Multi Objective Optimization of The Structural
Design of Double End Beam Load Cell
Hilman Syaeful Alam
1, Bahrudin
1, Demi Soetraprawata
11Technical Implementation Unit for Instrumentation Development, Indonesian Institute of Sciences, Jl. Sangkuriang Komplek LIPI Gedung 30 Bandung, 40135, Indonesia
Abstract—Multi objective optimization of the structural design of double end beam load cell had been performed using genetic algorithm. There were two design variable which was used to minimize the structural mass, to maximize the structural strength, and to maximize the level of measurement accuracy. Based on the optimization results, the weight reduction of the structure was successfully achieved of about 8.6%. Then the design safety factor reached 24 % larger than the yield failure criterion therefore the structure will certainly secure against the plastic deformation. The accuracy of measurement of the load cell was predicted to reach the category of high accuracy due to the amount of strain on the strain gauge cavity managed to exceed ± 1600 με.
Keywords—genetic algorithm; accuracy; safety factor; yield; strain.
I. INTRODUCTION
In the design process, quality and competitiveness of a product will usually have problems if faced with many requirements, for example low price, high efficiency, light weight and long life. But along with the rapid development in technology design, these challenges can be overcome by involving the multi-objective optimization techniques. These techniques can help the designer to solve various problems in the field of engineering design. With these advantages, multi-objective optimization techniques have been used to solve the design problems in some recent studies which are applied for the complex mechanical structures [1], mold of the automotive interior component [2], geometry of the groove micro-mixer [3], boiler combustion process of power plant [4], heating/cooling channels of injection molding [5], geometry of heat exchanger system [6], and airfoil of wind turbine [7].
One of the optimization methods that can solve the problem in the design process with many criteria is genetic algorithm. This method is inspired by the basic principles of the evolution theory. The first stage is a collection of randomly generated solution, called the initial population. Several solutions are then selected and combined to obtain a new solution with slightly different from the initial population. The solution that comes closest to a target placed in the population to produce more new solutions. The most appropriate solution is produced with several iterations and chosen as the most optimal solution. The advantage of genetic algorithm is easy to implement on a multi-criteria problems, such as the following of optimization studies: construction panel of a cargo ship [8], heat exchanger [9], insulation for buildings [10], heat sink of photovoltaic system
[11], oven heating system [12], and fixture for the machining process [13].
In this study, genetic algorithms will be applied to obtain optimum shape or geometry of the structure of the double-end beam load cell. The structure requires a number of requirements that must be met including strength, stiffness, weight, and measurement accuracy. Double end beam load cell is a kind of force measuring instrument which are popular and widely applied for industrial scales. This instrument is designed to change the force measurement into an electrical signal, which the force on the structure is measured by strain gauge and converted into electrical quantities.
II. MATERIALS AND METHODS A. Genetic Algorithm Concepts
The concepts of genetic algorithm in details was published by Kaya [13] and can be explained as follows. Genetic algorithms are randomly search techniques of optimum solution, which mimics the mechanism of natural evolution. The algorithm works on the population design. Populations evolve from generation to generation gradually through natural selection. The best individual of the population will transmit its nature to the next generation. The process of natural selection in the algorithm, replaced with artificial selection process based on computed fitness for each design. The term fitness is used to indicate the chromosome to survive in the population and basically is the objective function of the optimization problem. The chromosome that defines the characteristics of biological beings replaced by string numeric value that represents the design variables.
1) Individual representation
The initial and most important step in preparing for the solution of optimization problems in the genetic algorithm is to define a specific code of design variables and settings into a series of numerical values to be used as a chromosome. In the genetic algorithm, binary code that has limited length, i.e. ones and zeros. In a multi-parameter optimization problem, code the individual parameters are generally converted into a complete string which is shown in Fig. 1.
2015 International Conference on Automation, Cognitive Science, Optics, Micro Electro-Mechanical System, and Information Technology (ICACOMIT), Bandung, Indonesia, October 29–30, 2015
Fig. 1. Binary representation in genetic algorithms [13]. The length of the code string depends on the precision required where mapping binary code into real numbers is done in two stages:
Step 1: Determine the length of the code for � (i=1, ..., n):
� = (� ��− � ) × (1)
where is the precision needed ( , , , . . .). Code length for � are as follows:
��� = � + (2)
where,
< � < + (3)
The length of total string is given by:
� = ∑= �� (4)
Step 2: Mapping of binary string into a real number.
� = � +�� ��−�� �
− ∑= − (5)
where ∈ [ , ]
To produce the chromosomes, the first step is the calculation of the length of chromosome, then a random value in the range of (0.1) is used to form of chromosomes.
2) Parameter of Genetic Algorithm
The determination of parameters or operator is crucial for genetic algorithm optimization problems. The parameters consist of reproduction, crossover, mutation and others, and can be explained as follows:
Reproduction. Reproduction operators allow individuals to be able to copy the code that makes the requirements in the next iteration. After checking the suitability of the fitness value of each individual in the initial population, only a few individuals who have a high fitness values are considered in the reproduction. In the selection process, two individuals are randomly selected from the population, then the individuals who have a high suitability values is selected. The procedure is continued until the reproduction population size equivalent to the size of the population.
Crossover. Crossover is the next operation in genetic algorithms which exchange the information between any two selected individuals. This operation choses genes from parent chromosomes and create new offspring. As in the reproduction operator, there are a number of crossover operator in genetic algorithms. At the crossover operator, two individual strings are cut and the right side portion of two individuals are swapped to produce two new strings as shown in Fig. 2. To create the crossover operator, two individuals are randomly selected from the population then randomized again by a random number in the range {0,1}. If this random number is less than the probability of crossover, then the individuals are subjected to
the crossover operator, otherwise they are copied to new population. Probability crossover � is selected in the ranges of 0.6 to 0.9.
Fig. 2 The illustrations of crossover operation and mutation. Mutation. Mutation is the process of modifying the random string with a small probability value. These mutations alter 1 and 0 with a small mutation probability � . Mutation is need to maintain a diversity in the population, therefore it can keep the fall of all the solutions in the population. Fig. 2 shows the illustrations of mutation operation after crossover operation. In order to determine whether a chromosome is to be subjected to mutation, the random numbers are applied in the the range {0,1}. If the random number is less than the probability of a mutation, the selected chromosomes will be mutated. The mutation probability � should be chosen as small as possible with the range of 0.02 and 0.06.
Constraints Handling. In most applications of genetic algorithms for constraints optimization problems, penalty function method can be used in the objective function. This method uses the selection tournament operator which two solutions are compared with each other and must meet the following criteria:
- Any feasible solution is preferred to any infeasible solution. - Among two feasible solutions, the one having better fitness
value is preferred
- Among two infeasible solutions, the one having smaller constraint violation is preferred.
Elitist strategy. At this stage, some of the best individual solutions are copied into the next generation without applying genetic operators. Elitist strategy always clones the best individuals of the current generation and into the next generation. This will ensure the best design and will never be lost in the next generation.
B. Analysis of Initial Design
the finite element modeler software, i.e. ANSYS. While, the geometry of the structure which was analyzed for homogeneous and linear elastic with AISI 4340 material.
Fig. 3 Schematic of double end beam load cell.
Based on finite element simulation using ANSYS, the design safety factor that indicates the strength of a structure based on the yield criteria of the material due to a maximum loading of 20 kN is 1.59, so the structure was still safe against the plastic failure of the material because the equivalent Von Mises stress due to loading is still 59 percent below the yield strength of the material. Then based on the density of the AISI 4340 material, the structural mass in accordance with the design of beam geometry is 1.043 kg.
One factors affecting the level of measurement accuracy of the load cell is the strain amount on the structure which depends on the specifications of the strain gauge. In this study, the commercial strain gauge is planned to be use to convert mechanical strain into electrical signal. To produce the load cell design load which can be categorized into high accuracy class, the minimum amount of strain on strain gauge cavity is ± 1600 με as described in the specification of commercial strain gauge. Based on the results of finite element simulation, the strain response due to the maximum loading ranges below ± 1300 με for tension and compression direction, therefore it can be categorized as the medium accuracy.
C. Selection of Variable Design & Optimization
In this study, structural topology of double end beam load cell was optimized using genetic algorithm where the objective function of the optimization was based on the design requirements, i.e. minimizing the weight of the structures, maximizing the structural strength based on the design safety
factors, and maximizing the level of measurement accuracy based on the strain response of the strain gauge cavity on the beam. Fig. 4 shows two design variables which are selected as the input parameter of optimization where X is the diameter of strain gauge cavity and Y is the width of the beam. The value of the design variable is determined based on the input and output parameters which are shown in Table 1. The input parameters is determined from the initial dimensions of the structure, while the search region of the optimization is determined by selecting the upper and lower limits of the initial value of variable designs as a constraints in the optimization problem.
There are four objectives as an output variable that will be optimized according to the input of variable design, i.e. minimization of the structural mass, maximization of the safety factor, minimization of minimum strain, and maximization of maximum strain. The initial mass, A is 1.043 kg based on the multiplication of the material density and the structural volume. In the optimization, the structural mass is minimized without constraints. The initial safety factor is 1.5939 to the yield strength of material which is maximized with a minimum safety factor of 1.2. The initial minimum strain of -1026 �� is minimized while the initial maximum strain of 1037 �� is maximized. Both last of the objective function is analyzed with unconstraint optimization.
Fig. 4 Design variables in the optimization of double end beam load cell.
Table 1. Determination of design variables.
Input variable Output variable
X, mm Y, mm Mass, A (kg) SF, B Min �, C (�� Max �, D (��
Initial design 23.0 30.0 1.043 1.59 -1267.9 1289.9
Upper bound 25.3 33.0 - - - -
Lower bound 20.7 27.0 - 1.20 - -
III. RESULTS &DISCUSSIONS
The search techniques of optimum solution in genetic algorithms are randomized by resembling the evolution mechanism naturally therefore the initial population in the form of random population is generated for each chromosomes. The generation process in finding the optimum solution is done through the process of reproduction, crossover and mutation until it produces the best individual solution. Fig. 5(a) shows the iteration results for structural mass P4 wherein the optimum search value is performed by 200 generation to obtain the optimum value. The population looks very random in about 50% of the initial iteration, which continues to mutate to a converged solution produced in accordance with the criteria in equation (4) and (5).
The iteration results of the structural strength maximization P5 can be shown in Fig. 5(b). The optimization for the second objective function is performed by the constraints which is limited by the minimum safety factor of 1.2. Design safety
factor is the ratio between the yield strength of material and the equivalent stress due to loading. From the iteration results, it appears that around 50% of the generation, the safety factor is still over the constraint limit, but along with the increasing of the iteration number, the safety factor is always above the constraint limit. It shows that the constraint handling strategy goes well by the selection operator. The iteration results for minimum and maximum strain can be shown respectively in Fig. 5 (c) and (d). The pattern generated by a minimum strain looks the opposite of the pattern generated by a maximum strain. The greater the range generated between the minimum and maximum strain, the better the level of measurement accuracy of the load cell. The minimum and maximum amount of strain required to produce a load cell which has a high degree of accuracy is ± 1600 με, however it must have a minimum safety factor greater than 1.0, in order to the structure is able to withstand the plastic deformation..
(a) (b)
(c) (d)
Fig. 5 The iteration results of (a) the structural mass, (b) structural strength, (c) minimum strain, and (d) maximum strain.
The iteration results for two input variables are shown in Fig. 6. In general, the iteration results of the variable input is not much different from the iteration results on the objective function, where the population is randomly occur from the initial iteration of up to 50% of generation and achieve converging at 200 times generation, The search area of the optimum value was a variable input range to meet the parameters of the objective function. Input variable X is the diameter of strain gauge cavity with the search areas are in the range of 20.7 mm to 25.3 mm, wherein the diameter before optimization was initially designed at 23 mm. Based on the iteration results of X variable in Fig. 6(a), it seems that the search areas are always consistent in the range of 20.7 mm to 25.3 mm, which the search area will decrease in line with the increasing of generations. It is also happen for the second design variable, i.e. the width of the structure Y (Fig. 6 (b)), where the search optimum areas were designed at 27.0 mm of lower bound and 33 mm of upper bound.
The optimum solution for input and output variables can be shown in Table 2. The input variables X and Y was optimum respectively at 23.60 mm and 27.33 mm. Based on the optimization results of the output parameters, the weight of the load cell structure was successfully reduced to 0.95 kg from the initial design of 1.04 kg or reduced by 8.65%. The results was generated significantly from a reduction in the width of the structure (X), from the first by 30 mm to 27.33 mm. While the weight reduction from the additional diameter of strain gauge cavity (Y) is less than X, since the additional diameter is only about 0.6 mm. The optimum safety factor was successfully maintained above the lower bound and reaches 1.24. It means that 24 percent larger than the failure criterion of the plastic material therefore the structure will certainly secure against the plastic deformation. The optimum strain on the structure was slightly above ± 1600 με, therefore the structural design of the load cell was successfully achieve the desired target because it could be categorized in a high accuracy class of measurement.
(a) (b) Fig. 6 The iteration results for input variable (a) X and (b) Y.
Table 2. The optimum solutions
Input variable Output variable
X, mm Y, mm Mass, A (kg) SF, B Min �, C (�� Max �, D (��
Initial design 23.0 30.0 1.04 1.59 -1267.9 1289.9
Upper bound 25.3 33.0 - - - -
Lower bound 20.7 27.0 - 1.20 - -
Optimum solution 23.60 27.33 0.95 1.24 -1638.2 1635.4
20 40 60 80 100 120 140 160 180 200
19 20 21 22 23 24 25 26 27
Generation
X
(m
m
)
20 40 60 80 100 120 140 160 180 200
25 26 27 28 29 30 31 32 33 34 35
Generation
Y
(m
m
IV. CONCLUSIONS
The structural optimization of double end beam load cell using genetic algorithm which involved two design variables and four objective function successfully achieved a convergent solution. Based on the optimization results, the weight of the structure was reduced up to 8.6%. Then the design safety factor could be maximized up to 24 percent larger than the failure criterion of the plastic material therefore the structure would certainly secure against the plastic deformation. The amount of strain on the strain gauge cavity in both compression and tension could reach over ± 1600 με, therefore the measurement accuracy of the load cell was predicted to reach the category of high accuracy.
ACKNOWLEDGMENT
The author would like to thank to the Head of Technical Implementation Unit for Instrumentation Development, Indonesian Institute of Sciences which has provided a financial support through LIPI research thematic in 2015 (No. 07901450196013). Moreover thanks to the Head of Research Center for Power and Mechatronics, Indonesian Institute of Sciences, for facilitating the finite element simulation using ANSYS.
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![Fig. 1. Binary representation in genetic algorithms [13].](https://thumb-ap.123doks.com/thumbv2/123dok/264430.506087/2.595.353.516.95.236/fig-binary-representation-genetic-algorithms.webp)
