Chapter 6 Conclusions and suggestions
6.2 Suggestions
The suggestions of this thesis are as follows:
(1) Although much research has been devoted on the family of MMA approximation method, little research has been done on the artificial bounds which effects the convexity of approximate function. Therefore, in order to make the approximation more accuracy, the strategy for the determination of U and i L can be studied in i the future.
(2) Several studies have proposed some strategies for the determination of move limit for each design variable. In this thesis, all design variables have the same strategy of move limit. Therefore, in order to make the process more stable, the strategy of move limit can be studied in the future.
(3) The approximation method presented in this thesis is only applied to represent displacement, stress, and natural frequency. Therefore, other types of structural behaviors can be tested in the future.
REFERENCES
[1] A. G. M. Mitchell, “The limits of economy of material in framed structures,”
Philosophical Magazine, series 6, Vol. 8, No. 47, pp. 589-597, 1904.
[2] L. A. Schmit and B. Farshi, “Some approximation concepts for structural synthesis,” AIAA Journal, Vol. 12, No. 5, pp. 692-699, 1974.
[3] A. K. Noor and H. E. Lowder, “Structural reanalysis via a mixed method,”
Computers & Structures, Vol. 5, No. 1, pp. 9-12, 1975.
[4] R. T. Haftka and C. P. Shore, “Approximation methods for combined thermal/structural design,” NASA Technical Paper 1428, 1979.
[5] J. H. Starnes and R. T. Haftka, “Preliminary design of composite wings for buckling, strength, and displacement constraints,” Journal of Aircraft, Vol. 16, No.
8, pp. 564-570, 1979.
[6] C. Fleury and V. Braibant, “Structural optimization: a new dual method using mixed variables,” International Journal for Numerical Methods in Engineering, Vol. 23, No. 3, pp. 409-428, 1986.
[7] K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” International Journal for Numerical Methods in Engineering, Vol.
24, No. 2, pp. 359-373, 1987.
[8] C. Fleury, “First and second order convex approximation strategies in structural optimization,” Structural Optimization, Vol. 1, No. 1, pp. 3-10, 1989.
[9] T. Jiang, “A first order method of moving asymptotes for structural optimization,”
WIT Transactions on The Built Environment, Vol. 14, 1970.
[10] R. T. Haftka, J. A. Nachlas, L. T. Watson, T. Rizzo, and R. Desai, “Two-point constraint approximation in structural optimization,” Computer Methods in Applied
Mechanics and Engineering, Vol. 60, No. 3, pp. 289-301, 1987.
[11] G. M. Fadel, M. F. Riley, and J. M. Barthelemy, “Two point exponential approximation method for structural optimization,” Structural Optimization, Vol. 2, No. 2, pp. 117-124, 1990.
[12] L. P. Wang and R. V. Grandhi, “Efficient safety index calculation for structural reliability analysis,” Computers & structures, Vol. 52, No. 1, pp. 103-111, 1994.
[13] L. P. Wang and R. V. Grandhi, “Improved two-point function approximations for design optimization,” AIAA Journal, Vol. 33, No. 9, pp. 1720-1727, 1995.
[14] J. A. Snyman and N. Stander, “New successive approximation method for optimum structural design,” AIAA Journal, Vol. 32, No. 6, pp. 1310-1315, 1994.
[15] W. H. Zhang and C. Fleury, “A modification of convex approximation methods for structural optimization,” Computers & Structures, Vol. 64, No. 1-4, pp. 89-95, 1997.
[16] S. Xu and R. V. Grandhi, “Effective two-point function approximation for design optimization,” AIAA Journal, Vol. 36, No. 12, pp. 2269-2275, 1998.
[17] M. S. Kim, J. R. Kim, J. Y. Jeon, and D. H. Choi, “Efficient mechanical system optimization using two-point diagonal quadratic approximation in the nonlinear intervening variable space,” KSME International Journal, Vol. 15, No. 9, pp.
1257-1267, 2001.
[18] A. A. Groenwold, L. F. P. Etman, J. A. Snyman, and J. E. Rooda, “Incomplete series expansion for function approximation,” Structural and Multidisciplinary Optimization, Vol. 34, No. 1, pp. 21-40, 2007.
[19] J. R. Kim and D. H. Choi, “Enhanced two-point diagonal quadratic approximation methods for design optimization,” Computer Methods in Applied Mechanics and Engineering, Vol. 197, No. 6, pp. 846-856, 2008.
[20] C. Fleury and W. Zhang, “Selection of appropriate approximation schemes in multi-disciplinary engineering optimization,” Advances in Engineering Software, Vol. 31, No. 6, pp. 385-389, 1995.
[21] C. Zillober, “A globally convergent version of the method of moving asymptotes,”
Structural Optimization, Vol. 6, No. 3, pp. 166-174, 1993.
[22] M. Bruyneel, P. Duysinx, and C. Fleury, “A family of MMA approximations for structural optimization,” Structural and Multidisciplinary Optimization, Vol. 24, No. 4, pp. 263-276, 2002.
[23] L. Li and K. Khandelwal, “An adaptive quadratic approximation for structural and topology optimization,” Computers & Structures, Vol. 151, pp. 130-147, 2015.
[24] 邱求慧，結構最佳設計保守近似法之改良，臺大機械工程學研究所博士論文，
2000。
[25] 陳建元，兩點近似法於結構最佳化設計之應用，臺大機械工程學研究所碩士 論文，2002。
[26] 張耀仁，結構最佳化設計之準二次兩點保守近似法，臺大機械工程學研究所 碩士論文，2007。
[27] 陳奕璋，結構最佳化之指數移動漸進線近似法，臺大機械工程學研究所碩士 論文，2010。
[28] 陳俊傑，結構最佳化之新式混合兩點近似法，臺大機械工程學研究所碩士論 文，2012。
[29] 江奇鴻，結構最佳化之加強兩點指數近似法，臺大機械工程學研究所碩士論 文，2013。
[30] 王維德，結構最佳化之準二次移動漸近線近似法，臺大機械工程學研究所碩
士論文，2016。
[31] 柯浩宇，結構最佳化之兩點分段適應近似法，臺大機械工程學研究所碩士論 文，2016。
[32] C. Zhong and K. Saito, “Equivalent drop test modification for determination of cushioning performance,” Journal of Packaging Science & Technology, Vol. 19, No. 2, pp. 123-135, 2010.
Appendix A: User manual of integrated optimization program
The optimization program for this thesis is developed in Microsoft Visual Studio 2015 with integrating AutoCAD 2002, ANSYS 15.0, approximation theory, and mathematical optimization method. The detailed procedure of this program is introduced in Section 2.6, and the operation of this program is described in this appendix. Also, the 3-bar truss optimization is taken as example.
A-1 Included files
The necessary C++ files for this program are listed in Table A-1.
Table A-1 Necessary files for optimization program
File name Description
main.cpp Source file for structural optimization problem
Approximation.h Header file for the interface of an approximation class consists of the operations (base class)
ConstraintElimination.h Header file for the interface of a constraint elimination class consists of the operations (base class)
SearchDirection.h Header file for the interface of a search direction class consists of the operations (base class)
RangeFinder.h Header file for the interface of a range finder class consists of the operations (base class)
MinimumLocator.h Header file for the interface of a minimum locator class consists of the operations (base class)
Approximation.cpp Resource file for definition of approximation class member function
ConstraintElimination.cpp Resource file for definition of constraint elimination class member function
SearchDirection.cpp Resource file for definition of search direction class member function
RangeFinder.cpp Resource file for definition of range finder class member function
MinimumLocator.cpp Resource file for definition of minimum locator class member function
.cpp and .h files of derived classes derived
from base classes
Header file and resource file of derived classes derived from approximation, constraint elimination, search direction, range finder and minimum locator classes
The necessary files for finite element analysis in ANSYS are listed in Table A-2.
Table A-2 Necessary files for finite element analysis
File name Description
TS03.mac Macro file for settings of the finite element analysis (File name depends on the name of optimization problem)
If CAD model is needed in optimization problem, the necessary files for creating CAD model in AutoCAD are listed in Table A-3.
Table A-3 Necessary files for creating CAD model
File name Description
TS03.lsp LISP file for creating CAD model (File name depends on the name of optimization problem)
util.lsp LISP file of self-defined functions for creating CAD models
A-2 Program setting
Important program settings are shown as follows.
(a) Setting the path of AutoCAD and ANSYS (main.cpp) // path of AutoCAD
acad_bat("D:\\AutoCAD 2002\\acad.exe");
// path of ANSYS
ansys_bat("D:\\ANSYS Inc\\v150\\ANSYS\\bin\\winx64\\ansys150.exe");
(b) Defining the optimization problem (main.cpp)
// set the number of the total functions and design variables respectively DesignPoint point0(1 + 3, 2); // 1 is number of objective functions, // 3 is number of constraint functions, // 2 is number of design variables // initial design vector
point0 = { 2,1 };
// number of behavior constraints
Constraint design_constraint(point0.num_func - 1);
// upper bound and lower bound of each behavior constraint
design_constraint.make_behavior({ { -15.0,20.0 },{ -15.0,20.0 },{ -15.0,20.0 } });
(c) Selecting of the algorithms (main.cpp)
// Choose the approximation and optimization methods
Approximation* pApprox1 = new ConvexLinear; // approximation for the first iteration Approximation* pApprox2 = new TAMMA; // approximation for other iterations ConstraintElimination* pCE = new ExteriorPenalty3(10, 1); // method for optimization SearchDirection* pSD = new BFGS; // method for search direction
RangeFinder* pRF = new RangeFinder1; // method for range finder
MinimumLocator* pML = new GoldenSection; // method for minimum locator
(d) Integration with AutoCAD and ANSYS (main.cpp) bool run_cad = false; // Whether modeling by AutoCAD if (run_cad)
autocad_fileout(point0, "drawplate.lsp", "plate"); // Lisp file name and the drawing // function name for AutoCAD ansys_fileout(point0, "TS03.mac"); // the macro file name for ANSYS
(e) Constraint treatment (main.cpp) for (int i = 0; i < point0.num_func - 1; i++) {
if (point0.value.component[i + 1] >= 0.0)
treat.value.component[i + 1] /=design_constraint.behavior[i].second.component[0];
else
treat.value.component[i + 1] /= design_constraint.behavior[i].first.component[0];
treat.value[i + 1] -= 1.0;
for (int j = 0; j < point0.num_var; j++) { if (point0.value.component[i + 1] >= 0.0)
treat.sensitivity[i + 1][j] /= design_constraint.behavior[i].second.component[0];
else
treat.sensitivity[i + 1][j] /= design_constraint.behavior[i].first.component[0];
} }
(f) Move limit (main.cpp) // move limit definition
for (int i = 0; i < point0.num_var; i++)
move_limit[i] = { point0.variable.component[i] / 3.0 , point0.variable.component[i] * 3.0 };
(g) Defining the explicit objective function and gradient (Approximation.h)
#define explicit_objective {\
#define explicit_objective_gradient {\
if (index_func_nf==0){\
grad(0) = 20 * sqrt(2);\
grad(1) = 10.0;\
return *this;\
}\
}
A-3 Operation step
Operation steps of the program are listed in the following and shown in Fig. A-1.
(1) Open the folder “2_ThreeBarTruss”.
(2) Open “ThreeBarTruss.vcxproj”.
(3) Open “main.cpp”.
(4) Set the program as in Section A-2.
(5) Press “Ctrl+F5”.
(6) The result is recorded in “result.txt”.
Fig. A-1 Operation steps of optimization program
Appendix B: User manual of integrated sensitivity analysis program
The sensitivity analysis program for this thesis is developed in AutoCAD 2002 with integrating Excel 2013, and ANSYS 15.0. The detailed procedure of this program is introduced in Section 2.7, and the operation of this program is described in this appendix. Also, the gantry structure is taken as example.
B-1 Included files
The necessary files of this program are listed in Table B-1.
Table B-1 Necessary files for sensitivity analysis program
File name Description
acad.pgp The command aliases used by AutoCAD
XLAuto2002.arx ARX file of self-defined functions for reading and writing Excel files
Gantry.lsp LISP file for creating CAD model (File name depends on the name of structure)
util.lsp LISP file of self-defined functions for creating CAD models
Gantry.xlsx Excel file with defined design parameters (File name depends on the name of structure)
SW.bat Batch file for calling ANSYS to do self-weight analysis Modal.bat Batch file for calling ANSYS to do modal analysis
SW.mac Macro file for settings of the self-weight analysis Modal.mac Macro file for settings of the modal analysis
B-2 Program setting
Important program settings are shown as follows.
(a) Setting the path of start location for AutoCAD 2002
(1) Right click on the icon of AutoCAD 2002, and select “Properties”.
(2) Start in: “C:\Users\fong\Desktop\EX_Gantry\”.
(b) Setting the path of working directory and the name of Excel (Gantry.lsp) (setq f_path "C:/Users/fong/Desktop/EX_Gantry/"
fn_XLarx (strcat f_path "XLAuto2002.arx") fn_XL (strcat f_path "Gantry.xlsx")
)
(c) Setting the path of ANSYS (SW.bat and Modal.bat) SW.bat:
@echo off
"D:\ANSYS Inc\v150\ANSYS\bin\winx64\ansys150.exe"
-j ansys_opt -b -i SW.mac -o ANSYS.out Modal.bat:
@echo off
"D:\ANSYS Inc\v150\ANSYS\bin\winx64\ansys150.exe"
-j ansys_opt -b -i Modal.mac -o ANSYS.out
B-4 Operation step
Operation steps of the program are listed in the following and shown in Fig. B-1.
(1) Open the folder “EX_Gantry”.
(2) Open “Gantry.xlsx” to set the original design parameters and the rate of change for each parameter. Select variables from the parameters, and set “Active?” to 1.
Otherwise, set to 0. Also, select the structure analysis to be performed, and set
“Active?” to 1. Otherwise, set to 0.
(3) Open AutoCAD and load the “Gantry.lsp”. Enter “gantry” function in AutoCAD command window to perform parametric modeling. Moreover, enter “ansys”
function to perform structure analysis, and enter “sen” function to perform sensitivity analysis.
(4) The result is recorded in “Gantry.xlsx”.
Fig. B-1 Operation steps of sensitivity analysis program
Vitae
Name: Fong-Yuan Chen 陳峰遠 Date of birth: June 02, 1993
Phone: +886-2-2967-2479, +886-928-081-736
E-mail: [email protected], [email protected] Educational background:
1999-2005 New Taipei Municipal Banqiao Elementary School 新北市立板橋國民小學
2005-2008 New Taipei Municipal Zhongshan Junior High School 新北市立中山國民中學
2008-2011 Department of Drafting Technique, Taipei Municipal Daan Vocational High School
臺北市立大安高級工業職業學校製圖科
2011-2015 Department of Mechanical Engineering, National Taiwan University of Science and Technology
國立台灣科技大學機械工程系
2015-2017 Institute of Mechanical Engineering, National Taiwan University 國立臺灣大學機械工程學研究所