a simple method for group analysis and its application to a model of detonation a simple procedure is suggested to obtain extensions of the principal lie algebra for invariance transformations of a given family of equations by using an equivalence algebra an application to a qualitative model of detonation is performed

continuum approach for second order shape design sensitivity of three dimensional elastic solids a second order shape design sensitivity analysis method for three dimensional linear elastic solids is derived using a continuum approach with the material derivative to compute the second order shape design sensitivity a shape design acceleration field is defined which is similar to the shape design velocity field for the first order shape design sensitivity both direct differentiation and hybrid methods are presented in this paper the shape changes are assumed on the traction free boundary only a numerical method which can be implemented using established finite element analysis codes is developed to demonstrate feasibility and accuracy of the proposed second order shape design sensitivity analysis method

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

improved constraint aggregation methods abstract many constraints in design optimization problems take the form of bounds on a physical quantity over a domain these infinite dimensional constraints are often addressed through aggregation methods that approximate the bound in a differentiable manner however the most common constraint aggregation methods exhibit mesh dependence as the mesh spacing decreases the aggregation function diverges in addition there are no established methods to assess the impact of constraint aggregation on the optimized design to address these deficiencies we propose a new class of constraint aggregation method that we call induced aggregates we examine the properties of these aggregation methods describe their numerical implementation and test their numerical accuracy on a pressure loaded circular cylinder test case furthermore we propose a post optimality estimation technique to assess the impact of the constraint aggregation method on the design we use this method to examine the optimized mass obtained from two stress constrained mass minimization problems a variable thickness sheet problem and a wingbox design problem

equivalence groups for balance equations abstract we consider a system of balance equations and by using ovsiannikov s method construct the most general equivalence algebra moreover some special cases are presented

a modular approach for three dimensional shape optimization of structures an approach for shape optimization of three dimensional solid structures is described a major consideration in the development of this capability was the desire to use a commercially available finite element program such as nastran for analysis since nastran cannot be called a subroutine a system architecture of independently executable modules was developed in which sequential execution is controlled by job control language also sensitivities are not commonly available in commercial programs a material derivative approach was developed to obtain shape sensitivities by postprocessing finite element boundary stresses stored on files a parameterized surface description and mesh generation are provided using isoparametric mapping patches the feasible directions algorithm conmin is used for optimization several automotive related examples are used to evaluate the program s effectiveness

group classification of three dimensional boundary layer equations of a class of non newtonian fluids group classification of three dimensional boundary layer equations of a class of non newtonian fluids in which thee stress is an arbitrary function of rates of strain is made group classification of the equations with respect to shear stress is done by using equivalence transformations special cases such as arbitrary stress newtonian fluids and power law fluids are treated

on the use of hadamard expansions in hyperasymptotic evaluation of laplace type integrals ii complex variable we review and discuss the application of hadamard expansions to the hyperasymptotic evaluation of laplace integrals c ex p t f t dt x where for simplicity in this paper x is restricted to be a positive real variable the integration path c can be taken over both finite and semi infinite intervals in the complex plane in general these expansions take the form of compound expansions each associated with a different exponential level and involve absolutely convergent series containing the incomplete gamma function as a smoothing factor the early terms in each convergent expansion possess a rapid asymptotic like decay when the variable x is large with late terms that can be transformed into a rapid decay comparable with that of the early terms the hadamard expansion of the above integral when the phase function p t is linear is shown to depend significantly on the singularity structure of the amplitude function f t the application of the theory to laplace type integrals with quadratic cubic and nonpolynomial phase functions is considered in addition to the amplitude function the location of the saddle points satisfying p t 0 also plays a role in the detailed structure of the different exponential levels in the resulting hadamard expansion numerical examples are given to illustrate the accuracy that can be achieved with this new procedure

uniform asymptotics and zeros of a system of orthogonal polynomials defined via a difference equation abstract we study a class of sieved pollaczek polynomials defined by a second order difference equation three term recurrence relation the measure of orthogonality is determined by using the markov theorem and the perron stieltjes inversion formula and is shown consisting of an absolutely continuous part and a discrete part with infinitely many mass points uniform asymptotic approximations of these polynomials for large degree n are derived at a turning point n and a critical point n involving respectively the airy function ai and a 1 2 ai i 2 bi darboux s method the method of steepest descents and various uniform asymptotic techniques such as cubic transformations are used to derive the results asymptotic formulas for the least zeros the largest zeros and the zeros on both sides of n are also obtained several numerical examples are provided to compare the approximate zeros with the true values

preliminary group classification of equations vtt f x vx vxx g x vx a classification is given of equations vtt f v vx vxx g x vx admitting an extension by one of the principal lie algebra of the equation under consideration the paper is one of few applications of a new algebraic approach to the problem of group classification the method of preliminary group classification the result of the work is a wide class of equations summarized in table ii

optimal design for polymer extrusion part ii sensitivity analysis for weakly coupled nonlinear steady state systems the die design methodology presented in part i is extended to include performance measures that are functions of material residence time we solve a hyperbolic differential equation using the velocity field computed from the hele shaw pressure analysis to evaluate the material residence time in polymer melts the residence time governing equation lacks natural diffusion therefore we employ the streamline upwind petrov galerkin supg finite element method to compute a spatially stable residence time field in the design problem we derive design sensitivities for steady state nonlinear weakly coupled systems and include design variables that parameterize essential boundary conditions design sensitivities are derived via the direct differentiation and adjoint methods special consideration is given to the supg weighting function since it is a function of the design to demonstrate the methodology sheet extrusion dies are designed to simultaneously minimize the exit velocity variation and the exit residence time variation

an efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs a mathematical model for the prediction of fluid pressures is given by a time dependent diffusion equation application of the finite element method leads to a system of linear equations a complication is that the underground consists of layers with very large differences in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the cg method has been observed moreover a classical termination criterion is not valid in this problem after diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability for the ic preconditioner the same behavior is observed to annihilate the effect of the extreme eigenvalues a deflated cg method is used the convergence rate improves considerably and the termination criterion becomes again reliable finally a cheap approximation of the eigenvectors is proposed

uniform asymptotic expansions of the pollaczek polynomials two uniform asymptotic expansions are obtained for the pollaczek polynomials pn cos a b one is for 0 n 0 a b in terms of elementary functions and in descending powers of n the other is for n 2 in terms of a special function closely related to the modified parabolic cylinder functions in descending powers of n this interval contains a turning point and all possible zeros of pn cos in 0 2

semi analytical static nonlinear structural sensitivity analysis the paper discusses two uses of the semi analytical method for nonlinear sensitivity analysis first the application to noncritical response is analyzed it is shown that the semi analytical method is equivalent to a particular form of the overall finite difference approach it is concluded that the difference between overall finite differences and the semi analytical approach is blurred for the case of nonlinear static response next the application of the semi analytical method to calculating sensitivities of limit loads is discussed a formulation that is easy to implement in general purpose finite element programs is derived three examples are used to demonstrate the application of the formulation and to explore its accuracy

asymptotic distribution of zeros of polynomials satisfying difference equations we propose a way to find the asymptotic distribution of zeros of orthogonal polynomials pn x satisfying a difference equation of the form b x pn x c x n pn x d x pn x 0 we calculate the asymptotic distribution of zeros and asymptotics of extreme zeros of the meixner and meixner pollaczek polynomials the distribution of zeros of meixner polynomials shows some delicate features we indicate the relation of our technique to the approach based on the nevai dehesa ullman distribution

implementation of design sensitivity analysis for nonlinear elastic structures implementation of design sensitivity analysis into marc a general nonlinear finite element code is described follower forces and constraints treated via lagrange multipliers are also included in the implementation the sensitivity analysis is implemented in the most general way i e arbitrary design criteria and design parameters including shape sizing and material property variables can be defined through special user supplied subroutines local derivatives of all of the quantities needed for sensitivity analysis are calculated using the central differences allowing a very general implementation without sacrificing the accuracy design sensitivities to multiple parameters are implemented even though an option for multiple loading does not apply to nonlinear finite element analysis also design sensitivities can be calculated at any load level whereas the analysis calculations march on as if there was no interruption derivation of design sensitivities with constraints such as incompressibility or unilateral frictionless contact indicates that the same steps as in the implementation of regular displacement based nonlinear finite element method can be followed without any modifications a numerical example using the modified finite element code is presented to demonstrate the new capability

spreading of a thin film with suction or blowing including surface tension effects in this paper we derive a fourth order nonlinear partial differential equation modelling the effects of suction blowing on the evolution of the free surface of a thin viscous film on a porous base gravity and surface tension effects are included the suction blowing is modelled as an arbitrary function v n t r the lie group classification method is used to determine a first order quasi linear partial differential equation satisfied by v n t r this equation is coupled with the model equation and solved numerically new results are obtained which show how the free surface and the apparent contact angle vary with suction blowing

shape variable definition with c0 c1 and c2 continuity functions the present paper proposes a new technique for the definition of the shape design variables in 2d and 3d optimisation problems it can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the cad defining surfaces the proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori this capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process structural analysis fluid dynamic analysis crash analysis aerodynamic analysis etc

calculation of sensitivity derivatives in thermal problems by finite differences the transient thermal analysis of large aerospace vehicles such as the space shuttle requires models with thousands of degrees of freedom the design of thermal protection systems for such vehicles requires the repetitive thermal analysis of the vehicle and the calculation of sensitivities derivatives of the transient response with respect to design parameters when the number of design parameters is large the cost of the sensitivity calculations may become prohibitive the present work is concerned with reducing the computational cost and errors in sensitivity calculations with one of the simplest available methods the finite difference approach a technique for calculating simultaneously the nominal and perturbed solutions required for the finite difference derivative is proposed an expression for the optimum value of the perturbation parameter is obtained which permits reducing the computational error an insulated cylinder is used as an example which demonstrates that significant improvements in accuracy and efficiency may be obtained

design sensitivity analysis and shape optimization of structural components with hyperelastic material abstract a continuum based design sensitivity analysis dsa method is developed for structural components with hyperelastic incompressible material a mixed variational principle mvp and the total lagrangian formulation are used for nonlinear analysis effects of large displacements large strains and material nonlinearities are included in the analysis model using appropriate kinematics and constitutive relations the material property and shape dsa using both the direct differentiation method ddm and the adjoint variable method avm are discussed for shape dsa the material derivative concept is used to compute effects of the shape variation the boundary displacement and isoparametric mapping methods are employed to compute the design velocity field both hydrostatic pressure and structural stiffness are considered as constraints for design optimization which is carried out by integrating shape design parameterization design velocity computation dsa nonlinear analysis and the optimization method examples such as an engine mount and a bushing demonstrate the feasibility of the proposed optimization method for designing structural components using hyperelastic material

thin films with high surface tension this paper is a review of work on thin fluid films where surface tension is a driving mechanism its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations in general the introduction of surface tension into standard lubrication theory leads to a fourth order nonlinear parabolic equation pad h t pad x left c frac h 3 3 frac partial 3 h partial x 3 f h h x h xx right 0 label abeq1 where h h x t is the fluid film height for steady situations this equation may be integrated once and a third order ordinary differential equation is obtained appropriate forms of this equation have been used to model fluid flows in physical situations such as coating draining of foams and the movement of contact lenses in the introduction a form of the above equation is derived for flow driven by surface tension surface tension gradients gravity and long range molecular forces modifications to the equation due to slip the effect of two free surfaces two phase fluids and higher dimensional forms are also discussed the second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given the third section reviews analytical work on the model equations as well as the generalized lubrication equation pad h t pad x left h n h xxx right 0 in particular the discussion focusses on asymptotic results travelling waves stability and similarity solutions numerical work is also discussed while for analytical results the reader is directed to existing literature

efficient structural optimization for multiple load cases using adjoint sensitivities adjoint sensitivity calculation of stress buckling and displacement constraints may be much less expensive than direct sensitivity calculation when the number of load cases is large in general it is more difficult to implement the adjoint method than the direct method but it is shown that the use of the continuum adjoint method along with homogeneity conditions can alleviate the problem expressions for von mises stress and local buckling sensitivities for isotropic plate elements are derived computational efficiency of the adjoint method is sensitive to the number of constraints and therefore the adjoint method benefits from constraint lumping a continuum version of the kreisselmeier steinhauser functional is chosen to lump constraints the adjoint and direct methods are compared for three examples a truss structure a small high speed civil transport hsct model and a large hsct model these sensitivity derivatives are then used in optimization

uniform asymptotics of the pollaczek polynomials via the riemann hilbert approach the pollaczek weight is an example of the non szeg class in this paper we investigate the asymptotics of the pollaczek polynomials via the riemannhilbert approach in the analysis the original e

energy flow analysis and design sensitivity of structural problems at high frequencies the design sensitivity formulation of an energy finite element method is presented using the direct differentiation and adjoint variable methods the continuum method is used to derive the design sensitivity equation of the energy flow equation whereas the discrete method is used to calculate the variation of the coupling relation for design variables material property panel thickness and structural shape are taken into account in addition to the structural damping factor the design variable s effect on the power transfer coefficient is discussed in detail even if the system matrix equation is not symmetric the adjoint problem is solved using the same factorized matrix from response analysis design sensitivity results calculated from the proposed method are compared to the finite difference sensitivity results with a good agreement

structural reanalysis for general layout modifications a procedure for reanalysis of structures subjected to various layout modifications is presented the procedure is based on exact analysis of the initial design it is suitable for all types of layout changes including the general case in which some members and joints are added other members and joints are deleted and some joint coordinates are modified in the particular and challenging case of addition of joints where the number of degrees of freedom is increased a simple reanalysis of a modified initial design is first carried out further changes are then applied to the latter design it is shown that for changes in a limited number of members of a truss structure an exact solution is readily achieved for changes in numerous members of a general structure an approximate solution can be efficiently obtained high quality approximations are demonstrated for various types of layout modifications

shape design sensitivity analysis and optimization of elasto plasticity with frictional contact a shape design sensitivity analysis and optimization are proposed for the ine nitesimal elasto plasticity with a frictional contact condition rate independent plasticity is considered with a return mapping algorithm and a von mises yield criterion the contact condition is formulated using the penalty method and the modie ed coulomb friction law a continuum based shape design sensitivity formulation is developed for structural and frictional contact variational equations the direct differentiation method is used to compute the displacement sensitivity and the sensitivities of various performance measures are computed from the displacement sensitivity path dependency of the sensitivity equation due to the constitutive relation and friction is discussed it is shown that no iteration is required to solve the sensitivity equation response analysis and the proposed sensitivity formulation are implemented using the mesh free method where the mesh distortion problem can be resolved numerical examples show accurate results of the proposed method compared to the e nite difference method dife culties in the sensitivity formulation for the e nite deformation problem are discussed i introduction b ecauseoftherecentdevelopmentofcomputationalmechanics it is now possible to analyze practical examples of complicated structural problems many design engineers who are not satise ed with response analysis alone have keen interests in the methodology of the design for more than two decades signie cant research efforthasbeen focused on the rate of response with respect to the changes in structural shape under shape design sensitivity analysis dsa analysis of the design sensitivity information is the most important and costly procedure in the automated optimum design process it supplies useful quantitative information to the design engineer about the direction of the desired design change in a classical linear problem dsa research has proved the differentiability of the solution of the response analysis using the linear operator theory and has derived specie c sensitivity expressions for variousproblems 1 aresultworthy of attention in linear dsais that theoriginalresponseandthesensitivityoftheresponsebelongtothe samekinematicallyadmissiblespaceandhavethesameregularities owing to the development of the response analysis capability engineers have directed their interest to the nonlinear problems that are dealt with efe ciently because many design application problems are accompanied by plastic deformation the design sensitivity of nonlinear problems has been actively developed and many research results are reported in the procedure of nonlinear response analysis the projection called a return mapping of the elastic trial stress is carried out to satisfy the variational inequality vi through an iteration in the stress space 2 the dsa on the other hand computes the rate of change of the projected response in the tangential direction of the constraint set without iteration note that the sensitivity analysis is linear and can be computed without iteration even thoughtheresponseanalysisisnonlinear 3 unlikethenonlinearelastic problem the sensitivity equation of the plastic problem requires sensitivity of the stress and internal evolution variables at the previous time the sensitivity equation is solved at each time without iteration and the sensitivity of the stress and evolution variables are

design sensitivity analysis in dynamic thermoviscoelasticity with implicit integration design sensitivity equations for coupled thermoviscoelastic systems are discretized via the finite element method the approach is developed for structural systems by using the total lagrangian approach and the reference domain concept the discretization is based on implicit integration schemes the implementation of finite element and design sensitivity analysis utilizes the christensen naghdi free energy function and the direct differentiation approach which is most suitable for the systems under consideration a relationship between the discretized sensitivity equations and the equations for the original analysis is shown use of the same discretization for analysis and sensitivity analysis is emphasized partial derivatives of the right hand side required for sensitivity calculations are implemented via the central difference method which provides greater flexibility without sacrificing accuracy examples of analysis and design sensitivity analysis for a thermoviscoelastic non linear truss and a plate with a hole are given the calculated sensitivity results are verified by comparison with overall finite difference calculations