Introduction to optimization techniques - - PowerPoint PPT Presentation

___________________________________________________________________ Università degli studi di Modena e Reggio Emilia Master universitario di II livello interateneo in Ingegneria del Veicolo XVII edizione A.A. 2017/2018 ________________________________________________________________________

Introduction to optimization techniques

OUTLINE

Introduction Optimization Structural optimization Topology optimization Detailed discussion An example: design of a steel piston Referneces

2

INTRODUCTION

A different approach to design  The need of a systematic design by means of optimization techniques  Design freedom given by new manufacturing techniques Structural optimization techniques  Topology  Topometry  Topography  Size  Shape

3

OPTIMIZATION

In an optimization problem we seek values of the variables that lead to an optimal value of the function that is to be optimized. We have to define:  The optimization problem: what we want optimize  The variables: we have to describe the problem by parameters  The optimization function: the objective Furthermore, we have to consider the constraints and we have to use an optimization algorithm

4

OPTIMIZATION

5

variables (x) experiment or simulation

• bjective

y=f(x) constraints c(x) ≥ O

• ptimization

algorithm minimize

𝒚∈𝐸

𝑔(𝒚 subject to 𝑑(𝒚 ≥ 0

STRUCTURAL OPTIMIZATION

Variables (design space) FE domain (1 variable per element) Experiment or simulation Finite elements analysis Objective Mass or compliance minimization Design constraints Stiffness, displacements, modal,… Structural boundary conditions Loads, structural constraints,… Optimization algorithm Gradient based

6

STRUCTURAL OPTIMIZATION

Optimization method Variable Applicability Topology Element density (material distribution) Solid and shell elements Topometry Element thickness (thickness distribution) Shell elements Topography Element offset (bead patterns) Shell elements Size Component thickness (thickness distribution) Shell elements Shape Morphing weight factors Solid and shell elements

7

TOPOLOGY OPTIMIZATION

SIMP methods: solid isotropic material with penalization method The main scope of the method is to find the optimum material distribution in a structure. Finite Element analyses are performed assuming as a parameters vector the element-by-element relative material density which is allowed to vary with continuity: 𝒚 = 𝑦𝑗 ∈ 0,1 , ∀𝑗 = 1, … , 𝑂 where N is the number of finite elements in the structure. The density of the i-th element is given by: 𝜍𝑗 𝑦𝑗 = 𝑦𝑗𝜍∗ where ρ∗ is the full density of the material. The material density and the material stiffness are correlated.

8

TOPOLOGY OPTIMIZATION

The SIMP method assumes that the stiffness of the i-th element is given by: 𝐹𝑗 𝑦𝑗 = 𝑦𝑗

𝑞𝐹∗

where E∗ is the full stiffness of the isotropic material. Two parameters control the behaviour of the algorithm: the penalty factor p, and the sensitivity filter r. The penalty factor p ≥ 1 appears in equation. Its role is to make intermediate densities unfavourable in the optimized solution. Setting the filter r ≥ 1 the sensitivity of each element is averaged with the sensitivities of its surrounding elements within a radius equal r times the average mesh size, thus preventing the phenomenon of checkerboarding. The objective of the optimizations is usually the minimization of the mass of the structure for a given displacement target. Larger values of p and r despite reducing the performance of the structure in terms of

• bjective function, make the solution physically meaningful. Thus, it is

relevant to properly tune these parameters, also considering that inappropriate values of the parameters may affect the convergence of the optimization process negatively.

9

DETAILED DISCUSSION: p AND r

10

DETAILED DISCUSSION: MESH

11

Topology optimization results are always linked to the mesh density (element size). Different mesh density leads to a different solution. A finer mesh should lead to a more clear sctruture and to a better definition of its boundaries. Actually a finer mesh leads to a different structural layout.

DETAILED DISCUSSION: OPTISTRUCT

12

p  DISCRETE: the DISCRETE parameter influences the tendency for elements in a topology optimization to converge to a material density of 0 or 1. Low values of this parameter help the solution to converge but results could be too sparse in terms of density distribution. r  MINDIM: the MINDIM parameter specifies the minimum diameter of members formed in a topology

• ptimization. Unnecessary high values of the MINDIM

parameter could lead to suboptimal solutions.

AN EXAMPLE: DESIGN OF A STEEL PISTON

13

The aluminium piston

Advantages

• Low density
• Easy to manufacture

Disadvantages

• High thermal deformations
• High blow-by
• Low strength at high temperatures

AN EXAMPLE: DESIGN OF A STEEL PISTON

14

The steel piston

Advantages

• Low thermal deformation
• Low blow-by
• Low dead volume at the top land
• High strength at high temperatures
• Timing advance
• Turbocharge
• Detonation

Disadvantages ?

AN EXAMPLE: DESIGN OF A STEEL PISTON

15

The steel piston

Disadvantages

• High density

Lightness Strength Very thin features Manufacturing difficulties

AN EXAMPLE: DESIGN OF A STEEL PISTON

16

Selective Laser Melting

• Complex geometries
• Easy productive process

Manufacture technique

AN EXAMPLE: DESIGN OF A STEEL PISTON

17

Manufacture technique

Additive Manufacturing Design freedom Chance to improve piston design

AN EXAMPLE: DESIGN OF A STEEL PISTON

18

Structural analysis: post-processing, TDCC

AN EXAMPLE: DESIGN OF A STEEL PISTON

19

Topology Optimization: non-design space

Crown Piston rings Pin boss Skirt

AN EXAMPLE: DESIGN OF A STEEL PISTON

20

Topology Optimization: optimization domain

AN EXAMPLE: DESIGN OF A STEEL PISTON

21

3 different load cases = 3 different optimization processes

Top Dead Centre during Combustion (TDCC)

Topology Optimization

AN EXAMPLE: DESIGN OF A STEEL PISTON

22

TDCC and TDCI PT Topology Optimization: design constraints

AN EXAMPLE: DESIGN OF A STEEL PISTON

23

Redesign

AN EXAMPLE: DESIGN OF A STEEL PISTON

24

Conclusions

Form the aluminium piston… …. to the steel piston

REFERENCES

25

 M.P. Bendsøe, N. Kikuchi. Generating optimal topologies in structural design using a homogenization

• method. Computer methods in applied mechanics and engineering. 71 (1988) 197-224.

 D. Brackett, I. Ashcroft, R. Hague. Topology optimization for additive manufacturing. Proceedings of the Solid Freeform Fabrication Symposium. (2011) 348–362.  M. Cavazzuti, A. Baldini, E. Bertocchi, D. Costi, E. Torricelli, P. Moruzzi. High performance automotive chassis design: a topology optimization based approach. Struct Multidisc Optim. 44 (2011) 45–56.  M.P. Bendsøe, O. Sigmund. Topology optimization: theory, methods and applications, Springer, Berlin, 2004.  O. Sigmund, J. Petersson. Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization. 16 (1998) 68-75.  K. Zuo, L. Chen, Y. Zhang, J. Yang. Study of key algorithms in topology optimization. Int J Adv ManufTechnol. 32 (2007) 787-796.  S.B. Hu, L.P. Chen, Y.Q Zhang, J. Yang, S.T. Wang. A crossing sensitivity filter for structural topology

• ptimization with chamfering, rounding, and checkboard-free patterns. Struct Multidisc Optim. 37 (2009)

529-540.  A. Dìaz, O. Sigmund. Checkerboard patterns in layout optimization. Structural Optimization. 10 (1995) 40-45.  M. Zhou, Y.K. Shyy, H.L. Thomas. Checkerboard and minimum member size control in topology optimization. Struct Multidisc Optim. 21 (2001) 152–158.