AN H-ADAPTIVE TOPOLOGY OPTIMIZATION PROCEDURE

International Conference on Adaptive Modeling and Simulation ADMOS 2003

N.-E Wiberg and P. Díez (Eds) CIMNE, Barcelona, 2003

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AN H-ADAPTIVE TOPOLOGY OPTIMIZATION PROCEDURE

João C. A. Costa Jr.* and Marcelo K. Alves†

*† Departamento de Engenharia Mecânica Universidade Federal de Santa Catarina

Campus da Trindade, 88040-970, cp 476, Florianópolis, Brazil.

e-mail: *[email protected] and †[email protected].

Key words: Topology Optimization, Layout Optimization, Large scale, Mesh refinement, Error estimator, H-adpative.

Abstract. The objective of this work is to propose a new and competitive procedure for the determination of the optimum topology of moderated thick plates. We consider the problem to be subjected to mechanical loading. The basic idea behind the concept of optimum topology is the characterization of the body domain. In order to improve the definition of the material/void interface, reduce the effective number of design variables and bound the solution error, we employ an h-adaptivity scheme. The finite element refinement procedure is implemented by using a classical triangular finite element, which interpolates both the displacement and the relative density fields. The checkerboard instability problem is circumvented, by applying the local slope constrained method. In order to obtain the optimum topology we make use of the SIMP microstructure. The formulation of the optimization problem is defined by the minimization of the compliance of the structure subjected to a volume, side and slope constraints. The design variables are then the nodal relative density of the material, defined at each node of the finite element mesh. We consider the classical plate theory proposed by Reissner-Mindlin, and use a locking-free finite element, based on the DSG (Discrete Shear Gap) approach i.

João C. A. Costa Jr. and Marcelo K. Alves.

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1 INTRODUCTION With the objective of defining the optimum topology of structures, Bendsoe & Kikuchiii

proposed the HBO (Homogenization Based Optimization) method. In the HBO method, the topology optimization problem is transformed into a material redistribution problem, defined in the feasible domain. The effective properties of the composite material are estimated using the homogenization theory. The HBO concept has been used to solve minimum compliance problemsiii,iv,v,vi,vii; frequency problemsviii,ix; stability problem casesx and different approaches of optimum plate topology problemsiii,xi,xii,xiii,xiv,xv,xvi,xvii,xviii,xix. Simplified formulations have been proposed by: Mlejnek & Schirrmacherxx, Yang & Chuangxxi and Costa Jr. & Alvesxxii,xxiii. In the work of Costa Jr. & Alves, the material density is considered to be constant within the finite element and is defined as a design variable. The effective material properties are determined from a homogenized constitutive equation, which depends only on the relative density of the material density and was proposed by Geaxxiv.

The objective of this work is to develop a competitive computational procedure for the determination of the optimum topology of structures and components, subjected to mechanical loads. In order to assure the existence of a solution, we extend the space of admissible solutions through the introduction of a "porous material" concept. The basic idea behind this concept is the characterization of the body domain by a relative density measure, ( )xρ , which determines the material and void regions of the body domain. This idea can be

illustrated in Fig. 1 where we consider an initially simply connected domain and obtain, after the optimization procedure, a multiple connected domain. The topology optimization is solved as an optimal material distribution problem, where we consider a porous material whose material properties are obtained from the SIMP model.

The topology optimization problem considers the minimization of the compliance of the structure subject to a volume constraint and whose design variables are submitted to side and slope constraints. The Galerkin finite element discretization of the state equation employs a tri3 element, which interpolates both the displacement and the relative density fields. The checkerboard instability problem and the initial mesh dependency of the optimized topology, are circumvented by enforcing, to the relative density field, a local slope constraint.

The topology optimization scheme is combined with an h-adaptive procedure with the aim of improving the definition of the material contour of the optimum layout of the structure. The procedure consists in the solution of a sequence of layout optimization intercalated by a one step mesh refinement strategy, as illustrated in Figures 11a,b,c and d and explained in section 5. The finite element mesh considered in each layout problem is obtained from the previous refinement by the application of an h-adaptive scheme. Here we consider the total number of layout problems to be defined a priori, even though a global convergence criterion could be employed for the complete automation of the procedure. Based on our experience, we have verified that a sequence of 3 to 4 layout optimization followed by a one step h-adaptive refinement is sufficient to determine an acceptable improved solution, if we depart from a refined initial mesh. Notice that, since the size of the problem increases very rapidly, and the relative gain, in the quality of the layout optimization with respect to the previous solution


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decreases considerably after the third layout problem, we consider most effective to employ a fixed a priori total number of layout optimization steps.

Here, let Ω represent the body domain with boundary ∂Ω = Γ ∪Γu t , Γ ∩Γ =∅u t , where Γu denotes the part of the boundary with a prescribed displacement, i.e. =u u , and Γt denotes the part of the boundary with a prescribed traction load, i.e. σ =n t . Also, let b

denote the prescribed body force defined in Ω , ( ) 21o oniH v H= ∈ Ω = Γuv 0 and

oH H= + u be respectively the sets of admissible variations and displacements.

b

t

domain

u

finalinitialdomain

b

t

u Figure 1: Definition of the problem.

2 FORMULATION OF THE PROBLEM

2.1 Topology optimization problem The problem will be formulated initially as 3D solid and after we will apply the hypothesis

of moderated thick plates theories. In order to perform the topology optimization we will employ the composite material approach by considering the material to be a porous material. The homogenized constitutive equation of the effective material may be fully expressed in terms of the relative density of the porous material.

The topology optimization problem may be formulated as follows:

2.1.1 Objective function

( )min l uρ

(1)

2.1.2 Volume constraint

d VαΩ

Ω =∫ρ . (2)

Here V is volume of the body and α is a prescribed volume fraction.

2.1.3 Side constraints The design variable (relative density) is confined to:

[ ]0,1ρ ∈ (3)


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2.1.4 Stability constraints Here, we make use of the slope-constrained conditions proposed by Petersson and

Sigmundxxv. These conditions are employed in order to ensure the existence of a solution to the layout optimization problem and to eliminate the well-known checkerboard instability problemxxvi,xxvii, that occur in the Galerkin finite element discretization, when using a low order interpolation base function, in the approximation space. Thus,

2

2xC

xρ∂ ≤ ∂

(4)

and

2

2yC

yρ ∂ ≤ ∂

. (5)

Here, the constants xC and yC define the bounds for the components of the gradient of the relative density. These bounds are imposed component wise with the objective of properly imposing a symmetry condition, which may be used in some particular cases. The determination of these bounds and the procedure adopted for the enforcement of a symmetry condition will be explained in section 4.1.

The displacement field ( ) H∈u ρ is determined by solving the following state equation

( )( ) ( ), oa l H= ∀ ∈u v v vρ (6)

where

( ) ( ) ( ), Ha dΩ

= ⋅ Ω∫u v D ε u ε v (7)

and

( )l d dΩ Γ

= ⋅ Ω + ⋅ Ω∫ ∫t

v b v t v . (8)

Here H is the set of admissible displacements, oH is the subspace of the admissible variations of the displacement field, ε is the infinitesimal strain tensor and HD is the effective constitutive equation, associated with the “porous material”.

2.2 Microstructure model Here, the modeling of the material properties at points with an intermediate relative

density is based on a power law constitutive relation, known as the SIMP model. In this model the effective Young’s modulus at an intermediate relative density ( )E ρ , is modeled as:


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( ) oE Eηρ ρ= (9)

where oE represents the Young’s modulus of the fully dense material and η a penalization parameter. This penalization parameter penalizes regions with intermediated densities resulting in optimal Layouts with reduced intermediate density regions. Here, we consider, as suggested by Bendsoe and Sigmundxxviii, a penalty parameter 4η = . With 4η = , we can determine a microstructure that represents a physical realization of a composite material, whose homogenized constitutive equation reproduces the properties of the power law at the given point. The homogenized constitutive equation can be defined as:

( )H ρ = Dσ ε (10)

where , , , ,Txx yy xy xz yzσ σ σ σ σ=σ and , , , ,T

xx yy xy xz yzε ε γ γ γ=ε with

( )( ) [ ]

[ ] ( )11

22

0

0

H

H

H

ρρ

ρ

=

DD

D, (11)

( ) ( )( )

o11 2

oo

111

H E νρρ

νν

= − D , (12)

and

( ) ( )22

1 0 00 1 00 0 1

H Gρ ρ =

D . (13)

( )G ρ denotes the effective shear modulus, that is defined by

( ) ( )( )o2 1E

Gρ

ρν

=−

, (14)

and oν is the Poison parameter of the fully dense material.

3 CLASSICAL PLATE THEORY OF REISSNER-MINDLIN The eqn.(6) may be specialized for moderated thick plates where we make use of the

classical theory proposed by Reissner-Mindlin.

3.1 Kinematics hypothesis At this point, we assume the Reissner-Mindlin kinematics hypothesis for the displacement

field U , at a point q , given by:


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q x x y y z zu u u= + +U e e e (15)

where xe , ye and ze are the global Cartesian base vectors, with

( ) ( ) ( )( ) ( ) ( )

( ) ( )

, , , , ;, , , , ;

, , , .

x x

y y

z

u x y z u x y z x yu x y z v x y z x y

u x y z w x y

= += +

=

θθ (16)

As a result, the components of the infinitesimal strain tensor are given as:

o o

o

, , 0 ,

2 ,

and .

yxxx xx xx yy yy yy zz

yxxy xy xy xy

xz x yz y

u vz e z z e zx x y y

u v z zy x y x

w wx y

θθε χ ε χ ε

θθε γ γ χ

γ θ γ θ

∂∂ ∂ ∂= + = + = + = + =∂ ∂ ∂ ∂

∂ ∂ ∂ ∂= = + + + = + ∂ ∂ ∂ ∂ ∂ ∂= + = +∂ ∂

(17)

Moreover, by defining

2

2

2

2

2

2

, , , , ,

, , , , ,

and , , ,

h

h

h

h

h

h

Txx yy xy xx yy xy

Txx yy xy xx yy xy

Txz yz xz yz

N N N dz

M M M zdz

Q Q dz

−

−

−

= =

= =

= =

∫

∫

∫

N

M

Q

σ σ σ

σ σ σ

σ σ

(18)

we can obtain the generalized homogenized constitutive equations, given as: The generalized membrane loading

[ ] om m=N D e (19)

where o o o o, ,T

m xx yy xye e γ=e and [ ]11

22

0

0m

m

mD

=

DD , with

( )

( ) ( )o11 222

oo

1and

11m m

E hD G h

νρρ

νν

= = − D . (20)

The generalized bending loading

[ ]b=M D χ (21)


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where , ,Txx yy xyχ χ χ=χ and [ ]

11

22

0

0b

b

bD

=

DD , with

( )( ) ( )

3 3o11 22

2oo

1and

1 1212 1b b

E h hD Gνρ

ρνν

= = − D (22)

The generalized shear loading

[ ]b t=Q D γ (23)

where ,Tt xz yzγ γ=γ and [ ] ( ) 1 0

0 1b kG hρ =

D , with the correction parameter 5 6k = .

Moreover, by defining

o ˆz= +v v θ (24)

where o ˆ ˆ ˆ, ,T

u v w=v and ˆ ˆ ˆ, ,0T

y xθ θ=θ , by integrating eqn.(7) in z for 2 2[ , ]h hz −∈

and by usinging the set of eqns.(18), we derive the following expression

( ) ( ) ( ) ( ) o, m ta dΛ

= ⋅ + ⋅ + ⋅ Λ∫u v N e v M v Q vχ γ (25)

Also, by assuming b to be constant, by integrating eqn.(8) in z for 2 2[ , ]h hz −∈ , and by using eqns.(18), we derive:

( ) o o oˆ ˆ,t

l h d dΛ Γ

= ⋅ Λ + ⋅ + ⋅ Γ∫ ∫v b v N v Mθ θ (26)

where

2

2

, 1,h

hz dz

−= ∫N M t . (27)

4 DISCRETIZATION OF THE OPTIMIZATION PROBLEM The Reissner-Mindlin plate problem is solved with the application of the Galerkin

Finite Element Method, where we use a tri3 finite element, which interpolates the displacement fields ( ), , , ,x yu v w θ θ and the relative density ρ . With the objective of circumventing the shear locking phenomena, we make use of the DSG approach proposed by Bletzinger et ali. The resulting element is free of locking, pass the patch-test and show reduced sensitivity to mesh distortions. The computation time for the construction of the element stiffness matrix for pure bending is equivalent to that for membrane, which makes the topology optimization procedure competitive.

4.1 Determination of the bounds exC and e

yC


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Consider a generic element as shown in Fig. 2 where ( ), , 1,...,3i i ix y i= =x , are the

coordinates of the vertices of the tri3 element and ( ),m m mx y=x , the coordinates of the baricenter of the element. Now, let

min 1,3min i id == d . (28)

Then, we define

min

1e ex xC C

d= = . (29)

x2

x1

x3

d i

x m

d = x - x with 1, 2 and 3i=i mi

Figure 2: Element coordinates.

A modification of the bounds, i.e. of exC and e

yC , must be performed if at least one of the sides of the element coincide with an axis of symmetry. Here, we consider two possible cases, which are:

4.1.1 x - axis of symmetry Here, we consider that the side a-b of the element coincides with the x-axis of symmetry.

In this case, as illustrated in Fig. 3, we must have: 0v = and ( ) 0yρ∂ ∂ = . These conditions

are satisfied by setting 0eyC = .

a

b c

n e= y

x, u

y, v

Figure 3: Case with x-axis of symmetry.


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4.1.2 y - axis of symmetry Here, we consider that the side a-b of the element coincides with the y-axis of symmetry.

In this case, as illustrated in Fig. 4, we must have: 0u = and ( ) 0xρ∂ ∂ = . Again, these

conditions are achieved by setting 0exC = .

x, u

y, v

a

b

c

n e= x

Figure 4: Case with y-axis of symmetry.

4.2 Formulation of the discrete optimization problem

The discrete layout optimization problem may be stated as: Determine n∈ρ so that it is the solution of:

( ) ( )( )omin ,f uρ

ρ θ ρ (30)

subjected to:

( ) 1 0h dρ αΩΩ

= Ω− =∫ρ , (31)

( ) ( )2

2

2 11 0e

e xe

g Cxρ

β−

∂ = − ≤ ∂ ρ , (32)

and

( ) ( )2

2

21 0e

e ye

g Cyρ

β ∂ = − ≤ ∂

ρ , (33)

for 1,..., ee n= and ∈ρ X , with inf sup , 1,...,ni i i i nρ ρ ρ= ∈ ≤ ≤ =X ρ . Here, n represents the


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total number of nodes in the finite element mesh. Also, the non-dimensional objective

function is given by: ( ) ( ) ( )( )o o

o

1, ,f lβ

=u uθ ρ θ ρ , with ( ) ( )( )oo o o,lβ = u ρ θ ρ , for some

given initial nodal relative density vector oρ . In order to solve the problem, defined by eqs. (30), (31), (32) and (33), we apply the

Augmented Lagrangian method. As a result, the problem is formulated as a sequence of box constrained optimization problem. The general procedure may be summarized as: Set 0k = , k = 0λ , 0kµ = , 1.0erro = , kε e tol . While erro tol>

i. Solve the bound constrained minimization problem

( )min , , , ,k k kχ µ ε ∀ ∈Xρ λ ρ (34)

where

( ) ( ) ( )( ) ( ) ( )21 1, , , , ,2 2

k k k k k kj j jk k

j

f g h h= + Ψ + +∑x x xρ λ ρχ µ ε λ ε µε ε

(35)

with

( )( )( ) ( ) ( )

( ) ( )2

2 , se, ,

, se

k k k kj j j j jk k

j j j k k k kj j j

g g gg

g

ε λ ε λε λ

ε λ ε λ

+ ≥ − Ψ = − < −

ρ ρ ρρ

ρ. (36)

At this point, we define the solution by kρ . ii. Update of the Lagrangian multiplier

( )1 1max 0,k k kj j jk gλ λ

ε+ = +

ρ (37)

and

( )1 1k k kk hµ µ

ε+ = + ρ (38)

iii. Compute the error

1

1 1max

max 1,

k kj j

kjj

eλ λ

λ

+

+

− =

, (39)


11

1

2 1max 1,

k k

ke

µ µµ

+

+

−= , (40)

where

1 2max ,erro e e= (41)

iv. update penalty parameter

( )1

1 , , for some 0,1, otherwise

k kk crit

crit

ifγε ε ε γε

ε

++ < ∈=

(42)

• End The bound constrained optimization problem is solved by a Method of Moving

Asymptotes that is commonly referred as MMA, see Svanbergxxix.

5 DESCRIPTION OF THE COMBINED METHOD

5.1 General description The objective, at this point, is to present a general description of the procedure that

combines the layout optimization method with the well-known mesh refinement strategies. The most relevant characteristics of this new approach are:

i. Improvement of the resolution of the material boundary defining the optimal

topology; ii. A significant reduction of the mesh size dependency of the final optimal layout; iii. A considerable reduction in the total number of design variables relative to the

specified final resolution of the optimal topology; iv. A decrease of the error of the solution of the state equation.

In this work we set a priori the total number of h-refinement levels to be considered and

solve, at each level, a topology optimization problem. A general description of the procedure is given as follows:

1. Read the initial finite element data structure

2. Read the initial value of the design variables

3. For each h-refinement level, do:

3.1. Solve the topology optimization problem

3.2. Apply the mesh refinement procedure

3.2.1. Read the mesh data structures


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3.2.2. Refine the finite element mesh 3.2.2.1. Identify the elements that should be refined 3.2.2.2. Perform the refinement of these elements and introduce

the transition elements necessary to maintain the mesh compatibility 3.2.2.3. Apply a constrained Laplacian smoothing procedure in order to improve the mesh quality 3.2.3. Optimize the element nodal incidence in agreement with the scheme of storage of the linear system solver of source code. 3.2.4. Update the mesh and the data structure of the finite elements

a

b c

d

e

f

a

b c

Initial element Refined element

Figure 5: Scheme of refinement element.

Incompatible mesh Compatibilized mesh

d

e

f

a

b c

g

d

e

f

a

b c

g

Figure 6: Transition element.

5.2 Mesh Refinement Strategy The strategy consists basically in the identification of the set of elements that must be

refined, their refinement, as shown in Fig. 5, complemented by the introduction of transition elements, as shown in Fig. 6, in order to maintain the mesh compatibilityxxx,xxxi. The set of elements to be refined is determined with the usage of a pointer vector: Pref(i), i=1,…, en . The default value is Pref(i)=0, representing no refinement. However, if Pref(i)=1, then we proceed the refinement of the i-th element. This procedure may be described as follows:

i. Set Pref(i)=0, i=1,..., en . ii. For each element, we determine the relative density at the baricenter, i.e.,

, 1,...,bari ei nρ = . If 0.4bar

iρ ≥ , the element is defined as a “material” element and we set: Pref(i)=1. Otherwise, the element is denoted a non-material element. Here, we define the material boundary to be given by the common interface of two


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elements, one been a material element and the other a non-material element. Notice that, at this step, we identify the “material” elements that are refined.

iii. Determine the global and the elements average errors denoted respectively by GEΘ and EΘ , e=1,…, en . Now, for each element we verify if ( )E GE1 ηΘ > + Θ , for some given 1η > . If true, we set Pref(e)=1, i.e., the e-th element will be refined.

iv. Determine the quality measure Q of each element in the mesh, given by

max

63

AQL P

= (43)

where A is the area of the triangle, P is the one half of the perimeter of the triangle, max max , ,L ab ac bc= is the length of the element’s largest side.

Thus, for each element we verify: if ( ) 0.55Q e ≤ then we set: Pref(e)=1. v. Identify all the non-material elements that have a face common with the material

contour and set their pointer reference Pref(e)=1. Thus, all non-material elements having a material element neighbor are also refined.

vi. Perform an additional smoothing refinement criterion. Here, for each element

whose Pref(e)=0 we identify their neighbors. If the element has 2 or more neighbors who’s Pref(.)=1, then refine the given element, i.e., set Pref(e)=1. The objective here is to avoid having a given non-refined element having 2 or more neighbors to be refined. This may lead to the generation of sharp edges in the material contour or may generate internal void regions with a poor material contour definition. Thus, we refine these elements.

5.3 Conditional Laplacian smoothing procedure In order to improve the mesh, after the refinement step, we employ a constrained Laplacian

smoothing, which is illustrated in Fig. 7. Here, dn is the number of adjacent nodes associated with node nx .

x =n x ind

1

n = 1

ndxn

x5

x4 x

3

x2

x1

(a) before smoothing (b) after smoothing

xn

x5

x4 x

3

x2

x1

Figure 7: Laplacian Smooth

The Laplacian process is conditional since it will only be implemented if the mesh quality of the set of elements, as illustrated in Fig. 7, improves. The mesh quality of the set of


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elements is given by the quality of the worst element in the set. The measure of quality of a given element is given by eqn.(43).

5.4 Error Estimator criteria Here, we make use of the error estimator proposed by Zienkiewics and Zhuxxxii, which

is based on a gradient recovery technique by means the energy normxxxiii,xxxiv,xxxv,xxxvi. Let ρ be a given realizable nodal relative density of the problem. Then, the local displacement error may be defined as:

( ) ( ) ( )h= −e u uρ ρ ρ (44)

where ( )u ρ and ( )hu ρ are the exact and approximate solution respectively. Then, the energy norm may be written as:

( ) ( ) ( )( ) ( )( )2

EHe d

Ω

= ⋅ Ω∫D e eρ ρ ε ρ ε ρ (45)

Now, the local stress error may be expressed in terms of the local displacement error as follows:

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )H Hh hσ = − = − =e D u u D eρ σ ρ σ ρ ρ ε ρ ρ ρ ε ρ . (46)

Therefore, we are able to rewrite the energy norm as:

( ) ( ) ( )( ) ( ) ( ) ( )( )12

EH

h h d−

Ω

= − ⋅ − Ω ∫e Dρ σ ρ σ ρ ρ σ ρ σ ρ . (47)

Since the exact stress distribution is unknown, we approximate ( )σ ρ by an improved

solution ( )*σ ρ , which is more, refined then ( )hσ ρ . Thus, the error indicator is approximated by:

( ) ( ) ( )( ) ( ) ( ) ( )( )12 * *E

Hh h d

−

Ω

= − ⋅ − Ω ∫e Dρ σ ρ σ ρ ρ σ ρ σ ρ . (48)

5.4.1 Determination of ( )*σ ρ

In order to determine the improved solution, ( )*σ ρ , we apply the projection technique

proposedxxxii,xxxvii, which is based on the fact that the finite element solution ( ) ( )0h C∈ Ωu ρ

but the stress field is only piece-wise linear. The determination of ( )*σ ρ consists in the

solution of the least square minimization of the potential ( )ψ ρ , where

( ) ( ) ( )( ) ( ) ( )( )* *h h dψ

Ω

= − ⋅ − Ω∫ρ σ ρ σ ρ σ ρ σ ρ (49)


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here, ( )*σ ρ is interpolated within each element as

* *

1

n

i ik

N=

=∑s s (50)

where the components of *is are stress components evaluated at the i-th node of the element,

and iN are the same classical interpolation functions used for the interpolation of the displacement field.

Now, once ( )*σ ρ is determined, we may compute global average error GEΘ , as:

( ) ( )( ) ( ) ( ) ( )( )11 HG h h d

−

Ω

Θ = − ⋅ − Ω Ω ∫ Dσ ρ σ ρ ρ σ ρ σ ρ (51)

and EΘ the element average error, as

( ) ( )( ) ( ) ( ) ( )( )1* *1

e

He h h e

e

d−

Ω

Θ = − ⋅ − Ω Ω ∫ Dσ ρ σ ρ ρ σ ρ σ ρ . (52)

The strategy adopted to verify, if a given element must be refined, due to the error measure criteria, is given by: if ( )1e GηΘ > + Θ with 0η > , then we refine the element.

6 PROBLEM CASES Here, we present some problem cases with the objective of evaluating the performance of

the proposed procedure. In order to show the evolution of the refinement strategy we illustrate some of the intermediate optimal topologies and their associated finite element meshes. Moreover, for simplicity, we employ the same material to all problem cases. The material

Young Modulus oE =215.0 GPa and the Poisson’s ratio o 0,3ν = . The thickness of the plate is h=0.1 m.

6.1 Problem case (1): Here, we consider the problem illustrated in Figure 8, which consists of a plate clamped on

right edge and submitted to a prescribed transversal load P=100.0e+03N on the middle of the right edge. The optimal topology is subjected to a volume fraction constraint α=0.30.

The first sequence of meshes, resulting from the h-refinement procedure is described as follows: The initial mesh, with 780 elements and 431 nodes, is illustrated in Figure 11a,e; The second refined mesh, with 1862 elements and 991 nodes, is illustrated in Figure 11b,f; The third refined mesh, with 5395 elements and 2784 nodes, is illustrated in Figure 11c,g; The final refined mesh, with 17848 elements and 9054 nodes, is illustrated in Figure 11d,h. Remark that the values of the number of nodes and elements are relative to half symmetric of mesh, i. e., due to the symmetry conditions, we have discretized one half of the domain only.


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2.0 m

P

0.8 m

0.8 m

Figure 8: Definition of Problem Case (1).

6.2 Problem case (2): Here, we consider the problem illustrated in Figure 9, which consists of a plate clamped for

all edge and submitted to a prescribed transversal load P=200.0e+03N on the center of the plate structure. The optimal topology is subjected to a volume fraction constraint α=0.30.

The first sequence of meshes, resulting from the h-refinement procedure is described as follows: The initial mesh, with 1410 elements and 756 nodes, is illustrated in Figure 12a,d; The second refined mesh, with 3105 elements and 1635 nodes, is illustrated in Figure 12b,e; The final refined mesh, with 9060 elements and 4665 nodes, is illustrated in Figure 12c,f. Remark that the values of the number of nodes and elements are relative to a quarter symmetric of mesh, i.e., due to the symmetry conditions, we have discretized one quarter of the domain only.

2.0 m

P

2.0 m

2.0 m

2.0 m



17

6.3 Problem case (3): Here, we consider the problem illustrated in Figure 10, which consists of a plate

clamped for all edge and submitted to a prescribed transversal load P=200.0e+03N on the center of the plate structure. The optimal topology is subjected to a volume fraction constraint α=0.30.

The first sequence of meshes, resulting from the h-refinement procedure is described as follows: The initial mesh, with 980 elements and 531 nodes, is illustrated in Figure 13a,d; The second refined mesh, with 2286 elements and 1198 nodes, is illustrated in Figure 13b,e; The final refined mesh, with 6540 elements and 3349 nodes, is illustrated in Figure 13c,f. Remark that the values of the number of nodes and elements are relative to a quarter symmetric of mesh, i.e., due to the symmetry conditions, we have discretized one quarter of the domain only.

1.9 m

P

1.9 m

1.9 m

1.9 m

0.1 m 0.1 m

0.1 m

0.1 m



18

Figure 11: Results of Problem Case (1).


19



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7 CONCLUSION The proposed algorithm has shown to be effective to obtain the optimum topology of the

plate structures. The usage of a remeshing procedure is important to accelerate the determination of the solution to the problem. This allow us to determine the solution of a larger problem using as a starting point, the converged solution obtained with a smaller mesh.


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This has shown to be effective. The usage of a non uniform refinement has shown to decrease the number of design variables and this decrease becomes even more relevant if we refine the problem with a very large number of elements. Moreover, the introduction of an error estimator was shown to improve the quality of the optimum layout since the remeshing procedure resulted many times in large elements together with very small elements generation large approximation errors for the solution to the problem. Based on the problem cases presented, we did not verify dependency of the optimal layout with respect to the initial mesh, i.e., different initial meshes not presented different optimal layouts. The imposition of a local gradient constraint eliminated the initial mesh dependence. Moreover, once the initial mesh is defined, the resulting sequence of partial optimal topologies indicates a convergence to a final optimal layout, with the h-adaptive scheme. The proposed combined approach resulted is a very promising tool for the determination of the optimum topology of plate compliance problems. Notice that, even though the cost of evaluating the objective function is higher then the cost obtained with the use of a pixel type of element approach, the enormous decrease in the number of design variables compensates the increase in the evaluation cost of the objective function. Moreover, if the objective is to develop commercial design tools, it is a primary requirement that the material contour be undoubtedly defined and that the final optimum topology be, as much as possible, independent of the initial mesh.

8 ACKNOWLEDGMENTS The support of the CNPq – Conselho Nacional de Desenvolvimento Científico e

Tecnológico – is gratefully acknowledged. Grant Numbers: 547198/1997-0 and 140501/1991-1.

The support of Prof. Krister Svanberg, from the Royal Institute of Technology, Stockholm, Sweden, that graciously provided the MMA solver.

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AN H-ADAPTIVE TOPOLOGY OPTIMIZATION PROCEDURE

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