semi definite programming for topology optimization of trusses under multiple eigenvalue constraints abstract topology optimization problem of trusses for specified eigenvalue of vibration is formulated as semi definite programming sdp and an algorithm is presented based on the semi definite programming algorithm sdpa which utilizes extensively the sparseness of the matrices since the sensitivity coefficients of the eigenvalues with respect to the design variables are not needed the sdpa is especially useful for the case where the optimal design has multiple fundamental eigenvalues global and local modes are defined and a procedure is presented for generating optimal topology from the practical point of view it is shown in the examples that sdpa has advantage over existing methods in view of computational efficiency and accuracy of the solutions and an optimal topology with five fold fundamental eigenvalue is found without any difficulty

coupling of two and three dimensional composite shell elements in linear and non linear applications abstract a numerical model for the non linear analysis of laminates is presented the developed element is based on piecewise polynomial interpolation in thickness direction although the total number of degrees of freedom is comparable to a discretization with 3d brick elements this approach provides several advantages e g a simple mesh generation due to a 2d type data structure and a better bending behavior the developed isoparametric quadrilateral finite element allows prediction of the complete stress state furthermore a transition element is presented which is used to couple the developed element with 5 parameter shell elements several linear and non linear examples are presented to illustrate the performance of the developed numerical model

hierarchic models for laminated plates and shells the definition essential properties and formulation of hierarchic models for laminated plates and shells are presented the hierarchic models satisfy three essential requirements approximability asymptotic consistency and optimality of convergence rate aspects of implementation are discussed and the performance characteristics are illustrated by examples

a comparison of preconditioned iterative techniques using rapid operator application against direct solution methods the p version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation and high accuracy with relatively few degrees of freedom however the solution time and storage requirements for large 3 d problems can be very high when direct methods are used to factor global matrices iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed but may not converge rapidly enough we study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level this resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods we illustrate the effectiveness of this scheme on some large three dimensional problems in structural analysis

fifteen node tetrahedral elements for explicit methods in nonlinear solid dynamics abstract despite the ease in meshing and benefits for modeling flexure curved shapes etc second order tetrahedral elements are not contained in typical explicit solid dynamic programs this is primarily due to the lack of both a satisfactory consistent nodal loading distribution and mass lumping technique row summation lumping for instance produces negative vertex node masses for the popular ten node serendipity tetrahedron which also has zero vertex node loads resulting from a constant pressure on an element face this has led to piecewise composites of four node tetrahedrons to represent a ten node one in explicit codes in this paper truly second order fifteen node formulations for compressible and for nearly incompressible materials are presented and evaluated in addition to producing all positive nodal loads from a uniform traction row summation mass lumping for the fifteen node element is shown to produce all positive nodal masses performance is assessed in standard benchmark problems and practical applications using various elastic and elastic plastic material models and involving very large strains deformations severe distortions and contact impact comparisons are also made with several first order elements and second order hexahedral formulations the offered elements performed satisfactorily in all examples as recently found for second order hexahedral elements it is shown that the inclusion of face and centroidal nodes is vital for robust performance with row summation lumping and high order quadrature rules are crucial with explicit methods these second order elements are shown to be viable for practical applications especially using today s parallel computers whereas the reliable performance is generally attained at significant computational expense compared with first order and brick types these elements can be more computationally competitive in flexure and have the desirable trait that they are amenable to automatic tetrahedral meshing software

on the membrane locking of h p finite elements in a cylindrical shell problem we study the performance of high order h p finite clement schemes in approximating the nearly inexlensional deformation state of a thin cylindrical shell roof with both ends free severe locking is demonstrated for both quadratic and cubic elements and even for higher order if the grid is not aligned with the axis of the cylinder carefully designed selective reduced integration is shown to improve the accuracy of the finite element approximation at low and moderately high orders however so far this technique seems to work well only on aligned grids

complementary energy method for cylindrical shells using second order stress functions abstract a new stress function based complementary energy approach and a finite element model are developed for linear axisymmetrical deformations of circular cylindrical shells the three dimensional equilibrium equations and stress boundary conditions are satisfied a priori with second order stress functions without employing the lagrange multiplier method or penalty function procedure this can be achieved i by utilization of the fact that three out of the six coordinates of the second order stress function tensor can be set to zero and ii by using truncated polynomial expansions for stresses and stress functions in the thickness direction kirchhoff type kinematical assumptions are not applied the second order stress functions based equilibrium element developed satisfies the continuity requirement for surface tractions at the element boundaries and gives accurate solutions for thin and moderately thick shells results are compared with elasticity and shell theory solutions

elastic solutions for periodically layered strip with perfect bonding or with an interface crack a non homogenization approach to the analysis of a strip consisting of a finite number of isotropic elastic layers arranged periodically is presented the result is obtained by the use of the representative cell method based on the discrete fourier transform two types of problems are addressed in the problems of the first type the bonding between the layers is perfect the solution for this case is found in a closed form in terms of the laplace integrals the numerical results for different boundary conditions are given and a comparison with plate theory is carried out in the problems of the second type there is a flaw crack at one of the interfaces the solution in this case hinges on the analytic expression for the green function corresponding to a single interface dislocation in the uncracked strip by using this expression the crack problem is reduced to a singular integral equation the influence of the elastic mismatch and other problem parameters on the fracture characteristics is examined it is found that similar to the case of the periodic plane when the thinner layeres in the strip are stiffer further increase of their stiffness may lead to the increase or to the decrease of the absolute value of the stress intensity factor depending upon the elastic mismatch between the composite constituents

p refinement and p threads abstract p type refinement leads to exponential decay of numerical errors for sufficiently smooth solutions and has been used effectively in turbulence and structural mechanics simulations in the context of spectral and hp finite element discretizations however it induces a computational cost of o p d 1 in d dimensions which is higher than lower order methods in this paper we demonstrate that by employing multi threading within mpi processes we manage to counter balance the cost increase associated with p refinement this approach reduces effectively the wall clock time and keeps it essentially constant as the polynomial order is increased while achieving exponential convergence rate since the number of threads within mpi processes can be dynamically adjusted through thread library functions the algorithm can be readily adapted for dynamic p refinement the resulting hybrid mpi threads dual level parallelism is particularly suitable for modern supercomputers consisting of symmetric multiprocessor nodes we demonstrate this approach in simulations of two three dimensional fluid dynamics problems

p 2 continuous finite elements on tetrahedra with local mass matrix inversion abstract when using continuous finite element methods for unsteady simulations one must invert a global mass matrix even for an explicit time advancement scheme in this paper we develop a local mass matrix inversion procedure for p 2 c 0 tetrahedral elements and also show that unlike on triangles b t helenbrook siam j numer anal 47 2009 1304 1318 this procedure does not generalize to higher order for the p 2 basis it is demonstrated that the accuracy of the local inversion is better than any previous mass lumping scheme this allows the development of fast and low memory iterative algorithms as well as accurate explicit computations to be performed the efficiency of using the approach as an iterative relaxation scheme for solving the helmholtz equation is shown by comparing to more standard approaches such as p multigrid with jacobi relaxation lastly the local mass matrix inversion is used to march the unsteady heat equation explicitly with third order spatial accuracy

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

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

semidefinite programming for uncertain linear equations in static analysis of structures abstract this paper presents a method for computing a minimal bound that contains the solution set to the uncertain linear equations particularly we aim at finding a bounding ellipsoid for static response of structures where both external forces and elastic moduli of members are assumed to be imprecisely known and bounded by using the s lemma we formulate a semidefinite programming sdp problem which provides an outer approximation of the minimal bounding ellipsoid bounding ellipsoids are computed for nodal displacements of uncertain braced frames as the solutions of the presented sdp problems by using the primal dual interior point method

superconvergent extraction of flux intensity factors and first derivatives from finite element solutions a superconvergent method for the computation of the derivatives of the solution and the coefficients of the asymptotic expansion at singular points is presented for the laplace problem in two dimensions the algorithm utilizes the complementary weak form on a localized small domain mathematical analysis demonstrates the superconvergent behavior and numerical experiments support our analysis this method is well suited for anisotropic multi material singular interface problems

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

robustness analysis of trusses with separable load and structural uncertainties this paper discusses evaluation techniques of the robustness function of trusses which is regarded as one of measures of structural robustness under the uncertainties of member stiffnesses and external forces by using quadratic embedding of the uncertainty and the s procedure we formulate a quasiconvex optimization problem which provides lower bounds of the robustness functions a bisection method is proposed where we solve a finite number of semidefinite programming problems in order to obtain a global optimal solution to the proposed quasiconvex optimization problem the lower bounds of the robustness functions are computed for various trusses under several uncertainty circumstances

a matrix d alembert formula for coupled wave initial value problems if c is an invertible matrix in cr r the coupled wave equation initial value problem utt c2uxx 0 u x 0 f x and ut x 0 g x for x is studied a matrix d alembert formula for the closed form solution of the coupled wave equation is given the approach is based on the fourier transform the holomorphic matrix functional calculus and some elements of complex variable functions for the scalar case the proposed d alembert formula coincides with the classical one

multiparametric shell eigenvalue problems abstract the eigenproblem for thin shells of revolution under uncertainty in material parameters is discussed here the focus is on the smallest eigenpairs shells of revolution have natural eigenclusters due to symmetries moreover the eigenpairs depend on a deterministic parameter the dimensionless thickness the stochastic subspace iteration algorithms presented here are capable of resolving the smallest eigenclusters in the case of random material parameters it is possible that the eigenmodes cross in the stochastic parameter space this interesting phenomenon is demonstrated via numerical experiments finally the effect of the chosen material model on the asymptotics in relation to the deterministic parameter is shown to be negligible

goal oriented adaptive finite element methods for elliptic problems revisited a goal oriented a posteriori error estimation of an output functional for elliptic problems is presented continuous finite element approximations are used in quadrilateral and triangular meshes the algorithm is similar to the classical dual weighted error estimation however the dual weight contains solutions of the proposed patch problems the patch problems are introduced to apply clement and scott zhang type interpolation operators to estimate point values with the finite element polynomials the algorithm is shown to be reliable efficient and convergent

unidirectional fiber reinforced composite laminae an algorithm for the computation of the homogenized material properties of unidirectional fiber reinforced laminae is described the method of extraction is such that superconvergence is realized it is shown that the homogenized material properties are substantially independent of the number of rves an algorithm for the determination of the micromechanical solution over an rve localization is described and the conditions that predictors of failure initiation must satisfy are stated

ellipsoidal load domain shakedown analysis with von mises yield criterion a robust optimization approach summary the static formulation of elastic shakedown analysis based upon melan s lower bound theorem can essentially be viewed as a robust optimization problem this paper discusses an advantage that is enjoyed by taking this perspective specifically we assume the von mises yield criterion and an ellipsoidal load domain in this setting the shakedown analysis problem which is viewed as robust second order cone programming can be recast as semidefinite programming copyright 2016 john wiley sons ltd

ellipsoidal bounds for uncertain linear equations and dynamical systems abstract in this paper we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations the proposed technique is based on the combination of a quadratic embedding of the uncertainty and the s procedure this formulation leads to convex optimization problems that can be essentially solved in o n3 n being the size of unknown vector by means of suitable interior point barrier methods as well as to closed form results in some particular cases we further show that the uncertain linear equations paradigm can be directly applied to various state bounding problems for dynamical systems subject to set valued noise and model uncertainty

on capacity computation for symmetric polygonal condensers abstract making use of two different analytical numerical methods for capacity computation we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers these two methods are based respectively on the use of the lauricella function bezrodnykh and vlasov 2002 bezrodnykh 2016 64 65 and riemann theta functions bogatyrev 2012 grigoriev 2013 bogatyrev and grigor ev 2017 bogatyrev and grigor ev 2018 we apply these results to benchmark the performance of numerical algorithms which are based on adaptive h p finite element method hakula et al 2011 2013 2018 and boundary integral method tsuji 1959 jaswon and symm 1977 albrecht and collatz 1980

hierarchic models for laminated composites the principles governing the formulation of hierarchic models for laminated composites are discussed the essential features of the hierarchic models described herein are a the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness and b the exact solution of each model converges to the same limit as the exact solution of the corresponding problem of elasticity with respect to the laminate thickness approaching zero hierarchic models make the computation of any engineering data possible to an arbitrary level of precision within the framework of the theory of elasticity examples are presented

on a certain mixed variational theorem and a proposed application formulation d une equation variationnelle pour les deplacements et certaines contraintes en vue d une application aux plaques elastiques stratifiees isotropes et anisotropes

deep doubly curved multilayered shell theory a two dimensional theory is presented for the analysis of deep doubly curved multilayered shells the theory is based on a kinematical approach in which the continuity conditions for displacements and shear stresses at layer interfaces and on the bounding surfaces of the shell are exactly satisfied it also takes into account refinements of the shear and membrane terms by means of trigonometric functions as proposed previously for the transverse shear the accuracy of the proposed theory is assessed through investigation of significant problems for which an exact three dimensional elasticity solution is known first the bending of a three layered laminated symmetric cross ply rectangular plate simply supported along all edges submitted to a double sinusoidal transverse loading and second the bending of a circular cylindrical panel results obtained with the model are compared with those yielded by previous theories the sensitivity of the model to edge effects for a hard clamped free edge cylindrical panel is also examined

high order finite elements applied to the discrete boltzmann equation a discontinuous galerkin approach for solving the discrete boltzmann equation is presented allowing to compute approximate solutions for fluid flow problems based on a two dimensional high order finite element and an explicit euler time stepping scheme the d2q9 model is discretized and the results are compared to the exact solution of the navier stokes equation four numerical examples are considered including stationary and instationary problems with curved boundaries it is demonstrated that the proposed method allows to obtain the desired highly efficient exponential convergence copyright 2006 john wiley sons ltd

extreme structural response analysis of truss structures under material uncertainty via linear mixed 0 1 programming this paper concerns the extreme structural response analysis of truss structures taking the possible uncertainties of young s moduli of the bars into consideration the uncertainty of young s modulus is described by a non probability but bounded interval model with the use of this uncertain model it is shown that the structural responses such as nodal displacement and bar stress will attain their extreme values when young s moduli of the bars take either their upper or lower bound value based on this conclusion and simultaneous analysis and design formulation a linear mixed 0 1 programming problem can be constructed to find the extreme values of the structural response by the well established branch and bound approach with global optimality the proposed approach is applied to several test problems for the demonstration of its validity

hierarchical models of thin elastic structures overview and recent advances in error estimation and adaptivity abstract in the numerical approximation of hierarchical models of thin bodies there are two separate contributions to the total error a modeling error is the error due to dimensional reduction and b discretization error is the error due to the numerical approximation traditionally estimation of these errors has been performed in terms of abstract quantities like the global energy norm which does not give a clear representation of the error in the local features of the solution recently a new class of finite element error estimation techniques has emerged wherein errors in a simulation are measured not in norms but in quantities of interest to the analyst following a brief overview of the literature we focus on extending the theory of a posteriori estimation of the modeling error to local quantities of interest that have physical meaning independent estimates are obtained for the modeling and discretization errors in quantities of interest these local estimates are used to develop goal oriented adaptive modeling strategies in which the model is automatically adapted by controlling the two errors independently and effectively to obtain local quantities of interest to within a preset level of accuracy numerical examples are presented to demonstrate the accuracy of the error estimates and the adaptive strategy

a posteriori modeling error estimation for hierarchic plate models hierarchic plate models are dimensional reductions obtained by semidiscretization of the three dimensional plate problem in the transverse direction and energy projection we derive computable a posteriori estimators for the modeling error thus incurred under the assumption that the resulting two dimensional plate models are solved exactly the estimators are valid for homogeneous monoclinic materials for plates with unsmooth midsurfaces and for a wide class of variational edge conditions computable a posteriori bounds on the effectivity indices are also derived and sufficient conditions for the asymptotic i e as the plate thickness tends to zero and the spectral i e as the order of the plate model tends to infinity exactness of the estimators are given numerical examples are presented

hypersingular residuals a new approach for error estimation in the boundary element method summary this paper presents a new approach for a posteriori pointwise error estimation in the boundary element method the estimator relies upon evaluation of the residual of hypersingular integral equations and is therefore intrinsic to the boundary integral equation approach a methodology is developed for approximating the error on the boundary as well as in the interior of the domain extensive computational experiments have been performed for the two dimensional laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve moreover a reasonable estimate of the magnitude of the actual error is also predicted

spectral element method in time for rapidly actuated systems in this paper the spectral element se method is applied in time to find the entire time periodic or transient solution of time dependent differential equations the time periodic solution is computed by enforcing periodicity of the element set of particular interest are periodic forcing functions possessing high frequency content to maintain the spectral accuracy for such forcing functions an h refinement scheme is employed near the semi discontinuity without increasing the number of degrees of freedom time discretization by spectral elements is applied initially to a standard form of a set of linear first order differential equations subject to harmonic excitation and an excitation admitting rapid variation other case studies include the application of the se approach to parabolic and hyperbolic partial differential equations the first order form of these equations is obtained through semi discretization using conventional finite element spectral element and finite difference schemes element clustering h refinement is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation the convergence in time of the method is demonstrated in some cases machine precision is obtained with 25 degrees of freedom per cycle finally the method is applied to a weakly nonlinear problem with time periodic solution to demonstrate its future applicability to the analysis of limit cycle oscillations in aeroelastic systems

analysis of thick sandwich construction by a 3 2 order theory abstract this study is an extension of the 1 2 order plate theory to a higher order 3 2 theory based on the equivalent single layer assumptions the in plane and transverse displacement components are expressed as cubic and quadratic expansions through the thickness of the sandwich construction also the transverse stress component is assumed to vary as a cubic function through the thickness utilizing reissner s definitions for kinematics of thick plates the displacement components at any point on the plate are approximated in terms of weighted average quantities displacements and rotations that are functions of the in plane coordinates the undetermined coefficients defining the in plane and transverse displacement fields are then expressed in terms of the weighted average displacements and rotations and their derivatives by directly employing reissner s definitions and enforcing the zero transverse shear stress conditions on the upper and lower surfaces of the sandwich panel the coefficients defining the transverse stress component are obtained by requiring the transverse strain component which is expressed in terms of the unknown coefficients of the transverse stress component from a mixed constitutive relation to be the least squares equivalent of the kinematic definition of the transverse strain component the resulting expressions for the unknown coefficients of the transverse stress component are related to resultant strains and curvatures defined from kinematic relations the equations of equilibrium and boundary conditions of the sandwich plate based on the 3 2 higher order theory are derived by employing the principles of virtual displacements the robustness and accuracy of this 3 2 order plate theory are established through comparisons with exact solutions available in the literature the finite element implementation of the present 3 2 order plate theory is also discussed

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

robust design of structures using optimization methods the robust design of structures with stochastic parameters is studied using optimization techniques the first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second order perturbation method for the approximation of mean value and variance of the structural response in this framework the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task the robust design of structures is formulated as a multi criteria optimization problem in which both the expected value and the standard deviation of the objective function are to be minimized the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints the two criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm to demonstrate the applicability of the presented method numerical examples are given involving static and dynamic response

stress based hierarchic models for laminated composites abstract derivation of hierarchic sequence of equilibrium models for cylindrical bending of composite laminates is presented the levels of hierarchy correspond to the degree to which the second order compatibility equation of the two dimensional elasticity is satisfied the stress fields of the hierarchic models satisfy a priori the equilibrium equations and the stress boundary conditions of two dimensional elasticity and the continuity requirements for the transverse shear and normal stresses at the lamina interfaces are also a priori satisfied for each member of the hierarchy the numerical solution is based on the principle of minimum complementary energy the number of degrees of freedom in the finite element approximation is independent of the numbers of the layers in the laminate the results are obtained directly for stresses whereas the displacements are calculated in the post processing phase by integration numerical results with comparisons present the performance of the mathematical and numerical models proposed

mid node admissible spaces for quadratic triangular 2d finite elements with one edge curved a robust element distortion metric based on the new concept of mid node admissible spaces for two dimensional quadratic triangular finite elements is developed the metric is based on the jacobian determinant over the entire element without requiring that it actually be computed everywhere on the element the metric is relatively inexpensive to compute especially for mildly distoted elements the metric is able to detect elementsof poor quality that other distortion metrics fall to detect it also has the ability to approve elements of good quality regardless of the extent to which they may appear geometrically distorted copyright 2001 john wiley sons ltd

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

three dimensional mixed explicit implicit generalized galerkin spectral element methods for high speed turbulent compressible flows abstract in high speed flows the interactions of shock waves with turbulent boundary layers are important design considerations because of the complex flowfields resulting in increased adverse pressure gradients skin friction and temperatures unsteadiness and three dimensional flowfield structure are also characteristic of shock wave turbulent boundary layer interactions such physical phenomena require sophisticated numerical schemes in the solution of governing equations the purpose of this paper therefore is to introduce an accurate and efficient approach the mixed explicit implicit generalized galerkin spectral element method mei gg sem with legendre polynomial spectral elements in which flowfield dependent implicitness parameters provide automatically adequate computational requirements for compressible and incompressible flows or high speed and low speed flows this is in contrast to the traditional approach in which all speed regime analysis requires a separate hyperbolic elliptic pressure equation for pressure correction if the flow becomes incompressible in the mei gg sem scheme mesh refinements are carried out adaptively until shock waves are resolved followed then by the adaptive increase of legendre polynomial degrees until turbulence microscales are resolved in which the traditional turbulence modeling is no longer required aimed toward direct numerical simulation in order to demonstrate the validity of the theory and numerical procedure two dimensional flat plate and compression corner high speed flows are investigated followed by a three dimensional sharp leading edged fin for swept shock wave turbulent boundary layer interactions comparisons of the present study with experimental measurements and other numerical studies show favorable agreement

formulation of geometrically non linear problems in the spatial reference frame this paper presents a new formulation for geometrically non linear problems and their numerical treatment by the p version of the finite element method the formulation is characterized by a weak form based on the spatial representation of the equilibrium equations a deformed geometry mapped by the displacement field a natural description for non conservative loads that keeps the symmetry of the equations and a direct non incremental iterative algorithm to solve the equations the paper is concerned only with problems for which the solution is unique i e the load displacement curves are single valued and the strains are small examples are presented 1997 by john wiley sons ltd

a general framework for robust topology optimization under load uncertainty including stress constraints we present a worst case approach for topology optimization under load uncertainty based on a general problem formulation involving maxima of quadratic functions of an uncertain load vector as both

the p and hp finite element method for problems on thin domains abstract the p and hp versions of the finite element method allow the user to change the polynomial degree to increase accuracy we survey these methods and show how this flexibility can be exploited to counter four difficulties that occur in the approximation of problems over thin domains such as plates beams and shells these difficulties are 1 control of modeling error 2 approximation of corner singularities 3 resolution of boundary layers and 4 control of locking our guidelines enable the efficient resolution of these difficulties when a p hp code is available

matching experimental and three dimensional numerical models for structural vibration problems with uncertainties abstract the simulation model which examines the dynamic behavior of real structures needs to address the impact of uncertainty in both geometry and material parameters this article investigates three dimensional finite element models for structural dynamics problems with respect to both model and parameter uncertainties the parameter uncertainties are determined via laboratory measurements on several beam like samples the parameters are then considered as random variables to the finite element model for exploring the uncertainty effects on the quality of the model outputs i e natural frequencies the accuracy of the output predictions from the model is compared with the experimental results to this end the non contact experimental modal analysis is conducted to identify the natural frequency of the samples the results show a good agreement compared with experimental data furthermore it is demonstrated that geometrical uncertainties have more influence on the natural frequencies compared to material parameters and material uncertainties are about two times higher than geometrical uncertainties this gives valuable insights for improving the finite element model due to various parameter ranges required in a modeling process involving uncertainty

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

g1 interpolation and geometry reconstruction for higher order finite elements abstract the paper proposes a new g1 interpolation and geometry reconstruction scheme for regular surfaces approximated with unstructured quadrilateral grids the parametrization is based on c1 transfinite interpolation and sixth order polynomials the g1 conforming approximation of surfaces guarantees continuity of surface normals and it is critical for higher order finite and boundary element discretizations on curvilinear domains the proposed interpolation reconstruction scheme is illustrated with a number of numerical examples including the geometry of human head

application of hp adaptive be fe methods to elastic scattering abstract the investigation focuses on the development of hp adaptive methods for simulation of acoustical scattering problems the present formulation is based on steady state linear elasticity equations coupled with the burton miller integral equation representing the behavior of a surrounding acoustical medium the highlights of the proposed approach include a mathematically consistent variational formulation of the problem an hp adaptive fe be approximation based on standard galerkin methods an element by element solver a residual based a posteriori error estimate an hp adaptive strategy particular attention is given to asymptotic convergence and to a comparison of theory and numerical simulations

a model study of the quality of a posteriori error estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles abstract this paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems in this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles a completely numerical methodology for the analysis of the quality of estimators is presented results from the application of the methodology to the study of the quality of several well known error estimators are reported in subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals non uniform grids at boundaries grid interfaces and near singular points

dynamic computer simulation of concentrated hard sphere suspensions ii re analysis of mean square displacement data in view of recent theoretical results by cichocki and felderhof we reevaluate previously published simulation data for the mean square displacement of a particle in a hard sphere suspension without hydrodynamic interactions this new evaluation is consistent with the theoretically predicted algebraic long time tails we observe better agreement for dense systems and obtain values for the relaxation time almost twice as large as previously

rotational velocity autocorrelation function of interacting brownian particles recently hagen frenkel and lowe physica a 272 1999 376 have obtained interesting simulation data for the rotational velocity autocorrelation function of a selected particle in a dense suspension of interacting spherical brownian particles although a wide time interval was studied it was found that the correlation function had not yet settled down to its t 5 2 long time tail at the longest time considered in the following analysis the method of pade approximants is applied to the laplace transform of the correlation function a dierent value for the coecient of the long time tail is obtained than found by hagen et al by extrapolation a prediction is made for the behavior of the correlation function outside the range of presently available data c 2001 elsevier science b v all rights reserved

a refined first order shear deformation theory and its justification by plane strain bending problem of laminated plates abstract a refined first order shear deformation theory is proposed and used to solve the plane strain bending problem of both homogeneous plates and symmetric cross ply laminated plates in reissner mindlin s traditional first order shear deformation theory fsdt the displacement field assumptions include a linear inplane displacement component and a constant transverse deflection through the thickness these assumptions are retained in the present refined theory however the associated transverse shear strain derived from these displacement assumptions which is still independent of the thickness coordinate is endowed with new meaning the stress weighted average shear strain through the thickness the variable distribution of transverse shear strain is assumed in such a way that it agrees with the shear stress distribution derived from the integration of equilibrium equation this paper introduces the effective transverse shear stiffness of plates by assuming that the normalized distribution of through the thickness transverse shear stress remains unchanged regardless of geometrical configuration span to thickness ratio for plane strain bending problem which is justified by the exact elasticity solution without losing the simplicity of the displacement field assumptions of reissner mindlin s fsdt the present refined first order theory not only shows improvement on predicting deflections but also accounts for a variable transverse shear strain distribution through the thickness in addition all the boundary conditions equilibrium equations and constitutive relations are satisfied pointwise comparisons of deflection transverse shear strain and transverse shear stress obtained using the present theory are made with the exact results given by pagano

semidefinite programming for dynamic steady state analysis of structures under uncertain harmonic loads this paper presents semidefinite programming approaches for computing confidential bounds for the dynamic steady state responses of a damped structure subjected to uncertain driving loads we assume that amplitudes of harmonic driving loads obey a non probabilistic uncertainty model the semidefinite programming problems are formulated for finding confidential bounds for various characteristic amounts of dynamic steady state response including the modulus and phase angle of the complex amplitude of the displacement and stress numerical examples demonstrate that sufficiently tight bounds can be obtained by solving the presented semidefinite programming problems

an improved plate theory of 1 2 order for thick composite laminates abstract a new two dimensional laminate plate theory is developed for the linear elastostatic analysis of thick composite plates the theory employs equivalent single layer assumptions for the displacements transverse shear strains and transverse normal stress the inplane and transverse displacements are respectively linear and quadratic expansions through the laminate thickness where the low order expansion coefficients correspond to the variables of reissner s first order shear deformable theory the transverse shear strains and transverse normal stress are assumed to be quadratic and cubic respectively through the thickness they are expressed in terms of the kinematic variables of the theory by means of a least squares compatibility requirement for the transverse strains and explicit enforcement of exact traction boundary conditions on the top and bottom plate surfaces application of the virtual work principle results in the loth order equations of equilibrium and associated poisson boundary conditions a major advantage of this theory over other higher order theories lies in its perfect suitability for finite element approximation simple c 0 continuous interpolations for the kinematic variables of the first order theory and optionally c 1 interpolations for the two higher order displacements are needed to formulate effective and robust two dimensional plate elements capable of full three dimensional ply by ply strain and stress recovery within the framework of general purpose finite element codes in assessing the predictive capability of the theory analytic solutions for the problem of cylindrical bending are derived and compared with exact three dimensional elasticity results and those of the earlier version of the 1 2 order plate theory

cones of matrices and successive convex relaxations of nonconvex sets let f be a compact subset of the n dimensional euclidean space rn represented by finitely or infinitely many quadratic inequalities we propose two methods one based on successive semidefinite programming sdp relaxations and the other on successive linear programming lp relaxations each of our methods generates a sequence of compact convex subsets ck k 1 2 of rn such that a the convex hull of f subseteq c k 1 subseteq c k monotonicity b cap k 1 infty c k text the convex hull of f asymptotic convergence our methods are extensions of the corresponding lovasz schrijver lift and project procedures with the use of sdp or lp relaxation applied to general quadratic optimization problems qops with infinitely many quadratic inequality constraints utilizing descriptions of sets based on cones of matrices and their duals we establish the exact equivalence of the sdp relaxation and the semi infinite convex qop relaxation proposed originally by fujie and kojima using this equivalence we investigate some fundamental features of the two methods including a and b above

slow dynamics of linear relaxation systems abstract linear relaxation occurring in dielectrics viscoelastic fluids and many other systems is often characterized by a broad continuous spectrum we show that the relaxation behavior may be analyzed effectively by means of n point pade approximants applied in the complex plane of the square root of frequency the method leads to a compact analytic expression for the laplace transform of the relaxation function characterized by a small number of poles and their residues in the square root of frequency plane we study the method in detail for a model of diffusion in three dimensions with a single or double radial potential barrier and demonstrate its use in the analysis of the viscoelastic relaxation spectrum of polyisobutylene

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

locking free mixed hp finite element methods for curvilinear domains mixed hp finite element methods fems are robust with respect to poisson ratio locking when straight sided quadrilateral elements are used for the p version which uses fixed and generally large elements curved elements are necessary to represent curved domains accurately we show that mixed hp curved quadrilateral elements retain the robustness of their straight sided counterparts provided degenerate elements are avoided we also show that hp version triangles are robust we use these concepts to develop a locking free fem for geometrically nonlinear elasticity

geometrical validity of curvilinear finite elements in this paper we describe a way to compute accurate bounds on jacobian determinants of curvilinear polynomial finite elements our condition enables to guarantee that an element is geometrically valid i e that its jacobian determinant is strictly positive everywhere in its reference domain it also provides an efficient way to measure the distortion of curvilinear elements the key feature of the method is to expand the jacobian determinant using a polynomial basis built using bezier functions that has both properties of boundedness and positivity numerical results show the sharpness of our estimates

locking effects in the finite element approximation of plate models we analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d we identify h version schemes that show locking i e for which the asymptotic convergence rate deteriorates as d 0 and also show that the p version is free of locking in order to isolate locking erects from boundary layer erects which also arise as d 0 our analysis is carried out for the periodic case which is free of boundary layers we analyze in detail the lowest model of the hierarchy the well known reissner mindlin model and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the riessner mindlin case