axisymmetric solutions for the multi layered transversely isotropic piezoelectric medium by introducing the precise integration algorithm pia and the technique of the mixed variable formalism accounting for any number of layers variations of components of the axisymmetric piezoelectric field including elastic displacement vertical normal stress electric potential and electric displacement along the axis of symmetry in the cylindrical coordinate system are explored the piezoelectric field is induced by the vertical loadings of mechanical or electric types uniformly distributed over a circular region with the aid of the mixed variable formulations along with the hankel integral transform and the corresponding matrix algebraic operations the governing partial differential equations of equilibrium expressed in terms of displacements and electric potential are reduced to the first order ordinary differential matrix equations as a highly accurate algorithm the pia is provided to evaluate the obtained ordinary differential matrix equations in the transformed domain both mechanical and electrical quantities in the physical domain can be acquired by taking the inversion of the hankel integral transform an example as benchmark is illustrated to examine the applicability and performance of the proposed technique compared with results in the literature numerical examples are used to demonstrate the influence of the degree of the material anisotropy and stratified parameters

a two dimensional eshelby problem for two bonded piezoelectric half planes the present paper concerns a two dimensional eshelby problem for an inclusion of arbitrary shape embedded within one of two bonded dissimilar piezoelectric halfplanes the elastic and piezoelectric constants of the inclusion and its surrounding half plane are assumed to be the same a simple explicit solution is derived in terms of some auxiliary functions which can be determined using several related conformal mappings of the inclusion shape the obtained solution is exact provided that the expansions of all conformal mappings include only a finite number of terms on the other hand if an exact conformal mapping includes infinite terms a truncated polynomial mapping function should be used and then the method gives an approximate solution the existing solutions obtained in an earlier work for a homogeneous piezoelectric plane or half plane can be derived from the present solution as special cases in particular the closed form solutions are given for the eshelby problem of an arbitrarily shaped inclusion in a piezoelectric half plane with various mixed surface conditions such as rigid insulating surface or traction free conducting surface these solutions are used to study the effects of various surface electrical conditions on an electro elastic field in a piezoelectric half plane

an alternating direction method of multipliers for the numerical solution of a fully nonlinear partial differential equation involving the jacobian determinant we consider the dirichlet problem for a partial differential equation involving the jacobian determinant in two dimensions of space the problem consists in finding a vector valued function such that the determinant of its gradient is given pointwise in a bounded domain together with essential boundary conditions this problem was initially considered in dacorogna and moser ann inst h poincare anal non lineaire 7 1990 pp 1 26 and several theoretical generalizations have been derived since in this work we design a numerical algorithm for the approximation of the solution of such a problem for various kinds of boundary data the proposed method relies on an augmented lagrangian algorithm with biharmonic regularization and low order mixed finite element approximations an iterative method allows us to decouple the nonlinearity and the differential operators numerical experiments show the capabilities of the method for benchmarks and then for more demanding test problems an erratum has bee

non singular somigliana stress identities in elasticity the paper presents two fully equivalent and regular forms of the hypersingular somigliana stress identity in elasticity that are appropriate for problems in which the displacement field and resulting stresses is c1 continuous each form is found as the result of a single decomposition process on the kernels of the somigliana stress identity in three dimensions the results show that the use of a simple stress state for regularization arises in a direct manner from the somigliana stress identity just as the use of a constant displacement state regularization arose naturally for the somigliana displacement identity the results also show that the same construction leads naturally to a finite part form of the same identity while various indirect constructions of the equivalents to these findings are published none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular somigliana stress identity that arise at corners and edges these new findings specifically focus on the corner problem and establish that the previous requirements for continuity on the densities in the hypersingular somigliana stress identity are replaced by a sole requirement on displacement field continuity the resulting regularized and finite part forms of the somigliana stress identity leads to a regularized form of the stress boundary integral equation stress bie the regularized stress bie is shown to properly allow piecewise discontinuity of the boundary data subject only to c1 continuity of the underlying displacement field the importance of the findings is in their application to boundary element modeling of the hypersingular problem the piecewise discontinuity derivation for corners is found to provide a rigorous and non singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress bie the boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non conforming elements for the hypersingular equations the analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress bie that have been the basis of boundary element analysis previously published by the first author of the current paper

modal analysis of magneto electro elastic plates using the state vector approach the state vector approach is proposed to analyze the free vibration of magneto electro elastic laminate plates the extended displacements and stresses can be divided into the so called in plane and out of plane variables once the state equation for the out of plane variables is obtained a complex boundary value problem is converted into an equivalent simple initial value problem through the state equation the propagator matrix between the top and bottom interfaces of every layer can be easily derived the global propagator matrix can also be assembled using the continuity conditions it is obvious that the order of global propagator matrix is not related to the number of layers consequently this approach possesses certain virtues including simple formulation less expensive computation etc to test the formulation the developed solution is then applied to a simply supported multilayered plate constructed of piezoelectric and or piezomagnetic materials the natural frequencies and corresponding mode shapes are computed and compared with existing results furthermore the fundamental modes along with a couple of other higher modes which have never been reported in previous literature are presented therefore this completed set of frequencies and mode shapes can be used as benchmarks for future research in this field it is also believed that the approach could be useful in the analysis and design of smart structures constructed from piezoelectric piezomagnetic composites

on the general solutions of transversely isotropic elasticity abstract in this paper we give the generalized boussinesq galerkin general solution of transversely isotropic elasticity as well as its simplified forms in two special cases and we prove the completeness of the lekhnitskii hu nowacki solution and the elliott lodge solution in such cases that s 0 2 s 1 2 and s 2 2 are possibly equal to each other

a unified boundary element method for the analysis of sound and shell like structure interactions i formulation and verification a unified boundary element method bem is developed in this paper to model both the exterior acoustic field and the elastic shell like structure in a coupled analysis the conventional boundary integral equation bie for three dimensional 3d elastodynamics is applied to thin shell like structures which can have arbitrary shapes and small thicknesses the nearly singular integrals existing in the bie when applied to thin bodies are transformed to nonsingular line integrals and are evaluated accurately and efficiently for the exterior 3d acoustic domain the burton and miller composite bie formulation is employed to overcome the fictitious eigenfrequency difficulty fed and the thin shape breakdown tsb conforming c0 quadratic elements are employed in the discretization of the two sets of bies the developed bie formulations are valid for both radiation and scattering problems and for all wave numbers numerical examples using spherical and cylindrical shells including nonuniform thickness and nond

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

penny shaped crack in a piezoelectric cylinder surrounded by an elastic medium subjected to combined in plane mechanical and electrical loads the fracture problem of a penny shaped crack in a piezoelectric ceramic cylinder surrounded by an infinite elastic medium under in plane normal mechanical and electrical loads is considered with the electric continuous boundary conditions on the crack surface by using the potential theory and hankel transform a system of dual integral equations is obtained and expressed to a fredholm integral equation of the second kind the mechanical and electrical field equations and all sorts of field intensity factors of mode i are obtained and the numerical values of various field intensity factors for pzt 6b piezoelectric ceramic surrounded by several different elastic media are graphically shown for a uniform load and a ring shaped load respectively and the effects of the size of the piezoelectric cylinder and the elastic material properties on various field intensity factors are obtained

schwarzian derivatives projective structures and the weil petersson gradient flow for renormalized volume to a complex projective structure on a surface thurston associates a locally convex pleated surface we derive bounds on the geometry of both in terms of the norms and 2 of the quadratic differential of given by the schwarzian derivative of the associated locally univalent map we show that these give a unifying approach that generalizes a number of important well known results for convex cocompact hyperbolic structures on 3 manifolds including bounds on the lipschitz constant for the nearest point retraction and the length of the bending lamination we then use these bounds to begin a study of the weil petersson gradient flow of renormalized volume on the space cc n of convex cocompact hyperbolic structures on a compact manifold n with incompressible boundary leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of dn the double of n

analysis of shell like structures by the boundary element method based on 3 d elasticity formulation and verification in this paper the boundary element method bem for 3 d elastostatic problems is studied for the analysis of shell or shell like structures it is shown that the conventional boundary integral equation cbie for 3 d elasticity does not degenerate when applied to shell like structures contrary to the case when it is applied to crack like problems where it does degenerate due to the closeness of the two crack surfaces the treatment of the nearly singular integrals which is a crucial step in the applications of bies to thin shapes is presented in detail to verify the theory numerical examples of spherical and ellipsoidal vessels are presented using the bem approach developed in this paper it is found that the system of equations using the cbie is well conditioned for all the thickness studied for the vessels the advantages disadvantages and potential applications of the proposed bem approach to shell like structures as compared with the fem regarding modelling and accuracy are discussed in the last section applications of this bem approach to shell like structures with non uniform thickness stiffeners and layers will be reported in a subsequent paper 1998 john wiley sons ltd

a symplectic map between hyperbolic and complex teichmuller theory let s be a closed orientable surface of genus at least 2 the space th ml where th is the hyperbolic teichmuller space of s and ml is the space of measured geodesic laminations on s is naturally a real symplectic manifold the space cp of complex projective structures on s is a complex symplectic manifold a relation between these spaces is provided by thurston s grafting map gr we prove that this map although not smooth is symplectic the proof uses a variant of the renormalized volume defined for hyperbolic ends

integral representation and green s functions for 3d time dependent thermo piezoelectricity abstract a boundary integral method is developed for 3d time dependent quasistatic thermo piezoelectricity this time domain formulation involves only surface quantities as a result volume discretization is completely avoided green s functions are derived by resorting to radon transform and presented in integral form reductions to the case of isotropic dielectric solid have been worked out in detail and its corresponding solutions expressed in explicit form are consistent with the existing solutions

on finite element methods for fully nonlinear elliptic equations of second order for the first time we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard c 1 finite element method fem we consider throughout the paper two cases in parallel for convex bounded polyhedral domains in bf r n or for c 2 bounded domains in bf r 2 we prove stability and convergence for the corresponding conforming or nonconforming c 1 fem respectively the results for equations and systems of orders 2 and 2m and quadrature approximations appear elsewhere the classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator the consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated the stability has to be proved in an unusual way this is the hard core of the paper essential tools are linearization a compactness argument the interplay between the weak and strong form of the linearized operator and a new regularity result for solutions of finite element equations an essential basis for our proofs are davydov s results for c 1 fes on polyhedral domains in bf r n or of local degree 5 for c 2 domains in bf r 2 better convergence and extensions to bf r n for c 2 domains are to be expected from his forthcoming results on curved domains our proof for the second case in bf r n includes the first essentially as a special case the method applies to quasi linear elliptic problems not in divergence form as well a discrete newton method is shown to converge locally quadratically essentially independently of the actual grid size by the mesh independence principle

three dimensional eigenfunction analysis of a crack in a piezoelectric material abstract a three dimensional analysis of a semi infinite crack embedded in a transversely isotropic piezoelectric material was performed by means of the eigenfunction expansions method the results show the characteristic 1 r singular behavior of the stress tensor in the vicinity of the crack a similar behavior is also revealed for the electric field in addition the study demonstrates that coupling or decoupling of the mechanical and electrical variables is a function of the distance to the crack tip as well as of the crack orientation

a modified scaled boundary finite element method for three dimensional dynamic soil structure interaction in layered soil summary this paper is devoted to the analysis of elastodynamic problems in 3d layered systems which are unbounded in the horizontal direction for this purpose a finite element model of the near field is coupled to a scaled boundary finite element model sbfem of the far field the sbfem is originally based on describing the geometry of a half space or full space domain by scaling the geometry of the near field far field interface using a radial coordinate a modified form of the sbfem for waves in a 2d layer is also available none of these existing formulations can be used to describe a 3d layered medium in this paper a modified sbfem for the analysis of 3d layered continua is derived based on the use of a scaling line instead of a scaling centre a suitable scaled boundary transformation is proposed the derivation of the corresponding scaled boundary finite element sbfe equations in displacement and stiffness is presented in detail the latter is a nonlinear differential equation with respect to the radial coordinate which has to be solved numerically for each excitation frequency considered in the analysis various numerical examples demonstrate the accuracy of the new method and its correct implementation these include rigid circular and square foundations embedded in or resting on the surface of layered homogeneous or inhomogeneous 3d soil deposits over rigid bedrock hysteretic damping is assumed in some cases the dynamic stiffness coefficients calculated using the proposed method are compared with analytical solutions or existing highly accurate numerical results copyright 2011 john wiley sons ltd

tau function and chern simons invariant abstract we define a chern simons invariant for schottky hyperbolic 3 manifolds of infinite volume we then prove an expression relating the bergman tau function on a fiber space over the teichmuller space to the lifting of the function f defined by zograf on teichmuller space and a holomorphic function on this space which we introduce if the point in this space corresponds to a marked riemann surface x then this function is constructed from the renormalized volume and our chern simons invariant for the bounding 3 manifold of x given by schottky uniformization together with a regularized polyakov integral we also obtain a relation between the chern simons invariant and the eta invariant of the bounding 3 manifold with defect given by the phase of the bergman tau function of x

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

the scaled boundary finite element method alias consistent infinitesimal finite element cell method for elastodynamics the scaled boundary finite element method alias the consistent infinitesimal finite element cell method is developed starting from the governing equations of linear elastodynamics only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one no fundamental solution is necessary and thus no singular integrals must be evaluated general anisotropic material is analysed without any increase in computational effort boundary conditions on free and fixed surfaces and on interfaces between different materials are enforced exactly without any discretization this method is exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions for a bounded medium symmetric static stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption a stress singularity is represented very accurately as the condition on the boundary in the vicinity of the point of singularity is satisfied without spatial discretization

liouville action and weil petersson metric on deformation spaces global kleinian reciprocity and holography we rigorously define the liouville action functional for the finitely generated purely loxodromic quasi fuchsian group using homology and cohomology double complexes naturally associated with the group action we prove that classical action the critical value of the liouville action functional considered as a function on the quasi fuchsian deformation space is an antiderivative of a 1 form given by the difference of fuchsian and quasi fuchsian projective connections this result can be considered as global quasi fuchsian reciprocity which implies mcmullen s quasi fuchsian reciprocity we prove that the classical action is a kahler potential of the weil petersson metric we also prove that the liouville action functional satisfies holography principle i e it is a regularized limit of the hyperbolic volume of a 3 manifold associated with a quasi fuchsian group we generalize these results to a large class of kleinian groups including finitely generated purely loxodromic schottky and quasi fuchsian groups and their free combinations

the state vector methods for space axisymmetric problems in multilayered piezoelectric media abstract the state vector equations for space axisymmetric problems of transversely isotropic piezoelectric media are established from the basic equations using the hankel transform the state vector equations are reduced to a system of ordinary differential equations an analytical solution of the problems in the hankel transform space is presented in the form of the product of initial state vector and transfer matrix the transfer matrices are given for the three distinct eigenvalues applications of the solutions are discussed an analytical solution for the transversely isotropic semi infinite piezoelectric media subjected to concerted point loads on the surface z 0 is presented in the hankel transform space using transfer matrix and the continuity conditions at the layer interfaces the general solution formulation of n layered transversely isotropic piezoelectric media is given a selected set of numerical solutions is presented for a layered semi infinite piezoelectric solid

on the renormalized volume of hyperbolic 3 manifolds the renormalized volume of hyperbolic manifolds is a quantity motivated by the ads cft correspondence of string theory and computed via a certain regularization procedure the main aim of the present paper is to elucidate its geometrical meaning we use another regularization procedure based on surfaces equidistant to a given convex surface n the renormalized volume computed via this procedure is equal to what we call the w volume of the convex region n given by the usual volume of n minus the quarter of the integral of the mean curvature over n the w volume satisfies some remarkable properties first this quantity is self dual in the sense explained in the paper second it verifies some simple variational formulas analogous to the classical geometrical schlafli identities these variational formulas are invariant under a certain transformation that replaces the data at n by those at infinity of m we use the variational formulas in terms of the data at infinity to give a simple geometrical proof of results of takhtajan et al on the kahler potential on various moduli spaces

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

three dimensional piezoelectric boundary element method the boundary element method is applied to problems of three dimensional linear piezoelectricity the continuum equations for conservation of linear momentum and charge are combined into one governing equation for piezoelectricity a single boundary integral equation is developed from this combined e eld equation and green s solution for a piezoelectric medium green s function and its derivatives are derived using the radon transform and the resulting solution is represented by a line integral that is evaluated numerically using standard gaussian quadrature the boundary integral equation is discretized using eight node isoparametric quadratic elements resulting in a matrixsystem ofequations thesolution of theboundary problem forpiezoelectricmaterialsconsistsof elastic displacements tractions electric potentials and normal charge e ux densities the complete e eld solutions can be obtained once all boundary values have been determined the accuracy of this linear piezoelectric boundary element method is illustrated with two numerical examples the e rst involves a unit cube of material with an applied mechanical load thesecond exampleconsistsof a spherical holein an ine nitepiezoelectricbody loaded by a unit traction on itsboundary comparisonsare made to the analytical solution for the cube and an axisymmetric e nite element solution for the spherical hole the boundary element method is shown to be an accurate solution procedure for general three dimensional linear piezoelectric material problems i introduction t hebrotherspierreandjacquescuriediscoveredpiezoelectricity in 1880 during their investigation of crystals a coupling occurs between the electrical and mechanical e elds in these materials such that application of an external mechanical loading induces achangeinpolarizationproportionaltoit conversely applicationof an electric e eld produces a proportional material deformation this phenomenon was discovered in ceramics in the 1940s and the subsequent manufactured piezoceramic materialshave higher coupling properties than their naturally occurring predecessors this strong piezoelectric effect has made these materials an important part of the emerging technologies of smart materials and structures smart structures are those that incorporate actuators and sensors that are integrated into the structure and have structural functionality the sensors and actuators are used to ine uence the structure s states

three dimensional green s functions in anisotropic piezoelectric solids abstract explicit expressions for three dimensional extended green s displacements in general anisotropic piezoelectric solids are derived a very efficient procedure for the numerical evaluation of the derivatives of the extended green s displacements is also proposed numerical comparisons are carried out for a transversely isotropic piezoelectric solid for which exact closed form solutions are available it is found that the extended green s displacements and their derivatives obtained with the present explicit formulation are in perfect agreement with the exact closed form solutions these green s functions can be used in the boundary integral equations for piezoelectric solids of general anisotropy and for subsequent numerical solutions of these equations by means of the boundary element method

integral formulas for chebyshev polynomials and the error term of interpolatory quadrature formulae for analytic functions we evaluate explicitly the integrals 1 1 n t r t dt r 1 with the n being any one of the four chebyshev polynomials of degree n these integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with chebyshev abscissae when the function to be integrated is analytic in a domain containing 1 1 in its interior

identification of symmetry type of linear elastic stiffness tensor in an arbitrarily orientated coordinate system abstract we develop a method through the mirror plane mp to identify the symmetry type of linear elastic stiffness tensor whose components are given with respect to an arbitrarily oriented coordinate system the method is based on the irreducible decomposition of high order tensor into a set of deviators and the multipole representation of a deviator into a scalar and a unit vector set since a unit vector depends on two euler angles we can illustrate the mp normals of the elastic tensor as zeros of a characteristic function on a unit disk and identify its symmetry immediately which is clearer and simpler than the methods proposed before furthermore by finding the common mps of three unit vector sets using fortran recipes we can also analytically recognize the symmetry type first and then recover the natural coordinate system associated with the linear elastic tensor the structures of linear elastic stiffness tensors of real materials with all possible anisotropies are investigated in detail

evaluation of nearly singular integrals in boundary element contour and node methods for three dimensional linear elasticity this article addresses the issue of nearly singular integrals that arise in boundary element methods bems when one tries to evaluate displacements and stresses at points inside a body that lie close to its bounding surface an efficient new approach is proposed for the evaluation of these integrals the same issue is then discussed in the context of the boundary contour method bcm and the boundary node method bnm finally numerical results from both the bcm and the bnm are presented for an illustrative problem

a boundary integral formulation and 2d fundamental solutions for piezoelectric media abstract first based on the basic equations of piezoelectricity the boundary integral formulation is derived by utilizing the reciprocal work theorem second for the plane problem of piezoelectric media one general solution in terms of harmonic functions is derived and the fundamental solutions are therefore obtained numerical calculations by boundary element method bem are performed to give the stress concentration coefficient of an infinite piezoelectric plane with a circular hole as well as the stress and electric intensity factors of a central crack locating on an infinite plate results are found to agree well with the exact solutions

the general solution of three dimensional problems in piezoelectric media abstract the governing equations of three dimensional problems in transversely isotropic piezo electric material are composed of four second order partial differential equations in which the displacement components and electric potential functions are the essential unknowns in this paper the so called general solutions of these equations are obtained with the aid of a set of new potential functions which are used to express the components of displacements and the electric potential functions as an application of the general solution the problem of a half space piezoelectrics acted by a concentrated lateral shear force is solved

wave propagation in a piezoelectric rod of 6mm symmetry abstract the wave propagation in a piezoelectric rod of 6 mm symmetry is investigated by applying a 3 d piezoelectric elastic model a self adjoint method is introduced to solve this problem this method avoids calculating the generalized eigenvalue equation it completely draws the dispersion curves in the forms of quasi p wave quasi sv wave and quasi sh wave under the self adjoint boundary condition and it can evaluate the dispersion curves of all kinds of boundary conditions as an example the dispersion curves of plt 5h are completely drawn we also found the quasi sv wave has standing wave phenomenon in the plt 5h rod in addition the relation of dispersion curves among different boundary conditions is discussed and an experiment method is introduce to decide the dispersion curves for another boundary conditions

three dimensional time harmonic elastodynamic green s functions for anisotropic solids a method based on the radon transform is presented to determine the displacement field in a general anisotropic solid due to the application of a time harmonic point force the radon transform reduces the system of coupled partial differential equations for the displacement components to a system of coupled ordinary differential equations this system is reduced to an uncoupled form by the use of properties of eigenvectors and eigenvalues the resulting simplified system can be solved easily a back transformation to the original coordinate system and a subsequent application of the inverse radon transform yields the displacements as a summation of a regular elastodynamic term and a singular static term both terms are integrals over a unit sphere for the regular dynamic term the surface integration can be evaluated numerically without difficulty for the singular static term the surface integral has been reduced to a line integral over half a unit circle reductions to the cases of isotropy and transverse isotropy have been worked out in detail examples illustrate applications of the method

elastic and piezoelectric fields due to polyhedral inclusions abstract we derive explicit closed form expressions describing elastic and piezoelectric deformations due to polyhedral inclusions in uniform half space and bi materials our analysis is based on the linear elasticity theory and green s function method the method involves evaluation of volume and surface integrals of harmonic and bi harmonic potentials in case of polyhedra such integrals are expressed through algebraic functions our results generalize numerous studies on this subject and they allow to obtain fully analytical solutions for a number of physical and engineering problems in the limiting case of an infinite space our relations have an essentially more compact form than relations obtained by other authors we present solutions to classical mindlin and cherruti problems we describe the elastic relaxation of a misfitting polygonal quantum dot in bi materials assuming isotropic and vertically isotropic properties it is explained how to analyze non hydrostatic and non uniform inclusions we also study piezoelectric fields induced by inclusions in materials with cubic and hexagonal lattices among other results we have found that a cubic inclusion in an isotropic material reproduces fields of quantum dots in gaas 0 0 1 and gaas 1 1 1 depending on the orientation of the cube this suggests that one can qualitatively model crystals with different lattices by choosing an appropriate inclusion shape

solutions for transversely isotropic piezoelectric infinite body semi infinite body and bimaterial infinite body subjected to uniform ring loading and charge abstract based on the fundamental solutions for transversely isotropic piezoelectric materials the fundamental solutions of axisymmetric problems are derived by integration and explicit expressions for three possible cases of different characteristic roots and multiple roots are all presented in the case of s 1 s 2 s 3 s 1 based on the greens functions for semi infinite piezoelectric body and bimaterial infinite piezoelectric body the greens functions for axisymmetric problems of semi infinite body and bimaterial infinite body are obtained taking pzt 4 as an example numerical computations are conducted by use of the fundamental solutions to axisymmetric problems comparison of the calculated results with those of fem shows good agreement between them

three dimensional green s function and its derivative for materials with general anisotropic magneto electro elastic coupling explicit expressions of green s function and its derivative for three dimensional infinite solids are presented in this paper the medium is allowed to exhibit a fully magneto electro elastic mee coupling and general anisotropic behaviour in particular new explicit expressions for the first order derivative of green s function are proposed the derivation combines extended stroh formalism radon transform and cauchy s residue theory in order to cover mathematical degenerate and non degenerate materials in the stroh formalism context a multiple residue scheme is performed expressions are explicit in terms of stroh s eigenvalues this being a feature of special interest in numerical applications such as boundary element methods as a particular case simplifications for mee materials with transversely isotropic symmetry are derived details on the implementation and numerical stability of the proposed solutions for degenerate cases are studied

mindlin s problem for an anisotropic piezoelectric half space with general boundary conditions this paper considers mindlin s problem in an anisotropic and piezoelectric halfspace with general boundary conditions including 16 different sets of surface conditions the green s function due to

two dimensional piezoelectricity part i eigensolutions of nondegenerate and degenerate materials abstract general solutions of two dimensional piezoelectricity which yield all solutions of 2 d boundary values problems are obtained by combining four complex conjugate pairs of independent eigensolutions each containing an arbitrary analytic function the forms of representation are fundamentally different for 14 different classes of nondegenerate and degenerate piezoelectric materials as determined by the multiplicity and types of eigenvalues degenerate materials possess high order eigensolutions in which the eigenvectors of equal and lower orders are intrinsically coupled such coupling is nonexistent in nondegenerate cases including the well known and analytically simple case with no multiple eigenvalues the present analysis is drastically simplified by using the compliance based formalism instead of the stiffness based extended eshelby stroh formalism explicit expressions are obtained for the eigensolutions the pseudometrics and the intrinsic tensors characterizing piezoelectric materials of every type

general solution for eshelby s problem of 2d arbitrarily shaped piezoelectric inclusions abstract eshelby s problem of piezoelectric inclusions arises sometimes in exploiting the electromechanical coupling effect in piezoelectric media for example it intervenes in the nanostructure design of strained semiconductor devices involving strain induced quantum dot qd and quantum wire qwr growth using the extended stroh formalism the present work gives a general analytical solution for eshelby s problem of two dimensional arbitrarily shaped piezoelectric inclusions the key step toward obtaining this general solution is the derivation of a simple and compact boundary integral expression for the eigenfunctions in the extended stroh formalism applied to eshelby s problem the simplicity and compactness of the boundary integral expression derived make it much less difficult to analytically tackle eshelby s piezoelectric problem for a large variety of non elliptical inclusions in the present work explicit analytical solutions are obtained and detailed for all polygonal inclusions and for the inclusions characterized by jordan s curves and laurent s polynomials by considering the piezoelectric material gaas 110 the analytical solutions provided are illustrated numerically to verify the coincidence between different expressions and to clarify the jump across the boundary of the inclusion and the singularity around the corner of the inclusion

plane problems in piezoelectric media with defects abstract a two dimensional electroelastic analysis is performed on a transversely isotropic piezoelectric material containing defects a general solution is provided in terms of complex potentials with emphasis being placed on stress concentrations that arise in the vicinity of circular and elliptical holes it is shown that for this genre of problem both mechanical and electrical variables are responsible for the peak stresses

a convergent difference scheme for the infinity laplacian construction of absolutely minimizing lipschitz extensions this article considers the problem of building absolutely minimizing lipschitz extensions to a given function these extensions can be characterized as being the solution of a degenerate elliptic partial differential equation the infinity laplacian for which there exist unique viscosity solutions a convergent difference scheme for the infinity laplacian equation is introduced which arises by minimizing the discrete lipschitz constant of the solution at every grid point existence and uniqueness of solutions to the scheme is shown directly solutions are also shown to satisfy a discrete comparison principle solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation

a nonconforming morley finite element method for the fully nonlinear monge ampere equation in this paper we study finite element approximations of the viscosity solution of the fully nonlinear monge ampere equation det d 2 u f 0 using the well known nonconforming morley element our approach is based on the vanishing moment method which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and feng j sci comput 38 1 74 98 2009 the vanishing moment method approximates the monge ampere equation by the fourth order quasilinear equation epsilon delta 2u epsilon rm det d 2u epsilon f with appropriate boundary conditions we develop a finite element scheme using the n dimensional morley element introduced in wang and xu numer math 103 155 169 2006 to approximate the regularized fourth order problem in two and three dimensions and then derive optimal order error estimates

boundary element free method for fracture analysis of 2 d anisotropic piezoelectric solids this paper considers a 2 d fracture analysis of anisotropic piezoelectric solids by a boundary element free method a traction boundary integral equation bie that only involves the singular terms of order 1 r is first derived using integration by parts new variables namely the tangential derivative of the extended displacement the extended displacement density for the general boundary and the tangential derivative of the extended crack opening displacement the extended displacement dislocation density are introduced to the equation so that solution to curved crack problems is possible this resulted equation can be directly applied to general boundary and crack surface and no separate treatments are necessary for the upper and lower surfaces of the crack the extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions and the unknown weight functions are modelled using the moving least squares mls approximation the numerical scheme of the boundary element free method is established and an effective numerical procedure is adopted to evaluate the singular integrals the extended stress intensity factors sifs are computed for some selected example problems that contain straight or curved cracks and good numerical results are obtained copyright 2006 john wiley sons ltd

completeness and nonuniqueness ofgeneral solutions of transversely isotropic elasticity abstract in this paper we give general solutions of transversely isotropic elasticity their completenessand nonuniqueness are proved we point out that famous lekhnitskii hu nowacki solutions and elliott lodge solutions are complete if the elastic region is z convex

on hermitian holomorphic classes related to uniformization the dilogarithm and the liouville action metrics of constant negative curvature on a compact riemann surface are critical points of the liouville action functional which in recent constructions is rigorously defined as a class in a cech de rham complex with respect to a suitable covering of the surface we show that this class is the square of the metrized holomorphic tangent bundle in hermitian holomorphic deligne cohomology we achieve this by introducing a different version of the hermitian holomorphic deligne complex which is nevertheless quasi isomorphic to the one introduced by brylinski in his construction of quillen line bundles we reprove the relation with the determinant of cohomology construction furthermore if we specialize the covering to the one provided by a kleinian uniformization thereby allowing possibly disconnected surfaces the same class can be reinterpreted as the transgression of the regulator class expressed by the bloch wigner dilogarithm

fourier integral representation of the green function for an anisotropic elastic half space this paper describes the development of a fourier integral representation of the green function for an anisotropic elastic half space the representation for an isotropic material is integrated in closed form and shown to reduce to mindlin s solution an application of the anisotropic representation is made to deduce the exact displacement caused by a two dimensional periodic vertical force distribu tion applied to the interior of a half space with cubic material symmetry

green s function of anisotropic piezoelectricity in this paper we give an algebraic formula for the green s function of piezoelectricity that is we write the green s function of piezoelectricity using the eigenvectors and or the generalized eigenvectors of the piezoelectric version of stroh s eigenvalue problem moreover applying this formula to piezoelectricity of hexagonal symmetry 622 we obtain an explicit expression of the green s function

general solutions for coupled equations for piezoelectric media abstract three general solutions are obtained for the coupled dynamic equations for a transversely isotropic piezoelectric medium these solutions are expressed in terms of the two functions and f where satisfies a second degree partial differential equation and f a sixth degree partial differential equation respectively if the terms concerning the derivatives of time are removed the results become three general solutions for the corresponding equilibrium equations in which the function f can be represented by functions fi i 1 2 3 each of which satisfies a second degree partial differential equation by utilizing a generalized almansi theorem and the solution wang and zheng int j solids structures32 105 115 1995 obtained is proved to be consistent with one case of one of the three general solutions when the constants e15 e31 e33 0 the piezo electric coupling is absent then two of the solutions reduce to the elasticity general solutions for a transversely isotropie medium one of which is the result hu acta scientia sin 2 2 145 151 1953 obtained the other one has not been published last the solution in the limiting explicit form for the problem for a half space with concentrated loads at the boundary is obtained by utilizing the general solutions

a general algorithm for the numerical evaluation of nearly singular integrals on 3d boundary element a general numerical method is proposed to compute nearly singular integrals arising in the boundary integral equations bies the method provides a new implementation of the conventional distance transformation technique to make the result stable and accurate no matter where the projection point is located the distance functions are redefined in two local coordinate systems a new system denoted as a b is introduced here firstly its implementation is simpler than that of the polar system and it also performs efficiently then a new distance transformation is developed to remove or weaken the near singularities to perform integration on irregular elements an adaptive integration scheme is applied numerical examples are presented for both planar and curved surface elements the results demonstrate that our method can provide accurate results even when the source point is very close to the integration element and can keep reasonable accuracy on very irregular elements furthermore the accuracy of our method is much less sensitive to the position of the projection point than the conventional method

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

generating functional in cft and effective action for two dimensional quantum gravity on higher genus riemann surfaces we formulate and solve the analog of the universal conformal ward identity for the stress energy tensor on a compact riemann surface of genus g 1 and present a rigorous invariant formulation of the chiral sector in the induced two dimensional gravity on higher genus riemann surfaces our construction of the action functional uses various double complexes naturally associated with a riemann surface with computations that are quite similar to descent calculations in brst cohomology theory we also provide an interpretation of the action functional in terms of the geometry of different fiber spaces over the teichmuller space of compact riemann surfaces of genus g 1

green s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces by virtue of two systems of vector functions and the propagator matrix method greens functions for transversely isotropic piezoelectric functionally graded exponentially graded in the vertical direction and multilayered half spaces are derived it is observed that the homogeneous solution and propagator matrices for each functionally graded layer in the transformed domain are independent of the choice of the two systems of vector functions for a point force and point charge density applied at any location of the functionally graded half space greens functions are expressed in terms of one dimensional infinite integrals to carry out the numerical integral involving bessel functions an adaptive gauss quadrature approach is introduced and modified the piezoelectric functionally graded greens functions include those in the corresponding elastic functionally graded media as special results with the latter being also unavailable in the literature two piezoelectric functionally graded half space models are analyzed numerically one is a functionally graded pzt 4 half space and the other a coated functionally graded pzt 4 layer over a homogeneous batio3 half space the effects of different exponential factors on greens function components are clearly demonstrated which could be useful in the design and manufacturing of piezoelectric functionally graded structures 2004 elsevier ltd all rights reserved

alternate bem formulations for 2 and 3 d anisotropic thermoelasticity abstract direct boundary element formulations are presented for the elastostatic analysis of two and three dimensional general anisotropic media subject to known temperature distributions the thermal effects are incorporated in the analysis through a volume integral or alternatively using particular integrals a multi region boundary element approach is adopted in which particular integrals are derived assuming a quadratic temperature distribution in each sub region a multiple regression scheme is used for the best fit quadratic temperature distribution in each sub region for temperature data given at discrete points the multi region analysis capability of the numerical implementation enhances its capacity to model discontinuous temperature fields as well as piecewise homogeneous media such as laminated composites the formulations presented are incorporated in a general purpose system and are illustrated with examples

high order derivatives of green s functions in magneto electro elastic materials abstract three dimensional green s functions and their arbitrary order derivatives in general anisotropic magneto electro elastic materials are derived by using fourier transform they are analytical solutions expressed in line integral forms and can be evaluated by a standard numerical integration method with this method we can obtain results with high accuracy besides a numerical finite difference method is also given to evaluate the second order derivatives quickly when setting the appropriate material coefficients to zero the piezoelectric piezomagnetic and purely anisotropic elastic green s functions and their derivatives can all be obtained from the current solutions

recent developments in numerical methods for fully nonlinear second order partial differential equations this article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations pdes a relatively new subarea within numerical pdes due to their ever increasing importance in mathematics itself e g differential geometry and pdes and in many scientific and engineering fields e g astrophysics geostrophic fluid dynamics grid generation image processing optimal transport meteorology mathematical finance and optimal control numerical solutions to fully nonlinear second order pdes have garnered a great deal of interest from the numerical pde and scientific communities significant progress has been made for this class of problems in the past few years but many problems still remain open this article intends to introduce these current advancements and new results to the siam community and generate more interest in numerical methods for fully nonlinear pdes

potentials and chern forms for weil petersson and takhtajan zograf metrics on moduli spaces abstract for the tz metric on the moduli space m 0 n of n pointed rational curves we construct a kahler potential in terms of the fourier coefficients of the klein s hauptmodul we define the space s g n as holomorphic fibration s g n s g over the schottky space s g of compact riemann surfaces of genus g where the fibers are configuration spaces of n points for the tautological line bundles l i over s g n we define hermitian metrics h i in terms of fourier coefficients of a covering map j of the schottky domain we define the regularized classical liouville action s and show that exp s is a hermitian metric in the line bundle l i 1 n l i over s g n we explicitly compute the chern forms of these hermitian line bundles c 1 l i h i 4 3 tz i c 1 l exp s 1 2 wp we prove that a smooth real valued function s s i 1 n log h i on s g n a potential for this special difference of wp and tz metrics coincides with the renormalized hyperbolic volume of a corresponding schottky 3 manifold we extend these results to the quasi fuchsian groups of type g n

quantum liouville theory in the background field formalism i compact riemann surfaces using polyakov s functional integral approach and the liouville action functional defined in zt87c and tt03a we formulate quantum liouville theory on a compact riemann surface x of genus g 1 for the partition function x and correlation functions with the stress energy tensor components langle prod i 1 n t z i prod k 1 l bar t bar w k x rangle we describe feynman rules in the background field formalism by expanding corresponding functional integrals around a classical solution the hyperbolic metric on x extending analysis in tak93 tak94 tak96a tak96b we define the regularization scheme for any choice of the global coordinate on x for the schottky and quasi fuchsian global coordinates we rigorously prove that one and two point correlation functions satisfy conformal ward identities in all orders of the perturbation theory obtained results are interpreted in terms of complex geometry of the projective line bundle fancyscript e c lambda h c 2 over the moduli space mathfrak m g where c is the central charge and h is the hodge line bundle and provide the friedan shenker fs87 complex geometry approach to cft with the first non trivial example besides rational models

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

weakly singular stress bem for 2d elastostatics a weakly singular stress bem is presented in which the linear state regularizing field is extended over the entire surface the algorithm employs standard conforming c0 elements with lagrangian interpolations and exclusively uses gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space the linear state stress bie on which the algorithm is based has no free term so that the bem treatment of external corners requires no special consideration other than to admit traction discontinuities the self regularizing nature of the somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary a key observation is the relation between bie density functions and successful interpolation orders numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations additionally the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions copyright 1999 john wiley sons ltd

state vector approach to analysis of multilayered magneto electro elastic plates abstract the state vector equations for three dimensional orthotropic and linearly magneto electro elastic media are derived from the governing equations by eliminating x y xy b x b y d x and d y an efficient method is presented for analysis of multilayered magneto electro elastic plates the methodology is based on the mixed formulation in which basic unknowns are formed by collecting not only displacements electrical potential and magnetic potential but also some of stresses electrical displacements and magnetic induction as special case simply supported and multilayered rectangular plate is analyzed under the surface loading numerical results are presented graphically the procedure of numerical calculation shows that the formulation presented here is simple and direct

three dimensional green s functions for infinite anisotropic piezoelectric media abstract a direct effective and concise method is adopted in this paper to find out the green s functions for infinite anisotropic piezoelectric media the partial differential equations satisfied by the green s functions turn into a set of inhomogeneous algebraic equations after by using fourier transform then inverse transform the solutions of the algebraic equations the green s functions can be expressed by contour integral finally the explicit expression can be obtained for the green s functions by using residual theory the method demonstrated in this paper is easier to follow by people without knowledge of radon transform which has been used to obtain the green s functions by others

analysis of thin piezoelectric solids by the boundary element method abstract the piezoelectric boundary integral equation bie formulation is applied to analyze thin piezoelectric solids such as thin piezoelectric films and coatings using the boundary element method bem the nearly singular integrals existing in the piezoelectric bie as applied to thin piezoelectric solids are addressed for the 2 d case an efficient analytical method to deal with the nearly singular integrals in the piezoelectric bie is developed to accurately compute these integrals in the piezoelectric bem no matter how close the source point is to the element of integration promising bem results with only a small number of elements are obtained for thin films and coatings with the thickness to length ratio as small as 10 6 which is sufficient for modeling many thin piezoelectric films as used in smart materials and micro electro mechanical systems

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

completeness of general solutions for three dimensional transversely isotropic piezoelectricity abstract in this paper a general solution for three dimensional transversely isotropic piezoelectricity in terms of four quasi quadri harmonic functions is established first owing to complexity of the higher order equation it is difficult to obtain rigorous analytic solutions and in most cases this general solution is not applicable by virtue of the generalized almansi s theorem the simplified generalized lhn solution and e l solution expressed by lower order functions are achieved respectively by taking a decomposition and superposition technique in the absence of piezoelectric coupling these two simplified general solutions can be degenerated into those for transversely isotropic elasticity i e lhn and e l solutions more importantly the completeness of these two generalized solutions is proved if the domain is z convex no matter whether the characteristic roots are distinct or possibly equal to each other

a semi analytical algorithm for the evaluation of the nearly singular integrals in three dimensional boundary element methods the nearly singular integrals occur in the boundary integral equations when the source point is close to an integration element as compared to its size but not on the element in this paper the concept of a relative distance from a source point to the boundary element is introduced to describe possible influence of the singularity of the integrals then a semi analytical algorithm is proposed for evaluating the nearly strongly singular and hypersingular integrals in the three dimensional bem by using integration by parts the nearly singular surface integrals on the elements are transformed to a series of line integrals along the contour of the element the singular behavior which appears as factor is separated from remaining regular integrals consequently standard numerical quadrature can provide very accurate evaluation of the resulting line integrals the semi analytical algorithm is applied to analyzing the three dimensional elasticity problems such as very thin walled structures meanwhile the displacements and stresses at the interior points very close to its bounding surface are also determined efficiently the results of the numerical investigation demonstrate the accuracy and effectiveness of the algorithm

liouville action and holography on quasi fuchsian deformation spaces we study the liouville action for quasi fuchsian groups with parabolic and elliptic elements in particular when the group is fuchsian the contribution of elliptic elements to the classical liouville action is derived in terms of the bloch wigner functions we prove the first and second variation formulas for the classical liouville action on the quasi fuchsian deformation space we prove an equality expressing the holography principle which relates the liouville action and the renormalized volume for quasi fuchsian groups with parabolic and elliptic elements we also construct the potential functions of the kahler forms corresponding to the takhtajan zograf metrics associated to the elliptic elements in the quasi fuchsian groups

symbolic and numerical solution of the axisymmetric indentation problem for a multilayered elastic coating abstract this paper is concerned with a semi analytical approach to the solution of the axisymmetric indentation problem for a multilayered elastic half space the stress and displacement fields for each layer and the substrate are derived in closed form by using the papkovich neuber potentials and the hankel transform the bonded or sliding interface conditions between the sub layers are handled by the use of the appropriate transfer matrix and then the mixed boundary value problem is reduced to a fredholm integral equation symbolic and numerical computations of the solution are implemented in the symbolic software mathematica in the form of a fast and efficient numerical algorithm allowing rapid determination of the load displacement curves and composite elastic properties for an arbitrary rigid indenter shape a series of results for different indenters flat conical spherical and blunted conical punch shapes and different multilayered composites is presented and discussed the complete set of symbolic and numerical computations are provided as supplementary resources with the paper

three dimensional stress analysis of thin structures using a boundary element method with sinh transformation for nearly singular integrals abstract in this work a three dimensional 3d boundary element method was established with an efficient nonlinear coordinate transformation scheme namely sinh transformation to evaluate nearly singular integrals in boundary integral formulations second order quadrilateral surface elements were developed based on this method to accurately describe the geometry of thin structures the elastic behaviors of selected thin structures were then computed by using the 3d boundary element model to demonstrate the accuracy and efficiency of this approach a number of testing examples i e the 3d kirsch problem the thin spherical shell problem the ellipsoidal vessel problem with non uniform thickness and the hollow circular cylinder problem were numerically studied to test the established method remarkable accuracy and efficiency were found in the developed approach through the comparison to the numerical results and analytical solutions reported in the literature

convergent finite difference methods for one dimensional fully nonlinear second order partial differential equations this paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations pdes in 1 d the goal of the paper is to extend the successful framework of monotone consistent and stable finite difference methods for first order fully nonlinear hamilton jacobi equations to second order fully nonlinear pdes such as monge ampere and bellman type equations new concepts of consistency generalized monotonicity and stability are introduced among them the generalized monotonicity and consistency which are easier to verify in practice are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear hamilton jacobi equations the main component of the proposed framework is the concept of a numerical operator and the main idea used to design consistent generalized monotone and stable finite difference methods is the concept of a numerical moment these two new concepts play the same roles the numerical hamiltonian and the numerical viscosity play in the finite difference framework for first order fully nonlinear hamilton jacobi equations in the paper two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear pdes the first class contains lax friedrichs like methods which also are proved to be stable and the second class contains godunov like methods numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper

finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method the vanishing moment method was introduced by the authors in feng and neilan 2009 as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations pdes it is based on the simple idea of approximating a hard to handle fully nonlinear second order pde by a family parameterized by a small parameter e of easy to handle quasilinear fourth order pdes the primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic pdes which fulfill some structure conditions abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order pdes in a general setting the abstract framework is then applied to three prototypical nonlinear equations namely the monge ampere equation the equation of prescribed gauss curvature and the infinity laplacian equation numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology

trefftz indirect methods for plane piezoelectricity in this paper a unified theory for the general solutions of three dimensional elasticity is employed to transform the governing equations of plane piezoelectricity into three generalized laplace equations the latter equations are uncoupled and can be pursued separately by introducing three generalized complex variables the complete solution sets for plane piezoelectricity are constructed by adopting the constituents in the solution sets as the trial functions the trefftz collocation and the trefftz galerkin methods are formulated both methods fall into the category of trefftz indirect methods as the coefficients of their trial functions are non physical numerical examples are presented to illustrate the efficacy of the formulations copyright 2005 john wiley sons ltd

the boundary layer phenomena in two dimensional transversely isotropic piezoelectric media by exact symplectic expansion a separable variable method is introduced to find the exact homogeneous solutions of a two dimensional transversely isotropic piezoelectric media to handle general boundary conditions the usual method of separable variables for partial differentiation equations cannot be readily applicable due to the tangling of the unknowns and their derivatives introducing dual variables of stresses we obtain a set of first order hamiltonian equations whose eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions the solutions consist of two parts the first part is the derogative zero eigenvalue solutions of the saint venant type together with all their jordan chains the second part is the decaying non zero eigenvalue solutions describing the boundary layer effects the classical solutions are actually the zero eigenvalue solutions representing the simple extension bending equipotential field and the uniform electric displacement on the other hand the non zero eigenvalue solutions represent the localized solutions which are sensitive to the boundary conditions and are decaying rapidly with respect to the distance from the boundaries some rate of decay curves of the newly found non zero eigenvalue solutions are shown by numerical examples finally the complete boundary layer effects are quantified for the first time

green s functions for a two phase infinite piezoelectric plane first on the basis of the governing equations of piezoelectric materials a displacement function is introduced and the general solutions are derived second by employing the generalized almansi s theorem these solutions are further simplified i e all physical quantities are expressed in three displacement functions i i 1 2 3 which satisfy respectively three similar harmonic secondorder partial differential equations then the green s functions for point forces and point charge acting in the interior of a twophase infinite piezoelectric plane are given using the method of mirror image source the paper is concluded by a discussion of some special cases in which the point force and charge solutions for an infinite uniform piezoelectric plane and for a halfplane with free or fixed straight boundary are given

quasi fuchsian manifolds with particles we consider 3 dimensional hyperbolic cone manifolds which are convex cocompact in a natural sense with cone singularities along infinite lines such singularities are sometimes used by physicists as models for massive spinless point particles we prove an infinitesimal rigidity statement when the angles around the singular lines are less than any infinitesimal deformation changes either these angles or the conformal structure at infinity with marked points corresponding to the endpoints of the singular lines moreover any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone manifold structure these results hold also when the singularities are along a graph i e for interacting particles

the precise computation for wave propagation in stratified materials the problem of random wave travelling along stratified materials is important for earthquake engineering the soil is considered multi layered and located above the base rock whose material property is assumed to be much stiffer than the soil and the power spectrum density of the random excitation is given at the base rock the soils can be arbitrarily anisotropic for each layer the governing differential equations are derived in frequency and wavenumber domain and only a set of ordinary differential equations must then be solved the eigensolution expansion method is used to solve for the responses of the layers the precise integration algorithm in combination with the extended w w algorithm is applied to obtain all the eigensolutions of the ode ordinary differential equation thereafter the recently developed pseudo excitation method for structural random vibration is transplanted to the solution of the layered soil responses numerical results are given and compared with some classical results this shows the relevance of the presented method copyright 2004 john wiley sons ltd

eshelby s problem for two dimensional piezoelectric inclusions of arbitrary shape eshelby s problem for piezoelectric inclusions of arbitrarily shaped crosssection remains a challenging topic in this paper a simple method is presented to obtain an analytic solution for eshelby

complete and exact solutions of a penny shaped crack in a piezoelectric solid antisymmetric shear loadings this paper presents an exact analysis of the problem of a penny shaped crack in a transversely isotropic piezoelectric medium subjected to arbitrary shear loading that is antisymmetric with respect to the crack plane the analysis is based on the general solution of three dimensional piezoelasticity which is represented by four quasi harmonic displacement functions it is shown that these harmonics can be represented by one complex potential by using the previous results in potential theory an exact solution is obtained in particular for uniform shear and point shear loadings complete expressions for the elastoelectric field are derived in terms of elementary functions