on the elastic fields of an elliptical inhomogeneity under plane deformation a generalized and unified approach is presented for the two dimensional plane problem of an elliptical inhomogeneity in an isotropic elastic medium the analysis is based upon the use of conformal mapping and laurent series expansion of muskhelishvili s complex potentials the resulting elastic fields are derived explicitly in both transformed and physical planes for the inhomogeneity and the surrounding matrix the associated expressions are universal in the sense of being applicable to generalized geometries and applied loads the application of the general solution is illustrated by several examples and the resulting solutions are verified with those existing in the literature the results advanced here can be used as building blocks for dealing with more complex problems involving a wide variety of geometry and loading conditions

an optimization based approach for the design of pde solution algorithms we develop and analyze an optimization based approach for the robust and efficient solution of pde problems consisting of multiple physics operators with fundamentally different mathematical properties our approach relies on three essential steps decomposition of the original problem into subproblems for which robust solution algorithms are available integration of the subproblems into an equivalent pde constrained optimization problem and solution of the resulting optimization problem either directly as a fully coupled algebraic system or in the null space of the pde constraints this strategy gives rise to a general approach for synthesizing robust solvers for complex coupled problems from solvers for their simpler physics components

robust topology optimization accounting for spatially varying manufacturing errors abstract this paper presents a robust approach for the design of macro micro or nano structures by means of topology optimization accounting for spatially varying manufacturing errors the focus is on structures produced by milling or etching in this case over or under etching may cause parts of the structure to become thinner or thicker than intended this type of error is modeled by means of a projection technique a density filter is applied followed by a heaviside projection using a low projection threshold to simulate under etching and a high projection threshold to simulate over etching in order to simulate the spatial variation of the manufacturing error the projection threshold is represented by a non gaussian random field the random field is obtained as a memoryless transformation of an underlying gaussian field which is discretized by means of an eole expansion the robust optimization problem is formulated in a probabilistic way the objective function is defined as a weighted sum of the mean value and the standard deviation of the structural performance the optimization problem is solved by means of a monte carlo method in each iteration of the optimization scheme a monte carlo simulation is performed considering 100 random realizations of the manufacturing error a more thorough monte carlo simulation with 10000 realizations is performed to verify the results obtained for the final design the proposed methodology is successfully applied to two test problems the design of a compliant mechanism and a heat conduction problem

the stress concentration factor for slightly roughened random surfaces analytical solution abstract we derive an analytical solution to the stress concentration factor k t for slightly roughened random surfaces topology is assumed to possess gaussian distribution of heights and auto correlation length acl for our development we combine gao s first order perturbation method the hilbert transform and an energy conservation principal related to the parseval theorem the root mean square rms value of k t results in a function of the ratio rms roughness to acl the derived formula agrees with experimental results previously reported the results provide insight for more efficient design

stress singularities along a cycloid rough surface abstract the stress concentration along a rough surface is of importance for understanding the nucleation of misfit dislocations and cracks in heteroepitaxial thin films and for general flaw initiation at material surfaces exposed to environmental corrosion in order to seek a basic understanding on this issue a cycloid wavy surface subject to a uniform bulk stress is adopted as a model problem the elastic stress and displacement fields are determined using muskhelishvili s conformal mapping method it is shown that a cusped cycloid surface generates a crack like singular stress field within a thin surface layer under uniform tension this singularity is found to have identical strength to a row of periodic parallel cracks the path independent j integral requires that the average surface strain energy density be identical to that in the bulk implying that any stress magnification along a rough surface must be compensated by unloading along some complementary portions of the surface along the cusped cycloid surface the strain energy distribution becomes dirac singular at the cusp tips the rest of the surface being completely stress free thus the effect of cycloid cusps is to redistribute and concentrate all the surface strain energy per wavelength at a single cusp point and because of that we can claim that the cycloid surface is the most efficient stress concentrator at a fixed wavelength even at a moderate bulk stress level this concentration may be sufficient to cause failure or nucleation of defects the full evolution of a rough surface under stress and other corrosion mechanisms must be solved by numerical methods our analytic solutions for a cycloid surface are of significant value for guiding numerical computations and gaining insights into essential features of the evolution process from a global point of view we show that a cusped cycloid surface becomes energetically favorable once the surface wavelength exceeds a critical value determined from the competition between the strain energy and the surface energy from a local point of view analysis of surface diffusion behaviors along an almost cusped cycloid surface indicates that the cusps are stable once they develop the critical condition for formation of a cusped cycloid corresponds to the griffith energy balance being exactly satisfied at the cusp tips while the chemical potential remains nearly constant along the rest of the surface this implies spontaneous griffith brittle fracture at tension cusps if plastic relaxation is not present to relieve the stress singularity

on the unified approach to anisotropic and isotropic elasticity for singularity interface and crack in dissimilar media abstract proposed in this paper is the equivalence between anisotropic and isotropic elasticity for two dimensional deformation under certain conditions that is the isotropic elasticity can be reconstructed in the same framework of the anisotropic elasticity when the interface between dissimilar media lies along a straight line therefore many known solutions for an anisotropic bimaterial are valid for a bimaterial of which one or both of the constituent materials are isotropic the usefulness of the equivalence is that the solutions for singularities and cracks in an anisotropic isotropic bimaterial can easily be obtained without solving the boundary value problems directly the interaction solutions of singularities interfaces and cracks in infinite anisotropic bimaterial are summarized to be used for the cases of isotropic isotropic and anisotropic isotropic bimaterials conservation integrals also have the similar analogy between anisotropic and isotropic elasticity so that j integral and j based mutual integral m are expressed in the same complex forms for anisotropic and isotropic materials when both end points of the integration paths are on the straight interface the use of j and m integrals together with the present equivalence are exemplified to obtain energy release rate stress intensity factors and t stresses of interfacial cracks lying in the interface of anisotropic anisotropic isotropic isotropic or anisotropic isotropic solids

optimization based remap and transport a divide and conquer strategy for feature preserving discretizations this paper examines the application of optimization and control ideas to the formulation of feature preserving numerical methods with particular emphasis on the conservative and bound preserving remap constrained interpolation and transport advection of a single scalar quantity we present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization based remap obr of mass density the latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity subject to a system of linear inequality constraints the approximation of an exact mass update operator defines the target quantity which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds the latter are enforced by the system of linear inequalities in so doing the generic obr formulation separates accuracy considerations from the enforcement of physical properties we proceed to show how the generic obr formulation yields the recently introduced flux variable flux target fvft 1 and mass variable mass target mvmt 2 formulations of remap and then follow with a formal examination of their relationship using an intermediate flux variable mass target fvmt formulation we show the equivalence of fvft and mvmt optimal solutions to underscore the scope and the versatility of the generic obr formulation we introduce the notion of adaptable targets i e target quantities that reflect local solution properties extend fvft and mvmt to remap on the sphere and use obr to formulate adaptable conservative and bound preserving optimization based transport algorithms for cartesian and latitude longitude coordinate systems a selection of representative numerical examples on two dimensional grids demonstrates the computational properties of our approach

an optimization based atomistic to continuum coupling method we present a new optimization based method for atomistic to continuum atc coupling the main idea is to cast the latter as a constrained optimization problem with virtual dirichlet controls on the interfaces between the atomistic and continuum subdomains the optimization objective is to minimize the error between the atomistic and continuum solutions on the overlap between the two subdomains while the atomistic and continuum force balance equations provide the constraints separation rather then blending of the atomistic and continuum problems and their subsequent use as constraints in the optimization problem distinguishes our approach from the existing atc formulations we present and analyze the method in the context of a one dimensional chain of atoms modeled using a linearized two body potential with next nearest neighbor interactions

stress related topology optimization via level set approach abstract considering stress related objective or constraint functions in structural topology optimization problems is very important from both theoretical and application perspectives it has been known however that stress related topology optimization problem is challenging since several difficulties must be overcome in order to solve it effectively traditionally simp solid isotropic material with penalization method was often employed to tackle it although some remarkable achievements have been made with this computational framework there are still some issues requiring further explorations in the present work stress related topology optimization problems are investigated via a level set based approach which is a different topology optimization framework from simp numerical examples show that under appropriate problem formulations level set approach is a promising tool for stress related topology optimization problems

structural optimization under uncertain loads and nodal locations this paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes the second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements we first review the discrete form of the uncertain loads problem which has been previously solved using a weighted average of multiple load patterns with minor modifications we extend this solution to include loads described by continuous joint probability density functions we then proceed to the main contribution of this paper structural optimization under uncertainty in the nodal locations this optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix it is shown however that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads by expressing these equivalent loads in terms of continuous random variables we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper the optimization algorithms are developed in the context of minimum compliance maximum stiffness design simple examples are presented the results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design for structures containing thin substructures under axial loads it is shown that these uncertainties a are of first order significance influencing the linear elastic response quantities and b can affect designs by avoiding unrealistically optimistic and potentially unstable structures the additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem

robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis abstract topology optimization often leads to structures consisting of slender elements which are particularly sensitive to geometric imperfections such imperfections might affect the structural stability and induce large displacement effects in these slender structures this paper therefore presents a robust approach to topology optimization which accounts for geometric imperfections and their potentially detrimental influence on the structural stability geometric nonlinear effects are incorporated in the optimization by means of a total lagrangian finite element formulation in the minimization of end compliance geometric imperfections are modeled as a vector valued random field in the design domain the resulting uncertain performance of the design is taken into account by minimizing a weighted sum of the mean and standard deviation of the compliance in the robust optimization problem these stochastic moments are typically estimated by means of sampling methods such as monte carlo simulation however these methods require multiple independent nonlinear finite element analyses in each design iteration of the optimization algorithm an efficient solution algorithm which uses adjoint differentiation in a second order perturbation method is therefore developed to estimate the stochastic moments during the optimization two applications with structures that exhibit different types of structural instabilities are examined in both cases it is demonstrated by means of an extensive monte carlo simulation that the deterministic design is very sensitive to imperfections while the design obtained by means of the proposed method is much more robust

enhanced deterministic performance of panels using stochastic variations of geometry and material the effect of stochastic variation in material and geometric properties on structural performance is important for robust design knowledge of such effects can be acquired by applying variation pat

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

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

some problems in the antiplane shear deformation of bi material wedges abstract the antiplane shear deformation of a bi material wedge with finite radius is studied in this paper depending upon the boundary condition prescribed on the circular segment of the wedge traction or displacement two problems are analyzed in each problem two different cases of boundary conditions on the radial edges of the composite wedge are considered the radial boundary data are traction displacement and traction traction the solution of governing differential equations is accomplished by means of finite mellin transforms the closed form solutions are obtained for displacement and stress fields in the entire domain the geometric singularities of stress fields are observed to be dependent on material property in general however in the special case of equal apex angles in the traction traction problem this dependency ceases to exist and the geometric singularity shows dependency only upon the apex angle a result which is in agreement with that cited in the literature for bi material wedges with infinite radii in part ii of the paper antiplane shear deformation of bi material circular media containing an interfacial edge crack is considered as a special case of bi material wedges studied in part i of the paper explicit expressions are derived for the stress intensity factor at the tip of an edge crack lying at the interface of the bi material media it is seen that in general the stress intensity factor is a function of material property however in special cases of traction traction problem i e similar materials and also equal apex angles the stress intensity factor becomes independent of material property and the result coincides with the results in the literature

asymptotic models of dilute composites with imperfectly bonded inclusions the asymptotic scheme for the analysis of dilute elastic composites which includes circular inclusions with imperfect bonding at the interface is presented interface is characterized by a discontinuous displacement field across it linearly related to the tractions the problem of a linear elastic circular inclusion with generic loading condition at infinity is solved and used to analyze effective elastic moduli of composite materials effects due to the interaction of a small circular defect and a crack are investigated it is shown that interfacial stiffness has a strong effect on the crack path and therefore may be an important design parameter for composites

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

explicit feature control in structural topology optimization via level set method abstract the present paper aims to address a long standing and challenging problem in structural topology optimization explicit feature control of the optimal topology the basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

a stress concentration formula generating equation for arbitrary shallow surfaces abstract analytical understanding of how stress concentrates is invaluable an equation that generates stress concentration formulas is derived and shown to apply very well to a number of shallow irregularities on surfaces for the plane stress conditions and to a first order approximation under shallow conditions for any first order holder continuous surface function f x the derived equation is k t x 1 2 h f x where h is the hilbert transform and f x is the spatial derivative of f with respect to the independent variable it is shown that using this generating equation well known traditional results can be easily derived also a number of other stress concentration formulas for various cases are generated furthermore a second order approximation is introduced which shows the dependence of k t on not only the slope but also on the concavity of the surface the approach used herein can be extended to finding closed form solutions to other integral equations possessing similar kernels for applications such as the variation of the stress intensity factor due to an arbitrary crack front profile work in progress

a high order boundary integral method for surface diffusions on elastically stressed axisymmetric rods many applications in materials involve surface diffusion of elastically stressed solids study of singularity formation and long time behavior of such solid surfaces requires accurate simulations in both space and time here we present a high order boundary integral method for an elastically stressed solid with axi symmetry due to surface diffusions in this method the boundary integrals for isotropic elasticity in axi symmetric geometry are approximated through modified alternating quadratures along with an extrapolation technique leading to an arbitrarily high order quadrature in addition a high order temporal integration factor method based on explicit representation of the mean curvature is used to reduce the stability constraint on time step to apply this method to a periodic in axial direction and axi symmetric elastically stressed cylinder we also present a fast and accurate summation method for the periodic green s functions of isotropic elasticity using the high order boundary integral method we demonstrate that in absence of elasticity the cylinder surface pinches in finite time at the axis of the symmetry and the universal cone angle of the pinching is found to be consistent with the previous studies based on a self similar assumption in the presence of elastic stress we show that a finite time geometrical singularity occurs well before the cylindrical solid collapses onto the axis of symmetry and the angle of the corner singularity on the cylinder surface is also estimated

multilevel algorithms for ill posed problems in this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations convergence estimates are established for multilevel algorithms applied to tikhonov type regularization methods our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter the algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations

multiscale analysis of heterogeneous domain decomposition methods for time dependent advection reaction diffusion problems abstract domain decomposition methods which use different models in different subdomains are called heterogeneous domain decomposition methods we are interested here in the case where there is an accurate but expensive model one should use in the entire domain but for computational savings we want to use a cheaper model in parts of the domain where expensive features of the accurate model can be neglected for the model problem of a time dependent advection reaction diffusion equation in one spatial dimension we study approximate solutions of three different heterogeneous domain decomposition methods with pure advection reaction approximation in parts of the domain using for the first time a multiscale analysis to compare the approximate solutions to the solution of the accurate expensive model in the entire domain we show that a recent heterogeneous domain decomposition method based on factorization of the underlying differential operator has better approximation properties than more classical variational or non variational heterogeneous domain decomposition methods we show with numerical experiments in two spatial dimensions that the performance of the algorithms we study is well predicted by our one dimensional multiscale analysis and that our theoretical results can serve as a guideline to compare the expected accuracy of heterogeneous domain decomposition methods already for moderate values of the viscosity

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

inexact preconditioned conjugate gradient method with inner outer iteration an important variation of preconditioned conjugate gradient algorithms is inexact preconditioner implemented with inner outer iterations g h golub and m l overton numerical analysis lecture notes in math 912 springer berlin new york 1982 where the preconditioner is solved by an inner iteration to a prescribed precision in this paper we formulate an inexact preconditioned conjugate gradient algorithm for a symmetric positive definite system and analyze its convergence property we establish a linear convergence result using a local relation of residual norms we also analyze the algorithm using a global equation and show that the algorithm may have the superlinear convergence property when the inner iteration is solved to high accuracy the analysis is in agreement with observed numerical behavior of the algorithm in particular it suggests a heuristic choice of the stopping threshold for the inner iteration numerical examples are given to show the effectiveness of this choice and to compare the convergence bound

boundary velocity control of incompressible flow with an application to viscous drag reduction an optimal boundary control problem for the navier stokes equations is presented the control is the velocity on the boundary which is constrained to lie in a closed convex subset of h 1 2 of the boundary a necessary condition for optimality is derived computations are done when the control set is actually finite dimensional resulting in an application to viscous drag reduction

confidence structural robust optimization by non linear semidefinite programming based single level formulation structural robust optimization problems are often solved via the so called bi level approach this solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non smooth nature of the bi level formulation another problem associated with the traditional bi level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically in the present paper confidence single level non linear semidefinite programming nlsdp formulations for structural robust optimization problems under stiffness uncertainties are proposed this is achieved by using some tools such as s procedure and quadratic embedding for convex analysis the resulted nlsdp problems are solved using the modified augmented lagrange multiplier method which has sound mathematical properties numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively copyright 2010 john wiley sons ltd

robust topology optimization of structures under loading uncertainty a new efficient and accurate approach to robust structural topology optimization is developed in this work the objective is to minimize the expected compliance of structures under loading uncertainty the new approach is akin to a deterministic multiple load case problem where the load cases and weights are derived by numerical methods efficient and accurate methods to compute expected compliance and its derivatives are developed for both concentrated and distributed uncertain loads the monte carlo method and matrix decomposition are employed for concentrated loads and the karhunen loeve expansion and its orthogonal properties of random variables are used for distributed loads numerical examples using the modified solid isotropic material with penalization approach are provided to demonstrate the accuracy and efficiency of the proposed approach

optimization based additive decomposition of weakly coercive problems with applications we present an abstract mathematical framework for an optimization based additive decomposition of a large class of variational problems into a collection of concurrent subproblems the framework replaces a given monolithic problem by an equivalent constrained optimization formulation in which the subproblems define the optimization constraints and the objective is to minimize the mismatch between their solutions the significance of this reformulation stems from the fact that one can solve the resulting optimality system by an iterative process involving only solutions of the subproblems consequently assuming that stable numerical methods and efficient solvers are available for every subproblem our reformulation leads to robust and efficient numerical algorithms for a given monolithic problem by breaking it into subproblems that can be handled more easily an application of the framework to the oseen equations illustrates its potential

topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system abstract this paper proposes a topology optimization for a linear elasticity design problem subjected to an uncertain load the design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors the idea of aggregation used in the field of control is introduced to assess the value of the robust compliance the aggregation solution technique provides the direct relationship between the uncertain input load and output displacement as a small linear system composed of these vectors and the reduced size of a symmetric matrix in the context of a discretized linear elasticity problem using the finite element method introducing the constraint that the euclidean norm of the uncertain load set is fixed the robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix according to the rayleigh ritz theorem for symmetric matrices moreover the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix the proposed structural optimization method is implemented using topology optimization and the method of moving asymptotes mma the numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures

a general solution to the antiplane problem of an arbitrarily located elliptical hole near the tip of a main crack abstract a general solution is obtained to the problem of the interaction between a main crack and an arbitrarily located and oriented elliptical hole near its tip under mode iii loading conditions the analysis is based on the complex potentials for the antiplane problem and the superposition principle the stress intensity factor at the main crack is obtained in a general series form and approximate closed form solutions are also derived using a perturbation procedure the present solution is shown to coincide with taylor expansion of exact solutions for collinear elliptical holes with specific aspect ratios numerical examples are provided to show the effect of the geometry location and orientation of the microdefect on the stress intensity factor of the main crack the present work should provide a valuable insight into main crack microdefect interaction phenomena in brittle materials

on the dynamic interaction between a microdefect and a main crack a generalized theoretical approach is presented for the dynamic interaction between an arbitrarily located and oriented microdefect and a finite main crack subjected to a plane incident wave the analysis is based upon the use of integral transform techniques and an appropriate superposition procedure the resulting dynamic stress intensity factors k text i ast and k text ii ast at the main crack are obtained by solving the appropriate singular integral equations using chebyshev polynomials for different incident waves the resulting solution is verified by comparison with existing results and numerical examples are provided to show the effect of the location and orientation of the microdefect and the frequency of the incident wave upon k text i ast and k text ii ast of the main crack the results advanced here can be used as building blocks in the fields of micromechanics damage and non destructive characterization of defects in solids

a novel finite element for treating inhomogeneous solids summary this study is concerned with the development and implementation of a novel finite element which is capable of treating the problem of interacting circular inhomogeneities in heterogeneous solids the general form of the element which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix is derived explicitly using the complex potentials of muskhelishvili and the laurent series expansion method the strength of the proposed eight noded plane element is demonstrated by its ability to treat arbitrarily and periodically located multiple inhomogeneities under general loading coflditions using a limited number of elements assessment of the accuracy and efficiency of the devised element is obtained by comparing its performance against existing analytical and traditional finite element attempts the current element enables the determination of the local and effective elastic properties of composite materials with relative ease

singularities interacting with interfaces and cracks abstract solutions to singularities such as point force point moment edge dislocation and transformation strain spot embedded in bonded elastic blocks of dissimilar materials are found to relate to the solutions to the same singularities in an infinite homogeneous plane by a formula independent of the nature of the singularities this universal result is then used to analyze the interactions between singularities and interface cracks the complete solutions and stress intensity factors are presented for two important interface crack configurations

a new petrov galerkin smoothed aggregation preconditioner for nonsymmetric linear systems we propose a new variant of smoothed aggregation sa suitable for nonsymmetric linear systems the new algorithm is based on two key generalizations of sa restriction smoothing and local damping restriction smoothing refers to the smoothing of a tentative restriction operator via a damped jacobi like iteration restriction smoothing is analogous to prolongator smoothing in standard sa and in fact has the same form as the transpose of prolongator smoothing when the matrix is symmetric local damping refers to damping parameters used in the jacobi like iteration in standard sa a single damping parameter is computed via an eigenvalue computation here local damping parameters are computed by considering the minimization of an energy like quantity for each individual grid transfer basis function numerical results are given showing how this method performs on highly nonsymmetric systems

a boundary perturbation analysis for elastic inclusions and interfaces abstract a first order boundary perturbation method based on muskhelishvilli s complex variable representations is formulated for the two dimensional elasticity problem of a nearly circular inclusion embedded in an infinite dissimilar material universal relations which are independent of loading conditions are established among the solutions for a homogeneous infinite plane a perfectly circular inclusion and a slightly perturbed non circular inclusion it is shown that the solution to a circular inclusion can be constructed algebraically from that of an infinite homogeneous plane while the stress distribution along the interface circle leads to the perturbation solutions for a nearly circular inclusion explicit results arc given for inclusions with smooth polygon shapes by considering cosine wavy perturbations along a reference circle a similar analysts is carried out for a bimaterial interface whose shape deviates slightly from a straight line our perturbation results can be used to study elastically induced morphological perturbations of surfaces interfaces voids precipitates and inclusions in a stressed solid as an example we demonstrate that under sufficiently large stresses material surfaces become unstable against a range of diffusional perturbations bounded by two critical wavelengths also as suggested from our perturbation analysis and verified by a finite element calculation even slight surface undulation caused by an unstable morphological perturbation can result in substantial stress concentration along the surface which may cause plastic deformation or brittle fracture before the bulk stress reaches a critical level

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

the compensation method applied to a one product production inventory problem this paper considers a one product one machine production inventory problem demand requests for the product are governed by a poisson process with demand size being an exponential random variable the production facility may be in production or idle while in production the facility produces continuously at a constant rate the objective is to minimize system costs consisting of setup costs inventory holding costs and backorder costs given a two critical number policy the problem is analyzed as a constrained markov process using the compensation method a closed form expression is found for system cost as a function of the policy parameters this cost function may then be searched to find an optimal policy the production inventory problem is related to a server backlog process for an m m 1 system in which the server departs when the system empties and returns when the backlog exceeds a specified threshold the application of the compensation method to this server backlog process is discussed

confidence structural robust design and optimization under stiffness and load uncertainties abstract most of the structural behavior constraints involved in structural robust design and optimization are non convex in nature therefore if local optimality criteria based optimization algorithms are employed to find the worst case structural responses that are used for examining the feasibility of a given design it is highly possible that the optimization process will get stuck in a local optimum if this is the case the reliability of a robust design cannot be guaranteed at least theoretically the aim of the present paper is to develop some new formulations and the corresponding numerical algorithms to find the confidence optimal design under non probabilistic stiffness and load uncertainties to this end two bi level program formulations for confidence robust design are proposed in order to ensure the strict feasibility of the optimal solution in the lower level of program the constraints are imposed on the confidence upper bounds of the structural responses which can be obtained efficiently by solving some convex linear semi definite programs lsdps based on the sensitivity analysis of the lower level lsdp problem the upper level programs are then solved by employing the classical gradient based nonlinear optimization algorithms furthermore for the case of stiffness uncertainty a single level nonlinear semi definite programming nsdp formulation is also proposed and its mathematical properties are analyzed numerical examples show that confidence robust optimal design can be obtained via the proposed approaches effectively without resorting to too many computational efforts

level set topology optimization for mechanical metamaterials under hybrid uncertainties abstract this paper proposes a level set based robust topology optimization rto method for computational design of metamaterials under hybrid uncertainties e g auxetics with negative poisson s ratio where the young s modulus of the solid is described as a random variable while the poisson s ratio is regarded as an interval variable firstly the robust objective function is formulated by a combination of interval mean and interval variance of the deterministic objective function secondly the interval mean and interval variance are computed by a hybrid uncertain analysis approach termed as polynomial chaos chebyshev interval pcci method thirdly the design sensitivities of the robust objective function are obtained after the implementation of the pcci method finally a powerful parametric level set method plsm in conjunction with the numerical homogenization method is applied to achieve the robust topological design for the auxetic microstructure several numerical cases are used to demonstrate the effectiveness of the proposed method for the robust topology optimization problems this method is non intrusive and general and can be easily extended to a range of design problems of micro structured metamaterials

optimal control of production rate in a failure prone manufacturing system we address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost where certain cost rates are specified for both positive and negative inventories and there is a constant demand rate for the commodity produced the underlying theoretical problem is the optimal control of a continuous time system with jump markov disturbances with an infinite horizon discounted cost criterion we use two complementary approaches first proceeding informally and using a combination of stochastic coupling linear system arguments stable and unstable eigenspaces renewal theory parametric optimization etc we arrive at a conjecture for the optimal policy then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides singularity of the optimal control problem smoothness and validity of the dynamic programming equation etc to give a rigorous proof of optimality of the conjectured policy it is hoped that both approaches will find uses in other such problems also we obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number which we call the optimal inventory level if the current inventory level exceeds the optimal one should not produce at all if less one should produce at the maximum rate while if exactly equal one should produce exactly enough to meet demand we also give a simple explicit formula for the optimal inventory level

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

production control of a manufacturing system with multiple machine states the production control of a single product manufacturing system with arbitrary number of machine states failure modes is discussed the objective is to find a production policy that would meet the demand for the product with minimum average inventory or backlog cost the optimal production policy has a special structure and is called a hedging point policy if the hedging points are known the optimal production rate is readily specified assuming a set of tentative hedging points the simple structure of the optimal policy is utilized to find the steady state probability distribution of the surplus inventory or backlog once this function is determined the average surplus cost is easily calculated in terms of the values of the hedging points the average cost is then minimized to find the optimum hedging points

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

a gradient method approach to optimization based multidisciplinary simulations and nonoverlapping domain decomposition algorithms it has been shown recently that optimization based nonoverlapping domain decomposition algorithms are connected to many well known algorithms using a gradient type iterative strategy for the optimization problem we present further discussion on how to develop various algorithms that can integrate subdomain solvers into a solver for the problem in the whole domain in particular the algorithms we discuss can be used to develop efficient solvers of multidisciplinary problems which are constructed using existing subdomain solvers without the need for making changes in the latter

elastic study on singularities interacting with interfaces using alternating technique part i anisotropic trimaterial abstract schwarz neumann s alternating technique is applied to singularity problems in an anisotropic trimaterial which denotes an infinite body composed of three dissimilar materials bonded along two parallel interfaces linear elastic materials under general plane deformations are assumed in which the plane of deformation is perpendicular to the two parallel interface planes it is well known that if the solution is known for singularities in a homogeneous anisotropic medium the solution for the same singularities in an anisotropic bimaterial can be constructed by the method of analytic continuation it is shown here that the solution for singularities in a homogeneous medium may also be used as a base of the solution for the same singularities in a trimaterial the alternating technique is applied to derive the trimaterial solution in a series form whose convergence is guaranteed the solution procedure is universal in the sense that no specific information about the singularity is needed the energetic forces exerted on a dislocation due to interfaces are also evaluated from the trimaterial solution the trimaterial solution studied here can be applied to a variety of problems e g a bimaterial including a half plane problem a finite thin film on semi infinite substrate and a finite strip of thin film etc some examples are presented to verify the usefulness of the obtained solutions

sporadic and continuous clearing policies for a production inventory system under an m g demand process a production inventory system is filled continuously at rate 1 and satisfies demands of i i d random sizes that arrive at poisson times for this system we consider two clearing policies under sporadic review clearing takes place after a random time independent of the content process under continuous review the system is cleared as soon as the inventory level reaches some pre specified threshold we derive explicit results for the associated expected discounted cost functionals under both types of clearing

singularities interfaces and cracks in dissimilar anisotropic media for a non pathological bimaterial in which an interface crack displays no oscillatory behaviour it is observed that apart possibly from the stress intensity factors the structure of the near tip field in each of the two blocks is independent of the elastic moduli of the other block collinear interface cracks are analysed under this non oscillatory condition and a simple rule is formulated that allows one to construct the complete solutions from mode iii solutions in an isotropic homogeneous medium the general interfacial crack tip field is found to consist of a two dimensional oscillatory singularity and a one dimensional square root singularity a complex and a real stress intensity factors are proposed to scale the two singularities respectively owing to anisotropy a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields however defined does not recover the classical stress intensity factors as the bimaterial degenerates to be non pathological collinear crack problems are also formulated in this context and a strikingly simple mathematical structure is identified interactive solutions for singularity interface and singularity interface crack are obtained the general results are specialized to decoupled antiplane and in plane deformations for this important case it is found that if a material pair is non pathological for one set of relative orientations of the interface and the two solids it is non pathological for any set of orientations for bonded orthotropic materials an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized a complex variable representation is presented for a class of degenerate orthotropic materials throughout the paper the equivalence of the lekhnitskii and stroh formalisms is emphasized the article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

unreliable production inventory model with a two phase erlang demand arrival process a production inventory system is studied in which an unreliable machine is susceptible to failure following which it must be repaired to make it operative again the demand for the product is governed by a two phase erlang process and demand sizes are independent and identically distributed random variables a two critical number policy m m is used to control a machine s setups and shutdowns namely the machine is shut down whenever the inventory level reaches m and is resumed to operate only when the inventory level falls below the critical number m m m we obtain the steady state distribution of the inventory process and some performance measures of the process

robust structural topology optimization considering boundary uncertainties in classical deterministic topology optimization the effect of the possible boundary variations on the performance of the structure is not taken into account which may lead to designs that are very sensitive to manufacturing errors as a consequence the performance of the real structure may be far from optimal and even not meet the design requirements in the present paper structural topology optimization considering the uncertainty of boundary variations is considered via level set approach in order to make the optimal designs less sensitive to the possible boundary variations we choose the compliance and fundamental frequency of structure enduring the worst case perturbation as the objective function for ensuring the robustness of the optimal solution with use of the schwarz inequality the original bi level optimization problem is transformed to a single level optimization problem which can be solved efficiently numerical examples demonstrate the effectiveness of the proposed approach

robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices abstract this paper proposes an efficient approach to solving robust topology optimization problem of structures under loading uncertainty the objective is to minimize a weighted sum of the mean and standard deviation of structural compliance loading uncertainties can be in either concentrated loads or uniformly distributed loads by exploiting of the linear elastic nature of structure monte carlo sampling is completely separated from the topology optimization process thus accurate calculation of objective function becomes possible efficient sensitivity analysis method is developed and its computational cost is only linearly proportional to the number of uncertain loads the sensitivity analysis is also integrated into the density based topology optimization approach to solve the robust topology optimization problems the numerical examples demonstrate the effectiveness of the proposed approach the effect of uncertainty level probability distribution of uncertainty and different influence of loading magnitude and directional uncertainty on the robust designs are also shown by the numerical examples

optimal control formulation for determining optical flow an optimal control formulation for determining optical flow is presented the new framework differs from preceding approaches in that it does not require differentiation of the data and does combine optical flow with image reconstruction it can be considered as a control in the coefficients problem with a cost functional of tracking type a numerical algorithm that solves the optimality system consisting of hyperbolic and elliptic partial differential equations is presented numerical experiments demonstrate the effectiveness of the optimal control approach

v cycle convergence of some multigrid methods for ill posed problems for ill posed linear operator equations we consider some v cycle multigrid approaches that in the framework of bramble pasciak wang and xu 1991 we prove to yield level independent contraction factor estimates consequently we can incorporate these multigrid operators in a full multigrid method that together with a discrepancy principle is shown to act as an iterative regularization method for the underlying infinite dimensional ill posed problem numerical experiments illustrate the theoretical results

flexible conjugate gradients we analyze the conjugate gradient cg method with preconditioning slightly variable from one iteration to the next to maintain the optimal convergence properties we consider a variant proposed by axelsson that performs an explicit orthogonalization of the search directions vectors for this method which we refer to as flexible cg we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small we further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments depending on the eigenvalue distribution corresponding to the fixed reference preconditioner several situations have to be distinguished in some cases the convergence is as fast with truncated versions of the algorithm or even with the standard cg method whereas quite large variations are allowed without too much penalty in other cases the flexible variant effectively outperforms the standard method while the need for truncation limits the size of the variations that can be reasonably allowed

introducing loading uncertainty in topology optimization uncertainty is an important consideration in structural design and optimization to produce robust and reliable solutions this paper introduces an efficient and accurate approach to robust structural topology optimization the objective is to minimize expected compliance with uncertainty in loading magnitude and applied direction where uncertainties are assumed normally distributed and statistically independent this new approach is analogous to a multiple load case problem where load cases and weights are derived analytically to accurately and efficiently compute expected compliance and sensitivities illustrative examples using a level set based topology optimization method are then used to demonstrate the proposed approach

a boundary perturbation method for circularly periodic plates with a core fundamental frequencies of vibrating circularly periodic plates with a circular core have been determined analytically a boundary perturbation method is developed to extract the fundamental eigenvalue of the governing biharmonic boundary value problem the method is then applied to wavy and polygonal plates with clamped and simply supported outer boundary conditions clamped simply supported and free circular cores are considered approximate analytical formulations of the fundamental frequency for such plates with core are obtained

a rotated rigid ellipsoidal inclusion in an elastic medium a rigid ellipsoidal inclusion is embedded at arbitrary orientation in a homogeneous arbitrarily anisotropic elastic matrix and is translated infinitesimally by an externally imposed force we find directly the relation between the force and translation vectors and the stress strain and rotation concentrations over the ellipsoidal surface without having to solve the equations of equilibrium in the matrix or the fundamental ones of a point force we refer particularly then to a spheroid aligned along the axis of symmetry of a transversely isotropic matrix and subsequently to the full elastic field of a general ellipsoid in an isotropic matrix

exact solutions in torsion of composite bars thickly coated neutral inhomogeneities and composite cylinder assemblages the study of microgeometries amenable to exact solutions has been of considerable interest in the modern literature of composite media although several such solutions exist today in the context of

on analyticity of travelling water waves in this paper we establish the existence and analyticity of periodic solutions of a classical free boundary model of the evolution of three dimensional capillary gravity waves on the surface of an ideal fluid the result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem and it yields analyticity jointly with respect to the perturbation parameter and the spatial variables the travelling waves we find can be interpreted as resulting from the nonlinear interaction of two two dimensional wavetrains giving rise to a periodic travelling pattern our analyticity theorem extends the most sophisticated results known to date in the absence of resonance short crested waves which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves indeed we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations these observations motivate the derivation and use of an alternative stable formulation which in addition to providing our method of proof suggests new stabilized implementations of boundary perturbation algorithms

piezoelastic study on singularities interacting with circular and straight interfaces abstract the piezoelastic investigation for a circular inclusion embedded in a sandwich has been carried out each medium of the composite is assumed to be transversely isotropic with hexagonal symmetry which has an isotropic basal plane of x1x2 plane and a poling direction of x3 axis the electromechanical loadings considered in this paper include a point force and a point charge located in the middle layer of the sandwich an efficient procedure is established by combining the analytical continuation method and alternating technique to derive the general forms of the piezoelastic fields in terms of the corresponding problem numerical results are provided for a number of particular examples to study the influence of material combinations geometry and loading condition on both the mechanical and electric response