discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations abstract nonlinear nonstationary problems arising in elastodynamics have naturally a nonconservative form for these systems it is not possible to define a weak solution according to distribution theory in order to define the concept of discontinuous solutions for these systems one is confronted with multiplications of distributions this is unavoidable from a physical viewpoint since discontinuous solutions are needed to represent shock waves the theory of generalized functions of the second author in which one may multiply arbitrary distributions gives a concept of generalized solutions that may be discontinuous functions existence of such global solutions of the cauchy problem for a system of two equations is proved by a compactness argument from a convergent numerical scheme in the case under consideration the generalized solutions are bounded variation functions the results thus obtained agree with the results expected by the engineers and lead to the development of new numerical schemes

products and compositions with the dirac delta function the need to define pointwise products and compositions with distributions is pointed out in the context of the problems of renormalisation junction conditions and curved shock waves earlier definitions are briefly reviewed and new definitions are proposed using non standard analysis basic properties are established and some products and compositions with the delta distribution are explicitly evaluated with these definitions the domain of validity of the nonlinear differential equations of classical field theory can be extended to include rankine hugoniot equations are derived from the euler equations an immediate application to quantum field theory is pointed out

shape functions and integration formulas for three dimensional finite element analysis shape functions and numerical integration formulas for three dimensional finite element analysis as found in most finite element reference books are incomplete for example shape functions and integration formulas for a pyramid with a quadrilateral base are missing it is also difficult to find symmetric higher order integration formulas for triangular and tetrahedral elements in general these shape functions and integration formulas cannot be satisfactorily derived as degenerate cases of shape functions and integration formulas for hexahedral brick elements in this paper we present c continuous quadrilateral pyramid elements and integration formulas for two and three dimensional elements

analytic solutions for nonlinear differential equations on considere des equations non lineaires de la forme l a f z g z f z ou g est une fonction analytique en f z au voisinage de zero l est un operateur differentiel defini dans h 2 et a un operateur borne sur h 2

numerical vector green s function for s parameter measurement with waveport excitation we present a numerical vector green s function that directly links the object material property within the electric field volume integral equation vie to the scattered s parameters this green s function is particularly useful for microwave inverse scattering applications in which s parameter measurements are used to reconstruct the dielectric properties of unknown objects in an inhomogeneous background where the green s function does not have an analytic form unlike the dyadic green s function in a standard vie which relates the fields in the object domain vector to the scattered fields vector this vector green s function kernel relates the object domain fields vector to the measured s parameters scalar also this new vector green s function uses a waveport source method which accurately accounts for the measured scattered s parameters between antenna feeds of arbitrary impedance and modes a detailed derivation is presented and simulated and experimental examples are provided to validate this vector green s function for the prediction of scattered s parameters

nonstandard analysis and jump conditions for converging shock waves nonstandard analysis is an area of modern mathematics that studies abstract number systems containing both infinitesimal and infinite numbers this article applies nonstandard analysis to derive jump conditions for one dimensional converging shock waves in a compressible inviscid perfect gas it is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval predistributions of the heaviside function and the dirac delta measure are introduced to model the flow parameters across a shock wave the equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the flow parameters

a structured grid finite element method using computed basis functions structured e g rectangular meshes in finite element fe analysis offer several advantages but are not often used because of the error introduced in fitting material interfaces the nonconforming voxel fe method nvfem is a hierarchical structured method that systematically reduces the geometric error in an adaptive way but it requires large numbers of elements and therefore is expensive by computing basis functions that take into account the interface cutting through an element the number of elements needed is greatly reduced the resulting method cbf nvfem is applied to the computation of the radar cross section of free space scatterers results indicate that cbf nvfem is much more efficient than nvfem and is competitive with fem using unstructured meshes of tetrahedra

an augmented incomplete factorization approach for computing the schur complement in stochastic optimization we present a scalable approach and implementation for solving stochastic optimization problems on high performance computers in this work we revisit the sparse linear algebra computations of the parallel solver pips with the goal of improving the shared memory performance and decreasing the time to solution these computations consist of solving sparse linear systems with multiple sparse right hand sides and are needed in our schur complement decomposition approach to compute the contribution of each scenario to the schur matrix our novel approach uses an incomplete augmented factorization implemented within the pardiso linear solver and an outer bicgstab iteration to efficiently absorb pivot perturbations occurring during factorization this approach is capable of both efficiently using the cores inside a computational node and exploiting sparsity of the right hand sides we report on the performance of the approach on high performance computers when solving stochastic unit commitment problems of unpre

nonlinear generalized functions on manifolds in this work we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators this produces a generaliz

multiplications of distributions in elasticity and hydrodynamics it is shown how a mathematical theory of generalized functions in which the multiplications of distributions appearing in nonlinear equations of physics make sense gives new formulas and new numerical results the new methods shown here are quite general but since each particular problem requires its own study this paper is limited to elasticity and hydrodynamics in elasticity hooke s law gives systems in a nonconservative form the study of shock waves for these systems gives nonclassical multiplications of distributions of the form y y heaviside function dirac mass at the origin using this new mathematical tool new formulas are obtained more generally new numerical schemes in a first step ambiguous results are obtained then the ambiguity is removed in hydrodynamics a formulation is obtained that has a nonconservative form and is at the basis of efficient new numerical schemes strictly speaking the reader is not assumed to know anything either on distributions or on elasticity and hydrodynamics since the basic equations are recalled all computations done in this paper are rigorous from the mathematical viewpoint

hierarchical curl conforming nedelec elements for triangular prism cells new curl conforming hierarchical vector basis functions are developed for triangular prism cells these bases are constructed by taking the product of base vectors with scalar orthogonal polynomials and are shown to maintain linear independence as their polynomial order is increased the procedure for constructing these basis functions is described and specific functions are tabulated for orders up to 6 5 these bases are compatible with recently proposed hierarchical bases for tetrahedral and hexahedral cells and will maintain tangential continuity with those bases if used in a mesh containing multiple cell shapes

global analytic solutions of the radial nonlinear wave equation abstract the initial valued problem f z z 1 f z g z f z f 0 f 0 0 under suitable assumptions on the function g z f z is reduced by a well known method 2 to an operator equation of the form f n 1 f gle 1 in an abstract banach space h1 the operator n1 is a nonlinear compact and holomorphic operator defined in an open sphere centered at the origin of h1 the bar in the number means complex conjugation of and e1 is a known element in h1 the existence of a solution to the above operator equation means that there exists an analytic function f z n 1x bn zn f 0 0 which together with the first two derivatives converges absolutely for z 1 it is well known that such solutions exist under suitable assumptions on g z f z for sufficiently small here the existence of such analytic solutions is proved under more general assumptions on g z f z moreover for a given sufficiently small the theorems predict an upper bound of the solution f z for all z 1 the most important is that the existence theorems have a constructive character and can provide an answer to the question how small is the initial point f 0 as examples some equations of particular interest are studied

adaptive superposition of finite element meshes in non linear transient solid mechanics problems an s adaptive finite element procedure is developed for the transient analysis of 2 d solid mechanics problems with material non linearity due to progressive damage the resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user specified tolerances the spatial error is quantified by the zienkiewicz zhu error estimator and computed via superconvergent patch recovery while the estimation of temporal error is based on the assumption of a linearly varying third order time derivatives of the displacement field in conjunction with direct numerical time integration the distinguishing characteristic of the s adaptive procedure is the use of finite element mesh superposition s refinement to provide spatial adaptivity mesh superposition proves to be particularly advantageous in computationally demanding non linear transient problems since it is faster simpler and more efficient than traditional h refinement schemes numerical examples are provided to demonstrate the performance characteristics of the s adaptive method for quasi static and transient problems with material non linearity

hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements new vector finite elements are proposed for electromagnetics the new elements are triangular or tetrahedral edge elements tangential vector elements of arbitrary polynomial order they are hierarchal so that different orders can be used together in the same mesh and p adaption is possible they provide separate representation of the gradient and rotational parts of the vector field explicit formulas are presented for generating the basis functions to arbitrary order the basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning matrix assembly for the frequency domain curl curl equation is conveniently carried out by means of universal matrices application of the new elements to the solution of a parallel plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient for tetrahedral elements to order 4 in particular the full order elements have only the same asymptotic convergence rate as elements with a reduced gradient space such as the whitney element however further tests reveal that the optimum balance of the gradient and rotational components is problem dependent

generalized finite element method for vector electromagnetic problems the generalized finite element method gfem as introduced by babuska is a partition of unity framework for solving scalar partial differential equations pdes to date they have been applied extensively to the solution of elliptic and parabolic pdes this technique can be interpreted as a unified prescription of a host of well known techniques such as classical finite element methods that are based on tessellation point cloud techniques and element free galerkin methods the focus of this paper is to extend this technique to solve vector electromagnetic problems however modification of gfem to address this problem poses a novel set of challenges i the vector nature of the problem and the different continuity requirements on each component imply that basis functions developed should share similar characteristics ii the basis functions have to be able to represent divergence free electromagnetic fields in a source free region when they themselves are not divergence free these have been the primary impediments to extension of this technique in this paper we propose a method that may be used to overcome these obstacles and will demonstrate h and p convergence characteristics of the proposed method for a range of problems finally we will demonstrate convergence characteristics of this technique for analyzing scattering from a sharp wedge when the basis functions include analytical local field representation

a discrete adomian decomposition method for discrete nonlinear schrodinger equations we present a new discrete adomian decomposition method to approximate the theoretical solution of discrete nonlinear schrodinger equations the method is examined for plane waves and for single soliton waves in case of continuous semi discrete and fully discrete schrodinger equations several illustrative examples and mathematica program codes are presented

a 3 d interface enriched generalized fem for electromagnetic problems with nonconformal discretizations an interface enriched generalized finite element method igfem is introduced for efficient 3 d electromagnetic analysis of heterogeneous materials without using meshes that conform to the material microstructures which greatly lessens the burden of mesh generation the method assigns generalized degrees of freedom dofs at material interfaces to capture the discontinuities of the field and its derivatives the generalized dofs are supported by enriched vector basis functions vbfs which are constructed through a linear combination of the vbfs from the subelements several verification examples are provided to show that the igfem is not sensitive to the quality of the subelements and maintains the same level of solution accuracy and computational complexity as the standard finite element method fem based on conformal meshes the potential of the proposed igfem is demonstrated by simulating some engineering problems with complex periodic internal structures including composite materials with randomly distributed spherical particles and ellipsoidal inclusions and microvascular channels

microscopic profiles of shock waves and ambiguities in multiplications of distributions a rigorous mathematical method is introduced for studying discontinuous solutions for systems in nonconservative form the evidence of the agreement between the theoretical and numerical results is shown on simplified models of physics

on spiking models for synaptic activity and impulsive differential equations we illustrate the problems that can arise in writing differential equations that include dirac delta functions to model equations with state dependent impulsive forcing specifically difficulties arise in the interpretation of the products of distributions with discontinuous functions we suggest several methods to resolve these ambiguities such as using limiting sequences and asymptotic analysis with applications of the results given for discrete maps these suggestions are applied to a popular model describing synaptic connections in the brain

exact analytic solutions of schrodinger linear and nonlinear equations exact analytic solutions of schrodinger linear partial differential equations are obtained moreover the cubic nonlinear schrodinger equation is treated with the use of a well known functional analytic method and the existence of convergent power series solutions is proved from these solutions under certain initial conditions similar results as those presented in the literature are obtained

a 3d interface enriched generalized finite element method for weakly discontinuous problems with complex internal geometries an interface enriched generalized finite element method gfem is introduced for 3d problems with discontinuous gradient fields the proposed method differs from conventional gfem by assigning the generalized degrees of freedom to the interface nodes i e nodes generated along the interface when creating integration subdomains instead of the nodes of the original mesh a linear combination of the lagrangian shape functions in these integration subelements are then used as the enrichment functions to capture the discontinuity in the gradient field this approach provides a great flexibility for evaluating the enrichment functions including for cases where elements are intersected with multiple interfaces we show that the method achieves optimal rate of convergence with meshes which do not conform to the phase interfaces at a computational cost similar to or lower than that of conventional gfem the potential of the method is demonstrated by solving several heat transfer problems with discontinuous gradient field encountered in particulate and fiber reinforced composites and in actively cooled microvascular materials

the elastoplastic shock problem as an example of the resolution of ambiguities in the multiplication of distributions one dimensional collisions between two isotropic solids in which the equations of physics lead to multiplications of distributions are considered based on this example a general physicomathematical method to be adapted to each particular case is proposed to resolve the ambiguity inherent in such products this can be achieved with the aid of a new mathematical theory of generalized functions which permits dealing with mathematical phenomena of a microscopic nature that govern products of distributions having singularities at the same point this tool has recently been applied in various situations in continuum mechanics in which the equations of physics lead to heuristic products of distributions one obtains new algebraic formulas in the simplest cases and new numerical schemes in more general cases the key to the resolution of ambiguities lies in more precise statements of the laws of physics than are permitted within distribution theory and have no analog in classical analysis so that in general a resolution cannot be obtained from formal calculations

phased array antenna analysis with the hybrid finite element method a new analysis technique for infinite phased array antennas was developed and demonstrated it consists of the finite element method fem in combination with integral equation radiation conditions and a novel periodic boundary condition for 3 d fem grids accurate modeling of rectangular circular and circular coaxial feeds is accomplished by enforcing continuity between the fem solution and several waveguide modes across an aperture in the array s ground plane the radiation condition above the array is enforced by a periodic integral equation in the form of a floquet mode summation thus reducing the solution to that of a single array unit cell the periodic boundary condition at unit cell side walls is enforced through a matrix transformation that mathematically folds opposing side walls onto each other with a phase shift appropriate to the array lattice and scan angle the unit cell electric field is expanded in vector finite elements galerkin s method is used to cast the problem as a matrix equation which is solved by the conjugate gradient method a general purpose computer code was developed and validated for cases of open ended waveguides microstrip patches clad monopoles and printed flared notches showing that the analysis method is accurate and versatile

generalized solutions of nonlinear parabolic equations with distributions as initial conditions abstract we consider the nonlinear parabolic equation u t u u 3 0 in q 0 t t 0 open set in r d d 1 2 with the boundary condition u x t 0 on 0 t and the initial condition u x 0 x in where is the dirac mass at the origin of r it is known that this problem has no weak solution in any known classical sense within the distribution theory using a theory of generalized functions we obtain existence uniqueness and consistence results which describe mathematically the behaviour of the solutions ue obtained with smooth initial conditions u e x 0 e x e d and e when e 0

nonlinear generalized functions and jump conditions for a standard one pressure liquid gas model abstract the mixture of a liquid and a gas is classically represented by one pressure models these models are a system of pdes in nonconservative form and shock wave solutions do not make sense within the theory of distributions they give rise to products of distributions that are not defined within distribution theory but they make sense by applying a theory of nonlinear generalized functions to these equations in contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations the physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible and so they do not satisfy equally well their respective state laws on the shock waves this modelization gives well defined explicit jump conditions the great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results the jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results we expose with care the mathematical originality of the theory of nonlinear generalized functions an original abstract analysis issued by the leopoldo nachbin team on infinite dimensional holomorphy that permits to state mathematically physical facts that cannot be formulated within distribution theory and are the key for the removal of ambiguities that classically appear when one tries to calculate on multiplications of distributions that occur in the differential equations of physics

a simple model for a symmetrical theory of generalized functions v some applications abstract the theory of generalized functions is shown to eliminate ambiguities in products which are unavoidable in distribution theory the physical examples are the velocity of shock waves and the ito stratonovich dilemma

an interface enriched generalized fem for problems with discontinuous gradient fields summary a new generalized fem is introduced for solving problems with discontinuous gradient fields the method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges the proposed approach has several advantages over conventional generalized fem formulations such as a lower computational cost easier implementation and straightforward handling of dirichlet boundary conditions a detailed convergence study of the proposed method and a comparison with the standard fem are presented for heat transfer problems the method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard fem with conforming meshes various application problems are presented including the conjugate heat transfer problem encountered in microvascular materials copyright 2011 john wiley sons ltd

generalized solutions of parabolic initial value problems with discontinuous nonlinearities we proved recently that parabolic initial value problems with discontinuous nonlinearities have no unique weak solution in general but have a unique generalized solution in the sense of colombeau in this paper we study the relationship between generalized solutions and weak solutions

variational iteration method for solving cubic nonlinear schrodinger equation the variational iteration method is applied to solve the cubic nonlinear schrodinger cnls equation in one and two space variables in both cases we will reduce the cnls equation to a coupled system of nonlinear equations numerical experiments are made to verify the efficiency of the method comparison with the theoretical solution shows that the variational iteration method is of high accuracy

numerical solutions of one pressure models in multifluid flows flows of a mixture of several fluids can be described by one pressure models their formulation is formally simple and their efficiency has been recognized but from the mathematical viewpoint they present severe peculiarities they are not hyperbolic appear to be ill posed and are in nonconservative form so they show ambiguous products of distributions in case of shock waves a resolution of these ambiguities is proposed it gives jump conditions similar to the classical rankine hugoniot jump conditions of conservative systems this resolution attempts to solve the riemann problem and then the cauchy problem by godunov type schemes following nonconservative techniques initiated by a y le roux and collaborators