Topology Optimisation Using the Level Set...

Department of Mechanical Engineering

Topology Optimisation Using the Level Set Method

Dr H Alicia Kim


•  Chris Bowen, Chris Budd, Julian Padget

•  Dave Betts, Peter Dunning, Yi-Zhe Song, Joao Duro

•  Chris Brampton, Phil Browne, Kewei Duan, Caroline Edwards, Peter Giddings, Tom Makin, Vincent Seow, Phil Williams

Researchers


The Level Set Method

•  Front or boundary tracking method •  Commonly used in image processing, moving boundary problems,

multiphase problems, movies, etc… •  Level set topology optimisation since 2000 (Sethian and Wiegmann),

< 20,000 papers (Google Scholar)


Example: Level-Set (3D) Cantilever Beam with Vertical Load


Element-Based Method (SIMP) Cantilever Beam with Vertical Load


The Level Set Method

•  Update by solving discrete Hamilton-Jacobi equation

•  Naturally splits and merges holes

http://en.wikipedia.org/wiki/File:Level_set_method.jpg

φ(x)> 0, x ∈ΩS

φ(x) = 0, x ∈ ΓS

φ(x)< 0, x ∉ΩS

%

&

''

(

''

€

φ x( )

€

φik+1 = φi

k − Δt ∇φik Vn, i


Level Set Topology Optimisation Method

1.  Define the design problem.

2.  Finite element analysis to compute boundary shape sensitivities, .

Dunning and Kim (2011) FEA&D

ς (u) = Aε(u)ε(u)Where A = material property, ε = strain

ς €

φ x( ) > 0

€

φ x( ) = 0

€

φ x( ) < 0

€

ΩS


3.  Level set functions updated using a Hamilton-Jacobi equation

where , , λ = Lagrange multiplier for a constraint and Δt = iterative time step.

4.  λ is determined by Newton’s method. 5.  Gradient, Δφ is computed using the upwind finite

difference scheme and higher order weighted essentially non-oscillatory method (WENO).

6.  Check for convergence and iterate.

φik+1 = φi

k −Δt ∇φik Vn,i

Vn = λ −ς (u)

Level Set Topology Optimisation Method


How does it create a new hole?

Where to create a hole is not difficult, when to create is!


Previous Hole Creation Approaches

Methodology Challenge Start with a random number of holes.

Does not create holes; solutions dependent on the initial design.

Topological derivative to create a hole.

Does not link to shape derivative so optimisation of boundaries and hole creation are unrelated.

Topological derivatives are exclusively used.

Convergence can be slow.

Holes are created at regular intervals.

The selection of the interval is arbitrary and can slow the convergence.

Hole creation criteria based on stress or strain energy.

Heuristic, fundamentally does not link to shape derivative.


Our Approach of Creating a Hole

•  Introduce a secondary level set function, . •  Describes the additional fictitious dimension, bounded by

•  is updated using the Hamilton-Jacobi equation

where à establishes link between shape and topological optimisation.

•  Hole creation only when more optimal than shape optimisation.

φ

−h ≤ φ ≤ +h

φ

φik+1 = φi

k −ΔtVn,iVn = λ −ς (u)

Dunning and Kim (2013) IJNME


160×80, 50% volume, 106 iterations

Cantilevered Beam in 2D


Cantilevered Beam in 2D


Cantilevered Beam

• 45% volume constraint

• 0.5% difference in compliance

• Robust with respect to the initial design


Robust Topology Optimisation


Robust Topology Optimisation

•  Minimisation of expected and variance of performance

where

•  Topology optimisation + Uncertainties = conventional methods are

computationally intractable

J = ηw!

"#

$

%&E C[ ]+ 1−η

w2

!

"#

$

%&Var[C]

E[C]= C(u, f ) P( fi )i=1

n

∏ dff∫

Var[C]= C(u, f )2 P( fi )i=1

n

∏ df −f∫ E[C]2


We have shown:

•  The robust energy functional has an analytical minimum

•  Can be solved by a small set of auxiliary problems

•  Uncertainties in magnitude of loading

–  Expected compliance: (N+1) cases

–  Variance: (N+3) cases

•  Uncertainties in direction of loading (fix = fi cos θi and fiy = fi sin θi)

–  Expected compliance: (N+1) cases

–  Variance: 256 à 23, on-going.

∴E C[ ] = κ ijµiµ ji, j=1

m

∑ + κ iiσ i2

i=1

m

∑

Var[C( f )]= 4 κ i.kκ j,kµiµ jσ k2( )

k=1

m

∑j=1

m

∑i=1

m

∑ + 2 κ i. j2σ i

2σ j2( )

j=1

m

∑i=1

m

∑

C =C1(u, f ,θ )+ w1,iC2,i (u,1,µθi )+w2,i Cx,i (u,1,θx )+Cy,i (u,1,θy )( )!" #$i=i

n

∑


Example: Column under compression

A single load with uncertainty in direction, 20% volume

Deterministic solution

Initial design


Example: Column under compression

E[J]: Det. sol. = 525 Robust sol. = 377 (22%)



σθ=0.1 σθ=0.2 σθ=0.3


Example: Double hook

Deterministic solution E[C] = 2.36

Robust solution E[C] = 1.50

Uncertainties in magnitude: µ = 5.0, σ = 0.5 Uncertainties in direction of loading: µ = 3π/2, σ = 0.25 50% volume


Example: Beam

Deterministic Solution Initial Design µ1=1.0 µ2=1.0 µ3=1.0 σ1=0.5 σ2=0.1 σ3=0.2

160 x 80, 40% volume J = ηw!

"#

$

%&E C[ ]+ 1−η

w2

!

"#

$

%&Var[C]


J = ηw!

"#

$

%&E C[ ]+ 1−η

w2

!

"#

$

%&Var[C]

Example: Beam


Robust Solutions for Varying Weights

J = ηw!

"#

$

%&E C[ ]+ 1−η

w2

!

"#

$

%&Var[C]


Multidisciplinary Topology Optimisation


Aircraft Wing Aero-Structural Optimisation

'x

(a) - bars in main and intermediate groups of optimal design.

all bars of optimal design.

HSCT wing with rigid fuselage.

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Balabanov & Haftka 1996

39

Fig. 10 Aeroelastic divergence: large local deformations

Fig. 11 Optimum material distribution: elements with relative density !/!0 ! 0.50 (light) and !/!0 ! 0.75 (dark)

Table 7 Optimization problem for 3-D aeroelastic layoutoptimization

objective min!mass

subject to

liftL+"g(m"minitial)

Linitial"1.0! 0

lift/drag(L/D)

(L/D)initial"1.0! 0

skin stress "#max

#initialmax

+1.0! 0

aerodynamic lift, the lift–drag ratio, and the maximumvon Mises stress in the skin. The lift may only decrease asthe weight of the wing decreases, the lift–drag ratio needsto be larger/equal to the ratio of the initial configura-

Maute & Allen 2004

mm! are simulated by means of beam elements. As an ex-ample, calculation is carried out for a stiffener thickness of 5mm and a stiffener distance of 1000 mm.The geometric center of gravity of the inserted triangular

bubble is placed at a point determined by the positioningcriterion. The side of the triangle closest to the boundaryshall be parallel to the boundary. The coordinates of the cor-ner points of the bubble are guided along the curvilinearcoordinate systems of the stiffener boundaries. The sizes ofthe respective bubbles are kept variable by means of scalingfactors, and they are used as design variables in the optimi-zation process.For the first optimization, we consider Load case 1 "pull-

out after gliding flight!. The topology optimization classesare presented in Fig. 41. With unchanged volume, thecomplementary energy "as a measure of mean compliance!could be reduced from 61818.5 Nmm "class 1! to 60697.6Nmm "class 3!.LC 2 "tank pressure! is treated in a second optimization

process. The first two topology classes are presented in Fig.42. With unchanged volume, the complementary energycould be reduced from 57725.6 Nmm "class 1! to 17264.6Nmm "class 2!. In the present example, superposition of thepull-out load and of the tank-pressure load leads to the re-

sults for the pull-out load, as the sensitivities of the holepositions and their shapes are very low in the load case tankpressure.This example was part of the investigations undertaken in

the scope of the Research and Development project Dynaflexof the German Federal Ministry of Research and Technology,Bonn, Germany #27$.

5.2.3 Optimization using topological sensitivityFollowing the above concept a modified approach is underdevelopment, see Garreau et al #230$. Here, the optimal de-sign problem is to minimize a cost function f (%)!F(% ,u%), where u% is the solution to a partial differentialequation defined on the domain % "Fig. 30a!. Let & be thedomain penetrated by a small spherical hole of radius rh andcenter r!% . Then an asymptotic expansion of the function fcan be obtained in the following form:

f "%p"1!# f "%p!! f "rh!g"r!"'# f "rh!$ , (5.27)

where limrh!0 f (rh)!0 and f (rh)$0.The topological sensitivity provides an information for

creating a small hole located at r. Hence, the function g(r)can be used in a similar way as a descent direction in anoptimization process. This topological sensitivity was usedby Eschenauer and Schumacher #212$ and Schumacher #95$in the case of compliance minimization. Next, Sokolowskiand Zochowski #99$ gave some mathematical justifications tothis topological sensitivity in the plane stress state with freeboundary conditions on the hole, and generalized it to vari-ous cost functions "see also #65,100,277,315$!. In the proce-dure by Garreau, the topological sensitivity is derived in lin-ear elasticity for a large class of cost functions and boundaryconditions on the hole, using an adaptation of the adjointmethod and a domain truncation. Particularly, this methodcan handle a Dirichlet boundary condition and the optimiza-tion of a criterion like the von Mises stress hypothesis.

Principle of the procedure. The aim of topological sensitiv-ity analysis is to obtain an asymptotic expansion of a crite-rion with respect to the creation of a small hole, a ball. Thelatter is introduced into a domain %, where r!% and rB$0 is sufficiently small for the ball B(r,rB) to be included in%(rB!0), see Fig. 43. The perforated domain then is

%rB!%/B"r,rB!. (5.28)

The physical behavior of the ball depends on the bound-ary condition. In an analogy with the above-mentioned con-cept, the homogeneous Neumann condition "free boundary

Fig. 40 a! Principle sketch of a wing rib of an aeroplane; b! Loadcase 1: Pull-out after gliding flight; and c! Load case 2: Tank pres-sure "2 bar!

Fig. 41 Step-wise positioning and shape optimization of a wingrib in three topology classes; Load case 1: Pull-out after glidingflight

Fig. 42 Step-wise positioning and shape optimization of a wingrib in two topology classes; Load case 2: tank pressure

Appl Mech Rev vol 54, no 4, July 2001 Eschenauer and Olhoff: Topology optimization of continuum structures 371

Downloaded 29 Oct 2007 to 138.38.149.86. Redistribution subject to ASME license or copyright, see http://www.asme.org/terms/Terms_Use.cfm

Eschenauer, Becker & Schumacher 1998

Stanford, Beran & Bhatia, 2013

•  Topology optimization be used to explore alternative designs

•  Mostly applied to a pre-determined layout or an individual component

For Peer Review

22

Figure 12. Optimal wing topologies (above) and weight-stability Pareto front (below) for case D.

V. Conclusions

This paper has detailed efforts towards the use of topology optimization to improve the aeroelastic stability

characteristics of plate-like wings in subsonic flow. The aeroelastic model consists of a well-validated plate finite

element (DKT) coupled with a quasi-steady vortex lattice method. The destabilizing flight speed (either flutter or

static divergence) is found by directly solving for the critical eigenvalue of the aeroelastic Jacobian whose real part

becomes zero. The derivative of this critical flight speed is found with respect to the thickness of each finite element

via a perturbation technique for use in a gradient based optimization. The entire Pareto front detailing the trade-off

between critical flight speed and wing mass is computed using a weighted-sum approach, for a variety of wing

planforms. The following conclusions can be drawn:

1. During the optimization process, the critical flight speed is liable to switch between flutter and divergence

(though higher-mode flutter was not observed), providing a moderately-discontinuous design space. The

Page 22 of 25

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Submitted to Journal of Aircraft for Review

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Stanford & Beran, 2010


3D Level Set Topology Optimisation of a Wing

•  Objective to optimize 3D topology of wing box domain

•  Including aero-structural coupling is important:

o  Loading dependent on deformed shape of the wing

o  Analysis & sensitivity computation


Aerostructural Topology Optimisation

•  Minimize: Total structural compliance

•  Subject to: Lift ≥ Weight

•  Aerodynamic loading from single flight condition

•  Fixed angle of attack

€

Minimize : C u( ) = f (u)T u

Subject to : L u( ) ≥Wc +Wb

Ku = f (u) = fc +Qu

C = total compliance

u = structure displacement vector

f = total load vector

K = structure stiffness matrix

L = total lift

Wc = fixed aircraft weight

Wb = wing structure weight

fc = fixed load vector

Q = aerodynamic stiffness matrix


3D Level Set Topology Optimisation of a Wing

•  Aerodynamics: Doublet Lattice Method

•  Fluid-structure interaction: Finite Plate Spline (work conserved)

•  Structures: Finite Element Analysis

DLM mesh

FPS mesh

FEA mesh


Example: 3D Wing Box

•  Aspect ratio 6, Chord 2m

•  Wing box from leading edge to 80% chord

•  Wing box depth 15% Chord

•  Discretization: 32 × 120 × 6 elements

Design Compliance (Nm)

Weight (kN)

Lift (kN)

Initial 897.1 19.92 19.94 Optimum 870.5 19.88 19.88

Leading edge

Initial design x-section

Roo

t


Example: 3D Wing Box


Composite Tow Paths Optimisation


Advanced Composite Materials

Continuous Fiber Angles. Discontinuous Fiber Angles.


Composite Tow Paths Optimisation

• Our approach: use the level set method • Optimise the tow paths not the fibre angles • Ensures continuity of fibre angles • Initial solution: topological optimum with isotropic material • Single and multiple level set functions • Minimise compliance


Test Models

Cantilever Beam:

Plate Loaded Out of Plane:


Cantilever Beam Result

Initialisation

Final Solution

Elemental Solution

Initial Compliance

Solution Compliance

% Difference

FCS

Level Set Method

26.94 19.30 18.99% 93.12%

Elemental Method

34.42 16.22 - 66.0%


Plate Loaded Out of Plane

Initialisation

Final Solution

Compliance %

Difference FCS

Level Set Method 1.51 9.42% 76.31%

Elemental Method 1.38 - 68.71%

Elemental Solution


Multiple Level Set Function

Bridge Beam:


Bridge Beam Result

Initialisation

Final Solution

Elemental Solution

Initial Compliance

Solution Compliance

% Difference

FCS

Level Set Method

2.54 1.86 18.47% 85.95%

Elemental Method

4.89 1.57 - 84.10%


Conclusion

•  Level set topology optimisation method o  Numerically stable o  Reduced dependency on mesh and starting solutions o  Robust with the starting solution o  Well-defined boundaries guaranteed.

•  Challenging and structural multidisciplinary problems o  Aeroelasticity o  Stress, buckling o  Piezoelectric (actuated structures, energy harvesting) o  Poro-elastic microstructures o  Electromagnetic (antenna) o  Composite materials o  Resource oriented architecture

•  Robust optimisation for uncertainties


Our Approach of Creating a Hole

Dunning and Kim (2013) IJNME

•  A new hole is created when

•  Minimum hole size threshold: not if only one

and . •  is copied on to . •  , are reinitialised and

continued for another hole creation.

φ < 0

φ φφφ

φ i < 0 φ j > 0


Example: Bridge

J = ηw!

"#

$

%&E C[ ]+ 1−η

w2

!

"#

$

%&Var[C]

Deterministic Solution Expectancy = 1321

Variance = 1326x103

Initial Design µ = 0.1 / unit length σ = 0.04 / unit length

200 x 50, 50% volume


η  = 1.0 Expectancy = 635.1 Variance = 100.7×103



Example: Bridge

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