1 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Multidisciplinary System Design Optimization (MSDO) Structural Optimization & Design Space Optimization Lecture 18 April 7, 2004 Il Yong Kim 2 ? Massachusetts Institute of Technology – Dr. Il Yong Kim I. Structural Optimization II. Integrated Structural Optimization III. Design Space Optimization 3 ? Massachusetts Institute of Technology – Dr. Il Yong Kim 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. 4 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Structural Optimization minimize ( ) subject to ( ) 0 () 0 f g h S d ? x x x x BC’s are given Loads are given How to represent the structure? or Which type of design variables to use? <Q> Typically, FEM is used. (1) Size Optimization (2) Shape Optimization (3) Topology Optimization <A> min compliance s.t. m dm C ? 5 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Size Optimization Example f(x) : compliance g(x) : mass h(x) : state equation ? Design variables (x) x : thickness of each beam ? Number of design variables (ndv) ndv = 5 Beams (2-Dim) minimize ( ) subject to ( ) 0 () 0 f g h S d ? x x x x 6 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Shape Optimization Example ? Design variables (x) x : control points of the B-spline (position of control points) ? Number of design variables (ndv) ndv = 8 B-spline (2-Dim) minimize ( ) subject to ( ) 0 () 0 f g h S d ? x x x x f(x) : compliance g(x) : mass h(x) : state equation 7 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Topology Optimization Example ? Design variables (x) x : density of each cell (0 d U d 1) ? Number of design variables (ndv) ndv = 27 Cells (2-Dim) Domain shape is determined at the beginning minimize ( ) subject to ( ) 0 () 0 f g h S d ? x x x x f(x) : compliance g(x) : mass h(x) : state equation 8 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Structural Optimization Size optimization Shape optimization Topology optimization - Topology is given - Optimize boundary shape -Shape Topology - Optimize cross sections are given - Optimize topology 9 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Size Optimization Schmit (1960) - General approach to structural optimization - Coupling FEA & NL math. Programming - Simplest method - Changes dimension of the component and cross sections - Applied to the design of truss structures - Length of the members - Thickness of the members - Layout of the structure * Unchanged * Changed Ndv: 10~100 10 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Shape Optimization 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 - Design variables control the shape - Size optimization is a special case of shape optimization - Various approaches to represent the shape Nodal positions (when the FEM is used) Basis functions B-spline (control points) Radius of a circle Ellipsoid Bezier curve Etc… Ndv: 10~100 1 (, ,) n ii i xyzDI | 12 ? Massachusetts Institute of Technology – Dr. Il Yong Kim (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 Topology Optimization Ndv > 1000 14 ? Massachusetts Institute of Technology – Dr. Il Yong Kim 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 33 :* ?* : adm iiijij ZzdzFdzz )()( HV How to define the relation between the density and Young’s modulus? U E? 15 ? Massachusetts Institute of Technology – Dr. Il Yong Kim 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 16 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Topology optimization ? Homogenization method - Relationship between density and elastic modulus - Design variables : a 1 , a 2 , T 12 11 11 12 11 22 12 22 22 12 66 12 12 For 2-D elastic problem, Solid part area : (1 ) 0 0 00 (, ,) s aa d DD DD D DDaa VH T : :  : -?a o-? °° °° ?? ?? ?? °° °° ˉ?? ?ˉ? 3 * Review papers : Hassani B and Hinton E (1998) 1 1 y Y X x q 1-a 2 1-a 1 18 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Artificial material - Design variable : density Topology optimization ? Density approach 10, dd UU o n EE Density U Young’s modulus E/E 0 M a t er ia l c o s t 01 1 Low computational cost Simple in its idea 19 ? Massachusetts Institute of Technology – Dr. Il Yong Kim I. Structural Optimization II. Integrated Structural Optimization III. Design Space Optimization 20 ? Massachusetts Institute of Technology – Dr. Il Yong Kim 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 21 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? 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 Integrated Structural Optimization I 23 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? 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 Integrated Structural Optimization II 26 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? 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 Integrated Structural Optimization III 28 ? Massachusetts Institute of Technology – Dr. Il Yong Kim - 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 Integrated Structural Optimization 29 ? Massachusetts Institute of Technology – Dr. Il Yong Kim I. Structural Optimization II. Integrated Structural Optimization ? Structural Optimization Software III. Design Space Optimization 30 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Shape Optimization Software ? Cosmosworks 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 31 ? Massachusetts Institute of Technology – Dr. Il Yong Kim 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. 32 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Topology Optimization Software ? ANSYS Design domain Static Topology Optimization Dynamic Topology Optimization Electromagnetic Topology Optimization Subproblem Approximation Method First Order Method 33 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? Design Space Finite Element Analysis Software for engineering designers CAE Templates – Input files for ANSYS, NASTRAN, ABAQUS are generated Topology Optimization Software 34 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? MSC. Visual Nastran FEA Elements of lowest stress are removed gradually. Topology Optimization Software 35 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? Optishape Topology Optimization Software - Mass/Rigid Element are available in Topology Optimization. - Any type elements are available in Shape Optimization. 36 ? Massachusetts Institute of Technology – Dr. Il Yong Kim I. Structural Optimization II. Integrated Structural Optimization III. Design Space Optimization 37 ? Massachusetts Institute of Technology – Dr. Il Yong Kim 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? 38 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design space optimization What is the proper length of the chromosome ? 1 0 1 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 1 n=6 n=8 n=11 1. Which is the best length for a given design problem? 2. The longer, the better? 39 ? Massachusetts Institute of Technology – Dr. Il Yong Kim DSO formulation Dimension of the design vector x is to be determined. or The number of design variables is to be determined. minimize ( , ) subject to ( , ) 0 (,) 0 fn gn hn d x x x Applications - Topology optimization - Plate optimization - Eigenvalue problems - MEMS (MicroElectroMechanical Systems) Design 40 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design Space Optimization ? Problem statement of design space optimization 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 minimize ( ) subject to ( ) 0 () 0 (isfixed) f g h S S d ? x x x x Conventional Optimization minimize ( , ) subject to ( , ) 0 (, ) 0 () 0 () 0 (isvariable). fS gS hS GS HS S S d d ? x x x x Design Space Optimization Design space improvement is achieved by addition of new design variable. 41 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Beams B-spline Cells x : thickness of each beam ndv = 5 x : control points of the B-spline ndv = 8 x : density of each cell ndv = 27 What is the proper no. of design variables? Design space optimization ndv = ? ndv = ? ndv = ? 42 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design Space Optimization BC * Load Load : S(:) : Domain shape ,0))(;K(.s.t ))(;K(min d: : Sg Sf Design Space Topology Optimization given),is( 0)K(.s.t )K(min : dg f Topology Optimization 43 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design Space Optimization Domain Boundary Shape Optimization Domain Interior Topology Optimization Initial shape of the domain Boundary variation Domain shape change (New pixels have been created) It is impossible to obtain sensitivities because addition of a design variable is a discreet process 44 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design Continuation Method ? Procedure ? ? 1 · ¨ ¨ ? § ' ' w <w  ' '< ' '< w <w  w <w < n z znn dn dz zndn d z Y END Design Variable Convergence ? Design Variable Sensitivity Analysis Improve Design Variable Y N Structural Analysis Initial Design Space START Initial Design Variables N Design Space Sensitivity Analysis Design Space Convergence ? Improve Design Space ddz dx x x dx <w<w<  ww In order to improve design space, increase the number of design variables is increased 45 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design Continuation Method ? Design space change : Generating new design variable < xDesign variables : Feature of cells n+1 cells d dx < n n G ' '< n cells x n+1 (n+1 th cell) < x n (n th cell) Continuation path Some finite value of x n+1 Pivot phase Functional is not continuous when a design variable is created. ? It is impossible to obtain derivatives. 46 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design Continuation Method ? Pivot phase Pivot Phase Equivalent problem Same design status Same design space (Same no of design variable) x ? S New x ? S Old x ? S New N=14 N=25 Uo0 47 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? Topology Optimization 1)(0 ,)( , dd d: * 3 3 : * x MdxtoSubject dzFMinimize o ii U U n o i E E U Problem statement Relationship between E i and U E i : Intermediate Young’s modulus E o : Reference Young’s modulus n : exponent 3 * o o * c c dzF ii 0, 0, U U Directional variation of the objective fn ······ M Sensitivity Analysis 48 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Sensitivity Analysis ? Topology Optimization (Cont’d) 33 : o : o : c  : c dzDzdzz klijklijijij )()()()( 0, 0, HHHV U U 33 :* ?* : adm iiijij ZzdzFdzz )()( HV State equation (variational form) 3 3 * o : o oo ?* c : c  c  c adm ii klijklijijijijij ZzdzF dzDzzzzz 0, 0, 0,0, )]()()()()()([ U U UU HHHVHV Assume 0 0, c o U z ······ N 49 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? Topology Optimization (Cont’d) 33 *: ?: : adm iijij ZdFd OOOHOV )()( 33 : o * o o : c  * c c dDzdzF klijklijii )()( 0,0, 0, OHHUU U 3 : o o : c  dzDz klijklij )()(' 0, 0, HHU U Adjoint equation ······O Using the symmetry of the energy bilinear form and combining the eqn M, N , and O Because z and O are identical, Sensitivity Analysis 50 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? Topology Sensitivity analysis ? Finite Differencing Sensitivity analysis (Uo0) Numerical Results - Sensitivities 51 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Numerical Results - Sensitivities E = 210u10 9 N/m 2 Q = 0.3 'b = 0.01 Isoparametric 8-node plane element i th new design variable candidate ? Topology Sensitivity analysis New Design Variable No. FDM '< i /Gb Analytic < i ’/Gb (< i ’ /'< i u100)% 1 -116.6826 -116.6470 99.97 2 -8.5387 -8.5367 99.98 3 -1.6238 -1.6235 99.98 4 -2.3714 -2.3706 99.97 5 -3.6479 -3.6474 99.99 6 -3.9573 -3.9564 99.98 7 -3.4781 -3.4776 99.99 8 -3.2490 -3.2481 99.97 9 -3.0011 -3.0005 99.98 10 -2.7633 -2.7626 99.97 11 -2.5634 -2.5628 99.98 12 -2.2540 -2.2534 99.97 13 -2.2088 -2.2081 99.97 14 -3.5688 -3.5680 99.98 15 -5.8834 -5.8819 99.97 16 -6.7057 -6.7040 99.97 17 -5.8702 -5.8683 99.97 18 -3.7185 -3.7169 99.96 19 -2.0841 -2.0830 99.95 20 -1.1828 -1.1819 99.92 21 -1.6386 -1.6361 99.85 52 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Sensitivity analysis vs. Finite differencing Number of function evaluation per step Analytical sensitivity analysis (SA): 1 + 1 = 2 Finite differencing (FD): 1 + ndv Ex) Topology optimization with ndv = 500 (1) Number of function evaluations per step (2) Computing time (1 minutes per function evaluation, 10 steps) SA: 2 FD: 501 SA: 2 u 1 u 10 = 20 minutes FD: 501 u 1 u 10 = 5010 minutes = 3 days 11.5 hours 53 ? Massachusetts Institute of Technology – Dr. Il Yong Kim 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. DVO (Design Variable Optimization) DOT ANSYS Number and topology of design variable control (Outer loop) Design variable value control (Inner loop) 54 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Problem Statement 1)(0 )( dd d: * 3 3 : * x MdxtoSubject dzFMinimize o ii U U Design domain Load compliance mass density of cells 55 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Topology Optimization Results Traditional optimization Design Space Optimization Number of design variables (pixel) : 30u16 = 480 Number of design variables (pixel) d 30u16 = 480 56 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Topology Optimization Results Number of design variables Objective function history 340 360 380 400 420 440 460 480 500 400 600 800 1000 1200 1400 1600 Domain shape optimization Domain fixed O b je c t iv e f u n c t i n No of design variables 012345678 360 380 400 420 440 460 480 No of de s i gn v a ri a b l e s Main iteration (Outer iteration) Young’s modulus: 210u10 9 N/m 2 Poisson’s ratio: 0.3 Maximum thickness: 0.012 m, Minimum thickness: 0.005 m Initial design domain size: 30u12, Final domain area: 480 (30u16) 57 ? Massachusetts Institute of Technology – Dr. Il Yong Kim ? Procedure Design Variable Addition & Reduction Y START N Y END N Initial Design Space Design Variable Convergence ? Design Variable Sensitivity Analysis Improve Design Variable Structural Analysis Design Space Sensitivity Analysis Initial Design Variables Improve Design Space 1 Reduce the number of design variables Design Space Convergence ? Improve Design Space 2 Increase the number of design variables If U i |0, then the i th design variable is removed from the design set. 58 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design Variable Addition & Reduction ? Bridge problem 1)(0 ,)( , dd d: * 3 3 : * x MdxtoSubject dzFMinimize o ii U U Mass constraints: 35% SymmetricSolid Distributed loading Fixed boundary condition 59 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Design Variable Addition & Reduction ? DongJak Bridge L H H 60 ? Massachusetts Institute of Technology – Dr. Il Yong Kim global minimum * To find the minimum value * Design space optimization DSO - discussion * Optimization based on sensitivity analysis ndv = 100 (No of design variables) ndv = 30 ndv = 50 ndv = 80 ndv = 100 Local minimum obtained 61 ? Massachusetts Institute of Technology – Dr. Il Yong Kim DSO based on GA 1 2 3 4 0 100 200 300 400 500 600 700 Stage C h r o mo s o me le n g t h Chromosome length changes as generations progress 62 ? Massachusetts Institute of Technology – Dr. Il Yong Kim Summary I. Structural Optimization - Size optimization - Shape optimization - Topology optimization 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 63 ? Massachusetts Institute of Technology – Dr. Il Yong Kim 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. 197-224, 1988 2) R. J. Yang and C. H. Chuang, “Optimal topology design using linear programming,” Comp. Struct., Vol. 52 (2), pp.265-275, 1994 3) Y.M. Xie and G. P. Steven, “A simple evolutionary procedure for structural optimization,” Comput. And Struct., Vol. 49, pp. 885-896, 1993 4) 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. 707-717, 1998 5) N. Olhoff et al., “On Cad-integrated structural topology and design optimization,” Computer Meth. In Appl. Mech. Engng., Vol. 89, pp.259-279, 1991 6) 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.551-587, 1991 7) E. Hinton et al., “Integrating Structural Topology, Shape and Sizing optimization Methods,” Computational Mechanics, New Trends and Applications, Barcelona, Spain, 1998. 8) E. Ramm, K. Maute and S. Schwarz, “Adaptive Topology and Shape Optimization,” Computational Mechanics, New Trends and Applications, Barcelona, Spain, 1998 9) 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. 1979-2002, 2002..