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Hierarchical parallel processing for design optimization - a case study
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 5 (2018) 5117–5123
www.materialstoday.com/proceedings
ICMPC 2017
Hierarchical parallel processing for design optimization - a case study Basani Satish* P.S.S. Murthy a K. Eswaraiahb *,a
Assistant Professor, KITS Warangal, Telangana, India b Professor, KITS Warangal, Telangana, India
Abstract
The optimization methods can be applied to design large scale mechanical systems. One popular method for increasing the efficiency of large scale design optimization problem is a hierarchical decomposition of the problem into a number of sub problems each with its own objective function, constraints, and design variables. Then each and every sub problem is optimized by using a powerful tool known as genetic algorithm, to get required optimum solutions. The solutions obtained from sub problems are used to get the required global optimum solution of a large and complex design problem. A well known speed reducer is considered as an example to illustrate the application of the above parallel processing method. Speed reducer includes gear, pinion, two shafts, four bearings enclosed in housing. The design objective is to minimize the overall volume, where 7 variables and 25 constraints are considered. © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of 7th International Conference of Materials Processing and Characterization.
Keywords: Hierarchical decomposition; Genetic algorithms; Design optimization; Parallel processing
* Corresponding author. Tel.: 9642684331; E-mail address: [email protected] 2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of 7th International Conference of Materials Processing and Characterization.
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1. Introduction Design may require the coordination of many design and analysis procedures. Due to the decreasing life cycle of products, it is important to reduce the time and cost of product development. It has been identified that 7080 percent of final production cost is determined during the design stage. For this reason we have to optimize the design process. The optimization of design process reduces the complexity of design and improves the product quality and reduces the development time and cost. Optimization is the process of maximizing or minimizing desired objective function while satisfying the prevailing constraints. It is the act of obtaining the best results under the given circumstances. 1.1. Hierarchical Decomposition Practical design optimization problem often include discrete design variables, or are non linear and thus yield a large number of unwelcome local optimum solutions with in the design spaces. Design problems for machine products are generally hierarchically expressed [1]. The optimization methods can be applied to design large scale mechanical systems having hierarchical systems [2, 3]. With conventional product optimization methods, however, it is difficult to concurrently optimize all design variables of portions with in such hierarchical structures of large scale problems. The optimization of such large and complex systems is very difficult because for hierarchical structures many local optima always arise [4]. One popular method for increasing the efficiency of large scale design optimization problem is to divide it into smaller sub-problems and optimize each and every sub-problem to get global optimum solution easily [2]. A well known speed reducer [5] is considered as an example to illustrate the application of the above parallel processing method. 1.2. Hierarchical Parallel Processing A large scale machine system often has a general hierarchical structure. One way to reduce the complexity of large scale design problem is, dividing that problem into number of sub problems by applying decomposition process. The concept of decomposition simplifies the design process. During decomposition process we prepare a matrix which will show the relationship between the design variables and the constraints, and we decompose that matrix into number of separable sub matrices. Each and every sub matrix is useful for the formulation of a sub problem, each with its own constraints, decision variables and objective function. The hierarchical structure of large scale problem is as shown in Fig 1.
Middle-Level Problem 1
Bottom-Level Problem 1, 1 Bottom-Level Problem 1, 2
Top-Level Problem Middle-Level Problem 2
Fig 1: Hierarchical Parallel Processing
Bottom-Level Problem 2, 1 Bottom-Level Problem 2, 2
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After forming the hierarchical structure of the optimization problem, all sub problems are processed in parallel to get required optimum solution within less time, effort and with less complexity. 2. Flow Chart of GA Process The GA process shown in Figure.2 begins with the generation of random population. The number of design variables, substring length, maximum and minimum bound, population size, number of generations, design data and GA parameters is first decided and then the value of objective function is computed and store the best ones. Next the new population is generated by applying the selection, crossover and mutation operations. By this new population i.e., next generation is obtained. After this the values of the objective function is computed and store the best ones. Apply the above three operators on the new generation. The cycle is repeated up to number of generations equal to ‘n’. After the completion of above cycle the best string is obtained, which optimize the value of objective function.
Fig.2 Flow chart of GA Process
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3. Design Optimization of Speed Reducer We optimize the design of speed reducer [1,3,5] by dividing the problem into number of sub problems. If we consider the physical structure of a simple speed reducer, its main components are gear pinion pair mounted on shaft 1 & 2 respectively. Each shaft is supported by one bearing at each end. The system includes gear, pinion, shafts and bearings enclosed in a housing as shown in Figure 3.
Fig 3: Speed Reducer
3.1. Design Variables The design of the speed reducer is considered with the face width (X1), the module of the teeth (X2), the number of teeth on pinion (X3), the length of the first shaft between bearings (X4), the length of the second shaft between bearings (X5), diameter of the first shaft (X6), and the diameter of the second shaft (X7). The main problem is divided into two sub problems as shown in figures 4 and 5.
Fig 4: Sub-Problem 1
Fig 5 Sub-Problem 2
3.2 Design Constraints g1: Upper bound on the bending stress of the gear tooth; g2: Upper bound on the contact stress of the gear tooth; g3-g4: Upper bound on the transverse deflection of the shaft; g5-g6: Upper bounds on the stress of the shaft. g7-g23: Dimensional restrictions based on space; g24-g25: Design condition for the shaft;
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3.3 Objective function The mathematical representation of the problem is given by the following equations. The equations (2) and (3) is the mathematical representation of sub-problem 1 and sub-problem 2, where as the equation (4) is the main problem which is the summation of the equations (1), (2), and (3). The main optimization problem is written as Minimize f(X) = f1+f2+f3. Where f1= 0.7854X1X22 (3.3333X32+14.9334X3-43.0934) 2
3
(1) 2
f2= -1.508X1X6 +7.477 X6 +0.7854 X4 X6
(2)
f3=-1.508 X1 X72+7.477 X73+0.7854 X5 X72
(3)
Minimize f(x) = f1+f2+f3 = 0.7854X1X22(3.3333X32+14.9334X3-43.0934) -1.508X1X62+7.477 X63+0.7854 X4 X62 -1.508 X1 X72+7.477 X73+0.7854 X5 X72
(4)
4. Hierarchical Representation of Problem: If the concept of decomposition to the above problem is applied, the hierarchical structure is as shown in the Figure.6
Minimize F=f(x) Variables: X1, X2 X3 Sub to constraints: g1, g2, g7, g8, g9, g10, g11, g12, g13, g14, g15
Minimize F1=f2(x) Variables: X4, X6 Sub to constraints: g3, g5, g20, g21, g24, g16, g17
Minimize F1=f3(x) Variables: X5, X7 Sub to constraints: g4, g6, g18, g19, g22, g23, g25
Fig 6: Hierarchical decomposition of Speed Reducer
Sub Problem 1: Minimize f2= -1.508X1X62+7.477 X63+0.7854 X4 X62 Subjected to constraints: g24, g16, g17, g20, g21, g5, g3. Variables involved are X4, X6
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Sub Problem 2: Minimize f3=-1.508 X1 X72+7.477 X73+0.7854 X5 X72 Subjected to constraints: g25, g22, g23, g18, g19, g4, g6 Variables involved are X5, X7 Top Level Problem: Minimize f(x) = 0.7854X1X22(3.3333X32+14.9334X3-43.0934) -1.508X1X62+7.477 X63+0.7854 X4 X62 -1.508 X1 X72+7.477 X73+0.7854 X5 X72 Subjected to constraints: g1, g2, g7, g8, g9, g10, g11, g12, g13, g14, g15 Variables involved are X1, X2, X3 (Where the values of X4, X5, X6, X7 are known) By this optimization technique these sub-problems are processed in parallel to get optimum solution using genetic algorithms. 5. Results: Face width (x1) =35mm Teeth module (x2) = 7mm No. of Teeth (x3) =17 The distance between bearings 1 (x4) = 73.1mm The distance between bearings 2 (x5) = 77.2mm The diameter of shaft 1 (x6) = 33.6mm The diameter of shaft 2 (x7) = 52.9mm Volume = 2992.556396×103 mm3 6. Conclusions: The hierarchical parallel processing method presented here makes it possible to design the large scale and complex machine systems very easily and the following conclusions are made: 1. By this optimization technique the large scale problems can be decomposed into number of sub problems, which can be processed in parallel to get required global optimum solution. 2. The problems without any specialized knowledge and experience can be optimized. 3. As genetic algorithms are used, enhancement in efficiency of searching the optimum solution can be achieved. 4. The computational time can be reduced. 5. The complexity in developing the programs can also be reduced.
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Appendix A. Design Constraints: g1: 27 X1-1 X2-2 X3-1<=1 g2: 397.5 X1-1 X2-2 X3-2<=1 g3: 1.93 X2-1 X3-1 X43 X6-4<=1 g4: 1.93X2-1 X3-1 X53 X7-4<=1 g5: A1/B1<=1100 A1= {(0.745 X4 /X2 X3)2+16900000)}0.5 B1= 0.1 X6-1 g6: A2/B2<=850 A2= {(0.745 X6 /X2 X3)2+157500000)}0.5 B1= 0.1 X7-1 g7: X2 X3<=40 g8: 5<= (X1 /X2) <=12: g9 g10: 2.6<= X1 <=3.6: g11 g12: 0.7<= X2 <=0.8: g13 g14: 17<= X3 <=28: g15 g16: 7.3<= X4 <=8.3: g17 g18: 7.3<= X5 <=8.3: g19 g20: 2.9<= X6 <=3.9: g21 g22: 5.0<= X7 <=5.5: g23 g24: (1.5X6+1.9) X4-1 g25: (1.1X7+1.9) X5-1 References [1] S. Azarm and W. C. Li, June 1989, Vol 111/259. “Multi level design optimization using global monotonicity analysis”,Journal of mechanisms, transmissions, and automation in design.
[2] Masataka yoshimura and Izui, March 2004, Vol. 126/217. “Hierarchical parallel processes of genetic algorithms for design optimization of large scale problems”, Journal of Mechanical Design.
[3] R.S. Krishnamachary & Papalambros, December 1997, vol. 119/440. “Optimal Hierarchical decomposition synthesis using integer programming”, Journal of Mechanical Design.
[4] K. Tai and T.H. Chee, December 2000, vol. 122, “Design of Structures and compliant mechanisms by evolutionary optimization of morphological representations of topology”, Transactions of the ASME.
[5] Ming-Hua Lin, Jung-Fa Tsai, Nian-Ze Hu, and Shu-Chuan Chang, Volume 2013, Article ID 419043, 7 pages. “Design Optimization of a Speed Reducer Using Deterministic Techniques”, Mathematical Problems in Engineering.
ScienceDirect Materials Today: Proceedings 5 (2018) 5117–5123
www.materialstoday.com/proceedings
ICMPC 2017
Hierarchical parallel processing for design optimization - a case study Basani Satish* P.S.S. Murthy a K. Eswaraiahb *,a
Assistant Professor, KITS Warangal, Telangana, India b Professor, KITS Warangal, Telangana, India
Abstract
The optimization methods can be applied to design large scale mechanical systems. One popular method for increasing the efficiency of large scale design optimization problem is a hierarchical decomposition of the problem into a number of sub problems each with its own objective function, constraints, and design variables. Then each and every sub problem is optimized by using a powerful tool known as genetic algorithm, to get required optimum solutions. The solutions obtained from sub problems are used to get the required global optimum solution of a large and complex design problem. A well known speed reducer is considered as an example to illustrate the application of the above parallel processing method. Speed reducer includes gear, pinion, two shafts, four bearings enclosed in housing. The design objective is to minimize the overall volume, where 7 variables and 25 constraints are considered. © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of 7th International Conference of Materials Processing and Characterization.
Keywords: Hierarchical decomposition; Genetic algorithms; Design optimization; Parallel processing
* Corresponding author. Tel.: 9642684331; E-mail address: [email protected] 2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of 7th International Conference of Materials Processing and Characterization.
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Basani Satish et al./ Materials Today: Proceedings 5 (2018) 5117–5123
1. Introduction Design may require the coordination of many design and analysis procedures. Due to the decreasing life cycle of products, it is important to reduce the time and cost of product development. It has been identified that 7080 percent of final production cost is determined during the design stage. For this reason we have to optimize the design process. The optimization of design process reduces the complexity of design and improves the product quality and reduces the development time and cost. Optimization is the process of maximizing or minimizing desired objective function while satisfying the prevailing constraints. It is the act of obtaining the best results under the given circumstances. 1.1. Hierarchical Decomposition Practical design optimization problem often include discrete design variables, or are non linear and thus yield a large number of unwelcome local optimum solutions with in the design spaces. Design problems for machine products are generally hierarchically expressed [1]. The optimization methods can be applied to design large scale mechanical systems having hierarchical systems [2, 3]. With conventional product optimization methods, however, it is difficult to concurrently optimize all design variables of portions with in such hierarchical structures of large scale problems. The optimization of such large and complex systems is very difficult because for hierarchical structures many local optima always arise [4]. One popular method for increasing the efficiency of large scale design optimization problem is to divide it into smaller sub-problems and optimize each and every sub-problem to get global optimum solution easily [2]. A well known speed reducer [5] is considered as an example to illustrate the application of the above parallel processing method. 1.2. Hierarchical Parallel Processing A large scale machine system often has a general hierarchical structure. One way to reduce the complexity of large scale design problem is, dividing that problem into number of sub problems by applying decomposition process. The concept of decomposition simplifies the design process. During decomposition process we prepare a matrix which will show the relationship between the design variables and the constraints, and we decompose that matrix into number of separable sub matrices. Each and every sub matrix is useful for the formulation of a sub problem, each with its own constraints, decision variables and objective function. The hierarchical structure of large scale problem is as shown in Fig 1.
Middle-Level Problem 1
Bottom-Level Problem 1, 1 Bottom-Level Problem 1, 2
Top-Level Problem Middle-Level Problem 2
Fig 1: Hierarchical Parallel Processing
Bottom-Level Problem 2, 1 Bottom-Level Problem 2, 2
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After forming the hierarchical structure of the optimization problem, all sub problems are processed in parallel to get required optimum solution within less time, effort and with less complexity. 2. Flow Chart of GA Process The GA process shown in Figure.2 begins with the generation of random population. The number of design variables, substring length, maximum and minimum bound, population size, number of generations, design data and GA parameters is first decided and then the value of objective function is computed and store the best ones. Next the new population is generated by applying the selection, crossover and mutation operations. By this new population i.e., next generation is obtained. After this the values of the objective function is computed and store the best ones. Apply the above three operators on the new generation. The cycle is repeated up to number of generations equal to ‘n’. After the completion of above cycle the best string is obtained, which optimize the value of objective function.
Fig.2 Flow chart of GA Process
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3. Design Optimization of Speed Reducer We optimize the design of speed reducer [1,3,5] by dividing the problem into number of sub problems. If we consider the physical structure of a simple speed reducer, its main components are gear pinion pair mounted on shaft 1 & 2 respectively. Each shaft is supported by one bearing at each end. The system includes gear, pinion, shafts and bearings enclosed in a housing as shown in Figure 3.
Fig 3: Speed Reducer
3.1. Design Variables The design of the speed reducer is considered with the face width (X1), the module of the teeth (X2), the number of teeth on pinion (X3), the length of the first shaft between bearings (X4), the length of the second shaft between bearings (X5), diameter of the first shaft (X6), and the diameter of the second shaft (X7). The main problem is divided into two sub problems as shown in figures 4 and 5.
Fig 4: Sub-Problem 1
Fig 5 Sub-Problem 2
3.2 Design Constraints g1: Upper bound on the bending stress of the gear tooth; g2: Upper bound on the contact stress of the gear tooth; g3-g4: Upper bound on the transverse deflection of the shaft; g5-g6: Upper bounds on the stress of the shaft. g7-g23: Dimensional restrictions based on space; g24-g25: Design condition for the shaft;
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3.3 Objective function The mathematical representation of the problem is given by the following equations. The equations (2) and (3) is the mathematical representation of sub-problem 1 and sub-problem 2, where as the equation (4) is the main problem which is the summation of the equations (1), (2), and (3). The main optimization problem is written as Minimize f(X) = f1+f2+f3. Where f1= 0.7854X1X22 (3.3333X32+14.9334X3-43.0934) 2
3
(1) 2
f2= -1.508X1X6 +7.477 X6 +0.7854 X4 X6
(2)
f3=-1.508 X1 X72+7.477 X73+0.7854 X5 X72
(3)
Minimize f(x) = f1+f2+f3 = 0.7854X1X22(3.3333X32+14.9334X3-43.0934) -1.508X1X62+7.477 X63+0.7854 X4 X62 -1.508 X1 X72+7.477 X73+0.7854 X5 X72
(4)
4. Hierarchical Representation of Problem: If the concept of decomposition to the above problem is applied, the hierarchical structure is as shown in the Figure.6
Minimize F=f(x) Variables: X1, X2 X3 Sub to constraints: g1, g2, g7, g8, g9, g10, g11, g12, g13, g14, g15
Minimize F1=f2(x) Variables: X4, X6 Sub to constraints: g3, g5, g20, g21, g24, g16, g17
Minimize F1=f3(x) Variables: X5, X7 Sub to constraints: g4, g6, g18, g19, g22, g23, g25
Fig 6: Hierarchical decomposition of Speed Reducer
Sub Problem 1: Minimize f2= -1.508X1X62+7.477 X63+0.7854 X4 X62 Subjected to constraints: g24, g16, g17, g20, g21, g5, g3. Variables involved are X4, X6
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Sub Problem 2: Minimize f3=-1.508 X1 X72+7.477 X73+0.7854 X5 X72 Subjected to constraints: g25, g22, g23, g18, g19, g4, g6 Variables involved are X5, X7 Top Level Problem: Minimize f(x) = 0.7854X1X22(3.3333X32+14.9334X3-43.0934) -1.508X1X62+7.477 X63+0.7854 X4 X62 -1.508 X1 X72+7.477 X73+0.7854 X5 X72 Subjected to constraints: g1, g2, g7, g8, g9, g10, g11, g12, g13, g14, g15 Variables involved are X1, X2, X3 (Where the values of X4, X5, X6, X7 are known) By this optimization technique these sub-problems are processed in parallel to get optimum solution using genetic algorithms. 5. Results: Face width (x1) =35mm Teeth module (x2) = 7mm No. of Teeth (x3) =17 The distance between bearings 1 (x4) = 73.1mm The distance between bearings 2 (x5) = 77.2mm The diameter of shaft 1 (x6) = 33.6mm The diameter of shaft 2 (x7) = 52.9mm Volume = 2992.556396×103 mm3 6. Conclusions: The hierarchical parallel processing method presented here makes it possible to design the large scale and complex machine systems very easily and the following conclusions are made: 1. By this optimization technique the large scale problems can be decomposed into number of sub problems, which can be processed in parallel to get required global optimum solution. 2. The problems without any specialized knowledge and experience can be optimized. 3. As genetic algorithms are used, enhancement in efficiency of searching the optimum solution can be achieved. 4. The computational time can be reduced. 5. The complexity in developing the programs can also be reduced.
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Appendix A. Design Constraints: g1: 27 X1-1 X2-2 X3-1<=1 g2: 397.5 X1-1 X2-2 X3-2<=1 g3: 1.93 X2-1 X3-1 X43 X6-4<=1 g4: 1.93X2-1 X3-1 X53 X7-4<=1 g5: A1/B1<=1100 A1= {(0.745 X4 /X2 X3)2+16900000)}0.5 B1= 0.1 X6-1 g6: A2/B2<=850 A2= {(0.745 X6 /X2 X3)2+157500000)}0.5 B1= 0.1 X7-1 g7: X2 X3<=40 g8: 5<= (X1 /X2) <=12: g9 g10: 2.6<= X1 <=3.6: g11 g12: 0.7<= X2 <=0.8: g13 g14: 17<= X3 <=28: g15 g16: 7.3<= X4 <=8.3: g17 g18: 7.3<= X5 <=8.3: g19 g20: 2.9<= X6 <=3.9: g21 g22: 5.0<= X7 <=5.5: g23 g24: (1.5X6+1.9) X4-1 g25: (1.1X7+1.9) X5-1 References [1] S. Azarm and W. C. Li, June 1989, Vol 111/259. “Multi level design optimization using global monotonicity analysis”,Journal of mechanisms, transmissions, and automation in design.
[2] Masataka yoshimura and Izui, March 2004, Vol. 126/217. “Hierarchical parallel processes of genetic algorithms for design optimization of large scale problems”, Journal of Mechanical Design.
[3] R.S. Krishnamachary & Papalambros, December 1997, vol. 119/440. “Optimal Hierarchical decomposition synthesis using integer programming”, Journal of Mechanical Design.
[4] K. Tai and T.H. Chee, December 2000, vol. 122, “Design of Structures and compliant mechanisms by evolutionary optimization of morphological representations of topology”, Transactions of the ASME.
[5] Ming-Hua Lin, Jung-Fa Tsai, Nian-Ze Hu, and Shu-Chuan Chang, Volume 2013, Article ID 419043, 7 pages. “Design Optimization of a Speed Reducer Using Deterministic Techniques”, Mathematical Problems in Engineering.