design sensitivity analysis of hyperelastic structures using a meshless method a continuum based design sensitivity analysis method for hyperelastic structures is presented analysis is performed using a meshless method called the reproducing kernel particle method yeoh s energy density function is used to describe thehyperelastic structural behavior the meshless method eliminates mesh distortion orentanglement encountered in using e nite element analysis for large deformation structural analysis and structural shape design optimization both the adjoint variable and direct differentiation methods are developed for material and shape design variables an ine nitely long rubber tube a two dimensional rubber band and an engine mount are examples used to demonstrate the feasibility and accuracy of the method

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

topology bounds energy of knots and links in this paper we determine two quantities of geometric and topological character that were left undetermined in two previous results obtained by arnold arnold 1974 in proc summer school in dif

design sensitivity analysis for rate independent elastoplasticity abstract a new incremental direct differentiation method for design sensitivity analysis of structures with rate independent elastoplastic behavior is presented we formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions the sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation combined with the inherent advantages of implicit integration strategies this represents a significant improvement over previous sensitivity formulations for history dependent materials first order sensitivity expressions involving the complete set of design variables including shape design variables are derived for a generic response functional the reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step no iterations are needed in the sensitivity computations a numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem explicit sensitivities from the new method are confirmed by finite difference estimates

design sensitivity analysis of coupled thermoviscoelastic systems abstract both the direct differentiation and adjoint variable methods for design sensitivity analysis of transient dynamic arbitrarily nonlinear thermoviscoelastic coupled systems are presented in this paper the approach is based on the thermodynamic description of simple materials due to coleman large strains as well as arbitrary material nonlinearities are accounted for in the derivations the domain parametrization or reference volume concept is used allowing a uniform treatment of both the shape and sizing design sensitivity analysis analytical examples demonstrating the use of the derived equations are given this approach provides a solid basis for discretization of the developed equations for subsequent use in numerical computations

helicity and the calugareanu invariant the helicity of a localized solenoidal vector field i e the integrated scalar product of the field and its vector potential is known to be a conserved quantity under frozen field distortion of the ambient medium in this paper we present a number of results concerning the helicity of linked and knotted flux tubes particularly as regards the topological interpretation of helicity in terms of the gauss linking number and its limiting form the c lug reanu invariant the helicity of a single knotted flux tube is shown to be intimately related to the c lug reanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given helicity is decomposed into writhe and twist contributions the writhe contribution involving the gauss integral for definition see equation 4 8 which admits interpretation in terms of the sum of signed crossings of the knot averaged over all projections part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as intrinsic twist of the field lines in the flux tube around the knot see equations 5 13 and 5 15 the generic behaviour associated with the deformation of the knot through a configuration with points of inflexion points at which the curvature vanishes is analysed and the role of the twist parameter is discussed the derivation of the c lug reanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems

on the implementation of inelastic constitutive equations with special reference to large deformation problems abstract methods for finite element analysis of inelastic problems are reviewed problems with currently popular methods based on rate principles of virtual work are pointed out a formulation is presented based on the exact incremental virtual work equation the procedure for integration of the constitutive rate equations to obtain incremental constitutive equations is based on the assumption of a constant strain rate or straight strain path during the increment full newton raphson iteration is used to solve the resulting set of nonlinear equations the treatment of the constitutive equations in this newton raphson scheme leads to a linearized form which is substantially different from the classical rate equations the procedure is implemented in a finite element program and its effectiveness is shown in a number of examples

a discussion of stress rates in finite deformation problems abstract it has recently been shown that the finite elastic plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed while the solution for isotropic hardening gives a monotonically increasing stress this paper analyzes this response on the basis of continuum mechanical descriptions of the problem three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress in light of the equations for the evolution of the yield surface the hypoelastic solution to the simple shear problem for each of the three stress rates is presented it is shown that the use of the jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution an alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress

eigenvalue sensitivity analysis of planar frames with variable joint and support locations various sensitivity equations are developed in this study for eigenvalue sensitivity analysis of planar frames with variable joint and support locations the emphasis is placed on the derivation of the sensitivity equations by the continuum approach a variational form of an eigenvalue equation is first derived in which all of the quantities are expressed with respect to the local coordinate system the material derivative of this variational form is then sought to account for the changes in the member s length and orientation resulting from the perturbation of joint and support locations finally eigenvalue sensitivity equations are formulated in either domain quantities by the domain method or boundary quantities by the boundary method examples are presented to validate these derived sensitivity equations with analytical solutions as well as numerical results it is concluded that the sensitivity equation derived by the boundary method is computationally more efficient but less accurate than that of the domain method nevertheless both of them are superior to the conventional discrete analytical method in terms of computational efficiency nomenclature ait ei ij cross sectional area young s modulus and moment of inertia of member respectively b design variable j a line functional j and its total shape derivative k m global stiffness and mass matrices at m elementary stiffness and mass matrices dj pj length orientation angle and mass density of member respectively mit rij cosine and sine of member n total number of members n

adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems an adjoint approach is presented to formulate explicit sensitivities for a general functional associated with a nonlinear dynamic thermoelastic system uncoupled thermoelastic response and small deformation behavior are assumed the formulation uses the lagrange multiplier method to define an adjoint system the convolution operator to incorporate transient effects and domain parameterization to describe shape variations variations of the functional are expressed in explicit form with respect to perturbations of the design fields structural shape material properties applied loads prescribed boundary conditions and initial conditions the functional is defined in terms of the design fields and the implicit response fields displacement temperature strain temperature gradient stress heat flux vector reaction force and reaction surface flux in an example problem the finite element method is used to evaluate the real and adjoint responses and the shape sensitivities for a tank design problem the tank is modeled as a nonlinear quasistatic uncoupled thermoelastic system

shape sensitivity analysis and design optimization of linear thermoelastic solids in this study a numerical scheme is developed for shape sensitivity analysis and design optimization of linear quasistatic thermoelastic solids in this scheme the finite element method is used as the analyzer for analyzing stress temperature shape sensitivity and design velocity field based upon the method of material derivatives both the techniques of the direct differentiation method and the adjoint variable method are applied to derive the shape sensitivity equations the shape optimization formulations discussed here include boundary integrals of displacements and heat fluxes as well as domain integrals of stresses and areas numerical results show that the proposed scheme works well in terms of accuracy

implicit bem formulations for usual and sensitivity problems in elasto plasticity using the consistent tangent operator concept this paper presents boundary element method bem formulations for usual and sensitivity problems in small strain elasto plasticity using the concept of the local consistent tangent operator cto usual problems here refer to analysis of nonlinear problems in structural and solid continua for which simo and taylor first proposed the use of the cto within the context of the finite element method fem a new implicit bem scheme for such problems using the cto is presented first a formulation for sensitivity analysis follows it is shown that the sensitivity of the strain increment associated with an infinitesimal variation of some design parameter solves a linear problem which is governed by the converged value of the same global cto as the one that appears in the usual problem numerical results for both usual and sensitivity problems are shown for a one dimensional example they demonstrate the effectiveness of the present approach in particular accurate sensitivities with respect to material parameters e g exponent of the power type hardening law are obtained even with few integration cells and for large load increments

design sensitivity analysis for nonlinear thermal systems abstract the lagrange multiplier method and convolution theory are used to formulate design sensitivities for nonlinear transient thermal systems variations of a general design functional are expressed in explicit form with respect to variations in structural shape and the conventional design quantities i e material properties applied loads prescribed boundary conditions and initial conditions the design functional depends on these explicit quantities and the implicit repsonse fields temperature temperature gradient heat flux vector and reaction surface flux a finite element implementation of the methodology illustrates the formulation

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

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

a family of summation formulas on the fox wright function abstract by applying the triple form of gould hsu inversions to the pfaff saalschutz theorem a family of summation formulas on the fox wright function which implies several known 3 f 2 4 3 series identities due to chen and chu 2011 20 and numerous new hypergeometric series identities is established

the jones polynomial for fluid knots from helicity in this paper we prove that under ideal conditions the helicity of fluid knots such as vortex filaments or magnetic flux tubes provides a fundamentally new topological means by which we may associate a topological invariant the jones polynomial that is much stronger than prior interpretations in terms of gauss linking numbers our proof is based on an extension of the kauffman bracket polynomial for unoriented knots explicit calculations of the jones polynomial for the left and right handed trefoil knots and for the whitehead link via the figure of eight knot are presented for illustration this novel approach establishes a topological foundation of classical field theory in general and of mathematical fluid dynamics in particular by opening up new directions of work both in theory and applications

design sensitivity analysis and optimization of dynamic response abstract the paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems a key feature of the paper is the development of procedures to handle point wise state variable constraints difficulties with a previous treatment where such constraints were transformed to equivalent integral constraints are noted and explained from theoretical as well as engineering standpoints an alternate treatment of such constraints is proposed developed and evaluated in this treatment each point wise state variable constraint is replaced by several constraints that are imposed at all the local max points for the original constraint function the differential equations of motion are formulated in the first order form so as to handle more general problems the direct differentiation and adjoint variable methods of design sensitivity analysis to deal with the point wise constraints are presented with the adjoint variable methods there are two ways of calculating design sensitivity coefficients the first approach uses an impulse load and the second approach uses a step load for the corresponding adjoint equation since the adjoint variable methods are better for a large class of problems an efficient computational algorithm with these methods is presented in detail optimum results for several problems are obtained and compared with those available in the literature the new formulation works extremely well as precise optimum designs are obtained

structural design sensitivity analysis with general boundary conditions dynamic problem this paper presents and analyses methods for design sensitivity analysis of static response for structural systems when general boundary conditions are used during the analysis phase the design sensitivity analysis methods where the boundary conditions are used to reduce the equilibrium equation have been presented earlier in such procedures the reduced stiffness matrix is non singular this fact is used to show differentiability of the displacement vector when reduction procedures are not used the global structural stiffness matrix may be singular in such a case differentiability of the displacement vector may be questionable however it is shown that the displacement vector is still differentiable several forms of the structural design sensitivity equations are presented although these equations are derived with lagrange multiplier treatment of general boundary conditions it is shown that they are applicable with any treatment of general boundary conditions for example they can also be used with co ordinate reduction schemes therefore design sensitivity analysis can be carried out with any existing finite element analysis computer code moreover the procedures presented can be used to calculate sensitivity of a force of constraint can that is desired the formulations are kept quite general such that design dependent boundary conditions can be routinely treated an example is used to illustrate the procedures

implicit consistent and continuum tangent operators in elastoplastic boundary element formulations this paper presents an assessment and comparison of boundary element method bem formulations for elastoplasticity using both the consistent tangent operator cto and the continuum tangent operator con these operators are integrated into a single computational implementation using linear or quadratic elements for both boundary and domain discretizations this computational setting is also used in the development of a method for calculating the j integral which is an important parameter in nonlinear fracture mechanics various two dimensional examples are given and relevant response parameters such as the residual norm computational processing time and results obtained at various load and iteration steps are provided the examples include fracture problems and j integral evaluation finally conclusions are inferred and extensions of this work are discussed

implicit objective integration for sensitivity analysis in non linear solid mechanics incrementally objective integration schemes are proposed for the accurate and efficient determination of design sensitivity coefficients dscs for solid mechanics problems with both material and geometrical non linearities the derivation of these schemes are based on the direct differentiation of objective schemes that are used in stress analysis for problems of this class two widely used objective stress rates the jaumann rate and the green naghdi rate are considered here within the same setting with only minor changes of the integration scheme numerical results are presented for a simple shear problem with different material constitutive laws including a hypoelastic model and a unified isotropic viscoplastic model for these two objective rates the numerical results are compared with analytical solutions or direct integration solutions the close agreement among these solutions demonstrates the accuracy and efficiency of the proposed scheme

nonlinear structural design sensivitity analysis for path dependent problems part 1 general theory abstract two procedures direct differentiation and adjoint structure for design sensitivity analysis of nonlinear structural systems with path dependent effects are described in this paper continuum formulation is used and both geometric and material nonlinearities are included in the derivations incremental nonlinear analysis based on the total lagrangian concept is adopted the reference volume concept is used to unify the shape and nonshape design problems to show the generality of the theory the derived expressions are reduced to some special cases that have been treated previously in the literature in part 2 of the paper a few analytical example problems will be solved to demonstrate the use of the design sensitivity analysis theory the theory has also beendiscretized elsewhere using finite element approximations for implementation into structural analysis programs these developments offer a firm foundation for design optimization of complex structural systems

contiguous relations for the fox wright function abstract contiguous relations are a powerful tool for dealing with hypergeometric functions in this paper four different types of contiguous relations for the fox wright function are given we demonstrate the usefulness of our formulas by showing how they can be used to recover the famous hagen rothe convolution

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

design sensitivity analysis for parameters affecting geometry elastic viscoplastic material constant and boundary condition by consistent tangent operator based boundary element method abstract this paper presents a design sensitivity analysis method by the consistent tangent operator concept based boundary element implicit algorithm the design variables for sensitivity analysis include geometry parameters elastic viscoplastic material parameters and boundary condition parameters based on small strain theory perzyna s elastic viscoplastic material constitutive relation with a mixed hardening model and two flow functions is considered in the sensitivity analysis the related elastic viscoplastic radial return algorithm and the formula of elastic viscoplastic consistent tangent operator are derived and discussed based on the direct differentiation approach the incremental boundary integral equations and related algorithms for both geometric and elastic viscoplastic sensitivity analysis are developed a 2d boundary element program for geometry sensitivity elastic viscoplastic material constant sensitivity and boundary condition sensitivity has been developed comparison and discussion with the results of this paper analytical solution and finite element code ansys for four plane strain numerical examples are presented finally

momenta of a vortex tangle by structural complexity analysis a geometric method based on information from structural complexity is presented to calculate linear and angular momenta of a tangle of vortex filaments in euler flows for thin filaments under the so called localized induction approximation the components of linear momentum admit interpretation in terms of projected area by computing the signed areas of the projected graph diagrams associated with the vortex tangle we show how to calculate the two momenta of the system by complexity analysis of tangle diagrams this method represents a novel technique to extract dynamical information of complex systems from geometric and topological properties and provides a potentially useful tool to test the accuracy of numerical methods and investigate scale distribution of fluid dynamical properties of vortex flows

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

sizing design sensitivity analysis of non linear structural systems part ii numerical method a numerical method is presented to carry out sizing design sensitivity calculations outside established finite element analysis codes using postprocessing data only geometric as well as material non linearities are treated to demonstrate the accuracy of the proposed method numerical results are presented for structural systems with linear elastic material large displacements large rotations and small strains a distributed parameter approach to structural design sensitivity analysis is used to retain the continuum elasticity formulation throughout the derivation of design sensitivity results using this approach and an adjoint variable method design sensitivity computations are carried out for structural performance functionals stress and displacement are considered it is shown that computations can be performed with the same computational effort as for sizing design sensitivity analysis of linear structural systems accurate design sensitivity results are obtained for both linear and non linear structural systems without the uncertainty of numerical accuracy and high cost associated with the selection of finite difference perturbations also the method does not require differentiation of element stiffness and mass matrices in conventional finite element models

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

multiscale analytical sensitivity analysis for composite materials we describe a methodology aimed at determining the sensitivity of the global structural behaviour such deformation or vibration modes with respect to the local characteristics such as material constants of micro constituents an analytical gradient computation which involves the direct differentiation of the multiple scale strong forms with respect to the design parameters is developed comparison of the multiple scale sensitivity analysis masa to the central finite difference cfd approximation in terms of accuracy and computation efficiency is carried out we demonstrate the robustness of the masa approach compared to the cfd approximation which has been found to be highly sensitive to the choice of the step size whose optimal value is problem dependent copyright 2001 john wiley sons ltd

shape sensitivity analysis for structural problems with non linear material behaviour this paper describes some considerations around the analytical structural shape sensitivity analysis when the structural behaviour is computed using the nite element method with a non linear constitutive material model depending on the type of non linear behaviour two di erent approaches are proposed first a new direct non incremental formulation is proposed for material models characterized by the fact that the stresses at any time t can be expressed in terms of the strains at the same time t and in some cases the strains at a speci c past time t6 t6 t this is the case of elasticity linear as well as non linear perfect plasticity and damage models second a more classical incremental approach is proposed for general plasticity cases a special strategy is also proposed for material models with strain softening the quality and reliability of the proposed approaches are assessed through their application in di erent examples copyright 1999 john wiley sons ltd

variational principle for shape design sensitivity analysis in this paper the adjoint method of design sensitivity analysis is stated as a general variational principle that is applicable to a wide range of problems the principle gives a very simple and straightforward method of obtaining the design sensitivity expression for a functional dependent on the state fields the sensitivity expression involves certain adjoint fields and explicit design variations of the functional and the governing equations for the primary state fields the principle is stated in terms of an augmented functional that is defined by adding to the response functional whose sensitivity is desired the equilibrium equation of the primary problem then explicit design variation of the augmented functional gives the total design variation of the response functional stationarity of the augmented functional with respect to the state fields defines the adjoint problem for the response functional the principle is proved and applied to several classes of linear and nonlinear problems such as field problems structural problems and dynamic response problems it is shown that the design sensitivity expressions derived in the literature using long procedures are obtained quite routinely by use of the principle

first order design sensitivities for transient conduction problems by an adjoint method an adjoint approach is presented for the design sensitivity analysis of transient conduction problems variations of a general design functional are expressed in explicit form with respect to all design quantities i e material properties shape applied thermal loads including convection and initial conditions the methodology incorporates the mutual energy between the real and adjoint thermal problems and a geometric mapping to describe shape variations finite element implementation of the method is discussed and an example is provided some potential difficulties that might be encountered when using the adjoint method with the finite element method are addressed these involve the application of impulse loadings and dirac temperature fields in the adjoint loadings

objectivity of incremental constitutive relations over finite time steps in computational finite deformation analyses abstract computational techniques which preserve the objectivity of incremental constitutive relations for finite elastic or inelastic materials for finite time steps during which the material elements may undergo finite rotations are presented these techniques are useful in finite deformation finite rotations and finite strain computational analysis of two as well as three dimensional solids and structures

a sensitivity interpretation of adjoint variables in optimal design abstract the adjoint method of computing derivatives of cost and constraint functions with respect to design variables requires the calculation of certain adjoint variables until now the adjoint variables have been looked upon only as some intermediate vectors needed to calculate design derivatives in this paper they are shown to have an important significance they represent the sensitivity of the cost and constraint functions with respect to the loading or forcing function in the design problem a sensitivity theorem for the adjoint variables is presented for structural mechanical dynamic and distributed parameter systems these results offer some immediate practical advantages such as a method for computing influence coefficients for structural systems and a method for verifying debugging the analytical calculation of adjoint variables in development of a computer code

sensitivity analysis of linear elastic systems using domain parameterization and a mixed mutual energy principle abstract a unified approach for explicit structural design sensitivity analysis of linear elastic systems is presented first order sensitivity expression involving the complete set of design variable including shape design parameters are derived for general response functional both direct differentiation and adjoint variable methods are developed in a general variational context a domain parameterization method related to the mixed eulerian lagrangian kinematic description is introduced for the treatment of shape variation problems this approach is an alternative to the material derivative methods presented elsewhere in the literature the adjoint variable derivation is based on a mixed energy principle the mutual hu washizu energy principle which is formally stated and proved the design sensitivity derivation based on this principle unifies and extends several of the adjoint variable methods found in the literature the derivations presented here are extended to nonlinear systems in a subsequent paper the sensitivity expressions are valid for use with force displacement or mixed approximate solutions of the elasticity equations other formulations are generally only consistent with displacement solution methods numerical implementation of the design sensitivity methods is discussed and illustrative examples are included

computational method for optimization of structural shapes in the direct method of shape optimum design mesh regridding is a necessary step in avoiding the distortion of finite element grids the effect of internal nodal movements on the shape design sensitivity associated with the regridding process has been carefully examined in this paper it shows that to maintain computational accuracy the effects of internal nodal movement should be considered in shape design sensitivity analysis to take the internal nodal movement into account the numerical accuracy of the speed method is investigated a new approach which constructs the velocity field over the element domain is presented although it takes more computational time the new approach does improve the accuracy of shape design sensitivity

on the mellin transform of a product of two fox wright psi functions closed form representations in terms of well known special functions are deduced for the mellin transform of a product of two fox wright psi functions 1 1 an application concerned with the recent analysis of the critical behavior of a variety of distinct physical systems is provided

how tangled is a tangle abstract new measures of algebraic geometric and topological complexity are introduced and tested to quantify morphological aspects of a generic tangle of filaments the tangle is produced by standard numerical simulation of superfluid helium turbulence which we use as a benchmark for numerical investigation of complex systems we find that the measures used based on crossing number information are good indicators of generic behaviour and detect accurately a tangle s complexity direct measurements of kinetic helicity are found to be in agreement with the other complexity based measures proving that helicity is also a good indicator of structural complexity we find that complexity based measure growth rates are consistently similar to one another the growth rate of kinetic helicity is found to be twice that of energy

shape optimization in elasticity and elasto viscoplasticity by the boundary element method abstract this paper is concerned with the optimal shape design of solids undergoing small strain small rotation elasto viscoplastic deformation shape sensitivities for this class of problems are determined by using a direct differentiation approach dda to the governing boundary element method bem equations of the problem the standard bem and the sensitivity equations are discretized and solved numerically shape optimization is carried out by coupling the standard and sensitivity analyses with an optimizer the optimization algorithm chosen here uses sequential quadratic programming to obtain the desired optimal shape of a body in an iterative manner numerical solutions for optimal shapes of cutouts in plates two dimensional plane stress are presented the difference between optimal shapes of solids undergoing purely elastic and elasto viscoplastic deformation is shown clearly in these examples

nonlinear structural design sensitivity analysis for path dependent problems part 2 analytical examples abstract in part 1 of this paper a general theory of design sensitivity analysis for nonlinear structures with history dependent effects is given the theory is also reduced to several special cases to show its generality in the present paper three examples are used to analytically verify the theory both the adjoint structure method asm and the direct differentiation method ddm are demonstrated the examples are 1 reaction force sensitivity for an indeterminate beam 2 displacement sensitivity for several cases of an axial force element and 3 stress and displacement sensitivities for a visco elastic problem these examples provide insights for numerical implementation of the theory for more complex design problems

an exposition of the material derivative approach for structural shape sensitivity analysis abstract the material derivative approach for structural shape design sensitivity analysis is described in a unified manner using the continuum formulation both the direct differentiation and adjoint methods are derived and computer implementation aspects are discussed the basic purpose of the paper is to present the shape design sensitivity analysis as an application of the material derivative concept of continuum mechanics and the calculus of variation procedures for taking variations of integrals over the variable domains it is also shown that the use of a recently developed variational principle for design sensitivity analysis greatly simplifies derivation of the adjoint method the indicial notation is used and expressions derived by different authors are related to each other a basic understanding of the material derivative approach of shape sensitivity analysis has been gained which can lead to insights for computer implementations of the method

reduction of a class of fox wright psi functions for certain rational parameters abstract the fox wright psi function is a special case of fox s h function and a generalization of the generalized hypergeometric function in the present paper we show that the psi function reduces to a single generalized hypergeometric function when certain of its parameters are integers and to a finite sum of generalized hypergeometric functions when these parameters are rational numbers applications to the solution of algebraic trinomial equations and to a problem in information theory are provided a connection with meijer s g function is also discussed

on structural shape optimization using an embedding domain discretization technique summary this contribution presents a novel approach to structural shape optimization that relies on an embedding domain discretization technique the evolving shape design is embedded within a uniform finite element background mesh which is then used for the solution of the physical state problem throughout the course of the optimization we consider a boundary tracking procedure based on adaptive mesh refinement to separate between interior elements exterior elements and elements intersected by the physical domain boundary a selective domain integration procedure is employed to account for the geometric mismatch between the uniform embedding domain discretization and the evolving structural component thereby we avoid the need to provide a finite element mesh that conforms to the structural component for every design iteration as it is the case for a standard lagrangian approach to structural shape optimization still we adopt an explicit shape parametrization that allows for a direct manipulation of boundary vertices for the design evolution process in order to avoid irregular and impracticable design updates we consider a geometric regularization technique to render feasible descent directions for the course of the optimization this article is protected by copyright all rights reserved

tangent operators and design sensitivity formulations for transient non linear coupled problems with applications to elastoplasticity tangent operators and design sensitivities are derived for transient non linear coupled problems the solution process and the formation of tangent operators are presented in a systematic manner and sensitivities for a generalized response functional are formulated via both the direct differentiation and adjoint methods the derived formulations are suitable for finite element implementations analyses of systems with materials that exhibit history dependent response may be obtained directly by applying the analyses of transient non linear coupled systems rate independent elastoplasticity is investigated as a case study and a problem with an analytical solution is analysed for demonstration purposes

a family of integration algorithms for constitutive equations in finite deformation elasto viscoplasticity a two parameter family of incrementally objective integration schemes is proposed for the analysis of a broad range of unified rate dependent viscoplastic constitutive models in large deformation problems a similar scheme can be applied to rate independent solids as well these algorithms are a generalization of the mid point integration rule full linearization of the principle of virtual work is performed in an updated lagrangian framework together with a calculation of the consistent linearized moduli some details of the finite element implementation are given for plane strain and axisymmetric problems the method is compared with other objective integration schemes and is tested with several examples where large strains and rotations occur

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

variational method for design sensitivity analysis in nonlinear structural mechanics this paper describes a unified variational theory of design sensitivity analysis of linear and nonlinear structures for shape nonshape and material selection problems the concepts of reference volume and adjoint structure are used to develop the unified viewpoint a general formula for design sensitivity analysis is derived several analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use such analytical examples give insight into the application of the general formula to more complex problems that must be treated numerically adjoint state fields are also interpreted as sensitivities for the functional related to the primary structure

incremental finite element sensitivity analysis for non linear mechanics applications finite element formulations for structural sensitivity analysis of non linear systems with fixed overall shape are discussed both the direct differentiation and adjoint variable methods are employed the resulting sensitivity algorithms are consistent with time integration scheme adopted for solving equilibrium problem effectiveness and computational aspects of the procedures are discussed and compared numerical algorithms are shown to be readily implemented in existing finite element codes large scale examples illustrate the paper

a framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition part ii computational aspects computational aspects of the formulation of elastoplasticity proposed in part i of this work employing a multiplicative decomposition and based on the principle of maximum plastic dissipation are examined in detail in particular the implementation of the proposed hyperelastic j2 flow theory is shown to reduce to a trivial modification of the classical radial return algorithm which is amenable to exact linearization this results in a closed form expression for the consistent elastoplastic tangent moduli the algorithmic treatment of the general case is considered subsequently it is emphasized that objective rates and incrementally objective algorithms play no role in the present approach a number of numerical experiments are presented that illustrate the excellent performance of the proposed formulation

shape design sensitivity analysis of viscoplastic structures abstract design sensitivity analysis of nonlinear viscoplastic structures is developed in the continuum form starting from hamilton s principle constitutive models based on internal variable theory are incorporated in response analysis such models are described by a set of first order nonlinear differential equations with respect to time and some of them allow development of the yield surface the direct variation method is employed to obtain the sensitivity of the response discretization of the analysis and sensitivity equations and computer implementation aspects are discussed the discretized equations are implemented into a computer program and two numerical examples are solved to demonstrate the theory in the numerical examples only material nonlinearity under quasi static loading history is considered and the explicit integration method is employed to integrate the constitutive equations and their design variations the accuracy and efficiency of the developed sensitivity analysis are compared with the central finite difference method cyclic loading and loading rate effects are studied and discussed to show the general applicability of the developed sensitivity formulation