theoretical basis of the finite element iterative method for the eigenvalue problem in stationary cracks in a recently published paper a finite element based iterative method was introduced for the solution of the eigenvalue problem of stationary cracks 1 in this paper we give the theoretical basis of this iterative method and we show why it converges and how it could be extended to more complex fracture problems the cases of cracks at interfaces are illustrated

an efficient wave based prediction technique for dynamic plate bending problems with corner stress singularities the finite element method is commonly used for simulations of dynamic plate bending problems a major disadvantage of this method is its practical frequency limitation in that the computational loads become prohibitively large at higher frequencies a newly developed wave based prediction technique aims to relax this frequency limitation through an enhanced computational efficiency this paper discusses the application of the wave based method for the particular case where stress singularities appear in one or more corners of a polygonal plate domain in this case the conventional set of field variable expansion functions is extended with some special purpose functions which incorporate the corner point singularities the beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples

a general solution on stress singularities in an anisotropic wedge abstract this paper presents a general solution for determining the stress singularity order in an anisotropic wedge the order depends on the wedge angle boundary conditions and material properties the characteristic equation which governs the eigenvalues is established from the lekhnitskii s complex function method a domain of one quarter circle is proposed in which the contours of singularity order are plotted for all fiber orientations and a certain wedge angle the numerical results agree well with the open literature for special cases to reduce to strength of stress singularity at the wedge corner the fiber direction corresponding to minimum singularity order can be determined

a limited memory bfgs type method for large scale unconstrained optimization in this paper a new numerical method for solving large scale unconstrained optimization problems is presented it is derived from a modified bfgs type update formula by wei li and qi it is observed that the update formula can be extended to the framework of limited memory scheme with hardly more storage or arithmetic operations under some suitable conditions the global convergence property is established the implementations of the method on a set of cute problems indicate that this extension is beneficial to the performance of the algorithm

on similarity solutions of zabolotskaya khokhlov equation abstract the new closed form solutions of the 2 1 dimensional zabolotskaya khokhlov equation are constructed by using the similarity transformations method via lie group theory the zabolotskaya khokhlov equation has been reduced into a new partial differential equation with smaller number of independent variables further using the similarity transformations method the new partial differential equation is reduced into an ordinary differential equation the ordinary differential equation provides semi analytical solutions elastic behaviour of multisolitons and dromion annihilation of solutions are analyzed physically through their evolutional profiles

on the global convergence of the bfgs method for nonconvex unconstrained optimization problems this paper is concerned with the open problem of whether the bfgs method with inexact line search converges globally when applied to nonconvex unconstrained optimization problems we propose a cautious bfgs update and prove that the method with either a wolfe type or an armijo type line search converges globally if the function to be minimized has lipschitz continuous gradients

a tool for the analysis of quasi newton methods with application to unconstrained minimization the bfgs update formula is shown to have an important property that is inde pendent of the algorithmic context of the update and that is relevant to both constrained and unconstrained optimization the bfgs method for unconstrained optimization using a variety of line searches including backtracking is shown to be globally and superlinearly convergent on uniformly convex problems the analysis is particularly simple due to the use of some new tools introduced in this paper

hamiltonian principle based stress singularity analysis near crack corners of multi material junctions this paper presents a new method for the stress singularity analysis near the crack corners of a multi material junctions the stress singularities near the crack corners of multi dissimilar isotropic elastic material junctions are studied analytically in terms of the methods developed in hamiltonian system the governing equations of plane elasticity in a sectorial domain are derived in hamiltonian form via variable substitution and variational principle respectively both of the methods of global state variable separation and symplectic eigenfunction expansion are used to find the analytical solution of the problem the relationships among the state vectors in different material spaces are obtained by means of coordinate transformation and consistent conditions between the two adjacent domains the expression of the original problem is thus changed into a new form where the solutions of symplectic generalized eigenvalues and eigenvectors are needed the closed form of expressions is established for the stress singularity analysis near the corner with arbitrary vertex angles numerical results are presented with several chosen angles and multi material constants to show the potential of the new method proposed a semi analytical finite element is furthermore developed for the numerical analysis of crack problems

global convergence of a cass of quasi newton methods on convex problems we study the global convergence properties of the restricted broyden class of quasi newton methods when applied to a convex objective function we assume that the line search satisfies a standard sufficient decrease condition and that the initial hessian approximation is any positive definite matrix we show global and superlinear convergence for this class of methods except for dfp this generalizes powell s well known result for the bfgs method the analysis gives us insight into the properties of these algorithms in particular it shows that dfp lacks a very desirable self correcting property possessed by bfgs

asymptotic approach to analyze singular stress state in anisotropic multi material application to the rivets the problem of stress singularities due to multi material junctions is described in this paper a semi analytical approach is proposed based on an asymptotic model in the case of anisotropic linear elasticity in three dimensions the advantage of the present method is the quasi explicit knowledge of the stress and displacement fields around the junctions thus after a brief explanation of the approach leading to the mechanical fields around the singular line at the junction between different materials numerical results are presented concerning various configurations of an assembly with rivets included in a bi layer composite that illustrate this method

singular hypersingular and singular free electromagnetic fields at wedge tips in metamaterials the engineering response of metamaterials has a dramatic impact on the physics optics and engineering communities because they offer electromagnetic properties that are difficult or impossible to achieve with conventional materials in this paper an asymptotic analysis of the electromagnetic fields at multi material wedges composed of metamaterials is proposed this is made possible by removing the assumption of positive electric permittivities and magnetic permeabilities an hypothesis which usually applies to conventional materials exploring the whole range of variability of these electromagnetic properties it is shown that in addition to the classical real eigenvalues 0 1 leading to power law singularities of the type o r 1 as r 0 it is also possible to find imaginary eigenvalues leading to hypersingular solutions as well as nonsingular configurations for a suitable choice of the negative electric permittivities and magnetic permeabilities of the media moreover to fully characterize the asymptotic fields the analysis is not only restricted to the determination of the lowest real and complex eigenvalues but is also extended to the evaluation of the higher order nonsingular ones the obtained analytical results collected in synthetic diagrams are expected to have impact on the design of micro and nano electro mechanical systems

discrete time zhang neural network for online time varying nonlinear optimization with application to manipulator motion generation in this brief a discrete time zhang neural network dtznn model is first proposed developed and investigated for online time varying nonlinear optimization otvno then newton iteration is shown to be derived from the proposed dtznn model in addition to eliminate the explicit matrix inversion operation the quasi newton broyden fletcher goldfarb shanno bfgs method is introduced which can effectively approximate the inverse of hessian matrix a dtznn bfgs model is thus proposed and investigated for otvno which is the combination of the dtznn model and the quasi newton bfgs method in addition theoretical analyses show that with step size h 1 and or with zero initial error the maximal residual error of the dtznn model has an o tau 2 pattern whereas the maximal residual error of the newton iteration has an o tau pattern with tau denoting the sampling gap besides when h neq 1 and h in 0 2 the maximal steady state residual error of the dtznn model has an o tau 2 pattern finally an illustrative numerical experiment and an application example to manipulator motion generation are provided and analyzed to substantiate the efficacy of the proposed dtznn and dtznn bfgs models for otvno

the plane elasticity problem of an isotropic wedge under normal and shear distributed loading application in the case of a multi material problem the problem of a multi material composite wedge under a normal and shear loading at its external faces is considered with a variable separable solution the stress and displacement fields are determined using the equilibrium conditions for forces and moments and the appropriate airy stress function the infinite isotropic wedge under shear and normal distributed loading along its external faces is examined for different values of the order n of the radial coordinate r the proposed solution is applied to the elastostatic problem of a composite isotropic k materials infinite wedge under distributed loading along its external faces applications are made in the case of the two materials composite wedge under linearly distributed loading along its external faces and in the case of a three materials composite wedge under a parabolically distributed loading along its external faces

nonlinear conjugate gradient methods with structured secant condition for nonlinear least squares problems in this paper we deal with conjugate gradient methods for solving nonlinear least squares problems several newton like methods have been studied for solving nonlinear least squares problems which include the gauss newton method the levenberg marquardt method and the structured quasi newton methods on the other hand conjugate gradient methods are appealing for general large scale nonlinear optimization problems by combining the structured secant condition and the idea of dai and liao 2001 20 the present paper proposes conjugate gradient methods that make use of the structure of the hessian of the objective function of nonlinear least squares problems the proposed methods are shown to be globally convergent under some assumptions finally some numerical results are given

the order of stress singularities for bonded and disbonded three material junctions abstract in plane solutions are given for the order of the stress singularity at an internal point in an elastic isotropic solid where three wedges of different materials meet the three interfaces are either all perfectly bonded or a disbond is introduced along one interface one feature of this three material junction that is not present for the corresponding two material case is that each interface is geometrically different and therefore the singular behavior is dependent on which interface is disbonded each material and geometrical combination therefore gives rise to four problems one with perfect bonding at all interfaces and three cases of an interface disbond numerical results are presented for selected three material junctions new results for two material junctions and wedges that serve as special cases for the present study are also presented

two new conjugate gradient methods based on modified secant equations following the approach proposed by dai and liao we introduce two nonlinear conjugate gradient methods for unconstrained optimization problems one of our proposed methods is based on a modified version of the secant equation proposed by zhang deng and chen and zhang and xu and the other is based on the modified bfgs update proposed by yuan an interesting feature of our methods is their account of both the gradient and function values under proper conditions we show that one of the proposed methods is globally convergent for general functions and that the other is globally convergent for uniformly convex functions to enhance the performance of the line search procedure we also propose a new approach for computing the initial steplength to be used for initiating the procedure we provide a comparison of implementations of our methods with the efficient conjugate gradient methods proposed by dai and liao and hestenes and stiefel numerical test results show the efficiency of our proposed methods

finite element analysis of anisotropic materials with singular inplane stress fields abstract a finite element formulation is developed for the analysis of singular stress states at material and geometric discontinuities in anisotropic materials loaded inplane the displacement field of the sectorial element is quadratic in the angular coordinate direction and exponential in the radial direction measured from the singular point the formulation of yamada and okumura 1983a b is extended to take into account the anisotropy of the material the stress and displacement fields are obtained when the order of the stress singularity is real as well as complex when the order of the stress singularity is complex it is shown that the angular variation of the stress and displacement fields can be expressed in an infinite number of ways results for the displacement and stress fields obtained when the order of the stress singularity is complex can be made to match already published results once a similarity transformation is applied the simplicity and accuracy of the formulation are demonstrated by comparison to several analytical solutions for both isotropic and anisotropic multi material wedges and junctions with and without disbonds the nature and rate of convergence associated with the element suggests that it could be used in developing enriched elements for use with standard elements to yield accurate and computationally efficient solutions to problems having complex global geometries

stress singularities in anisotropic multi material wedges and junctions abstract the general problem of stress singularities in anisotropic multi material wedges and junctions due to geometric and material discontinuities is investigated the anisotropic wedges and junctions discussed here comprise two or three materials with different geometric configurations and different material properties the interfaces in multi material junctions are either all perfectly bonded or a disbond is introduced along one interface the order of stress singularities is obtained by solving the characteristic equations that are developed from the boundary conditions of the problem numerical examples for the order of stress singularities are given for the general problem including two and three material wedges a crack terminating at an interface between two materials bonded and disbonded two and three material junctions as well an extra root of the characteristic equations in addition to three roots corresponding to higher singular stresses is observed for the cases of disbonded anisotropic two and three material junctions the stress singularities affected by geometric configurations and material properties of anisotropic materials are also investigated

enrichment of finite elements with numerical solutions for singular stress fields enriched 2 d and 3 d finite elements are formulated for analysis of solids having multi material junction and wedge configurations that create singular stress fields due to the material and or geometric discontinuities the order and angular variation of the displacements associated with the singular fields are determined from a separate special finite element eigenanalysis and used in the enrichment process the use of these numerically determined singular fields allows enriched elements to be developed for complex configurations for which analytical fields are not available in addition to this added flexibility of application the current formulation applies to elements that may or may not be in direct contact with the singular point this allows multiple layers of enriched elements to be used around the singular point and traditional mesh refinement studies to be carried out in the enriched element region previous enriched formulations have not provided this important capability for cases where analytical fields are available such as cracked solids the performance of elements developed with the current approach is shown to be equivalent to that obtained using analytically enriched elements mesh refinement techniques using enriched elements are described that allow accurate stress distributions and generalized stress intensity factors to be directly determined the importance of high order numerical integration use of multiple layers of enriched elements and proper choice of the size of the enriched region are demonstrated by comparison to existing solutions for solids with cracks application of enriched element modelling to a 2 d bi material wedge and a 3 d stepped thickness anisotropic composite laminate is demonstrated 1997 john wiley sons ltd

decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problems abstract this paper presents the formulation of piezoelectric elasticity under generalized plane deformation derived from the three dimensional theory there are four decoupled classes in the generalized plane deformation formulation i e when l 3 l 2 0 l 3 l 3 0 l 3 l 2 0 or l 3 l 3 l 2 0 only the inplane fields of the first class and the antiplane field of the second class include the piezoelectric effect several examples of wedge problem often encountered in smart structures such as sensors or actuators are studied to examine the stress singularity near the apex of the structure the bonded materials to the pzt 4 wedge are pzt 5 graphite epoxy or aluminum conductor the influencing factors on the singular behavior of the electro elastic fields include the wedge angle material type poling direction and the boundary and interface conditions the numerical results of the first case are compared with xu s graphs and some comments are made in detail in addition some new results regarding the antiplane stress singularity of the second class are obtained via the case study the coupled singularity solutions under generalized plane deformation are also investigated to seek the conditions of the weakest or vanishing singular stress fields

the global convergence of partitioned bfgs on problems with convex decompositions and lipschitzian gradients the main purpose of this paper is the extension of powell s 1976 global convergence result to the partitioned bfgs method introduced by griewank and toint 1982 even in the unpartitioned case the original result is strengthened because the search directions need not be computed exactly and the gradient is only required to be lipschitzian rather than differentiable using the functional of byrd and nocedal 1987 a strong form of r superlinear convergence is obtained if the element functions are uniformly convex and their gradients are strictly differentiable at the minimizerx only in order to deal with the possibility of singular functions we utilize a damping of the bfgs update that becomes inactive if the problem turns out to be regular nearx

global convergence of the nonmonotone mbfgs method for nonconvex unconstrained minimization in this paper we propose a new nonmonotone armijo type line search and prove that the mbfgs method proposed by li and fukushima with this new line search converges globally for nonconvex minimization some numerical experiments show that this nonmonotone mbfgs method is efficient for the given test problems

new quasi newton methods for unconstrained optimization problems abstract many methods for solving minimization problems are variants of newton method which requires the specification of the hessian matrix of second derivatives quasi newton methods are intended for the situation where the hessian is expensive or difficult to calculate quasi newton methods use only first derivatives to build an approximate hessian over a number of iterations this approximation is updated each iteration by a matrix of low rank in unconstrained minimization the original quasi newton equation is bk 1sk yk where yk is the difference of the gradients at the last two iterates in this paper we first propose a new quasi newton equation b k 1 s k y k in which y k is decided by the sum of yk and aksk where ak is some matrix then we give two choices of ak which carry some second order information from the hessian of the objective function the three corresponding bfgs type algorithms are proved to possess global convergence property the superlinear convergence of the one algorithm is proved extensive numerical experiments have been conducted which show that the proposed algorithms are very encouraging

a limited memory optimization method using the infinitely many times repeated bns update and conjugate directions abstract to improve the performance of the limited memory variable metric l bfgs method for large scale unconstrained optimization repeating of some bfgs updates was proposed e g in al baali 1999 2002 since the repeating process can be time consuming the suitable extra updates need to be selected carefully we show that for the limited memory variable metric bns method matrix updating can be efficiently repeated infinitely many times under some conditions with only a small increase of the number of arithmetic operations the limit matrix can be written as a block bfgs update vlcek and luksan 2018 which can be obtained by solving of some low order lyapunov matrix equation the resulting method can be advantageously combined with methods based on vector corrections for conjugacy see e g vlcek and luksan 2015 global convergence of the proposed algorithm is established for convex and sufficiently smooth functions numerical experiments demonstrate the efficiency of the new method

a new variable order singular boundary element for two dimensional stress analysis a new variable order singular boundary element for two dimensional stress analysis is developed this element is an extension of the basic three node quadratic boundary element with the shape functions enriched with variable order singular displacement and traction fields which are obtained from an asymptotic singularity analysis both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy the enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour integral methods numerical examples involving crack notch and corner problems in homogeneous materials and bimaterial systems show the singular element s great versatility and accuracy in solving a wide range of problems with various orders of singularities the stress intensity factors which are obtained agree very well with those reported in the literature copyright 2002 john wiley sons ltd

a partitioned psb method for partially separable unconstrained optimization problems in this paper we propose a partitioned psb method for solving partially separable unconstrained optimization problems by using a projection technique we construct a sufficient descent direction under appropriate conditions we show that the partitioned psb method with projected direction is globally and superlinearly convergent for uniformly convex problems in particular the unit step length is accepted after finitely many iterations finally some numerical results are presented which show that the partitioned psb method is effective and competitive

a double parameter scaled bfgs method for unconstrained optimization abstract a double parameter scaled bfgs method for unconstrained optimization is presented in this method the first two terms of the known bfgs update formula are scaled with a positive parameter while the third one is scaled with another positive parameter these parameters are selected in such a way as to improve the eigenvalues structure of the bfgs update the parameter scaling the first two terms of the bfgs update is determined by clustering the eigenvalues of the scaled bfgs matrix on the other hand the parameter scaling the third term is determined as a preconditioner to the hessian of the minimizing function combined with the minimization of the conjugacy condition from conjugate gradient methods under the inexact wolfe line search the global convergence of the double parameter scaled bfgs method is proved in very general conditions without assuming the convexity of the minimizing function using 80 unconstrained optimization test functions with a medium number of variables the preliminary numerical experiments show that this double parameter scaled bfgs method is more efficient than the standard bfgs update or than some other scaled bfgs methods

a new backtracking inexact bfgs method for symmetric nonlinear equations a bfgs method in association with a new backtracking line search technique is presented for solving symmetric nonlinear equations the global and superlinear convergences of the given method are established under mild conditions preliminary numerical results show that the proposed method is better than the normal technique for the given problems

an adaptive three term conjugate gradient method based on self scaling memoryless bfgs matrix abstract due to its simplicity and low memory requirement conjugate gradient methods are widely used for solving large scale unconstrained optimization problems in this paper we propose a three term conjugate gradient method the search direction is given by a symmetrical perry matrix which contains a positive parameter the value of this parameter is determined by minimizing the distance of this matrix and the self scaling memoryless bfgs matrix in the frobenius norm the sufficient descent property of the generated directions holds independent of line searches the global convergence of the given method is established under wolfe line search for general non convex functions numerical experiments show that the proposed method is promising

a modified bfgs type quasi newton method with line search for symmetric nonlinear equations problems abstract in this paper using approximate gradient of the norm square metric function we present an inexact mbfgs method with line search for solving symmetric nonlinear equations which is a generalization of the mbfgs method proposed by li and fukushima 2001 for solving smooth unconstrained optimization the used approximate gradient of the proposed method only requires computing two residuals at every iteration and is easy to obtain by utilizing symmetric structure of the system we establish global convergence of the proposed method and report some numerical results

conjugate gradient methods based on secant conditions that generate descent search directions for unconstrained optimization conjugate gradient methods have been paid attention to because they can be directly applied to large scale unconstrained optimization problems in order to incorporate second order information of the objective function into conjugate gradient methods dai and liao 2001 proposed a conjugate gradient method based on the secant condition however their method does not necessarily generate a descent search direction on the other hand hager and zhang 2005 proposed another conjugate gradient method which always generates a descent search direction in this paper combining dai liao s idea and hager zhang s idea we propose conjugate gradient methods based on secant conditions that generate descent search directions in addition we prove global convergence properties of the proposed methods finally preliminary numerical results are given

global convergence of a new hybrid gauss newton structured bfgs method for nonlinear least squares problems in this paper we propose a hybrid gauss newton structured bfgs method with a new update formula and a new switch criterion for the iterative matrix to solve nonlinear least squares problems we approximate the second term in the hessian by a positive definite bfgs matrix under suitable conditions global convergence of the proposed method with a backtracking line search is established moreover the proposed method automatically reduces to the gauss newton method for zero residual problems and the structured bfgs method for nonzero residual problems in a neighborhood of an accumulation point a locally quadratic convergence rate for zero residual problems and a locally superlinear convergence rate for nonzero residual problems are obtained for the proposed method some numerical results are given to compare the proposed method with some existing methods

spectral scaling quasi newton methods with updates from the one parameter of the broyden family in this paper based on the spectral scaling secant condition w y cheng d h li spectral scaling bfgs method journal of optimization theory and applications 146 2010 305 319 we propose spectral scaling one parameter broyden family methods which allow for negative values of the parameter we show that the proposed methods possess some good properties such as quadratic termination property and single step convergence rate not inferior to that of the steepest descent method when minimizing an n dimensional quadratic function under appropriate conditions we establish the global convergence of the proposed methods for uniformly convex functions numerical results from problems in the cute test set show that the proposed methods are promising

stress singularity analysis in three dimensional bonded structure abstract stress analysis for joints between two isotropic materials is performed using three dimensional boundary element method with rongved s fundamental solution rongved l 1955 force interior to one of two joined semi infinite solids in second midwestern conf on solid mechanics pp 1 13 which satisfies boundary conditions at the interface the stress distribution around the vertex of the joint is precisely examined and the order of stress singularity obtained in three dimensional analysis is compared with that in two dimensional analysis it is shown that the order of the stress singularity along the stress singularity line which is the intersection of the interface with the free surface varies from the value in plane strain to that in plane stress as the opposite side free surface is approached the order of stress singularity in the spherical coordinates centered at the vertex is next investigated when singularity lines with a singularity order meet with each other at the vertex the order of stress singularity is larger than that for plane strain and plane stress when the vertex angle between two side free surfaces is varied the order of stress singularity is also altered in particular a bound at which singularity disappears at the plane for dundurs composite parameters is examined for a cubic dissimilar material it is demonstrated that the bound varies little in comparison with the one obtained from the two dimensional theory

invariant solutions and conservation laws to nonconservative fp equation we generate conservation laws for the one dimensional nonconservative fokker planck fp equation also known as the kolmogorov forward equation which describes the time evolution of the probability density function of position and velocity of a particle and associate these where possible with lie symmetry group generators we determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied as the fokker planck equation is evolution type no recourse to a lagrangian formulation is made moreover we obtain invariant solutions for the fp equation via potential symmetries

a conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing abstract cg descent is a state of the art algorithm to solve large scale unconstrained minimization problems however research activities on cg descent in some other scenarios are relatively fewer in this paper by combining with the projection method of solodov and svaiter we extend cg descent to solve large scale nonlinear convex constrained monotone equations the proposed method does not require the jacobian information even though it does not store any matrix at each iteration it thus has the potential to solve large scale non smooth problems under some mild conditions we show that the proposed method converges globally primary numerical results illustrate that the proposed method works quite well moreover we also extend this method to solve the l 1 norm regularized problems to decode a sparse signal in compressive sensing performance comparisons show that the proposed method is practical efficient and competitive with the compared ones

a limited memory quasi newton trust region method for box constrained optimization by means of wolfe conditions strategy we propose a quasi newton trust region method to solve box constrained optimization problems this method is an adequate combination of the compact limited memory bfgs and the trust region direction while the generated point satisfies the wolfe conditions and therefore maintains a positive definite approximation to the hessian of the objective function the global convergence and the quadratic convergence of this method are established under suitable conditions finally we compare our algorithms iwtral and ibwtral with an active set trust region algorithm astral xu and burke 2007 on the cutest box constrained test problems presented by gould et al 2015 numerical results show that the presented method is competitive and totally interesting for solving box constrained optimization

two modified hs type conjugate gradient methods for unconstrained optimization problems abstract based on the modified secant equation we propose two new hs type conjugate gradient formulas their forms are similar to the original hs conjugate gradient formula and inherit all nice properties of the hs method by utilizing the technique of the three term hs method in zhang et al 2007 15 without the requirement of truncation and convexity of the objective function we show that one with wolfe line search and the other with armijo line search are globally convergent moreover under some mild conditions the linear convergence rate of the two modified methods is established the numerical results show that the proposed methods are efficient

on the singularities of the thermo electro elastic fields near the apex of a piezoelectric bonded wedge based on the generalized lekhnitskii formulation and mellin transform the thermo electro elastic fields of a piezoelectric bonded wedge are investigated in this paper from the potential theory in a wedge shaped region a general form of the temperature change is proposed as a particular solution in the generalized lekhnitskii formulation the emphasis is on the singular behavior near the apex of the piezoelectric bonded wedge including singularity orders and angular functions which can be computed numerically the interface between two materials can be either perfectly bonded namely type a so that the continuity of electric displacements holds or a thin electrode namely type b so that the electric potential is grounded case studies of pzt 5h pzt 4 and graphite epoxy pzt 4 bonded wedges reveal that in most cases the type b continuity condition has more severe singularities than type a due to the mixed boundary point of the electrostatics at the apex of the wedge the results of this study show that the reduction or disappearance of singularity orders is possible through the appropriate selection of poling fiber orientations and wedge angles

descent directions of quasi newton methods for symmetric nonlinear equations in general when a quasi newton method is applied to solve a system of nonlinear equations the quasi newton direction is not necessarily a descent direction for the norm function in this paper we show that when applied to solve symmetric nonlinear equations a quasi newton method with positive definite iterative matrices may generate descent directions for the norm function on the basis of a gauss newton based bfgs method d h li and m fukushima siam j numer anal 37 1999 pp 152 172 we develop a norm descent bfgs method for solving symmetric nonlinear equations under mild conditions we establish the global and superlinear convergence of the method the proposed method shares some favorable properties of the bfgs method for solving unconstrained optimization problems a the generated sequence of the quasi newton matrices is positive definite b the generated sequence of iterates is norm descent c a global convergence theorem is established without nonsingularity assumption on the jacobian preliminary numerical results are reported which positively support the method

stress singularities in an anisotropic body of revolution abstract to fill the gap in the literature on the application of three dimensional elasticity theory to geometrically induced stress singularities this work develops asymptotic solutions for williams type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials the cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution so the problems under consideration are very complicated the eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three dimensional equilibrium equations in terms of the displacement components the correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed form characteristic equations for an isotropic body of revolution and using the commercial finite element program abaqus for orthotropic bodies of revolution thereafter the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries made of a single material or bi materials under different boundary conditions

hierarchy of conservation laws of diffusion convection equations we introduce notions of equivalence of conservation laws with respect to lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems we also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials to construct conservation laws we develop and apply the most direct method which is effective to use in the case of two independent variables admitting possibility of dependence of conserved vectors on a number of potentials we generalize the iteration procedure proposed by bluman and doran wu for finding nonlocal potential conservation laws as an example we completely classify potential conservation laws including arbitrary order local ones of diffusion convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations

the dai liao nonlinear conjugate gradient method with optimal parameter choices minimizing two different upper bounds of the matrix which generates search directions of the nonlinear conjugate gradient method proposed by dai and liao two modified conjugate gradient methods are proposed under proper conditions it is briefly shown that the methods are globally convergent when the line search fulfills the strong wolfe conditions numerical comparisons between the implementations of the proposed methods and the conjugate gradient methods proposed by hager and zhang and dai and kou are made on a set of unconstrained optimization test problems of the cuter collection the results show the efficiency of the proposed methods in the sense of the performance profile introduced by dolan and more

edge singularities in anisotropic composites abstract the stress singularity at the vertex of an anistropic wedge has the form r f r as r 0 where 0 f is a real function of the polar coordinates r in many cases f is independent of r the explicit form of f r depends on the eigenvalues of the elasticity constants called p here and on the order of singularity k when k is real k if k is complex is the real part of k the p s are all complex and consist of 3 pairs of complex conjugates which reduce to i when the material is isotropic the function f depends not only on p and k it also depends on whether p and k are distinct roots of the corresponding determinants in this paper we present the function f r in terms of p and k for the cases when p and k are single roots as well as when they are multiple roots the relationship between the complex variable z introduced in the analysis and the polar coordinates r is interpreted geometrically after presenting the form of f for individual cases a general form of f is given in eqn 74 we also show that the stress singularity at the crack tip of general anisotropic materials has the order of singularity 1 2 which is at least a multiple root of order 3 the implication of this on the form f r is discussed

the global convergence of a modified bfgs method for nonconvex functions abstract the standard bfgs method plays an important role among the quasi newton algorithms for constrained un constrained optimization problems however dai 2003 constructed a counterexample to demonstrate that this method may fail for non convex functions with inexact wolfe line searches and mascarenhas 2004 showed the nonconvergence of this method and other methods in the broyden family even with exact line search techniques these works motivate us to try to find another way to obtain another quasi newton method with better convergence based on the standard bfgs formula in this paper four approaches are used in the designed algorithm 1 a modified weak wolfe powell line search technique is introduced 2 if one defined condition is satisfied then the search direction and its associated step length are accepted and the next point is designed 3 otherwise a parabolic will be presented and it is regarded as the projection surface to avoid using the failed direction and the next point x k 1 is generated by a projection technique and 4 to easily obtain the global convergence of the proposed algorithm the projection point is not used in the current bfgs update but rather for all the next iterations the new algorithm possesses global convergence for general functions under the inexact modified weak wolfe powell line search technique and it is shown that other methods in the broyden class also have this property numerical results are reported to show the performance of the given algorithm and other similar methods

singularities in 2d anisotropic potential problems in multi material corners real variable approach an analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane the concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi material corner problems explicit forms of eigenequations for evaluation of singularity exponent in the case of multi material corners are derived both for all combinations of homogeneous neumann and dirichlet boundary conditions at faces of open corners and for multi material planes with singular interior points perfect transmission conditions at wedge interfaces are considered in both cases it is proved that singularity exponents are real for open anisotropic multi material corners and a sufficient condition for the singularity exponents to be real for anisotropic multi material planes is deduced a case of a complex singularity exponent for an anisotropic multi material plane is reported apparently for the first time in potential theory simple expressions of eigenequations are presented first for open bi material corners and bi material planes and second for a crack terminating at a bi material interface as examples of application of the theory developed here analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces and also for a special case of a crack perpendicular to a bi material interface a numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi material interface is presented

globally convergent modified perry s conjugate gradient method abstract conjugate gradient methods are probably the most famous iterative methods for solving large scale optimization problems in scientific and engineering computation characterized by the simplicity of their iteration and their low memory requirements in this paper we propose a new conjugate gradient method which is based on the mbfgs secant condition by modifying perry s method our proposed method ensures sufficient descent independent of the accuracy of the line search and it is globally convergent under some assumptions numerical experiments are also presented

a finite element approach to three dimensional singular stress states in anisotropic multi material wedges and junctions abstract a finite element formulation is developed to determine the order and angular variation of singular stress states due to material and geometric discontinuities in anisotropic materials the formulation applies to any two dimensional geometry that is prismatic in the third direction and has three dimensional displacement fields in some special cases the three dimensional fields become uncoupled antiplane and inplane fields and this formulation yields the uncoupled results the formulation provides for the determination of the asymptotic stress and displacement fields present at interior singular points of three dimensional structures the displacement field of the sectorial finite element is quadratic in the angular coordinate direction and asymptotic in the radial direction measured from the singular point the formulation of yamada and okumura 1983 hybrid and mixed finite element methods pp 325 343 wiley chichester for inplane problems is adapted for this purpose the simplicity and accuracy of the formulation are demonstrated by comparison with several analytical solutions for both isotropic and anisotropic multi material wedges and junctions the nature and speed of convergence associated with the element suggests that it could be employed in developing two dimensional and three dimensional enriched elements for use along with standard elements to yield accurate and computationally efficient solutions to problems having complex global geometries leading to singular stress states

influence of the intermediate material on the order of stress singularity in three phase bonded structure the plane problem for dissimilar materials composed of three isotropic homogeneous wedges with arbitrary angles under surface tractions is analysed using the theory of elasticity the order of singularity in stress fields near the apex of a three phase bonded structure is investigated it is first demonstrated how the order of stress singularity varies with several combinations of material with different properties in cases where the angles of wedge are of 3 3 3 and 2 2 2 when the materials and wedge angles of the side regions are identical the order of stress singularity is additionally studied in detail with the wedge angle of the intermediate region varying in cases where the total angle of the bonded wedge geometry is fixed and not fixed a method reducing the stress singularity in a two phase bonded structure is finally proposed

improved hessian approximation with modified secant equations for symmetric rank one method symmetric rank one sr1 is one of the competitive formulas among the quasi newton qn methods in this paper we propose some modified sr1 updates based on the modified secant equations which use both gradient and function information furthermore to avoid the loss of positive definiteness and zero denominators of the new sr1 updates we apply a restart procedure to this update three new algorithms are given to improve the hessian approximation with modified secant equations for the sr1 method numerical results show that the proposed algorithms are very encouraging and the advantage of the proposed algorithms over the standard sr1 and bfgs updates is clearly observed

a finite element analysis of the singular stress fields in anisotropic materials loaded in antiplane shear a finite element formulation is developed to determine the order and angular variation of singular stress states at material and geometric discontinuities in anisotropic materials subject to antiplane shear loading the displacement field of the sectorial element is quadratic in the angular co ordinate direction and asymptotic in the radial direction measured from the singular point the formulation of yamada and okumura14 for in plane problems is adapted for this purpose the simplicity and accuracy of the formulation are demonstrated by comparison to several analytical antiplane shear solutions for both isotropic and anisotropic multi material wedges and junctions with and without disbonds the nature and speed of convergence of the eigensolution suggests that the solution presented here could be used in developing enriched elements for accurate and computationally efficient evaluation of stress intensity factors in problems having complex global geometries

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

a globally convergent bfgs method with nonmonotone line search for non convex minimization in this paper we propose a modified bfgs broyden fletcher goldfarb shanno method with nonmonotone line search for unconstrained optimization under some mild conditions we show that the method is globally convergent without a convexity assumption on the objective function we also report some preliminary numerical results to show the efficiency of the proposed method

an efficient dai liao type conjugate gradient method by reformulating the cg parameter in the search direction equation abstract based on the modified bfgs method proposed by li and fukushima 2001 we present a new value of the parameter t in dai liao conjugate gradient method under mild assumptions we establish the global convergence property of the proposed method numerical results on some test problems in the cutest library illustrate computational efficiency of the new method

on hager and zhang s conjugate gradient method with guaranteed descent abstract in this paper we point out that hager zhang family of conjugate gradient methods are based on the spectral scaling secant equation are really members of dai liao type of gradient methods with common constant parameters we propose a possible efficient parameter for hager zhang s conjugate gradient method with guaranteed descent cg descent and present a conjugate gradient method cg descent1 the cg descent1 method preserves the same global convergence properties as cg descent method of hager and zhang and is numerically tested compared with cg descent method and other conjugate gradient methods considering the difference of the search directions approximation to the search direction of the memoryless bfgs method we give the reason why the proposed method is more efficient than cg descent method under some conditions and further find an optimal conjugate gradient method and an approximation optimal conjugate method in hager zhang family of conjugate gradient methods

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

some more similarity solutions of the 2 1 dimensional blp system some more similarity solutions of the coupled boiti leon pempinelli system have been obtained by using similarity transformations method via lie group theory the solutions are attained under the restrictions imposed on the functions involved in infinitesimals then solutions are investigated through the graphical representation in different cases the cases depend upon some suitable choice of arbitrary functions involved there i e in the solutions to get physically meaningful structures analytical solutions are supplemented by numerical simulations with appropriate choice of arbitrary functions and constants eventually profiles show single soliton elastic multisolitons quasi periodic parabolic and asymptotic nature of solutions

a globally convergent bfgs method for nonlinear monotone equations without any merit functions r since 1965 there has been significant progress in the theoretical study on quasi newton methods for solving nonlinear equations especially in the local convergence analysis however the study on global convergence of quasi newton methods is relatively fewer especially for the bfgs method to ensure global convergence some merit function such as the squared norm merit function is typically used in this paper we propose an algorithm for solving nonlinear monotone equations which combines the bfgs method and the hyperplane projection method we also prove that the proposed bfgs method converges globally if the equation is monotone and lipschitz continuous without differentiability requirement on the equation which makes it possible to solve some nonsmooth equations an attractive property of the proposed method is that its global convergence is independent of any merit function we also report some numerical results to show efficiency of the proposed method

singular functions for heterogeneous composites with cracks and notches the use of equilibrated singular basis functions abstract in this paper a novel method is presented to solve problems with weak singularities in two dimensional heterogeneous media using equilibrated singular basis functions this especially includes crack problems in composites of functionally graded material types the method is mainly presented in a boundary formulation although it may be found quite useful in other mesh based or mesh less approaches as the extended finite element method xfem the present paper considers harmonic and elasticity problems the most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem to this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach after developing the formulation in a mapped polar co ordination some primary basis functions generated from chebyshev polynomials and trigonometric functions along with corresponding weight functions are employed the numerical examples either selected from the well known literature or solved by well established techniques will demonstrate the capability of the method in problems related to composite materials

a modified bfgs method and its global convergence in nonconvex minimization in this paper we propose a modification of the bfgs method for unconstrained optimization a remarkable feature of the proposed method is that it possesses a global convergence property even without convexity assumption on the objective function under certain conditions we also establish superlinear convergence of the method

finite element evaluation of free edge singular stress fields in anisotropic materials a finite element formulation based on the work of yarnada and okumura 14 is presented to determine the order of singularity and angular variation of the stress and displacement fields surrounding a singular point on a free edge of anisotropic materials emphasis is placed on the computational aspects of this method when applied to configurations including fully bonded multi material junctions intersecting a free edge as well as materials containing cracks intersecting a free edge the study shows that the singularity of the three dimensional stress field may be accurately determined with a relatively small number of elements only when a proper level of numerical integration is used the method is applied to isotropic and orthotropic materials with a crack intersecting a free edge and an anisotropic three material junction intersecting a free edge the efficiency and accuracy of the method indicates it could be used to develop a numerical solution for the singular field that could in turn be used to create free edge enriched finite elements

global convergence of online limited memory bfgs global convergence of an online stochastic limited memory version of the broyden fletcher goldfarb shanno bfgs quasi newton method for solving optimization problems with stochastic objectives that arise in large scale machine learning is established lower and upper bounds on the hessian eigenvalues of the sample functions are shown to suffice to guarantee that the curvature approximation matrices have bounded determinants and traces which in turn permits establishing convergence to optimal arguments with probability 1 experimental evaluation on a search engine advertising problem showcase reductions in convergence time relative to stochastic gradient descent algorithms

soliton solutions of blmp equation by lie symmetry approach abstract shallow water wave equations are usually described by korteweg de vries kdv type equations in this paper we have used lie transformation group theory to solve 2 1 dimensional boiti leon manna pempinelli blmp equation we have obtained some exact solutions of blmp equation in the explicit form through similarity reduction all the reported results are expressed in closed form and analysed physically through their evolution profiles the physical analysis reveals that the nature of solutions is parabolic quasi periodic multisoliton and asymptotic

contact symmetries and solutions by reduction of partial differential equations the correspondence between a contact symmetry of a second order pde e and a point symmetry of the equivalent first order system s is used to determine a class of solutions for e which may not always be invariant under contact transformation we name these solutions pseudo invariant because they are determined from a point symmetry of s via a reduction method invariant solutions under the related contact transformation are included in the family of pseudo invariant solutions

a new globalization technique for nonlinear conjugate gradient methods for nonconvex minimization abstract it is well known that the hs method and the prp method may not converge for nonconvex optimization even with exact line search some globalization techniques have been proposed for instance the prp globalization technique and the grippo lucidi globalization technique for the prp method in this paper we propose a new efficient globalization technique for general nonlinear conjugate gradient methods for nonconvex minimization this new technique utilizes the information of the previous search direction sufficiently under suitable conditions we prove that the nonlinear conjugate gradient methods with this new technique are globally convergent for nonconvex minimization if the line search satisfies wolfe conditions or armijo condition extensive numerical experiments are reported to show the efficiency of the proposed technique