Multidisciplinary System Design Optimization (MSDO)

Transcription

1 Multidisciplinary System Design Optimization (MSDO) Structural Optimization & Design Space Optimization Lecture 18 April 7, 2004 Il Yong Kim 1

2 I. Structural Optimization II. Integrated Structural Optimization III. Design Space Optimization 2

3 Structural Optimization * Definition - An automated synthesis of a mechanical component based on structural properties. - A method that automatically generates a mechanical component design that exhibits optimal structural performance. 3

4 Structural Optimization minimize f ( x) subject to g( x) 0 h() x 0 x S? Typically, FEM is used. BC s are given Loads are given <Q> How to represent the structure? or Which type of design variables to use? <A> (1) Size Optimization (2) Shape Optimization (3) Topology Optimization min compliance s.t. m m C 4

5 Size Optimization Example Beams (2-Dim) minimize f ( x) subject to g( x) 0 h() x 0 x S Design variables (x) x : thickness of each beam f(x) : compliance g(x) : mass h(x) : state equation Number of design variables (ndv) ndv = 5 5

6 Shape Optimization Example B-spline (2-Dim) minimize f ( x) subject to g( x) 0 h() x 0 x S Design variables (x) x : control points of the B-spline (position of control points) f(x) : compliance g(x) : mass h(x) : state equation Number of design variables (ndv) ndv = 8 6

7 Topology Optimization Example Cells (2-Dim) minimize f ( x) subject to g( x) 0 h() x 0 x S Domain shape is determined at the beginning Design variables (x) x : density of each cell (0 1) Number of design variables (ndv) ndv = 27 f(x) : compliance g(x) : mass h(x) : state equation 7

8 Structural Optimization Size optimization Shape optimization Topology optimization - Shape are given Topology - Optimize cross sections - Topology is given - Optimize boundary shape - Optimize topology 8

9 Size Optimization - Simplest method - Changes dimension of the component and cross sections - Applied to the design of truss structures Schmit (1960) - General approach to structural optimization - Coupling FEA & NL math. Programming * Changed - Length of the members - Thickness of the members * Unchanged - Layout of the structure Ndv: 10~100 9

10 Shape Optimization - Design variables control the shape - Size optimization is a special case of shape optimization - Various approaches to represent the shape Zolesio (1981), Haug and Choi et al. (1986) Univ. of Iowa - A general method of shape sensitivity analysis using the material derivative method & adjoint variable method 10 Nodal positions (when the FEM is used) Basis functions n i1 ( xyz,, ) i i B-spline (control points) Radius of a circle Ellipsoid Bezier curve Etc Ndv: 10~100

11 Topology Optimization (1) The evolutionary method Xie and Steven (1993) (2) The homogenization method Bendsoe and Kikuchi (1988) Univ. of Michigan (3) Density approach Yang and Chuang (1994) * cell-based approach Ndv >

12 Topology optimization Homogenization method / Density approach (1) Design variables: density of each cell (2) The constitutive equation is expressed in terms of Young s modulus ij ij ( z ) ( z) d F i z i d z Z adm How to define the relation between the density and Young s modulus?? E 14

13 Topology optimization Homogenization method - Infinitely many micro cells with voids - The porosity of this material is optimized using an optimality criterion procedure - Each material may have different void size and orientation 15

14 Topology optimization Homogenization method - Relationship between density and elastic modulus - Design variables : a 1, a 2, y Y For 2-D elastic problem, Solid part area : (1 aa ) d 11 D11 D D12 D D D D( a, a, ) 1 2 s a 2 1-a 1 1 θ x X * Review papers : Hassani B and Hinton E (1998) 16

15 Topology optimization Density approach Artificial material - Design variable : density Low computational cost Simple in its idea Young s modulus E/E 0 1 Material cost 0 Density 1 E n Eo,

16 I. Structural Optimization II. Integrated Structural Optimization III. Design Space Optimization 19

17 Integrated Structural Optimization Motivation 1. Shape optimization - Small number of design variables - Smooth definite results - Topology remains unchanged (Cannot make holes in the design domain) 2. Topology optimization - Extremely large number of design variables - Non smooth indefinite results - Intermediate densities between void and full material unrealistic Integrate shape optimization and topology optimization 20

18 Integrated Structural Optimization I On CAD-integrated structural topology and design optimization - N. Olhoff, M. P. Bensoe and J. Rasmussen (1991) - Interactive CAD-based structural system for 2-D - Topology optimization CAD Shape optimization - Topology optimization : Homogenization method (HOMOPT) - Shape optimization : CAOS (Computer Aided Optimization of Shapes) - CAD : Commercial CAD system AutoCAD - The designer decides the initial shape for shape optimization interactively with the results of the topology optimization 21

19 Integrated Structural Optimization II Integrated Topology and Shape Optimization in Structural Design -M. Bremicker, M. Chirehdast, N. Kikuch and P. Y. Papalambros, (1991) - 3-phase design process Phase I : Generate information about the optimum topology Phase II : Process and interpret the topology information Phase III : Create a parametric model and apply standard optimization - ISOS (Integrated Structural Optimization System) -Image processing scheme instead of interactive scheme 23

20 Integrated Structural Optimization III Integrating Structural Topology, Shape and Sizing Optimization Methods - E. Hinton, J. Sienz, S. Bulman, S. J. Lee and M. R. Ghasemi (1998) - Interface : Interactive CAD data structure Automatic image processing - Topology optimization : Evolutionary method Homogenization method - Shape optimization : Mathematical programming Genetic Algorithm - Shape optimization / Size optimization for 2-D elastic problems -FIDO-TK 26

21 Integrated Structural Optimization - Communications between SO and TO are not easy. - The designer must provide many control parameters for optimization. The optimal solutions highly depend on the user defined parameters - Computationally very expensive. Less expensive integrated scheme: design space optimization 28

22 I. Structural Optimization II. Integrated Structural Optimization Structural Optimization Software III. Design Space Optimization 29

23 Cosmosworks Shape Optimization Software Optimize parts and assemblies, whether for constraints such as static, thermal, frequency or buckling, or for objectives such as mass, volume or load factors. Initial design Optimal design 30

24 Topology Optimization Software Altair OptiStruct Topology, shape, and size optimization capabilities can be used to design and optimize structures to reduce weight and tune performance. 31

25 Topology Optimization Software ANSYS Static Topology Optimization Dynamic Topology Optimization Electromagnetic Topology Optimization Subproblem Approximation Method First Order Method Design domain 32

26 Topology Optimization Software Design Space Finite Element Analysis Software for engineering designers CAE Templates Input files for ANSYS, NASTRAN, ABAQUS are generated 33

27 Topology Optimization Software MSC. Visual Nastran FEA Elements of lowest stress are removed gradually. 34

28 Optishape Topology Optimization Software - Mass/Rigid Element are available in Topology Optimization. - Any type elements are available in Shape Optimization. 35

29 I. Structural Optimization II. Integrated Structural Optimization III. Design Space Optimization 36

30 Design Space Optimization - The optimum solution depends on the optimization method used. e.g) gradient based search, GA, Simulated annealing, etc - But it also depends on the selection of the design variables. (objective functions and constraints given) <Q> - What is the proper number of design variables for the given problem? - What is the proper layout of the design variables? 37

31 What is the proper length of the chromosome? n= n= n=11 1. Which is the best length for a given design problem? 2. The longer, the better? Design space optimization 38

32 DSO formulation minimize f( x, n) subject to g( x, n) 0 h( x, n) 0 Dimension of the design vector x is to be determined. or The number of design variables is to be determined. Applications - Topology optimization - Plate optimization - Eigenvalue problems - MEMS (MicroElectroMechanical Systems) Design 39

33 Design Space Optimization Problem statement of design space optimization Conventional Optimization minimize f ( x) subject to g( x) 0 h( x) 0 x S ( S is fixed ) S{N, T N, {x 1, x 2,, x N }} N : Number of design units T N : Topology of design units {x 1, x 2,, x N } : Remaining features of design units Design Space Optimization minimize f( x, S) subject to g( x, S) 0 h( x, S) 0 GS ( ) 0 H( S) 0 x S ( S is variable ). Design space improvement is achieved by addition of new design variable. 40

34 What is the proper no. of design variables? Beams B-spline Cells x : thickness of each beam x : control points of the B-spline x : density of each cell ndv = 5 ndv = 8 ndv = 27 ndv =? ndv =? ndv =? Design space optimization 41

35 Design Space Optimization Topology Optimization min f ( ) s.t. g( ) 0 ( is given), Design Space Topology Optimization min s.t. f ( ; S( )) g( ; S( )) 0, S() : Domain shape Load Load BC 42

36 Design Space Optimization Domain Boundary Shape Optimization Initial shape of the domain Boundary variation Domain Interior Topology Optimization Domain shape change (New pixels have been created) It is impossible to obtain sensitivities because addition of a design variable is a discreet process 43

37 Design Continuation Method Procedure START Initial Design Space Initial Design Variables In order to improve design space, increase the number of design variables is increased Structural Analysis Improve Design Space Improve Design Variable Design Variable Sensitivity Analysis d dz dx x x dx 44 N N Design Variable Convergence? END Y Design Space Sensitivity Analysis Design Space Convergence? Y d dn n n n z z dz dn z n z

38 Design Continuation Method Design space change : Generating new design variable Some finite value of x n+1 d dx n n n cells n+1 cells Pivot phase x n+1 (n+1 th cell) Design variables : Feature of cells x x n (n th cell) Continuation path Functional is not continuous when a design variable is created. It is impossible to obtain derivatives. 45

39 Pivot phase Design Continuation Method Equivalent problem Same design status Pivot Phase Same design space (Same no of design variable) N=14 N=25 0 x S Old x S New x S New 46

40 Topology Optimization Sensitivity Analysis Problem statement Relationship between E i and Minimize Subject to F i i z d, ( x) d M o, 0 ( x) 1 E E i o n E i : Intermediate Young s modulus E o : Reference Young s modulus n : exponent Directional variation of the objective fn F i z i, 0, 0 d 47

41 Sensitivity Analysis Topology Optimization (Cont d) State equation (variational form) ij ij ( z ) ( z) d [ ij ( z F i z i d z Z adm ij ij ij ij ijkl kl, 0) ( z) ( z) ( z, 0) ( z) D, 0 ( z)] F i z i, 0 d z Z d adm Assume 0 z, 0 ij ij ij ijkl kl ( z, 0) ( z) d ( z) D, 0 ( z) d 48

42 Topology Optimization (Cont d) Adjoint equation Sensitivity Analysis ij ij ( ) ( ) d i F d Z adm Using the symmetry of the energy bilinear form and combining the eqn,, and i i ij ijkl kl, 0 F z, 0d ( z) D, 0 ( ) Because z and are identical, d ij ijkl kl ', 0 ( z) D, 0 ( z) d 49

43 Numerical Results - Sensitivities Topology Sensitivity analysis Finite Differencing Sensitivity analysis (0) 50

44 Numerical Results - Sensitivities Topology Sensitivity analysis 51 E = N/m 2 = 0.3 b = 0.01 Isoparametric 8-node plane element i th new design variable candidate New Design Variable No. FDM i /b Analytic i /b ( i / i 100)%

45 Sensitivity analysis vs. Finite differencing Number of function evaluation per step Analytical sensitivity analysis (SA): = 2 Finite differencing (FD): 1 + ndv Ex) Topology optimization with ndv = 500 (1) Number of function evaluations per step SA: 2 FD: 501 (2) Computing time (1 minutes per function evaluation, 10 steps) SA: = 20 minutes FD: = 5010 minutes = 3 days 11.5 hours 52

46 Optimization Scheme Computer program structure DSO (Design Space Optimization) [Step 1] Read output file from DVO. [Step 2] Select new design variable candidates. [Step 3] Calculate sensitivities for new design variable candidates. [Step 4] Select new design variables. [Step 5] Improve design space. [Step 6] Write input file for DVO. Number and topology of design variable control (Outer loop) DVO (Design Variable Optimization) Design variable value control (Inner loop) DOT ANSYS 53

47 Problem Statement Design domain Load Minimize Subject to F i z i d ( x) d M 0 ( x) 1 o compliance mass density of cells 54

48 Topology Optimization Results Traditional optimization Design Space Optimization Number of design variables (pixel) : 3016 = 480 Number of design variables (pixel) 3016 =

49 Topology Optimization Results Initial design domain size: 3012, Final domain area: 480 (3016) No of design variables Main iteration (Outer iteration) Number of design variables Objective functin Domain fixed Domain shape optimization No of design variables Objective function history Young s modulus: N/m 2 Poisson s ratio: 0.3 Maximum thickness: m, Minimum thickness: m 56

50 Design Variable Addition & Reduction Procedure START Initial Design Space Initial Design Variables Structural Analysis Improve Design Variable Design Variable Sensitivity Analysis Improve Design Space 2 Increase the number of design variables N Design Variable Convergence? Y Improve Design Space 1 Reduce the number of design variables If i 0, then the i th design variable is removed from the design set. Design Space Sensitivity Analysis 57 N Design Space Convergence? Y END

51 Design Variable Addition & Reduction Bridge problem Solid Symmetric Minimize F i i z d, Subject to ( x) d M o, 0 ( x) 1 Distributed loading Fixed boundary condition Mass constraints: 35% 58

52 Design Variable Addition & Reduction DongJak Bridge H H L 59

53 DSO - discussion * To find the minimum value * Design space optimization ndv = 30 Local minimum obtained global minimum ndv = 100 (No of design variables) ndv = 50 ndv = 80 * Optimization based on sensitivity analysis ndv =

54 DSO based on GA Chromosome length changes as generations progress Chromosome length Stage 61

55 I. Structural Optimization - Size optimization - Shape optimization - Topology optimization Summary II. Integrated Structural Optimization - Integration of size and shape optimization III. Design Space Optimization - The number of design variables is considered as a design variable - Effect of adding new design variables is determined at the pivot phase 62

56 References 1) M. O. Bendsoe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, comp. Meth. Appl. Mech. Engng, Vol. 71, pp , ) R. J. Yang and C. H. Chuang, Optimal topology design using linear programming, Comp. Struct., Vol. 52 (2), pp , ) Y.M. Xie and G. P. Steven, A simple evolutionary procedure for structural optimization, Comput. And Struct., Vol. 49, pp , ) B.Hassani and E. Hinton, A review of homogenization and topology optimization I - homogenization theory for media with periodic structure, Comput. Mech., Vol. 69 (6), pp , ) N. Olhoff et al., On Cad-integrated structural topology and design optimization, Computer Meth. In Appl. Mech. Engng., Vol. 89, pp , ) M. Bremicker, M. Chirehdast, N. Kikuch and P. Y. Papalambros, Integrated Topology and Shape Optimization in Structural Design, Mech. Struct. & Mach., Vol. 19 (4), pp , ) E. Hinton et al., Integrating Structural Topology, Shape and Sizing optimization Methods, Computational Mechanics, New Trends and Applications, Barcelona, Spain, ) E. Ramm, K. Maute and S. Schwarz, Adaptive Topology and Shape Optimization, Computational Mechanics, New Trends and Applications, Barcelona, Spain, ) Il Yong Kim and Byung Man Kwak, Design Space Optimization Using a Numerical Design Continuation Method, International Journal for Numerical Methods in Engineering, Vol. 53, pp ,