Topological Machining Fixture Layout Synthesis Using Genetic Algorithms

Transcription

1 Topological Machining Fixture Layout Synthesis Using Genetic Algorithms Necmettin Kaya Uludag University, Mechanical Eng. Department, Bursa, Turkey Ferruh Öztürk Uludag University, Mechanical Eng. Department, Bursa, Turkey Abstract Deformation of the workpiece may cause dimensional problems in machining. Supports and locators are used in order to reduce the error caused by elastic deformation of the workpiece. The optimization of support and locator locations is a critical problem to minimise the geometric error in workpiece machining. In this paper, the application of genetic algorithms (GAs) to the fixture layout optimisation is presented to handle fixture layout problem. A genetic algorithm based approach is developed to optimise fixture layout through integrating ANSYS running in batch mode to compute the objective function values for each generation. A case study is given to illustrate the application of proposed approach. Introduction Fixtures are used to locate and constrain a workpiece during a machining operation, minimizing workpiece and fixture tooling deflections due to clamping and cutting forces is critical to ensuring accuracy and production-quality results in the machining operation. Traditionally, machining fixtures are designed and manufactured through trial-and-error, which prove to be both expensive and time-consuming to the manufacturing process. To ensure a workpiece is manufactured according to specified dimensions and tolerances, it must be appropriately located and clamped, making it imperative to develop tools that will eliminate costly and time-consuming trial-and-error designs. Proper workpiece location and fixture design are crucial to product quality in terms of precision, accuracy and finish of the machined part. Theoretically the locating principle can satisfactorily locate all prismatic shaped workpieces. This method provides the maximum rigidity with the minimum number of fixture elements. To position a part from a kinematic point of view means constraining the six degrees of freedom of a free moving body (three translations and three rotations). Three supports are positioned below the part to establish the location of the workpiece on its vertical axis. Locators are placed on two peripheral edges and are intended to establish the location of the workpiece on the x and y horizontal axes. Properly locating the workpiece in the fixture is vital to the overall accuracy and repeatability of the manufacturing process. Locators should be positioned as far apart as possible and should be placed on machined surfaces wherever possible. Supports are usually placed to encompass the center of gravity of a workpiece and are positioned as far apart as possible to maintain its stability. The primary responsibility of a clamp in fixture is to secure the part against the locators and supports. Clamps should not be expected to resist the cutting forces generated in the machining operation. For a given number of fixture elements, the machining fixture synthesis problem is the finding optimal layout or positions of the fixture elements around the workpiece. In this paper, a method for fixture layout optimization using genetic algorithms is presented. The optimization objective is to search for a 2D fixture layout that minimizes the maximum elastic deformation of the edges on which locators are positioned. ANSYS program has been used for calculating the deflection of the part under clamping and cutting forces. Movement of the cutting forces is also taken into account. A case study is given to illustrate the proposed approach. In literature, several fixture layout optimization approaches have been presented by researchers. The fixture layout synthesis problem is described in [1]. Locator and clamp positions specified by the node numbers. Optimization of support positions in machining of mechanical part using GAs is given in [2]. GA integrated

2 with ANSYS is used in [3,4,5]. An approach for shape, topology and configuration optimization is presented in [3]. It is shown that how ANSYS can be integrated into the MATLAB environment. The GA is implemented in MATLAB. A new designing procedure based on GA for the rotor and the stator of a universal motor is presented in [4]. Fitness was estimated in ANSYS. The optimal design of magnetic microactuator using the genetic algorithm is given in [5]. Computing the magnetic force is done in ANSYS FE program. Previous works do not include the real-coded genetic algorithm applied to fixture layout synthesis. A chromosome library approach is applied to reduce computation time in this study. Procedure Genetic algorithms are robust, stochastic and heuristic optimization methods based on biological reproduction process [6]. There are several optimization techniques that are used in the context of engineering design optimization. Genetic algorithm is one such technique and is a search strategy based on the rules of natural genetic evolution. The traditional approach to optimization uses a derivative-based approach. Examples are Newton-Raphson and conjugate gradient techniques. The scheme of these algorithms is; hold one solution at a time, look locally to see what direction to move in (via the gradient of the function at the current solution), select the new current solution after deciding how far to move along that path. These algorithms work remarkably well for a great majority of practical problems. There are at least three situations where genetic algorithms are useful: The objective function is not smooth (i.e., not differentiable). There are multiple local optima. There are a large number of parameters. If the objective function does not have a derivative, then clearly a derivative-based algorithm can not be used. Derivative-based algorithms take no allowance of multiple optima. They go to a local optimum near to where they start. If there are likely to be only a few local optima, then using several random starts may be enough to solve the problem. Derivative-based algorithms often progress slowly when there is a large number of parameters. If the gradient is being evaluated numerically, then each iteration of the optimization requires as many function evaluations as there are parameters. The space to be searched is vast, so a large number of iterations will probably be required. In general, derivative-based algorithms work better as they approach the optimum. So a reasonable strategy for many problems with a lot of parameters is to start with a genetic algorithm, then use the best solution from the genetic algorithm as the start for a derivative-based optimizer. The Simple Genetic Algorithm The idea behind the GA is to simulate the behavior of natural evolutionary selection. Although there exist many different algorithms, the basic structure is still the same (See Figure 1). Genetic algorithm is an iterative optimization procedure. Algorithm is started with a set of solutions (represented by chromosomes) called population. Solutions from one population are taken and used to form a new population. This is motivated by a hope, that the new population will be better than the old one. Solutions which are selected to form new solutions (offspring) are selected according to their fitness - the more suitable they are the more chances they have to reproduce. This is repeated until some condition (for example number of populations or improvement of the best solution) is satisfied.

3 Figure 1. A flowchart of working principle of a genetic algorithm Outline of the basic GA is given as follows: 1. [Start] Generate random population of n chromosomes (suitable solutions for the problem). 2. [Fitness] Evaluate the fitness f(x) of each chromosome x in the population. 3. [New population] Create a new population by repeating following steps until the new population is complete. [Selection] Select two parent chromosomes from a population according to their fitness (the better fitness, the bigger chance to be selected). [Crossover] With a crossover probability crossover the parents to form a new offspring (children). If no crossover was performed, offspring is an exact copy of parents. [Mutation] With a mutation probability mutate new offspring at each locus (position in chromosome). [Accepting] Place new offspring in a new population. 4. [Replace] Use new generated population for a further run of algorithm. 5. [Test] If the end condition is satisfied, stop, and return the best solution in current population. 6. [Loop] Go to step 2. Population size means how many chromosomes are in population (in one generation). If there are too few chromosomes, GA have a few possibilities to perform crossover and only a small part of search space is explored. On the other hand, if there are too many chromosomes, GA slows down.

4 Defining the Chromosome (Coding) The chromosome should in some way contain information about solution which it represents. The most used way of encoding is a binary string. The chromosome then could look like this: Chromosome 1 : Chromosome 2 : Each chromosome has one binary string. Each bit in this string can represent some characteristic of the solution. Or the whole string can represent a number. Of course, there are many other ways of encoding. This depends mainly on the solved problem. For example, one can encode directly integer or real numbers, sometimes it is useful to encode some permutations and so on. Reproduction A reproduction operator allows individual strings to be copied for possible inclusion in the next generation. The chance that a string will be copied is based on the string's fitness value. There are many different type of reproduction operators; tournament selection, fitness-proportional selection, ranked selection etc. In tournament selection, two individuals are chosen from the population at random. Then the string which has best fitness value is selected. Crossover Crossover selects genes from parent chromosomes and creates a new offspring. The simplest way how to do this is to choose randomly some crossover point and everything before this point copy from a first parent and then everything after a crossover point copy from the second parent. Crossover can then look like this ( is the crossover point): Chromosome 1 : Chromosome 2 : Offspring 1 : Offspring 2 : Crossover can be rather complicated and very depends on encoding of the encoding of chromosome. Specific crossover made for a specific problem can improve performance of the genetic algorithm. Mutation After a crossover is performed, mutation take place. This is to prevent falling all solutions in population into a local optimum of solved problem. Mutation changes randomly the new offspring. For binary encoding, a few randomly chosen bits switched from 1 to 0 or from 0 to 1. Mutation can then be following: Original offspring 1 : Original offspring 2 : Mutated offspring 1 : Mutated offspring 2 :

5 Crossover and mutation probability There are two basic parameters of GA; crossover probability and mutation probability. Crossover probability says how often will be crossover performed. If there is no crossover, offspring is exact copy of parents. If there is a crossover, offspring is made from parts of parents' chromosome. If crossover probability is 100%, then all offspring is made by crossover. If it is 0%, whole new generation is made from exact copies of chromosomes from old population. Crossover is made in hope that new chromosomes will have good parts of old chromosomes and maybe the new chromosomes will be better. However it is good to leave some part of population survive to next generation. Mutation probability says how often will be parts of chromosome mutated. If there is no mutation, offspring is taken after crossover (or copy) without any change. If mutation is performed, part of chromosome is changed. If mutation probability is 100%, whole chromosome is changed, if it is 0%, nothing is changed. Mutation is made to prevent falling GA into local extreme, but it should not occur very often, because then GA will in fact change to random search. Analysis There is no direct analytical relationship exist between the machining error and the fixture layout. The finite element method can be used to compute the machining error due to workpiece elastic deformation and the GA can be applied to reduce the error. Since the GA deals with only the design variables and objective function value for a particular fixture layout, no gradient or auxiliary information is needed [1]. In this paper, GA is applied to optimization of locator positions in a 2D fixture layout problems. A 2D fixture layout and a slot milling operation is shown (See Figure 2). Although two dimensional problem is considered for simplicity, it is also applicable to three dimensional problems. The workpiece edges are used for locating and clamping. 2D equivalent of the fixture layout method is to be used for this workpiece. Therefore three locators and two clamps are to be used here. Two locators are positioned on the longest edge and a locator is placed perpendicular to longest edge. Two clamps are used to hold the workpiece against the locators. Clamp forces F 1 and F 2 are selected as 1000 N and 500 N respectively. Figure 2. Fixture layout and end milling path The boundary conditions are specified as zero displacement in the y direction for locator 1 and locator 2, zero displacement in the x direction at locator 3. Finite element model and boundary conditions are given (See Figure 3).

6 Figure 3. Finite element model and boundary conditions The machining forces are first applied at two nodes. This procedure has repeated for all machining surface nodes along the slot nodes. Specific instants in the machining process are chosen to minimize computational effort. Machining forces are moved step by step (15 step) and analysis results (displacements) are stored at each step in a text file. Machining forces for machining of slot are computed as Fs=572 N (+y direction) and Fv=200 N (-x direction). The Genetic Algorithm Applied to Fixture Layout Optimization Fixture layout optimization is implemented using a developed program written in Delphi language named GenFix. Displacement values are calculated in ANSYS program. The ANSYS FEM model is able to perform batch jobs when giving a parameter vector, the complete simulation, including geometric modeling, meshing and post-processing of the simulation results is performed in a fully automatic manner. GenFix sends locator positions (L 1, L 2, L 3 ) to ANSYS. After analyzing the model, ANSYS writes displacements to a text file (See Figure 4). Fitness values are computed from the displacement values. For every given locator positions (or chromosome), a total of 15 analysis results (displacements) are stored in text file further fitness computation. GA input parameters are chosen as follows: Number of iteration :150, Population size : 40, Crossover probability : 85%, Mutation probability: 2%. Locator positions L 1, L 2 and L 3 are encoded as real-valued. In many GA based methods developed have used FEA node numbers to represent the locations for fixture elements as well as integer encoding. This paper uses real-valued coordinates to represent the location of locators. The use of FEA node numbers rather than real-valued coordinates to represent the locator positions, results in a significant roughening of the position of the fixture evaluation procedure. A single chromosome consists of 39 bits representing locator positions. For example refers to L 1 =44.71, L 2 =4.03, L 3 = The length of string depends on the number of parameters and accuracy of the solution desired. In order to reduce the computation time, chromosomes and fitness values are stored in a library for further evaluation. GenFix first checks if current chromosome's fitness value has been calculated. If not, locator positions are sent to ANSYS, otherwise fitness values are taken from the library.

7 Figure 4. The flowchart of the proposed methodology and ANSYS interface Elitism approach is used in this study. Elitism is name of method, which first copies the best chromosome to new population. The rest is done in same way. Elitism can very rapidly increase performance of GA, because it prevents losing the best found solution. The optimization formulation is as follows; Minimise the maximum nodal displacement of edges on which locators are positioned, subject to 0 < L 1, L 2, L 3 < 50. Analysis Results & Discussion GA optimization tool (GenFix) runs for 150 generations with a population size of 40 on Pentium III 1.2 GHz processor. This corresponds with 6000 ANSYS FE evaluations. Since the ANSYS has computed only new chromosome values, so run time takes about 2 hours. Some chromosomes from the initial population and fitness values are shown in Table 1.

8 Table 1. Some initial chromosomes, design variables and fitness values Chromosome L1 mm L2 mm L3 mm Fitness mm ,71 4,033 45,61 0, ,195 24,51 31,25 0, ,92 10,35 31,75 0, ,03 41,04 13,23 0, ,31 41,34 40,76 0, ,32 28,68 2,225 0, ,106 18,89 40,90 0, ,53 30,20 27,84 0, ,78 40,13 44,70 0,5111 After 150 iterations, the best fitness value is found as mm. Variation of maximum displacement with number of generations is given (See Figure 5). It can be seen that considerable improvement in the objective function is obtained. Figure 5. The convergence of GA All of the chromosomes are the same in the population at the end of the generation. The best chromosome can be seen in Table 2. Table 2. The best chromosome, design variables and fitness value Chromosome L1 mm L2 mm L3 mm Fitness mm ,76 13,49 30,74 0,1298

9 Figure 6. Optimum locator positions (mm) As can be seen from the figure (See Figure 6), Locator 1, Locator 2 and Locator 3 are positioned nearly at the opposite of the machining area in order to hold the machining forces and clamping forces. Thus maximum displacement value is decreased to minimum value of mm. Conclusion In this paper, an evolutionary optimization technique of fixture layout optimization is presented. ANSYS has been used for FE calculation of fitness values. It is seen that the combined genetic algorithm and FE method approach seems to be powerful approach for present type problems. GA approach is particularly suited for problems where there does not exist a well-defined mathematical relationship between the objective function and the design variables. The results prove the success of the application of GAs for the fixture layout optimization problems. References 1) K. Krishnakumar, S. N. Melkote, Machining Fixture Layout Optimization Using the Genetic Algorithm, International Journal of Machine Tools & Manufacture, v.40, 2000, pp ) J.L. Marcelin, Genetic Search Applied to Selecting Support Positions in Machining of Mechanical Parts, International Journal of Advanced Manufacturing Technology, v.17, 2001, pp ) H. Langer, T. Pühlhofer, H. Baier, An Approach for Shape and Topology Optimization Integrating CAD Parameterization and Evolutionary Algorithms, 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 4-6 September 2002, Atlanta, Georgia, USA. 4) B. Benedicic, T. Kmecl, G. Papa, B. K. Seljak, Evolutionary Optimization of a Universal Motor, IECON'01, 27th Annual Conference of the IEEE Industrial Electronics Society, ) C.H. Ko, J.C. Chiou, Optimal Design of the Magnetic Microactuator Using the Genetic Algorithm, Journal of Magnetism and Magnetic Materials, v.263, 2003, pp ) D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison- Wesley, Reading, MA, 1989.