OPTIMAL TOPOLOGY DESIGN USING H-ADAPTIVE MESH REFINEMENTS João Carlos Arantes Costa Júnior*

Marcelo Krajnc Alves‡

Universidade Federal de Santa Catarina, Departamento de Engenharia Mecânica, Campus da Trindade – CEP 88040-970, CP 476, Florianópolis-SC, Brazil, arantes@grante.ufsc.br* and krajnc@grante.ufsc.br‡. Abstract. The objective of this work is to propose a new and competitive procedure for the determination of the optimum topology of 2D structures. The basic idea behind the concept of optimum topology is the characterization of the layout 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 proposed an h-adaptive 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 consists in the minimization of the compliance of the structure, subjected to a volume, side and slope constraints. The design variables are defined as the nodal relative densities of the material, defined at each node of the finite element mesh. Keywords: Topology optimization, Large scale, Mesh refinement, FEM, H-adaptivity.

1 INTRODUCTION With the objective of defining the optimum topology of structures, Bendsoe & Kikuchi (1988) 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 by using the homogenization theory. The HBO concept has been used to solve minimum compliance problems by Suzuki & Kikuchi (1991), Díaz & Bendsoe (1992), Tenek & Hagiwara (1993), Bendsoe et. al. (1995) and Krog & Olhoff (1999). Simplified formulations have been proposed by: Mlejnek & Schirrmacher (1993), Yang & Chuang (1994) and Costa Jr. & Alves (2000 and 2002). In the works of Costa Jr. & Alves (2000 and 2002), the material density is considered to be constant within the finite element and is defined as the design variable. Moreover, the effective material properties are determined from a homogenized constitutive equation, which depends only on the relative density of the material and is based on the model proposed by Gea (1996). 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 Figs. 8a-c and explained in section 4. 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 optimizations 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 decreases considerably after the third layout problem, we consider most effective to employ a fixed a priori total number of layout optimization steps.

2 DEFINITION OF THE PROBLEM 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 , the part of the boundary with a prescribed traction load, i.e. =n tσ . 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

u Initial

domain

b

u Final

domain

t

t

Figure 1: Definition of the problem.

2.1 Formulation of the Problem Here, we consider initially a general 3D solid and after, we particularize to a plane stress condition. Thus, the initial formulation of the topology optimization problem is stated as: Determine the relative density ( ) ( )1,ρ W ∞∈ Ωx that solves

( )min l uρ

(1)

subjected to:

[1] Volume constraint

od VαΩ

Ω =∫ρ , (2)

here oV is the volume of the body and α is a prescribed volume fraction.

[2] Side constraints The design variable (relative density) is confined to:

[ ]0,1ρ ∈ . (3)

[3] Stability constraints Here, we make use of the slope-constrained conditions proposed by Petersson & Sigmund (1998). 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 problem, see Bendsoe (1995) and Sigmund & Petersson (1998), which occur in the Galerkin finite element discretization, when using a low order interpolation base function, in the approximation space. Thus,

2

2xC

xρ∂⎛ ⎞ ≤⎜ ⎟∂⎝ ⎠

and 2

2yC

yρ⎛ ⎞∂

≤⎜ ⎟∂⎝ ⎠. (4)

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, when required. The determination of these bounds and the procedure adopted for the enforcement of a symmetry condition will be explained in section 3.1. The displacement field ( ) H∈u ρ is determined by solving the following state equation

( )( ) ( ) o, a l Hρ = ∀ ∈u v v v (5)

where

( ) ( ) ( ), Ha dΩ

= ⋅ Ω∫u v D ε u ε v (6)

and

( )l d dΩ Γ

= ⋅ Ω + ⋅ Ω∫ ∫t

v b v t v . (7)

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 constitutive equation, is modeled as:

H ηρ=D D (8)

where D represents the constitutive equation of the fully dense material. Here, as suggested by Bendsoe & Sigmund (1999), we considered 4η = what ensures the existence of a microstructure, which represents a physical realization of a composite material, whose homogenized constitutive equation reproduces the properties of the power law. 3 DISCRETIZATION OF THE OPTIMIZATION PROBLEM The structural problem is solved with the application of the Galerkin Finite Element Method, where is employed a tri3 finite element, which interpolates both the displacement components and the relative density ( ), ,u v ρ .

3.1 Determination of the bounds exC and e

yC

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 . (9)

Then, we define

min

1e ex yC C

d= = . (10)

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 e

xC and eyC , must be performed if at least one of the

sides of the element coincide with an axis of symmetry. Here, we considered two possible cases, which are: 3.1.1 x - axis of symmetry At this point, we consider that the side ab of the element coincides with the x -axis of symmetry. In this case, as illustrated in Fig. 3, we must have: 0v = and 0yρ∂ ∂ = . In this case, the imposition of the latter condition is achieved by setting 0e

yC = .

a b

c

n e=y

x, u

y, v

x, u

y, v

a

b

c

n e= x

3.1.2 y - axis of symmetry

Figure 4 - Symmetry enforcement (y-axis).

Figure 3 - Symmetry enforcement (x-axis).

At this point, we consider that the side bc 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, the imposition of the latter condition is achieved by setting 0e

xC = . 3.2 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 (11)

subjected to

( ) 1 0h dρ αΩΩ

= Ω− =∫ρ , (12)

( ) ( )2

2

2 11 0e

e xe

g Cxρ

β−

⎧ ⎫∂⎪ ⎪⎛ ⎞= − ≤⎨ ⎬⎜ ⎟∂⎝ ⎠⎪ ⎪⎩ ⎭ρ and ( ) ( )

22

21 0e

e ye

g Cyρ

β

⎧ ⎫⎛ ⎞∂⎪ ⎪= − ≤⎨ ⎬⎜ ⎟∂⎝ ⎠⎪ ⎪⎩ ⎭ρ , (13)

for 1,..., ee n= and ∈ρ X , with inf sup , 1,...,n

i i i i nρ ρ ρ= ∈ ≤ ≤ =X ρ . Here, n represents the

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

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

o

1f lβ

=u u ρ , with ( )( )oo olβ = u ρ , for some given initial nodal

relative density vector oρ . In order to solve the problem, defined by eqs. (11), (12) and (13), is applied the Augmented Lagrangian method. As a result, the problem is formulated as a sequence of box constrained optimization problems. The general procedure may be summarized as: • Set 0k = , k = 0λ , 0kµ = , 1.0error = , kε e tol . • • While error tol> i. Solve the bound constrained minimization problem

( )min , , , ,k k kχ µ ε ∀ ∈Xρ λ ρ (14)

where

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

k k k k k kj j jk k

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

ε ε (15)

with

( )( )( ) ( ) ( )

( ) ( )2

2 , if, ,

, if

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. (16)

At this point, the solution is defined by kρ . ii. Update of the Lagrangian multiplier

( )1 1max 0,k k kj j jk gλ λ

ε+ ⎧ ⎫= +⎨ ⎬

⎩ ⎭ρ (17)

and

( )1 1k k kk hµ µ

ε+ = + ρ (18)

iii. Compute the error

1

1 1max

max 1,

k kj j

kjj

eλ λ

λ

+

+

⎧ ⎫−⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

and 1

2 1max 1,

k k

ke

µ µ

µ

+

+

−= , (19)

where

1 2max ,erro e e= (20)

iv. update penalty parameter

( )1

1 , , for some 0,1, otherwise

k kcritk

crit

ifγε ε ε γε

ε

++ ⎧ < ∈⎪= ⎨

⎪⎩ (21)

• • End The bound constrained optimization problem is solved by a memory less projected quasi-Newton method, see Ni & Yuan (1997). 4 DESCRIPTION OF THE COMBINED METHOD 4.1 General description The objective, at this point, is to present a general description of the proposed procedure that combines the layout optimization method with the well-known mesh refinement strategies. The most relevant characteristics of this new approach are: o Improvement of the resolution of the material boundary defining the optimal topology; o A significant reduction of the mesh size dependency of the final optimal layout; o A considerable reduction in the total number of design variables, relative to the specified

final resolution of the optimal topology, when compared with the traditional pixel approach that uses uniformly refined meshes;

o A decrease of the error of the solution of the state equation. In this work, the total number of h-refinement levels is defined a priori and is solved, 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 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 order to reduce the band-width of the associated with the linear system of equations 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. 4.2 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 compatibility, see George & Borouchaki (1997) and Carey (1997). 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 the i-th element is refinement. This procedure is described as follows: 1. Set, initially, Pref(i)=0, i=1,..., en . 2. For each element, determine the relative density at the baricenter, i.e., , 1,...,bar

i ei nρ = . If 0.4bar

iρ ≥ , the element is defined as a “material” element and Pref(i)=1. Otherwise, the element is denoted a “non-material” element. Here, the material boundary is defined as the union of the common interfaces of two elements, one been a “material” element and the other a “non-material” element. Notice that, all the “material” elements are refined.

3. Determine the global and the elements average errors denoted respectively by GΘ and

eΘ , e=1,…, en . Now, for each element we verify if ( )1e GϕΘ > + Θ , for some given 0ϕ > . If true, we set Pref(e)=1, i.e., the e-th element will be refined.

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

max

63

AQL P

= (22)

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. 5. 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.

6. 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.

4.3 Conditional Laplacian smoothing procedure In order to improve the mesh, after the refinement step, is employed a constrained Laplacian smoothing, which is illustrated in Fig. 7. Here, dn is the number of adjacent nodes associated with node nx .

Figure 7: Laplacian smoothing scheme.

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 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. (22). 4.4 Error Estimator criteria Here, we make use of the error estimator proposed by Zienkiewics & Zhu (1990), which is based on a gradient recovery technique by means the energy norm, see Babuska et al. (1986), Georges & Shephard (1991), Bugeda (1991) and Wilberg, & Abdulwahab (1997). 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 (23)

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 . (24)

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 . (25)

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 . (26)

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 . (27)

Determination of ( )*σ ρ

In order to determine the improved solution, ( )*σ ρ , is applied the projection technique proposed by Zienkiewicz & Zhu (1990 and 1991), which is based on the fact that the finite element solution ( ) ( )0h C∈ Ωu ρ but the stress field is only piece-wise continuous. The

determination of ( )*σ ρ consists then 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 (28)

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

* *

1

mn

j jj

N=

= ∑s s , (29)

where *s is a vector representation of the stress tensor and *js is the stress vector evaluated at

the j -th node of the element, mn is the number of nodes of the element, and jN are the same classical interpolation functions used for the interpolation of the displacement field. Now, once ( )*σ ρ is determined, we may compute the global average error GΘ , as:

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

−

Ω

⎡ ⎤Θ = − ⋅ − Ω⎣ ⎦Ω ∫ Dσ ρ σ ρ ρ σ ρ σ ρ (30)

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 . (31)

The strategy adopted to verify, if a given element must be refined, due to the error measure criteria, is given by: if ( )1e GϕΘ > + Θ with 0ϕ > , then we refine the element. 5 PROBLEM CASES For simplicity, the same material is employed in all cases, where the Young Modulus

oE =215.0 GPa and the Poisson’s ratio o 0.3ν = .

5.1 Problem 1 Here, the problem consists of a plate clamped at the inner circular arc, submitted to a prescribed vertical load 660.0 10 N= ×P on the middle of the right edge, as illustrated in Fig. 8a, and subjected to a volume fraction constraint 0.15α = (classical Mitchell frame problem). The initial mesh, with 2152 elements and 1165 nodes, is illustrated in Fig. 8a; the second refined mesh, with 4418 elements and 2311 nodes, is illustrated in Fig. 8b; the final refined mesh, with 10748 elements and 5497 nodes, is illustrated in Fig. 8c.

0.01

0.208

0.406

0.604

0.802

1.00

min.

max.

Density

c)

0.01

0.208

0.406

0.604

0.802

1.00

min.

max.

Density

b)

2.0 m

2.0 m

1.0 m

1.5 m 4.0 m

+ P

0.01

0.208

0.406

0.604

0.802

1.00

min.

max.

Density

a)

Some typical topologies obtained for this problem are presented in Hassani & Hinton (1999), and in the work of Reynolds et al. (1999). They all have different topologies, but they employ different approaches or different material models. The advantage of our approach is the reduced number of members (bars) and the superior definition of the resolution of the final optimal layout.

5.2 Problem 2 Here, the problem consists of an L-shaped cantilever, clamped at the upper edge. The cantilever is subjected to a concentrated load 660.0 10 N= ×P at the middle of the right edge and to a prescribed volume fraction α=0.25, as illustrated in Fig. 9a. The sequence shows the initial mesh in Fig. 9a with 4550 elements and 2387 nodes; the second refined mesh in Fig. 9b with 11199 elements and 5775 nodes; the final refined mesh in Fig 9c, with 31165 elements and 15875 nodes.

Figure 8: Results of the problem case 1.

Here, again, we can see a well-defined final optimal topology. The description of the associated optimal shape is described with very fine details. The ability to generate the associated optimal shape is one of the main advantages of the proposed procedure. 6 CONCLUSION The proposed combined approach of topology optimization with h-refinement procedure has shown to be very effective and robust in generating a well-defined optimal topology with a refined material contour definition. The incorporation of the h-adaptive scheme has improved considerably the rate of convergence to the optimum topology of the domain, for a given final optimal resolution, when compared with the traditional pixel approach that employs uniformly refined meshes. Thus a significant reduction in the total number of design variables where necessary in order to generate a final optimal topology with a well-defined optimal shape. Moreover, we can verify, in both examples, that the proposed procedure

Figure 9: Results of the problem case 2.

generates a sequence of optimal topologies that tends to converge to a final optimal layout. Notice that the optimal topologies, in the intermediate steps, have basically the same topology, what indicates a small initial mesh dependency of the final optimal topology. The difference occurs only in the initial mesh, but is due to the usage of a very coarse initial mesh. The results show that the proposed combined procedure of h-adative mesh refinement with topology optimization methods may be a very promising tool for the determination of the optimum layout of 2D compliance problems. 7 ACKNOWLEDGMENTS o The support of the CNPq – Conselho Nacional de Desenvolvimento Científico e

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

8 REFERENCES Babuska, I., Zienkiewicz, O.C., Gago, J. & Oliveira, A., 1986, “Accuracy estimates and

adaptive refinements in finite element computations”, John Wiley & Sons, ASIN 0-471-90862-2.

Bendsoe, M.P., 1995,“Optimization of Structural Topology, Shape, and Material”, Berlin Heidelberg: Springer-Verlag, 271p.

Bendsoe, M.P., Díaz, A.R., Lipton, R. & Taylor, J.E., 1995, “Optimal design of material properties and material distribution for multiple loading conditions”, Int. J. Numer. Meth. Engrg., 35, 1449-1470.

Bendsoe, M.P. & Kikuchi, N., 1988, “Generating optimal topologies in structural design using a homogenization method”, Comput. Methods Appl. Mech. Engrg., 71, 197-224.

Bendsoe, M.P. & Sigmund, O., 1999, “Material interpolation schemes in topology optimization”, Archive of Applied Mechanics, 69, 635-654.

Bugeda, G., 1991, “Estimación y corrección del error en el análisis estructural por el MEF”. Centro Internacional de Metodos Numericos en Ingenieria, Gran Capitán s/n, 08034 Barcelona, España.

Carey, G.F., 1997, “Computational Grids: Generation, Adaptation, and Solution Strategies”, Washington: Taylor & Francis, 496 p.

Costa Jr., J.C.A. & Alves, M.K., 2003, “Layout optimization with h-adaptivity of structures”, Int. J. Numer. Meth. Engng., 58(1), 83-102.

Costa Jr., J.C.A. & Alves, M.K., 2002, “Topology optimization with h-adaptivity of thick plates”. In: GIMC-third joint conference of italian group of computational mechanics and ibero-latin american association of computational methods in engineering, Italy, electronic publication - CDRom media.

Díaz, A.R. & Bendsoe, M.P., 1992, “Shape optimization of structures for multiple loading condition using a homogenization method”, Struct. Optim., 4, 17-22.

Gea, H.C., 1996, “Topology optimization: a new microstructure-based design domain method”, Computers & Structures, 61(5), 781-788.

George, P.L. & Borouchaki, H., 1997, “Triangulation de Delaunay et Maillage: Application aux Éléments Finis”, Paris: Hermes, 432 p.

Georges, M.K. & Shephard, M.S., 1991, “Shephard automated adaptive two-dimensional system for the hp-version of the finite element method”, Int. J. Numer. Meth. Engng., 32, 867-893.

Hassani, B. & Hinton, E., 1999, “Homogenization and Structural Topology Optimization”, eds Springer-Verlag London Limited, 268 p.

Krog, L.A. & Olhoff, N., 1999, “Optimum topology and reinforcement design of disk and

plate structures with multiple stiffness and eigenfrequency objectives”, Computers & Structures, 72, 535-563.

Mlejnek, H.P. & Schirmacher, R., 1993, “An engineer’s approach to optimal material distribuition and shape finding”, Comput. Methods Appl. Mech. Engrg., 106, 1-26.

Ni, Q. & Yuan, Y., 1997, “ A subspace limited memory quasi-Newton algorithm for large scale nonlinear bound constrained optimization”, Mathematics of Computation, 66, 220, 1509-1520.

Petersson, J. & O. Sigmund, 1998, “Slope constrained topology optimization”, Int. J. Numer. Meth. Engng., 41, 1417-1434.

Reynolds, D., McConnachie, J., Bettess, P., Christie, W.C. & Bull, J.W., 1999, “Reverse adaptivity – a new evolutionary tool for structural optimization”, Int. J. Numer. Meth. Engng., 45, 529-552.

Sigmund, O. & Petersson, J., 1998, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh dependencies and local minima”. Struct. Optim., 16, 68-75.

Suzuki, K. & Kikuchi, N., 1991, “A homogenization method for shape and topology optimization”, Comput. Methods Appl. Mech. Engrg., 93, 291-318.

Tenek, L.H., & Hagiwara, I., 1993, “Static and vibrational shape and topology optimization using homogenization and mathematical programming”, Comput. Methods Appl. Mech. Engrg., 109, 143-154.

Wilberg, N.E. & Abdulwahab, 1997, “Error estimation with postprocessed finite element solutions”, Computers & Structures, 64(1-4), 113-137.

Yang, R.J. & Chuang, C.H., 1994, “Optimal topology design using linear programming”, Comp. Struct., 52(2), 265-275.

Zienkiewicz, O.C. & Zhu, J.Z., 1990, “The three R's of engineering analysis and error estimation and adaptivity”, Comput. Methods Appl. Mech. Engrg., 82, 95-113.

Zienkiewicz, O.C. & Zhu, J.Z., 1991, “Adaptivity and mesh generation”, Int. J. Numer. Meth. Engng., 32, 783-810.