TOPOLOGY OPTIMIZATION: AN INTRODUCTION. Pierre DUYSINX LTAS Automotive Engineering Academic year

Transcription

1 TOPOLOGY OPTIMIZATION: AN INTRODUCTION Pierre DUYSINX LTAS Automotive Engineering Academic year

2 LAY-OUT Introduction Topology problem formulation Problem statement Compliance minimization Homogenization Perimeter constraints Sensitivity analysis Optimality criteria Filtering techniques Conclusion 2

3 INTRODUCTION 3

4 What is topology? 4

5 STRUCTURAL & MULTIDISCIPLINARY OPTIMISATION TYPES OF VARIABLES a/ Sizing b/ Shape c/ Topology (d/ Material) TYPES OF OPTIMISATION structural multidisciplinary structural aerodynamics, thermal, manufacturing

6 Topology optimization One generally distinguishes two approaches of topology optimization: Topology optimization of naturally discrete structures (e.g. trusses) Topology optimization of continuum structures (eventually after FE discretization) 6

7 Why topology optimization? CAD approach does not allow topology modifications Zhang et al A better morphology by topology optimization (Duysinx, 1996) 7

8 Problem formulation Abandon CAD model description Optimal topology is given by an optimal material distribution problem Functional definition of the component Ability of the component to fulfill a certain function E.g. withstand given load for some boundary conditions Preliminary design approach! 8

9 TOPOLOGY PROBLEM FORMULATION 9

10 TOPOLOGY OPTIMIZATION PROBLEM Formulated as the optimal distribution of material within a given design domain Discretize the design domain into Finite Element to evaluate structural responses Discretize the optimal material distribution using constant value element by element 10

11 TOPOLOGY OPTIMIZATION Optimal material distribution formulation (Bendsøe & Kikuchi, 1988) Optimal topology without any a priori Fixed mesh Design variables = Local density parameters Continuous interpolation law of effective properties Homogenization SIMP (power law) Solid: i = 1 E i = E 0 i = 0 Design domain where the material properties have to be distributed Void: i = 0 E i = 0 i = 0 Bruyneel & Duysinx (2004) 11 Duysinx (1996)

12 NUMERICAL FORMULATION Finite elements (F.E.) discretisation Density in each element = design variable Discrete variable 0-1 problem Continuous variable problem with penalisation

13 TOPOLOGY OPTIMIZATION PROBLEM Avoid 0/1 problem by considering all possible composite materials with variable density from void (0) to solid (1) Homogenization law of mechanical properties for any volume fraction (density) of materials SIMP model E =µ³e Efficient solution of optimization problem based on sensitivity analysis and mathematical programming algorithms Well-posed ness of problem requires extending the design space (G-closure) or restricting the design space via filtering or perimeter constraints E * p E 0 13

14 DESIGN PROBLEM FORMULATION The fundamental problem of topology optimization deals with the optimal material distribution of continuum structure subject to a single static loading. In addition one can assume that the structure is subject to homogeneous boundary conditions on G u. The principle of virtual work writes 14

15 DESIGN PROBLEM FORMULATION A typical topology optimization problem is to find the best subset of the design domain minimizing the volume or alternatively the mass of the structure, while achieving a given level of functional (mechanical) performance. Following Kohn (1988), the problem is well posed from a mathematical point of view if the mechanical behaviour is sufficiently smooth. Typically one can consider : Compliance (energy norm) A certain norm of the displacement over the domain A limitation of the maximum stress 15

16 DESIGN PROBLEM FORMULATION Compliance performance: The mechanical work of the external loads Using finite element formulation Limitation of a given local stress measure s(x) over a subdomain W 2 excluding some neighborhood of singular points related to some geometrical properties of the domain (reentrant corners) or some applied loads 16

17 DESIGN PROBLEM FORMULATION The average displacement (according to a selected norm) over the domain or a subdomain W 1 excluding some irregular points If one considers the quadratic norm and if the finite element discretization is used, one reads Assuming a lumped approximation of the matrix M, one can the simplified equivalent quadratic norm of the displacement vector 17

18 DESIGN PROBLEM FORMULATION The choice of the compliance is generally the main choice of designers. At equilibrium, the compliance is also the strain energy of the structure, so that compliance is the energy norm of the displacements giving rise to a smooth displacement field over the optimized structure. One can interpret the compliance as the displacement under the loads. For a single local case it is the displacement under the load. 18

19 DESIGN PROBLEM FORMULATION The choice of the compliance is generally the main choice of designers. The sensitivity of compliance is easy to calculate. Being self adjoined, compliance is self adjoined and it does not require the solution of any additional load case. Conversely local stress constraint call for a important amount of additional load case to compute the local sensitivities. One can find analytical results providing the optimal bounds of composites mixture of materials for a given external strain field. The problem is known as the G-closure 19

20 DESIGN PROBLEM FORMULATION Finally the statement of the basic topology problem writes: Alternatively it is equivalent for particular bounds on the volume and the compliance to solve the minimum compliance subject to volume constraint 20

21 PROBLEM FORMULATION For several load cases, worst case approach Where k is load case index, K is the stiffness matrix of FE approximated problem, g k, and q k are the load case and generalized displacement vector for load case k and (x) is the local density V is the volume min 0 ( x) 1 max T k s. t. V ( x) dx V W k g q k 21

22 PROBLEM FORMULATION Maximum strength problem: min ( x) 0 ( x) 1 W eq s. t. s ( ( x), x) s if ( x) 0 l dx eq Where s ( ( x), x) is a strength criterion predicting the end of the elastic linear regime in the material and in the microstructure Investigation of overstressing in the microstructure due to micro porosities based on rank-2 materials * eq ( ) eq / q s s s l 22

23 DESIGN PROBLEM FORMULATION Minimize compliance s.t. Given volume (bounded perimeter) (other constraints) Minimize the maximum of the local failure criteria s.t. Given volume (bounded perimeter) (other constraints) Maximize eigenfrequenclies s.r. Given volume (bounded perimeter) (other constraints)

24 TOPOLOGY OPTIMIZATION Optimal material distributions are not easy problems: Mesh dependency of numerical solutions Natural tendency to generate microstructures and composite solutions Mathematical difficulties: non existence and uniqueness of a solution

25 TOPOLOGY OPTIMIZATION Two solutions: 1/ Homogenisation Method: Extend the design space to all porous composites of variable density Introduce a family of composites of variable density Use homogenization theory to compute their effective properties 2/ Perimeter method: Filter method:

26 TOPOLOGY OPTIMIZATION Two solutions: 1/ Homogenisation Method: Extend the design space to all porous composites of variable density 2/ Perimeter method: Filter method: Restrict solution design space and eliminate chattering designs from design space

27 HOMOGENIZATION METHOD Select one family of microstructures whose geometry is fully parameterized in terms of a set of design variables G closure : optimal microstructure (full relaxation) Suboptimal microstructures (partial relaxation)

28 HOMOGENIZATION METHOD Problem of optimal distribution of porous material The topology problem becomes formally similar to a sizing optimization problem with continuous variables in terms of local density parameters and angles of orthotropy

29 HOMOGENISATION METHOD Each point (E.F.) is made of a porous microstructure whose geometric parameters become the design variables Introduction of optimal or sub-optimal microstructures 29

30 MATERIAL LAWS Layered materials (rank-2 laminates) Rectangular porosities Power law model (SIMP model) Hashin composite spheres Halpin-Tsai model Porous Uni-Directional (fibre reinforced) composite 30

31 HOMOGENIZATION METHOD Investigation of the influence of the selected microstructure upon the optimal topology : 1 orthotropic v.s. isotropy 2 penalization of intermediate properties 31

32 POWER LAW MODEL (SIMP) Simplified model of a microstructured material with a penalisation of intermediate densities Stiffness properties: <E> = x p E 0 < > = x 0 0 x 1, p>1 Strength properties: <s l > = x p s l 0 32

33 POWER LAW MODEL (SIMP) Prescribing immediately a high penalization may introduce some numerical difficulties: Optimization problem becomes difficult to solve because of the sharp variation of material properties close to x=1 Optimization problem includes a lot of local optima and solution procedure may be trapped in one of these. To mitigate these problems, one resorts to the so-called continuation procedure in which p is gradually increased from a small initial value till the desired high penalization. Typically: p(0) = 1.6 p(k+1) := p(k) + Dp after a given number of iterations or when a convergence criteria is OK 33

34 SENSITIVITY ANALYSIS Study of the derivatives of the structure under linear static analysis when discretized by finite elements. The study is carried out for one load case, but it can be easily extended to multiple load cases. Equilibrium equation of the discretized structure: q generalized displacement of the structure K stiffness matrix of the structure discretized into F.E. g generalized load vector consistent with the F.E. discretization 34

35 SENSITIVITY ANALYSIS Let x be the vector of design variables in number n. The differentiation of the equilibrium equation yields the sensitivity of the generalized displacements: The right hand side term is called pseudo load vector Physical interpretation of the pseudo load (Irons): load that is necessary to re-establish the equilibrium when perturbating the design. 35

36 SENSITIVITY ANALYSIS A central issue is the calculation of the derivatives of the stiffness matrix and of the load vector. In some cases the structure of the stiffness matrix makes it easy to have the sensitivity of the matrix with respect to the design variable For thin walled structures (bars, membranes): It comes 36

37 SENSITIVITY ANALYSIS For bending elements, one can write: It comes In topology optimization using SIMP model: The stiffness matrix And its derivatives 37

38 SENSITIVITY OF COMPLIANCE The compliance is defined as the work of the applied load. It is equal to the twice the deformation energy The derivative of the compliance constraint gives: Introducing the value of the derivatives of the generalized displacements: 38

39 SENSITIVITY OF COMPLIANCE The expression of the sensitivity of the compliance writes Generally the load vector derivative is zero (case of no body load), it comes: In fact, we could have got this result also by using the virtual load approach: 39

40 OPTIMALITY CRITERIA One considers the fundamental problem of compliance minimization subject to a volume constraint Compliance is an implicit, non linear function of the density variables. To do so one can use either the virtual work principle make a first order Taylor expansion with respect to the inverse (reciprocal) variables. 40

41 OPTIMALITY CRITERIA Compliance is the strain energy at equilibrium Decompose into FE element contributions Stiffness matrix dependency on density variables 41

42 OPTIMALITY CRITERIA For isostatic structures, for which the load vector remains constant, we have also that the element displacement vector of an inverse function of the design variable: Therefore the following quantity remain constant for isostatic structures. For hyperstatic structures, it does not remain constant but it is reasonable to assume that generally it does change too much from one iteration to another 42

43 OPTIMALITY CRITERIA Therefore one can define: c i is constant for a statically determinate structure. We get an explicit expression which is a good approximation of the compliance in the neighborhood of the current design point: 43

44 OPTIMALITY CRITERIA It can be noticed from now that the same expression can be obtained if we perform a first order Taylor expansion of the compliance with respect to the intermediate variables The Taylor expansion of the compliance writes 44

45 OPTIMALITY CRITERIA One can demonstrate that Let s define the coefficient so that we get exactly the same explicit expression of the compliance 45

46 OPTIMALITY CRITERIA We can demonstrate that this last expression is exactly the same as the first one because, the coefficients c i are exactly the same as the ones we defined previously using the engineering approach based on the decomposition of the strain energy We have 46

47 OPTIMALITY CRITERIA The fundamental compliance minimization problem Using approximation concepts, we get an explicit (sub)-problem that can be solved more efficiently may be at the price of an repeating iteratively the process of generating of creation of the subproblems: 47

48 OPTIMALITY CRITERIA Let introduce the Lagrange multiplier $\lambda$ and let's shape the Lagrange function Stationary conditions (from KKT conditions) It gives 48

49 OPTIMALITY CRITERIA The stationarity conditions gives: If c i >0, the x i variable is called as active. If c i <0, the Langrangian function is monotonic increasing, so that the minimum is The variable is said passive. 49

50 OPTIMALITY CRITERIA To identify the Lagrange variable, one substitutes the x i (l) by its value into the volume constraint that is satisfied as an equality constraint: It comes And with 50

51 OPTIMALITY CRITERIA We have It can be solved analytically or numérically Finally the optimized value is reused into the primal variables 51

52 OPTIMALITY CRITERIA For statically determinate case, the c i remain constant and one structural analysis and reach the optimum. For statically indeterminate case c i is not constant, one has use an iterative scheme. And the iteration scheme 52

53 OPTIMALITY CRITERIA If we remember that It follows The quantity has the meaning of strain energy unit volume 53

54 OPTIMALITY CRITERIA For active variables (i.e. c i >0), For passive variables (i.e. c i <0), This reminds the iteration scheme proposed by Bendsoe and Kikuchi (1988) 54

55 FILTERING MATERIAL DENSITIES To avoid mesh dependency and numerical instabilities like checkerboards patterns, one approach consists in restricting the design space of solutions by forbidding high frequency variations of the density field. Initially Ole Sigmund (1994, 1997) introduced a filter of the densities The convolution factors (weight factors) are given by 55

56 FILTERING MATERIAL DENSITIES Later it was demonstrated that filtering sensitivities should be better replaced by filtering the densities The convolution factors are given by Recently Lazarov et al. demonstrated that the density filter is equivalent to solving the associated Helmotz equation With the following Neuman boundary conditions 56

57 PERIMETER METHOD Ambrosio and Buttazzo, (1993): Adding a perimeter constraint to optimal material problem regularizes the design problem Haber, Jog, and Bendsoe (1996): Perimeter method in topology optimization Efficient numerical solution: Duysinx (1996), Beckers (1997) Extension from 2D problems to 3D and axi-symmetric geometries 57

58 PERIMETER METHOD Eliminate chattering designs with a bounded perimeter Restriction of the design space Similar to the filter method (Sigmund) 58

59 PERIMETER METHOD The structural complexity of the structure is controlled with a bound over the perimeter An efficient numerical strategy has been elaborated to cater with the difficult perimeter constraint 59

60 NUMERICAL SOLUTION OF LARGE SCALE TOPOLOGY PROBLEMS 60

61 NUMERICAL SOLUTION OF TOPOLOGY OPTIMIZATION PROBLEMS Optimal material distribution = very large scale problem Large number of design variables: Number of restrictions: 1 10 (for stiffness problems) (for strength problem with local constraints) Solution approach based on the sequential programming approach and mathematical programming Sequence of convex separable problems based on structural approximations Efficient solution of sub problems based on dual maximization Major reduction of solution time of optimization problem Generalization of problems that can be solved 61

62 SEQUENTIAL CONVEX PROGRAMMING APPROACH Direct solution of the original optimisation problem which is generally non-linear, implicit in the design variables is replaced by a sequence of optimisation sub-problems by using approximations of the responses and using powerful mathematical programming algorithms 62

63 SEQUENTIAL CONVEX PROGRAMMING APPROACH Two basic concepts: Structural approximations replace the implicit problem by an explicit optimisation sub-problem using convex, separable, conservative approximations; e.g. CONLIN, MMA Solution of the convex sub-problems: efficient solution using dual methods algorithms or SQP method. Advantages of SCP: Optimised design reached in a reduced number of iterations: 10 to 20 F.E. analyses Efficiency, robustness, generality, and flexibility, small computation time Large scale problems in terms of number of design constraints and variables 63

64 OPTMISATION ALGORITHMS CONLIN (version 2.1) software Convex approximations: CONLIN, MMA DSQA, MMA 2nd order, Power law Dual solvers Pure dual Primal-dual (e-relaxation technique for stress constraints) 64

65 Example: Genesis of a structure with topology optimization 65

66 COMPLIANCE MINIMIZATION 66

67 Optimization of a maximum stiffness bicycle frame Load cases 1000N y x -1000N -150N 150N Conventional design -5000N -7500N Optimum topology Non conventional design 67

68 Design of a Crash Barrier Pillar (SOLLAC) 68

69 Design of a Crash Barrier Pillar (SOLLAC) 69

70 Topology Optimization of a Parasismic Building (DOMECO) 70

71 3D cantilever beam problem E=100 N/m², n= x 32 x 4 = 2560 F.E.s No perimeter constraint 71

72 3D cantilever beam problem Perimeter = 1400 Perimeter =

73 Sandwich panel optimization Geometry of the sandwich panel reinforcement problem Optimal topology 73

74 Sandwich panel optimization Geometry of the sandwich panel reinforcement problem Optimal topology 74

75 PLATE AND SANDWICH PLATE MODELS (a) (b) (c) (d)

76 USING SIMP MODEL FOR TOPOLOGY OPTIMISATION OF PLATES AND SHELLS is replaced by: PHYSICAL MEANING OF DENSITY VARIABLE:

77 Discrete solution of topology problem Discrete 0/1 solution was studied by M. Beckers (1997) Bridge structure: discrete 0/1 topology solution 77

78 Acknowledgments This work was partly supported by the Walloon Region of Belgium and SKYWIN (Aerospace Pole of Competitiveness of Wallonia), through the project VIRTUALCOMP (contract RW 6293) And by INTERREG 4a ASTE (Automotive Sustainable Training Euregio) 78