槽形托輥帶式輸送機設(shè)計（含CAD圖紙源文件）

槽形托輥帶式輸送機設(shè)計（含CAD圖紙源文件）,槽形托輥帶式,輸送,設(shè)計,CAD,圖紙,源文件 ORIGINAL ARTICLEPseudo-constructal theory for shape optimizationof mechanical structuresJean Luc MarcelinReceived: 10 January 2007 /Accepted: 1 May 2007 /Published online: 25 May 2007#Springer-Verlag London Limited 2007Abstract This work gives some applications of a pseudo-constructal technique for shape optimization of mechanicalstructures. In the pseudo-constructal theory developed inthis paper, the main objective of optimization is only theminimization of total potential energy. The other objectivesusually used in mechanical structures optimization aretreated like limitations or optimization constraints. Twoapplications are presented; the first one deals with theoptimization of the shape of a drop of water by using agenetic algorithm with the pseudo-constructal technique,and the second one deals with the optimization of the shapeof a hydraulic hammers rear bearing.Keywords Shapeoptimization.Constructal.Geneticalgorithms1 IntroductionThis paper introduces a pseudo-constructal approach toshape optimization based on the minimization of the totalpotential energy. We are going to show that minimizing thetotal potential energy of a structure to find the optimalshape might be a good idea in some cases. The reference tothe constructal theory can be justified in some way for thefollowing reasons.According to Bejan 1, shape and structure spring fromthe struggle for better performance in both engineering andnature; the objective and constraints principle used inengineering is the same mechanism from which thegeometry in natural flow systems emerges. Bejan 1 startswith the design and optimization of engineering systemsand discovers a deterministic principle for the generation ofgeometric form in natural systems. This observation is thebasis of the new constructal theory. Optimal distribution ofimperfection is destined to remain imperfect. The systemworks best when its imperfections are spread around so thatmore and more internal points are stressed as much as thehardest working parts. Seemingly universal geometricforms unite the flow systems of engineering and nature.Bejan 1 advances a new theory in which he unabashedlyhints that his law is in the same league as the second law ofthermodynamics, because a simple law is purported topredict the geometric form of anything alive on earth.Many applications of the constructal theory weredeveloped in fluids mechanics, in particular for theoptimization of flows 210. On the other hand, thereexists, to our knowledge, little examples of applications insolids or structures mechanics. So we have at least half ofthe references to papers in fluid dynamics (most of the sameauthor), because the constructal method was developed firstby the same author, Adrian Bejan, with only references topapers in fluid dynamics. The constructal theory rests onthe assumption that all creations of nature are overalloptimal compared to the laws which control the evolutionand the adaptation of the natural systems. The constructalprinciple consists of distributing the imperfections as wellas possible, starting from the smallest scales to the largest.The constructal theory works with the total macroscopicstructure starting from the assembly of elementary struc-tures, by complying with the natural rules of optimaldistribution of the imperfections. The objective is theresearch of lower cost.Int J Adv Manuf Technol (2008) 38:16DOI 10.1007/s00170-007-1080-2J. L. Marcelin (*)Laboratorie Sols Solides Structures 3S, UMR CNRS C5521,Domaine Universitaire,BP n53,38041 Grenoble Cedex 9, Francee-mail: Jean-Luc.Marcelinujf-grenoble.frHowever, a global and macroscopic solution for theoptimization of mechanical structures having least cost asthe objective can be very close to the constructal theory,from where the term pseudo-constructal comes. Theconstructal theory is a predictive theory, with only onesingle principle of optimization from which all rises. Thesame applies to the pseudo-constructal step which is thesubject of this article. The single principle of optimiza-tion of the pseudo-constructal theory is the minimizationof total potential energy. Moreover, in our examplespresented hereafter, the pseudo-constructal principle willbe associated with a genetic algorithm, with the resultthat our optimization will be very close to the naturallaws.The objective of this paper is thus to show how thepseudo-constructal step can apply to the mechanics of thestructures, and in particular to the shape optimization ofmechanical structures. The basic idea is very simple: amechanical structure in a balanced state corresponds to aminimal total potential energy. In the same way, an optimalmechanical structure must also correspond to a minimaltotal potential energy, and it is this objective which mustintervene first over all the others. It is this idea which willbe developed in this article.Two examples will be presented thereafter. The idea tominimize total potential energy in order to optimize amechanical structure is not brand new. Many papers alreadydeal with this problem. What is new, is to make thisapproach systematic. The only objective of optimizationbecomes the minimization of energy.In Gosling 11, a simple method is proposed for thedifficult case of form-finding of cablenet and membranestructures. This method is based upon basic energyconcepts. A truncated strain expression is used to definethe total potential energy. The final energy form isminimized using the Powell algorithm. In Kanno andOhsaki 12, the minimum principle of complementaryenergy is established for cable networks involving onlystress components as variables in geometrically nonlinearelasticity. In order to show the strong duality between theminimization problems of total potential energy andcomplementary energy, the convex formulations of theseproblems are investigated, which can be embedded into aprimal-dual pair of second-order programming problems. InTaroco 13, shape sensitivity analysis of an elastic solid inequilibrium is presented. The domain and boundary integralexpressions of the first and second-order shape derivativesof the total potential energy are established. In Warner 14,an optimal design problem is solved for an elastic rodhanging under its own weight. The distribution of the cross-sectional area that minimizes the total potential energystored in an equilibrium state is found. The companionproblem of the design that stores the maximum potentialenergy under the same constraint conditions is also solved.In Ventura 15, the problem of boundary conditionsenforcement in meshless methods is solved. In Ventura15, the moving least-squares approximation is introducedin the total potential energy functional for the elastic solidproblem and an augmented Lagrangian term is added tosatisfy essential boundary conditions.The principle of minimization of total potential energy isin addition at the base of the general finite elementsformulation, with an aim of finding the unknown optimalnodal factors 16.2 The methods usedIn the pseudo-constructal theory developed in this paper,the main objective of optimization is only the minimizationof total potential energy. The other objectives usually usedin mechanical structures optimization are treated here likelimitations or optimization constraints. For example, onemay have limitations on the weight, or to not exceed thevalue of a stress.The idea which will be developed in this paper is thusvery simple. A mechanical structure is described by twotypes of parameters: variables known as discretizationvariables (for example, degrees of freedom in displacementfor finite elements method), and geometrical variables ofdesign (for example parameters which make it possible todescribe the mechanical structure shape). Total potentialenergy depends on an implicit or explicit way of determin-ing discretization and design variables at the same time.One thus will carry out a double optimization of themechanical structure, compared to the discretization anddesign variables; the objective being to minimize totalpotential energy overall. Clearly, the problem of optimiza-tion of a mechanical structure will be addressed by thefollowing approach:Objective: to minimize total potential energyVariables of optimization: concurrently determiningdiscretization variables (in the case of a traditional useof the finite element method in mechanics of struc-tures), and design variables describing the shape of thestructureOptimization limitations:Weight or volumeDisplacements or strainsStressesFrequenciesThe problem of optimization of a mechanical structurewill be solved in the following way, while reiterating on2Int J Adv Manuf Technol (2008) 38:16these stages, if needed (according to the nature of theproblem):Stage 1Minimization of the total potential energy of themechanical structure compared to the only dis-cretization variables of the structure (degrees offreedom in finite elements). It acts here as anoptimization without optimization limitations.The only limitations at this stage are of purelymechanical origin, and relate to the boundaryconditions and to the external efforts applied tothe structure.In this stage 1, the design variables remain fixed, andone obtains the implicit or explicit expressions of thedegrees of freedom according to the design variables(which can be the variables which make it possible todescribe the shape, in the case of a shape optimization, forexample). One will see in the examples of the followingpart that these expressions can be explicit or implicit andwhich is the suitable treatment following the cases. In thecase of a finite elements method of calculation, this stage 1is the basis of finite elements calculation to obtain thedegrees of freedom of the mechanical structure. Indeed, infinite elements, displacements with the nodes of themechanical structure mesh are obtained by minimizationof total potential energy 16.Stage 2The expressions of the degrees of freedom of themechanical structure according to the designvariables obtained previously are then injected intothe total potential energy of the mechanicalstructure (one will see in the second example ofthe followingparthowone treatsthecasewherethedegrees of freedom are implicit functions of thedesign variables). One then obtains an expressionof the total potential energy which depends only onthe design variables (in explicit or implicit form).Stage 3One then carries out a second and new minimi-zation of the total potential energy obtained in thepreceding form, but this time compared to thedesign variables while respecting the technolog-ical limitations or the optimization constraints ofthe problem. This method can be applied withmore or less facility according to the nature of theproblem. It is clear, for example, that if thediscretization variables can be expressed in anexplicit way according to the design variables, thesetting in of stages 2 to 3 is immediate, andwithout iterations.If the discretization variables cannot be expressed in anexplicit way according to the design variables, or if thetopology of the structure is not fixed, or if the behavior isnot linear, it will be necessary to proceed by successiveiterations on stages 1 to 3. It is the case of the examplespresented in the following part, and one will see on thisoccasion which type of strategy one can adopt for theseiterations. To summarize, in the pseudo-constructal step, themain objective is only the minimization of total potentialenergy, the other possible objectives are treated likelimitations or optimization constraints.The optimization method used for our examples is GA(genetic algorithm), as described in 17. Examples withsimilar instructional value can also be found in manybooks, e.g. in 18. This evolutionary method is veryconvenient for our pseudo-constructal method. The authorhas worked extensively in GAs and published in somereputed journals on this topic 1931. As the topic of GAsis still relatively new in the structural mechanics commu-nity, we provide here some details of exactly what is usedin this GA. A multiple point crossover is used rather than asingle point crossover. The selection scheme used at eachgeneration is entirely stochastic. For our examples, thenumber of generations is equal to that used for conver-gence. The results provided for our examples wereconsistently reproduced by using different seeds in theGA. It has been proved that a rather standard geneticalgorithm is sufficient for our examples.3 ExamplesEven though potential energy may be a good measure forsome optimizations, potential energy is not what gives theshape to a water droplet, nor defines the optimal shape for ahammer, which is why potential energy is not the onlyobjective; but the optimization problem is a multiobjectiveone and the objective functions for the two examples arethen clearly formulated.3.1 Example 1: optimization of the shape of a drop of waterThe first test example is the optimization of the shape of adrop of water (Fig. 1). This problem is equivalent to anequal resistance tank calculated by the membrane theory.The objective is to see if the pseudo-constructal theorygives the natures optimum design.3.1.1 The methods usedThe geometry of the drop of water is defined by thegenerating line of a thin axisymmetric shell. This line isdescribed by successive straight or circular segmentsdescribed in a given sense and defined by input data ofmaster point coordinates and radius values. The initial dataare a set of nodal points connected by straight segments.Each nodal point is identified by its two cylindricalInt J Adv Manuf Technol (2008) 38:163coordinates (r, z), and a real R which represents the radiusof the circle tangent to the two straight segments intersect-ing at the point. The other computer calculations give thecoordinates of any boundary point and especially thetangent points necessary to define the circular arc lengths.The design of the drop of water is described by three arcs ofcircles as indicated in Fig. 1.Analysis is performed by the finite element method withthree-node parabolic elements using the classical Love-Kirchoff shell theory. An automatic mesh generator createsthe finite element mesh of each straight or circular segmentconsidered as a macro finite element.The objective is to obtain a shape for the drop of watergiving rise to a minimum total potential energy (which isthe main objective) and an equal resistance tank (which isthe only constraint or limitation of the problem).In fact, for the drop of water problem, the goal is a multi-objective one, the two objectives ( f1=minimum totalpotential energy and f2=equal resistance) are combined ina multi-objective: f=f1+f2.The constraint or limitation of the problem is taken intoaccount by a penalization of the total potential energy asindicated in Marcelin et al. 19.3.1.2 The resultsThe design of the drop of water is described by three arcs ofa circle (Fig. 1). Their centers and radius are the designvariables. So, there are nine design variables: r1, z1, R1 forcircle 1; r2, z2, R2 for circle 2; and r3, z3, R3 for circle 3.In the genetic algorithm, each of these design variables iscoded by three binary digits.The tables of coding-decoding will be the following:For r1:For z1:For R1:For r2:For z2:For R2:For r3:For z3:For R3:All these binary digits are put end to end to form achromosome length of 27 binary digits.GA is run for a population of 30 individuals, a numberof generations of 50, a probability of crossing of 0.8, and aprobability of mutation of 0.1.The optimal solution corresponds to the chromosome100 100 011 011 010 011 100 011 101which gives the solution of Fig. 1, for which:r1=18, z1=17,and R1=0.065r2=13.75, z2=12.2 and R2=7.7r3=4.1, z3=21.4 and R3=21It is very close to the natures optimal solution for theshape of a drop of water. The model of the water dropmodelled by three arcs of a circle is imperfect. However,the constructal theory optimizes the imperfections, and1052025r510 20z3 211515Fig. 1 Optimization of the shape of a drop of water0000010100111001011101111616.51717.51818.51919.50000010100111001011101111515.51616.51717.51818.50000010100111001011101110.0500.0550.0600.0650.0700.0750.0800.0850000010100111001011101111313.2513.513.751414.2514.514.750000010100111001011101111212.112.212.312.412.512.612.70000010100111001011101117.47.57.67.77.87.988.10000010100111001011101113.73.83.944.14.24.34.400000101001110010111011121.121.221.321.421.521.621.721.800000101001110010111011118.51919.52020.52121.5224Int J Adv Manuf Technol (2008) 38:16finds the nearest solution to that of nature. So, theconstructal principle consists of distributing the imperfec-tions as well as possible.3.2 Example 2: optimization of the shapeof an axisymmetric structureIn this part, the very localized optimization of the rearbearing of a hydraulic hammer is presented. The bearing inquestion (Fig. 2) breaks after relatively few cycles ofoperation.For axisymmetric structures, analysis is performed bythe finite element method in which the special character ofa GA optimization process has been considered to ease thecalculations and to save computer time. First, because just afew parts of the structure must often be modified, thesubstructure concept is used to separate the “fixed” and the“mobile” parts. The fixed parts are calculated twice: once atthe beginning and also at the end of the optimizationprocess. Only the reduced stiffness matrices of thesesubstructures are added to the matrices of the mobile parts.Related to this division, an automatic generator creates thefinite element mesh of each substructure considered as amacro finite element. These macro elements are eithertriangular (six nodes) or quadrilateral (eight nodes).Following a well-known technique, the same subdivisionis used in the parent space to obtain the mesh itself, whichis obviously made out of the same types of elements.During the optimization process this mesh is controlled anda new discretization can be chosen if necessary.To summarize, the optimization problem is the following:Objective function Minimization of the total potentialenergy. It is important to note that another importantobjective (the minimization of the maximum value of theVon Mises equivalent stress along the mobile contour) istaken here as a constraint of the problem. This secondobjective is necessary to achieve the minimization of therear bearing of the hydraulic hammer .Design variables The design variables are radius r andwidth X near the radius (Fig. 2).Constraints The side constraints are established in such away that only small changes in geometry are allowed. Theytake into account the technological constraints. They areincluded in the coding of the design variables. Anotherimportant constraint is that the maximum value of the VonMises equivalent stress along the mobile contour must notexceed a certain value. The constraints are taken intoaccount by a penalization of the total potential energy asindicated in 19.The tables of coding-decoding are the following:For r:For X:All these binary digits are put end to end to form achromosome length of eight binary digits.The GA is run for a population of 12 individuals, anumber of generations of 30, a probability of crossing of0.5, and a probability of mutation of 0.06.The optimal solution corresponds to the chromosome1101 1000which gives the solution of Fig. 2, for which:r=1.95, X=6.0The automatic optimization of the shape of this producthas,simplybyasmallmodificationofshape,whichisdifficultto predict other than by calculation (increased radius,decreased width), considerably improved the mechanicaldurability of the bearing: the over-stress being reduced by50%.4 DiscussionThe two examples in this paper may prove the truth of thepseudo-constructal theory. The first one was the shapeoptimization ofanaxisymmetricmembranedrop-shaped shellConstraint: the bearing should not penetrate into the casing during deformation Hydraulic hammers rear bearingCASINGr=1.95r= 1.5initial shapefinal shapeX Xmobile