University of Liège
Faculty of Applied Sciences
Aerospace and Mechanical Department
Contributions to handle maximum size constraints in density-based topology optimization
Eduardo Fernández* , Maxime Collet* , Simon Bauduin* , Etienne
Lemaire$ , Pierre Duysinx*
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Eduardo FERNANDEZ - ULG YIC 2017 2
Maximum size in the design process
INTRODUCTION
Optimized Cantilever beam under local damage scenarios , Jansen et al (2014).
Cantilever beam with maximum size and minimum gap control.
Design domain Minimum size Maximum size Indirect desired properties
Aesthetic reasons
Guest’s approach (2009)
Condition satisfied if
Guest (2009) Constraint:
Guest JK, Imposing maximum length scale in topology optimization, Struct Multidisc Optim, 37, 463-473, 2009.
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Eduardo FERNANDEZ - ULG YIC 2017 4
Maximum size formulation
Local Constraint : Compliance minimization with
maximum size constraints
Compliance minimization with maximum size constraints
Matrix with neighbors information
Voids vector with penalization
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Constraints distribution and aggregation
Iteration of the MBB beam with
maximum size constraints Satisfied
Violated
Aggregation functions: p-mean & p-norm
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Eduardo FERNANDEZ - ULG YIC 2017 8
Aggregation functions: p-mean & p-norm
Aggregation
Aggregation of maximum size constraints
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Eduardo FERNANDEZ - ULG YIC 2017 10
Aggregation of maximum size constraints
P-mean curves
Modification of the sensitivities to improve convergence
Addressed problems in linear elasticity
Compliance Minimization
Heat conduction
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Eduardo FERNANDEZ - ULG YIC 2017 12
MBB-beam with different mesh sizes
6144 Elements 55296 Elements P-mean P-norm Sept. 14, 2017
New test region
Circular
region Annular
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Eduardo FERNANDEZ - ULG YIC 2017 14
New test region
Circular
Annular Circular
Annular
Compliance minimization with maximum size Continuation procedure : SIMP Min imu m Siz e m siz e Time/Iter = Ref. Design Domain
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Eduardo FERNANDEZ - ULG
• Maximum size constraints where aggregated
using p-mean and p-norm functions.
• With p-norm is not possible to get
mesh-indenpent solutions.
• The ring-shaped region reduces the amount of
holes in the design
• The aggregation reduces the computation time
compared to a highly constrained problem and allows to solve 3D problems in a reasonable time.
YIC 2017 16
Concluding remarks
Acknowledgement
