AIAA 2004-4558
10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 30 August - 1 September 2004, Albany, New York
Hybrid Cellular Automata: a biologically-inspired structural optimization technique Andr´es Tovar∗
Neal Patel†
Amit K. Kaushik‡
Gabriel A. Letona§
John E. Renaud¶
Department of Aerospace and Mechanical Engineering University of Notre Dame, Notre Dame, Indiana, 46556.
Brian Sanders Air Force Research Laboratory, Wright-Patterson AFB, Ohio, 45433.
In this investigation the hybrid cellular automaton (HCA) method for structural synthesis is extended to facilitate simultaneous topology and shape optimization. The HCA methodology has been developed for application to continuum structures. The development of this methodology has been inspired by the biological process of bone remodeling. In bone remodeling, only those elements located on the surface of the mineralized structure can be modified. In the HCA methodology implemented in this research only surface elements are allowed to change density during the structural synthesis process. The HCA method combines local design rules based on the cellular automaton paradigm and finite element analysis. Closed-loop control is used to modify the mass distribution on the internal and external surfaces of the design domain to find an optimum structure. The local control maintains a balance between mass and rigidity. The new methodology effectively combines elements of topology optimization and shape optimization into a single tool. Three classes of test problems are used to illustrate the method’s efficacy.
I.
Introduction
Shape optimization involves determining the optimal profile or boundary of the structure. Two of the most common approaches for shape optimization are: the basis vector and the grid perturbation approach. The basis vector approach requires the definition of different trial designs called basis vectors. The design variables are the weighting parameters that define the participation of each basis vector in the design process. On the other hand, the grid perturbation approach requires the definition of perturbation vectors. These vectors perturb or deform the boundary of the design domain. The design variables are the values that determine the amount of the perturbation during the optimization process. Topology optimization strives to achieve the optimal distribution of material within a finite volume (design domain) that maximizes certain mechanical performance under specified constraints. In a continuum, the design domain is discretized into a large number of elements representing finite portions of material. The topology optimization algorithm selectively removes and relocates these elements to achieve optimum performance. Numerical methods, available in some commercial packages, parametrize the material distribution problem into a set of continuum design variables. These design variables depend on the type of material model used in the optimization algorithm. The nature of the material model characterizes the different approaches to topology optimization. The most commonly referenced approaches are the homogenization approach1, 2, 3 and the SIMP (solid isotropic material with penalization) approach.4, 5 For an overview see, e.g., Eschenauer & Olhoff 6 and Rozvany.7 ∗ Graduate
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1 of 15 American Institute of Aeronautics andInc., Astronautics Copyright © 2004 by John E. Renaud. Published by the American Institute of Aeronautics and Astronautics, with permission.
In the SIMP approach, 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 as a function of the relative density, xi , using a power law. This is Ei (xi ) =
xpi E0
ρi (xi ) =
xi ρ0
(p ≥ 1) (0 ≤ xi ≤ 1),
(1) (2)
where ρ0 is the density of the base material, ρi is a variable density, and p is a penalization power. This power is used to penalize intermediate densities, which generally leads to a black and white structure. For an overview on this method see, e.g., Bendsøe & Sigmund.8 In topology optimization, the number of elements and, hence, the number of design variables depends 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 the classical gradient-based optimization methods might be impractical. This has motivated the implementation of efficient, specialized numerical methods. Some of the most common methods include approximation techniques,9, 10, 11 methods of moving asymptotes (MMA)12 and optimality criteria.13, 14, 15 Numerical instabilities such as checkerboarding and mesh dependency are commonly found in topology optimization. Checkerboarding refers to regions where the solid elements (black) and voids (white) alternate forming a checkerboard pattern. Mesh dependency refers to obtaining qualitatively different topologies for different mesh sizes. Usually, image filtering techniques, gradient constraint and perimeter control strategies are used to deal with these numerical instabilities.16 The purpose of these techniques is to smooth the spatial variation of the design variables to avoid instabilities; however, convergence delays and intermediate densities are associated with the use these techniques. In Tovar et al.33 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 a hybrid cellular automaton (HCA) method with local control rules. In this approach, the design domain is discretized into a regular lattice of CAs. Each CA 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. This approach is inspired in control models proposed in bone remodeling simulations.17, 18 In Tovar et al.,33 the power-law approach is used as the material model. In this investigation, the HCA methodology is extended to include both aspects of topology optimization and shape optimization. Only surface elements on the structure can be modified by the modified HCA algorithm. The modified HCA methodology in effect performs topology optimization using only those elements on the surface. This can effectively emulate shape optimization techniques starting from a known structural genus. In addition, the use of an on-off control strategy with intermediate densities allows the modified HCA method to synthesize topologies with a different genus than the initial design. This work has been inspired by the biological process of bone remodeling in which only surface elements are remodeled.
II.
Cellular automaton paradigm
The time evolution of physical quantities is often governed by non-linear partial differential equations. In many cases, the solutions of these dynamic systems can be very complex and strongly sensitive to initial conditions. Cellular automata (CAs) provide an alternative method to describe, understand and simulate the behavior of complex systems.19 CAs are a computational technique that has been used to simulate biological phenomena for over 60 years. CAs were used as early as 1946 by Weiner and Rosenblunth to describe the operation of the heart muscle.20 John von Neumann formalized the CA theory at the end of the 1940s.21 CAs are an idealization of a physical system in which space and time are discrete.22 CA models are composed of a regular lattice of T cells. A cell or CA, at discrete position r, is defined by a set of states, S = {S1 , . . . , Sn } . The evolution of each state, Si (r, t + 1), is governed by a local rule defined as Si (r, t + 1) = Ri (S(r, t), S(r + ∆1 , t), . . . , S(r + ∆N , t), S(r, t − 1), . . . , S(r, t − T )),
(3)
where r + ∆j designates the cells belonging to a given neighborhood of the CA and T is a time span that T accounts for historic data. The set of rules, R = {R1 , . . . , Rn } , is identical for all CAs and is applied 2 of 15 American Institute of Aeronautics and Astronautics
(a)
(b)
(c)
(d)
(e)
Figure 1. CA Neighborhoods. (a) Empty (N = 0), (b) Von Neumann (N = 4), (c) Moore (N = 8), (d) MvonN (N = 12), and (e) Extended (N = 24).
simultaneously to all cells. The CA neighborhood is an idealization of the radius of action of the local rule. The neighborhood does not have any restrictions on size or location, except that it is the same for all the CAs. Since the computations are limited to neighborhoods and the local rules are identical for the whole lattice, CAs have been proven to have an inherent massive parallel computation capability. In practice, the size of the neighborhood is often limited to the adjacent cells but can also be extended. Figure 1 depicts some common neighborhood layouts. The most commonly used are the von Neumann layout that includes four neighboring cells (N = 4) and the Moore layout that includes eight neighboring cells (N = 8). Another possible layout is the so-called MvonN composed of twelve cells (N = 12). The neighborhood can also be reduced down to an empty layout (N = 0), or extended as much as the model requires. In addition to the layouts described above, this work makes use of an extended neighborhood that includes 24 cells (N = 24). The von Neumann neighborhood is used in this study unless specified otherwise. 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. Figure 2 depicts different types of boundary conditions obtained by extending the design domain. A fixed boundary is defined so the neighborhood is completed with cells having a pre-assigned fixed state. An adiabatic boundary condition is obtained by duplicating the value of the cell in an extra virtual neighbor. In a reflecting boundary, the state of the opposite neighbor is replicated by the virtual cell. Periodic boundary conditions are used when the design domain is assumed to be wrapped in a torus-like shape. This work makes use of fixed boundary conditions where the extra cells are considered empty spaces without physical or mechanical properties.
X
X
(a)
0
X X
(b)
X
(c)
X
(d)
Figure 2. CA Boundaries. (a) Periodic, (b) Fixed, (c) Adiabatic and (d) Reflecting.
III.
Cellular automata for structural synthesis
CA models have inspired different techniques for structural synthesis. A basic CA-like approach was developed by Inou et al. wherein the design variables were the elastic moduli of the cells, which were iteratively updated using a heuristic function.23, 24 This function operated in each CA using the difference between the current stress, evaluated by the finite element method, and a target stress. Evolutionary rules based on the growing/reforming procedure were used to fine tune the structure. Cells with low elastic modulus are removed while cells with high elastic modulus create a new cell in the empty surrounding space. This approach led to structures that are similar to the ones observed in bird bones.24 Even though this is not necessarily a topology optimization algorithm it illustrates the application of an evolutionary CA rule. More recently, the concept of a CA model for topology optimization was presented by Kita & Toyoda.25 In their approach, thickness was used as the design variable. The local design rule was derived from the optimality condition of a multiobjective function. This multi-objective function minimizes the weight of the
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structure and the deviation between the yield stress and the equivalent stress in a Moore neighborhood for each CA. The finite element method was used to evaluate these stresses in each iteration. The algorithm required hundreds of iterations to achieve convergence even in simple test problems. The CA model presented by Tatting & G¨ urdal26 is implemented with a simultaneous analysis and design (SAND) approach. In their work, both field and design variables are simultaneously updated. Hundreds of thousands of iterations are required to achieve convergence;27 however, the algorithm can make use of massive parallel computing. Therefore, the overall computational time can be reduced compared to techniques based on the finite element method; however, the convergence can deteriorate as the number of elements increase. This occurs because the (field variable) information propagates slowly. Multigrid and full multigrid acceleration strategies have been shown to mitigate this problem.28 In a recent publication, Abdalla & G¨ urdal29 demonstrated the SAND-CA approach in application to column design for buckling. The difference between a hybrid cellular automaton (HCA) and a CA method is that the HCA makes use of global information, while the CA just uses local information. In the present context, an HCA method makes use of the finite element method, while a CA method uses local equilibrium equations. The residual between external work and internal energy in an HCA method is zero in every iteration, while in a CA approach the residual is iteratively reduced to zero. As reported in the literature, the number of iterations in a HCA approach can vary from several tens27 to several hundreds25 while in a CA approach the number of iterations is measured in hundreds of thousands;27 however, the time for convergence in both approaches might be comparable. This work presents a new HCA method where different controllers are implemented in the local design rules. No gradient information is required in the design process. Numerical instabilities such as checkerboarding and mesh dependency can be avoided using CA parameters. In comparison to other finite-element-based methods, the HCA presented in this work can dramatically reduce the number of iterations.
IV.
Hybrid cellular automaton method
In the HCA method,33 the state of each cell, S i , is defined by design variables S Di (e.g., geometry and material properties) and field variables S F i (e.g., stress and strain functions). This is S Di . (4) Si = SF i Local design rules modify the design variables and lead the field variables toward a target, S ∗F i . The HCA algorithm, illustrated in Fig. 3, is described as follows: Step 1 Definition of the design domain, load conditions, initial design and material properties. Step 2 Evaluation of the field variables by global structural analysis using the finite element method. Step 3 Evaluation of the local equivalent field variables, S¯P i , using the CA neighborhood. Step 4 Comparison of the averaged field variables, S¯P i , with the target or optimum values SP∗ i . Step 5 Application of a local design rule to modify the design variables. Step 6 Check for convergence. The final topology is obtained when there is no change in the design variables; otherwise, go to Step 2. The design variables for each CA, S Di , are defined according to the material model. This work makes use of the power-law approach. The relative density, xi , defined in Eq. (2), is used as the design variable for each CA. The field variables, S F i , and the targets, S ∗F i , depend on the definition of the optimization problem. An optimum structure is designed to use the minimum amount of material, while supporting the external loads. Strain energy, U , is a measure of the mechanical energy stored in the structure while undergoing deformation. The more rigid the structure, the lower the strain energy the structure can store. The strain energy in a discretized domain can be expressed as U=
Ne
Ui vi ,
i=1
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(5)
Start Local design rule
Structural analysis FEM no
?
yes
End
Figure 3. Hybrid cellular automaton algorithm with local control rules. The iterative process involves finite element analysis, application of a local design rule and a convergence test. The local design rule seeks to satisfy Eq. (7) using a control strategy.
where Ui is the strain energy density (SED) in the CA and vi is the volume of the CA. The SED Ui is the local indicator of the the structure’s rigidity and, consequently, the field variable used in the present work. In this way, the state of each CA is defined by xi Si = . (6) Ui Following the principles of a fully stressed design, the local optimization problem can be stated as min
xi
s.t.
¯i − U ∗ = 0 U i 0 ≤ xmin ≤ xi ≤ 1.0. i
xi
(7)
For a discussion of optimality using a fully stressed design approach see, e.g., Patnaik & Hopkins.30 In the ¯i is the average SED in the CA neighborhood. previous expression, Ui∗ is a pre-defined local SED target and U This is N Ui + j=1 Uj ¯i = , (8) U N +1 ¯i toward the target U ∗ , the HCA where Uj corresponds to the SED of a neighboring cell. In order to drive U i algorithm implements local design rules based on control theory. The iterative process converges when the ¯i = 0 or that the CAs are saturated. Saturation design does not change. The convergence implies that U ∗ − U in Eq. (7), is required occurs when the relative density, xi , reaches one of the bounds. The lower bound, xmin i −3 to avoid singularity in the stiffness matrix for the FEM. In this work, xmin = 10 is used. i
V.
Local design rules
Biological structures and materials are continually adapting to changes in their physical environment. In bones, for example, it has been widely accepted that mineral tissue is resorbed in regions exposed to a low level of mechanical stimuli, whereas new bone is deposited where the level is high. This process of functional adaptation is thought to enable bone to perform its mechanical functions with a minimum of mass. Many theoretical models for bone remodeling use the concept of an error signal as part of a strategy to simulate bone structural adaptation.17, 18 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. For an overview of bone remodeling computational models see, e.g., Martin et al.31 and Hart.32 Bone tissue that is mainly composed of collagen, mineral and water. From the physiological standpoint, besides the mineralized matrix, bone also contains bone cells. According to their function, bone cells may 5 of 15 American Institute of Aeronautics and Astronautics
be divided into osteoclasts, which resorb bone, or osteoblasts, which form new bone, and osteocytes, which sense mechanical stimuli. When the osteocytes detect a deviation of the mechanical stimulus level from the equilibrium state, they send a signal to the osteoblasts and osteoclasts to remodel the local bone surface. No structural changes occur in the interior of the bone matrix. Osteoclasts and osteoblasts are formed only if there is contact with the marrow. In Tovar et al.,33 the HCA methodology did not account for the effect that structural changes can only occur on the surfaces of the mineralized bone tissue. The HCA methodology did however make use of local control rules to emulate the response of bone to mechanical stimuli. In this investigation, the HCA methodology is modified to allow only surface elements to be modified during the structural synthesis process. This modification has been implemented to facilitate the simulation of the bone remodeling process. In HCA algorithm, the design domain is composed of CAs that apply a local design rule using concepts from control theory. The local rules seek to minimize the deviation between a target SED, Ui∗ , and the ¯i . The amount of material in the final structure is indirectly determined by the target SED. averaged SED, U The local design rule can be expressed as (t+1)
where
(t)
(t)
= xi + ∆xi ,
(9)
(t) ¯i ). ∆xi = f (Ui∗ − U
(10)
xi
33
In a previous work, Tovar et al. used four control strategies: two-position, proportional, integral and derivative control. This work makes use of a two-position controller, but the model can be extended to the other types of control. With a two-position control strategy, the local change in the relative density is one of two discrete values. ¯i and U ∗ . In this work, the two-position control These values depend on the sign of the deviation between U i is defined as (t) ¯ (t) − Ui∗ ), (11) ∆xi = kf sgn(U i
where ¯ (t) sgn(U i
−
Ui∗ )
=
+1.0 if −1.0 if
¯ (t) > U ∗ U i i ¯ (t) < U ∗ , U i i
(12)
¯i = U ∗ . In this work, a value and kf is a positive constant value. No change in relative density is required if U i of kf = 1.0 is used by default. The value of kf defines the number of discrete states of the automatons. For example, if the initial design is a solid piece of material, xi = 1.0, then a value kf = 1.0 will define two states (black and white). If the change in density is not constrained to a particular region, the algorithm will not reach convergence. In the process of bone remodeling the structural changes are localized in the surfaces of the mineralized tissue. In this work, the structural changes are restricted to automata that satisfy a surface condition. An automaton is considered to be part of the surface of the structure if it is not in the middle of a completely void space or a completely solidified material. In each iteration, an automaton checks the state of the design variables in its neighborhood. If the average of the relative density in its neighborhood is 1.0, , it means that the automaton is in the middle of a solid region. On the other hand, if the average is xmin i then the automaton is in the middle of an empty region. The surface condition is satisfied when < xmin i
N 1 xj < 1.0, N j=1
(13)
where N is the number of (adjacent) neighbors. The smallest (radial) neighborhood in two dimensions, in the Cartesian coordinate system used in this investigation, is the von Neumann neighborhood (N = 4). This neighborhood will be used for the surface condition. If an automaton does not satisfy this condition, its density will not be modified. Notice that the surface condition is independent of the relative density of the (central) automaton.
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VI.
Implementation
In the HCA approach developed by Tovar et al.,33 the finite element analysis and the application of the local design rule are two independent steps. Furthermore, the mesh in the finite element model and the lattice in the cellular automaton framework can have independent configurations. In this way, the HCA approach allows for the use of commercial, well-developed finite element software. In this work, commercial package Femlab from Comsol, Inc. was incorporated into the HCA algorithm. The finite element mesh can be made more fine or coarse depending on the geometry and load conditions of the model. The cellular automaton lattice can be as fine as required without affecting the resolution of the mesh in places where it is not required. The connection between the finite element mesh and the automaton lattice is the mapping of the new material properties after each design step and the mapping of the field variables after each analysis (Fig. 4). After the application of the local design rule, design variables are mapped from the lattice to the mesh. After each structural analysis, strain energy density is mapped from the mesh onto the lattice. The retrieved field variables are taken from the positions that correspond to the centroid of each (square) automaton. Femlab’s postinterp.m function is used to calculate the strain energy density at these positions by interpolating the values from the closest finite element nodes.
Initial design p
E(0) = x(0) E0 Structural Analysis (FEMLAB)
x(t)
SE(x(0))
SE*
Update material distribution using HCA rule ∆x = f(SE-SE*)
x(t+1) = |x(t+1) – x(t)|
