Multi-Criteria Multi-Material Topology Optimization

of Laminated Composite Structures

Erik Lund∗, Leon S. Johansen, Christian G. Hvejsel and Esben L. Olesen

Department of Mechanical Engineering, Aalborg University,

Pontoppidanstraede 101, DK-9220 Aalborg East, Denmark

The design of laminated composite shell structures is a challenging problem, and inorder to obtain a cost effective design, it is desirable to have a general computer aided toolthat can generate a high performance topology in the initial design phase. In this workthe potential of using the so-called Discrete Material Optimization (DMO) approach formulti-material topology optimization is studied for multi-criteria design where conflictingdesign criteria such as mass, compliance, buckling load factors and local failure criteria aretaken into account. Examples are given for a generic main spar from a wind turbine blade.

Nomenclature

C Constitutive matrixxi Design variablexi Lower limit on design variablexi Upper limit on design variable∆xi Perturbation of design variablewi, wiWeight functionne Number of element candidate materialsnl Number of layer candidate materialsp Penalization powerhε DMO convergence measureN l,tot

conv Number of converged layers in all elementsN l,tot Total number of layers in all elementsM Mass, kgC Compliance, N·mF Inverse safety factor against material failures Safety factor against material failureK Global stiffness matrixD Global displacement vectorF Global load vectorKσ Global stress stiffness matrixf Objective functionfk Function values associated with the objective functiongj Constraint functiongj Upper limit for constraint functionJ Number of constraint functionsn0 Number of function values associated with the objective functionI Total number of design variablesM Mass constraint, kgSubscript

i Design variable number

∗Professor, Department of Mechanical Engineering, Aalborg University.

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12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 10 - 12 September 2008, Victoria, British Columbia Canada

AIAA 2008-5897

Copyright © 2008 by Erik Lund, Leon Johansen, Christian Gram Hvejsel and Esben Lindgaard Olesen. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

j Eigenvalue number, constraint function numberk Layer number, objective function value numberSuperscript

e Elementl LayerSymbols

β Bound parameter in bound formulation∆ Incrementλ1 Linear buckling load factor (lowest eigenvalue)Φ1 Corresponding buckling eigenvectorε Vector of strainsσ Vector of stresses

I. Introduction

The use of laminated composite structures with Glass or Carbon Fiber Reinforced Polymers (GFRP/CFRP)is popular for lightweight constructions due to their superior strength and stiffness characteristics. In orderto fully exploit the weight saving potential of these multilayered structures, it is necessary to tailor thelaminate layup and behavior to the given structural needs. This calls for the use of advanced structuraloptimization tools.An example of a challenging lightweight structure made of multi-material laminates is a wind turbine blade.It typically consists of several materials including GFRP/CFRP along with foam materials and balsa tree,bonded together by a resin and stacked in a number of layers and groups. In order to obtain a cost effectivedesign, it is desirable to have a general computer aided tool that can generate a high performance topology inthe initial design phase. Thus, the design problem considered in this work consists of optimal distribution ofthese different materials in multi-layered composite shell structures, taking different structural performancecriteria into account. The design parametrization method applied in this work is denoted Discrete MaterialOptimization (DMO), and it enables the use of gradient based optimization formulations that can be solvedusing mathematical programming, see.1–4

The paper addresses the challenging problem of multi-material topology optimization of fiber reinforcedlaminated composite structures with focus on multi-criteria design. The idea behind the DMO methodis given in Section II, and next the analysis and design sensitivity analysis methods used are outlined inSection III. The use of the DMO method is illustrated for design criteria including mass, compliance, linearbuckling load factors and local criteria like failure indices, and the mathematical programming method usedis described in Section IV. For complicated engineering structures like wind turbine blades several of thesecriteria are conflicting quantities, which makes it even more difficult to solve the topology optimizationproblems. This will be illustrated for several multi-criteria design studies of a generic main spar in SectionV.

II. The discrete material optimization approach

The basic idea in the Discrete Material Optimization approach is to formulate an optimization problem usinga parametrization that allows for efficient gradient based optimization on real-life problems while reducingthe risk of obtaining a local optimum solution when solving the discrete material distribution problem.The approach is related to the mixed materials strategy suggested by Sigmund and co-workers5,6 for multi-phase topology optimization, where the total material stiffness is computed as a weighted sum of candidatematerials. By introducing differentiable weighting functions for the material interpolation, the topologyoptimization problem is converted to a continuous problem that can be solved using standard gradient basedoptimization techniques.In the present context this means that the stiffness (or density) of each layer of the laminated composite willbe computed from a weighted sum of a finite number of candidate constitutive matrices, each representing agiven lay-up of the layer. Consequently, the design variables are no longer the fiber angles or layer thicknessesbut the scaling factors (or weight functions) on each constitutive matrix in the weighted sum.As in topology optimization the parametrization of the DMO formulation is invoked at the finite element

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level. The laminated composite shell structures considered are analyzed using layered shell finite elementswhere the element constitutive matrix, Ce, for a single layered laminate structure may in general be expressedas a sum over the element number of candidate material configurations, ne:

Ce =

ne

∑

i=1

wiCi = w1C1 + w2C2 + · · · + wneCne , 0 ≤ wi ≤ 1 (1)

Each candidate material is characterized by a constitutive matrix Ci, and the weight functions wi must allhave values between 0 and 1 in order to be physically allowable. Furthermore, in case of having, e.g., a mass

constraint M , it is necessary that the sum of the weight functions is 1.0, i.e.,∑ne

i=1 wi = 1.0. If this demandis not satisfied, the computed mass M cannot be compared to the prescribed mass constraint M , as the massdensity ρ is computed using the weight functions.The objective of the optimization is then to drive the influence of all but one of these constitutive matricesto zero for each ply by driving all but one weight function to zero. As such, the methodology is very similarto that used in topology optimization.Several different parametrization schemes have been developed, see details in,1,2 and the most effectiveimplementation is briefly outlined here. For each element a number of design variables xe

i , i = 1, . . . , ne isdefined, and the weight functions are defined as

wi =wi

∑ne

k=1 wk

, i = 1, . . . , ne where wi = (xei )

p

ne

∏

j=1; j 6=i

(

1 − (xej)

p)

, 0 < xi ≤ xi ≤ xi < 1 (2)

As an example, in case of three candidate materials the weight functions wi are given as

w1 = (xe1)

p(1 − (xe2)

p)(1 − (xe3)

p), w2 = (xe2)

p(1 − (xe1)

p)(1 − (xe3)

p), w3 = (xe3)

p(1 − (xe1)

p)(1 − (xe2)

p) (3)

To push the design variables xei towards 0 or 1, the SIMP method known from topology optimization has

been adopted by introducing the power p to penalize intermediate values of xei . The power p is typically set

to 1 or 2 in the beginning of the optimization process and then increased by 1 for every 10 design iterationsuntil p is 3 or 4. Moreover, the term (1−xe

j)j 6=i is introduced such that an increase in xei results in a decrease

of all other weight functions. Finally, the weights have been normalized in order to satisfy the constraint thatthe sum of the weight functions is 1.0. Note that the expression in Eq. (2) means that complicated additionalconstraints on the design variables xe

i are avoided and only simple box constraints have to be dealt with.It should be noted that the weight functions described in Eq. (2) require a lower limit xi > 0 and an upperlimit xi < 1 on the design variables. These lower and upper limits are needed in cases where two (or more)equally good candidate materials have been proposed, as the corresponding design variables then will obtainthe same value. The lower limit xi is typically set to 0.01 and the upper limit xi to 0.99.The initial values of the design variables, xi, may in principle be any set of numbers between 0 and 1but in general the values should be chosen such that the initial weighting is uniform, i.e. wi = wj for alli, j = 1, . . . , ne. In this way no candidate material is favored a priori.In case of multilayered laminated structures the interpolation given above is simply used for all layers, i.e.,the layer constitutive matrix Cl

k for layer k is computed as

Clk =

nl

∑

i=1

wiCi (4)

where nl is the number of candidate materials for the layer.In general, the design variables xi may be associated with each finite element of the model or the numberof design variables may be reduced by introducing patches, covering larger areas of the structure. This isa valid approach for practical design problems since laminates are typically made using fiber mats coveringlarger areas.In order to describe whether the optimization has converged to a satisfactory result, i.e. whether a singlecandidate material has been chosen in all layers and all other materials have been discarded, a DMO con-vergence measure is defined. For each layer in each element the following inequality is evaluated for all nl

weight factors, wi:

wi ≥ ε√

w21 + w2

2 + · · · + w2nl (5)

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where ε is a tolerance level, typically 95-99.5 %. If the inequality Eq. (5) is satisfied for one of the weightfactors wi in the layer it is flagged as converged. The DMO convergence measure, hε, is then defined as theratio of converged layers in all elements N l,tot

conv to the total number of layers in all elements N l,tot:

hε =N l,tot

conv

N l,tot(6)

The DMO convergence measure is denoted h99.5 if the tolerance level is 99.5 % (and so forth) and fullconvergence, i.e. h99.5 = 1, simply means that all layers in all elements have a single weight factor contributingmore than 99.5 % to the Euclidian norm of the weight factors.

III. Analysis and design sensitivity analysis

The analysis of the general laminated composite structure is in this work based on eight node stabilizedlayered solid shell finite elements with 24 nodal displacement and 7 internal degrees of freedom, see theelement description in.7 All materials are assumed to behave linearly elastic.The analyses involved are static stress analysis and linear buckling analysis in order to determine displace-ments, compliance, failure indices and the buckling load factor. Thus, the analyses performed are

KD = F (7)

and(K − λjKσ)Φj = 0, j = 1, 2, . . . (8)

Here K is the global stiffness matrix, D is the global displacement vector and F the global load vector. Inthe buckling analysis problem Kσ is the global stress stiffness matrix, the eigenvalues λj are assumed orderedby magnitude, such that λ1 is the lowest eigenvalue, i.e., buckling load factor, and Φ1 is the correspondingeigenvector. The linear buckling analysis assumes that the structure is perfect with no geometrical imper-fections and the buckling load found will typically be an upper limit for the real value. In this paper this isof minor importance as the focus is on demonstrating the DMO method for multi-criteria problems.In order to optimize the structure using the DMO approach gradients must be made available. The designvariables are termed xi, i = 1, . . . , I, and in the following subsections the design sensitivity analysis (DSA)methods used are described.

A. DSA of compliance

The compliance C of the structure is defined as the work done by the applied loads F at the equilibriumstate and is computed as

C(D) = DT F (9)

By differentiating Eqs. (7) and (9) and assuming design independent loads, the need for displacementsensitivities can be eliminated, resulting in the well-known expression for the compliance sensitivity

dC

dxi

= −DT dK

dxi

D (10)

For the DMO material parametrization used in this work where the geometry is fixed and only the material ischanged, the stiffness matrix derivative dK/dxi only involves the derivative of the element layer constitutivematrix Cl

k with respect to xi, which is easily done analytically by differentiating the interpolation functionsdescribed by Eqs. (1) and (2).

B. DSA of local failure criteria

Failure F is defined layer-wise in each element similar to the stiffness interpolation in Eq. (1), i.e. the failuremeasure Fk for layer k is computed as

Fk =

nl

∑

i=1

wiFi (11)

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where wi is defined in Eq. (2) and Fi is the failure measure of the i’th candidate material. Assuming linearelasticity and proportional loading the safety factor against failure s is defined as the load scaling factor thatleads to failure. Here, the failure measure is taken to be the inverse safety factor against failure computedusing, e.g., the Tsai-Wu criterion,8–10 i.e., F = 1

s. The inverse safety factor of each candidate material i is

evaluated on basis of the stresses in it, i.e.σi = Ciε (12)

evaluated at the top and bottom of each layer in the elements.Each layer failure measure is an implicit function of the displacement field, i.e. Fk = Fk(D(x),x). Thefailure measure sensitivities dFk

dxi

are evaluated using a first order forward difference approximation

dFk

dxi

≈Fk(x + ∆xi,D + ∆D) − Fk(x,D)

∆xi

(13)

where ∆D ≈dDdxi

∆xi. The perturbation ∆xi of the DMO design variable xi is set to 0.001. The displacement

sensitivity dDdxi

is obtained analytically as shown next. The static equilibrium equation (7) is differentiatedwith respect to a design variable xi:

KdD

dxi

=dF

dxi

−dK

dxi

D (14)

where the load sensitivity dF/dxi is zero for the DMO material design variables used, unless volume forcesare considered. The displacement sensitivity is computed by solving Eq. (14) for each design variable xi,reusing the factorized global stiffness matrix K from the analysis in case of using a direct solver.

C. DSA of buckling load factor

In case of including the buckling load factor in the optimization problem, gradients are computed by thedirect differentiation approach. In case of a simple, i.e. distinct, eigenvalue λj in the buckling analysis, theeigenvalue sensitivity is given as (11–15)

dλj

dxi

= ΦTj

(

dK

dxi

− λj

dKσ

dxi

)

Φj (15)

where it has been assumed that the eigenvectors have been Kσ-orthonormalized, such that ΦTj KσΦj = −1.

In case of multiple eigenvalues the eigenvectors are not unique, which complicates the sensitivity analysis andoptimization due to the non-differentiability of the eigenvalues. In such situations the sensitivity analysisdescribed in,16 see also,4 is used together with the optimization algorithm developed in.17 The details areomitted here for brevity.The stress stiffness matrix is an implicit function of the displacement field, i.e. Kσ = Kσ(D(x),x), andthe computation of the stress stiffness matrix derivative thus becomes the time consuming part of the DSAas all finite elements give contributions (as opposed to the stiffness matrix derivative that only receivescontributions from elements directly linked to the design variable xi considered).The displacement sensitivities dD/dxi are computed from Eq. (14), and the stress stiffness matrix sensitivitydKσ/dxi in Eq. (15) is computed by a central difference approximation at the element level, i.e.

dKσ

dxi

≈Kσ(x + ∆xi,D + ∆D) − Kσ(x − ∆xi,D − ∆D)

2∆xi

(16)

where ∆D ≈dDdxi

∆xi as described above for local criteria. Alternatively, the stress stiffness matrix sensitiv-ities can be computed using

dKσ

dxi

=∂Kσ

∂xi

+∂Kσ

∂D

dD

dxi

(17)

but this requires quite a lot of additional derivations and programming to implement which is the mainreason for using Eq. (16).The well-known inaccuracy problem associated with semi-analytical design sensitivity analysis of slenderbeam, plate and shell structures only appears in case of shape design variables (see18,19 for details) and noinaccuracy problems have been observed for the difference approximations used in case of DMO materialdesign variables.

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IV. Optimization formulation

The multi-criteria optimization problem is in this work solved using mathematical programming, i.e., thedesign optimization problem is written in a general form as

Minimize f(x)

x

Subject to

gj(x) ≤ gj j = 1, . . . , J

xi ≤ xi ≤ xi, i = 1, . . . , I

(18)

where the objective function f is to be minimized subject to certain constraint functions gj , j = 1, . . . , J .The objective and constraint functions considered are compliance, mass, linear buckling load factors andlocal criteria like failure indices.In case of having a scalar objective like compliance or mass, Eq. (18) is solved directly using the Method ofMoving Asymptotes (MMA) by Svanberg.20

In case of maximizing the buckling load factor a number of eigenvalues is typically considered in the opti-mization problem in order to take into account the possibility of crossing eigenvalues (mode switching) andcreation of multiple eigenvalues. Similarly, a number of failure criteria is taken into account when the objec-tive is to minimize the maximum failure index in the structure. Thus, the objective function of a min-maxproblem can be written as

f(x) = f( fk(x) ) = max( fk(x) ), k = 1, . . . , n0 (19)

The min-max optimization problem is solved using a bound formulation, see,21 where an additional scalardesign variable β is introduced:

Minimize β

x, β

Subject to

fk(x) ≤ β k = 1, . . . , n0

gj(x) ≤ gj j = 1, . . . , J

xi ≤ xi ≤ xi, i = 1, . . . , I

(20)

The min-max problem is thus transformed into the problem of minimizing a bound β subject to the constraintfk(x) ≤ β. The parameter β has replaced a non-differentiable functional and is to be minimized over aconstraint set in an enlarged design space. The points of non-differentiability correspond to “corners” in theconstraint set of the enlarged space, where the corners arise from intersections of differentiable constraints.Again, MMA is used for solving the mathematical programming problem.The bound formulation thus provides a simple solution to the non-differentiability problem in connectionwith min/max and max/min problems, and it also makes it possible to handle the optimization problemin a uniform way regardless of the blend of local, integral and min/max criteria. Weighting factors can beused on the individual criteria functions, if wanted. It should be noted that any of the constraint functionsgj , j = 1, . . . , J , might as well be vector functions but this causes no problems.In case of eigenvalue optimization involving multiple eigenvalues, the sensitivity analysis described in16 canbe used together with the optimization algorithm developed in.17 However, for the examples considered inthis paper, it has not been necessary to apply this more complicated approach.

V. Multi-criteria optimization of generic main spar

The potential of using the DMO approach for multi-criteria design of a challenging engineering structure isstudied for a simplified model of a wind turbine blade. Some designs of wind turbine blades basically consistof two structural components, the main spar and the aerodynamic shell as illustrated on Figure 1 (A). Themain spar carries most of the flapwise bending loads whereas the shell carries most of the edgewise bendingloads. In this study the main spar is subjected to the most critical static load case, which is the flapwise

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Assembly

Leading edge

Trailing edge

Suction side shell

Main spar

Pressure side shell

Flapwise bending

(A)

Pressure load

Local buckling

(B)

Figure 1. Generic wind turbine main spar. A: The two main structural components of the blade. B: Bucklingcollapse of blade and geometrically nonlinear finite element analysis of main spar.

bending load that arises when the turbine has been brought to a standstill owing to high wind and the bladeis hit by the 50 year extreme wind.In case of such a flapwise bending load the structure may collapse due to local buckling at the compressiveside of the main spar as illustrated on Figure 1 (B). In order to estimate the load factor due to local bucklingaccurately, the buckling analysis should be performed on the deformed geometry, but as the main focus ofthis paper is to address multi-criteria design only linear analyses are used.The candidate materials to use for the multi-material topology optimization are given by Table 1.

Table 1. Data for candidate materials

E-glass/epoxy E-glass/epoxy Foam

(UD) (0◦/90◦ biax) Divinycell H130

Ex 45.0 GPa 24 GPa 160.0 MPa

Ey 10.0 GPa 24 GPa -

νxy 0.30 0.11 0.45

Gxy 5.0 GPa 4.5 GPa -

Gxz 5.0 GPa 4.5 GPa -

Gyz 5.0 GPa 4.5 GPa -

Xt 1100.0 MPa 84 MPa 4.2 MPa

Xc 675.0 MPa 260 MPa 2.6 MPa

Yt 35.0 MPa 84 MPa -

Yc 120.0 MPa 260 MPa -

S 80.0 MPa 60 MPa 2 MPa

ρ 2000 kg/m3 1870 kg/m3 130 kg/m3

Failure Tsai-Wu Tsai-Wu Max. principal

criterion stress

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It is chosen to use a DMO parametrization with the following 6 candidate materials: E-glass/epoxy (UD -unidirectional) at 0◦, 45◦, -45◦ and 90◦, the biax material oriented at 45◦ and the H130 foam material. Theorientation of the E-glass/epoxy is given relative to the length direction of the main spar. The foam materialcan only be chosen in interior layers of the structure.

Ø0.86

0.07

0.4

0

0.42

0.020.02

0.1

3

0.225

Pressu

reloa

d

Patch1

Patch2

Patch3

Patch4

Patch5

Patch6

Patch7

Patch8

Tip section Mid section Root section

Figure 2. Definition of the generic wind turbine main spar. It is a generic model without twist. The length ofthe model is 15 meter. The root section has a length of 3 meter and the mid section which is the design areahas a length of 9 m. A linear interpolation between the three cross sections shown is used. The tip sectionhas a constant cross section.

The actual geometry of the generic main spar is given on Figure 2. As the main purpose of the example isto illustrate the use of the DMO method, a simplified geometry compared to the true geometry is used. Thefinite element model consists of 1856 layered solid shell finite elements and the resulting flapwise bending loadof 280 KN is applied as a pressure load at the tip section. The root section is modeled using 4 equal thicknesslayers with orientations -10◦/10◦/10◦/-10◦, and the tip section used for load introduction is modeled usinga symmetric layup of 8 layers with 45◦/-45◦ orientations.The mid section of the blade is chosen as design area and it is modeled using 10 layers of equal thickness.For simplicity, only 8 patches are used for the DMO parametrization, i.e., the 8 patches shown on Figure 2have the same DMO parametrization. The number of design variables is then reduced to 464 (8 patches eachhaving 58 design variables). The reduction of the design space makes it more difficult to obtain a distinctchoice of material everywhere but it makes it easier to present the optimization results in tables as done inthe following.

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A. Minimum compliance design with mass constraint

First the simplest problem of maximizing the stiffness of the structure, i.e. minimizing the compliance, witha mass constraint is solved. It is chosen that approximately 1/5 of the design domain should be filled withfoam material which leads to the mass constraint M = 1408 kg.

Table 2. Result of minimum compliance design of generic main spar with a mass constraint. Layer 1 is theinner layer whereas layer 10 is the outer layer.

Layer Patch 1 Patch 2 Patch 3 Patch 4 Patch 5 Patch 6 Patch 7 Patch 8

1 0◦ 0◦ 0◦ 0◦ 0◦ 0◦/45◦/-45◦ 0◦ 0◦/45◦/-45◦

2 0◦ 0◦ 0◦ 0◦ foam foam foam foam

3 0◦ 0◦ 0◦ 0◦ foam foam foam foam

4 0◦ 0◦ 0◦ 0◦ foam foam foam foam

5 0◦ 0◦ 0◦ 0◦ foam foam foam foam

6 0◦ 0◦ 0◦ 0◦ foam foam foam foam

7 0◦ 0◦ 0◦ 0◦ foam foam foam foam

8 0◦ 0◦ 0◦ 0◦ foam foam foam foam

9 0◦ 0◦ 0◦ 0◦ foam foam foam foam

10 0◦ 0◦ 0◦ 0◦ 0◦ 0◦/45◦/-45◦ 0◦ 0◦/45◦/-45◦

The results of the minimum compliance design with a mass constraint are given in Table 2. If the UDmaterial has been selected, then only the angle of the UD material is given. In case of a non-unique choiceof candidate material, the different materials with a significant weight function associated are listed. Forexample, in patches 6 and 8 in layers 1 and 10 the optimization ended up with three candidate materials:UD at 0◦, 45◦ and -45◦ all having approximately the same weight factor. The iteration history can be seenon Figure 3.

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

0 20 40 60 80 100 0

0.2

0.4

0.6

0.8

1

1.2

Obj

ectiv

e, c

ompl

ianc

e [N

m]

Nor

mal

ized

mas

s co

nstr

aint

Iteration number

Iteration history for minimum compliance design with mass constraint

ComplianceNormalized mass constraint

Figure 3. Iteration history for minimum compliance problem with only mass constraint.

The DMO convergence measure h95 is 0.98, see definition in Eq. (6). As the loading case is flapwise bending,the flanges of the main spar are main loaded in tension/compression whereas the webs are subjected to

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shear. As expected the foam material is put in the webs but it is surprising that the biax material (whichcorresponds to layers of 45◦/-45◦) is not selected in patches 5-8. The reason for this is the parametrizationinto large patches used which makes it more difficult to make a distinct choice of material. In patches 6 and8 in layers 1 and 10, the optimization ended up with UD at 0◦, 45◦ and -45◦, and the reason for not selectingthe biax material is that parts of these patches give a significant contribution to the global bending stiffness,making it favorable to include the UD 0◦ material. If the same optimization example is executed with anelement-wise parametrization (with 68672 design variables), the material selection for the outer layer is asillustrated in Figure 4.

Figure 4. The orientation of the selected FRP candidate materials for the outer layer (number 10) is shownin case of using elementwise parametrization of the mid section for minimum compliance design. The biaxmaterial is selected at the middle of the webs whereas UD 0◦ is selected in the rest of the design domain. Onlyhalf of the main spar is shown.

It is seen that now the biax material is selected in the middle of the webs whereas UD at 0◦ is selected inthe rest of the domain. The interior distribution of material is similar to the result obtained using patchparametrization.

B. Minimum compliance design with mass and buckling constraints

Next buckling constraints are added to the design problem. The buckling load factor should be larger than1.25 while still fulfilling the same mass constraint as before. The three lowest eigenvalues of the bucklingproblem are taken into account in the optimization formulation in order to take into account the possibilityof crossing eigenvalues (mode switching) and creation of multiple eigenvalues.

Table 3. Result of minimum compliance design of generic main spar with mass and buckling constraints

Layer Patch 1 Patch 2 Patch 3 Patch 4 Patch 5 Patch 6 Patch 7 Patch 8

1 -45◦/45◦ -45◦/45◦(0◦) 0◦ 0◦ 0◦ 45◦(-45◦/90◦/0◦) 0◦ -45◦(45◦/90◦)

2 -45◦/45◦(0◦) 0◦(-45◦/45◦) 0◦ 0◦ foam(0◦) foam foam(0◦) foam

3 0◦(-45◦/45◦) 0◦ 0◦ 0◦ foam(0◦) foam foam(0◦) foam

4 0◦ 0◦ 0◦ 0◦ foam(0◦) foam foam(0◦) foam

5 0◦ 0◦ 0◦ 0◦ foam foam foam foam

6 0◦ 0◦ 0◦ 0◦ foam foam foam foam

7 0◦ 0◦ 0◦ 0◦ foam foam foam foam

8 0◦ 0◦ 0◦ 0◦ foam foam foam foam

9 0◦(90◦) 0◦ 0◦ 0◦ foam foam foam foam

10 0◦(90◦) 0◦ 0◦ 0◦ 0◦ 0◦(-45◦/45◦/0◦) 0◦ 0◦(-45◦/45◦/0◦)

The DMO convergence measure h95 for this problem is 0.86, and as also seen from the results presented inTable 3, the material selection is less distinct. This is also expected as maximum stiffness and maximumbuckling resistance are two conflicting design criteria. In order to suppress the lowest buckling modes that

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are similar to the buckling mode shown on Figure 1 (B), the glass fiber material in the flanges should mainlybe oriented in the transverse direction, i.e. at 90◦, which is the opposite of the best orientation in case ofcompliance design. Thus, the results presented in Table 3 should be seen as a compromise between the twoconflicting demands, and the value of the buckling constraint determines the weighting between them.In several places of the design domain a dominant but not distinct candidate material has been selected.In Table 3 this has been indicated by using parentheses. For example, in layer 1 in patch 8 the notation-45◦(45◦/90◦) means that UD -45◦ has far the largest weight factor but UD at 45◦ and 90◦ also have non-zeroweight factors. It is noted that the results are not completely symmetric as expected as patch 6 and 8 arenot identical in layer 1.From an engineering point of view, the non-uniqueness of the results in part of the design domain maynot be a big problem as the weight factors give information about the relative importance of the candidatematerials. Thus, the results in Table 3 can be used as a starting point for a more detailed description of thelayups.The iteration history of the optimization again documents good performance of the gradient based approachusing the bound formulation described in Section IV, see Figure 5. The initial design is infeasible w.r.t. bothconstraints, but after 10 iterations the constraints are fulfilled. At the end of the optimization the threelowest eigenvalues of the buckling problem obtain the values λ1 = 1.251, λ2 = 1.253 and λ3 = 1.254, i.e.,the lowest eigenvalues are very close.

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

0 20 40 60 80 100 0

0.2

0.4

0.6

0.8

1

1.2

Obj

ectiv

e, c

ompl

ianc

e [N

m]

Nor

mal

ized

con

stra

int

Iteration number

Iteration history for minimum compliance design with mass and buckling constraint

ComplianceNormalized mass constraint

Normalized buckling constraint

Figure 5. Iteration history for minimum compliance problem with mass and buckling constraints.

Similar results and performance of the DMO approach are obtained if the design problem is changed tomaximization of the lowest buckling load factor with compliance and mass constraints.

C. Minimum failure design

Finally, ongoing work related to optimization w.r.t. failure is illustrated. Multi-material failure analysisand optimization have been implemented in our optimization code MUST,22 but in the following the foammaterial is removed as candidate material. The Tsai-Wu criteria is used for the failure evaluation, seematerial data in Table 1.One major issue when solving local criteria problems in combination with topology optimization is the largenumber of design variables and criteria functions. For the example shown this problem is reduced by usingthe 8 patches whereby the number of design variables for the 5 candidate materials becomes 400. It is chosento include the largest 100 failure indices for each patch such that the total number of criteria functionsbecomes 800.

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The result of the optimization is shown in Table 4.

Table 4. Result of minimum failure design of generic main spar

Layer Patch 1 Patch 2 Patch 3 Patch 4 Patch 5 Patch 6 Patch 7 Patch 8

1 0◦ 0◦ 0◦(-45◦) 0◦(90◦) -45◦/45◦ 0◦/biax 0◦ 0◦/-45◦

2 0◦ 0◦ 0◦(-45◦) 0◦(90◦) -45◦/45◦ 0◦(biax) 0◦ 0◦/-45◦

3 0◦ 0◦ 0◦(-45◦) 0◦/biax -45◦(45◦/0◦) 0◦ 0◦ 0◦(-45◦/biax)

4 0◦ 0◦ 0◦(-45◦) 0◦/biax 45◦/0◦ 0◦ 0◦ 0◦

5 0◦ 0◦ 0◦(-45◦) 0◦/biax 0◦(-45◦) 0◦(-45◦) 0◦ 0◦

6 0◦ 0◦ 0◦(-45◦) 0◦/biax 0◦(-45◦) 0◦(-45◦) 0◦ 0◦(45◦)

7 0◦ 0◦ 0◦(-45◦) 0◦(-45◦) 0◦(-45◦) 0◦/-45◦ 0◦ 0◦(biax/45◦)

8 0◦ 0◦ 0◦(-45◦) 0◦ 0◦ 0◦(-45◦) 0◦ 0◦/biax(45◦)

9 0◦ 0◦(biax) 0◦(-45◦) 0◦(-45◦) 0◦ 0◦/-45◦ 0◦ biax/0◦(45◦)

10 0◦ 0◦/biax 0◦(-45◦) 0◦(-45◦) 0◦ 0◦(-45◦) 0◦ biax(0◦/45◦)

The DMO convergence measure h95 is reduced to 0.56 for this example, i.e., the material selection is muchless distinct as also seen in Table 4. In several places of the design domain more than one material isselected, and in even more places one candidate material has been found but several other materials havea non-zero weight factor associated as indicated by parentheses around these additional materials. A moredistinct choice of material can be obtained if using a finer parametrization but the current limitation isthe treatment of many design variables together with many local criteria when solving the mathematicalprogramming problem. Experience shows that it is necessary to take a large portion of the largest failureindices into account when solving the min-max problem of minimizing the maximum failure index.The iteration history is seen on Figure 6 which indicates that a sufficient number of failure indices has beenincluded in the mathematical programming problem due to the relatively smooth convergence.

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100

Obj

ectiv

e, fa

ilure

inde

x

Iteration number

Iteration history for minimization of failure

Max. failure index

Figure 6. Iteration history for minimum failure design.

VI. Conclusion

The paper has focused on using the so-called Discrete Material Optimization (DMO) approach for multi-criteria multi-material topology optimization of laminated composite shell structures. The design criteriaconsidered have been mass, compliance, linear buckling load factors and local criteria like failure indices.

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The DMO approach has been described and the performance studied for three different design cases of ageneric wind turbine blade main spar. The main idea of the DMO approach is to enable the solution of multi-material topology problems by converting the underlying discrete problem to a continuous problem whereweight functions are used for obtaining a distinct material choice. For the three design cases studied it hasbeen found that the method may give a distinct choice of material, especially if non-conflicting design criteriaare used. In case of contradicting design criteria the material distribution becomes a compromise betweenthe different demands, and in many places of the design domain a non-converged solution is obtained. Froman engineering point of view these results may anyway be of much importance as a preprocessing tool in theearly design phase as the weight functions associated with the candidate materials indicate the importanceof the different materials available.The problem of using local criteria like failure indices of the laminated composite structure in the DMOapproach is ongoing work, and even though promising results w.r.t. convergence of the objective functionhave been demonstrated, the method needs further investigation in order to obtain a more distinct materialchoice.

Acknowledgments

The work has been supported by the Danish Research Council for Technology and Production Sciences,grant on “Multi-Material Design Optimization of Composite Structures”.

References

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