further results for enhanced strain methods with isoparametric elements abstract in this contribution the enhanced strain finite element method is investigated further with particular attention given to the analysis of the method for isoparametric elements it is shown that the results established earlier b d reddy and j c simo stability and convergence of a class of enhanced strain methods siam j numer anal 32 1995 in press for affine equivalent meshes carry over to the case of isoparametric elements that is the method is stable and convergent provided that a set of three criteria is satisfied and for plane elements at least convergence is at the same rate as in the standard method a procedure for recovering the stress is shown to lead to an approximate stress which converges at the optimal rate to the actual stress with regard to computations a new basis for enhanced strains is introduced and numerical results are presented for this basis as well as for existing bases the results for enhanced strain methods are distinguished by the fact that the constant appearing in asymptotic error estimates is much smaller for these methods than for standard approaches

positive solutions of some singular m point boundary value problems at non resonance this paper investigates the existence of positive solutions for second order singular m point boundary value problems at non resonance a necessary and sufficient condition for the existence of c 0 1 as well as c 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem our nonlinearity f t x may be singular at x t 0 and or t 1

toward a universal adaptive finite element strategy part 3 design of meshes abstract in this third paper in our series the issue of designing h p meshes which are optimal in some sense is addressed criteria for h p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees of freedom an optimization algorithm is developed based on these criteria and is applied to several model one dimensional and two dimensional elliptic boundary value problems numerical results indicate that the approach can lead to exponential rates of convergence

on numerical cubatures of singular surface integrals in boundary element methods we present and analyze methods for the accurate and efficient evaluation of weakly cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3 the class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three dimensional domains by the boundary integral equation method the possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point this rules out weighted quadrature approaches with a fixed singular weight for weakly singular integrals it is shown that duffy s triangular coordinates leadalways to a removal of the kernel singularity also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero these are based on the rabinowitz richter estimates in connection with an asymptotic estimate of domains of analyticity in 2 it is further shown that the modified extrapolation approach due to lyness is in the weakly singular case always applicable corresponding error and asymptotic work estimates are presented for the weakly singular as well as for cauchy and hypersingular integrals which e g arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates these rules are hierarchically constructed from finite part integration formulas in radial and gaussian formulas in angular direction again we show how the rabinowitz richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size

economical symmetrical quadrature rules for complete polynomials over a square domain it is of interest in numerical analysis to develop symmetrical quadrature rules for integration of complete polynomial functions over a square domain with minimum computational effort gaussian product quadrature rules integrate such functions with maximum effort symmetrical quadrature rules are developed and presented for integration of complete polynomial functions up to 21st order with minimum computational effort

the optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron abstract the main result of this paper is the computation of the mean optimal symmetrical interpolation points in the tetrahedron up to degree 9 this interpoation set has the smallest lebesgue constant known today

kelvin voigt equations perturbed by anisotropic relaxation diffusion and damping abstract the purpose of this work is the analysis of the existence and uniqueness of weak solutions to a kelvin voigt problem wherein the viscous and relaxation parts of the stress tensor are given by distinct power laws we assume that the viscous and relaxation terms may be fully anisotropic and that the momentum equation is perturbed by a damping term which may also be fully anisotropic in the particular case of considering this problem with a linear and isotropic relaxation term we prove the existence of global and local weak solutions the uniqueness of weak solutions is established in this case as well for the full anisotropic problem we show how all the anisotropic exponents of nonlinearity and all anisotropic coefficients must interact in order to be established global and local in time a priori estimates

proof of a conjectured asymptotic expansion for the approximation of surface integrals georg introduced a new kind of trapezoidal rule and midpoint rule to approximate a surface integral over a curved triangular surface and conjectured the existence of an asymptotic expansion for this approximation as the subdivision of the surface gets finer the purpose of this paper is to prove the conjecture

a numerical integration scheme for finite element method based on symbolic manipulation this paper is concerned with the improvement of the efficiency and accuracy of the usual numerical integration procedures for the finite element method a modified numerical integration scheme based on both the symbolic manipulation system and the numerical integration formula is proposed its efficiency for the integration of complicated rational or irrational expressions is demonstrated through several numerical examples to verify its effectiveness in the calculation of the finite element stiffness matrix a simple numerical example is also presented

a necessary and sufficient condition for the existence of positive solutions of singular super linear m point boundary value problems abstract this paper investigates the existence of positive solutions of singular super linear m point boundary value problems of fourth order ordinary differential equations a necessary and sufficient condition for the existence of c 2 0 1 positive solutions as well as c 3 0 1 positive solutions is given by means of the fixed point theorems on cones

new basis functions and computational procedures for p version finite element analysis new basis functions and solution procedures for p version finite element analysis are described they are used in a highly efficient p version finite element solver for linear elastostatics and dynamics which has been used in an industrial environment for over two years using two sample applications it is shown that using the techniques proposed here p version finite element analysis can have a substantially lower computational cost for given accuracy than standard finite element methods this makes the industrial applicability of p version finite element analysis much wider than is commonly believed

solvability of singular second order m point boundary value problems abstract let f 0 1 r 2 r be a function satisfying caratheodory s conditions and 1 t e t l 1 0 1 let i 0 1 a i r i 1 m 2 0 1 2 m 2 1 be given this paper is concerned with the problem of existence of a c 1 0 1 solution for the m point boundary value problem x f t x t x t e t 0 t 1 0 1 x 0 0 x 1 i 1 m 2 a i x i the proof of our main result is based upon the leray schauder continuation theorem

on some modification navier stokes equations abstract this article investigates some modification of the navier stokes equations of type as the modification that was suggested by lions quelques methodes de resolution des problemes aux limites non lineares dunod gauthier villaris paris 1969 in the form u j t i 1 n d i u j p i 2 d i u j i 1 n u i d i u j p x j h j t x t x q r n 1 div u 0 t x q 0 t t 0 0 u j 0 x 0 x u 0 0 t under p i p i 1 i n n 2 in this we prove the existence theorem for the different pi max 2 3 2 n and on some additional conditions i e of the pi p 4 in the isotropic nonlinearity case we prove the uniqueness theorem for the considered problem

kelvin voight equation with p laplacian and damping term existence uniqueness and blow up abstract in this paper we consider the initial boundary value problem for a generalized kelvin voight equation with p laplacian and a damping term v t v v p x t div v t div v p 2 v v m 2 v div v 0 here v x t is the velocity field p x t is the pressure is the viscosity kinematic coefficient and is the viscosity relaxation coefficient is a length scale parameter characterizing the elasticity of the fluid the coefficient and the exponents p m are given constants under appropriate conditions on the data we prove the existence and uniqueness of the global and local weak solutions under several assumptions on the exponents p m the coefficients and specified initial data a finite time blow up and the behavior of the solutions for large times are also established

necessary and sufficient conditions for the existence of positive solutions of fourth order multi point boundary value problems abstract this paper investigates the existence of positive solutions of singular multi point boundary value problems of fourth order ordinary differential equation with p laplacian a necessary and sufficient condition for the existence of c 2 0 1 positive solution as well as pseudo c 3 0 1 positive solution is given by means of the fixed point theorems on cones

integration by table look up for p version finite elements on curved tetrahedra abstract we develop a method of evaluating element level matrices and vectors associated with p version finite element computations on tetrahedral meshes with curved boundaries the procedure uses optimal interpolation formula q chen i babuska approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle comput methods appl mech engrg 128 1995 405 417 q chen i babuska approximate optimal polynomial interpolation points in the tetrahedron comput methods appl mech engrg 137 1997 89 94 for non polynomial portions of integrands followed by exact integration exact integrals of product combinations of polynomial interpolants and shape functions are stored so that different integrals may be efficiently evaluated by a table look up we present error analysis for our scheme applied to curvilinear tetrahedral meshes with exact geometric representation of the boundary of the domain as defined in a geometric modelling system numerical results are presented for problems involving the helmholtz equation

a straightforward structure to construct shape functions for variable p order meshes this paper describes a method for specifying variable p order meshes based on the topological hierarchy of mesh entities defining the finite elements a decomposition of p version mesh entity shape functions into a product of a function defined over the mesh entity and a blending function defined over the elements that the mesh entity bounds is presented the decomposed shape functions allow the effective construction of the shape functions and derivatives needed to construct element matrices for variable p enrichment

positive solutions of multi point boundary value problems at resonance abstract we establish new results on the existence of positive solutions for some multi point boundary value problems at resonance our results are based on a recent leggett williams norm type theorem due to o regan and zima we also derive a new result for a three point problem previously studied by several authors

object oriented finite elements ii a symbolic environment for automatic programming abstract the object oriented approach is getting more and more attention in the finite element community the object oriented approach leads to faster prototyping of new codes and better reusability in this context the authors research group has systematically developed a methodology of coding finite elements 1 4 in a companion paper 5 the key features of an interactive quasi automatic environment for the development of a new numerical finite element model for the solution of an initial boundary value problem have been presented in this article the detailed description of the proposed environment which is integrated in a smalltalk environment is described

quadrature free implementation of discontinuous galerkin method for hyperbolic equations a discontinuous galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions this approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent in the discontinuous galerkin method test problems include the linear and nonlinear one dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity and two dimensional linear euler solutions on unstructured grids

high order x fem and levelsets for complex microstructures uncoupling geometry and approximation in this contribution a strategy is proposed for uncoupling geometrical description and approximation with the x fem the strategy is based on an uniform coarse mesh that defines a high order approximation of the mechanical fields and an adapted mesh that defines the geometrical features by means of levelsets the connection between the geometry and the approximation is obtained by sharing the quadtree trees of the two meshes numerical examples involving level set based parts convergence studies mechanical computations and numerical homogenization show good promise for this approach

positive solution of singular dirichlet boundary value problems for second order differential equation system this paper investigates the existence of positive solutions of singular dirichlet boundary value problems for second order differential system a necessary and sufficient condition for the existence of c 0 1 c 0 1 positive solutions as well as c1 0 1 c1 0 1 positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems our nonlinearity fi t x1 x2 may be singular at x1 0 x2 0 t 0 and or t 1 i 1 2

approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle the main results of this paper are the analysis of the quality of approximation of polynomial interpolation and the computation of the approximate optimal interpolation points in the triangle we introduce various norms for the interpolation operator computational results indicate that for a given polynomial degree the set that minimizes the mean l2 norm of the interpolation operator is close to the smallest lebesgue constant interpolation set in particular for the triangle this set gives the smallest lebesgue constant currently known

efficient symmetrical cubature rules for complete polynomials of high degree over the unit cube the fully symmetrical cubature rules developed and presented are at present the most economically available for numerical integration over the unit cube of complete polynomial functions up to and including nineteenth order conventional gaussian triple product integration formulae by comparison require more than twice as many integration points and weights

necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems abstract necessary and sufficient conditions are obtained for the existence of symmetric positive solutions to the boundary value problem u p 1 u f t u u u t 0 1 u 2 i 0 u 2 i 1 0 i 0 1 applications of our results to the special case where f is a power function of u and its derivatives are also discussed moreover similar conclusions for a more general higher order boundary value problem are established our results extend some recent work in the literature on boundary value problems for ordinary differential equations the analysis of this paper mainly relies on fixed point theorems on cones

on the existence of explicit hp finite element methods using gauss lobatto integration on the triangle spectral element simulations on quadrilaterals and hexahedra rely on the gauss lobatto gl integration rule to enable explicit simulations with optimal spatial convergence rates in this work it is proved that a similar integration rule does not exist on triangles the following properties of the rule are sought a p 1 p 2 2 point integration rule capable of exactly integrating the space given by cal t 2p 1 equiv x my n 0 leq m n m n leq2p 1 where p is an integer integration points located at each of the triangle vertices p 1 integration points located on each side and p 1 p 2 2 integration points located in the interior of the element the proof hinges on the fact that the existence of such a rule implies the existence of a nodal basis with an approximate diagonal mass matrix that can be inverted to obtain exact galerkin projections of functions in cal t p 1 the proof shows that vertex functions of a basis having this property exist and are unique but on a triangle these functions are not nodal and therefore the gl rule does not exist in spite of this the existence of the vertex functions indicates that there may be a nonnodal basis that has the above property this basis would enable explicit hp finite element simulations on the triangle with optimal spatial accuracy the methodology developed in the paper gives insight into a possible way to find such a basis

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

a comparison of transformation methods for evaluating two dimensional weakly singular integrals accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique in this paper a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of laplace s equation in three dimensions and poisson s equation in two dimensions both are two dimensional weakly singular integrals and are evaluated using in a product fashion methods which have recently been used for evaluating one dimensional weakly singular integrals arising in the boundary element method the methods used are based on various polynomial transformations of conventional gaussian quadrature points where the transformation polynomial has zero jacobian at the singular point methods which split the region of integration into sub regions are considered as well as non splitting methods in particular the newly introduced and highly accurate generalized composite subtraction of singularity and non linear transformation approach gssnt is applied to various two dimensional weakly singular integrals a study of the different methods reveals complex relationships between transformation orders position of the singular point integration kernel and basis function it is concluded that the gssnt method gives the best overall results for the two dimensional weakly singular integrals studied

object oriented finite element analysis of thermo hydro mechanical thm problems in porous media the design implementation and application of a concept for object oriented in finite element analysis of multi field problems is presented in this paper the basic idea of this concept is that the underlying governing equations of porous media mechanics can be classified into different types of partial differential equations pdes in principle similar types of pdes for diverse physical problems differ only in material coefficients local element matrices and vectors arising from the finite element discretization of the pdes are categorized into several types regardless of which physical problem they belong to i e fluid flow mass and heat transport or deformation processes element ele objects are introduced to carry out the local assembly of the algebraic equations the object orientation includes a strict encapsulation of geometrical geo topological msh process related fem data and methods of element objects geometric entities of an element such as nodes edges faces and neighbours are abstracted into corresponding geometric element objects ele geo the relationships among these geometric entities form the topology of element meshes ele msh finite element objects ele fem are presented for the local element calculations in which each classification type of the matrices and vectors is computed by a unique function these element functions are able to deal with different element types lines triangles quadrilaterals tetrahedra prisms hexahedra by automatically choosing the related element interpolation functions for each process of a multi field problem only a single instance of the finite element object is required the element objects provide a flexible coding environment for multi field problems with different element types here the c implementations of the objects are given and described in detail the efficiency of the new element objects is demonstrated by several test cases dealing with thermo hydro mechanical thm coupled problems for geotechnical applications copyright 2006 john wiley sons ltd

generalized gaussian quadrature rules over two dimensional regions with linear sides abstract this paper presents a generalized gaussian quadrature method for numerical integration over triangular parallelogram and quadrilateral elements with linear sides in order to derive the quadrature rule a general transformation of the regions r1 x y a x b g x y h x and r2 x y a y b g y x h y where g x h x g y and h y are linear functions is given from x y space to a square in space s 0 1 0 1 generlized gaussian quadrature nodes and weights introduced by ma et al in 1997 are used in the product formula presented in this paper to evaluate the integral over s as it is proved to give more accurate results than the classical gauss legendre nodes and weights the method can be used to integrate a wide class of functions including smooth functions and functions with end point singularities over any two dimensional region bounded by linear sides the performance of the method is illustrated for different functions over different two dimensional regions with numerical examples

a reduced integration solid shell finite element based on the eas and the ans concept geometrically linear problems in this paper a new reduced integration eight node solid shell finite element is presented the enhanced assumed strain eas concept based on the hu washizu variational principle requires only one eas degree of freedom to cure volumetric and poisson thickness locking one key point of the derivation is the taylor expansion of the inverse jacobian with respect to the element center which closely approximates the element shape and allows us to implement the assumed natural strain ans concept to eliminate the curvature thickness and the transverse shear locking the second crucial point is a combined taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking free hourglass stabilization without rank deficiency hence the element requires only a single integration point in the shell plane and at least two integration points in thickness direction the formulation fulfills both the membrane and the bending patch test exactly which has to the authors knowledge not yet been achieved for reduced integration eight node solid shell elements in the literature owing to the three dimensional modeling of the structure fully three dimensional matenal models can be implemented without additional assumptions

generalized kelvin voigt equations with p laplacian and source absorption terms abstract this study considers the blow up in finite time and large time behavioral properties of solutions to the initial boundary value problem for generalized kelvin voigt equations with p laplacian and source absorption terms v t v v p div d v q 2 d v t d v p 2 d v v m 2 v div v x t 0 x t q t where d v 1 2 v v t is the rate of the strain tensor v x t is the velocity field p x t is the pressure is the viscosity kinematic coefficient and is the viscosity relaxation coefficient the coefficient and the exponents p m are given constants two different cases are considered in the case where 0 with an absorption term we prove that the solutions of the associated problem exhibit exponential and power decay and in the case where 0 with nonlinear source term the solutions blow up in a finite time under suitable assumptions on the exponents p q m and the coefficients with certain initial data

quadrature rules for rational functions summary it is shown how recent ideas on rational gauss type quadrature rules can be extended to gauss kronrod gauss turan and cauchy principal value quadrature rules numerical examples illustrate the advantages in accuracy thus achievable

on the application of two symmetric gauss legendre quadrature rules for composite numerical integration over a triangular surface this paper first presents a gauss legendre quadrature rule for the evaluation of i t f x y d x d y where f x y is an analytic function in x y and t is the standard triangular surface x y 0 x y 1 x y 1 in the two space x y we transform this integral into an equivalent integral s f x y frac x y d d where s is the 2 square in space 1 1 we then apply the one dimensional gauss legendre quadrature rules in and variables to arrive at an efficient quadrature rules with new weight coefficients and new sampling points then a second gauss legendre quadrature rule of composite type is obtained this rule is derived by discretising t into three new triangles tic i 1 2 3 of equal size which are obtained by joining centroid of t c 1 3 1 3 to the three vertices of t by use of affine transformations defined over each tic and the linearity property of integrals leads to the result i underover i 1 3 tic f x y d x d y frac 1 3 t g x y d x d y where g x y i 1n n f xic x y yic x y and x xic x y and y yic x y refer to affine transformations which map each tic into t the standard triangular surface we then write t g x y d x d y s g x y frac x y d d and a composite rule of integration is thus obtained we next propose the discretisation of the standard triangular surface t into n2 right isosceles triangular surfaces ti i 1 1 n2 each of which has an area equal to 1 2 n2 units we have again shown that the use of affine transformation over each ti and the use of linearity property of integrals lead to the result t f x y d x d y underover i 1 n n ti f x y d x d y frac 1 n2 t h x y d x d y where h x y i 1n n f xi x y yi x y and x xi x y y yi x y refer to affine transformations which map each ti in x y space into t a standard triangular surface t in the x y space we can now apply the two rules earlier derived to the integral t h x y d x d y this amounts to application of composite numerical integration of t into n2 and 3n2 triangles of equal sizes respectively we can now apply the rules which are derived earlier to the evaluation of the integral t f x y d x d y and each of these procedures converges to the exact value of the integral t f x y d x d y for sufficiently large value of n and the convergence is much faster for higher order rules we have demonstrated this aspect by applying the above composite integration method to some typical integrals 2007 elsevier inc all rights reserved

analysis of some quadrilateral nonconforming elements for incompressible elasticity in this work some nonconforming elements on arbitrary quadrilateral meshes in solving incompressible elastic equations are analyzed a uniform optimal convergence rate is established at the incompressible limit nu 0 5 for both displacement and stresses or pressure in the case of incompressible flow

a local projection operator for quadrilateral finite elements this note studies the approximation error of a local projection operator on polynomials of total degree k defined on quadrilaterals among other applications this projection operator permits to derive easily error estimates for quadrature formulas

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

geometry representation issues associated with p version finite element computations this paper addresses issues related to accurate geometry representation for p version finite elements on curved three dimensional domains specific options to account for domain geometry information during element level computation are identified accuracy requirements on the geometry related approximations to preserve the optimal rate of finite element error convergence for second order elliptic boundary value problems are given an element geometric mapping scheme based on blending the exact shape of the domain boundary is described that can either be used directly during element integrations or used to construct element level geometric approximations of required accuracy smoothness issues of the rational blends on simplex topologies are discussed and a numerical example based on the solution of poisson s equation in three dimensions is presented to illustrate the impact of the rational blends on the optimal rate of finite element error convergence

a new triangular and tetrahedral basis for high order hp finite element methods in this paper we describe the foundations of a new hierarchical modal basis suitable for high order hp finite element discretizations on unstructured meshes it is based on a generalized tensor product of mixed weight jacobi polynomials the generalized tensor product property leads to a low operation count with the use of sum factorization techniques variable p order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two and three dmensional computational domains employing standard meshes

integration of rational functions of bivariate polynomial numerators with linear denominators over a 1 1 square in a local parametric two dimensional space abstract this paper is concerned with explicit formulas and algorithms for computing integral of rational function of bivariate polynomial numerators with linear denominators over a 1 1 square in the local parametric space these integrals arise in finite element formulations of second order partial differential equations explicit evaluation of these integrals produce analytical finite element relations provided that the original element geometry is restricted to a linear convex quadrilateral we have also presented two algorithms either of these can be used to compute n n 1 2 integrals of 0th to nth order bivariate polynomial numerator whenever n 1 such integrals of order 0 zero to n in one of the variates are known by explicit integration formulas the analytical quadrature formulas from cubic to quintic monomial numerator in one of the vari tes due to different element geometry are for clarity and reference summarized in tabular forms all the explicit formulas are constructed with three simple functions of element nodal values they can be easily coded and save much computation time besides their inherent merit on numerical accuracy associated with analytical integration we have also derived some integration formulas for the product of global derivatives which has applications to second order linear partial differential equations in a variety of disciplines finally an application example to compute the torsional constant k for an equilateral triangular cross section is also considered for which we have explained the detailed computational scheme

numerical integration with taylor truncations for the quadrilateral and hexahedral finite elements for general quadrilateral or hexahedral meshes the finite element methods require evaluation of integrals of rational functions instead of traditional polynomials it remains as a challenge in mathematics to show the traditional gauss quadratures would ensure the correct order of approximation for the numerical integration in general however in the case of nested refinement the refined quadrilaterals and hexahedra converge to parallelograms and parallelepipeds respectively based on this observation the rational functions of inverse jacobians can be approximated by the taylor expansion with truncation then the gauss quadrature of exact order can be adopted for the resulting integrals of polynomials retaining the optimal order approximation of the finite element methods a theoretic justification and some numerical verification are provided in the paper

some geometrical results and estimates for quadrilateral finite elements abstract the concept of the equivalent parallelogram associated with a quadrilateral is introduced the quadrilateral may be regarded as a perturbation of this parallelogram which is most conveniently described by making use of properties of the isoparametric map which defines the quadrilateral the equivalent parallelogram generates a natural means of defining a regular family of quadrilaterals this definition is used together with other properties to obtain in a relatively simple manner estimates in appropriate seminorms or norms of the isoparametric map and its jacobian for use in the determination of finite element interpolation error estimates

analytical integral formulae related to convex quadrilateral finite elements abstract analytical quadrature formulae for rational functions integrated over finite elements with quadrilateral geometry are presented definitively for second order differential equations these formulae produce algebraic finite element relations associated with the isoparametrical or subparametrical rectangular interpolations

positive solutions of singular emden fowler boundary value problem with negative exponent and multiple impulses abstract in this paper we first give the method of lower and upper solutions for the existence of positive solutions to a singular impulsive boundary value problem then we obtain a necessary and sufficient condition for the existence of positive solutions of emden fowler singular impulsive boundary value problem by means of the method of lower and upper solutions

the method of lower and upper solutions for fourth order singular m point boundary value problems abstract this paper investigates the existence of positive solutions for fourth order singular m point boundary value problems firstly we establish a comparison theorem then we define a partial ordering in c 2 0 1 c 4 0 1 and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c 2 0 1 as well as c 3 0 1 positive solutions our nonlinearity f t x y may be singular at x y t 0 and or t 1

a fast method of numerical quadrature for p version finite element matrices a new technique of numerical quadrature especially suited for quadrilateral and hexahedron p version finite element matrices is presented this new technique separates the integrand into two parts and numerically operates on each part separately the objective of this scheme is to minimize the computational cost of integrating the entire element matrix as opposed to minimizing the cost of integrating a single function the efficiency of the new technique is compared with gaussian quadrature and found to take a small fraction of the computational effort

a necessary and sufficient condition for positive solutions for fourth order multi point boundary value problems with p laplacian abstractinthispaper weconsidertheexistenceofpositivesolutionsforaclassofsingularfourth order p laplacianequationwithmulti point boundary conditions by means of a monotone iterative technique a necessary and suf cient condition for the existence of apseudo c 3 0 1 positive solution is established also the existence and uniqueness of the pseudo c 3 0 1 positive solution theiterative sequence of the solution and an error estimation for the pseudo c 3 0 1 positive solution are obtained c 2007 elsevier ltd all rights reserved msc 34b15 34b25keywords necessary and suf cient condition p laplacian pseudo c 3 0 1 positive solutions cone 1 introductionin this paper we consider the existence of positive solutions for the singular fourth order multi point boundaryvalue problem with p laplacian p u 00 t 00 f 0 1u 0 m x 2i 1 a i u u 1 0 u 00 0 m x 2i 1 b u 00 u 00 1 0 1 1 where p t t p 2 t p 2 0 a i b i i

a necessary and sufficient condition for the existence of positive solutions to singular sublinear three point boundary value problems abstract this paper investigates the existence of positive solutions for second order singular sublinear three point boundary value problems a necessary and sufficient condition for the existence of c 0 1 as well as c 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem also the uniqueness iterative methods convergence rate and computational methods of the c 1 0 1 positive solutions are studied our nonlinear term may be singular at t 0 and or t 1

quadrature rules for numerical integration based on haar wavelets and hybrid functions in this paper haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration this approach is the generalization and improvement of the methods siraj ul islam et al 2010 9 where the numerical methods are only applicable to the integrals with constant limits apart from generalization of the methods 9 the new approach has two major advantages over the classical methods based on quadrature rule i no need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically ii mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n the new methods are more efficient the novel methods are compared with existing methods and applied to a number of benchmark problems accuracy of the methods are measured in terms of absolute errors

finite element method for solving problems with singular solutions abstract numerical treatment of the elliptic boundary value problem with nonsmooth solution by the finite element method is discussed the nonsmoothness could have its origin in the unsmooth boundary or the differential equation this paper which is a survey of the recent results elaborates among others on the method of auxiliary mapping the partition of unity finite element method and the hp version of fem in three dimensions numerical examples illustrate mathematical results an

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

symmetric gaussian quadrature formulae for tetrahedronal regions abstract quadrature formulae of degrees 2 to 6 are presented for the numerical integration of a function over tetrahedronal regions the formulae presented are of gaussian type and fully symmetric with respect to the four vertices of the tetrahedron

numerical integrations over an arbitrary quadrilateral region in this paper double integrals over an arbitrary quadrilateral are evaluated exploiting finite element method the physical region is transformed into a standard quadrilateral finite element using the basis functions in local space then the standard quadrilateral is subdivided into two triangles and each triangle is further discretized into 4xn 2 right isosceles triangles with area 12n 2 and thus composite numerical integration is employed in addition the affine transformation over each discretized triangle and the use of linearity property of integrals are applied finally each isosceles triangle is transformed into a 2 square finite element to compute new n 2 extended symmetric gauss points and corresponding weight coefficients where n is the lower order conventional gauss legendre quadratures these new gauss points and weights are used to compute the double integral examples are considered over an arbitrary domain and rational and irrational integrals which can not be evaluated analytically

object oriented finite elements iii theory and application of automatic programming in a first paper 1 the authors have developed the key features of an object oriented environment for the symbolic derivation of linear initial boundary value problems the approach was illustrated on a simple example of an elastic bar in dynamics in a companion paper 2 the structure of the proposed environment was described at length in this article the key features of the automatic generation of finite elements in a symbolic object oriented environment are given a few examples of derivation and automatic programming are developed and numerically tested first a classical galerkin formulation of stokes flow is compared to a galerkin least squares stabilization formulation then various stabilized formulations are tested on a simple scalar diffusion model equation

object oriented finite elements i principles of symbolic derivations and automatic programming abstract the development of a numerical finite element model for a given initial boundary value problem goes essentially through the derivation of a weak form its discretization and finally the derivation of the matrix forms these steps can to a large extent be automated except for the ones which require intelligence or some form of decision making from the developer the article outlines the essential steps of an interactive quasi automatic approach and illustrates it in the example of a linear uniaxial bar formulation for dynamics the implementation is done in smalltalkv a detailed description of the proposed environment is described in a companion article

analytical integration formulae for linear isoparametric finite elements exact and approximate analytical expressions can be derived for integrals arising in finite element methods employing isoparametric linear quadrilaterals in two space dimensions with bilinear basis functions the formulae associated with rectangular elements arbitrarily oriented in space can be shown to be a special case the proposed method provides considerable savings in computational effort in comparison with a numerical method that employs gaussian quadrature procedures in addition the method when applied to a quadrilateral inscribable in a circle can be shown to produce better accuracy than the associated 2 2 gaussian quadrature formulae

quadrature over curved surfaces by extrapolation in this paper the authors describe and justify a method for integrating over curved surfaces this method does not require that the jacobian be known explicitly this is a natural extension of extrapolation or romberg integration for planar squares or triangles 9 refs

necessary and sufficient conditions for the existence of solution to three point bvp abstract we study the existence of a solution of the three point boundary value problem p x x g x h x 0 0 x x we give sufficient and necessary conditions on g and h in order for p to have a real solution our conditions are of landesman lazer type and depend on the real values of and

on the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation it is shown in this paper that the optimal approximation property of bilinear or trilinear finite elements would be retained when the affine mapping is used to replace the q1 mapping on each element if the grids are refined nestedly the new method truncates the quadratic and cubic terms in reference mappings and produces constant jacobians and jacobian matrices this would avoid a shortcoming of the quadrilateral and hexahedral elements where the integrals of rational functions have to be computed or approximated numerical tests verify the analysis

control of spurious modes in the nine node quadrilateral element the nine node quadrilateral is much more accurate than its eight node relative if elements are non rectangular however the nine node element may display spurious low energy modes under 2 2 gauss quadrature these modes can be controlled by using the 3 3 quadrature rule but at added cost this paper investigates two inexpensive control devices that can be used with 2 2 gauss quadrature

a stable affine approximate finite element method the notion of the affine figure closest to a given quadrilateral can be given a precise mathematical definition the resulting figure is referred to as the equivalent parallelogram associated with a quadrilateral equipped with such a concept it is then feasible to consider finite element approximations in which the original quadrilateral elements are replaced by their equivalent parallelograms the idea is appealing not least because of the resulting economy arising from computations performed on an element generated by an affine map furthermore numerical experiments reported recently indicate that highly efficient and accurate schemes result when such a concept is combined with the enhanced strain method or the method of incompatible modes the purpose of this work is to analyze finite element schemes resulting from approximation of quadrilaterals by their equivalent parallelograms the focus is on low order bilinear elements and the analysis is carried out in the context of linear elasticity for standard approximations as well as for those which use enhanced strains the affine approximation applies only to the element map and the primary unknown the displacement vector in the context of elasticity is approximated by conventional piecewise bilinear functions the analysis confirms convergence at the optimal rate provided that the deviations of the quadrilaterals from their equivalent parallelograms are at most o h

necessary and sufficient conditions for the existence of symmetric positive solutions of multi point boundary value problems abstract we study the nonlinear boundary value problem with multi point boundary condition u p 1 u f t u u u t 0 1 u 2 i 0 u 2 i 1 j 1 m a i j u 2 i t j i 0 1 necessary and sufficient conditions are obtained for the existence of symmetric positive solutions of this problem using fixed point theorems on cones applications of our results to the special case where f is a power function of u and its derivatives are also discussed moreover similar conclusions for a more general higher order boundary value problem are established our results extend some recent work in the literature on boundary value problems for ordinary differential equations

object oriented finite elements iv symbolic derivations and automatic programming of nonlinear formulations reference epfl article 102770view record in web of science record created on 2007 04 23 modified on 2016 08 08