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ORIGINAL ARTICLEDeformation control through fixture layout designand clamping force optimizationWeifang Chen & Lijun Ni & Jianbin XueReceived: 2 February 2007 /Accepted: 4 July 2007# Springer-Verlag London Limited 2007Abstract Workpiece deformation must be controlled in thenumerical control machining process. Fixture layout andclamping force are two main aspects that influence thedegree and distribution of machining deformation. In thispaper, a multi-objective model was established to reducethe degree of deformation and to increase the distributinguniformity of deformation. The finite element method wasemployed to analyze the deformation. A genetic algorithmwas developed to solve the optimization model. Finally, anexample illustrated that a satisfactory result was obtained,which is far superior to the experiential one. The multi-objective model can reduce the machining deformationeffectively and improve the distribution condition.Keywords Fixturelayout.Clampingforce.Geneticalgorithm.Finiteelementmethod1 IntroductionFixture design is an important procedure in manufacturingengineering. It is critical to machining accuracy. Aworkpiece should be constrained in a fixture duringmachining with fixture elements such as locators, clamps,and supports. The positions of locators, clamps andsupports should be strategically designed and appropriateclamping forces should be applied. The fixture elementscan be placed anywhere within the candidate regions on theworkpiece surfaces. Clamping force must be large enoughto hold the workpiece during machining. Typically, it reliesheavily on the designers experience to choose the positionsof the fixture elements and to determine the clampingforces. Thus there is no assurance that the resultant solutionis optimal or near optimal for a given workpiece.Consequently, the fixture layout and the clamping forceoptimization become two main aspects in fixture design.The positions of locators and clamps, and the values ofclamping force should be properly selected and calculatedso that the workpiece deformation due to clamping andcutting force is minimized and uniformed.The objective of fixture design is to find an optimallayout or positions of the fixture elements around theworkpiece and optimal clamping force. In this paper, amulti-objective optimization method is presented for thefixture layout design and clamping force optimization.The objective is two folded. One is to minimize themaximum elastic deformation of the machined surfaces,and another is to maximize the uniformity of deforma-tion. The ANSYS software package is used to calculatethe deformation of the workpiece under given clampingforce and cutting force. A genetic algorithm is devel-oped, and the direct search toolbox of MATLAB isemployed to solve the optimization problem. Finally, acase study is given to illustrate the application of theproposed approach.2 Literature reviewWith the wide applications of optimization methods inindustry, fixture design optimization has gained moreinterests in recent years. Fixture design optimizationincludes fixture layout optimization and clamping forceoptimization. King and Hutter presented a method forInt J Adv Manuf TechnolDOI 10.1007/s00170-007-1153-2W. Chen:L. Ni:J. Xue (*)College of Mechanical and Electronical Engineering,Nanjing University of Aeronautics and Astronautics,No. 29, Yudao Street,Nanjing 210016, Chinae-mail: optimal fixture layout design using a rigid body model of thefixture-workpiece system 1. DeMeter also used a rigidbody model for the analysis and synthesis of optimalfixture layouts and minimum clamping force 2. Hepresented a finite element method (FEM) based supportlayout optimization procedure with computationally attrac-tive qualities 3. Li and Melkote used a nonlinearprogramming method and a contact elasticity model tosolve the layout optimization problem 4. Two years later,they presented a method for determining the optimalclamping force for a multiple clamp fixture subjected toquasi-static machining force 5. They also presented anoptimal synthesis approach of fixture layout and clampingforce that considers workpiece dynamics during machining6. A combined fixture layout and clamping forceoptimization procedure was presented. Other researchers7, 8 used the FEM for fixture design and analysis. Cai etal. 9 extended the work of Menassa and DeVries 8toinclude synthesis of fixture layout for sheet metal assembly.Qin et al. 10 established an elastic contact model betweenclamp and workpiece to optimize the clamping force withan objective to minimize the position error of theworkpiece. Deng and Melkote 11 presented a model-based framework for determining the minimum requiredclamping force, which ensures the dynamic stability of afixtured workpiece during machining.Most of the above studies used nonlinear programmingmethods, which seldom gave global or near-global opti-mum solutions. All of the fixture layout optimizationprocedures must start with an initial feasible layout. Inaddition, solutions obtained from these models are verysensitive to the initial feasible fixture layout. The problemof fixture design optimization is nonlinear because there isno direct analytical relationship between the objectivefunction and design variables, i.e. between the machinedsurface error and the fixture parameters (positions of locatorand clamp, and clamping forces).Previous researchers had shown that genetic algorithm(GA) was a useful technique in solving such optimiza-tionproblems.WuandChan12usedtheGAtodetermine the most statically stable fixture layout. Ishikawaand Aoyama 13 applied GA to determine the optimalclamping condition for an elastic workpiece. Vallapuzha etal. 14 used spatial coordinates to encode in the GA basedoptimization of fixture layout. They also presented themethodology and results of an extensive investigation intothe relative effectiveness of the main competing fixtureoptimization methods, which showed that continuous GAyielded the best quality solutions 15. Krishnakumar andMelkote 16 developed a fixture layout optimizationtechnique that used GA to find the fixture layout thatminimized the deformation of the machined surface due toclamping and cutting force over the entire tool path.Locator and clamp positions were specified with nodenumbers. Krishnakumar et al. 17 presented an iterativealgorithm that minimized the workpiece elastic deformationfor the entire cutting process by alternatively varying thefixture layout and clamping force. Lai et al. 18 set up ananalysis model that treated locator and clamps as the samefixture layout elements for the flexible part deformation.Hamedi 19 discussed a hybrid learning system that usednonlinear FEA with a supportive combination of artificialneural network (ANN) and GA. The ANN was used tocalculate workpiece maximum elastic deformation, the GAwas used to determine the optimum clamping forces.Kumar 20 proposed to combine the GA and ANN todevelop a fixture design system. Kaya 21 used the GAand FEM to find the optimal locators and clampingpositions in 2D workpiece and took chip removal effectsinto account. Zhou et al. 22 presented a GA based methodthat optimized fixture layout and clamping force simulta-neously. Some of the studies did not consider theoptimization of the layout for entire tool path. Some ofthe studies used node numbers as design parameters.Some of the studies addressed fixture layout or clampingforce optimization methods but not both simultaneously.And there were few studies taking friction and chipremoval into account. The effects of chip removal andfrictional contact cannot be neglected for achieving amore realistic and accurate workpiece-fixture layoutverification analysis 23, so it is essential to take chipremoval effects and friction effect into account to achieve abetter machining accuracy.In this paper, the friction and chip removal are takeninto account to achieve the minimum degree of themaximum deformation of the machined surfaces underclamping and cutting force and to uniform the deforma-tion. A multi-objective optimization model is established.An optimization process based on GA and FEM ispresented to find the optimal fixture layout and clampingforce. Finally, the result of the multi-objective optimiza-tion model is compared with the single objectiveoptimization method and the experience method for a lowrigidity workpiece.3 A multi-objective optimization model for fixturedesignA feasible fixture layout has to satisfy three constraints.First, the locators and clamps cannot apply tensile forces onthe workpiece. Second, the Coulomb friction constraintmust be satisfied at all fixture-workpiece contact points.The positions of fixture element-workpiece contact pointsmust be in the candidate regions. For a problem involving pfixture element-workpiece contacts and n machining loadInt J Adv Manuf Technolsteps, the optimization problem can be mathematicallymodeled as followsmin max 1jj; 2jj; :; jC12C12C12C12; :; njjC0C1 s ; j 1; 2; :; n 1Subject tom FnijjC21F2ti F2hiq2FniC21 0 3pos i2Vi; i 1; 2; :; p 4where jrefers to the maximum elastic deformation at amachining region in the j-th step of the machiningoperation,Xnj1j C0C0C16C172C30nvuut is the average of jFniis the normal force at the i-th contact point is the static coefficient of frictionFti; Fhiare the tangential forces at the i-th contact pointpos(i) is the i-th contact pointV(i) is the candidate region of the i-th contact point.The overall process is illustrated in Fig. 1 to design afeasible fixture layout and to optimize the clamping force.The maximal cutting force is calculated in cutting modeland the force is sent to finite element analysis (FEA) model.Optimization procedure creates some fixture layout andclamping force which are sent to the FEA model too. InFEA block, machining deformation under the cutting forceand the clamping force is calculated using finite elementmethod under a certain fixture layout, and the deformationis then sent to optimization procedure to search for anoptimal fixture scheme.4 Fixture layout design and clamping force optimization4.1 A genetic algorithmGenetic algorithms (GA) are robust, stochastic and heuristicoptimization methods based on biological reproductionprocesses. The basic idea behind GA is to simulate “survivalof the fittest” phenomena. Each individual candidate in thepopulation is assigned a fitness value through a fitnessfunction tailored to the specific problem. The GA thenconducts reproduction, crossover and mutation processesto eliminate unfit individuals and the population evolvesto the next generation. Sufficient number of evolutions ofthe population based on these operators lead to anincrease in the global fitness of the population and thefittest individual represents the best solution.The GA procedure to optimize fixture design takesfixture layout and clamping force as design variables togenerate strings which represent different layouts. Thestrings are compared to the chromosomes of naturalevolution, and the string, which GA find optimal, ismapped to the optimal fixture design scheme. In this study,the genetic algorithm and direct search toolbox of MATLABare employed.The convergence of GA is controlled by the populationsize (Ps), the probability of crossover (Pc)andtheprobability of mutations (Pm). Only when no change inthe best value of fitness function in a population, Nchg,reaches a pre-defined value NCmax, or the number ofgenerations, N, reaches the specified maximum number ofevolutions, Nmax., did the GA stop.There are five main factors in GA, encoding, fitnessfunction, genetic operators, control parameters and con-straints. In this paper, these factors are selected as what islisted in Table 1.Since GA is likely to generate fixture design strings thatdo not completely restrain the fixture when subjected tomachining loads. These solutions are considered infeasibleand the penalty method is used to drive the GA to a feasiblesolution. A fixture design scheme is considered infeasible orunconstrained if the reactions at the locators are negative, inother words, it does not satisfy the constraints in equations(2)and(3). The penalty method essentially involvesMachiningProcess ModelFEAOptimizationprocedurecutting forcesfitnessOptimization resultFixture layout andclamping force Fig. 1 Fixture layout and clamp-ing force optimization processTable 1 Selection of GAs parametersFactors DescriptionEncoding RealScaling RankSelection RemainderCrossover IntermediateMutation UniformControl parameter Self-adaptingInt J Adv Manuf Technolassigning a high objective function value to the scheme thatis infeasible, thus driving it to the feasible region insuccessive iterations of GA. For constraint (4), when newindividuals are generated by genetic operators or the initialgeneration is generated, it is necessary to check up whetherthey satisfy the conditions. The genuine candidate regionsare those excluding invalid regions. In order to simplify thechecking, polygons are used to represent the candidateregions and invalid regions. The vertex of the polygons areused for the checking. The “inpolygon” function inMATLAB could be used to help the checking.4.2 Finite element analysisThe software package of ANSYS is used for FEAcalculations in this study. The finite element model is asemi-elastic contact model considering friction effect,where the materials are assumed linearly elastic. As shownin Fig. 2, each locator or support is represented by threeorthogonal springs that provide restrains in the X, Y and Zdirections and each clamp is similar to locator but clampingforce in normal direction. The spring in normal direction iscalled normal spring and the other two springs are calledtangential springs.The contact spring stiffness can be calculated accordingto the Herz contact theory 8 as followskiz16RC3iEC32i9C16C1713fiz13kiz kiy6EC3i2C0vfiGfi2C0vwiGwiC16C17C01C1 kiz8:5wherekiz, kix, kiyare the tangential and normal contactstiffness,1RC3i1Rwi1Rfiis the nominal contact radius,1EC3i1C0V2wiEwi1C0V2fiEfiis the nominal contact elastic modulus,Rwi, Rfiare radius of the i-th workpiece andfixture element,Ewi, Efiare Youngs moduli for the i-thworkpiece and fixture materials,wi, fiare Poisson ratios for the i-th workpieceand fixture materials,Gwi, Gfiare shear moduli for the i-th workpieceand fixture materials and fizis thereaction force at the i-th contact point inthe Z direction.Contact stiffness varies with the change of clampingforce and fixture layout. A reasonable linear approximationof the contact stiffness can be obtained from a least-squaresfit to the above equation.The continuous interpolation, which is used to applyboundary conditions to the workpiece FEA model, isFig. 2 Semi-elastic contact model taking friction into accountSpring positionFixture element position12345678 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 32 33 34 3536 37 38 39 40 41 4243 44 45 46 47 48 49Fig. 3 Continuous interpolationFig. 4 A hollow workpieceTable 2 Machining parameters and conditionsParameter DescriptionType of operation End millingCutter diameter 25.4 mmNumber of flutes 4Cutter RPM 500Feed 0.1016 mm/toothRadial depth of cut 2.54 mmAxial depth of cut 25.4 mmRadial rake angle 10Helix angle 30Projection length 92.07 mmInt J Adv Manuf Technolillustrated in Fig. 3. Three fixture element locations areshown as black circles. Each element location is surroundedby its four or six nearest neighboring nodes. These sets ofnodes, which are illustrated by black squares, are 37, 38,31 and 30, 9, 10, 11, 18, 17 and 16 and 26, 27, 34, 41,40 and 33. A set of spring elements are attached to each ofthese nodes. For any set of nodes, the spring constant iskijdijPk2hidikki6wherekijis the spring stiffness at the j-th node surrounding thei-th fixture element,dijis the distance between the i-th fixture element and thej-th node surrounding it,kiis the spring stiffness at the i-th fixture elementlocation.iis the number of nodes surrounding the i-th fixtureelement location.For each machining load step, appropriate boundaryconditions have to be applied to the finite element model ofthe workpiece. In this work, the normal springs areconstrained in the three directions (X, Y, Z)andthetangential springs are constrained in the tangential direc-tions (X, Y). Clamping forces are applied in the normaldirection (Z) at the clamp nodes. The entire tool path issimulated for each fixture design scheme generated by theGA by applying the peak X, Y, Z cutting forces sequentiallyto the element surfaces over which the cutter passes 23.In this work, chip removal from the tool path is takeninto account. The removal of the material during machiningalters the geometry, so does the structural stiffness of theworkpiece. Thus, it is necessary to consider chip removalaffects. The FEA model is analyzed with respect to toolmovement and chip removal using the element deathtechnique. In order to calculate the fitness value for a givenfixture design scheme, displacements are stored for eachload step. Then the maximum displacement is selected asfitness value for this fixture design scheme.The interaction between GA procedure and ANSYS isimplemented as follows. Both the positions of locators andclamps, and the clamping force are extracted from realstrings. These parameters are written to a text file. Theinput batch file of ANSYS could read these parameters andcalculate the deformation of machined surfaces. Thus thefitness values in GA procedure can also be written to a textfile for current fixture design scheme.It is costly to compute the fitness value when there are alargenumberofnodesinanFEMmodel.Thusitisnecessaryto speed up the computation for GA procedure. As thegeneration goes by, chromosomes in the population aregetting similar. In this wo