Mixed-Discrete Structural Optimization with Distributed ... - CiteSeerX

Mixed-Discrete Structural Optimization with Distributed Advanced Evolution Strategies H. Grill, D. Hartmann Institute of Computational Engineering, GK Computational Structural Dynamics, Ruhr-University Bochum, Bochum, Germany

Abstract This paper describes the application of distributed Advanced Evolution Strategies (AES) in the field of mixed-discrete structural optimization. AES are direct optimization methods based on the use of the three partially probabilistic evolution operators recombination, mutation and selection in an evolving population of competing individuals in the design space. Advanced features of the AES include self-adaption of strategy parameters to achieve on-line tuning of the optimization process and a flexible selection scheme that allows scalable lifetimes of individuals. The property of using a population of coexisting but inherently independent individuals allows the efficient realization in distributed computing environments. This approach results in a drastic reduction of response time and an improved applicability for large scale problems. Introduction While continuous optimization methods are well established and frequently used, mixed-discrete and discrete optimization methods are still in a developing state and represent an important current research topic. Several methods have been proposed to solve the rather general case of mixed-discrete nonlinear programming problems. Only some approaches are mentioned here. E.g. Bremicker, Papalambros and Loh apply a sequential linear programming technique linked with a branch and bound algorithm (Bremicker et al., 1990). Salajegheh makes use of a penalty function approach in combination with approximation concepts (Salajegheh, 1995). Ringertz and Groenwold et.al. proceed from a continuous solution by rounding variables to discrete values (Ringertz, 1988; Groenwold et al., 1995). Rajeev and Krishnamoorthy and Hajela employ Genetic Algorithms (Rajeev and Krishnamoorthy, 1992; Hajela, 1990), Bennage and Dhingra use the Simulated Annealing method (Bennage and Dhingra, 1995) and Cai (Cai, 1995) and Grill and Hartmann (Grill and Hartmann, 1997) use Evolution Strategies. In the following an approach to the solution of mixed-discrete nonlinear programming problems for optimum structural design based on Evolution Strategies will be presented. Spe-

cial attention is paid to improved functionality of the strategy and efficient use of distributed computing environments. Definition of the Optimization Problem In this current research project the goal of the optimization task is to minimize the weight of structures where the structural response due to external loads is simulated through a Finite Element model. The structural response includes static as well as dynamic behavior. The formulation of the considered mixed-discrete optimization problem can be stated as:

min f f (x) x with

8 g (x; r(x)) < j S=: xk xkL

 2 

0 Dk xk

x2Sg

j

(1)

9



xkU

j = 1; m = Dk = (dk ; : : : ; dkqk ); k = 1; nd k = nd + 1; n ; 1

(2)

where the following notation is used:

f (x) x

gj (x; r(x)) m r(x) Dk qk

: : : : : : :

Objective function Vector of design variables Constraint functions Number of constraints Structural response Set of values of discrete design variable xk Number of discrete values of k–th design variable

xkL

:

xkU

:

nd

:

n

:

Lower limit of contin. design variable xk Upper limit of contin. design variable xk Number of discrete design variables Overall number of design variables

Objective function f and constraint functions gj are collapsed into an augmented objective function through an exterior penalty function method according to Eq.3,

(x; r) = f (x) + p

m X i=1

gi (x; r(x))




+

(3)

where p is a scalar penalty parameter which is increased, e.g. doubled in each generation and gi+(x; r(x)) = max(0; gi(x; r(x))). This penalty formulation is used here because: 

the initial design is not needed to be feasible,



random moves into the infeasible domain, which can be very important e.g. in the case of disjoint solution spaces don’t pose any algorithmical difficulties and



no gradient information are needed in the evolution process.

The augmented objective function (x; r) will be used as the quality criterion for the optimization process with Evolution Strategies (ESs).

Standard and Advanced Evolution Strategies Based upon Darwin’s observations and the modern theory of organic evolution, ESs represent a powerful and versatile tool for the solution of optimization tasks. They do not rely on restrictive mathematical properties of the underlying optimization model like continuity, differentiability or unimodality. Basically, the more difficult a problem is, the more appropriate and necessary the use of this type of solution method becomes, as many other methods only give unsatisfactory results or simply fail. ESs are direct search and optimization methods employing the collective learning capability of a population P of simultaneously existing individuals. Each of these individuals represents one point in the often topologically complex and highly dimensional search space. In the series of generations the population evolves towards better solutions through the application of the randomized evolution operators mutation m and recombination r and the deterministic selection operator s. Mutation creates new information through random changes of individuals, recombination realizes the exchange of existing information between individuals and selection drives the population towards improved quality, measured in terms of the augmented objective function given in Eq.3 by neglecting lower quality individuals from the reproduction in next generations. An explanation of ESs is given here based on the description of the evolution operators mutation, recombination and selection. Mutation A new individual is generated according to

xo = xp + N(0;  )

(4)

where xp is the parent individual, xo is the offspring individual and N(0; ) is a vector of normally distributed random numbers with the expectation values 0 and the standard deviations . This is depicted in Fig. 1.

Probability Density

2

N

x

σ = 0.5 σ=1 σ=2

xo lines of constant PD

xp N

x1

Figure 1: Mutation The standard deviation  can be seen as a stepwidth of the optimization process in the search space. Standard ESs usually employ a hard-coded rule for stepwidth-adaption, the so called 1=5-success-rule which is derived from convergence investigations of two simple analytical optimization models. To achieve greater flexibility and a problem-oriented adaption capability compared to standard ESs, AESs incorporate the stepwidths  as strategy variables which are subject to variations during the course of the generations. The mutation m is performed in two

steps, mutation of the strategy variables ms

o;i = p;i eN


;

(0 )

(5)

and mutation of the design variables md

xo;i = xp;i + N (0; o;i)

(6)

with each individual x represented as

x = [(x1 ; 1 ); (x2 ; 2 ); : : : ; (xn ; n )]

(7)

This approach accomplishes a two level learning behavior, the so called object-level-learning of the design variables and the meta-level-learning of the strategy variables (Schwefel and Rudolph, 1995). The advantage of this two-level method is that no exogenous interference or problem-specific knowledge of the user is needed to fine-tune this important strategy characteristic. Recombination The evolution operator recombination uses the information of two or more parent individuals for the generation of a new offspring individual. Different recombination schemes can be accomplished, for example, through 

stochastic or deterministic choice of the parents,



the number of parents,



intermediary or discrete combination of the information from the parent individuals.

As opposed to standard ESs the AESs also apply the recombination operator to the strategy variables to support the generation of favorable strategy properties of the evolving individuals. Selection Applying the selection operator determines the subset of highest quality individuals from the current population as the parent population for the next generation. There are two basic selection strategies, where  denotes the number of individuals of the parent population and  denotes the number of individuals of the offspring population: 

( + )-strategy: Selection of the new parent population from the current offspring and parent population,



(; )-strategy: Selection of the new parent population only from the current offspring population.

In AES an additional selection scheme can be used: 

(; ; )-strategy: The individuals of the parent population have limited, predefined lifetime of  generations. After the expiration of this lifetime a parent individual can not be selected as a parent of the next generation step anymore.

Thus the (; ; )-strategy can be seen as a scalable selection scheme between the (; )strategy with a maximum lifetime of one generation, (; 1; ) = (; ), and the ( + )-strategy with unlimited lifetime of the parent individuals, (; inf ; ) = ( + ). This strategy profitably combines the stability of the ( + )-strategy and the diversity retaining behavior of the (; )-strategy.

Distributed Evolution Strategies Because of the high computational demands of structural optimization tasks with ES the efficient use of distributed computational resources is of significant importance. The goal of the present project is a flexible and portable system exhibiting advantageous runtime properties in different distributed computing environments. The logical structure of the distributed ES is a master-slave configuration, where the single master task is responsible for the overall coordination and the execution of the evolution steps whereas the slave tasks take over the numerically intensive part of the structural analysis and constraint evaluation. The implementation is based on the message-passing paradigm using PVM as the basic communication device. PVM is preferred over MPI because of it’s better suitability for heterogeneous computing environments (Geist et al., 1996). The optimization program has been ported to and used on different distributed computing systems ranging from commonly available heterogeneous workstation clusters to state-of-the-art parallel computers. Examples 252-Bar Tower The first application example is the minimal weight design of the spatial truss structure shown in Fig. 2 subject to time dependent loads at the outer nodes of the arms and the top nodes of the mast. The definition of the optimization model includes 55 discrete cross section design variables and 16 continuous geometry design variables maintaining symmetry with respect to the x-z and the y-z plane. Stress, deformation and buckling properties are limited through constraints which are evaluated at each time step in the applied direct integration scheme of the equation of motion1.

Figure 2: 252-Bar Tower The optimization is performed with a (25; 25; 200)-strategy where a final weight of 6124.55 units is obtained after 288 generations starting from an initial weight of 14589.4 units. The his1

A more detailed description of this example can be found in (Grill, 1997)

tory of the calculation is depicted in Fig. 3(a) showing the augmented objective function and weight of the best individual (BI) and the corresponding mean value for the parent population (PP), and in Fig. 3(b), showing the mean stepwidth respectively. 20000

4 aug. objective function (BI) weight (BI) aug. objective function (PP) weight (PP)

18000

mean stepwidth (BI) mean stepwidth (PP)

3.5 3

16000

2.5

14000

2

12000

1.5 10000

1 8000

0.5 6000

0 0

50

100

150 generations

200

250

300

0

50

(a) augmented objective function and weight

100

150 generations

200

250

300

(b) mean stepwidth

Figure 3: 252-Bar Tower

To assess the runtime characteristics of the application several calculations have been performed in different environments and with varying numbers of slave tasks. Results are given in Tab. 1 for a heterogeneous workstation cluster (WSC) consisting of common workstations (HP 9000/7xx, HP C160, SGI Indigo, Sun Sparc) connected with standard 10BaseT/10Base2 Ethernet and a 20-CPU SGI Origin 2000 (SGI). More detailed information about the computing environments, the testing procedures and the runtime properties can be found in (Grill, 1997; Grill and Hartmann, 1997). Table 1: Runtime Characteristics for 252-Bar Tower system WSC

SGI

slave procs 1 2 5 10 1 2 5 10

runtime (s) 23799 7433 3778 3055 24603 12220 4995 2469

speedup 1 3.2 6.3 7.8 1 2.0 4.93 9.97

efficiency 1 1.6 1.25 0.78 1 1.0 0.99 1.0

The data in Tab. 1 show very good scalability behaviour of the distributed application. For the cluster of workstations (WSC) even superlinear speedup is achieved for up to five slave tasks which is due to the heterogeneity of the environment. For the parallel computer environment (SGI) almost linear speedup is achieved for up to ten slave tasks. Rod-Stiffened Panel The second example is the minimal weight design of the rod-stiffened panel subject to a static loading case shown in Fig. 4. The structure is made of 144 plane elements and 109 truss

elements. The plane elements are grouped into twelve regions (P1 - P12) and the truss elements are grouped into nine stiffeners (S1 - S9) in the optimization model. Corresponding to these groupings twelve design variables corresponding to the thickness of the panel regions and nine design variables corresponding to the cross sections of the stiffeners are considered in the optimization. Both the thickness and the cross section design variables are defined to take on only discrete values. Constraints are imposed to limit the maximum von-Mises stress of the plane elements and the maximal normal stress of the truss elements 2. S1

S2

600

P9

S3

P10

P5

P6

P1

P2

S4

P11

S5

S6

P12

P7

P3

P8

S7

P4

S8

y x

F

S9

1200

Figure 4: Rod-Stiffened Panel This example is solved with a (15; 25; 120)-strategy. Starting from a weight of 50.79 units a final weight of 23.23 units is reached after 78 generations. The history of the optimization is given in Fig. 5(a) showing augmented objective function and weight for the best individual (BI) and the parent population (PP) and in Fig. 5(b), showing the mean stepwidth. 55

3 aug. objective function (BI) weight (BI) aug. objective function (PP) weight (PP)

50

mean stepwidth (BI) mean stepwidth (PP) 2.5

45 2 40 1.5 35 1 30 0.5

25 20

0 0

10

20

30

40 50 generations

60

70

80

0

10

(a) augmented objective function and weight

20

30

40 50 generations

60

70

80

(b) mean stepwidth

Figure 5: Rod-Stiffened Panel For this example calculations have also been performed in different computing environments and with different numbers of slave tasks. Similar scalability behavior as in the previous example of the 252-bar tower was obtained. 2

A more detailed description of this example can be found in (Grill, 1997)

Conclusions In this paper a method for solving mixed-discrete structural optimization problems based on the use of Advanced Evolution Strategies was presented. Advanced Evolution Strategies are robust, highly problem independent and well suited for realization in distributed computing environments. The developed optimization tool can be applied to different structural optimization tasks and shows promising scaling and runtime behavior in the employed computing environments. This makes it an attractive approach to solving complex, mixed-discrete structural optimization problems where other, less compute-intensive methods fail.

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