closed form green s dyadics for a class of media with axial bi anisotropy the green s dyadics are derived for a new combined nonreciprocal and uniaxial bi anisotropic medium with material dyadics of the form spl mu sup spl alpha spl epsiv sup t spl xi u sup spl xi u sub z u sub z and spl zeta sup spl zeta u sub z u sub z the result can be expressed as an infinite series of exponential integrals it is investigated for which media this series truncates to a closed form expression and the result is checked with known green s dyadics for special cases of the medium parameters

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

governing equations and general solutions of plane elasticity of one dimensional quasicrystals plane elasticity theory of one dimensional quasicrystals dealing with all point groups is investigated systematically the governing equations of elastic fields and their general solutions are derived by the complex variable functions method as an example the elastic fields of a straight dislocation along the quasiperiodic axis of an orthorhombic quasicystal are calculated the relevant singularity of the stress field for the dislocation in the quasicrystals is also discussed

some explicit expressions of extended stroh formalism for two dimensional piezoelectric anisotropic elasticity abstract since the extended stroh formalism for two dimensional piezoelectric anisotropic elasticity preserves essential features of stroh formalism for pure elastic materials it becomes important to get the corresponding explicit expressions of some important matrices frequently appeared in stroh formalism in this paper explicit expressions are obtained for the fundamental matrix n material eigenvector matrices a and b and barnett lothe tensors l s and h although the explicit expressions are presented under the generalized plane strain and short circuit condition by suitable replacement of the material constants they are still valid for the other two dimensional states to provide a clear picture of these expressions two typical examples are presented which are piezoelectric ceramics with two different poling axes

a priori error estimates for space time finite element discretization of parabolic optimal control problems part i problems without control constraints in this paper we develop a priori error analysis for galerkin finite element discretizations of optimal control problems governed by linear parabolic equations the space discretization of the state variable is done using usual conforming finite elements whereas the time discretization is based on discontinuous galerkin methods for different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters the paper is divided into two parts in the first part we develop some stability and error estimates for space time discretization of the state equation and provide error estimates for optimal control problems without control constraints in the second part of the paper the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable

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

a first order perturbation analysis of a non ideal crack in a piezoelectric material abstract in this paper a first order perturbation analysis is carried out on a symmetrically perturbed non ideal crack for three kinds of electric boundary conditions namely permeable impermeable and conducting crack boundary condition by using the extended stroh formula the two domain problems are reduced to standard riemann hilbert problems and the singular integral equations of the internal electric field inside the permeable crack are solved the stress and electric intensity factors seifs are determined to the first order of accuracy the results indicate that for a symmetrically perturbed non ideal crack the electro mechanical loading at infinity does not affect the first order solution for the mode i intensity factor for general piezoelectric materials the energy release rate and the seifs are determined by remote mechanical loads only and the perturbation effect on the seifs and energy release rate is small the electric field distribution inside crack is constant for the zeroth order solution and quadratic for the first order solution which is different from the constant electric field distribution for an ideal permeable crack the internal electric concentration near the crack tip caused by the perturbation reveals that the dielectric inside the crack probably breaks down before the matrix does when the matrix is subjected to a not too high electro mechanical load at infinity the seifs and the energy release rate are also given for the non ideal crack under the impermeable and conducting electric boundary condition respectively for all three kinds of electric boundary conditions the lateral stresses 11 13 have no contribution to the seifs to the first order of accuracy

numerical sensitivity analysis for the quantity of interest in pde constrained optimization in this paper we consider the efficient computation of derivatives of a functional the quantity of interest which depends on the solution of a pde constrained optimization problem with inequality constraints and which may be different from the cost functional the optimization problem is subject to perturbations in the data we derive conditions under with the quantity of interest possesses first and second order derivatives with respect to the perturbation parameters an algorithm for the efficient evaluation of these derivatives is developed with considerable savings over a direct approach especially in the case of high dimensional parameter spaces the computational cost is shown to be small compared to that of the overall optimization algorithm numerical experiments involving a parameter identification problem for navier stokes flow and an optimal control problem for a reaction diffusion system are presented which demonstrate the efficiency of the method

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 the interplay between interior point approximation and parametric sensitivities in optimal control this paper is concerned with the sensitivities of function space oriented interior point approximations in parameter dependent optimization problems for an abstract setting that covers control constrained optimal control problems the convergence of interior point sensitivities to the sensitivities of the optimal solution is shown error bounds for lq norms are derived and illustrated with numerical examples ams msc 2000 90c51 90c31 49m30

a multigrid method for constrained optimal control problems we consider the fast and efficient numerical solution of linear quadratic optimal control problems with additional constraints on the control discretization of the first order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints the complementarity conditions are treated by a primal dual active set strategy that serves as outer iteration at each iteration step a kkt system has to be solved here we develop a multigrid method for its fast solution to this end we use a smoother which is based on an inexact constraint preconditioner we present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying elliptic pde problem furthermore when combined with a nested iteration the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the pde solution

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

optimal control of the stokes equations a priori error analysis for finite element discretization with postprocessing an optimal control problem for 2d and 3d stokes equations is investigated with pointwise control constraints this paper is concerned with the discretization of the control by piecewise constant functions the state and the adjoint state are discretized by finite element schemes in the paper a postprocessing strategy is suggested which allows for significant improvement of the accuracy

exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack based on the complex potential approach the two dimensional problems in a piezoelectric material containing an elliptic hole subjected to uniform remote loads are studied the explicit closed form solutions satisfying the exact electric boundary condition on the hole surface are given both inside and outside the hole when the elliptic hole degenerates into a crack the field intensity factors are obtained it is shown that the stress intensity factors are the same as that of isotropic material while the electric displacement intensity factor depends on both the material properties and the mechanical loads but not on the electric loads in other words the uniform electric loads have no influence on the field singularities it is also shown that the impermeable crack assumption used previously to simply the electric condition is not valid to crack problems in piezoelectric materials

new developments concerning piezoelectric materials with defects abstract the problem of a piezoelectric body with an elliptic cavity is revisited within the frame work of two dimensional electro elasticity in contrast to our previous formulation the present analysis is based on the use of exact electric boundary conditions at the rim of the hole thus avoiding the common assumption of electric impermeability expressions for the elastic and electric variables induced inside and outside the cavity are derived in closed form in terms of complex potentials comparisons between present exact and previous approximate models are effected to establish the limitations associated with the assumption of the impermeable hole in particular when it becomes a slit crack

ciarlet raviart mixed finite element approximation for an optimal control problem governed by the first bi harmonic equation the ciarlet raviart mixed finite element approximation is constructed to solve the constrained optimal control problem governed by the first bi harmonic equation the optimality conditions consisting of the state and the co state equations is derived also the a priori error estimates are analyzed in the analysis of the a priori error estimates the improved convergent rate of the higher order than existed results is proved some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

least squares finite element methods for optimization and control problems for the stokes equations the approximate solution of optimization and control problems for systems governedby the stokes equations is considered modern computational techniques for such problems are predominantly based on the application of the lagrange multiplier rule while penalty formulations even though widely used in other settings have not enjoyed the same level of popularity for this class of problems a discussion is provided that explains why naively defined penalty methods may not be practical then practical penalty methods are defined using methodologies associated with modern least squares finite element methods the advantages with respect to efficiency of penalty leasts squares methods for optimal control problems compared to methods based on lagrange multipliers are highlighted a tracking problem for the stokes system is used for illustrative purposes

consistency and recovery in electroelasticity part i standard finite elements developing finite elements for electroelastic analysis requires a special care as the coupling in the discrete equations depends on the matching between the approximations assumed for mechanical and electrical variables to provide a formal basis to this intuitive remark a notion of consistency is established and a rigorous consistency analysis is presented for the standard finite element model based on assumed displacement and electric potential in this way specific analytical requirements are obtained which serve as a guide to select the interpolation functions for primary variables moreover the analysis shows that violating consistency can be reflected as spurious outcomes upsetting the local distributions of secondary variables indeed this undesirable effect is shown to be typical of the standard approach if stress and electric flux density are computed via the constitutive equations to cure the trouble an alternative recovery procedure is devised based on the consistency analysis the procedure is variationally correct and reconstitutes in a consistent manner the distributions of stress and electric flux density

analysis of stress and electric fields in a rectangular piezoelectric body with a center crack under anti plane shear loading abstract the singular stress and electric fields in a rectangular piezoelectric ceramic body containing a griffith center crack under anti plane shear loading are obtained by the theory of linear piezoelectricity fourier transforms and fourier sine series are used to reduce the problem to a pair of dual integral equations which is expressed to a fredholm integral equation of the second kind numerical values on the stress intensity factor and the energy release rate are obtained to show the influence of the electric field

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

saint venant end effects for plane deformations of linear piezoelectric solids the recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity in this paper we investigate the decay of saint venant end effects for plane deformations of a piezoelectric semi infinite strip first of all we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid just as in the mechanical case not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non trivial state of plane deformation the governing system of four second order partial differential equations for the two in plane displacements and electric potential are overdetermined in general sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation the resulting traction boundary value problem with prescribed surface charge is an oblique derivative boundary value problem for a coupled elliptic system of three second order partial differential equations the special case of a piezoelectric material transversely isotropic about the poling axis is then considered thus the results are valid for the hexagonal crystal class 6mm the geometry is then specialized to be a two dimensional semi infinite strip and the poling axis is the axis transverse to the longitudinal direction we consider such a strip with sides traction free subject to zero surface charge and self equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity a formulation of the problem in terms of an airy type stress function and an induction function is adopted the governing partial differential equations are a coupled system of a fourth and third order equation for these two functions on seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second order ordinary differential equations this problem is the piezoelectric analog of the well known eigenvalue problem arising in the case of an anisotropic elastic strip it is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti symmetric deformations and electric fields assuming completeness of the eigenfunctions the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part numerical results are given for pzt 5h pzt 5 pzt 4 and ceramic b it is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials

analytic green dyadic for a class of nonreciprocal anisotropic media applying dyadic algebra analytic expressions are derived for green dyadics corresponding to a class of anisotropic media with permittivity and permeability dyadics satisfying the condition spl mu spl tau spl epsiv sup t this class of media contains the affinely isotropic media as a reciprocal subclass when the parameter dyadics are symmetric as a novel aspect green dyadics are thus obtained for certain nonreciprocal anisotropic media with nonsymmetric parameter dyadics

optimal control of the convection diffusion equation using stabilized finite element methods in this paper we analyze the discretization of optimal control problems governed by convection diffusion equations which are subject to pointwise control constraints we present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case moreover the in general different approaches optimize then discretize and discretize then optimize coincide for the proposed discretization scheme this allows for a symmetric optimality system at the discrete level and optimal order of convergence

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

the two dimensional electroelasticity problems for multiconnected bodies situated under electric potential difference action abstract solutions of the two dimensional electroelasticity problems for infinitely long piezoelectric cylinders with cavities and cracks under electrical potential difference action are obtained using the lekhnitskii s generalized complex potentials herewith piezoelectric bodies under consideration are free from mechanical loads some or all openings and maybe the exterior boundary of the cylinders are fully covered by thin electrodes voltage differences are supplied on the electroded surfaces the rest boundary surfaces are charge free in this case the complex potentials are investigated boundary conditions are satisfied by the least squares method there are carried out numerical investigations of behaviour of some basic electroelastic characteristics in a circular hollow cylinder and in an infinite body with two longitudinal cavities and a charge free plane crack new electromechanical regularities of influence of material properties geometrical characteristics of considered regions on values of electroelastic state characteristics and the stress electric displacements and tensions intensity factor k 1 are determined

time harmonic two and three dimensional green s dyadics for general uniaxial bianisotropic media a rigorous study of the green s dyadics in two and three dimensions for general uniaxial bianisotropic media is made the green s dyadics are derived in the time harmonic regime a general uniaxial bianisotropic medium is described by eight parameters in general it is impossible to derive a closed form representation of the green s dyadics in three dimensions in one important special case however where one relation is satisfied between the eight parameters it is possible to obtain a closed form representation

an efficient meshless technique for the solution of transversely isotropic two dimensional piezoelectricity in this paper a new meshless technique is presented for the solution of two dimensional piezoelectricity problem the technique derivation is based on the solution of the corresponding analog equation by transforming the original set of differential equations into three poisson equations with unknown right hand side terms boundary discretisation of the resulting integral equations is eliminated by the use of the method of fundamental solutions the right hand side terms are represented in the solution as particular solutions expressed in terms of radial basis functions the problem solution is then rewritten in its new form which involves complementary solution and particular solution the governing partial differential operator for piezoelectricity is applied on the obtained solution form and forced to be satisfied at a set of domain points whereas the prescribed boundary conditions are satisfied at another set of boundary points the proposed technique is implemented into computer code where several numerical examples with different boundary conditions are tested the results demonstrated excellent agreement with those obtained from analytical and fem solutions

some problems in plane thermopiezoelectric materials with holes piezoelectric materials have recently attracted considerable attention due to their potential use in intelligent structural systems in this paper we treat the plane problem of thermopiezoelasticity with various holes and subject to coupled mechanical electric and thermal loads an analytical solution is obtained by applying the technique of conformal mapping and some identities in the stroh formalism the solution has a simple unified form for various holes such as ellipse circle triangle and square by way of the solution the expressions for the energy release rate and stress intensity factors of cracks are presented numerical results for concentration coefficients of stress and electric displacement along the hole boundary are given to assess the acceptability of the proposed method

a level set approach for the solution of a state constrained optimal control problem state constrained optimal control problems for linear elliptic partial differential equations are considered the corresponding first order optimality conditions in primal dual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary interface between the active and inactive sets as optimization variable the new algorithm is based on the level set methodology the speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization encouraging numerical results attained by the new algorithm are reported on

a partition of unity method for the displacement obstacle problem of clamped kirchhoff plates a partition of unity method for the displacement obstacle problem of clamped kirchhoff plates is considered in this paper we derive optimal error estimates and present numerical results that illustrate the performance of the method

a c0 interior penalty method for the dirichlet control problem governed by biharmonic operator an energy space based dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a c0 interior penalty method is proposed and analyzed an abstract a priori error estimate is derived under the minimal regularity conditions the abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular further an optimal order l2 norm error estimate is derived numerical experiments illustrate the theoretical findings

the boundary contour method for piezoelectric media with linear boundary elements this paper presents a development of the boundary contour method bcm for piezoelectric media first the divergence free property of the integrand of the piezoelectric boundary element is proved secondly the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and green s functions computer methods in applied mechanics and engineering 1998 158 65 for piezoelectric media the bcm is applied to the problem of piezoelectric media finally numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method bem the numerical results of the bcm coincide very well with the exact solution and the feasibility and efficiency of the method are verified copyright 2003 john wiley sons ltd

the primal dual active set method for nonlinear optimal control problems with bilateral constraints the primal dual active set method has proved to be an efficient numerical tool in the context of diverse applications so far it has been investigated mainly for linear problems this paper is devoted to the study of global convergence of the primal dual active set method for nonlinear problems with bilateral constraints utilizing the close relationship between the primal dual active set method and semismooth newton methods local superlinear convergence of the method is investigated as well

interaction integrals for fracture analysis of functionally graded piezoelectric materials this paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials conservation integrals of j type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium the additional terms play an important role of ensuring domain independence of the presented interaction integrals comparison of the numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples

an all at once approach for the optimal control of the unsteady burgers equation we apply an all at once method for the optimal control of the unsteady burgers equation the nonlinear burgers equation is discretized in time using the semi implicit discretization and the resulting first order optimality conditions are solved iteratively by newton s method the discretize then optimize approach is used because it leads to a symmetric indefinite saddle point problem numerical results for the distributed unconstrained and control constrained problems illustrate the performance of the all at once approach with semi implicit time discretization

a posteriori error estimation for a pde constrained optimization problem involving the generalized oseen equations we derive globally reliable a posteriori error estimators for a linear quadratic optimal control problem involving the generalized oseen equations as state equations control constraints are also considered the corresponding local error indicators are locally efficient the assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods when stabilized methods are considered no a priori relation between the stabilization terms for the state and adjoint equations is required if a lower bound for the inf sup constant is available a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained we illustrate the theory with numerical examples

a further investigation of green s functions for a piezoelectric material with a cavity or a crack abstract in this paper the green s functions of an infinite two dimensional piezoelectric material containing an elliptical cavity are re investigated by introducing exact electric boundary conditions on the hole boundary and the corresponding modified solutions are obtained by setting 0 the dielectric constant in the cavity to be zero the modified green s functions are returned to the conventional ones lu and williams 1998 furthermore the hoop stresses along the hole boundary under the action of a set of generalized concentrated forces are obtained when the hole is reduced to a slit crack the expressions of generalized stress intensity factors are also provided

a partition of unity method for a class of fourth order elliptic variational inequalities abstract we consider a partition of unity method pum for a class of fourth order elliptic variational inequalities on convex polygonal domains that include obstacle problems of simply supported kirchhoff plates and elliptic distributed optimal control problems with pointwise state constraints as special cases by including singular functions in the local approximation spaces we are able to show that the partition of unity method converges optimally numerical results that corroborate the theoretical estimates are also presented

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

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

regularized state constrained boundary optimal control of the navier stokes equations abstract the numerical solution of the dirichlet boundary optimal control problem of the navier stokes equations in presence of pointwise state constraints is investigated two different regularization techniques are considered first a moreau yosida regularization of the problem is studied optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved a source representation of the control combined with a lavrentiev type regularization strategy is also presented the analysis concerning optimality conditions and convergence of the regularized solutions is carried out in the last part of the paper numerical experiments are presented for the numerical solution of each regularized problem a semi smooth newton method is applied

a feasible active set method for strictly convex quadratic problems with simple bounds a primal dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of kunisch and rendl siam j optim 14 2003 pp 35 52 based on a guess of the active set a primal dual pair x alpha is computed that satisfies stationarity and the complementary condition if x is not feasible the variables connected to the infeasibilities are added to the active set and a new primal dual pair x alpha is computed this process is iterated until a primal feasible solution is generated then a new active set is determined based on the feasibility information of the dual variable alpha strict convexity of the quadratic problem is sufficient for the algorithm to stop after a finite number of steps with an optimal solution computational experience indicates that this approach also performs well in practice

a coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate part ii numerical solution and applications this two part contribution presents a novel and efficient method to analyze the two dimensional 2 d electromechanical fields of a piezoelectric layer bonded to an elastic substrate which takes into account the fully coupled electric and mechanical behaviors in part i we have obtained a system of governing integro differential equations for the structure via a variational principle this part presents a numerical solution algorithm of the integro differential equations and the numerical results of some applications a numerical algorithm for solving the system of four integro differential equations with strongly singular kernels is developed the convergence of the numerical algorithm is discussed the numerical results suggest that the fully coupled electromechanical analysis is helpful for a better understanding of the performance of the piezoelectric sensor and actuator the interfacial normal stress is much higher than the interfacial shear stress suggesting that the interfacial normal stress causes a delamination initiation

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

detection of multiple flaws in piezoelectric structures using xfem and level sets abstract an iterative procedure to solve the inverse problem of detecting multiple voids in piezoelectric structure is proposed in each iteration the forward problem is solved for various void configurations and at each iteration the mechanical and electrical responses of a piezoelectric structure is minimized at known specific points along the boundary to match the measured data the extended finite element method xfem is employed for determining the responses as it allows the use of a fixed mesh for varying void geometries the numerical method based on combination of classical shape derivative and of the level set method for front propagation used in structural optimization is utilized to minimize the cost function the results obtained show that this method is effectively able to determine the number of voids in a piezoelectric structure and its corresponding locations and shapes

an exact correspondence between plane piezoelectricity and generalized plane strain in elasticity we consider an anisotropic body bounded by a cylindrical surface which is infinitely long in the axial direction suppose the body is loaded in such a way that the field variables do not vary along the generators an exact correspondence is established between the plane piezoelectric equations and those of generalized plane strain in elasticity in particular we show that by setting a linkage between the two sets of material constants any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa the assertation is true for rectilinearly as well as for cylindrically anisotropic solids the equivalence is found for the most general anisotropic case which links the fields between a monoclinic piezoelectric body of class m and a fully anisotropic triclinic elastic solid a few degenerate systems are also identified in addition these correspondences can be extended to inhomogeneous media applied to composite materials or polycrystalline aggregates they imply that results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors and vice versa we also demonstrate that the connection can be used to extend the scope of the invariant stress theorem proved by cherkaev lurie and milton in illustration we present solutions for the plane problem of an elliptical inhomogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity lekhnitskii9s complex potential approach together with the conformal mapping technique are employed general solutions of the fields inside the inhomogeneity and the matrix are obtained the results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem

solving elliptic control problems with interior point and sqp methods control and state constraints abstract we study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints both boundary control and distributed control problems are considered with boundary conditions of dirichlet or neumann type by introducing suitable discretization schemes the control problem is transcribed into a nonlinear programming problem necessary conditions of optimality are discussed both for the continuous and the discretized control problem it is shown that the recently developed interior point method loqo of 35 is capable of solving these problems even for high discretizations four numerical examples with dirichlet and neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang bang controls

feedback optimal control of distributed parameter systems by using finite dimensional approximation schemes optimal control for systems described by partial differential equations is investigated by proposing a methodology to design feedback controllers in approximate form the approximation stems from constraining the control law to take on a fixed structure where a finite number of free parameters can be suitably chosen the original infinite dimensional optimization problem is then reduced to a mathematical programming one of finite dimension that consists in optimizing the parameters the solution of such a problem is performed by using sequential quadratic programming linear combinations of fixed and parameterized basis functions are used as the structure for the control law thus giving rise to two different finite dimensional approximation schemes the proposed paradigm is general since it allows one to treat problems with distributed and boundary controls within the same approximation framework it can be applied to systems described by either linear or nonlinear elliptic parabolic and hyperbolic equations in arbitrary multidimensional domains simulation results obtained in two case studies show the potentials of the proposed approach as compared with dynamic programming

a family of second order methods for convex ell _1 l1 regularized optimization this paper is concerned with the minimization of an objective that is the sum of a convex function f and an ell 1 l1 regularization term our interest is in active set methods that incorporate second order information about the function f to accelerate convergence we describe a semismooth newton framework that can be used to generate a variety of second order methods including block active set methods orthant based methods and a second order iterative soft thresholding method the paper proposes a new active set method that performs multiple changes in the active manifold estimate at every iteration and employs a mechanism for correcting these estimates when needed this corrective mechanism is also evaluated in an orthant based method numerical tests comparing the performance of three active set methods are presented

semidiscretization and error estimates for distributed control of the instationary navier stokes equations optimal control problems governed by the two dimensional instationary navier stokes equations and their spatial discretizations with finite elements are investigated a concept of semi discrete solutions to the control problem is introduced which is utilized to prove existence and uniqueness of discrete controls in neighborhoods of regular continuous solutions furthermore an optimal error estimate in terms of the spatial discretization parameter is given

multigrid solution of a distributed optimal control problem constrained by the stokes equations in this work we construct multigrid preconditioners to accelerate the solution process of a linear quadratic optimal control problem constrained by the stokes system the first order optimality conditions of the control problem form a linear system the kkt system connecting the state adjoint and control variables our approach is to eliminate the state and adjoint variables by essentially solving two stokes systems and to construct efficient multigrid preconditioners for the schur complement of the block associated with the state and adjoint variables these multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the stokes system in particular the number of conjugate gradient iterations is shown to decrease as the resolution increases a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems

electroelastic analysis of an anti plane shear crack in a piezoelectric ceramic strip the electroelastic analysis of a piezoelectric strip with an anti plane shear crack whose surfaces are parallel to the strip boundaries is made four cases of combined electromechanical loadings applied at the strip boundaries are considered for the case of a crack lying at the center of a strip the electroelastic field field intensity factors and energy release rate can be determined in explicit form via solving a resulting singular integral equation for other cases they can be given in terms of an auxiliary function which is obtained as a power series of via solving a resulting fredholm integral equation using the iterative method 2 being the ratio of the crack length over the strip width some numerical results are presented graphically to show the influence of crack length and crack position on the normalized energy release rate

analysis of a central crack normal to a piezoelectric orthotropic interface this study is concerned with the general solution of the field intensity factors and energy release rate for a griffith crack normal to the interface between a rectangular piezoelectric ceramic and two same rectangular orthotropic materials of finite length under combined in plane electrical and anti plane mechanical loading both electrically continuous and impermeable crack surface conditions are considered employing fourier transforms and fourier series the problem is reduced to dual integral forms the solution to the resulting expressions is expressed in terms of fredholm integral equation of the second kind analytical solutions are provided for a number of cases to study the influence of material combinations geometry crack surface boundary and loading conditions the work also examines the influence of the crack boundary conditions on the results of the stress intensity factors and the energy release rate

superconvergence properties of optimal control problems an optimal control problem for a two dimensional 2 d elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise constant functions the state and the adjoint state are discretized by linear finite elements approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state it is proved that these approximations have convergence order h2

semismooth newton and augmented lagrangian methods for a simplified friction problem in this paper a simplified friction problem and iterative second order algorithms for its solution are analyzed in infinite dimensional function spaces motivated from the dual formulation a primal dual active set strategy and a semismooth newton method for a regularized problem as well as an augmented lagrangian method for the original problem are presented and their close relation is analyzed local as well as global convergence results are given by means of numerical tests we discuss among others convergence properties the dependence on the mesh and the role of the regularization and illustrate the efficiency of the proposed methodologies

error estimates for the discretization of the velocity tracking problem in this paper we are continuing our work casas and chrysafinos siam j numer anal 50 5 2281 2306 2012 concerning a priori error estimates for the velocity tracking of two dimensional evolutionary navier stokes flows the controls are of distributed type and subject to point wise control constraints the discretization scheme of the state and adjoint equations is based on a discontinuous time stepping scheme in time combined with conforming finite elements in space for the velocity and pressure provided that the time and space discretization parameters tau and h h respectively satisfy tau le ch 2 ch2 error estimates of order mathcal o h 2 o h2 and mathcal o h frac 3 2 frac 2 p o h32 2p with p 3 p 3 depending on the regularity of the target and the initial velocity are proved for the difference between the locally optimal controls and their discrete approximations when the controls are discretized by the variational discretization approach and by using piecewise linear functions in space respectively both results are based on new duality arguments for the evolutionary navier stokes equations

a framework for the adaptive finite element solution of large scale inverse problems since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging efficient methods are indispensable in this paper we will introduce a framework for the efficient solution of such problems this comprises the use of adaptive finite element schemes solvers for the large linear systems arising from discretization and methods to treat additional information in the form of inequality constraints on the parameter to be recovered the methods to be developed will be based on an all at once approach in which the inverse problem is solved through a lagrangian formulation the main feature of the paper is the use of a continuous function space setting to formulate algorithms in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed this entails that steps such as the description of a newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions on the other hand this approach avoids the dependence of finite dimensional norms on the mesh size making individual steps of the algorithm comparable even if they used differently refined meshes numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10 000 parameters

multiple collinear cracks in a piezoelectric material abstract in this paper an exact solution of multiple collinear cracks in piezoelectric material is obtained the permittivity of air environment is considered two cases have been studied in the first case the permittivity of air is far less than that of piezoelectric material therefore the electric induction in the air is negligible in the second case the permittivity of air is comparable with that of piezoelectric material the problem is deduced into riemann hilbert problem and solved by the way the electric boundary conditions are discussed this result demonstrates that the consideration of air in crack gap reduces the stress intensity factor

non commutative discretize then optimize algorithms for elliptic pde constrained optimal control problems abstract in this paper we analyze the convergence of several optimize then discretize and discretize then optimize algorithms based on either a second order or a fourth order finite difference discretization for solving elliptic pde constrained optimal control problems to ensure the convergence of a discretize then optimize algorithm one well accepted criterion is to design the discretization scheme such that the resulting discretize then optimize algorithm commutes with the corresponding optimize then discretize algorithm in other words both algorithms should give rise to exactly the same discrete optimality system however such a restrictive criterion is not trivial to fulfill by investigating a distributed control problem governed by an elliptic equation we first show that enforcing such a stringent condition of commutative property is only sufficient but not necessary for achieving the desired convergence we then introduce some suitable h 1 semi norm penalty regularization terms to recover the lost convergence due to the inconsistency caused by the loss of commutativity numerical experiments are carried out to verify our theoretical analysis and also validate the effectiveness of our proposed regularization techniques

infinite sequence of parallel cracks in an anisotropic piezoelectric solid in this paper the problem of an infinite sequence of parallel cracks in an infinite extended piezoelectric solid is analyzed a system of singular integral equations is formulated for general anisotropic piezoelectric materials of which the kernel functions developed are in complex form for commonly used transversely isotropic piezoelectric materials the kernel functions are given in real forms furthermore the obtained real kernel functions may be reduced respectively to those kernel functions for purely elastic and purely electric problems when the coupled mechanical and electric effects disappear the system of singular integral equations is solved numerically and the coupling effects of the mechanical and electric phenomena are presented by the generalized stress intensity factors

on the dynamic behavior of a functionally graded piezoelectric strip with periodic cracks vertical to the boundary abstract the dynamic anti plane problem for a functionally graded piezoelectric strip containing a periodic array of parallel cracks which are perpendicular to the boundary is considered integral transforms techniques are employed to reduce the problem to the solution of singular integral equations numerical results are presented to show the influences of geometry electromechanical combination factor and material gradient parameter on the fracture behavior

analysis and finite element approximation of optimal control problems for the stationary navier stokes equations with distributed and neumann controls we examine certain analytic and numerical aspects of optimal control problems for the stationary navier stokes equations the controls considered may be of either the distributed or neumann type the functionals minimized are either the viscous dissipation or the l4 distance of candidate flows to some desired flow we show the existence of optimal solutions and justify the use of lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced we study the regularity of solutions of this system then we consider the approximation by finite element methods of solutions of the optimality system and derive optimal error estimates

piezoelectric solid with an elliptic inclusion or hole the two dimensional problem of an elliptic hole in a solid of anisotropic piezoelectric material is studied the stroh formalism is adopted here real form solutions are obtained along the hole boundary in the case of an arbitrarily prescribed vector field on the hole surface for an elliptic rigid inclusion of electric conductor subjected to a line force a torque and a line charge a real form solution at the interface is obtained finally general solutions for an elliptic piezoelectric inclusion with uniform loading at infinity are investigated

investigation of the dynamic behavior of two parallel symmetric cracks in piezoelectric materials use of non local theory in this paper the dynamic behavior of two parallel symmetric cracks in piezoelectric materials under harmonic anti plane shear waves is investigated by use of the non local theory for permeable crack surface conditions to overcome the mathematical difficulties a one dimensional non local kernel is used instead of a two dimensional one for the problem to obtain the stress occurs near the crack tips by means of the fourier transform the problem can be solved with the help of two pairs of dual integral equations that the unknown variables are the jumps of the displacement along the crack surfaces these equations are solved using the schmidt method numerical examples are provided contrary to the previous results it is found that no stress and electric displacement singularity is present near the crack tip the non local elastic solutions yield a finite hoop stress near the crack tip thus allowing for a fracture criterion based on the maximum stress hypothesis the finite hoop stress at the crack tip depends on the crack length the frequency of the incident wave the distance between two cracks and the lattice parameter of the materials respectively contrary to the impermeable crack surface condition solution it is found that the dynamic electric displacement for the permeable crack surface conditions is much smaller than the results for the impermeable crack surface conditions the results show that the dynamic field will impede or enhance crack propagation in the piezoelectric materials at different stages of the dynamic load

primal dual active set strategy for a general class of constrained optimal control problems an algorithm for efficient solution of optimal control problems with pointwise constraints is introduced it is based on an active set strategy using primal as well as dual variables under certain assumptions a decrease of an appropriately chosen merit function and convergence of the algorithm are proved numerical examples for a parabolic optimal control problem demonstrate the efficiency of the proposed method for control constrained problems

the behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes in this paper the behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes subjected to an anti plane shear loading is investigated by using schmidt method by using the fourier transform the problem can be solved with the help of two pairs of dual integral equations these equations are solved using the schmidt method this process is quite different from that adopted previously the normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions numerical examples are provided to show the effect of the geometry of the interacting cracks the thickness and the materials constants of the piezoelectric layer upon the stress and the electric displacement intensity factor of the cracks contrary to the impermeable crack surface condition solution it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions

on stabilized mixed methods for generalized stokes problem based on the velocity pseudostress formulation a priori error estimates abstract in this paper we present an augmented mixed formulation applied to generalized stokes problem and uses it as state equation in an optimal control problem the augmented scheme is obtained adding suitable least squares terms to the corresponding velocity pseudostress formulation of the generalized stokes problem to ensure the existence and uniqueness of solution at continuous and discrete levels we prove coerciveness of the corresponding augmented bilinear form and using approximation properties of the respective discrete subspaces we deduce the optimal rate of convergence as by product and considering the associated optimal control problem we derive error estimates for the approximated control unknown finally we present several numerical examples confirming the theoretical properties of this approach

a priori error estimates for space time finite element discretization of parabolic optimal control problems in this article we summarize recent results on a priori error estimates for space time finite element discretizations of linear quadratic parabolic optimal control problems we consider the following three cases problems without inequality constraints problems with pointwise control constraints and problems with state constraints pointwise in time for all cases error estimates with respect to the temporal and to the spatial discretization parameters are derived the results are illustrated by numerical examples

two dimensional piezoelectricity part ii general solution green s function and interface cracks abstract the complete solution space of a piezoelectric material is the direct sum of several orthogonal eigenspaces one for each distinct eigenvalue each one of the 14 different classes of piezoelectric materials has a distinct form of the general solution expressed in terms of the eigenvectors of the zeroth and higher orders and a kernel matrix containing analytic functions when these functions are chosen to be logarithmic one obtains in a unified way green s function of the infinite space as a single 8 8 matrix function g for the various load cases of concentrated line forces dislocations and a line charge this expression of green s function is valid for all classes of nondegenerate and degenerate materials with an appropriate choice of the parameters it reduces to the solution of a half space with concentrated line forces at a boundary point and with dislocations in the displacements as another application eigenvalues and eigensolutions are obtained for the bimaterial interface crack problem

robust error estimates for the finite element approximation of elliptic optimal control problems in this paper we consider the finite element approximation of an elliptic optimal control problem based on an assumption on the adjoint state of the continuous problem with a small parameter which represents a regularization of the bang bang type control problem we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented numerical experiments confirm our theoretical results

edge stabilization for the generalized stokes problem a continuous interior penalty method in this note we introduce and analyze a stabilized finite element method for the generalized stokes equation stability is obtained by adding a least squares penalization of the gradient jumps acro

l infty estimates for approximated optimal control problems an optimal control problem for a two dimensional elliptic equation is investigated with pointwise control constraints this paper is concerned with discretization of the control by piecewise linear functions the state and the adjoint state are discretized by linear finite elements approximation of order h in the l infty norm is proved in the main result

trefftz solutions for piezoelectricity by lekhnitskii s formalism and boundary collocation method in this paper a solution procedure for plane piezoelectricity is developed by trefftz boundary collocation method starting with the general plane piezoelectricity solution derived by lekhnitskii s formalism the basic sets of trefftz functions which satisfy the homogeneous governing equations are derived moreover special sets of trefftz functions are derived for impermeable elliptical voids impermeable sharp cracks and permeable sharp cracks with arbitrary orientations with respect to the material poling direction the functions in the special sets satisfy not only the homogeneous governing equations but also the boundary conditions at the peripheries of the pertinent defects by adopting trefftz functions as the trial functions multi domain trefftz boundary collocation method is formulated numerical examples are presented to illustrate the efficacy of the formulation copyright 2005 john wiley sons ltd

a locking free fem in active vibration control of a timoshenko beam in this paper we analyze the numerical approximation of an active vibration control problem of a timoshenko beam in order to avoid locking we focus on the finite element method used to compute the beam vibration to minimize it optimal order error estimates are obtained for the control variable which is the amplitude of secondary forces modeled as dirac s delta distributions these estimates are valid with constants that do not depend on the thickness of the beam in order to assess the performance of the method numerical tests are reported

fracture in magnetoelectroelastic materials using the extended finite element method static fracture analyses in two dimensional linear magnetoelectroelastic mee solids is studied by means of the extended finite element method x fem in the x fem crack modeling is facilitated by adding a discontinuous function and the crack tip asymptotic functions to the standard finite element approximation using the framework of partition of unity in this study media possessing fully coupled piezoelectric piezomagnetic and magnetoelectric effects are considered new enrichment functions for cracks in transversely isotropic mee materials are derived and the computation of fracture parameters using the domain form of the contour interaction integral is presented the convergence rates in energy for topological and geometric enrichments are studied excellent accuracy of the proposed formulation is demonstrated on benchmark crack problems through comparisons with both analytical solutions and numerical results obtained by the dual boundary element method copyright 2011 john wiley sons ltd