carried by the material points are transferred to a background mesh where equations of motion are discretized and solved then the solution on the background mesh is used to update the membrane material points this process of combining lagrangian and eulerian features is standard in mpm the modification for membranes involves merely an implementation of the constitutive equation in a local normal tangential coordinate system it is shown that this procedure does in fact provide adequate resolution of membranes with thicknesses that can vary substantially from that of the background mesh spacing a general formulation is given but the implementation is in a two dimensional code that provides a proof of principle numerical examples including a spring pendulum and a string with initial slack are used to illustrate the method the string with slack uses an additional modification of the membrane constitutive equation that allows wrinkles to be modeled at low computational cost presented also are examples of two disks impacting pinching a membrane and rebounding a difficult problem for standard finite element codes these simulations require a relaxation of the automatic no slip contact algorithm in mpm the addition of the capability to model membranes and the new contact algorithm provide a significant improvement over existing methods for handling an important class of problems copyright 1999 john wiley sons ltd

time discontinuous material point method for transient problems abstract this paper presents a time discontinuous material point method tdmpm for transient problems such as the wave propagation and impact responses in solids by dividing the continuous time domain into discrete time intervals the weak form of the tdmpm is established by considering the discrete grid based governing equations constraint and discontinuity conditions the displacement and velocity fields in a time interval are interpolated with the piecewise cubic and linear functions respectively by substituting the assumed displacement and velocity fields into the weak form a novel computational framework for the grid displacements and velocities at the discrete time instants is constructed in the new formulations the displacement field at each time instant remains to be continuous whereas the velocity field at the time instant becomes discontinuous these unique features ensure the tdmpm could properly capture the discontinuous characteristics and control the spurious numerical oscillations two numerical examples under the impact loading are used to verify the proposed method two representative impact problems are then presented for further verification and demonstration besides the corresponding contact algorithm adopted in the tdmpm is shown to be capable of capturing the correct contact behavior with higher fidelity and less computational cost than the mpm the presented results illustrate that the tdmpm could successfully control the spurious numerical oscillations associated with transient simulations

flip a method for adaptively zoned particle in cell calculations of fluid flows in two dimensions a method is presented for calculating fluid flow in two dimensions using a full particle in cell representation on an adaptively zoned grid the method has many interesting properties among them an almost total absence of numerical dissipation and the ability to represent large variations in the data the method is described using a standard formalism and its properties are illustrated by supersonic flow over a step and the interaction of a shock with a thin foil

an evaluation of explicit time integration schemes for use with the generalized interpolation material point method the stability and accuracy of the generalized interpolation material point gimp method is measured directly through carefully formulated manufactured solutions over wide ranges of cfl numbers and mesh sizes the manufactured solutions are described in detail the accuracy and stability of several time integration schemes are compared via numerical experiments the effect of various treatments of particle size are also considered the hypothesis that gimp is most accurate when particles remain contiguous and non overlapping is confirmed by comparing manufactured solutions with and without this property

the optimization of convergence for chebyshev polynomial methods in an unbounded domain by using the method of steepest descent i have compared the suitability of three different methods for solving problems in a semi infinite or infinite domain using chebyshev polynomials exponential mappings are uniformyly bad domain truncation and algebraic mapping both work well but each is superior for a different category of problems when the solution is an entire function then domain truncation is best it is always simpler to apply than algebraic mapping and always at least as accurate much more accurate for functions which decay very rapidly for singular functions on the other hand algebraic mapping is better because it is less sensitive to the mapping scale factor l permitting a better compromise in resolving both the singularity and the exponential decay for both types of problems i give simple explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy for the model entire function exp azk on a semi infinite interval 0 the optimum domain size is l 0 896 na 1k and the smallest attainable error is roughly e n 0 896 k where n is the number of chebyshev polynomals used similar formulas for singular functions handled via algebraic mapping are given in 3 41 to 3 45 below

discrete hamilton s equations for viscous compressible fluid dynamics abstract lagrange s and hamilton s equations are used extensively in numerical modeling of rigid body dynamics and continuum solid dynamics problems the use of energy methods in viscous compressible flow problems has been by contrast rather limited largely confined to the development of basic balance laws in partial differential equation form however finite element interpolation of the modeled flow field allows for the direct application of the discrete form of hamilton s equations to viscous compressible fluid dynamics in eulerian frames the resulting model is a true energy formulation developed without reference to the partial differential balance equations which underlie conventional finite difference weighted residual finite element and finite volume methods

decoupling and balancing of space and time errors in the material point method mpm summary the material point method mpm is a computationally effective particle method with mathematical roots in both particle in cell and finite element type methods the method has proven itself extremely useful in solving solid mechanics problems involving large deformations and or fragmentation of structures problem domains which are sometimes problematic for finite element type methods recently the mpm community has focused significant attention on understanding the basic mathematical error properties of the method complementary to this thrust in this paper we show how spatial and temporal errors are typically coupled within the mpm framework in an attempt to overcome the challenge to analysis that this coupling poses we take advantage of mpm s connection to finite element methods by developing a moving mesh variant of mpm which allows us to use finite element type error analysis to demonstrate and understand the spatial and temporal error behaviors of mpm we then provide an analysis and demonstration of various spatial and temporal errors in mpm and in simplified mpm type simulations our analysis allows us to anticipate the global error behavior in mpm type methods and allows us to estimate the time step where spatial and temporal errors are balanced larger time steps result in solutions dominated by temporal errors and show second order temporal error convergence smaller time steps result in solutions dominated by spatial errors and hence temporal refinement produces no appreciative change in the solution based upon our understanding of mpm from both analysis and numerical experimentation we are able to provide to mpm practitioners a collection of guidelines to be used in the selection of simulation parameters that respect the interplay between spatial grid resolution number of particles and time step copyright c 2009 john wiley sons ltd

simulation of elastic filaments interacting with a viscous pulsatile flow abstract motivated by the need to understand the motion of ruptured atherosclerosis plaque fragments in a flowing blood we simulated flexible elastic filaments interacting with a viscous pulsatile flow in two dimensions using the immersed boundary method our simulations show that 1 a flexible horizontal filament positioned perpendicularly to the flow is a stable configuration and 2 hydrodynamical forces prevent adhesion of filaments but mutual interaction of vortical fields yields clustering of filaments into groups that may be called aggregation

modified deflation algorithm for the solution of singular problems ii nonlinear multipoint boundary value problems abstract various kinds of iterative methods have been proposed for the solution of nonlinear multipoint boundary value problems mpbvp s however it is necessary for these methods that the adjusting matrix which corresponds to the jacobian of nonlinear equations is nonsingular at the solution in this paper an algorithm for the singular solution of nonlinear mpbvp s which is an extension of the modified deflation algorithm for the singular root of nonlinear algebraic equations developed by the author is presented according to the present method the singular solution can ultimately be reduced to the usual simple solution and both convergency and accuracy can greatly be improved the effectiveness of the present method is shown by solving two illustrative examples

a subdivision approach to the construction of approximate solutions of boundary value problems with deviating arguments abstract using the ideas employed in the construction of subdivision algorithms we offer here a high accuracy algorithm to compute numerical solutions for two point boundary value problems of differential equations with deviating arguments numerical examples are included to demonstrate the fast convergence and high accuracy of the algorithm this paper is a further development to our previous works for solving various types of boundary value problems

a numerical method for suspension flow abstract peskin s j comput phys 25 220 1977 immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles as before the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations as a result one set of equations holds in the entire computational domain eliminating the need to apply boundary conditions on the surface of suspended objects the new method computes the force density by discretizing the stress strain constitutive equations for an elastic solid on a grid using data provided by clusters of lagrangian points this approach clearly specifies the material properties of the suspended objects a simple data structure for the lagrangian points makes it easy to model suspended solids with arbitrary shape and size the method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids with theory and analysis the capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems

approximations by orthonormal mapped chebyshev functions for higher dimensional problems in unbounded domains this paper is concerned with approximation properties of orthonormal mapped chebyshev functions omcfs in unbounded domains unlike the usual mapped chebyshev functions which are associated with weighted sobolev spaces the omcfs are associated with the usual non weighted sobolev spaces this leads to particularly simple stiffness and mass matrices for higher dimensional problems the approximation results for both the usual tensor product space and hyperbolic cross space are established with the latter particularly suitable for higher dimensional problems

finite difference methods for boundary value problems of differential equations with deviating arguments abstract we propose a convergent finite difference method to construct an approximate solution of the boundary value problems with deviating arguments the method developed takes into account the continuity limitations of the solutions of these problems to demonstrate the effectiveness of the proposed method numerical evidence is also included

implicit dynamics in the material point method abstract a time implicit discretization is derived and validated for the material point method mpm the resulting non linear discrete equations are solved using newton s method combined with either the conjugate gradient method or the generalized minimum residual method these newton krylov solvers are implemented in a matrix free fashion for numerical efficiency a description of the algorithms and evaluation of their performance is presented on all test problems if the time step is chosen appropriately the implicit solution technique is more efficient than an explicit method without loss of desired features in the solutions in a dramatic example time steps 10 000 times the explicit step size are possible for the large deformation compression of a cylindrical billet at 1 2 the computational cost

a partitioned modeling approach with moving jump conditions for localization abstract although many investigations have been performed on localization problems there still exist some pressing issues specifically it is difficult to show in general well posed governing equations based on the essential features of localization phenomena a partitioned modeling approach is proposed here via moving jump conditions for localization problems by taking the initial point of localization as that point where the type of the governing differential equation changes i e a hyperbolic to an elliptic type for dynamic problems and an elliptic to another elliptic type for static problems a moving boundary between localized and non localized deformation zones is defined through jump forms of conservation laws across the boundary as a result localization problems might be considered in the same category as shocks in fluids and solidification in heat transfer to illustrate the proposed procedure one dimensional analytical solutions are given with an emphasis on the definition of boundary conditions and the experimental means to determine model parameters associated with localization future research is then discussed on an extension to general cases

enhancement of the material point method using b spline basis functions the mpm material point method enhanced with b spline basis functions referred to as bsmpm b spline mpm is developed and demonstrated using representative quasi static and dynamic example problems smooth b spline basis functions could significantly reduce the cell crossing error as known for the original mpm a gauss quadrature scheme is designed and shown to be able to diminish the quadrature error in the bsmpm analysis of large deformation problems for the improved accuracy and convergence especially with the quadratic b splines moreover the increase in the order of the b spline basis function is also found to be an effective way to reduce the quadrature error and to improve accuracy and convergence for plate impact examples it is demonstrated that the bsmpm outperforms the generalized interpolation material point gimp and convected particle domain interpolation cpdi methods in term of the accuracy of representing stress waves thus the bsmpm could become a promising alternative to the mpm gimp and cpdi in solving certain types of transient problems

efficient spectral sparse grid methods and applications to high dimensional elliptic problems we develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by smolyak s construction based on a nested quadrature more precisely we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis and by using the aforementioned fast transform we construct two very efficient sparse spectral galerkin methods for a model elliptic equation in particular the chebyshev legendre galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low order sparse grid methods based on finite elements or wavelets and can be efficiently solved by a suitable sparse solver ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms

monotone iterative methods for a general class of discrete boundary value problems abstract in this paper we shall offer comparison results as well as monotone iterative schemes for the construction of solutions to a very general class of discrete boundary value problems the discrete system we consider includes in particular the n th order prototype systems finite as well as infinite discrete delay equations and discrete integral equations further the boundary conditions we consider include the initial terminal periodic and transport type problems numerical examples illustrating the usefulness of the proposed schemes to a variety of boundary value problems are also included

axisymmetric form of the material point method with applications to upsetting and taylor impact problems abstract the material point method is an evolution of particle in cell methods which utilize two meshes one a material or lagrangian mesh defined over material of the body under consideration and the second a spatial or eulerian mesh defined over the computational domain although meshes are used they have none of the negative aspects normally associated with conventional eulerian or lagrangian approaches the advantages of both the eulerian and lagrangian methods are achieved by using the appropriate frame for each aspect of the computation with a mapping between the two meshes that is performed at each step in the loading process the numerical dissipation normally displayed by an eulerian method because of advection is avoided by using a lagrangian step the mesh distortion associated with the lagrangian method is prevented by mapping to a user controlled mesh furthermore explicit material points can be tracked through the process of deformation thereby alleviating the need to map history variables as a consequence problems which have caused severe numerical difficulties with conventional methods are handled fairly routinely examples of such problems are the upsetting of billets and the taylor problem of cylinders impacting a rigid wall numerical solutions to these problems are obtained with the material point method and where possible comparisons with experimental data and existing numerical solutions are presented

an enhanced immersed structural potential method for fluid structure interaction within the group of immersed boundary methods employed for the numerical simulation of fluid structure interaction problems the immersed structural potential method ispm was recently introduced gil et al 2010 1 in order to overcome some of the shortcomings of existing immersed methodologies in the ispm an incompressible immersed solid is modelled as a deviatoric strain energy functional whose spatial gradient defines a fluid structure interaction force field in the navier stokes equations used to resolve the underlying incompressible newtonian viscous fluid in this paper two enhancements of the methodology are presented first the introduction of a new family of spline based kernel functions for the transfer of information between both physics in contrast to classical ibm kernels these new kernels are shown not to introduce spurious oscillations in the solution second the use of tensorised gaussian quadrature rules that allow for accurate and efficient numerical integration of the immersed structural potential a series of numerical examples will be presented in order to demonstrate the capabilities of the enhanced methodology and to draw some key comparisons against other existing immersed methodologies in terms of accuracy preservation of the incompressibility constraint and computational speed

the material point method in large strain engineering problems abstract the material point method is a variant of the finite element method formulated in an arbitrary lagrangian eulerian description of motion two kinds of spatial discretization are utilized in the method the motion of material points representing subregions of the analysed continuum is traced against a background of the computational element mesh the mesh can be chosen arbitrarily so that the main disadvantage of the finite element method related to mesh distortion is avoided application of the method to some large strain industrial problems as granular flow in a silo as well as geomechanical and plastic forming problems are shown

implicit formulation of material point method for analysis of incompressible materials abstract material point method mpm is a particle based numerical technique with application to solid mechanics problems involving large deformations recently mpm has been successfully applied in many engineering applications including solid and solid fluid interaction problems due to its capability to model complex history dependent material behaviours however the conventional explicit mpm approach exhibits numerical limitations in incompressible or near incompressible material behaviour problems we study an implicit treatment of the pressure in mpm algorithm to simulate material incompressibility avoiding artificial pressure oscillations and significantly small time steps present in the explicit mpm approach applying chorin s projection method to the velocity pressure coupled equations an elliptic equation is solved for the pressure which is then used to solve the divergence free velocity field the results of dam break problem show no spurious pressure oscillations and large time steps compared to the traditional explicit mpm the ability of the present implicit mpm formulation to model incompressible materials will be an additional advantage in simulating materials with complex constitutive equations which exhibit a time dependent memory

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

improved convected particle domain interpolation method for coupled dynamic analysis of fully saturated porous media involving large deformation abstract based on the u p form governing equations and the convected particle domain interpolation technique the improved convected particle domain interpolation based material point method ccpdi is developed in this paper for the coupled dynamic and contact analysis of fully saturated porous media involving large deformation the numerical artifact noises due to material points crossing computational grid boundaries that usually occur in large deformation are eliminated by using the smoother interpolation functions presented in the convected particle domain interpolation technique the discrete equations are derived in the framework of the generalized interpolation material point method the boundary load tracking algorithm and a modified contact algorithm are proposed based on the definition of particle domains to apply accurately the surface loads on the motive boundaries and to capture correctly the contact time behaviors in large deformation problems respectively simulations of several representative one and two dimensional problems are presented to demonstrate the accuracy and effectiveness of the proposed methods compared to those obtained by using the coupling material point method and or finite element method the simulation results illustrate that the proposed ccpdi method can be successfully used in simulating the large deformation coupled dynamic responses of the solid skeleton and fluid phase in fully saturated porous media and the large deformation impact between solid and saturated porous bodies

chebyshev rational spectral and pseudospectral methods on a semi infinite interval a weighted orthogonal system on the half line based on the chebyshev rational functions is introduced basic results on chebyshev rational approximations of several orthogonal projections and interpolations are established to illustrate the potential of the chebyshev rational spectral method a model problem is considered both theoretically and numerically error estimates for the chebyshev rational spectral and pseudospectral methods are established preliminary numerical results agree well with the theoretical estimates and demonstrate the effectiveness of this approach

optimal transportation meshfree approximation schemes for fluid and plastic flows we develop an optimal transportation meshfree otm method for simulating general solid and fluid flows including fluid structure interaction the method combines concepts from optimal transportation theory with material point sampling and max ent meshfree interpolation the proposed otm method generalizes the benamou brenier differential formulation of optimal mass transportation problems to problems including arbitrary geometries and constitutive behavior the otm method enforces mass transport and essential boundary conditions exactly and is free from tension instabilities the otm method exactly conserves linear and angular momentum and its convergence characteristics are verified in standard benchmark problems we illustrate the range and scope of the method by means of two examples of application the bouncing of a gas filled balloon off a rigid wall and the classical taylor anvil benchmark test extended to the hypervelocity range

a fully multidimensional positive definite advection transport algorithm with small implicit diffusion in numerical modeling of physical phenomena it is often necessary to solve the advective transport equation for positive definite scalar functions numerical schemes of second or higher order accuracy can produce negative values in the solution due to the dispersive ripples lower order schemes such as the donor cell or lax friedrichs or higher order schemes with zeroth order diffusion added produce no ripples but suffer from excessive implicit diffusion in the last ten years a possible resolution of this dilemma has been developed in the form of hybrid schemes in which the advective fluxes are given as a weighted average of a first order positive definite scheme s fluxes and a higher order scheme s fluxes the difference in determination of the weights in the calculation of the average advective fluxes has led to different hybrid schemes two main hybrid type schemes have been developed one the so called flux corrected transport fct method was originated by boris and book i 31 and generalized by zalesak 141 the other was developed by harten and zwas 6 9 in the form of the self adjusting hybrid schemes sahs method both methods were constructed to deal effectively with shocks and contact discontinuities solutions of the advection transport equation obtained by using fct or sahs maintain positive definiteness of the initial condition and as can be seen from presented tests zalesak 141 harten 6 be very accurate unfortunately application of these methods to the modeling of complex multidimensional hydrodynamical systems like atmospheric phenomena is rather limited due to the excessive computer time required furthermore in many hydrodynamical systems

sparse spectral approximations of high dimensional problems based on hyperbolic cross hyperbolic cross approximations by some classical orthogonal polynomials functions in both bounded and unbounded domains are considered in this paper optimal error estimates in proper anisotropic weighted korobov spaces for both regular hyperbolic cross approximations and optimized hyperbolic cross approximations are established these fundamental approximation results indicate that spectral methods based on hyperbolic cross approximations can be effective for treating certain high dimensional problems and will serve as basic tools for analyzing sparse spectral methods in high dimensions

an efficient binning scheme with application to statistical crack mechanics summary a computational scheme is developed for sampling based evaluation of a function whose inputs are statistically variable after a general abstract framework is developed it is applied to initialize and evolve the size and orientation of cracks within a finite domain such as a finite element or similar subdomain the finite element is presumed to be too large to explicitly track each of the potentially thousands or even millions of individual cracks in the domain accordingly a novel binning scheme is developed that maps the crack data to nodes on a reference grid in probability space the scheme which is clearly generalizable to applications involving arbitrary numbers of random variables is illustrated in the scope of planar deformations of a brittle material containing straight cracks assuming two random variables describe each crack the cracks are assigned uniformly random orientations and non uniformly random sizes their data are mapped to a computationally tractable number of nodes on a grid laid out in the unit square of probability space so that gauss points on the grid may be used to define an equivalent subpopulation of the cracks this significantly reduces the computational cost of evaluating ensemble effects of large evolving populations of random variables copyright 2015 john wiley sons ltd

an energy consistent material point method for dynamic finite deformation plasticity energy consistency for the material point method mpm is examined for thermodynamically consistent hyperelastic plastic materials it is shown that mpm can be formulated with implicit three field variational finite element algorithms which dissipate energy and conserve momentum for that class of material models with a consistent mass matrix the resulting overall numerical method inherits the energy dissipative and momentum conserving properties of the mesh solution thus the proposed mpm algorithm satisfies by construction a time discrete form of the second law of thermodynamics properties of the method are illustrated in numerical examples

multi scale methods in this paper four multiple scale methods are proposed the meshless hierarchical partition of unity is used as a multiple scale basis the multiple scale analysis with the introduction of a dilation parameter to perform multiresolution analysis is discussed the multiple field based on a 1 d gradient plasticity theory with material length scale is also proposed to remove the mesh dependency difficulty in softening localization problems a non local smoothing particle integration procedure with its multiple scale analysis are then developed these techniques are described in the context of the reproducing kernel particle method results are presented for elastic plastic one dimensional problems and 2 d large deformation strain localization problems to illustrate the effectiveness of these methods copyright 2000 john wiley sons ltd

a hybridized discontinuous galerkin framework for high order particle mesh operator splitting of the incompressible navier stokes equations abstract a generic particle mesh method using a hybridized discontinuous galerkin hdg framework is presented and validated for the solution of the incompressible navier stokes equations building upon particle in cell concepts the method is formulated in terms of an operator splitting technique in which lagrangian particles are used to discretize an advection operator and an eulerian mesh based hdg method is employed for the constitutive modeling to account for the inter particle interactions key to the method is the variational framework provided by the hdg method this allows to formulate the projections between the lagrangian particle space and the eulerian finite element space in terms of local i e cellwise l 2 projections efficiently furthermore exploiting the hdg framework for solving the constitutive equations results in velocity fields which excellently approach the incompressibility constraint in a local sense by advecting the particles through these velocity fields the particle distribution remains uniform over time obviating the need for additional quality control the presented methodology allows for a straightforward extension to arbitrary order spatial accuracy on general meshes a range of numerical examples shows that optimal convergence rates are obtained in space and given the particular time stepping strategy second order accuracy is obtained in time the model capabilities are further demonstrated by presenting results for the flow over a backward facing step and for the flow around a cylinder