Some mathematical challenges in additive manufacturing: … · 2020. 9. 21. · structural...

Some mathematical challenges in additivemanufacturing: modelling, simulation and

optimization

Gregoire ALLAIRE, M. Bihr, B. Bogosel, M. Boissier, C.Dapogny, F. Feppon, A. Ferrer, P. Geoffroy-Donders, M.

Godoy, L. Jakabcin, O. Pantz

CMAP, Ecole Polytechnique

LJLL, Paris, September 18, 2020

G. Allaire, et al. Mathematics of additive manufacturing

Outline

I - Introduction: additive manufacturing

II - What can be achieved ?

III - Models of the manufacturing process

IV - Thermal constraints from additive manufacturing instructural optimization

V - Multi-physics optimization

VI - Optimization of shape and laser path

VII - Optimization of lattice structures

VII - Conclusion and perspectives

A ”hot” topic with a lot of room for new ideas...

Sofia project: Add-Up, Michelin, Safran, ESI, etc.


I - Additive Manufacturing (a.k.a. 3-d printing)

Structures built layer by layer

No topological constraints on the built structures


Additive manufacturing

Various materials: plastic, polymer, metal, ceramic...

We focus on metallic additive manufacturing

Various processes: wire, direct energy deposition (DED), layerby layer...

We focus on powder bed techniques


Metallic additive manufacturing

Metallic powder melted by a laser or an electron beam.


Metallic additive manufacturing


AddUp machine at LURPA (thanks to C. Tournier)


II - What can be achieved ?

Very different from classical techniques (molding, milling)

No topological constraints on the built structures

Very complicated structures: new applications, new designs

Lattice (porous) materials

Functionally graded materials


Examples from LURPA (thanks to C. Tournier)

comb-shaped structure lattice structure


Examples from SAFRAN (thanks to M. Bihr)

Academic example (MBB beam and its support)


Multi-physics applications (F. Feppon, Safran)

New multi-physics designs can be built. For example:

heat exchanger turbine blade with internal cooling


Lattice structures

3-d printing enables structures made of composite materials (calledlattice materials).



Work of Denis Solas, ICMMO, Orsay, Paris-Saclay.

Texture cristallographique et anisotropie

Pilotage de l anisotropie en fabrication additive par SLM

44

44

44

111212

121112

121211

C00000

0C0000

00C000

000CCC

000CCC

000CCC

C

1211

44

CC

C2A

εCσ g :

C11

(GPa)

C12

(GPa)

C44

(GPa)

A E <100>

(GPa)

E <110>

(GPa)

E <111>

(GPa)

Molybdène 457.7 160,9 111,2 0,707 394.4 312,9 292,8

Chrome 350,0 67,8 100,8 0,714 328.0 266,2 250,4

Tungstène 501,0 198,0 151,0 0,997 388,8 388,0 387,7

Aluminium 108,2 61,3 28,5 1,225 63,9 72,6 76,1

Nickel 244,0 158,0 102,0 2,372 119,8 200,6 258,9

Fer alpha 231.4 134.7 116.4 2,407 132,0 220,4 283,3

Cuivre 168,4 121,4 75,4 3,190 66.7 130.3 191,1 Anisotrope

6/26/2020 Séminaire FAPS - Pilotage de l'anisotropie - Denis SOLAS 7

Anisotropie lastiq e d n monocristal





Work of Denis Solas, ICMMO, Orsay, Paris-Saclay.



One can optimize the material properties (anisotropy) bycontrolling the laser path, its speed and power.

PhD thesis of Mathilde Boissier (co-supervised with C.Tournier, LURPA): simultaneous optimization of the path andof the shape

PhD thesis of Abdelhak Touiti (co-supervised with F. Jouve,LJLL): simultaneous optimization of the anisotropy and of theshape


Some failures of additive manufacturing...

Thermal stresses and deformations:


Other failure: overhang limitation

The angle between the structural boundary and the build directionhas an impact on the quality of the processed shape.


Other failure: overhang limitation

Example of a bad 3-d printing due to overhangs.


Other failure: residual stresses

Strong deformation after separation from the baseplate


Constraints in Additive Manufacturing

Constraints are required to avoid failures in the fabrication process

almost horizontal overhang surfaces cannot be built

metal melting → large temperatures → thermal residualstresses and thermal deformations

deformations of the structure may stop the powder depositionsystem

minimal time (or energy) for completion

removing the powder (no closed holes)

bad metallurgical properties (for example, porosities)


What do we need ?

good models at different length-scales

multi-physics models

model reduction and/or HPC

optimization

and new ideas !


III - Models of the manufacturing process

Microscopic model: heat exchange, phase change, fluid mechanicsin the melt pool, granular media for the powder coating...(Spears & Gold, 2016)


Microscopic models

For example: to simulate the ”keyhole” phenomenon.


Macroscopic models

Microscopic models are too computationally intense to be used inoptimization loops.

Macroscopic models ignore small details and a lot of physics...

Two examples

thermo-mechanical model

inherent strain model


Thermo-mechanical model

Heat equation:

ρ∂T

∂t− div(λ∇T ) = Q(t) in (0, tF )× D

T = Tinit on (0, tF )× Γbaseλ∇T · n = −He(T − Tinit) on (0, tF )× (∂D \ Γbase)T (t = 0) = Tinit in D

Thermoelastic quasi-static equation:

− div(σ) = 0 and σ = σel + σth in (0, tF )× D,

σel = Ae(u) and σth = K (T − Tinit) Id,

Material parameters ρ, λ,A,K are different for solid or powder.Source term Q(t) = beam spot, traveling on the upper layer.Weak coupling: first, solve the heat equation, second,thermoelasticity.


Path of the source term Q(t)

path


Inherent strain model

No heat equation !


− div(σ) = 0 and σ = σel + σinh in (0, tF )× D,

σel = Ae(u) and σinh tabulated from experiments


Macroscopic models: layer by layer process

Additive manufacturing involves a layer by layer process.We must take this process into account.


Layer by layer modeling

For a final shape Ω, define intermediate shapes Ωi of increasingheight hi

Ωi = x ∈ Ω such that xd ≤ hi 1 ≤ i ≤ n.


IV - Thermal constraints from additive maufacturing

A structure Ω is optimized for its final use with a constraint on itsbehavior during the manufacturing process.

Two different state equations:

1 for the objective function of the final shape Ω,

2 for the additive manufacturing constraint on eachintermediate shape Ωi .

G. Allaire, L. Jakabcin, Taking into account thermal residual

stresses in topology optimization of structures built by additive

manufacturing, M3AS 28(12), 2313-2366 (2018).


1st state equation for the final shape

For a given applied load f : ΓN → Rd ,

− div (A e(ufinal)) = 0 in Ωufinal = 0 on ΓD(

A e(ufinal))

n = f on ΓN(

A e(ufinal))

n = 0 on Γ

Objective function: compliance

J(Ω) =

∫

ΓN

f · ufinal dx ,


2nd state equation for the intermediate shapes

Heat equation:

ρ∂T

∂t− div(λ∇T ) = Q(t) in (ti−1, ti )× Di

T = Tinit on (ti−1, ti )× Γbaseλ∇T · n = −He(T − Tinit) on (ti−1, ti )× (∂Di \ Γbase)T (t = ti−1) = Tinit in Di \ Di−1


− div(σ) = 0 and σ = σel + σth in (ti−1, ti )× Di ,

σel = Ae(u) and σth = K (T − Tinit) Id,


Notations

1 Each layer i is built between time ti−1 and ti , 1 ≤ i ≤ n.

2 Build chamber D, vertical build direction ed .

3 Intermediate domains Di = x ∈ D such that xd ≤ hi.

4 Final shape Ω and intermediate shapes Ωi = Ω ∩ Di .

5 Mixture Di = Ωi ∪ Pi of solid and powder.


Thermo-mechanical objective

The objective function is

J(Ω) =

n∑

i=1

∫ ti

ti−1

∫

Di

j(u, σ,T ) dx dt

where (u, σ,T ) is the displacement, stress and temperature fieldsfor the intermediate shapes. A constraint on the compliance ofthe final shape is imposed

C (Ω) =

∫

Ω

f · ufinal dx ≤ C (Ωref ),

where ufinal is the elastic displacement for the final shape

− div (A e(ufinal)) = f in Ω

The shape derivative of J(Ω) is computed by an adjoint method.


Adjoint problems

Example for an objective j(u) (without T and σ for simplicity).Elasticity adjoint equation: no ”backward effect”

− div (e(η)) = −j ′(u) in (ti−1, ti )× Di

Adjoint heat equation: backward in time, from i = n to 1,

ρ∂p

∂t+ div(λ∇p) = K divη in (ti−1, ti )× Di

p = 0 on (ti−1, ti )× Γbaseλ∇p · n = −Hep on (ti−1, ti )× (∂Di \ Γbase)p(t = tn) = 0 in Dn

Reversed order of coupling: first, solve the adjoint elasticity,second, the adjoint heat equation.


Test case: minimize the thermal stresses

Half MBB beam (2-d).

Full model with 20 layers and 5 time steps per layer.

Minimize the deviatoric part of the stress σD = 2µe(u)D

J1(Ω) =n

∑

i=1

∫ ti

ti−1

∫

D

|σD |2 dx dt

Constraints on volume (fixed) and compliance.

Initial design: optimal design for compliance minimization.


Initial (top) and final (bottom) shape


Convergence history (thermal stress)

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 20 40 60 80 100 120 140 160 180 200

"objective.data"


Plot of thermal stress√

∫ T

0 |σD |2(x) dt

initial

final


V - Multi-physics optimization

Heat exchangers design (F. Feppon, Safran): two networks of fluidchannels, one hot, one cold in a conducting solid. The heatexchange is maximized with a constraint on the pressure drop.

Two Navier-Stokes equations coupled with the heat equation.


Multi-physics optimization

F. Feppon, G. Allaire, C. Dapogny, P. Jolivet, Topologyoptimization of thermal fluid-structure systems using body-fitted

meshes and parallel computing, J. Comp. Phys., 417 (2020).

Exactly meshed domains with mmg3d, non-mixing constraint for thetwo fluids, FreeFem++ with domain decomposition


VI - Optimization of shape and laser path

PhD thesis of Mathilde Boissier (co-supervised with C. Tournier).Optimize the laser path Γ in the domain Σ to build ΣS :

minΓ

J (Γ) =

∫

Γ

ds such that Cφ(T ) = CM(T ) = 0,

with the temperature T solution of

−∇ · (λ∇T (x)) + β(T (x)− Tinit) = PδΓ(x) inΣ,λ∂nT (x) = 0 on ∂Σ,

and the constraints

Cφ(T ) =

∫

ΣS

[

(Tφ − T (x))+]2

dx CM(T ) =

∫

Σ

[

(T (x)− TM(x))+]2

dx


Optimization of laser path only

Initialization (left), optimal design(right).Temperature: blue (cold), red (hot).


Optimization of laser path only


Coupled optimization of shape and laser path

Half cantilever (for symmetry)


VII - Optimization of lattice structures

Lattice materials are periodic structures, with macroscopicallyvarying parameters of the type

A(

x ,x

ǫ

)

where y → A(x , y) is periodic and x → A(x , y)describes themacroscopic variations. Resurrection of the homogenization theoryin optimal design !


References

Joint work with P. Geoffroy-Donders and O. Pantz:Computers & Mathematics with Applications, 78, 2197-2229(2019).J. Comp. Phys., 401, 108994 (2020).

See also:J. P. Groen and O. Sigmund, Homogenization based topology

optimization for high resolution manufacturable microstructures,

International Journal for Numerical Methods in Engineering,113(8):1148-1163, 2018.

Pionneering paper:O. Pantz and K. Trabelsi, A post-treatment of the homogenization

method for shape optimization, SIAM J. Control Optim.,47(3):1380–1398, 2008.


Modelling issues for lattice materials

For manufacturing reasons, a single microscopic scale isallowed. No sequential laminates !

Choice of the period (square, rectangle, triangle, hexagon...).

Choice of a parametrized cell (rectangular or ellipsoidal hole).

Orientation of the cell is crucial because optimalmicrostructures are known to be anisotropic !

No existence of optimal designs. It can be seen numericallyfor a ”bad” choice of the cell...


Example: rectangular hole in a square cell(Bendsoe-Kikuchi)

m2

m1

Γint

y1y2

Cell parameters: m1,m2 and angle α (applied to the cell).Homogenized properties: A∗(m1,m2, α).Good choice because it is close to the optimal rank-2 laminate.

Remark: the same ideas apply to other geometries.


A three-step approach for optimization

1 Pre-compute (off-line). the homogenized propertiesA∗(m1,m2, α) for all values of the parameters.

2 Apply a simple parametric optimization process to thehomogenized problem with design variables m1,m2, α, varyingin space.

3 Choose a lengthscale ǫ and reconstruct a periodic domainA(

x , xǫ

)

approximating the optimal A∗.(This is the delicate step of the approach !)


Orientation/reconstruction issue

The most delicate point is the combined problem of orientation ofthe microstructure and reconstruction of a macroscopically varyingperiodic lattice: the entire cell is rotated by an angle α.It implies that the periodic grid must be deformed accordingly.

Regular grid (left), orientation field (middle), distorted grid (right).


1st step: pre-computing the homogenized properties

Compute the homogenized properties A∗(m1,m2) for a discretesampling of 0 ≤ m1,m2 ≤ 1 (with fixed 0 orientation).If the cell is rotated by an angle α (in 2− d), then thehomogenized properties are given by

A∗(m1,m2, α) = R(α)TA∗(m1,m2, 0)R(α)

where R(α) is the fourth-order tensor defined by :

∀ξ ∈ Ms2 R(α)ξ = Q(α)T ξQ(α)

where Q(α) is the rotation matrix of angle α.


(A∗0(m))1111 (A∗

0(m))2222

(A∗0(m))1122 (A∗

0(m))1212

Isolines of the entries of the homogenized tensor A∗ and theirgradient (small arrows) depending on m1 (x-axis) and m2 (y -axis).


2nd step: parametric optimization of the homogenizedproblem

The homogenized equation in a box D (containing the latticeshape) is

div σ = 0 in D,σ = A∗(m1,m2, α)e(u) in D,u = 0 on ΓD ,σ · n = g on ΓN ,σ · n = 0 on Γ = ∂D \ (ΓD ∪ ΓN).

We consider compliance minimization with a weight constraint

minm1,m2,α

J(A∗) =

∫

ΓN

g · u ds .


Bridge test case

D

ΓNΓD ΓD


Results for the bridge

Density Cell orientation

m1 m2


Regularity issues for the optimal orientation

Caution: α or α+ π are the same orientation. Singularities appearnear the corners and at the bottom middle...


3rd step: reconstruction of an optimal periodic structure

We computed an optimal homogenized design (with anunderlying modulated periodic structure).

Let us project it to obtain a lattice material !

This is a post-processing step.

We have to choose a lengthscale ε for this projection step.


Projection with orientation α

Main idea (Pantz and Trabelsi): find a map ϕ = (ϕ1, ϕ2) fromD into R

2 which distorts a regular square grid in order to orientateeach square at the optimal angle α.Geometrically (in 2-d), the gradient matrix ∇ϕ should beproportional to the rotation matrix defined by

Q(α) =

(

cosα − sinαsinα cosα

)

.

In other words, there should be a (scalar) dilation field r such that

∇ϕ = erQ(α) in D.

This equation can be satisfied only if α satisfies a conformalitycondition.


Conformality condition

Lemma. Let α be a regular orientation field and D be a simplyconnected domain. There exists a mapping function ϕ and adilatation field r satisfying ∇ϕ = erQ(α) if and only if

∆α = 0 in D.

Notation. For a vector field u = (u1, u2) its curl is defined ascurlu = ∇∧ u = ∂u2

∂x1− ∂u1

∂x2, where ∧ is the 2-d cross product of

vectors.

Proof. curl∇ϕ = 0, thus ∇r =(

− ∂α∂x2

, ∂α∂x1

)T

and ∆α = 0.


Is the orientation angle α harmonic ?

Since α is a stress eigen-direction, it has no reason of beingharmonic !

Even worse, α is not smooth at some places...

Conclusion: we regularize the angle α and make it harmonic by avariational approach.


Regularized orientation α for the bridge case


Angle difference between optimized and regularizedorientations

The regularization occurs mainly in areas where density is close to0 or to 1, i.e. where the homogenized material is almost isotropicand the orientation has no significant impact.


Map |ϕi | (isolines) and the orientation vectors ai (arrows)for the bridge case

|ϕ1| and a2 (left) |ϕ2| and a1 (right)


Projection of a regular grid through the map ϕ for thebridge case


Reconstruction for several ε in the case of the bridge

ε = 0.4 ε = 0.2

ε = 0.1 ε = 0.05


A final post-processing/cleaning of the latticereconstruction

There are disconnected components of the lattice structure tobe removed.

There are too thin members.

A final post-processing is made to cure these defects.


Post-processed structures for several ε

ε = 0.4 ε = 0.2

ε = 0.1 ε = 0.05


Cantilever case


L-beam


3-d generalization

m3

m1

m2

e1

e2

e3

Cell orientation by a direct rotation matrix (ω1, ω2, ω3).

No more conformality property (Liouville theorem).

The map ϕ is computed direction by direction with 3 dilationfields:

∀i ∈ 1, 2, 3 ∇ϕi = eriωi

Cubes are transformed in rectangles...


3-d projection: construction of the cell from Yi(mi)

e1

e2

e3

Y0(m) = ∪1≤i<j≤3 (Yi (m) ∩ Yj(m))


3-d cantilever Yi(mi)


3-d cantilever


3-d bridge and mast


VII - Conclusions and perspectives

No limits for modeling ! Many possible variants...

Further work on support optimizationG. Allaire, B. Bogosel, Optimizing supports for additive

manufacturing, SMO 58(6), 2493-2515 (2018).G. Allaire, M. Bihr, B. Bogosel, Support optimization in

additive manufacturing for geometric and thermo-mechanical

constraints, SMO, 61, pp. 2377-2399 (2020).

Real experiments on building such structures.


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