Topology Optimization of Truss Structures using Cellular Automata ...

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

Topology Optimization of Truss Structures using Cellular Automata with Accelerated Simultaneous Analysis and Design Henry Cort´ es1,a , Andr´ es Tovar1,a , Jos´ e D. Mu˜ noz1,b , Neal M. Patel2 , John E. Renaud2 (1) a. Department of Mechanical and Mechatronic Engineering, b. Department of Physics National University of Colombia - Carrera 30 # 45-03, Bogot´ a, Colombia Emails: [email protected], [email protected], [email protected] (2) Department of Aerospace and Mechanical Engineering University of Notre Dame - Notre Dame, Indiana 46556, USA Emails: [email protected], [email protected]

Abstract Cellular automata (CAs) have been used as an alternative to represent macroscopic behavior of continuum systems governed by non-linear partial differential equations (PDEs). This task is generally accomplished using simple rules that represent the micromechanics of the system. CA models are composed of regular lattice of cells or automata. Each automaton may change its state at discrete, fixed times (iterations) according to a local rule. The premise behind a CA model is that an overall global behavior can be computed based on local rules operating on cells that only know local conditions. These rules depend on the present states of the cell and its neighbors within a certain proximity. CAs have been proven to have an inherent massive parallel computation capability which makes CAs an efficient computational tool for large systems. This work introduces a new strategy that uses CA rules for analysis and design, and is applied to solving topology optimization problems of truss structures. Analysis rules are derived from local equilibrium equations. They govern cell displacements, where each cell is a nodal point of the truss structure. The displacements represent the field variables of the analysis. Design rules are derived from optimality conditions. The cross sectional areas of the truss members connected to each cell are used as design variables. In this work, an accelerated convergence technique, based on prediction using gradient information, is formulated and implemented. The computational efficiency of this approach is demonstrated through two-dimensional truss structures. Keywords: Topology optimization, cellular automata, simultaneous analysis and design (SAND), truss structures design.

1. Introduction The conventional techniques to do analysis and design for the industry are based on highly flexible finite element implemented in software of numerical analysis. This techniques are useful for accomplishing that, but the need to accomplish it repeatedly during the design cycle does it necessary to automatize the design process, leading to the intensive use of optimization algorithms to drive this task. Nevertheless, the use of optimization together with stages of analysis for finite elements bring computational requests increasingly to fulfill. On one side the increase on the speed and memory capacity of the modern computers has induced them to make increasingly more complex analysis that usually require a lot of degrees of freedom, turn hard to accomplish this in the design stage due to the large numbers of repetitive analysis and the great computational cost. On another side the structures have been increased in complexity, and therefore the grade of the analytical complexity of the analysis too, turning the algorithms often less and less efficient due to the complexity of the solution techniques. To work out great part of the problems mentioned, the CAs were combined with the finite element method (FEM) to solve topology optimization problems of continuous structures. This approach have been referred to as the hybrid cellular automaton (HCA) method. The HCA method uses FEM for structural analysis and local CA rules for design. This technique have been studied in recent investigations, being used to develop algorithms for reproducing the bone remodeling process [1] and design of compliant mechanisms [2]. The HCAs have offered substantial improvements in the case of efficiency and robustness, striking out problems such as the obtaining of structures with chessboard patterns, among others. The use of FEM to perform the stress analysis and the CAs for optimizing also have been investigated by other authors [3],[4] which dealing with problems such as the not obvious mathematical relationship between the automaton rule and the optimization problem, and the difficulty of introduce the stress constraint conditions to the optimization problem. In the same foundations the evolutionary structural optimization (ESO) scheme [5] was developed. The HCA, ESO and similar schemes add certain improvements to the optimization stage but keep on needing to accomplish a complete analysis by FEM before accomplishing an iteration in the design stage. Furthermore, the FEM has been required

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for the increasing demand of analysis of complex structures and physical phenomena. This has demanded the use of multiple processors and in general of parallel computing. The use of the FEM on parallel computing1 [6] have appended disadvantages such as a decreasing rate of processing speeds with additional processors. In parallel to these contributions, other investigations have taken away whose mayor goal is to avoid complete stress analysis while it accomplishes adding an inherent parallelism. A possible solution to these troubles is to attack the governing differential equations of a solid in terms of both, the field variables and the design variables. The conventional methods of variational calculus have been used for domains with simple geometries, such as bars, beams and shafts. For example treating the cross-sectional area distribution along the beam length as a continuous variable, it is possible to derive an optimality condition that can be used to solve both, the optimal area distribution and the corresponding displacement fields. For problems with complex geometries, solution of the design problem may be performed by formulating the constraints of the original optimization problem and the discrete equilibrium equations of the structural analysis model as equality constraints. This approach is called simultaneous analysis and design (SAND) [7], and requires only the solution of one nonlinear optimization problem without the need to complete structural analysis at every design stage. On this scheme, the analysis variables such as displacements and stresses are treated as unknown variables of the optimization problem in addition to the conventional sizing design variables. The SAND scheme using CA has been studied previously [8] resulting in a promising method to reduce the complexity of the system and improve the efficiency of the numerical solution process. For these reasons this last study is the foundation of the present research. The Cellular automata emulate systems in nature which exhibit sophisticated collective information-processing through local rules of interaction with their nearest neighbors. The rules can involve well known concepts such as force equilibrium, or momentum conservation, or may be specific equations relating values of state variables at various points in the discrete system to those at neighborhood sites. However there could be problems where such relations may not be readily apparent. On this case the use of a genetic algorithm based optimization process has been studied [9] to derive the rules for local interaction required in evolving the cellular automata. In previous studies, it has been observed [10] that the CA convergence rate deteriorates considerably as the lattice is refined. The deterioration of the convergence rate is due to slow propagation of information across the domain; that is, as the cell spacing is reduced, more iterations are needed to update all cells to achieve full convergence in the domain. In order to treat this hitch, new techniques were formulated, one of them [11] uses a genetic algorithm (GA) in order to find a CA with discrete state space, having the smallest possible lattice size and the smallest possible number of discrete states, owing to that the CA with discrete state space evolves much faster than the CA with continuous state space. Another recent technique to accelerate convergence uses a Multigrid (MG) acceleration scheme for CA algorithms [12]. (Specifically for combined analysis and design problems). The present investigation attempts to increase the speed of the structural analysis stage to accelerate the convergence of the SAND technique keeping the distinctive simplicity of the CA scheme. The idea is to make use of the selforganization feature that possesses the CA. The computational efficiency of this approach will be demonstrated through two-dimensional truss structures.

2. Cellular automaton paradigm The evolution in the time of physical quantities is often governed by non-linear partial differential equations. This nonlinearities, solutions of these dynamical systems can be very complex and strongly sensitive to initial conditions. Cellular automata (CA) models provide an alternative method to describe, understand and simulate the behavior of complex systems. CAs are an idealization of physical system in which space and time are discrete. CA models are composed of a regular lattice of cells. Each cell may be in one of a finite number of states. A cell or automaton may change its state at fixed discrete time steps according to a local rule. The premise under a CA model is that an overall global behavior can be computed by local rules over cells that only know local conditions. This rule depends on the present states of the cell and its neighbors within a certain proximity. Since the computations are limited to neighborhoods and the local rules are identical for the whole lattice, CAs have been proved to have an inherent massive parallel computation capability [13]. CAs were used as early as 1946 by Weiner and Rosenblunth [14] to describe the operation of heart muscle, John von Neumann, who was involved in the design of the first digital computer, is certainly the one who formalized the CA theory at the end of 1940s [15]. In physics the CA has been used to simulate dynamic phenomena such as traffic flow [16],[17], population dynamics [18], fluid particle flow, crack propagation, crystal structures, microstructure evolution, among many other applications [19],[13]. The question that arises next concerns is the development of techniques to asses the range of a model´s validity and its limitations. 1 the

substructuring concept in finite element analysis

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2.1. Components A basic CA is composed of: (1) a regular lattice of cells in a d -dimensional space; (2) a set of K discrete variables, i.e., states, S(r, t) = [S1 , . . . , Sk ] associated with each cell located in r, at the discrete time t = 0, 1, 2, . . .; and (3) a set of K rules R = [R1 , . . . , RK ] which specifies the time evolution of the states as

Sk (r, t + 1) = Rk (S(r, t), S(r + δ1 , t), . . . , S(r + δj , t)),

(1)

where r + δj designating the cells belonging to a given neighborhood of the cell r. In the simplest case, the state of each cell consists of a single bit S1 (r, t) of information. In the above definition, the set of rules R is identical for all sites and is applied simultaneously to each one of them, leading to a synchronous dynamic. In other words, the rule is homogenous, and does not depend on the position r. However, spatial (or temporal) inhomogeneities can be introduced to model a particular characteristic (i.e., boundary conditions, random phenomena, etc). An asynchronous updating scheme is useful to model events which do not necessarily occur in parallel. The form of the cellular space directly reflects the physical dimensions of the problem under consideration. Usually, the CA models are incorporated in one, two or three dimensions. Three sample lattice structures representing twodimensional cellular spaces, are shown in Fig. 1. The rectangular lattice configurations are often used, but other two-dimensional cell geometries, such as triangular and hexagonal lattices, are also employed. Frisch, Hasslacher and Pomeau (1986) [20], for example, used a regular two-dimensional lattice of triangular cells for a cellular automaton fluid model, while Wolfram(1986) used the hexagonal ones [21]. The neighborhoods do not have any restriction in size or location, except that it is the same for all cells. In practice, the size of the neighborhood is often limited to the adjacent cells but can also be extended. Figure 2 depicts some common neighborhood layouts.

(a) Rectangular lattice

(b) Triangular lattice

(c) Hexagonal lattice

Figure 1: Two-dimensional lattices

(a) Empty (N = 0)

(b) Von Neumann (N = 4)

(c) Moore (N = 8)

Figure 2: CA neighborhoods To define the evolution rule for a cell located on the boundary of the design domain, the design domain can be extended in different ways. A fixed boundary is defined so the neighborhood is completed with cells having a pre-assigned fixed state. Other boundary types are adiabatic .This work makes use of fixed boundary conditions where extra cells are considered empty spaces without physical or mechanical properties.

2.2. CA Representation of two-dimensional truss structures A regular array of truss elements that form a rectangular domain. Each node is connected to the neighboring nodes (a ”ground truss” configuration). The Fig. 3(a) shows an example of this configuration type. The computational cells are composed of a node and the eight truss members connected to it at every 45o orientation. The neighborhood is, therefore, composed of the eight adjacent nodes (Fig. 3(b)). This is the Moore neighborhood. To describe structural domains, selective assignment of cell properties are sufficient. Therefore, boundaries are defined by setting the member areas for cells outside of the domain to be zero, which effectively removes those truss members. The vertices of the truss are the center points of cells and the domain of each cell extends halfway between the neighboring vertices in each direction.

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uc

vc

Cell

Cell

(a) A truss structure in the CA scheme

(b) An CA cell and its neighborhood

Figure 3: A CA representation of two-dimensional truss structures. For the cell so defined, its state may be expressed in terms of field variables ({u, v},{Fx , Fy }) and design variables ({A1 , A2 , . . . , A8 , E}) associated with the cell. Thus, the state of a cell is represented by

Sk = {[u, v], [Fx , Fy ], [A1 , A2 , . . . , A8 , E]}k ,

(2)

where u and v are the displacements of the cell measured from its no-formed position, Fx and Fy are the forces applied to the cell along the x and y coordinates, respectively, A1 , A2 , . . . , A8 represents the cross-sectional area of members into the cell, and E is the modulus elasticity of every member. The expression (2) represent the state of every cell in the domain. The update rule will be applied to all cells in the domain to change some of the state variables. It is possible to change specific variables in expression (2) based on different criteria, accord to field (analysis, design) will be updated. For the displacement update, the objective is to determine the location of a cell point given the displacement of all the neighboring points. For the update of the cross-sectional areas, the objective is to obtain the minimal cross-sectional area of each member to achieve the strength condition without overcoming the allowable strength of the material.

3. Methodology Two sets of rules (analysis rules and design rules) are used to compute an optimal solution to a given topology optimization problem. Where the final results are a set of truss bars with the minimum size required to support the applied forces. One sample rule for analysis and design was presented before by G¨ urdal and Tatting (2000) [8], and was adopted in the present work.

3.1. Evolutionary rule for analysis Within a cell, each truss member (indexed relative to the cell by k = 1, . . . , 8) has an elastic modulus E , a cross-sectional area Ak , length Lk , and an orientation angle θk from the cell center. Denote the displacement of the k th truss members near end from the original cell center by (u, v ), and the displacement of the far end from the neighboring cell’s center by (uk , vk ). These neighboring displacements are taken as fixed when the CA rule calculates the displacements (u, v ) for each cell. The strain εk , and force fk within each member are calculated from these properties and displacements by

εk = ∆k /Lk = (Ldk − Lk )/Lk ,

(3)

fk = EAk εk .

(4)

Where the extension ∆k can be calculated by one of two forms mainly, depending on the way the strains are computed. For the linear case, each member strain is computed using the projection of the displacements at the two ends of the member onto its no-formed state. This approach result in the following expression for the member extension,

∆k = (uk − u) cos θk + (vk − v) sin θk ,

(5)

which is a linear function of the unknown displacements. However its simplicity involve some error if any of the members undergo substantial rotations with respect to their original orientation. To overcome this deficiency, nonlinear straindisplacement relations must be used. The use of nonlinear relations result on a set of two coupled highly nonlinear equations that need a numerical iterative solution. This nonlinear solution require longer computation time than the linear algorithm and an increase on the complexity level of the algorithm formulation.

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Due to simplicity required for a first analysis of the accelerated convergence technique, the linear case of the equation (5) is selected for this work solving problems with small displacements. The CA’s technique suggests that applying a local rule that leads to a local equilibrium, the structure as a whole also tends towards same behavior, obtaining so a global equilibrium. Taking this into account, when minimizing the total potential energy for the bars that are members of an individual cell, is getting equilibrium equations. In this way, the node will move by itself to minimize the potential energy of the local system. The total potential energy for a cell is given by

Π = U + W,

(6)

where: U is the total strain energy, and W is the potential of work (the work done by external forces). Thus, developing the Eq. (6) and using the relations (5) and (3), it follows that total potential energy for a given cell results in

Π=

8 X

2 1 2 (EAk Lk εk )

8 X

− Fx u − Fy v =

k=1

1 1 2 EAk Lk L2k [(uk

− u) cos θk + (vk − v) sin θk ]2 − Fx u − Fy v.

(7)

k=1

To minimize the potential energy Π, we setting the partial derivatives of the potential energy with respect to cell displacements to zero

∂Π = 0, ∂u doing it for the horizontal displacement case we have that 8 X

1 Lk EAk [(uk

∂Π = 0, ∂v ∂Π ∂u

(8)

= 0 lead to

− u) cos θk + (vk − v) sin θk ] cos θk + Fx = 0.

(9)

k=1

Replacing the expresions (5) and (3) into Eq. (9) we have 8 X

1 Lk EAk ∆Lk

cos θk + Fx = 0,

or

8 X

EAk εxk + Fx = 0.

k=1

k=1

Is easy to see that here we get the equilibrium equations for the x direction as follow. 8 X

fxk + Fx = 0

(10)

k=1

Similarly, doing

∂Π ∂v

= 0 lead to corresponding equilibrium equations for the y direction as the Eqs. (11) and (12) show. 8 X

1 Lk EAk [(uk

− u) cos θk + (vk − v) sin θk ] sin θk + Fy = 0

(11)

k=1 8 X

fyk + Fy = 0

(12)

k=1

Where: fxk y fyk : are the horizontal and vertical components of internal force in the k th member (k =1,. . . ,8). Fx y Fy : are the horizontal and vertical components of external force applied to the cell. The force components fxk and fyk can be calculated from the internal force in the truss members using orientation of the k th member θk as follow.

fxk = fk cos θk ,

fyk = fk sin θk

(13)

The equations (10) and (12), or its equivalents Eqs. (9) and (11) respectively, conform a system of two equations with two unknowns u and v , where the values of uk and vk are taken from the previous iteration (t-1).

3.2. Accelerated convergence technique In previous studies, it has been observed [10] that the CA convergence rate deteriorates considerably as the lattice is refined. The deterioration of the convergence rate is due to slow propagation of information across the domain; that is,

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as the cell spacing is reduced, more iterations are needed to update all cells to achieve full convergence in the domain. In order to treat this hitch, were formulated new techniques, one of them [11] use a genetic algorithm (GA) to search the optimum CA’s parameters (lattice size, number of discrete states) before the final run. Other recent technique for accelerate convergence use a Multigrid (MG) acceleration scheme for CA algorithms [12]. (Specifically for combined analysis and design problems). Keep the distinctive simplicity of the CA scheme on the accelerating of the convergence in the SAND technique, is the main feature of the novel accelerated convergence technique formulated in the present research. 0.015

0.016 0.014

1

0.012

x 10

NDA OddNA PairNA DforA EDA

0.01 −3

x 10

Odd data Pair data

v

1.25

v

1.3

0.01

0.8

0.008

0.6

v

0.005

(2)

v

1.2

−3

1.15

0.006

0.4

0.004

0.2

(1)

1.1 1.05 1 50

0 0

500

1000

1500

55

2000

60 t

65

2500

0.002

70

0 0

3000

t

(a) without accelerating

0 500

2

1000

4

6 1500

t

8 t 2000

10

12 2500

14 15 3000

(b) comparison between with and without accelerating

Figure 4: A typical behavior in the vertical displacement of an node in a CA implementation for structural analysis. The part (b) show a comparison between the structural analysis data without accelerating OddNA (Odd iterations non accelerated), PairNA (pair iterations non accelerated); and with accelerating NDA (Normal Data after each Accelerating), DforA (Data for do the accelerating), EDA (Extrapolated data in the Accelerating). Looking at the vertical displacement of a cell (node) on a CA implementation for structural analysis (Fig. 4(a)), it is easy to see that there are two evident paths around the main path; corresponding the data for the odd iterations (odd data) and the data for the pair iterations (pair data) respectively. This kind of behavior has been observed in many other cases of CA implementations. As because both, the odd data and the pair data paths are very closed to the main path (average tendency) either can be used to describe the real behavior. To reduce the great time needed to get the equilibrium state (final configuration), we propose the use of gradient information to realize consecutive extrapolations for the displacements (horizontal and vertical) in each cell. To do it only one set of data (odd data or pair data) are required as was indicated before. As show Fig. 4(b), the strategy consist in take the gradient information of two consecutive data of the DforA set (Data for Accelerating) and do the prediction of the future state (EDA, Extrapolated Data in the Accelerating) various iterations forward (arrow (1)), further(s) iteration(s) (the first NDA after each EDA) is required to achieve a real state and stabilize the abrupt change in the tendency. This process is consecutively repeated until the convergence. Moreover, Fig. 4(b) show an distinctive improvement, achieving the final state with a great reduction in the number of iterations (t). The extrapolated data fbt+T (for the EDA set) are computed from two previous data {t − 2, ft−2 } and {t, ft } as show the following linear extrapolation:

ft − ft−2 fbt+T = [(t + T ) − (t − 2)] + ft−2 = 21 (ft − ft−2 )(T + 2) + ft−2 , t − (t − 2)

(14)

where t is the actual iteration, t − 2 is the iteration corresponding to the previous DforA, T is the number of iterations to skip; ft and ft−2 are the displacements for iterations t and t − 2 respectively. This is, f represent one of its component displacements (u or v ).

3.3. Evolutionary rule for optimization The design variables for the structural optimization problem are the cross-sectional areas of the members connecting the cell to its neighboring cells. The cross sectional update equation used on this work is based on a fully stressed design (FSD) formulation [22]. A FSD

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formulation states that “For the optimum design each member of the structure that is not at its minimum gage is fully stressed under at least one of the design load conditions,” in other words it is assumed that a truss member needs no more area than the least required to support the allowable stress (σall ). The stress in the truss members are computed from the displacement of a cell and its neighbors by

σk = Eεk ,

k = 1, . . . , 8,

(15)

where the strains are computed from Eq. (3) and (5). The ratio technique is used to resize truss structural elements in the FSD approach. The ratio technique assume a force into the member that keep constant and thus the relation between stress and cross-sectional area follow a typical fk behavior as show the Fig. 5. The stress can be expressed as σk = A . In the same way, the equilibrium condition k

σk∗ can be approximated to σk∗ =

fk∗ , A∗ k

where fk∗ and A∗k represent the internal force and the cross-sectional area at the

equilibrium state. Hence, the ratio σk∗ /σk can be expressed as constant force, that is fk∗ = fk , thus

A∗k = Ak

∗ σk σk

=

fk∗ Ak , fk A∗ k

and keeping in mind the assumption of

σk . σk∗

(16)

Using Eq. (16), the updating rule can be written as (t+1)

Ak

(t) (t) σk

= Ak

σall

,

(17)

σallowable

σ

where t is a discrete time step and σall is the allowable stress of the material. Since the propagation of displacement takes certain time (iterations) on reach to some cells, the internal force at those

Ak

(t)

Ak

(t+1)

A (t)

Figure 5: Ratio technique scheme. To resize a truss member, the previous cross-sectional area Ak is afected by a ratio (t+1) to achieve the new area Ak which is the optimum area to support the allowable stress σallowable . places will be zero. In such case, the cross-sectional area become zero, and therefore, these cells will not take part in future iterations. Consequently on this work, we lessening the member area re-sizing along with a small lower limit AL is implemented for the cross-sectional area of cells.

3.4. Algorithm The Fig. 6 show the flow chart for the Accelerated SAND using CA for topology optimization of two-dimensional truss structures. The algorithm is composed of three stages. The first stage is structural analysis, where the cell displacements are updated by the CA structural analysis rule (AR(x(t))) given by the Eqs. (9 and 11) or Eqs. (10 and 12). These update displacements are penalized to fulfill the restrictions. The second stage is accelerating convergence, where the gradient information is taken (first and second data) to make the forward prediction by a linear extrapolation (Eq. 14), and these results are penalized too. The third stage is optimization, where the CA design rule (Eq. 17) is applied to make periodical resizing each N iterations to guarantee the displacements propagation through the structure. The convergence reaches when the maximum cell displacement (Dxmax .) and the maximum change on elements area (DAmax .) are less than δx and δA respectively.

4. Implementation The following example show the Accelerated SAND using CA (ASAND) algorithm. That is a 10-bar truss example, and

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Start

x(0)=0 Ak(0)=AL ; AOut of Domain=0

Structural Analysis

Update Cells Displacement using CA Structural Analysis Rule x (t+1)=AR(x (t))

Penalize displacements to fulfil restricctions x (t+1)=P0 *x (t+1)

Accelerating convergence Yes

Take gradient information? t=t* or t=t*+2?

Is the first data? t=t*?

No

No Second data

Yes

Take information

Take information: t, x (t) Do the prediction (linear extrapolation)

t, x (t)

t*=t+2 x E (t)=f (t*,x (t*); t, x (t), T)

Penalize displacements to fulfil restricctions x (t+1)=P0 *x E (t)

D xm

m a x (t)| . m a x (t+1)-x . a x =|x .

Make the resizing? t is multiple of N? D Ak m

No

D xm

No

Yes

Optimization Update Areas using CA Design Rule

. . a x =|A . k m a x (t)| k m a x (t+1)-A

A k (t+1)=DR(A k (t))

δx ?

a x< .

and D Am

a x