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/ 45Pierre DUYSINX
University of LiègeFaculty of Applied Sciences
Aerospace and Mechanical Department
Topology Optimization
of Components Fabricated by Additive
Manufacturing
Eduardo Fernández Sánchez*, Simon Bauduin*, Pablo Alarcón*, Ioanna Koutla*, Maxime Collet*,+, Etienne Lemaire$, Pierre Duysinx*.
Orsay, Dec. 13, 2018 1
* University of Liège, Aerospace and Mechanical Engineering Department, Liège, Belgium.$ Samtech – Siemens, Liège, Belgium.
+ SAFRAN AERO BOOSTER, Liège, Belgium
I3D-Metal
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/ 45Pierre DUYSINX
• Introduction
• Topology optimization
• Maximum Size & Minimum Gap
• Overhanging Angle
• Conclusion and Perspectives
Orsay, Dec. 13, 2018I3D-Metal 2
Table of Content
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 3
Motivation : Topology Optimization
Variables 𝝆𝒆 , 0 ≤ 𝜌𝑒 ≤ 1
Max. Performance
s.t. DesignConstraints
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 4
Motivation : Topology Optimization
Variables 𝝆𝒆 , 0 ≤ 𝜌𝑒 ≤ 1
Max. Performance
s.t. DesignConstraints
Topology optimization: a new design tool that offers innovative design ideas.
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 5
Motivation : Topology Optimization & Additive Manufacturing
Courtesy by SAMTECH S.A.
Additive ManufacturingTopology optimization
Mass : - 40%Max. Stress : - 40%
Tomlin & Meyer (2011)
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 6
Motivation : Topology Optimization & Additive Manufacturing
• Large enthusiasm in research and industrial community for the new developments
of the innovative, all-digital approach combing additive manufacturing and
topology optimization.
• Topology optimization is
a powerful (re-) design
tool to suggest new
concepts taking
advantages of AM
• Additive Manufacturing
offers more freedom but
novel manufacturing
constraintsTomlin & Meyer (2011)
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 7
Additive Manufacturing: EBM, SLM
Focus on two Additive Manufacturing
Technologies: Laser Beam Melting (LBM)
and Electron Beam Melting (EBM):
– Large choice of materials (Steel,Aluminum, Titanium…)
– Good mechanical properties
– Various widths of layer deposition(20-100μm)
– Good precision
– Capability to realize high geometricalcomplexities.
https://www.manufacturingguide.com/en/electron-beam-melting-ebm
Electron Beam Melting
https://3dprint.com/5505/3d-printed-steel-arup/
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 8
Additive Manufacturing – Current Limitations
In FRED Project assessment of constraints caused by metallic additive manufacturing (LBM, EBM):
• Minimum and maximum width of walls
• Minimum size of canals (powder evacuation & insertion of tools)
• Overhanging angle
• Part orientation
• Surface state
• Post machining of working surfaces
• Dimensions precision
• No closed cavities
• Thermal constraints
• Support structure needed and removed…
Meunier (2015) Meunier (2015)
4 mm
http://www.qualifiedrapidproducts.com/?p=2193 https://hvm.catapult.org.uk
https://3dprint.com/146259/swanson-aerotech-metal-am/
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 9
Topology Optimization must adapt to A.M.
Leary et al. (2014)
Lazarov & Wang (2017)
Stiffest design for a given volume
Problem including maximum size
constraints
Small Cavities Narrow channels
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 10
Topology Optimization : Formulation
Topology optimization: a creativity tool for engineers
Zhang et al. 1993
Modification of geometry model parameter
A better topological layout (Duysinx, 1996)
Modification of the component nature
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/ 45Pierre DUYSINX
• Introduction
• Topology optimization
• Maximum Size & Minimum Gap
• Overhanging Angle
Orsay, Dec. 13, 2018I3D-Metal 11
Table of Content
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 12
An optimal material distribution
• Concept of optimal material distribution (Bendsoe et Kikuchi, 1988)
• Bitmap representation of the structure
• Functional definition of the design
– The best components has to fulfill a
service function subject to ressources
(and specifications)
• We are clearly in preliminary design
approach
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 13
Topology Optimization : Formulation
• Definition of a design domain that contains the
structure
• Discretization of the domain into Finite Elements
to evaluate the mechanical or physical responses
• Applications of boundary conditions and load
cases
• Discretization of the material distribution: constant
density per element) = the design variables
• Optimization algorithm to solve the maximization
problem: prediction of new improved design
characterized by a set of density variable map
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 14
A First Example: Genesis of a Structure
Max. Stiffness
s.t. VolumeConstraint
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 15
Topology Optimization : Formulation
• To circumvent the 0/1 nature of the problem, one can introduce a continuous interpolation by considering porous microstructures whose density of ranging from void to full solid
• The simplest model: a power law model of the physical properties in terms of the relative density (SIMP)
• Unfortunately the problem remains ill-posed from a mathematical point of view. The solution depends on the mesh scale
• Restricting the solutions space
• Perimeter
• Filtering the solution (image)
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/ 45Pierre DUYSINX
• Controlling the perimeter enables a
control of the geometry:
– Control of the shape complexity
– Guarantee of the solution stability
Orsay, Dec. 13, 2018I3D-Metal 16
Control of the solution: perimeter control
Duysinx, DCAMM (1996), Zhang & Duysinx, C&S (2003)
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/ 45Pierre DUYSINX
• To avoid the mesh dependency of the solutions, Sigmund (1994, 1997) has
proposed to use filtering techniques inspired from image processing to
eliminate fast varying solutions
• Density filter
Orsay, Dec. 13, 2018I3D-Metal 17
Control of the solution: Filtering techniques
The convolution factors are limited by a given characteristic ball around the design point
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/ 45Pierre DUYSINX
• To obtain 0/1 solutions , Guest et al. (2014) modifies the density filter with a Heaviside function such that if xe>0, the Heaviside gives a physical value of the density equal to ‘1’ and if the xe=0, the Heaviside gives a density ‘0’.
Smooth approximation:
– For b→ 0, the filter gives the original filter
– For b→ infinity, the function reproduces the max operator, that is the density becomes 1 if there is any element in the neighborhood that is non zero.
Orsay, Dec. 13, 2018I3D-Metal 18
Control of the solution: Filtering techniques
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 19
Control of the solution: Filtering techniques
• Three field topology optimizationscheme (Wang et al., 2011):• → Design field• → filtered field• → a physical field
Filtering:
Heaviside:
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 20
Application: Aircraft engine pylon
• Minimization of compliance
14 load charges
– Static linear FE with SAMCEF
– SAMCEF TOPOL
– CONLIN solver
– Continuous interpolation law.
Courtesy of Samtech and Airbus Industries
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/ 45Pierre DUYSINX
• Introduction
• Topology optimization
• Maximum Size & Minimum Gap
• Overhanging Angle
• Conclusion and Perspectives
Orsay, Dec. 13, 2018I3D-Metal 21
Table of Content
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 22
Maximum Size Control
Condition satisfied if
Guest (2009)Constraint: amount of void
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 23
Maximum Size Formulation
Local Constraint :
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 24
Constraints evaluation and aggregation
• Linear Filtering :
• Heaviside Projection:
• Voids vector :
• Local constraints :
• Const. Aggregation :
≈ 2,3 % of the time spent in one iteration
(pmean)
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 25
Local constraints distribution
2 Iterations of the MBB beam with
maximum size constraints
Fernandez-Sanchez et al., submited to SMO
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 26
Interpretation of the parameter in Maximum Size
Guest K, (2009)Problematic
condition
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 27
Minimum gap constraint based on the Maximum Size
Constraint to impose the
Maximum size
Constraint to move away the solid members:
Minimum gap
In 3D
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 28
Minimum gap constraint based on the Maximum Size
Constraint to impose the
Maximum size
Constraint to move away the solid members:
Minimum gap
Compliance minimization with maximum size and
minimum gap
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 29
2D Test Cases
Minimum gap constraint reduces the appearance of small cavities in the optimized designs.
MBB beam
Compliant Mechanism
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 30
3D Test CaseHeaviside Projection +Maximum Size + Minimum Gap
V*=40%
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/ 45Pierre DUYSINX
• Introduction
• Topology optimization
• Maximum Size & Minimum Gap
• Overhanging Angle
• Conclusion and Perspectives
Orsay, Dec. 13, 2018I3D-Metal 31
Table of Content
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/ 45Pierre DUYSINX
• Two approaches:– Oriented filter can provide designs free of minimizing overhanging angles
(Gaynor and Guest, 2016):
➔ New projection approach using orthotropic weighting function
– Mimicking the layer deposition and prevent the unsupported layers (Bauduin, 2016)
Orsay, Dec. 13, 2018I3D-Metal 32
Overhanging angle
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/ 45Pierre DUYSINX
• Revisit filtering techniques using orthotropic weighting functions
• Superformula by Gelis (2003)
Orsay, Dec. 13, 2018I3D-Metal 33
Overhanging angle using GELIS formula
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/ 45Pierre DUYSINX
• MBB Beam problem
Orsay, Dec. 13, 2018I3D-Metal 34
Overhanging angle using GELIS formula
No over-angle control: 5 violations
Over-angle control: 4 violations Over-angle control: 3 violations
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/ 45Pierre DUYSINX
• Overhanging parts are not supported by the powder
➔ high deflections under their self weight
• Given a current design model, generate a sequence of models obtained by
applying masks at different stages of the fabrication process.
• Boundary conditions of slave models are related to fabrication conditions
Orsay, Dec. 13, 2018I3D-Metal 35
Overhanging angle using self weight constraint
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/ 45Pierre DUYSINX
• MBB Beam example
Orsay, Dec. 13, 2018I3D-Metal 36
Overhanging angle using self weight constraint
4 associated self weight models
2 associated self weight models
1 associated self weight model
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/ 45Pierre DUYSINX Orsay, Dec. 13, 2018I3D-Metal 37
Overhanging angle - comparison
Compliance (Nm)
Overhanging constraint violation
CPU time (hour)
No overhanging control 194.9260 5 1
Superformula 1 184.9758 4 1
Superformula 2 299.7548 2 2
Selfweight : 1 layer 186.7574 5 21
Selfweight : 2 layers 189.0491 3 32
Selfweight : 4 layers 223.1763 2 44
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/ 45Pierre DUYSINX
• Super formula filtering approach :– Yields different optimized solution with reduced overhanging angle
issues.
– Nearly no increase of CPU time.
– Unprintable without additional support structures
• Self weight approach– Promotes self supporting structures.
– Represent more precisely the real process.
– Provides a finer control over the overhanging angle and direct printability is increased.
– Computational time increases significantly.
Orsay, Dec. 13, 2018I3D-Metal 38
Overhanging angle - comparison
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/ 45Pierre DUYSINX
• Introduction
• Topology optimization
• Maximum Size & Minimum Gap
• Overhanging Angle
• Conclusion and Perspectives
Orsay, Dec. 13, 2018I3D-Metal 39
Table of Content
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/ 45Pierre DUYSINX
• Topology optimization enables to suggest innovative designs that
take advantage of enhanced capabilities of additive manufacturing
technologies
• Generating designs ready to be printed require accounting for the
specific manufacturing constraints of AM
• Satisfactory solutions for
• Minimum size
• Maximum size
• Minimum gap
Orsay, Dec. 13, 2018I3D-Metal 40
Conclusions
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/ 45Pierre DUYSINX
• Some additive manufacturing constraints can be formulated as
variants the maximum size constraints proposed by Guest (2009).
• Encouraging progress for
– Overhanging angles
– Overhanging constraints are not strictly satisfied
– CPU time dramatically increases once adjoin models are used
Orsay, Dec. 13, 2018I3D-Metal 41
Conclusions
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/ 45Pierre DUYSINX
• New research directions:
– Thermal associated problems
– Optimal supporting of components (structural and thermal criteria)
– Infill materials: self supported structures
– Power removal: No closed cavities…
• Thermomechanical analysis to predict the presence of residual stresses and microcracks
• Improving the computational issues– From 2D to 3D
– Metamodeling technique
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Conclusions & Perspectives
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• Considering microstructures with intermediate densities?
• From technological reasons, sometimes difficult to fabricate a continuous range of cellular materials →discrete valued problem
• Formulate the optimization problem as a discrete ‘n’ materials selection problem
– Void / Solid
– Some porous materials from a catalog
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Conclusions & Perspectives
Solid
Void
Lattice 1
Lattice 2
Lattice m
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Acknowledgments
Acknowledgement
This work was supported by the AERO+ project funded by the Plan Marshall 4.0 and the Walloon Region of Belgium.
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/ 45Pierre DUYSINX
Pierre DUYSINX
Automotive Engineering
Aerospace and Mechanics Engineering
University of Liège
Allée de la découverte 13A, building B52
4000 Liège Belgium
Email: [email protected]
Tel +32 4 366 9194
Fax +32 4 366 9159
url: www.ingveh.ac.be
www.am.uliege.be
Orsay, Dec. 13, 2018I3D-Metal 45
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