elastoplastic truss design using a displacement based optimization abstract a displacement based optimization dbo method is applied to truss design problems with material nonlinearities to explore feasibility and verify efficiency of the method as compared to traditional structural optimization minimum weight truss sizing problems with various path independent elastoplastic laws including elastic perfectly plastic linear strain hardening and ramberg osgood models are investigated this type of material nonlinearity allows us to naturally extend the linear elastic truss sizing in the dbo setting to nonlinear problems to implement the methodology a computer program that uses the commercially available optimizer dot by vr d and imsl linear programming solver by visual numerics is developed several test problems are successfully solved using the dbo approach and solutions are compared to those available in the literature demonstrating significant reduction of computational time in comparison to the traditional structural optimization method in particular the dbo approach is found to be suitable for truss topology design since the method allows member areas to have cross sectional areas equal to zero exactly

structural optimization a new dual method using mixed variables a new and powerful mathematical programming method is described which is capable of solving a broad class of structural optimization problems the method employs mixed direct reciprocal design variables in order to get conservative first order approximations to the objective function and to the constraints by this approach the primary optimization problem is replaced with a sequence of explicit subproblems each subproblem being convex and separable it can be efficiently solved by using a dual formulation an attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs examples of application to real life aerospace structures are offered to demonstrate the power and generality of the approach presented

analysis of a unilateral contact problem taking into account adhesion and friction abstract in this paper we investigate a contact problem between a viscoelastic body and a rigid foundation when both the effects of the irreversible adhesion and of the friction are taken into account we describe the adhesion phenomenon in terms of a damage surface parameter according to fremond s theory and we model unilateral contact by signorini conditions and friction by a nonlocal coulomb law all the constraints on the internal variables as well as the contact and the friction conditions are rendered by means of subdifferential operators whence the highly nonlinear character of the resulting pde system our main result states the existence of a global in time solution to a suitable variational formulation of the related cauchy problem it is proved by an approximation procedure combined with time discretization

coalgebras in functional programming and type theory this is a survey article on the use of coalgebras in functional programming and type theory it presents the basic theory underlying the implementation of coinductive types families and predicates it gives an overview of the application of corecursive methods to the study of general recursion formal power series tabulations of functions on inductive data it also sketches some advanced topics in the study of the solutions to non guarded corecursive equations and the design of non standard type theory

optimal material selection using branch and bound techniques a methodology for structural synthesis is presented in which the optimal material selection for the truss structures is treated in terms of 0 1 variables structural member sizes and material selection variables are treated simultaneously as design variables optimization is carried out by generating and solving a sequence of explicit approximate problems using a branch and bound strategy intermediate design variables and intermediate response quantities are used to enhance the quality of the approximate design problems numerical results for example problems are presented to illustrate the efficiency of the design procedure set forth

a quasistatic viscoelastic frictional contact problem with multivalued normal compliance unilateral constraint and material damage abstract the goal of this paper is to deal with a mathematical model which describes the quasistatic frictional contact between a viscoelastic body and a foundation we are interested in the foundation which is made of a hard material covered by a thin layer composed of a rigid crust and a viscoplastic material with finite thickness the mechanical damage caused by excessive stress or strain is emerged in the viscoelastic body during the contact process we provide the variational formulation of the model which is a system consisting of a history dependent variational hemivariational inequality a fully nonlinear integral equation and a parabolic variational inequality then in order to solve this complex system we deliver abstract results on a class of history dependent variational hemivariational inequalities finally our abstract results are applied to derive the unique weak solvability of the frictional contact problem under consideration

structural optimization using beam elements techniques for handling beam elements in structural optimization are investigated the complexities of evaluating the geometrical properties of a section stress recovery and design variable selection are successfully handled by the concept of an element library use of equivalent stress and separation of sensitivity variables and design variables program implementation is based on the use of an existing finite element analysis package and a numerical optimizer the quality of linearization of the behavior constraint functions is also investigated

a consistent model coupling adhesion friction and unilateral contact a model considering both unilateral contact coulomb friction and adhesion is presented in the framework of continuum thermodynamics the contact zone is considered as a material boundary and the local constitutive laws are derived by choosing two specific surface potentials the free energy and the dissipation potential because of the non regular properties of these potentials convex analysis is used to derive the local behavior laws from the state and the complementary laws the adhesion is characterized by an internal variable introduced by fremond which represents the intensity of adhesion the continuous transition from a total adhesive condition to a possible pure frictional one is enforced by using elasticity coupled with damage for the interface non penetration conditions and coulomb law are strictly imposed without using any penalty the variational formulation for quasistatic problems is written as the coupling between an implicit variational inequality a variational inequality and a differential equation an increment formulation is proposed an existence result under a condition on the friction coefficient is given a numerical method is derived from the incremental formulation and various algorithms are implemented they solve a sequence of minimization problems under constraints the model is used to simulate a micro indentation experiment conducted to characterize the behavior of fiber matrix interface in a ceramic composite identification of the constitutive parameters is discussed

a quasistatic frictional problem with normal compliance a friction and contact problem will be considered for an elastic solid body which under the influence of volume and surface forces is in contact with a rigid foundation

efficient accurate reanalysis for structural optimization the combined approximations method developed recently is an efficient reanalysis method providing high quality results through the use of this approach the computed terms of a series expansion are used as basis vectors in a reduced basis expression by solving a reduced system of equations first and second order approximations were demonstrated in previous studies for small structures the efficiency and the accuracy of the method are improved and results are illustrated for larger structures by the utilization of a gram schmidt orthogonalization procedure new set of basis vectors is generated and normalized such that the reduced system of equations becomes uncoupled the advantage in using the latter vectors is that all expressions for evaluating the displacements are explicit functions of the design variables consequently additional vectors can be considered without modifying the calculations that have already been carried out in addition the uncoupled system is more well conditioned some considerations related to the efficiency of the solution process and the accuracy of the results are discussed and the effect of various parameters on the accuracy is studied numerical results are demonstrated for several medium and large scale structures it is shown that accurate and efficient approximations are achieved for very large changes in the design

high quality approximation of eigenvalues in structural optimization a new function for approximating natural frequency constraints during structural optimization is presented the rayleigh quotient approximation rqa presented here icreases the quality of the approximate frequency constraint by approximating the modal strain and kinetic energies instead of the frequency eigenvalue itself

design sensitivity metric for structural dynamic response an approach to study the sensitivity of dynamic responses for structural systems with respect to a set of design parameters is presented the sensitivity is evaluated by considering the behavior of the system response when the design parameters vary within a given region of the design space the sensitivity is computed by means of approximate responses which are evaluated using approximation concepts the approximation is based on modal analysis and it is valid for general underdamped linear systems intermediate design variable and intermediate response quantity concepts are used to enhance the accuracy of the approximation in this approach modal energies and mode shapes are chosen as intermediate response quantities and they are approximated in terms of selected intermediate system parameters the approximation of these quantities requires a standard eigenvalue and eigenvector sensitivity analysis numerical results that illustrate the usefulness and effectiveness of the method are presented great insight into the behavior of the system can be gained using this methodology

three dimensional shape optimization using the boundary element method a practical design sensitivity calculation technique of displacements and stresses for three dimensional bodies based on the direct differentiation method of discrete boundary integral equations is formulated in detail then the sensitivity calculation technique is applied to determine optimum shapes of minimum weight subjected to stress constraints where an approximated subproblem is constructed repeatedly and solved sequentially by the mathematical programming method the shape optimization technique suggested here is applied to determine optimum shapes of a cavity in a cube and a connecting rod

combining global and local approximations a method based on a linear approximation to a scaling factor designated the global local approximation gla method is presented and shown capable of extending the range of usefulness of derivative based approximations to a more refined model the gla approach refines the conventional scaling factor by means of a linearly varying rather than constant scaling factor the capabilities of the method are demonstrated for a simple beam example with a crude and more refined fem model 6 refs

structural optimization by multilevel decomposition a method for decomposing an optimization problem into a set of subproblems and a coordination problem that preserves coupling between the subproblems is described the decomposition is achieved by separating the structural element optimization subproblems from the assembled structural optimization problem each element optimization and optimum sensitivity analysis yields the cross sectional dimensions that minimize a cumulative measure of the element constraint violation as a function of the elemental forces and stiffness the assembled structural optimization produces the overall mass and stiffness distributions optimized for minimum total mass subject to constraints that include the cumulative measures of the element constraint violations extrapolated linearly with respect to the element forces and stiffnesses the method is introduced as a special case of a multilevel multidisciplinary system optimization and its algorithm is fully described for two level optimization for structures assembled of finite elements of arbitrary type numerical results are given as an example of a framework to show that the decomposition method converges and yields results comparable to those obtained without decomposition it is pointed out that optimization by decomposition should reduce the design time by allowing groups of engineers using different computers to work concurrently on the same large problem

kripke semantics for higher order type theory applied to constraint logic programming languages abstract we define a kripke semantics for intuitionistic higher order logic with constraints formulated within church s theory of types via the addition of a new constraint base type we then define an executable fragment hohh c of the logic a higher order logic programming language with typed abstraction implication and universal quantification in goals and constraints and give a modified model theory for this fragment both formal systems are shown sound and complete for their respective semantics we also solve the impredicativity problem in prolog semantics namely how to give a definition of truth without appealing to induction on subformula structure in the last section we give a simple semantics based conservative extension proof that the language hohh c satisfies a uniformity property along the lines of 39

new concepts of transferred and potential transferred forces in structures abstract new concepts of transferred force and potential transferred force from the loaded surface to the support surface for a structure are introduced the basic ideas of these new types of forces are explained using a general continuum model for the structure numerical procedures to calculate these forces are presented several example problems are solved and a procedure to depict transferred and potential transferred forces as force transfer diagrams is developed these forces and transfer diagrams show relative importance of various parts of the structure that can be useful in redesign or optimum design of the structure

comparison of local and global approximations for reliability estimation an approach based on approximation concepts to estimate system reliability is presented limit state functions are approximated explicitly in terms of the random variables of the system these approximations are then used in conjunction with monte carlo simulation to estimate system reliability local and global approximations are studied and compared in terms of efficiency and accuracy for their potential use in reliability based optimization example problems are presented to illustrate the ideas set forth

signorini type variational inequality with state dependent discontinuous multi valued boundary operators abstract we consider an elliptic variational inequality of signorini type in a bounded domain r n of the form find u k and 2 u u such that a u v u s v u d 0 v k where a is a quasilinear elliptic operator in divergence form k is the nonempty closed convex set representing the thin obstacle signorini problem on s and w 1 p l p denotes the trace operator the multi valued boundary operator is generated by the multifunction s 2 s s where r r r is a function such that s r s is supposed to be locally lipschitz for all r r while r r s is allowed to be discontinuous and 2 r s stands for clarke s generalized gradient of with respect to its second argument the novelty of this paper is that the multifunction r 2 r s may discontinuously depend on r in a certain specified way which gives rise to a new class of discontinuous nonmonotone multi valued boundary operators that is rich in structure to cover a wide range of multi valued constitutive laws on the contact surface s such as e g multi valued adhesion and friction laws that are not necessarily of subdifferential type our main goal is to provide an analytical framework to prove existence enclosure and comparison results for this new class of multi valued boundary variational inequalities that includes the theory of variational hemivariational inequalities as special case

alternative approximation for stresses in plate structures an accurate nonlinear approximation for stresses caused by membrane and bending loads in plate structures is presented although the approximation requires only a first order sensitivity analysis it is more accurate than a simple linear taylor series expansion of the stresses with respect to the design variables or their reciprocals and is exact for statically determinate structures it requires less computation both in approximate analysis and the corresponding sensitivity analysis than other approximations with the same order of accuracy the use of this alternative approximation for plate stresses in the approximation concepts approach to structural optimization results in very rapid design convergence examples of the structural optimization of plate structures subject to stress constraints are presented to show the accuracy of this approximation

elasto plastic contact problems with heat exchange and fatigue abstract we deal with a one dimensional temperature dependent model for fatigue accumulation in a moving visco elasto plastic material in contact with an elasto plastic obstacle the problem for the unknown displacement and temperature is formulated using hysteresis operators as solution operators of the underlying variational inequalities the existence result for this problem consisting of the momentum and energy balance equations and an evolution equation for the fatigue is obtained using a priori estimates established for the space discretized problem the uniqueness result follows from the lipschitz continuity of the nonlinearities

alternative approximation concepts for space frame synthesis a structural synthesis methodology for the minimum mass design of 3 dimensionall frame truss structures under multiple static loading conditions and subject to limits on displacements rotations stresses local buckling and element cross sectional dimensions is presented a variety of approximation concept options are employed to yield near optimum designs after no more than 10 structural analyses available options include a formulation of the nonlinear mathematcal programming problem in either reciprocal section property rsp or cross sectional dimension csd space b two alternative approximate problem structures in each design space and c three distinct assumptions about element end force variations fixed element design element linking and temporary constraint deletion features are also included the solution of each approximate problem in either its primal or dual form is obtained using conmin a feasible directions program the frame truss synthesis methodology is implemented in the compass computer program and is used to solve a variety of problems these problems were chosen so that in addition to exercising the various approximation concepts options the results could be compared with previously published work

improved transient response approximation for general damped systems a new approximation for peak transient dynamic displacements is presented this approximation is based on modal analysis and is valid for general damped structures which includes the case of actively controlled structures intermediate design variable and intermediate response quantity concepts are used to enhance the accuracy of the approximation when this high quality approximation is used within the context of the approximation concepts approach to design synthesis the overall efficiency of the process expected to be improved by reducing the number of complete analyses required for convergence numerical results that illustrate the effectiveness of the method are presented

a homogenization method for shape and topology optimization abstract shape and topology optimization of a linearly elastic structure is discussed using a modification of the homogenization method introduced by bendsoe and kikuchi together with various examples which may justify validity and strength of the present approach for plane structures

optimality conditions of the hybrid cellular automata for structural optimization the hybrid cellular automaton method has been successfully applied to topology optimization using a uniform strain energy density distribution approach in this work a new set of design rules is derived from the first order optimality conditions of a multi objective problem in this new formulation the final topology is derived to minimize both mass and strain energy in the hybrid cellular automaton algorithm local design rules based on the cellular automaton paradigm are used to efficiently drive the design to optimality in addition to the control based techniques previously introduced a new ratio technique is derived in this investigation this work also compares the performance of the control strategies and the ratio technique

load paths visualization in plane elasticity using load function method abstract the paper proposes a load function method for load path visualization and load flow calculation in plane elasticity problems the load functions are directly derived from 2d equilibrium equations the stresses are written in terms of the load functions and the load flow is calculated using the load function gradient the load functions are defined using generalized beltrami representation a constructive proof is given for the existence of load functions a set of poisson s equations in terms of load functions and stress components are developed and an efficient numerical procedure to solve them is discussed numerical results show the proposed load function method is able to define the load paths and obtain the load flow using an airy stress function for problems with available closed form solution and also can be easily integrated into the numerical approaches to calculate load paths for complex problems

reduced basis approximations of structural displacements for optimaldesign a new solution procedure for improved approximate reanalyses of structures using results of a single precise analysis is presented the proposed procedure is based on combining the computed terms of a series expansion used as high quality basis vectors with coefficients of a reduced basis expression the latter coefficients can readily be determined for each trial design by various criteria the results are measured by the errors in satisfying the analysis equations and are compared in terms of the quality of the approximations the proposed approach is suitable for various types of design variables and can be used with a general finite element model a general reanalysis procedure is introduced and the physical significance of first order approximations is demonstrated several numerical examples illustrate the effectiveness of the solution process it is shown that high quality approximations of displacements can be obtained with a small computational effort for very large changes in the design variables

two level approximation concept in structural synthesis a new concept for complex structural optimization problems is presented based on investigation of the mechanical nature of the studied problem a set of basic mechanical characteristics can be introduced as the generalized variables a high quality explicit first level approximate problem fa is formed by first order taylor series expansion of the behaviour constraints in terms of the generalized variables the fa is explicit but highly non linear with respect to design variables therefore a second level approximation is introduced to solve the fa by considering a sequence of second level approximate problems sa in the design variable space this approach is a two level approximation concept its application to space frame synthesis shows its good grasp of the mechanical nature of the problem the computational results for several examples show that the method presented in this paper is very efficient

approximate reanalysis based on the exact analytic expressions the first author has recently given the exact analytic expressions of the inverse of the stiffness matrix the nodal displacements and the stress resultants in linear elastic structures composed of prismatic elements for structures of constant geometry the expressions are explicit in terms of the unimodal stiffnesses of the components of the structures however the expressions are intractable in their exact form due to their inordinate length it all has to do with the number of statically determinate substructures embedded in common engineering structures this paper describes some preliminary results obtained from approximate analysis models for the internal forces using truncated expressions that are similar in form to the exact analytic ones the approach is illustrated with numerical examples he classical methods of structural analysis methods are numerical techniques that compute the structural response to external loads for a given configuration of the structure and for given stiffnesses of its elements very often engineers are faced with the need to reanalyze structures for the same geo metrical configuration and for the same applied loads but with modified stiffnesses such instances occur in structural design and in particular in optimal structural design where one usually tests many candidate stiffness distributions before an acceptable structure is found 1 structural reanalysis is also present in the analysis of framed structures when the p d effect is accounted for 2 since the axial load in a member modifies its flexural rigidity such structures are solved itera tively at a given iteration the flexural stiffnesses of the members are computed based on the axial loads of the previ ous iteration the procedure is repeated until convergence when the stiffnesses of the individual members change the analysis procedure must in principle be repeated with the modified parameters there is no formula for the structural response in terms of the rigidities of its members to alleviate the computational cost of multiple reanalysis engineers have over the years developed reanalysis techniques that use results of previous analysis to predict the structural response when some or all of the rigidities are modified 3 4 noteworthy is the reciprocal approximation 1 which expresses the nodal dis placements as a linear taylor polynomial in terms of the reciprocals of the member stiffnesses recently the first author has written a series of papers 5 7 describing the analytic solution of the structural analysis prob lem for structures composed of uniform prismatic members of given geometry and assuming linear elastic behavior the analytic expressions are exact equations for the inverse of the stiffness matrix for the nodal displacements and for the internal forces in the members explicit in terms of the cross sectional structural characteristics of the components of the structure however what seems to be eldorado for the struc tural designer an exact formula for the structural response

on the algebraic structure of declarative programming languages we develop an algebraic framework logic programming doctrines for the syntax proof theory operational semantics and model theory of horn clause logic programming based on indexed premonoidal categories our aim is to provide a uniform framework for logic programming and its extensions capable of incorporating constraints abstract data types features imported from other programming language paradigms and a mathematical description of the state space in a declarative manner we define a new way to embed information about data into logic programming derivations by building a sketch like description of data structures directly into an indexed category of proofs we give an algebraic axiomatization of bottom up semantics in this general setting describing categorical models as fixed points of a continuous operator

algorithm for modification of parameters in vibrating systems concluding remarks a global local approximation based on a linear approximation to a scaling factor has been presented the approximation permits us to use a global approximation based on a simple model of our problem to extend the range of usefulness of derivative based approximations to a more refined model the method was demonstrated for a simple beam example with a crude and more refined finite element model

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

on some existence and uniqueness results in contact problems with nonlocal friction the workwe present is a mathematical study of signorini s problem with friction i e the problem of equilibrium of a linearly elastic body in contact with a rigid foundation until recently the questions of existence and uniqueness of solutions to such problems were open recently two ideas have appeared which have led to some positive results the first presented by netas jarusek haslinger 11 involves a concept of small friction and provides for the development of an existence result for a very special case of a uniformly loaded infinite strip the second established by duvaut 5 is based on the idea of a nonlocal friction law which enables one to establish an existence result for any friction and also to establish uniqueness for small friction in this paper we present some new results in this direction and a new proof of a non local existence theory our approach differs considerably from the proof outlined in 5 instead of using fixed point arguments we develop a theory of perturbed variational inequalities the origin of which can be found in 9 and establish a generalization for the class of nonlinear variational inequalities characterizing the non local signorini problem this approach should prove to be very useful in studies of approximations of friction prohlems

a theory of observables for logic programs we define a semantic framework to reason about properties of abstractions of sld derivations the framework allows us to address problems such as the relation between the top down operational semantics and the bottom up denotational semantics the existence of a denotation for a set of definite clauses and their properties compositionality w r t various syntactic operators correctness minimality and precision using abstract interpretation techniques to model abstraction allows us to state very simple conditions on the observables which guarantee the validity of several general theorems

new approximation method for stress constraints in structural synthesis a new approximation method for dealing with stress constraints in structural synthesis is presented in previous methods it has been common to create an approximation of the actual constraints with respect to the design variables and then use this with optimization to produce a new candidate design in the present method the finite element nodal forces are instead approximated and these are used to create an explicit but often nonlinear approximation to the original problem the principal motivation is to create the best approximation possible in order to reduce the number of detailed finite element analyses needed to reach the optimum the method is shown to be quite simple while providing significant efficiency gains in the overall optimization task also the method produces such high quality approximations to the original problem that the need for move limits during the approximate optimization stage is greatly reduced examples are offered and compared with published results to demonstrate the efficiency and reliability of the proposed method it is concluded that there are considerable gains yet to be achieved through continued careful investigation of approximation techniques

focusing and polarization in linear intuitionistic and classical logics a focused proof system provides a normal form to cut free proofs in which the application of invertible and non invertible inference rules is structured within linear logic the focused proof system of andreoli provides an elegant and comprehensive normal form for cut free proofs within intuitionistic and classical logics there are various different proof systems in the literature that exhibit focusing behavior these focused proof systems have been applied to both the proof search and the proof normalization approaches to computation we present a new focused proof system for intuitionistic logic called ljf and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing we also use ljf to design a focused proof system lkf for classical logic our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in girard s lc and lu proof systems

new approximation for steady state response of general damped systems a new approximation for steady state dynamic displacements which captures the nonlinearities associated with resonance is presented this approximation is based on modal analysis and is valid for general damped structures which include the case of actively controlled structures intermediate design variable and intermediate response quantity concepts are used to enhance the accuracy of the approximations when these high quality approximations are used within the context of the approximation concepts approach the overall efficiency of the design synthesis process is improved by reducing the number of complete analyses required for convergence numerical results that illustrate the effectiveness of the method are presented

shape optimization with multiple loading conditions and mesh refinement the capability to handle multiple static loading conditions has been added to two dimensional shape optimization with adaptive mesh refinement the strain energy density variations for each mesh were normalized and then maximum values from all loading conditions were used for refinement several sample test cases were considered a two dimensional bracket with two in plane loading cases and a folded stamped sheet metal part representing an upper suspension control arm and a folded stamped sheet metal part representing a brake pedal support bracket

approximation method for configuration optimization of trusses an efficient method for truss configuration optimization is presented elastic trusses are designed for minimum weight by varying the areas of the members and the location of the joints constraints on member stresses and euler buckling are imposed and multiple loading conditions are considered the method presented here utilizes an approximate structural analysis based on first order taylor series expansions of the member forces the force approximation succeeds in reducing the degree of coupling between the sizing and geometry design variables and thereby increases the accuracy of the approximation a numerical optimizer uses information from the approximate structural analysis as it minimizes the weight of the truss

topological optimization of compliant adaptive wing structure load path based topology optimization is used to synthesize a compliant adaptive aircraft wing leading edge which deforms in a prescribed way when subject to a single point internal actuation the load path based optimization method requires the specification of a parent lattice increasing the complexity of this lattice means the number of parameters required for a complete representation of the structure in the topology optimization becomes prohibitive although it is desirable to enable a full exploration of the design space a new method based on graph theory and network analysis is proposed which enables a substantial reduction in the required number of parameters to represent the parent lattice the results from this load path based approach are compared with those obtained from the better known density based topology optimization method

an optimization design method for large scale structures in this paper based on the mechanical property of structures a unified approximate formula with high precision suitable to several structural response functions and various types of finite elements is introduced first and then an algorithm for solving the nonlinear program is presented this algorithm is not only effective for solving the program constructed by this unified approximate formula but also easy to change the objective function and to solve the structural optimization problem with dual or multiple objective functions