Folding Wing

8/18/2019 Folding Wing

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SIMP based Topology Optimization of a Folding

Wing with Mixed Design Variables

Xiaohui Wang1

School of Astronautics

Beijing University of Aeronautics

and Astronautics

Beijing, 100191, China

[email protected]

Zhiwei LinChina Great Wall Industry

Corporation


Renwei XiaSchool of Astronautics

Beijing University of Aeronautics

and Astronautics


Abstract —A several cycles of design-verification-modification

are needed during traditional design, which is very time-

consuming. Topology optimization is rapidly expanding in the

recent years in the field of structural mechanics. Currently it is

widely applied to structural design. The combination of topology

optimization and sizing and/or shape optimization has been

adopted for flight vehicle structural design, which results possible

better designs. In the paper, continuum structural topology

optimizations as well as size optimization are combined for a

folding wing structural design. A structure optimization problem

is established using Solid Isotropic Material with Penalization

(SIMP) method, it consists of the thickness variable of the skin

and the relative element density variables of the tip part of the

folding wing. The design objective is to minimize the compliance

of the whole structure, and the constraints are structural mass

constraints and stress constraints. The problem was solved based

on commercial software package HyperWorksTM

. The resulting

design shows that a new tip structure which satisfies the

requirements of stiffness and strength was reconstructed.

Keywords—folding wing; structural design; topology

optimization; optimization method

I. I NTRODUCTION

Structural optimization uses the computer to automaticallyfind the optimal structural size or shape and topological form based on modern mechanical and mathematical numericalmethod, under the condition of the specified design conditions(such as displacement, stress, dynamic of structuralcharacteristic requirements)[1].. Structural optimization has been widely applied in aeronautics, astronautics, shipbuildingand other fields. Usually structural optimization was dividedinto three levels according to the variable type: sizing

optimization, shape optimization and topology optimization[2],respectively corresponding to three different design stages:detailed design stage, basic design stage and conceptual designstage.

Structural topology optimization is one of the frontier

topics of the structural optimization, improving the topological

form of the structure can greatly improve the performance of

the structure and reduce the weight of the structure. According

to the research object, topology optimization can be divided

into topology optimization of discrete structures (such as truss,

rigid frame, strengthening tendon-plate and other framework

structure) and topology optimization of continuum structures

(such as 2D plate and shell, 3D solid)[3].The topology

optimization of continuum structures has been studied more inthe recent years, and has been applied in many project fields,

such as missile and aircraft structure design[2][3][4][5].

Many scholars put forward to combine the topology

variables and the size or shape variables of optimization to

solve project problems, such as the optimization design of

wing structural layout and so on, under decomposition strategy

[6][7] (Topology variables are optimized before size or shape

variables) or coordinating optimization strategy[2] (All kinds

of variables are optimized at the same time). But

decomposition strategy may limit the design space, so

collaborative optimization is more likely to find possible better

design[8].The rest of the paper is organized as follows: in Section II,

the Solid Isotropic Microstructure with Penalization method is briefly discussed. The optimization method used in the designis given In Section III. The detailed design procedure for afolding wing design is shown in Section IV, followed byconclusions.

II. SOLID ISOTROPIC MATERIAL WITH PENALIZATION

Topology optimization is introduced by Maxwell to analyzethe minimum weight truss under the stress constraints in 1954.In 1964, Dorn proposed the ground structure method bringingnumerical methods into the field of topology optimization. In1988, Bendsoe and Kikuchi proposed Homogenization Methodand create a new situation for topology optimization of

continuum structures.

The idea of Solid Isotropic Microstructure with

Penalization (SIMP) method is first proposed by Bendsoe

under different names, for intermediate densities. SIMP was

developed by Zhou and Rozvany independently. SIMP is a

technique for topology optimization of overwhelming

advantages. SIMP is currently the most popular finite element

based topology optimization method for composite continua.

SIMP usually treats element relative density as design

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variables, and then transforms integer optimization problem

from 0 to 1 into a problem of continuous variable

optimization.

The most popular interpolation model in SIMP is density

- stiffness power time relationship model (SIMP interpolation

model), namely the relationship between element elastic

modulus and element density is exponential

0

p

i E x E =

where xi is the relative density of element (0 xi 1), and p is

penalty factor, and E 0 and E are elastic modulus before and

after penalty respectively. By setting a penalty factor p more

than 1 (penalty factor is valued from 3 to 5), one can penalize

the stiffness matrix of intermediate density element, so that to

reduce the number of intermediate density element, to obtain

the approximated discrete design.

In addition to SIMP interpolation model, there is Rational

Approximation of Material Properties (RAMP) interpolation

model, which admits the following expression:

01 (1 )

i

i

x E E

p x=

+ −

Note that RAMP interpolation model can achieve a similar penalty result to SIMP interpolation model.

III. OPTIMIZATION METHOD

There are many optimization methods available to solve

topology optimization problem, such as Optimality Criteria

(OC), Mathematics Programming (MP), Genetic Algorithm

(GA) and some other simulating biology intelligent

algorithms. The sequence planning of mathematical

programming method, such as Sequence Linear Programming

(SLP), Sequential Quadratic Programming (SQP) and

Sequence Convex Programming (SCP) and so on, is combinedwith approximate problems of structural optimization at

present. It has been widely applied in solving the problem of

topology optimization because of its high efficiency and

accuracy. A general structural optimization problem can be

expressed as:

{ }1 2, , ,

( )

. . ( ) 0 1,

1,

T

n

j

L U

i i i

Find X x x x

Min f X

s t g X j m

x x x i n

=

< = =

(1)

where X is the design variables, and n is the number of design

variables, and and are the lower limit and upper limit of

xi, and f(x) is objective function, and g i(x) is the constraint

function of structural character, and m is the number of

constraints.

We adopt the mathematical programming with

approximation problems to solve the optimization problem (1).

The detailed steps are given as follows [9]:

1) Finite element based method to analyze corresponding

physical problem.

2) Convergence judgment.

3) Design sensitivity analysis.

4) Use the sensitivity information to establish

approximation model to solve approximation optimization

problem and update the design variables.

5) Return to 1).The flow chart of the above optimization process is shown below.

Figure. 1. The process flow of the optimization

It needs sensitivity analysis while establishing the explicit

approximation model of the problem (1). For the finite

element equation, it meets:

KU P =

Where K is the stiffness matrix, and U is the element node

displacement vector, and P is element node load vector.

Calculate the partial derivative to the design variables X

of both sides: K U P

U K X X

∂ ∂ ∂+ =

∂ ∂ ∂

1( )

U P K K U

X X X

−∂ ∂ ∂=

∂ ∂ ∂-

The general constraint functions of structural feature g(x)

can be described as the functions of displacement vector U :

( ) T g X Q U =

So the sensitivity of the constraint function is:

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1( )T T

T T g Q U Q P K U Q U Q K U X X X X X X

−∂ ∂ ∂ ∂ ∂ ∂= + = +

∂ ∂ ∂ ∂ ∂ ∂-

Solving the optimization problem by using the above

method directly is known as the direct method which isapplicable to the optimization problems with a large number

of constraints but with relatively few variables, such as the

sensitivity problems of the shape optimization and the size

optimization. For the optimization problems with few

constraints but a large number of variables, such as topology

optimization, it is prefer to adopt another method where the socalled adjoint variable E is introduced, and E meets the

constraint as follows:

KE Q=

So the sensitivity of the constraint function reads as:

( )T

T g Q P K U E U

X X X X

∂ ∂ ∂ ∂= +

∂ ∂ ∂ ∂-

Once the sensitivity of structure response is obtained, one

needs to conduct the Taylor expansion for the structure

response by means of one of the following approximation

methods:

Linear approximation:

0 00

( ) ( )n

j

j j i ii i

g g X g x x

x=

∂= +

∂ -

Inversion approximation:

2

0 00 0

1 1( ) ( )

n j

j j ii i i i

g g X g x

x x x=

∂=

∂ - -

Convex approximation:

0 00

( ) ( )n

j

j j ij i ii i

g g X g c x x

x=

∂= +

∂ -

Where

0

1, 0

0

j

i

ij

ji

i i

g when

xc

g xwhen

x x

∂≥

∂=

∂<

∂

In addition, there are other methods to establish the

approximation model, such as the secondary multipoint

approximation method[10], the response surface method and

neural network method etc.For the explicit approximation problem, we can applyfeasible direction method (CONMIN) and the dual method(CONLIN).

IV. OPTIMIZATION DESIGN OF A FOLDING WING

A folding wing is a design feature of aircraft to reducethe size, save space, increase the carrying capability of

vehicles [11]. Due to the specialty of the structure and

mechanism of this kind of aircraft, the requirement of the

structure mass, stiffness and strength is stricter than that ofnon-folding design. The conventional design method is usually

based on the engineer’s experiences, by means of a several ofcycles, i.e. design -- verification -- revision, and finally a

structural design is obtained to meet all the design

requirements. This method, however, is very time consuming,

and also requires the designers have certain design

experiences. Here we adopt the topology optimization for the

folding wing structural concept design, in order to obtain thestructural arrangement form owning the best path of force

transfer. It is significant for the structural design of the aircraft

which can effectively shorten the design period. With the

development of computer technology, several software packages are available to aid the structural topology design,

for example, OptiStruct by Altair.Figure 2 shows a folding wing structure and mechanism,

we can find there are folding mechanism and locking

mechanism inside of wing root; When wing surface unfolds to

the unfolding state from folding state, the locking mechanism

begins to work, pushes a locking part to the connectors to

connect the wing root with the connectors, so that the wing

surface will be locked in the unfolding state. Wing tip, wingroot and the connectors are all made of titanium alloy, while

supporting shaft and locking part are made of 45Cr. Accordingto the requirement of design, the weight of wing tip cannot be

more than 3 kg, and the maximum displacement of the tipcannot be more than 4.5 mm.

Figure. 2. The folding wing structure and mechanism

Figure. 3. The finite element model of the folding wing

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We established the finite element model of the folding

wing based on HyperMesh, solid is made of tetrahedronelement, while plate is made of triangular element. Because

the force between two parts is transferred through the

interface, so we need to define the contact between two parts

to simulate the unfolding state of the folding wing accurately,

Figure 3 provided the finite element model of the folding

wing. Every node of the wing root bottom has been restrainedof three degrees of translational freedom, and the equivalent

uniformly distributed pressure imposed on the wing surface

pressure is 0.15 Mpa.In this article, the wing tip will be treated as the

topological design area in concept design, while the skin

thickness of outer surface in the wing tip will be treated as adesign variable. We combined the topology variables with

sizing variables, and then optimized them at the same time, in

order to obtain the optimal structure form inside of wing tip

and the optimal outer skin thickness, thus we obtained the

optimal monolithic structure. As for the structural design of

the wing tip, we adopt SIMP to establish a structural

optimization problem that contains topology variables andskin thickness variable, it can be described in mathematical

form as follows:

{ }1 2, , , ,

( )

. . ( ) 0 1,

0 1 1,

T

n

j

i

L U

Find X x x x t

Min f X

s t g X j m

x i n

t t t

=

=

=

(2)

where X is continuous variable, and xi is the topology variable,

i.e. element relative density, and n is the number oftopological variables, and t is the skin thickness variable of

outer surface, and t L

and t U is the lower and upper limit of the

thickness respectively, and f(x) is the optimization objectivefunction, which means the strain energy of the structure,

shown in equation (3), and g i(x) is the constraint of structural

feature, and m is the number of constraints, including thestructural weight, element stress and the node displacement,shown in the expression (4).

1

0

1 1

1 1 1( )

2 2 2

1( )

2

N T T T

i i i

i

n N n p T T

i i i i i i

i i

f X F U U KU u k u

x u k u u k u

=

−

= =

= = =

= +

(3)

where, f(x) is the total strain energy of the structure, namelyflexibility, and F and U is the node force and displacement

vector of the finite element respectively, and K is the total

stiffness matrix before optimization, and N is total number ofthe structural elements, k i is the element stiffness matrix, andk 0 is the element stiffness matrix without penalty in topology

design area.

tip

m

3.2kg

1070 1, ,

4 1, ,6k

m

Mpa m n

u mm k

σ

≤

≤ =

≤ =

(4)

where mtip is the weight of the wing tip, mσ is element Mises

stress, uk is the mode of the six node’s displacement (The six

nodes are in the front or the outer margin edge of the wing tip,

circled in Figure 2).

We established the optimization model according to theexpression (2) in HyperMesh, treating the relative density of

the node in wing tip and the skin thickness as design variables.The skin thickness varies from 0.1 mm (t L) to 3 mm (t U ). And

penalty factor p is set to 3, defining the stripper constraint in

the normal direction of the wing surface. Finally we adopt

optimization solver OptiStruct to solve the problem.

Figure. 4. Optimal topology form of the structure

We obtain the optimal topology form of wing tip after an

iterative with 17 steps, the last step of the optimization

iteration meets the constraints. Figure 4 provides the Get theoptimal element relative density distribution of the wing tip

structure, namely the materials distribution of the structure.

We can easily find that the number of the node with middledensity is few, the material distribution has obvious limits, and

it is advantageous to transform the result into an actualstructure. The optimal skin thickness is 1.901 mm, rounded to

be 1.9 mm.

We redesigned the wing tip structure according to theobtained optimal topology form. We removed the materials of

intermediate density, and simplified the optimal topology

solution to adjust to the requirements of machining, and

considered the structure support the wing surface to meet therequirements of aerodynamic shape. Figure 5 shows the

profile structure of wing tip according to the optimal topologydesign. The skin thickness of the surface is 1.9 mm. The

weight of the wing tip is 2.994 kg after removing some

material inside according to the topology form, satisfying the

requirement of structural weight.

We re-establish the model of the wing tip structure

according to the topology optimization results, and analyze toverify the stiffness and strength of the wing structure. All the

parts are made of tetrahedron element, and the loading

condition is same as that in optimization design. Figure 6 provides the displacement nephogram of the folding wing

under the uniformly distributed pressure according to the static

analysis. We can find that the maximum displacement of wingtip is 4.361 mm meeting the requirement of stiffness; Figure 7

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provides the vonMises stress nephogram of the folding wingunder the uniformly distributed pressure. We can find that the

maximum stress is 875.5 Mpa, located in the interface betweenthe wing tip and the wing root, it is within the strength limit

and owns a great safety margin. The structure is safe.

Figure. 5. Profile structure in the optimal topology design

Figure 6. The displacement nephogram of the folding wing under the

uniformly distributed pressure

Figure. 7. The von Mises stress nephogram of the folding wing under the

uniformly distributed pressure

According to the above results, we found that the designedstructure under the optimal topology form meets therequirements of weight, stiffness and strength.

V. CONCLUSIONS

In this the work, we have applied the structural topologyoptimization method to the folding wing tip structure design.We have obtained the optimal topology form from finiteelement based topological optimization, which provides areference for the design of the wing tip structure. The adoptedmethod treats the skin thickness as the design variables, andoptimizes the design variables and the wing tip topology

simultaneously. It improves the degree of freedom for thedesign and helps the designer to obtain a better structure form.The method can be applied to the design of similar structure,such as the wing and the fuselage of aircraft, treating the skinsize variables of the wing and the fuselage as design variablesto the topology optimization during the conceptual designstage. Application of this method to such a topologyoptimization with mixed design variables will bring more

prominent effect.

ACKNOWLEDGMENT

This work is supported by the Fundamental Research Fundsfor the China Central Universities under grant No.YWF-12-

LZGF-101.

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