Master universitario di II livello interateneo in Ingegneria del Veicolo XVII edizione A.A. 2017/2018 ________________________________________________________________________ Introduction to optimization techniques ___________________________________________________________________ Università degli studi di Modena e Reggio Emilia
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 objective y=f( x ) variables experiment or ( x ) simulation constraints minimize 𝑔(𝒚 c ( x ) ≥ O 𝒚∈𝐸 subject to 𝑑(𝒚 ≥ 0 optimization algorithm 5
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 Solid and shell elements (material distribution) Topometry Element thickness Shell elements (thickness distribution) Topography Element offset Shell elements (bead patterns) Size Component thickness Shell elements (thickness distribution) Shape Morphing weight Solid and shell elements factors 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 = 𝑦 𝑗 ∈ 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: 𝜍 𝑗 𝑦 𝑗 = 𝑦 𝑗 𝜍 ∗ 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: 𝑞 𝐹 ∗ 𝐹 𝑗 𝑦 𝑗 = 𝑦 𝑗 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 objective 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 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. 11
DETAILED DISCUSSION: OPTISTRUCT 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 optimization. Unnecessary high values of the MINDIM parameter could lead to suboptimal solutions. 12
AN EXAMPLE: DESIGN OF A STEEL PISTON The aluminium piston Advantages • Low density • Easy to manufacture Disadvantages • High thermal deformations • High blow-by • Low strength at high temperatures 13
AN EXAMPLE: DESIGN OF A STEEL PISTON The steel piston Advantages • Low thermal deformation • Timing advance • Low blow-by • Turbocharge • Low dead volume at the top land • Detonation • High strength at high temperatures Disadvantages ? 14
AN EXAMPLE: DESIGN OF A STEEL PISTON The steel piston Disadvantages • High density Lightness Very thin features Strength Manufacturing difficulties 15
AN EXAMPLE: DESIGN OF A STEEL PISTON Manufacture technique Selective Laser Melting • Complex geometries • Easy productive process 16
AN EXAMPLE: DESIGN OF A STEEL PISTON Manufacture technique Additive Design Chance to improve Manufacturing freedom piston design 17
AN EXAMPLE: DESIGN OF A STEEL PISTON Structural analysis: post-processing, TDCC 18
AN EXAMPLE: DESIGN OF A STEEL PISTON Topology Optimization: non-design space Crown Piston rings Skirt Pin boss 19
AN EXAMPLE: DESIGN OF A STEEL PISTON Topology Optimization: optimization domain 20
AN EXAMPLE: DESIGN OF A STEEL PISTON Topology Optimization 3 different load cases = 3 different optimization processes Top Dead Centre during Combustion (TDCC) 21
AN EXAMPLE: DESIGN OF A STEEL PISTON Topology Optimization: design constraints TDCC and TDCI PT 22
AN EXAMPLE: DESIGN OF A STEEL PISTON Redesign 23
AN EXAMPLE: DESIGN OF A STEEL PISTON Conclusions Form the aluminium piston… …. to the steel piston 24
REFERENCES 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 optimization 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. 25
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