thermomechanically coupled sensitivity analysis and design optimization of functionally graded materials this paper presents a systematic numerical technique for performing sensitivity analysis of coupled thermomechanical problem of functionally graded materials fgms general formulations are presented based on finite element model by using the direct method and the adjoint method in the modeling of spatial variances of material properties the graded finite element method is employed to conduct the heat transfer analysis and structural analysis and their sensitivity analysis the design variables are the volume fractions of fgms constituents and structural shape parameters the design optimization model is then constructed and solved by the sequential linear programming slp numerical examples are presented to demonstrate the accuracy and the applicability of the present method

a block equation solver for large unsymmetric matrices arising in the boundary integral equation method an equation solver for the large unsymmetric systems of linear equations arising in the application of the boundary integral equation method to problems of linear elasticity in homogeneous or piecewise homogeneous solids is presented the solver uses gaussian elimination the advantages of the solver over many existing eliminational or iterational methods are that it ignores many of the large groups of zeros that occur in piecewise homogeneous work and also restrains growth of the number of non zero matrix entries memory requirements and computation times therefore are reduced for many piecewise homogeneous problems

preconditioned krylov solvers for bea the performance of a number of preconditioned krylov methods is analysed for a large variety of boundary element formulations low and high order element two dimensional 2 d and three dimensional 3 d regular singular and hypersingular collocation and symmetric galerkin single and multi zone thermal and elastic continuous and discontinuous boundary formulations with and without condensation are considered preconditioned conjugate gradient cg solvers in standard form and a form effectively operating on the normal equations cgn generalized minimal residual gmres conjugate gradient squared cgs and stabilized bi conjugate gradient bi cgstab krylov solvers are employed in this study both the primitive and preconditioned matrix operators are depicted graphically to illustrate the relative amenability of the alternative formulations to solution via kryiov methods and to contrast and explain their computational performances a notable difference between 2 d and 3 d bea operators is readily visualized in this manner numerical examples are presented and the relative conditioning of the various discrete bea operators is reflected in the performance of the krylov equation solvers a preconditioning scheme which was found to be uncompetitive in the collocation bea context is shown to make iterative solution of symmetric galerkin bea problems more economical than employing direct solution techniques we conclude that the preconditioned krylov techniques are competitive with or superior to direct methods in a wide range of boundary formulated problems and that their performance can be partially correlated with certain problem characteristics

shape design sensitivity analysis for geometrically and materially nonlinear problems by the boundary element method abstract this paper is concerned with the determination of design sensitivity coefficients dscs for fully nonlinear problems i e problems with both material and geometric nonlincarities in solid mechanics the clastic strains arc assumed to be small but the nonclastic plastic or viscoplustic strains as well as rotations can be arbitrarily large dscs here refer to rates of change of response variables such as displacements or stresses in a deforming solid body with respect to parameters called design variables that define the initial undeformed shape of the body these dscs are history dependent the direct differentiation approach dda of the relevant derivative boundary element method dbem formulation is employed to obtain the dscs a general computer program is developed for the analysis of two dimensional plane strain and plane stress problems numerical results are obtained for illustrative problems and arc compared against direct solutions the results from the two methods agree almost perfectly for these problems this research has potential applications in the optimization of manufacturing processes such as extrusion rolling or forging

free vibration analysis of cylindrical helical springs by the pseudospectral method abstract the pseudospectral method is applied to the free vibration analysis of cylindrical helical springs the displacements and the rotations are approximated by the series expansions of chebyshev polynomials and the governing equations are collocated the number of collocation points is chosen to be less than the number of the expansion terms to handle the boundary condition numerical examples are provided for fixed fixed free free fixed free and hinged hinged boundary conditions the results show good agreement with those of the transfer matrix method and the dynamic stiffness method the formulation of the pseudospectral method is straightforward and shows an exponential rate of convergence with mesh refinement

boundary integral equations for recovery of design sensitivities in shape optimization a new formulation for obtaining design sensitivities in shape optimization has been developed the formulation is based on a direct application of the material derivative concept to the appropriate boundary integral equations for displacements and stresses in an elastic solid as a check on accuracy the approach was applied to a uniformly loaded infinite plate containing an elliptical hole in this case the availability of an analytical solution made it possible to calculate errors exactly furthermore a wide range of stress states could be considered simply by varying the aspect ratio of the elliptical hole for convenience the design sensitivities were calculated with respect to changes in the major axis numerical convergence was established by comparing results based on four successive boundary meshes in general the predictions for both displacement and stress sensitivities were remarkably accurate

design sensitivities in problems involving material and geometric nonlinearities abstract the subject of this paper is the determination of design sensitivity coefficients dscs for linear elasticity and nonlinear both material and geometric problems in solid mechanics dscs here refer to rates of change of response variables such as displacements or stresses in a deforming solid body with respect to parameters called design variables that control the initial undeformed shape of the body in nonlinear problems these dscs are history dependent the direct differentiation approach dda of the relevant derivative boundary element method dbem formulation is employed here to obtain the dscs modelling of sharp corners on the boundary of a body is given special attention here numerical results are presented for illustrative examples the numerically obtained sensitivities even for stresses on the boundary of a body are shown in general to be very accurate typically it is difficult to obtain dscs for boundary stresses accurately by other methods yet they are very useful for optimal shape design of bodies ongoing research is currently focused on numerical calculation of dscs for large deformation problems and use of these dscs in process optimization an important future application area of this work is optimal design of manufacturing processes such as extrusion or forging

a new approach to 2 d eigenvalue shape design a new approach to solve problems in the optimal shape design of eigenvalues is presented the idea behind the approach is based on a combination of the known material derivative method4 7 and a specially adapted green s functions formulation 14 in the current study numerical testing has been limited to two dimensional shape optimization the two test cases involve the first eigenvalues of boundary value problems which model i free vibrations of a fixed membrane and ii natural frequencies of clamped thin plates the basic algorithm can easily be adapted to serve other types of boundary value problems dealing with the highest eigenvalues that occur in applied mechanics computational aspects of the proposed approach such as accuracy of the obtained sensitivities cost of computation and so on are discussed in references 16 in preparation

an iterative solution scheme for systems of boundary element equations in this paper an iterative scheme of first order is developed for the purpose of solving linear systems of equations in particular systems that are derived from boundary integral equations are investigated the iterative schemes to be considered are of the form ex k 1 dx k d where e and d are square matrices it will be assumed that e is a lower matrix i e the coefficients above the central diagonal are zero it will be shown that by considering matrix d embedded in a vector space and reducing its size with respect to a chosen metric that convergence rates can be substantially improved equation ordering and parameter matrices are used to reduce the magnitude of d a number of examples are tested to illustrate the importance of the choice of metric equation ordering and the parameter matrix computation times are determined for both the iterative procedure and gauss elimination indicating the usefulness of iteration which can be orders of magnitude faster

cross sectional analysis of composite beams including large initial twist and curvature effects an asymptotically exact methodology based on geometrically nonlinear three dimensional elasticity is presented for cross sectional analysis of initially curved and twisted nonhomogeneous anisotropic beams the analysis is subject only to the restrictions that the strain is small relative to unity and that the maximum dimension of the cross section is small relative to the wavelength of the deformation and to the minimum radius of curvature and or twist the final one dimensional strain energy per unit length exhibits asymptotically correct second order dependence of the initial curvature and twist parameters cross sectional constants of the one dimensional theory are obtained via finite element discretization over the cross sectional plane numerical results obtained for both isotropic and composite beams are compared with published results from special purpose analyses for initially twisted straight beams as well as initially curved untwisted beams the agreement with previously published results is excellent

theoretical and experimental analysis of the stop band behavior of elastic springs with periodically discontinuous of curvature the advanced design of springs with tailored transmission characteristics is proposed the existence of frequency stop bands in the transmission characteristics is theoretically predicted and experimentally validated the underlying mechanism of generating stop band behavior is the periodic discontinuity of the curvature the theoretical analysis is performed in the framework of the floquet theory for an infinitely long spring experimental results are obtained for the springs consisting of a small number of repeated periodicity cells

boundary element shape optimization system for aircraft structural components the design sensitivities are obtained economically by implicit differentiation of the boundary integral equations the mesh generation and regeneration is done using a parametric and auxiliary geometry concept that allows the original mesh to remain adequate for a wide range of subsequent evolving geometries as the optimization proceeds

metric geometry in homogeneous spaces of the unitary group of a c algebra part i minimal curves abstract we study the metric geometry of homogeneous spaces p of the unitary group of a c algebra a modulo the unitary group of a c subalgebra b where the invariant finsler metric is induced by the quotient norm of a b the main results include the following in the von neumann algebra context for each tangent vector x at a point p there is a geodesic t 0 x which is obtained by the action on of a 1 parameter group in the unitary group of a this geodesic is minimizing up to length 2

orthogonality to matrix subspaces and a distance formula abstract we obtain a necessary and sufficient condition for a matrix a to be birkhoff james orthogonal to any subspace w of m n c using this we obtain an expression for the distance of a from any unital c subalgebra of m n c

quasi static and dynamic response of viscoelastic helical rods in this study the dynamic behaviour of cylindrical helical rods made of linear viscoelastic materials are investigated in the laplace domain the governing equations for naturally twisted and curved spatial rods obtained using the timoshenko beam theory are rewritten for cylindrical helical rods the curvature of the rod axis effect of rotary inertia and shear and axial deformations are considered in the formulation the material of the rod is assumed to be homogeneous isotropic and linear viscoelastic in the viscoelastic material case according to the correspondence principle the material constants are replaced with their complex counterparts in the laplace domain ordinary differential equations in scalar form obtained in the laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem in the solutions the kelvin model is employed the solutions obtained are transformed to the real space using the durbin s numerical inverse laplace transform method numerical results for quasi static and dynamic response of viscoelastic models are presented in the form of graphics

iterative solution techniques in boundary element analysis iterative techniques for the solution of the algebraic equations associated with the direct boundary element analysis bea method are discussed continuum structural response analysis problems are considered employing single and multi zone boundary element models with and without zone condensation the impact on convergence rate and computer resource requirements associated with the sparse and blocked matrices resulting in multi zone bea is studied both conjugate gradient and generalized minimum residual preconditioned iterative solvers are applied for these problems and the performance of these algorithms is reported included is a quantification of the impact of the preconditioning utilized to render the boundary element matrices solvable by the respective iterative methods in a time competitive with direct methods to characterize the potential of these iterative techniques we discuss accuracy storage and timing statistics in comparison with analogous information from direct sparse blocked matrix factorization procedures matrix populations that experience block fill in during the direct decomposition process are included with different degrees of preconditioning iterative equation solving is shown to be competitive with direct methods for the problems considered

a characterization of minimal hermitian matrices abstract we describe properties of a hermitian matrix m m n c having minimal quotient norm in the following sense m m d for all real diagonal matrices d m n c here denotes the operator norm we show a constructive method to obtain all the minimal matrices of any size

dynamic stiffness formulation free vibration and wave motion of helical springs a coil spring can only be treated as a simple massless force element at low frequencies the effects of internal resonances leading to significant dynamic stiffening for an automotive suspension spring this occurs at frequencies as low as about 40 hz this paper presents an efficient method for calculating the dynamic stiffness of a helical coil spring the partial differential equations of motion are used to derive the relation between wavenumber and frequency along with the associated wave shapes by expressing the response in terms of these waves the dynamic stiffness matrix is assembled natural frequencies are obtained from the reduced stiffness matrix allowing for different boundary conditions making use of the wittrick williams algorithm the results of the dynamic stiffness method are compared with those of the transfer matrix method and the finite element method the nature of the wave propagation is also investigated although at low frequencies four wave types propagate above a particular frequency only two propagating waves remain these are composite waves which are excited by both axial and transverse motion for lower values of helix angle an intermediate frequency range exists where six propagating waves can occur

sensitivity analysis formulation for three dimensional conduction heat transfer with complex geometries using a boundary element method a special boundary integral formulation had been proposed to analyse many engineering problems of conduction heat transfer in complex three dimensional geometries closely spaced surface and circular hole in infinite domain or simple modification of it by rezayat and burton one example of such geometries is the mold sets in the injection molding process in this paper an efficient and accurate approach for the design sensitivity analysis dsa is presented for these kinds of problems in the similar complex geometries using the direct differentiation approach dda based on the above special boundary integral formulation the present approach utilizes the implicit differentiation of the boundary integral equations with respect to the design variables radii and locations of circular holes to yield the sensitivity equations a sample problem heat transfer of injection molding cooling system is solved to demonstrate the accuracy of the present sensitivity analysis formulation although the techniques introduced here are applied to a particular problem in heat transfer of injection molding cooling system their potential application is quite broad

variational approach by means of adjoint systems to structural optimization and sensitivity analysis ii structure shape variation abstract for a linear elastic structure the first variation of an arbitrary stress strain and displacement functionals corresponding to variation of shape of external boundaries or interfaces is derived by using the solutions for primary and adjoint systems the application to optimal design is next presented and the relevant optimally conditions are derived from general expressions the path independent integrals used in fracture mechanics are rederived as a particular case of general expressions

a bem sensitivity and shape identification analysis for acoustic scattering in fluid solid problems in this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems the formulation developed utilizes the boundary integral equation of the helmholtz equation for the external problem and the cauchy navier equation for the internal elastic problem the sensitivities are obtained by the implicit differentiation technique examples are presented to demonstrate the accuracy of the proposed formulations 1998 john wiley sons ltd

sensitivity analysis and optimal design in charge transport problems the focus of this paper is the accurate and efficient determination of sensitivities of electrostatic potential and charge density in charge transport problems and the use of these sensitivities to carry out optimal design direct differentiations of the boundary integral formulation of poisson s equation for charge conservation and of the non linear partial differential equation for current continuity are carried out to obtain equations satisfied by the sensitivities methods for solving the sensitivity equations are discussed a numerical implementation of the methods is validated through several examples it is demonstrated that the design sensitivity coefficients dscs of the quantities of interest in charge transport are obtained accurately and that optimal design problems can be solved efficiently using these dscs

conforming boundary elements in plane elasticity for shape design sensitivity the structural design sensitivity analysis of a two dimensional continuum using conforming continuous boundary elements is investigated implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices a singular formulation of the boundary element method is used implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis the behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated a rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices thus avoiding the problems associated with their numerical integration a formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented the design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results numerical data comparing the performance of conforming and non conforming formulations in the calculation of design sensitivities are also presented the accuracy of the present results is demonstrated through comparisons with existing analytical results

axisymmetric thermoelastic shape sensitivity analysis and its application to turbine disc design a general method for shape design sensiti vityi analysis sdsa as applied to an axisymmetric thermoelasticity problem is presented using the material derivative concept and the adjoint variable method the sensitivity of a general functional composed of thermal and mechanical quantities is considered the method for deriving the sensitivity formula is based on standard direct thermal and elastic boundary integral equation formulation it is then applied to obtain explicit formulas for a representative displacement and stress constraint imposed on a sector of the boundary results of numerical implementation are presented for weight minimization of a turbine disc under thermomechanical loading the sensitivities of the displacement and stress constraint calculated by the formulas are compared with those by finite differences optimum shape obtained under the thermomechanical loading is discussed with that under the mechanical loading only clearly showing the practical importance of the sdsa of thermoelastic systems

free vibration analysis of non cylindrical helices with variable cross section by using mixed fem the objective of the study is to present a mixed finite element fe formulation for the free vibration analysis of non cylindrical helices with variable cross section non cylindrical helix geometry was well approximated through the variable arc lengths and linear variations in curvatures via shape functions the mixed fe model for non cylindrical helices was developed by using the exact cylindrical helix geometry and corresponding field equations the element and consistent mass matrices are attained under these assumptions the element matrix is derived based on the timoshenko beam theory and the effects of rotary inertia are involved into the mass matrix although the formulation is quite simple the generated element is capable to provide highly accurate solutions for the conical barrel and hyperboloidal geometries of non cylindrical helices the results of presented fe model based on an approximated geometry approach are in a good agreement with the other studies in the actual literature some original examples were generated and solved for the literature as well

best approximation by diagonal compact operators abstract we study the existence and characterization properties of compact hermitian operators c on a hilbert space h such that c c d for all d d k h h or equivalently c min d d k h h c d dist c d k h h where d k h h denotes the space of compact self adjoint diagonal operators in a fixed base of h and is the operator norm we also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal

coarse division transform based preconditioner for boundary element problems this paper is devoted to the development of efficient preconditioners for an iterative solution of equation sets arising in the boundary element method bem a standard collocation system of equations is transformed to a new basis associated with an auxiliary coarse division model for boundary unknowns and solved in this basis new systems have rapidly decreasing coefficients and by neglecting a large number of them it is possible to construct readily invertible sparse preconditioners for iterative procedures the specific features of the transformed matrix can be attributed to the analytical properties of integral equations although the transformation is based on an auxiliary coarse division model it does not require any additional operations with boundary elements all manipulations necessary to construct the mapping are performed on the level of algebraic equations numerical experiments included in the paper confirm a high rate of convergence of the developed iterative scheme

axisymmetric viscoplastic deformation by the boundary element method abstract this paper presents a boundary element formulation and numerical implementation of the problem of small axisymmetric deformation of viscoplastic bodies while the extension from planar to axisymmetric problems can be carried out fairly simply for the finite element method fem this is far from true for the boundary element method bem the primary reason for this fact is that the axisymmetric kernels in the integral equations of the bem contain elliptic functions which cannot be integrated analytically even over boundary elements and internal cells of simple shape thus special methods have to be developed for the efficient and accurate numerical integration of these singular and sensitive kernels over discrete elements the accurate determination of stress rates by differentiation of the displacement rates presents another formidable challenge a successful numerical implementation of the boundary element method with elementwise called the mixed approach or pointwise called the pure bem or bem approach determination of stress rates has been carried out a computer program has been developed for the solution of general axisymmetric viscoplasticity problems comparisons of numerical results from the bem and fem for several illustrative problems are presented and discussed in the paper it is possible to get direct solutions for the simpler class of problems for cylinders of uniform cross section and these solutions are also compared with the bem and fem results for such cases

symmetric galerkin boundary formulations employing curved elements accounts of the symmetric galerkin approach to boundary element analysis bea have recently been published this paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency a new symmetric galerkin theoretical formulation for both the harmonic heat conduction and the biharmonic elasticity problem that employs regularized singular and hypersingular boundary integral equations bies is presented the novel use of regularized bies in the galerkin context is shown to allow straightforward incorporation of curved isoparametric elements a symmetric reusable intrinsic sample point risp numerical integration algorithm is shown to produce a galerkin i e double integration strategy that is competitive with its counterpart i e singular integration procedure in the collocation bea approach when the time saved in the symmetric equation solution phase is also taken into account this new formulation is shown to be capable of employing hypersingular bies while obviating the requirement of c 1 continuity a fact that allows the employment of the popular continuous element technology the behavior of the symmetric galerkin bea method with regard to both direct and iterative equation solution operations is also addressed a series of example problems are presented to quantify the performance of this symmetric approach relative to the more conventional unsymmetric bea in terms of both accuracy and efficiency it is concluded that appropriate implementations of the symmetric galerkin approach to bea indeed have the potential to be competitive with if not superior to collocation based bea for large scale problems

partitioning boundary integral equations and exact green s functions discretization of boundary integral equations leads to large full systems of algebraic equations in practice partitioning is a method for solving such systems by breaking them down into smaller systems it may be viewed merely as a technique from linear algebra however it is profitable to view it as arising directly from partitions of the boundary these partitions could be natural such as two separate boundaries but they need not be we investigate partitioning in the context of multiple scattering of acoustic waves by two sound hard obstacles the ideas extend to other physical situations specifically we make a connection between partitioning and the use of the exact green s function for a single obstacle in isolation this suggests computing the action of this green s function once and for all storing it perhaps on a compact disc and then using it to solve other problems in which the second obstacle is altered one example of this approach is computing the stress distribution around a cavity of a standard but complicated shape inside a structure whose shape is varied the theoretical foundation for these ideas is given as well as a connection with the use of generalized born series for multiple scattering problems important distinctions between the partitioning green s function idea in this paper and seemingly similar ideas such as substructuring multi zoning and domain decomposition are made

boundary element shape sensitivity analysis for thermal problems with nonlinear boundary conditions implicit differentiation of the discretized boundary integral equations governing the conduction of heat in solid objects subjected to nonlinear boundary conditions is shown to generate an accurate and economical approach for the computation of shape sensitivities for this class of problems this approach involves the employment of analytical derivatives of boundary element kernel functions with respect to shape design variables a formulation is presented that can consistently account for both temperature dependent convection and radiation boundary conditions several iterative strategies are presented for the solution of the resulting sets of nonlinear equations and the computational performances examined in detail multizone analysis and zone condensation strategies are demonstrated to provide substantive computational economies in this process for models with either localized nonlinear boundary conditions or regions of geometric insensitivity to design variables a series of nonlinear example problems are presented that have closed form solutions 39 refs

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

free vibration analysis of cylindrical helical springs with noncircular cross sections abstract the free vibration analysis of cylindrical helical springs with noncircular cross sections is carried out by means of an analytical study in the governing equations of motion of a spring all displacement functions and a generalized warping coordinate are defined at the centroidal principal axis the effects of the rotational inertia axial and shear deformations including torsion related warping deformations are also considered in the formulations explicit analytical expressions that give the vibrating mode shapes are derived by rigorous application of the symbolic computing package mathematica and the muller root search method is used to determine the natural frequencies numerical examples are provided for springs with elliptic rectangular and equilateral triangular cross sections and subjected to clamped clamped and clamped free boundary conditions the natural frequencies are presented for a range of geometric parameters in the case of elliptical wires results are presented for the aspect ratio a b ranging from 3 5 to 5 3 the helix pitch angle from 5 to 12 5 the number of active turns n from 6 to 12 and the ratio of cylinder radius to minor axis r 2a ranging from 20 3 to 50 3 validation of the proposed model has been achieved through comparison with a finite element model using three dimensional solid elements and the results available in published literature which in these cases indicates a good correlation

some characterizations of minimal compact normal operators abstract in this note we consider properties of a compact normal operator a such that a a d for all d d k h or equivalently a dist a d k h where denotes the usual operator norm and d k h is the subalgebra of diagonal compact operators in a fixed orthonormal basis of the hilbert space h

design sensitivity coefficients for elasto viscoplastic problems by boundary element methods this paper is concerned with accurate and efficient determination of design sensitivity coefficients dscs for solid mechanics problems with small strains and rotations but with material non linearities present elasto plastic or elasto viscoplastic problems this approach is based on direct differentiation dda of the relevant derivative boundary element method dbem formulation of the problem analytical differentiation of the dbem equations leads to singular integral equations for the dscs with weakly logarithmically for 2 d singular kernels which are easy to deal with also stress components and their sensitivities are obtained on the boundary of a body with great accuracy these quantities are typically difficult to obtain accurately from finite element methods fem a computer program for general two dimensional plane strain and plane stress problems has been developed based on the above formulation numerical results are presented for some sample problems and these are compared against direct solutions the agreement between the dbem and direct solutions is excellent for these examples

an application of trefftz method to the sensitivity analysis of two dimensional potential problem this paper presents a new formulation for the sensitivity analysis with the help of the trefftz boundary type solution procedure direct differentiation of the expressions of the field variables leads to the expressions of the sensitivities since the field variables are approximated by linear combination of regular t complete functions the analytical expressions of the sensitivities are also regular sensitivity analysis schemes with respect to variations of dimensions of the object and boundary conditions are explained in detail two dimensional potential problems in the bounded region are considered as a numerical example in order to confirm its validity

a study of the efficiency of iterative methods for linear problems in structural mechanics the efficiency of iterative methods in linear structural mechanics is studied the efficiency concerns the calculation time the numerical accuracy and the core storage needed we state that iterative methods are effective in connection with hierarchical improvement of a primary approximation three iterative methods are studied the conjugate gradient method preconditioned by a modified incomplete factorization matrix the same method preconditioned by a matrix obtained from natural factors on elemental level and a jacobi integration preconditioned by viscous relaxation split in an element by element way we make comparisons with direct methods gaussian elimination and factorization by use of natural factors

the importance of diagonal dominance in the iterative solution of equations generated from the boundary element method the unsymmetric matrix equations generated from the boundary element method bem can be solved iteratively with convergence to the correct solution guaranteed if the boundary element system of equations can be first transformed into an equivalent diagonally dominant system a transformation is presented which selectively annihilates terms in the coefficient matrix of the system ax b until an equivalent row diagonally dominant system if available is obtained the new row diagonally dominant system is well suited for use with jacobi and gauss seidel point iterative equation solvers the diagonal dominizing transformation presented in this work is not suitable for large systems of equations but is useful as a research tool for studying the importance of diagonal dominance in the iterative solution of equations generated from the bem a simple laplacian problem is used to examine the structure of the bem equations and to introduce the diagonal dominizing transformation the importance of diagonal dominance is shown by comparing iterative convergence of positive definite symmetric positive definite and diagonally dominant systems of bem equations obtained from a plane strain elasticity problem

minimal matrices and the corresponding minimal curves on flag manifolds in low dimension abstract in general c algebras elements with minimal norm in some equivalence class are introduced and characterized we study the set of minimal hermitian matrices in the case where the c algebra consists of 3 3 complex matrices and the quotient is taken by the subalgebra of diagonal matrices we thoroughly study the set of minimal matrices particularly because of its relation to the geometric problem of finding minimal curves in flag manifolds for the flag manifold of four mutually orthogonal complex lines in c 4 it is shown that there are infinitely many minimal curves joining arbitrarily close points in the case of the flag manifold of three mutually orthogonal complex lines in c 3 we show that the phenomenon of multiple minimal curves joining arbitrarily close points does not occur

flaw identification using the boundary element method in this paper a new boundary element formulation is presented for the identification of the location and size of internal flaws in two dimensional structures an introduction to inverse analysis is given with special reference to methods of flaw identification along with a brief review of the optimization methods employed both the standard boundary element and the dual boundary element method are presented with the dual boundary element method proposed as the basis for the new formulation the flaw identification method is presented along with the computation of the boundary displacement and traction derivatives and the specialized analytical integration used for cracked boundaries examples are given to demonstrate the accuracy of the sensitivity values and the performance of flaw location

supports for minimal hermitian matrices abstract we study certain pairs of subspaces v and w of c n we call supports that consist of eigenspaces of the eigenvalues m of a minimal hermitian matrix m m m d for all real diagonals d for any pair of orthogonal subspaces we define a non negative invariant called the adequacy to measure how close they are to form a support and to detect one this function is the minimum of another map f defined in a product of spheres of hermitian matrices we study the gradient hessian and critical points of f in order to approximate these results allow us to prove that the set of supports has interior points in the space of flag manifolds

semianalytical structural sensitivity formulation in boundary elements developpement d une methode semi analytique qui combine la differenciation implicite des equations integrales frontieres discretisees et une perturbation a une variable etude de milieux continus a symetrie axiale et elastiques plans 2d incluant des forces de volume de type gravite et centrifuge l integration numerique d une nouvelle classe de noyaux de sensibilite des solutions fondamentales est evitee et la mise en uvre est facile presentation de resultats numeriques pour l etude de l effet des perturbations dans les variables du modele

a boundary contour formulation for design sensitivity analysis in two dimensional linear elasticity abstract a formulation for computing first order shape design sensitivities in two dimensional 2 d linear elastostatics by the boundary contour method bcm along with a numerical implementation using quadratic boundary elements is presented in this paper here the direct differentiation approach is analytically applied to the appropriate boundary contour equations in order to derive the sensitivities of all the physical quantities displacements tractions and stresses on the boundary as well as those for displacements and stresses inside the body under consideration the nonsingular formulation of the bcm is used for computing the boundary displacements and boundary stresses at off contour regular points a regular boundary point is a point on the boundary where it is locally smooth an off contour point lies inside a boundary element their corresponding sensitivities are obtained in a straightforward manner from the resulting regular sensitivity formulation also the stress sensitivities at the boundary nodes can be recovered easily from the global displacement shape functions described in a cartesian coordinate system finally through three numerical examples for which analytical solutions exist it is shown that the bcm can provide remarkably accurate numerical results for shape sensitivities

the direct numerical solution of a volterra integral equation arising out of viscoelastic stress in materials abstract in this paper the convergence of a class of linear multistep methods for a volterra integral equation arising out of viscoelastic stress in materials is analysed and the theoretical orders of convergence are verified by numerical results

a comparison of direct and preconditioned iterative techniques for sparse unsymmetric systems of linear equations in this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric sparse systems of linear algebraic equations these problems occur in finite difference and finite element simulations of semiconductor devices and fluid flow problems we consider five iterative methods that appear to be the most promising for this class of problems the biconjugate gradient method the conjugate gradient squared method the generalized minimal residual method the generalized conjugate residual method and the method of orthogonal minimization each of these methods was tested using similar preconditioning incomplete lu factorization on a set of large sparse matrices arising from finite element simulation of semiconductor devices results are shown where we compare the computation time and memory requirements for each of these methods against one another as well as against a direct method that uses lu factorization to solve these problems the results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large sparse systems of equations and can offer significant savings in storage and cpu time

a generalized sor method for dense linear systems of boundary element equations in this paper an iterative scheme of first degree is developed for the purpose of solving linear systems of boundary element equations of the form hx c where h is a dense square nonsingular matrix the iterative scheme considered is d omega h sl x k 1 d omega h u x k omega c noindent where omega h u and omega h s1 are defined as the upper triangular and strictly lower triangular terms of omega h respectively the parameter matrix omega is selected to minimize the frobenius norm d omega h u f mathematical arguments and numerical experiments are presented to show that minimizing d omega h u f provides for faster convergence numerical tests are performed for systems of boundary element equations generated by three dimensional potential and elastostatic problems computation times are determined and compared against those for gaussian elimination and gauss seidel iteration

design sensitivity coefficients for linear elastic bodies with zones and corners by the derivative boundary element method abstract the subject of this paper is the efficient and accurate determination of design sensitivity coefficients dscs in an elastic solid with zones and corners direct differentiation of the relevant derivative boundary element method dbem formulation of the problem is carried out here corners and zones in a body are treated carefully with conforming elements a numerical implementation of the method is carried out with isoparametric quadratic boundary elements the power of this approach is demonstrated through several numerical examples it is shown that the dscs of the various mechanical quantities of interest are obtained accurately and efficiently in one example even the dsc of the stress at a corner with respect to the corner angle is obtained very accurately by this method

boundary formulations for shape sensitivity of temperature dependent conductivity problems used in concert with the kirchhoff transformation implicit differentiation of the discretized boundary integral equations governing the conduction of heat in solids with temperature dependent thermal conductivity is shown to generate an accurate and economical approach for computation of shape sensitivities for problems with specified temperature and heat flux boundary conditions a linear problem results for both the analysis and sensitivity analysis in problems with either convection or radiation boundary conditions a non linear problem is generated several iterative strategies are presented for the solution of the resulting sets of non linear equations and the computational performances examined in detail multi zone analysis and zone condensation strategies are demonstrated to provide substantive computational economies in this process for models with either localized non iinear boundary conditions or regions of geometric insensitivity to design variables a series of non linear example problems is presented that have closed form solutions exact anaytical expressions tor the shape sensitivities associated with these problems are developed and these are compared with the sensitivities computed using the boundary element formulation

transient thermoelasticity and other body force effects in boundary element shape sensitivity analysis several new formulations for structural design sensitivity analysis dsa of elastic solids are presented that consistently account for steady and transient thermal loading along with centrifugal and gravity loading the approach employs implicit differentiation of the discretized boundary integral equations and facilitates the production of accurate and efficient computational procedures that obviate the need for matrix factorizations in the dsa step for centrifugal gravity and general thermal loading a new dsa concept employing a particular integral technique is developed for centrifugal gravity and steady thermal effects a surface integration approach is also presented and the two methods compared alternative strategies for modelling the behaviour of temperature distributions of evolving shapes undergoing shape optimization and the ramifications of these strategies in the dsa process are also discussed it is shown that for certain classes of thermoelastic problems shape optimization can be performed without the need to incorporate thermal analysis or dsa into the overall process a thermoelastic surface stress sensitivity analysis formulation is presented that does not require additional numerical integration a new thermoelastic sharp corner traction component consistent sensitivity analysis formulation is also presented this technique is shown to work for any initial stress distribution numerical sensitivity results are presented for many problems with analytical solutions

a boundary element strategy for elastostatic inverse problems involving uncertain boundary conditions in this paper a convenient strategy is developed to find solutions for a class of uncertain boundary value problems by the boundary element method bem such problems are ill posed but ill conditioning of the associated algebraic systems of equations can be controlled to a large extent and useful boundary data can be obtained despite ill conditioning interior data of not only sufficient quantity but also of good quality at good locations contribute to good solutions our strategy permits the condition number of the algebraic systems as a function of interior data locations to be monitored such that measured data from displacement sensors and or strain sensors at locations found to be good ones for the problem at hand can be used the present strategy is based upon the concept of a green s function library through partitioning of the bem algebraic system algebraic systems are solved using least squares via singular value decomposition svd the library idea takes advantage of modern data storage and retrieval technology and permits the process of repeated trials in order to determine good data sensor locations to be done quickly and efficiently several numerical examples are given to demonstrate the strategy some examples examine the consequences of errors in measured data copyright 1999 john wiley sons ltd

a unified boundary element method for the analysis of sound and shell like structure interactions ii efficient solution techniques efficient solution methods are investigated in this paper for solving the linear system of equations resulting from the recently developed boundary element method bem for the coupled structural acoustic analysis s h chen and y j liu j acoust soc am 106 pt 1 1247 1254 1999 an iterative solver namely the quasiminimal residual method qmr is selected among others and found to be very favorable over the direct solver for solving the linear systems of equations with complex coefficients generated by the structural acoustic bem four problem dependent preconditioning schemes are developed to facilitate or accelerate the convergence of the iterative solver a new effective preconditioner specially designed for frequency sweep analysis is also presented in this paper with this preconditioner the iterative solver has been found to be stable in a frequency sweep analysis and can converge much faster than the direct solver the double precision arithmetic is also found very useful in improvi

reanalysis for structural modifications in boundary element responseand design sensitivity analysis a reanalysis technique is presented for obtaining changes in response variables due to changes in design parameters with the boundary element method the approach is considered for static response analysis and design sensitivity analysis two iterative schemes one based on total quantities and another based on incremental quantities are developed numerical results are obtained for analyses using 1 single zone discretization 2 multizone discretization and 3 multizone discretization with condensation

some characterizations of the trace norm triangle equality abstract in this paper we investigate some equalities for the trace class operators on a hilbert space for two trace class operators a and b we get some equivalent conditions for a 1 b 1 a b 1 where a 1 is the trace norm of the trace class operator a particularly we show that a 1 b 1 a b 1 if and only if a b a b for two trace class operators a and b this condition is related to a result of ando and hayashi moreover some characterizations of the equality a 1 a 1 a a 1 and other relevant results are given

solving large and sparse linear least squares problems by conjugate gradient algorithms abstract let a e m n with m n and rank a n and b e m 1 be given assume that an approximation to x a b ata 1atb is to be calculated this problem is transformed into an equivalent problem with a symmetric and positive definite matrix c d 1 rt 1atar 1d 1 where d is a diagonal matrix and r is an upper triangular matrix the conjugate gradient cg algorithm is applied in the solution of the system of linear algebraic equations cy d y drx d d 1 rt 1atb if d r i then the cg algorithm is in fact applied to the system of normal equations atax atb and the speed of convergence could be very slow if d and r are obtained by some kind of orthogonalization dr qta with q e m n satisfying qtq i d is often equal to i and q is never used in the cg algorithm and if the calculations are performed without rounding errors then c i and the cg algorithm converges in one iteration only even if the orthogonalization process is carried out with rounding errors the matrix c is normally close to the identity matrix i and the cg algorithm is quickly convergent however for large m and n the orthogonal decomposition is an expensive process both in regard to storage and in regard to computing time therefore it may be profitable to calculate an incomplete orthogonal decomposition this is achieved by introducing a special parameter t a drop tolerance such that all elements which in the course of the computations become smaller in absolute value than t are removed numerical examples are given to illustrate that the cg algorithm applied to system cy d and used with incomplete factors d and r is very efficient for some classes of problems it should be emphasized that matrix c is never calculated explicitly the whole work is carried out by the use of a d and r only

sensitivity analysis for the dynamic response of thermoviscoplastic shells of revolution abstract a computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric fully coupled thermoviscoplastic response of shells of revolution the analytical formulation is based on reissner s large deformation shell theory with the effects of large strain transverse shear deformation rotatory inertia and moments turning around the normal to the middle surface included the material model is chosen to be viscoplasticity with strain hardening and thermal hardening and an associated flow rule is used with a von mises effective stress a mixed formulation is used for the shell equations with the fundamental unknowns consisting of six stress resultants three generalized displacements and three velocity components the energy balance equation is solved using a galerkin procedure with the temperature as the fundamental unknown spatial discretization is performed in one dimension meridional direction for the momentum and constitutive equations of the shell and in two dimensions meridional and thickness directions for the energy balance equation the temporal integration is performed by using an explicit central difference scheme leap frog method for the momentum equation a predictor corrector version of the trapezoidal rule is used for the energy balance equation and an explicit scheme consistent with the central difference method is used to integrate the constitutive equations the sensitivity coefficients are evaluated by using a direct differentiation approach numerical results are presented for a spherical cap subjected to step loading the sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness mass density young s modulus two of the material parameters characterizing the viscoplastic response and the three parameters characterizing the thermal response time histories of the response and sensitivity coefficients are presented along with spatial distributions of some of these quantities at selected times

comparison of shape design sensitivity analysis formulations via material derivative adjoint variable and implicit differentiation techniques for 3 d and 2 d curved boundary element this paper reviews and compares two different approaches to continuum structural shape design sensitivity analysis the material derivative adjoint variable method and the implicit differentiation method using 3 d and 2 d bem employing identical bea models curved element technology numerical integration and equation solving techniques the similarities and differences of both methods are highlighted significant theoretical and computational aspects associated with both methods are presented and accuracy and computational efficiency characteristics are demonstrated advantages and disadvantages of both techniques are identified the characterization provided can aid in the prudent selection between these sensitivity analysis techniques based on the characteristics of the proposed application

an iterative solution strategy for boundary element equations from mixed boundary value problems abstract an iterative solution strategy is presented for mixed boundary value problems from the direct boundary element method where both dirichlet and neumann constraints are prescribed on the boundary in the mixed boundary value problem constraints are imposed upon the bem system of equations hu gq to ax b the a matrix is fully populated unsymmetric and made up of columns from h and g corresponding to the neumann and dirichlet boundary conditions respectively in this work the coefficient matrix a is rearranged to place the columns from h in the left most columns of a and the columns from g in the right most columns of a the diagonal submatrix in a containing terms from g is then reduced by elimination while the diagonal submatrix containing terms from h is retained for iteration jacobi gauss seidel adn conjugate gradient normal iterative solvers are considered convergence of the proposed solution strategy is studied using four two dimensional potential and elasticity problems

the green s matrix and the boundary integral equations for analysis of time harmonic dynamics of elastic helical springs helical springs serve as vibration isolators in virtually any suspension system various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions the method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time harmonic dynamics of elastic springs in the framework of bernoulli euler beam theory in this paper the derivations of the green s matrix of the somigliana s identities and of the boundary integral equations are presented the vibrational power transmission in an infinitely long spring is analyzed by means of the green s matrix the eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations in the course of analysis the essential features and advantages of the method of boundary integral equations are highlighted the reported analytical results may be used to study the time harmonic motion i

linear dynamics of elastic helical springs asymptotic analysis of wave propagation helical springs serve as vibration isolators in virtually any suspension system a variety of theories to describe the dynamic behaviour of these structural elements which involves interaction of flexural torsion and longitudinal waves can be found in the literature alongside this various approximate methods are employed to determine the eigenfrequencies of vibrations of springs in this paper the validity ranges of alternative theories are assessed by comparison of the location of the dispersion curves this paper also contains a rigorous asymptotic analysis of the exact dispersion equation with two small parameters being employed it allows for the identification of significant regimes of linear wave motion in a helical spring in each of these regimes simple formulae for wavenumbers are obtained by the dominant balance method and their validity ranges are checked against direct numerical solution mode shapes associated with each wavenumber are also analysed

boundary integral equation method for shape optimization of elastic structures a general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation using the material derivative concept and adjoint variable method the problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field the method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary the accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem among the several numerical implementations tested the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy a smooth characteristic function is found to be better than a plateau function for localization of the stress constraint optimal shapes for the two problems are presented to show numerical applications

defect of a unitary matrix abstract we analyze properties of a map f sending a unitary matrix u of size n into a doubly stochastic matrix b f u defined by bi j ui j 2 for any u we define its defect determined by the dimension of the image df t u u of the space t u u tangent to the manifold of unitary matrices u at u under the tangent map df corresponding to f the defect of u equal to zero for a generic unitary matrix gives an upper bound for the dimension of a smooth orbit a manifold stemming from u of inequivalent unitary matrices mapped into the same doubly stochastic matrix b f u we demonstrate several properties of the defect and prove an explicit formula for the defect of the fourier matrix fn of size n in this way we obtain an upper bound for the dimension of a smooth orbit of inequivalent unitary complex hadamard matrices stemming from fn it is equal to zero iff n is prime and coincides with the dimension of the known orbits if n is a power of a prime two constructions of these orbits are presented at the end of this work

simulating deformation of objects with multi materials using boundary element method in this paper an algorithm for simulating the elastostatic deformation of multiple objects with different material properties using boundary element method is introduced by tessellating the surface of a geometric model into elements and classifying all the element nodes into different groups with different attributes and partitioning the stiffness matrix into several sub matrices according to these attributes a compact expression for the unknown variables is obtained comparing with the direct matrix inversion method the dimension of the system matrix in the expression has been effectively reduced besides this expression shows that the deformation of a multi component object can be simulated in a way similar to that of a single component object the capacitance method is adopted for evaluating the deformation of the object experimental results illustrate that the proposed method is practical and efficient copyright 2007 john wiley sons ltd

direct differentiation formulation for boundary element shape sensitivity analysis of axisymmetric elastic solids abstract an efficient direct differentiation method by continuum approach is presented for the shape design sensitivity analysis of axisymmetric elastic solids the method is purely based on the axisymmetric boundary integral equation formulation a new boundary integral equation for sensitivity analysis is derived by taking material derivative to the same integral identity that was used in the adjoint method as sensitivities in terms of the shape design variables are directly calculated by solving the new bie the singular boundary condition which may arise in the adjoint method is avoided in the present direct method to validate the theoretical formulation numerical implementation is done for three examples for a hollow disk and a thick walled cylinder problem with analytic solution the sensitivities by present method are compared with analytic sensitivities for an example of a spherical vessel being rather practical the sensitivities are compared with those by finite differences

iterative methods in solving navier stokes equations by the boundary element method the solution of navier stokes equations of time dependent incompressible viscous fluid flow in planar geometry by the boundary domain integral method bdim is discussed the introduction of a subdomain technique to fluid flow problems is considered and improved in order to maintain the stability of bdim to avoid problems with flow kinematics computation in the sudomain mesh a segmentation technique is proposed which combines the original bdim with its subdomain variant and preserves its numerical stability in order to reduce the computational cost of bdim which greatly depends on the solution of systems of linear equations iterative methods are used conjugate gradient methods conjugate gradients squared and an improved version of the biconjugate gradient method bicgstab together with the generalized minimal residual method are used as iterative solvers different types of preconditioning from simple jacobi to incomplete lu factorization are carried out and the performance of chosen iterative methods and preconditioners are reported test examples include backward facing step flow and flow through tubular heat exchangers test computation results show that bdim is an accurate approximation technique which together with the subdomain technique and powerful iterative solvers can exhibit some significant savings in storage and cpu time requirements

boundary element implicit differentiation equations for design sensitivities of axisymmetric structures abstract design sensitivity analysis of axisymmetric elastic media is formulated using boundary elements the kernels for sensitivity matrices are obtained through implicit differentiation of the corresponding boundary element elasticity kernels the singular terms are obtained by applying the boundary displacements and tractions and their respective sensitivities for the rigid body motion mode and the inflation mode the equations for the recovery of sensitivities of axisymmetric boundary stresses are presented as a check on accuracy the approach is applied to a series of examples for which analytical elasticity solutions are available the predictions for both displacement and stress sensitivities are accurate additional examples are provided to demonstrate the versatility of the present approach