Hybrid cellular automata with local control rules - CiteSeerX

6th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil

Hybrid cellular automata with local control rules: a new approach to topology optimization inspired by bone functional adaptation Andr´ es Tovar1a , Wilson I. Quevedo1b , Neal M. Patel2c , John E. Renaud2d (1) Department of Mechanical and Mechatronic Engineering. Universidad Nacional de Colombia. Cr. 30 45-03, Bogot´ a, Colombia. (a) Assistant Professor, [email protected] (b) Graduate Research Assistant, [email protected]. (2) Department of Aerospace and Mechanical Engineering. University of Notre Dame, Notre Dame, Indiana 46556, USA. (c) Graduate Research Assistant, [email protected] (d) Professor, [email protected].

1. Abstract In previous investigations, the functional adaptation process in bones was modeled using the hybrid cellular automaton (HCA) algorithm. In this algorithm, the structural analysis is performed using the finite element method, while the cellular communication within the bone structure and its adaptation process are modeled using cellular automaton (CA) principles. This model of bone functional adaptation also demonstrated to be an effective technique for topology optimization of continuum structures. The optimization problem used for the bone adaptation process was defined as minimizing mass while maximizing strength. Using optimality conditions, it was possible to demonstrate the existence of a local equilibrium state where the bone structure is adapted to the environment and no further structural change is required. Inspired by phenomenological approaches to simulating bone functional adaptation, the HCA algorithm makes use of control rules in order to obtain an optimal configuration. Four different control strategies has been implemented, including two-position, proportional, integral and derivative control. The combination of proportional, integral and derivative (PID) control has shown improved convergence in comparison to other topology optimization approaches. This algorithm has been successfully applied to a variety of two- and three-dimensional structural models. The objective of this investigation is to enhance the flexibility of the HCA algorithm. This work introduces a new methodology for setting various types of constraints in the HCA optimization problem formulation. These constraints include mass, deflection, stress and strain energy density. Different strategies based on control theory and optimality conditions are implemented to achieve fast and reliable convergence. 2. Keywords: Topology optimization, cellular automata, hybrid cellular automata, control, constrained optimization. 3. Introduction Topology optimization involves the optimal distribution of material within a design domain. Initially, the design domain comprises a large number of elements. The topology optimization process finds an optimal structure by selectively removing unnecessary elements. The design variables in the optimization problem depend on the type of material model used in the structural analysis. The most commonly referenced approaches are referred to as the homogenization approach [1, 2, 3] and the solid isotropic material with penalization (SIMP) approach [4, 5, 6]. In topology optimization, the number of elements and, hence, the number of design variables depend on the size of the design domain and the desired resolution of the final structure. Even the design of a small mechanical component might involve thousands of design variables. In addition, the cost of a function call increases with the number of elements. Therefore, the use of classical gradient-based optimization methods might be impractical. This has motivated the implementation of specialized numerical methods such as approximation techniques [7, 8, 9], methods of moving asymptotes (MMA) [10], optimality criteria [11, 12, 13], genetic algorithms [14, 15], the so-called evolutionary structural optimization (ESO) approach [16, 17] and cellular automaton (CA) techniques [18, 19, 20, 21, 22, 23]. The methodology developed in this research is referred to as the hybrid cellular automaton (HCA) algorithm. In conventional CA methods, a global analysis of field states is not performed. In this research, the HCA method makes use of finite element analysis to evaluate the field states, i.e., the strain energy densities. In this context, the work of Kita and Toyoda [19] can be considered a hybrid method since they use finite element analysis to update the stress states during each iteration of their

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algorithm. The HCA method is a finite element-based approach and, therefore, it reduces the residual between external work and internal energy to zero at every iteration. Conversely, in the SAND-CA implementation of Tatting and G¨ urdal [20], the residuals are iteratively reduced to zero by the optimizer. In Tovar et al. [23], a new approach to topology optimization was developed. This approach reduces numerical instabilities by using cellular automaton (CA) principles. This method is referred to as the hybrid cellular automaton (HCA) method with local control rules. In this approach, the design domain is discretized into a regular lattice of cells. Each cell locally modifies the design variables according to a design rule. This rule drives the local strain energy density (SED) to a local SED target using a control strategy. Recently, Tovar et al. [24] derived a new set of design rules from the Karush-Kuhn-Tucker (KKT) optimality conditions of a multi-objective optimization problem. In this formulation, the design process seeks to minimize both mass and strain energy. In addition to the control-based techniques previously introduced, they derived a new ratio technique that drives the design to optimality. This ratio technique is based on the one traditionally used to design truss structures in the fully stressed design (FSD) approach. This investigation introduces a new methodology for setting various types of constraints in the structural optimization problem. These constraints include mass, deflection, stress and strain energy density. The strategy to satisfy the constraints is based on control theory. The performance of this strategy is demonstrated through two- and three-dimensional problems. 4. Hybrid cellular automata The hybrid cellular automaton (HCA) method is intended to solve complex structural optimization problems in engineering. The premise of the HCA method is that complex static and dynamic problems can be decomposed into a set of simple local rules that operate over a large number of CAs that only know local conditions. 4.1 Biological background The capacity of bone tissue to alter its mass and structure in response to mechanical demands has been recognized for decades. The notion of a relationship between form and function in bones produced by mechanical stimulus is commonly known as the “Wolff’s Law”. In his classic publication from 1870, Julius Wolff wrote: “Only static usefulness and necessity or static superfluity determine the existence and location of every bony element and, consequently of the overall shape of the bone.” [25] Later, this theory developed into the paradigm that bones are formed as mechanically optimal structures of maximum strength and minimum weight [26]. Many theoretical models use the concept of an error signal to simulate the bone adaptation process. [27, 28, 29, 30] These models imply the existence of an equilibrium state (or zero error condition) where the bone structure is adapted to the environment and no remodeling is required. Deviations from the equilibrium state initiate the remodeling activity. It has been widely accepted that bone tissue is resorbed in regions exposed to low mechanical stimulus, whereas new bone is deposited where the stimulus is high. This process of functional adaptation is thought to enable bone to perform its mechanical function with a minimum of mass. From the physiological standpoint, in addition to the mineral tissue, bone also contains bone cells. They make up a small percentage of the volume of bone but play a critical role in the adaptation of its structure. According to their function, bone cells may be divided into osteoblasts, which form new bone, and osteoclasts, which resorb old bone. Osteoclasts are giant multi-nucleated cells that secrete acids and enzymes to break down the mineralized bone matrix. They erode bone structure as they make their way through the bone matrix. On the other hand, osteoblasts are cubicoidal single-nucleated cells that synthesize and deposit uncalcified new bone matrix or osteoid (Fig. 1). The process in which old bone is replaced by new tissue is referred to as remodeling. It is thought that bone structural adaptation occurs through the remodeling process. There are two other types of bone cells which are derived from the osteoblasts. They are the bone lining cells and the osteocytes. Bone lining cells are inactive osteoblasts that cover all available bone surface. Osteocytes are osteoblasts encased in bone matrix. They are also believed to be sensor cells within bone, sensing mechanical stimuli. Osteocytes sit in cavities called lacunae and communicate with each other and with osteoblasts by tunnels called canaliculi. They create a communication network that is believed

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to transmit mechanical signals that eventually cause the remodeling of the bone tissue (Fig. 2). Bone marrow Osteoblasts Osteoclasts

Osteoid Osteocytes Bone tissue

Figure 1: Bone cells. Picture from the educational resource materials of the American Society for Bone and Mineral Research.

Figure 2: Cellular network of osteocytes. Picture from Hokkaido University Graduate School of Medicine.

4.1 Functional adaptation model In this work, a regular lattice of cells (cellular automata) is proposed to model the connected cellular network in bones. Each cell represents an osteocyte with its surrounding mineralized tissue. Based on the average density of osteocytes within the mineralized tissue, a lattice of 1 × 1 × 1 mm3 contains 25 × 25 × 25 cells [31] (Fig. 3). The osteocytes establish communication with the surrounding neighbors. This connection can be represented by a three-dimensional, radially symmetrical layout. Based on ˆ = 18 [31](Fig. biological observations, the most appropriate one is composed of eighteen neighbors, N 4).

Figure 3: Cellular automaton model of one cubic millimeter composed of 15625 cellular automata.

Figure 4: Cellular automaton neighborhood in the bone model composed of 18 neighbors.

In the proposed algorithm, the state of the i-th cell is defined by a design variable xi and a state variable yi . A local evolutionary rule Ri modifies the design variable until the state variable reaches an optimum value yi∗ . In this application, the design variable represents the amount of mass surrounding the osteocyte while the state variable represents its mechanical stimulus. The evolutionary rule defined in a classical cellular automaton (CA) algorithm only makes use of local information, i.e., the states of the neighboring cells. However, in order to obtain the mechanical stimulus yi for each cell, the proposed hybrid cellular automaton (HCA) algorithm makes use of the finite element method. In order to model the intercellular communication, the HCA algorithm defines the effective mechanical stimulus value y¯i as
 Nˆ yi + k=1 ηk yk y¯i = ηi , (1) ˆ +1 N where ηk (t) represents the efficiency of the transmission of the error signal to the i-th cell from its k-th neighbor, and ηi (t) represents the strength of that signal. These values account for physical deterioration 3

in the intercellular communication due to microcracks, aging and apoptosis [31]. In bone functional adaptation, the cells that are allowed to modify their mass are the ones on the surface that are in contact with the bone marrow (Fig. 5). The local rules used to identify these cells are referred to as the surface conditions [31].

Figure 5: Bone functional simulation using hybrid cellular automata. The cells that are allowed to modify their mass are the ones in contact with the bone marrow.

4.2 Topology optimization algorithm In topology optimization, the material model implemented in the HCA algorithm is the artificial density. This model was formalized by Bendsøe [4] and later coined as the solid isotropic material with penalization (SIMP) model [6]. In this way, the material properties within each element are assumed to be constant. Normally, a continuous relative density is used as a design variable. The elastic modulus of each element Ei is modeled by a power-law function of the relative density xi . This can be expressed as Ei

=

xpi E0

ρi

=

xi ρ0

(p > 1) (0 ≤ xi ≤ 1),

(2) (3)

where E0 is the elastic modulus of the solid material, ρ0 is the density of the solid material, and ρi is a variable density. The power p is used to penalize intermediate relative density values and drive the design to a black and white structure. This model has been also used for decades to relate bone’s apparent density with its Young’s modulus [32, 33, 34]. In this algorithm it is possible to define an effective design variable x ¯i as
 Nˆ xi + k=1 xk , (4) x ¯i = ˆ +1 N which is the equivalent to applying an image filter. As shown by Tovar et al. [24], it might be compuˆ = 0 to determine the effective value for the design variable, this tationally more efficient to define N is x ¯i = xi . In topology optimization the efficiencies ηk (t) and ηi (t) in Eq. (1) are not considered. In the same way, no scale is imposed in the topology optimization models. The size of the cells does not necessarily represent a physical quantity. Also, the cell’s neighborhood does not have any restrictions on size or location, except that it is the same for all the cells. Topology optimization in general does not require the use of surface conditions; however, their use can be applied to shape optimization [35]. The HCA algorithm, illustrated in Fig. 6, can be described as follows: Step 1. Define the design domain, material properties, load conditions and initial design x(0). Step 2. Evaluate the field variables y(t) using finite element method. Step 3. Apply the evolutionary rule and update the design variables x(t + 1). Step 4. Check for convergence. The final topology is obtained when the convergence criterion is satisfied; otherwise, the iterative process continues from Step 2. 4.3 Evolutionary rules Tovar et al. [24] presented two approaches to derive an evolutionary rule that drives the system to an optimum configuration. These approaches are referred to as the ratio technique and the control strategies. The ratio technique is based on the fully stressed design (FSD) formulation which has been traditionally used to design truss structures. In this approach, the iterative change in the relative density of each element is given by 1  ∗  p−1 yi , (5) xi (t + 1) = x¯i (t) y¯i 4

Start

?

x(0)

Initial design

x(t) Structural analysis

y(t) FE mesh Evolutionary rule

x(t+1) = R(x(t), y(t)) no

CA lattice

Convergence? yes End

Mass update

Final design

Figure 6: The hybrid cellular automaton algorithm. where t is a discrete time step and p > 1. The control strategies are inspired by error-based phenomenological models used to simulate the functional adaptation of the bone structure [30, 29]. In this approach, one defines a local error signal ei (t) as ei (t) = yi (t) − yi∗ ,

(6)

and an effective error signal given as e¯i (t) =

ei (t) +

Nˆ

k=1 ek (t)

ˆ +1 N

.

(7)

The equilibrium in a CA is determined by the condition ei (t) = 0. When this condition does not hold true, a local rule modifies the design variable xi to restore the equilibrium condition. In control theory, the simplest strategy is the two-position control [36]. Using this controller, the change in mass is a piecewise constant function that can be expressed as ⎧ if e¯i (t) > 0, +cF T dxi (t) ⎨ 0 if e¯i (t) = 0, (8) = ⎩ dt −cR if e ¯ (t) < 0, i T R where cF T and cT are positive constants. The superscripts F and R refer to the processes of formation and resorption. A more complex strategy makes use of the proportional-integral-derivative (PID) control. With the PID controller, the change in mass can be expressed as t d¯ ei (t) dxi (t) = cP e¯i (t) + cI , (9) e¯i (τ )dτ + cD dt dt 0

where cP , cI and cD are positive constants respectively referred to as proportional, integral and derivative gains. The evolutionary rules are now implemented into the hybrid cellular automaton algorithm. The PID control requires tunning of the proportional, integral and derivative gains. A wrong selection of the gain values might decrease the stability margins of the convergence. In the ratio technique, no gains need to be adjusted. In this sense, this technique is simpler to implement; however, the convergence takes longer in comparison to the PID control rules. 5

In order to compare the different control strategies, the change in the mass of the structure is used as a convergence criterion. The iterative optimization process converges when no further change in mass is possible. This state can be expressed as ∆M (t) = M (t) − M (t − 1) ≈ 0.

(10)

Numerical experience with the HCA algorithm has shown that, in some applications, ∆M (t) displays a cyclic behavior in which a small change in mass is followed by a bigger change. To avoid premature convergence, the convergence criterion is defined by using the average change in two consecutive iterations. This yields |∆M (t)| + |∆M (t − 1)| ≤ ε, (11) 2 where ε is a small fraction of the total mass of the solid structure. In this application, the fraction value is defined as ε = 0.001 × M0 , where M0 is the maximum mass of the structure. 5. Unconstrained minimization 5.1 Multi-objective problem Let the optimum structure be of maximum stiffness and minimum weight. This condition can be expressed as a global, multi-objective optimization problem in which stiffness and mass are conflicting functions. Maximizing stiffness is equivalent to minimizing the strain energy. Following the procedure used by Saxena and Ananthasuresh [13], the multi-objective optimization problem can be stated as c(x) = f (U ) + g(M )

min x s.t.

0 ≤ x ≤ 1,

(12)

where f (U ) is a function of the strain energy U and g(M ) is a function of the mass M . The design variable xi varies between the boundaries 0 and 1. In practice, the lower boundary of xi is not zero but a small positive value; for example, 1 × 10−3 . This condition guards against the singularity of the stiffness matrix during the application of the FEM. 5.2 Optimality conditions As demonstrated by Tovar et al. [24], the optimality condition for an interior point, i.e., 0 < xi < 1, is given by ∂U/∂xi ∂g(M )/∂M . (13) =− ∂M/∂xi ∂f (U )/∂U A conventional way to define f (U ) and g(M ) in Eq. (12) is as follows: f (U ) = ω

U U0

(14)

and

M , (15) M0 where U0 and M0 respectively represent the strain energy and mass of the solid design domain, i.e., when xi = 1. The coefficient ω balances the relative weight of the ratios U/U0 and M/M0 in the objective function, where 0 ≤ ω ≤ 1. In a discrete design domain, the strain energy and mass of the structure can be defined as the sum of the strain energies and masses of the discrete elements. This is g(U ) = (1 − ω)

U= and M=

N

ui =

N

xpi u0 ,

i=1

i=1

N

N

mi =

i=1

i=1

6

xi m0 ,

(16)

(17)

where u0 is the solid element strain energy and m0 is its mass. It can be shown that ui ∂U = −pv0 . ∂xi xi

(18)

where v0 is the constant volume of an element and ∂M = m0 . ∂xi If one defines the state variable yi as yi =

ui xi

(19)

(20)

and its optimum value yi∗ as

(1 − ω) ρ0 U0 , (21) ω p M0 then the optimality condition for an interior point can be simplified as yi = yi∗ . It can be demonstrated that the optimality condition when xi = 0 can be expressed as yi ≤ yi∗ , and the optimality condition when xi = 1 can be expressed as yi ≥ yi∗ . The final topology of the structure is determined by the selection of the weight coefficient ω, i.e., the value of yi∗ . The combinations of all the solutions forms the Pareto frontier. yi∗ =

5.3 Pareto optimality The design procedure begins with the definition of the optimization problem. Let us consider the optimum design of a two-dimensional, Michell-type structure in a continuum design domain. The design domain has an area of 50×25 mm2 and its thickness is 1 mm. The design domain is discretized into 50×25 identical CAs. One of its lower corners is restrained to prevent vertical and horizontal displacement, while the displacement of the opposite lower corner is constrained in the vertical direction. A vertical load of 100 N is applied in the middle of the lower edge (Fig. 7). The mechanical properties of the isotropic material correspond to the ones of cortical bone: Young’s modulus E = 20 GPa and Poisson’s ratio ν = 0.3 are considered.

Figure 7: Michell-type structure. The structural optimization problem in Eq. (12) considers two conflicting objectives: minimize strain energy U and minimize mass M . This problem has an infinite number of solutions depending on the selection of the weight coefficient ω. Each solution is denoted as a Pareto optimum. The set of all Pareto optima forms the Pareto frontier. The Pareto frontier provides an insight into the optimization problem that cannot be obtained by looking at a single point in the design space. The complete Pareto frontier can be obtained by varying the weight coefficient ω from 0 to 1. This coefficient determines the equilibrium value yi∗ according to Eq. (21) and, therefore, the final topology (Figs. 8 and 9). 6. Constrained minimization Using the one-to-one correspondence between final mass and the value of the weight coefficient ω, it is possible to solve optimization problems involving a mass constraint. Let us consider for example min x s.t.

U/U0 M/M0 ≤ mf , 0 ≤ x ≤ 1, 7

(22)

Figure 8: Topologies of different Pareto points, ω = 0.05, ω = 0.25 and ω = 0.50 respectively. 3.5

ω=0.05

min ω U/U0 + (1-ω) M/M0

3.0

U/U0

2.5 2.0 1.5 ω=0.25

1.0

ω=0.50

ω=0.75

0.5

ω=1.00

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

M/M0

Figure 9: The set of all Pareto optimal forms the Pareto frontier which is limited on the right-hand side by ω = 1, for which M/M0 = 1 and U/U0 = 1, and limited on the left-hand side by ω = 0, for which M/M0 = 0 and U/U0 = ∞. For an interior point, the Lagrangian function for this optimization problem is given by   M U + λm − mf , L= U0 M0

(23)

where λm is the Lagrange multiplier associated with the mass constraint. If one defines the state variable yi as in Eq. (20) and its optimum value as yi∗ = λm

ρ0 U 0 , p M0

(24)

then the optimality condition for an interior point can be simplified as before. Note that λm =

(1 − ω) . ω

(25)

In order to find the value of λm that satisfies the active mass constraint, this work introduces an outer loop to the HCA algorithm. This outer loop searches for the value of λm or ω. Let us define the mass constraint as M/M ∗ ≤ 1, where M ∗ = mf M0 . Since ω is proportional to the final mass, then the iterative updating rule can be defined as ω(t + 1) = ω(t)

M∗ . M

(26)

Numerical experience has shown that limiting the change of ω provides a better convergence. In this work ω(t + 1) 0.8 ≤ ≤ 1.2. (27) ω(t) For the example presented above (Fig. 7), let us define the mass constraint as M/M0 ≤ 0.40. Figures 10 and 11 depict the results. The solution were obtained using the PID control strategy which is superior 8

Figure 10: Intermediate topologies for t = 3, t = 5, t = 7 and t = 17. to the other approaches as it converge in less iterations. In this two-dimensional problem, the Moore ˆ = 8) was used. neighborhood (N The control of the strain energy is equivalent to the one used for the mass. The same approach has been used for maximum deflection and maximum stress. ω

Μ/Μ0

1.00 0.80 0.60 0.40 0.20 0.00 0

5

10

15

t

Figure 11: Convergence plot for mass control.

5. Conclusions The HCA method presented in this investigation is a new approach to topology optimization for continuum structures. This non-gradient-based method makes use of the finite element method for structural analysis and the cellular automaton paradigm to apply local evolutionary rules. This algorithm follows optimization principles derived from the KKT conditions. Evolutionary rules presented in this research include the ratio technique and control strategies. The HCA method does not require the use of image filtering techniques, gradient constraint or perimeter control strategies to prevent numerical instabilities such as checkerboarding. The computational efficiency of the iterative process is given by the type of evolutionary rule applied. The convergence in this method is determined by local evolutionary rule. The rules modify the design variables in order to achieve the zero-error condition between the effective value of the state variables and their optimum value. The effective value of the state variable is defined as the average value in the neighborhood of the cell. The state variable and its optimum value are derived from the KKT conditions of the structural optimization problem to be solved. This investigation developed the corresponding expressions for both unconstrained and constrained optimization problems. This work demonstrates the implementation and performance of the HCA algorithm through a two-dimensional sample problem. 7. Acknowledgments Support for this research has been provided by the Colombian Institute for Development of Science and Technology “Francisco Jose de Caldas” – COLCIENCIAS, the Fulbright Program, the Honda Initiation Grant HIG 2004 and the Defense Advanced Research Projects Agency DARPA.

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8. References [1] Bendsøe, M. P. and Kikuchi, N., “Generating optimal topologies in optimal design using a homogenization method,” Comp. Meth. Appl. Mech. Engrg, Vol. 17, 1988, pp. 197–224. [2] Theocaris, P. S. and Stavroulakis, G. E., “Optimal material design in composites: An iterative approach based on homogenized cells,” Comput. Methods Appl. Mech. Engrg., Vol. 169, 1999, pp. 31– 42. [3] Allaire, G., Shape Optimization by the Homogenization Method , Vol. 146 of Applied mathematical sciences, Springer, 2002. [4] Bendsøe, M. P., “Optimal shape design as a material distribution problem,” Struct. Optim., Vol. 1, 1989, pp. 193–200. [5] Zhou, M. and Rozvany, G. I. N., “The COC algorithm, part II: Topological, geometry and generalized shape optimization,” Comput. Methods Appl. Mech. Engrg., Vol. 89, 1991, pp. 197–224. [6] Rozvany, G. I. N., “Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics,” Structural and Multidisciplinary Optimization, Vol. 21, No. 2, 2001, pp. 90–108. [7] Schmit, L. A. and Farsi, B., “Some approximation concepts for structural synthesis,” AIAA J., Vol. 12, No. 5, 1974, pp. 692–699. [8] Schmit, L. A. and Miura, H., “Approximation concepts for efficient structural synthesis,” NASA CR-2552 , March 1976. [9] Vanderplaats, G. N. and Salajegheh, E., “A new approximation method for stress constraints in structural synthesis,” AIAA J., Vol. 27, No. 3, 1989, pp. 352–358. [10] Svanberg, K., “The method of moving asymptotes – a new method for structural optimization,” Int. J. Numer. Meth. Engrg., Vol. 24, 1987, pp. 359–373. [11] Venkayya, V. B., “Optimality criteria: a basis for multidisciplinary design optimization,” Comp. Mech., Vol. 5, 1989, pp. 1–21. [12] Rozvany, G. I. N., Zhou, M., and Gollub, W., “Continuum-type optimality criteria methods for large finite-element systems with a displacement constraint.” Struct. Optim., Vol. 2, No. 2, 1990, pp. 77–104. [13] Saxena, A. and Ananthasuresh, G. K., “On an optimal property of compliant topologies,” Struct. Multidisc. Optim., Vol. 19, 2000, pp. 36–49. [14] Fanjoy, D. and Crossley, W., “Using a genetic algorithm to design beam corss-sectional topology for bending, torsion, and combined loading,” Structural Dynamics and Material Conference and Exhibit , AIAA, Atlanta, GA, April 2000, pp. 1–9. [15] Jakiela, M. J., Chapman, C., Duda, J., Adewuya, A., and Saitou, K., “Continuum structural topology design with genetic algorithms,” Comput. Methods Appl. Mech. Engrg., Vol. 186, 2000, pp. 339–356. [16] Xie, Y. M. and Steven, G. P., Evolutionary Structural Optimization, Springer, 1993. [17] Xie, Y. M. and Steven, G. P., “A simple evolutionary procedure for structural optimization,” Comput. Struct., 1993, pp. 885–896. [18] Inou, N., Uesugi, T., Iwasaki, A., and Ujihashi, S., “Self-organization of mechanical structure by cellular automata,” Fracture and Strength of Solids, Vol. 145, No. 9, 1998, pp. 1115–1120.

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[19] Kita, E. and Toyoda, T., “Structural design using cellular automata,” Struct. Multidisc. Optim., Vol. 19, 2000, pp. 64–73. [20] Tatting, B. and G¨ urdal, Z., “Cellular automata for design of two-dimensional continuum structures,” Proceedings of 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2000. [21] Hajela, P. and Kim, B., “On the use of energy minimization for CA based analysis in elasticity,” Struct. Multidisc. Optim., Vol. 23, 2001, pp. 24–33. [22] Abdalla, M. M. and G¨ urdal, Z., “Strctural design using cellular automata for eigenvalue problems.” Struct. Multidisc. Optim., Vol. 26, No. 3, 2004, pp. 200–208. [23] Tovar, A., Niebur, G. L., Sen, M., and Renaud, J. E., “Bone structure adaptation as a cellular automaton optimization process,” 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, California, April 2004. [24] Tovar, A., Patel, N. M., Agarwal, H., and Renaud, J. E., “Optimality of the hybrid cellular automata,” 1st AIAA Multidisciplinary Design Optimization Specialist Conference, Austin, Texas, April 2005. [25] Wolff, J., The Law of Bone Remodelling, Springer-Verlag, Berlin, 1986. [26] Huiskes, R., “If bone is the answer, then what is the question?” J. Anat., Vol. 197, 2000, pp. 145– 156. [27] Cowin, C. S. and Hegedus, D. H., “Bone remodeling I: A theory of adaptive elasticity,” J. Elasticity, Vol. 6, 1976, pp. 313–326. [28] Frost, H. M., “A determinant of bone architecture. The minimum effective strain,” Clin. Orthop., Vol. 175, 1983, pp. 286–292. [29] Carter, D. R., “Michanical loading history and skeletal biology,” J. Biomech., Vol. 20, 1987, pp. 1095–1105. [30] Huiskes, R., Weinans, H., Grootenboer, J., Dalstra, M., Fudala, M., and Slooff, T. J., “Adaptive bone remodelling theory applied to prosthetic-design analysis,” J. Biomech., Vol. 20, 1987, pp. 1135– 1150. [31] Tovar, A., Bone Remodeling as a Hybrid Cellular Automaton Optimization Process, Doctor of philosophy, University of Notre Dame, Aerospace and Mechanical Engineering, 2004. [32] Carter, D. R. and Hayes, W. C., “The bahavior of bone as a two-phase porous structure,” J. Bone Joint Surgery, Vol. 59-A, 1977, pp. 964–962. [33] Currey, S. C., “The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone,” J. Biomech., Vol. 21, 1988, pp. 131–139. [34] Rice, J. C., Cowin, S. C., and Bowman, J. A., “On the dependence of the elasticity and strenght of cancellous bone on apparent density,” J. Biomech., Vol. 21, 1988, pp. 155–168. [35] Tovar, A., Patel, N. M., Kaushik, A. K., Letona, G. A., and Renaud, J. E., “Hybrid Cellular Automata: a biologically-inspired structural optimization technique,” 10th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Albany, New York, August 2004. [36] Shearer, J. L., Kulakowski, B. T., and Gardner, J. F., Dynamic modeling and control of engineering systems, Prentice Hall, 2nd ed., 1997.

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