an integrated system for optimal structural synthesis and remodelling

0011-i94918J 53.w - .oa Pergamon Prtrs Ltd.

AN INTEGRATED SYSTEM FOR OPTIMAL STRUCTURAL SYNTHESIS AND REMODELLING

B. FRASADI

Ford Motor Company, P.O. Box 2053. Dearborn, MI 48121. U.S.A.

(Received 18 November 1983: received for publicnrion 19 January 1984)

Abstract-The paper presents an integrated system approach to efficient structural qynthesis. it is based on a production analysis program (EAL), two generat purpose optimization programs (CONMIN and NEWSUMT) and several problem-inde~ndent processors (some available with PARS and a few recently developed on EAL). The approach makes it possible to resize structures which can be ‘*genericaIIy” modelled. The paper describes the advantages of following this integrated approach, the process of incorporating PARS (Programs for Analysis and Resizing of Structures) into EAL (Engineering Analysis Language) to form EAL/PARS and some of its newly added capabilities. An advanced synthesis system with enhanced generality and flexibility is achieved by taking advantage of the EAL higher order language commands and data base. The EAL modular structure and the ease with which new processors can be added to EAL make this system ~ten~iaily adaptable to a wide spectrum of structural op~imi~tion probtems.

1. INTRODUCTION

One simple approach to extend the flexibility of a structural synthesis system (SSS) is by using a networking scheme and an organization which uses a particular computer operating system (OS) as a connecting network[l-51. Another approach (the second approach) is to utilize the programming features of the parent analysis code to build or extend beyond the existing capabilities through an integration of the information that communicates through the same data base@-91. Though a large measure of flexibility can be achieved easily and quickly through the first approach-it often requires user’s intervention or problem-dependent commands for adequate flexibility. The OS based process cannot reach the level of efficiency the second approach could have achieved if carefully interiinked via a data base network. The question of computational efficiency [combined CPU/(~/O) operations] is very critical in structural optimization because several analyses have to be repeated and each may be extremely costly and computer time-intensive.

Lack of availability of the analysis source codes (proprietorship) coupled with a growing demand to find solutions for new and more difficult problems had been a most compelling cause for combining analysis and optimization programs based on the first approach. This does not result, however, into a system that yields maximum efficiency (both CPU + I/O operations). The second approach, because of its integrated network, provides a more efiicient system than the first approach. PARS[8,9] ;Yas one such integrated development, which utilized the programming features of its parent program, SPAR[ IOJ, to build processors that communicate with the same data base.

In 1978, when PARS was first developed, programs were added to SPAR using the special language (FORTRAN”caIlable routines) to perform optimization, aerodynamic and aeroetastic analysis. PARS then evolved as an efficient modular system

tEngineering and Research Staff.

within a single data base program. However, because of the construction of SPAR[IOI, its parent analysis code, the capabilities of PARS were limited in scope to a set of pre-defined constraints and design variables. It lacked the desired level of generality for providing (a) a huge repertoire of finite elements for resizing, (b) a number of design variables such as materials, sections, geometry, etc. (c) a number of constraints such as deflection, stress, frequency, modeshape, etc. and (d) a number of constraint approximation strategies. It also lacked the desired level of user’s flexibility in (i) controlling linking of design variables set, (ii) setting up one’s own optimization procedure, (iii) controlling rate of convergence, and (iv) defining or local processing of data in addition to the existing scope of the program (to temporary alter or extend its basic function.

The recent availability of anew finite element analysis (higher order) language EAL(I I] has provided the necessary tools. The proprietary nature of the EAL, fortunately, does not come in the way of developing any additional capabilities foreign to EAL (because of its internal construction). With its help an efficient structural synthesis system (SSS) can be developed within a single program (common data base) which can also satisfy all the aforementioned criteria.

The purpose of this paper is to describe the development of such a system, EAL/PARS-an interim report on an accomplished development in the last five years or so. A blueprint of an efficient SSS layout, on which EALjPARS is based, is also described. In particular, it discusses how EAL/PARS system is integrated, what its capabilities are, how this system communicates, how much user friendly it is and why it results into an efficient and flexible system.

2. STRUCTUR4L REDESIGN OBJECTIVES

The objectives of the structural designers/analysts for redesign can be viewed as composed of accom- plishing one or more of the following:

2(a) Design ranking or local (interim) modljications Often a designer is interested in:

8’7

828 B. PR.GtD

ii) improving an existing design by, for example, increasing its strength, stiffness, etc.; or

(ii) determining a group change in design variables that can cause a large design improvement (local).

2(b) Optimal remodeling The problem in this category can be formulated as

to find a solution of the model design parameters CCD which would drive most of the stipulated functions into their acceptable (prescribed) ranges:

wherefcould be a function representing displacement U, stress g, frequency i and modeshape amplitude 4 and also the weight w at one or more reference points on the structure. In the case of displacement, e.g. eqn (1) can be expressed as

u?‘” <u,,(D) < u;” ‘/ (2)

where subscripts i and j are the reference notations for node number and the direction, respectively. Equation (2) may also include the gage limits on the design variables u.

2(c) Structural optimization The problems of this category can be posed as to

find a vector of design variables UED that minimizes an objective function F(G) while satisfying constraint equations g,(v). In a mathematical notation:

F(u)-+minimum (3)

subject to

g,(c) > 0 j = I. m. (4)

In most cases, F(c) is the weight of the structure and g,(c) are various behavioral (deflection, stress, frequency, modeshape, etc.) or side constraints. They may acquire, however, different meanings in different optimizing situations.

Development of a computer software to support a number of structural redesign objectives as demanded by the above example sets is not easy. It poses a unique challenge of choosing a software structure and an integration scheme for combining various entities of a design process.

3. CHOICE OF AN INTEGRATION SCHE,ME

Choosing an integration scheme can be very important in determining how efficient and how flexible (or general) a resulting software system will be. The basic components, that are involved in a typical integration scheme, are shown in Fig. 1 by a flow chart in which an analysis procedure is labelled as “analyzer”; a search algorithm as an “optimizer” and the sensitivity procedure as a “sensor”. Though, there can be several possible ways[3] for integrations; some depending heavily on the modes of integrations, and some just on the manner of computer implemen- tation-the three most advanced concepts are re- viewed here and illustrated in Fig. 2.

The discussion begins with a “Rigid System Net- working (RSN)” which is the least flexible but potentially the most efficient in execution and simplest to

SECONDARY LOOPS (ZNO LEVEL1

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O-A INTERFACE

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Fig. 1. The basic components involved in a typical integration scheme, their order of distributions and

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use. it then proceeds to the concept of “Operating System Networking (OSN)” for improving Aexibiiity and then finally describes an “Adaptive System Net- working (ASN)” which ranks very well in computational eticicncy and program Rexibiiity.

3. I Rigid system networking (RS:V) Figure 2(b) shows the inner workings of a simple-

minded integration approach. It is set up for a predetermined scope of application with efficiency as a prime concern[S]. It is difficult, however, in this system (RSN) to easily alter the scope of applications to other structures or constraints of interest. Input data handling utiIities[l?] provide the major source ofcontroll~ng the program Function in a most efficient way.

3.2 Operating system networking ((XV) In order to increase the flexibility of the system

more than what is provided by RSN approach, an operating system networking was suggested in Ref. [ 131. This approach, as shown in Fig. 2(a). appears to be more direct and logically suitable for connecting systems with incompatible data bases or structures. In OSN (which is also called “programming system”[4]) a user must furnish problem- dependent code modules in addition to input data in contrast to RSN.

There are three basic drawbacks to this approach. First, a connecting network tan executive software) is required to carry out additional computational (data transfer and to monitor the optimization) process. In PROSSS[13], for example, this accomplished by using CDC/BOS (Network Operating System) through a set of system-dependent commands. Thus, the program becomes dependent on its parent operating system and does not remain portable. Second, one has to modify O-A and A-O processors whenever one wishes to solve a different problem (or class of problems). This might be very inconvenient and time consuming (elapsed real time). Third, such a “third party” ar- rangement may also result in an inef~cient operation (computationwise). It would be extremely difficult to creditably match its efficiency (CPU -t I/O operations) with that of equivalent integrated approach-where all the major linkings were accomplished internally through an analysis command language (such as EAL) and where the same data base as the analysis program was utifized.

3.3 Adaptable system networking (ASN) Thus far, two approaches, namely RSN and OSN,

to software integration have been described, the former ranking well in efficiency and the latter in program flexibility. The question is how one can achieve the same level of generality as OSN and still retain a competitive edge when it comes to efficiency. Such an approach, which is called herein “Adaptable System Networking (ASN)“, is suggested in block diagram of Fig. 2(c). It is used here to form a basis of developing the present EALjPARS system. The approach utilizes the advanced “programming” concept of EAL, as described in Ref.[l l] and its muiti- level analysis language structure in its most efficient combinations to build this system within a frame- work of a common data base. The three levels of language-structure which are crucial to achieving this

degree of efficiency and generality (arranged in an increasing order of sophistication) are:

Level (I) consists of the EAL “FORTRAN” ljke

command language and the capabilities of performing a number of 1.0 operations between the user. data base and the program. With its use, one can define any reai, integer or alphanumeric stcng of names into an array of registers, and can store their assigned or computed values into their respective locations on the information complex. This also enables one to perform a variety of control and computational tasks including test branching and transferring to a fixed or varlabie label statement.

Lerc/ (rrj is comprised of the language facilities of tailing a number of ‘*canned” instructions from inside an executable module. such as retrieving the intrinsic stiffness or displacement matrices (using LSK or LSU), performing matrix or vector operations (addition, multiplication, inversion, etc.). It may include facilities for non-routine jobs, such as solution of a system of equations~eigensolutions with modified (user perturbed) stiffness or mass matrices or for retrieving pre-stored set of control images from master libraries of generically applicable procedures.

Lr~l (1/l) is made up of the facilities of calling from within a program a number ofefficient (fortran- callable) data handling routines for doing several independent program development tasks such as writing a new processor. modifying existing ones. These routines allow to perform data commu- nications to and fro between the common data base and the user processors in central memory.

The use of a higher degree of sophistication (e.g. Level III) results into a better efficient system than the use of Level f or II facilities. Nevertheless, they all are based on efficient data communication and core management techniques[l?] so that, for any problem application. no more than a minimum amount of core, is used.

In order to build an efficient structural synthesis system (SSS), the three most important ingredients required are (a) an approach of system’s distribution, (b) an approach of networking, and (c) a data base manager (DBM). In the previous sections, we have described three different approaches of networking and finally chose Adaptive System Networking (ASN) for EALjPARS because of its inherent generality and efficiency. There exists several similar approaches to system’s distribution, but they have not been formally recognized by any particular acro- nyms. In order to provide maximum flexibility in EALjPARS, we propose here a new scheme of distribution called “Efficient Distributed Programming” (EDP) to go hand-in-hand with our chosen networking approach-ASN. Eflicient distributed programming is a programming concept for arranging, distributing and separating the functionally independent or semi-independent elements of the system under construction. Adaptive System Networking was a concept of combining, connecting and pack- aging these system elements together. ASN was described earlier, EDP will be dealt with next.

Data Base Manager is an essential ingredient for any data management-based system. EAL contains

An integrated system for optimal structural synthesis and remodelhng 831

its own DBM, which is independent of the OS and provides a broad range of efficient data handling utilities to perform a variety of data communication and management tasks. To make efficient use of central memory, it maintains a TOC (Table of Con- tents) of each library and the master directory of libraries[ 121. The same DBM is used also for developing this (EAL/PARS) system.

4.1 Ejkient distributed programming (EDP) In EDP, a peripheral organization of the system is

allowed to obtain a most flexible system configuration, and a hierarchical distribution of the organization is employed to gain maximum efficiency. The first order of distribution is composed of six basic functionally independent components of the overall design process:

A. A-O Interface (Controller) B. Structural Analysis (Analyzer) C. Constraint Processing (Constructor) D. Design Sensitivity (Sensor) E. Optimization Procedure (Optimizer) F. O-A Interface (Monitor)

All these components (A-F) are essential to carry out a synthesis task (see Fig. 1) and, hence, this level of distribution is called primary-loop. The second order of distribution (called secondary-loop) is ob- tained by subdividing these components into a number of secondary components (see Fig. 1) such that each may be separated from its primary counterpart as being at least functionally semi-independent. They are described separately in Section 6. In most applications, two orders of divisions are sufficient. For additional flexibility, a third order of distribution (as shown by dotted lines in Fig. 1) may be used.

5. FEATURES OF THE EAL/PARS SYSTEM

One way of viewing the qualities of a software system is to consider the elements of efficiency and generality along with its accompanied design functions or capabilities in a three-dimensional matrix, Fig. 3. The blocks of the matrix along its three axes represent the available tools or features to achieve an individual design objective. The blocks along the horizontal axis representing the elements of generality are arranged in the declining order of flexibility. These bfocks are: common data base; moduiar inter- active; programming features: generic modeling; reset control; data manipulation; print (output) control and finally, the logical command structure such as loop, branch or jump. The user may opt to write a road map using the blocks for solving a problem. As an alter- native, system provides pre-defined library of procedures for standard design objectives in a way that could be efficient and general in a stand-alone mode. It also provides a number of utilities. which can be helpful in meeting a number of “nonstandard” objectives, such as

(a) To generate new design capabilities or execution sequences, which do not come as standard packages, or

(b) To further increase the efficiency of the existing (standard) library of procedures by exploiting the structural characteristics which may be very specific to a problem set.

The integrated organization of this system allows a user to introduce such changes with minimum effort, time and memory resources.

6. COMPONENTS AND ORGANIZATIONS

This section describes the organization of SSS layout used in EAL/PARS, its principal components and their primary functions. The SSS layout

FEATURE MATRIX FOR EALIPARS

ELEMENTS OF GENERALITY OR FLEXIBILITY

Fig. 3. Feature matrix for EALjPARS system.

is configured as an array of upright-tree-structures, each structure representing an element of the design- prowess. Five separate tree-structures are shown in Fig. 4 corresponding to the five principal components of SSS. The sixth component as described in Section 4.1 has not been shown. The ramification of these components and their connections are shown in Fig. 1 through appropriate branching and sectioning. The three ingredients of SSS, namely ASN, EDP and DBM. have been explained in Section 4.

6.1 A -0 inrerfuce (controiler) The function of this component is to identify the

design variables and to define their inter-relationship with the interface model (finite element analysis in this case). The design variables have been categorized into sections, materials, geometrical and topologicaf types for their utilization during “design sensitivity” computations (it will be shown later how this division helps to increase the efficiency of the design computations). The design variables are also further classified here as “primary” and “secondary” variables. Primary variables are considered to be the independent parameters of the structural system- thus, limiting the size of the problem to this number. Primary variables can then be used more efficiently during sensitivity computations, and secondary variables when it comes to updating the FEA model. Secondary variables are, thus, not independent but are linked to the primary variables either through a common grouping, geometrical compatibility or some user-defined relationships. P, S and L are short notations for (Pfrimary, (Sjecondary and (L)inking, respectively-used in Fig. 4. The classifications (P, S and L) also provide the flexibility of automatic linking for most commonly used design variables (such as thickness or area) and providing the user with the option of defining or creating his own linking relationship for less frequently used design variables (such as geometry, nodal coordinates or topology).

There is no need to write a new processor or to modify an existing one, the available utilities unique to EAL aid the user in this type of adaptation. In general terms, the process is called “generic mod- elling.” This capability is available with all the EAL processors. The advantage comes because one may employ variable names for parameters that need to be changed. Arithmetic relationships governing the de- pendence of various parameters can also be defined. Their appropriate values are not substituted or re- placed until at the time of actual executions. This pro-

vides for a great flexibility during both initial de~~t~on of the basic model and for the definition of design variables and their appropriate linking.

6.2 Structural unalysis (analyzer) The required analysis capability (analyzer) is pro-

vided by the user in one of the following ways: user constructing one of his own specialized execution sequences desired for the structure under study through a library of available processors or; user calling the required standard runstreams for the desired analysis (called rigid formats) from master libraries of ” generically” applicable procedures.

6.3 Constr~~~it processing ~co~sfrucl~r)

Several new constraint processing concepts are

available for increased efficiency. Elimination of side constraints is considered as a by-product of design variable linking process. Side constraints are not treated with other behavioral constraints and are handled indirectly[l5] to save computational time in processing. Two other features which are useful in this context are (a) partitioning of the constraints and (b) folding of the active sets. In an earlier paper[lS], it was recognized that:

(a) Constraints of different types (if mixed together) generally would not provide a good and conservative estimate when approximated along a line search.

(b) All the constraints in an active set need not all be considered “individually” for structural design.

Two concepts were, consequently, introduced; (a) to partition the constraints according to their types and (b) to derive a cumulative constraint function 52. The cumulative function R should effectively weigh the contribution of the individual constraints without affecting their overall characteristics compared to its member constraints. Such a measure was proposed as[l5];

where

k> = -gi if g;* Gg;<g,- GE-

g, ifg,‘>g,ag,- 26:’ (6)

where g- and g+ represent some closely spaced bounds on g, constituting a packet of a feasible or an infeasible space, & is the number of active constraints for a given type and b is a real number which the user controls. L - and L + represent a small portion of the constraint sets on either side of the constraint boundary g, = 0. The constraints within this portion are not collapsed, that is

n, = g E- <g,<e‘.

6.4 Resign sensitivity (sensor) The selection of a design sensitivity procedure is an

integral part of this package. It may be recognized that the efficient way of computing design derivatives varies with the types of constraints (stress, deflection, frequency, modeshape, amplitude, etc), types of finite elements or the types of the design variables (section, materials, geometry, topology, etc.) involved. This is discussed in Ref.[l6] in ample detail. The system is capable of recognizing these types of constraints, elements and design variables. It can logically select a method which is most appropriate and efficient for the problem at hand.

In general, it may be pointed out that analytical differentiation of the finite element equations at the element stiffness level is potentially the most efficient way of computing the constraint derivatives. In this, the derivatives of an implicit constraint are usually expressed as (consider for example, displacement)

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where u,,, is the displacement vector, [KO] is the stiffness and Q:, can be looked upon as a pseudo-force vector for load 1 and design variable i. Special considerations are often taken[l6] in computing the r.h.s. of eqn (7) depending upon whether the product of design variables and load cases is smaller than the number of active constraints or vice versa (to be com- putationally efficient). In either case, however, the computation of JKiJzc, is required and as said earlier. it is most efficient to base this computation on an element level. The procedural difference in the computation of the stiffness derivatives for different design variables can be important. Two cases are shown here.

6.4.1 Sectional or material. Design Variables. For this class of variables, the computation of design sensitivity for most element types has not been difficult smce their stiffness matrices are explicit (linear or nonlinear) functions of the design variables; the derivatives of the stiffness matrix can be expressed

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where k, represents the element stiffness matrix and 1, the corresponding thickness.

6.4.2 Geometr;v or topological-design variables. These types of design variables are not element-related (such as coordinate points, width, height, etc.). It is more difficult to find the derivatives of their corresponding stiffness matrices. A systematic (semi-analytical) approach was shown[l6] to result by taking a finite difference at stiffness matrix level for only those groups of elements which were affected. These groups were ones controlled by the particular variable with respect to which the derivatives were being computed. The gradient computation is still based on eqns (7) and (8): however, in this case Jk,/Jv, is computed numeri- cally

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6.5 Optimization procedure (optimizer) An optimization procedure herein means an or-

dered group of various entities forming a Nonlinear Programming Process (NLP). The entities are shown in Fig. 4. These entities are considered important here because a set of chosen assemblage of entities could be more efficient for one class of structures and less efficient for the others. The EAL/PARS provides this capability of selecting an ordered set on a limited basis; the user can also form one of his own.

6.5. I Search algorithms. Two types of search algorithms are now available in EAL/PARS: (a) NEWS- UMT and (b) FEASIBLE. NEWSUMT is a search algorithm which uses a modified Newton method

with a variable penalty function formulation[l7] for SUMT. It requires only the first derivatives of the constraints. The method has been applied to a number of probl:ms[9. 151. FEASIBLE is developed ex- clusively to work eticiently with EAL and PARS system. It uses the method of feasible direction derived from CON,LfIN[ 181 but incorporates some additional useful characteristics of structural approximations. sensitivity and program constructions to be more efhcient and general purpose than CONMIN.

6.5.1 E.rpiicit power constraint approsimation. EAL,PARS system uses a parametric power form of the constraint approximation to simulate a variety of contruint types. It is based on a most recent development described in Ref.[l9]. The previously available forms of approximations (SLA: simple linear approximation; SIA: simple inverse approximation. see Fig. -1) were not general enough to simulate constraints for different analysis needs (e.g. static, vibration, buckling. etc.) in various types of structures of interest and accurate enough to perform well. Consequently. an “open-ended” library of approxi- mate forms (GLA. GIA: generalized linear. inverse approximation; GPA: generalized power approximation and CHA: generalized hybrid approximation. see Fig. 1) vvere developed to provide both flexibility in choic: and generality in applications. “Open- ended” forms, the terms introduced in Ref.[l9], are those which can supply a continuous dial-in-form of constraint approximations for increasing conservative- ness and/or accuracy. Selections of one or more of these forms can be initiated or progressively changed at the request of the user or the optimization procedure. if so required.

7. II.I.CSTR.ATIONS

Three examples arc used herein to illustrate the basic ideas behind each of the objectives for structural redesign described in Section 2 and some of the pertinent features of EAL/PARS.

7. I Objectiw (20), e.wmple: u composite wheel This example is used to show how the generic

design sensitivity EFC (Execution-flow control) pre- dicts design ranking and group change. The procedure is described in Ref.[l6]. Figure 5 shows a typical finite element model of a wheel generated when any one or more of the design parameters listed rn Fig. 6 was varied “generically”. In the case of stillness based design-deflection at the tip of the loading arm, and during strength based design-a critical Von mises stress tor a standard rotary fatigue test was used as performance indicators. The results of the “relative unit changes” for the base design are summarized in Table I. Table I also shows the associated rank number for the stiffness and strength designs when they were considered individually and when they were equally weighted. The results are presented for two design situations when derivatives are taken with respect to design variables (function dominant) and when they are taken with respect to mass (weight dominant). Thus, depending upon a particular design improvement situation, one can use any one of the rankings to suit a given design goal. The rankings also suggest the most and least effective locatIons to redistribute materials in the wheel.

An integrated system for optimal structural synthesis and remodelling 835

Fig. 5. A typical finite element model of an automobile wheel on EAL.

7.2 Objectice (2b), example: a connecting rod This example is the one previously shown in

Ref.[ZO] to illustrate one of the uses of the optimal remodeling technique. The design problem was to obtain an optimal profile of a connecting rod in the region of its connection with the eye (see Fig. 7) SO

DESIGN VARIABLE

DESCRIPTION INITIAL VALUE ( INCIIES 1

LONG-SPOKE SPIOER TOP THICK

LONG-SPOKE SIDE THICKNESS

WIDTH OF THE CCOLING HOLE AT RIM

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Fig. 6. Design variables, locations and their initial values used for the design sensitivity.

that the stress concentration effects are minimized. The details of the procedure and the problem are described in Ref.[20]. A fifth degree polynomial was used to describe the profile of the eye-end connection from point-A to point-D. A straight line segment was used between point-D and point-E.

y = a, + a,,~ + aPx2 + a,x’ + a4x4 + a,xS; x I <x < 13

y = 5.825; O~XGXl. (II)

Three geometrical parameters x 1, ~2, y3, see Fig. 7, were employed as design variables to control the shapes. Continuity of curves and slopes at the two ends were sufficient to evaluate a’s in terms of the design variables. A generic modeling procedure was used to automatically regenerate the finite element model as the design variables change. A typical finite element model is shown in Fig. 7. For the initial starting design (x 1 = 0.2, y2 = 4.5, and y3 = 3.0) the stress intensity factor (SIF = cr,Jcr,,) was 2.36. The strategy was to use the fifteen outer fiber stresses along the transition area (A to E) as indicators for the design remodelling process. Theoretical details are contained in Ref.[20]. The procedure converged in 7 interactions; the final stress intensity factor was (SIF = 1.59). The SIF profiles for a few typical iterations are shown in Fig. 8.

7.3. Objective (2c), example: an unsymmetrical truss This example was intentionally chosen to illustrate

the execution control sequences for a typical applica-

B. F’RASAD

&I integrated system for optimal structural synthesis and remodelling 837

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CONNECTING ROD

FINITE ELEMENT MODEL

Fig. 7. Objective (2b) example, a connecting rod showing design variables and FE model.

tion when both geometrical and sectional parameters are chosen as design variables. The intention was to present the input flexibility, data flow and features of EALfPARS and not to burden with excess details.

The problem was to minimize the weight of a 2-bar unsymmetrical truss subject to stress and displacement constraints. The loads, geometry and design variables are shown in Fig. 9. There were 2 design variables, area of the truss A,(A, = 2A,) and height H. In order to avoid poor convergence due to mix of design variables, the height variable was scaled by 100. The starting point and side constraints were as folfows:

2.5 I I I I I SYMBOL ITERATION

a e 0 2.0 iz r

t 0.5 1 I I I I I I

2.0 4.0 6.0 a.0 IO.0 12.0 14.0 X-DIR POSITION

Fig. 8. Iteration histories and stress intensity factors for the connecting rod example.

&e-----&-b4

Fig. 9. Objective (2~) example, an unsymmetrical 2-bar truss (design variables. A, and H).

Variable Starting Minimum Maximum Al 3.00 0.1 100.0

Height 1.75 1.5 2.5

The finite element input representing the generic model for EAL used for this problem is given in Table 2. The corresponding optimization input for EAL/PARS is given in Table 3. The allowable stresses of 104kN and an arbitrary displacement mag- nitude were specified to make the optimization problem stress critical. The “constraint selection” input in Table 3 shows the road map for the selection of the design sensitivity procedure (an override option of EAL/PARS). With the data supplied in a triad (s,D,c), s indicates the required design sensitivity option (1,2,3, etc.) for each (u,c): design variable (v), and constraint (c) combination. Following this, “NEW- TON” processor, which is based on a SUMT, and Newton’s method[l’l] is run with some of the optimization routing furnished by its newly assigned reset values. “LOOP” specifies the maximum number of iteration cycles allowed for a run if the normal procedure did not converge.

For the actual design run on EAL/PARS, the procedure did converge in 5 iterations. The final designs were A I = 4.478 and H = 2.0, which were very close to the computed optimal solution (A I = 4.48, H = 2.00) derived in Ref.[20].

Table 2. EAL data representing generic model for Z-bar truss example

CXQT ul l (29 GENERIC TRS2 0 0) TRS2 l XQT 111 l CALLt29 SET VARIABLES 0 0) +XQT TM

fA2=2.+Al IR- ffEIGET'100.

START 3 UTERIAL CONSTANTS

1 0.207+6 0.3 0.78186-E JOINT LOCATIONS

1 -400. 0, 0. 2 100. 0. 0. 3 0. l B- O.

-- CONSTRAINT CASE 1 ZERO 3 4 5 6: 1.3 ZERO 1 2: i;2

BC 1 *Al* 2 'b2=

+XQT ELD E23

GROW 1' ELEMEW WITR AREA Al NSECT-1

13 GROUP 2' ELEllRNT WITS ARBA ~2 NSECT=2

2 3 ‘RETDRN l TRS2 '(29 REGISTERS TRS2 0 01 REGISTERS

ILIBl=1 lM=3. IEEIGilT=1.75

*REGISTERS

8. CONCLUSIONS AND REMARKS

A Structural Synthesis System (SSS) layout for EALiPARS, which is an offspring of combining three novel ingredients: an efficient approach of networking (ASN); a general-flexible approach of distribution (EDP) and an efficient way of communication (DBM), is described. The effcient networking is made possible by using the system’s multi-level language structure ofthe parent’s analysis program, EAL. The approach of distribution, EDP, results in large flexibility of the optimization procedure organization and versatility of the formulation for constraints and design variables. The flexibiIity for diverse utilizations such as imposition of constraints on any calculable re- sponse functions and definitions of design variables to any generic parameter used for the FEA model is achieved by combining in a modular manner, two state of the art optimization algorithms NEWSUMT and FEASIBLE, a production-level structural analysis program EAL, and a “generic” sensitivity procedure. The user needs to define mainly the options for generic variables, constraints, design variables, sensitivity and optimization control parameters through a user-supplied and problem-dependent runstream. The remain- der is automati~~ly handled by the respective execution-flow controls of each tree-structure (Fig. 4).

To achieve maximum efficiency, selection -of the order of execution-flow control in a tree-structure is an integral part of this package. The use+defined runstream options and reset control features can, if desired, be used to override this pre-defined order of execution-flow. This marrying of the automated and override features of the EALJPARS system results into a vast measure of flexibility required in research use and adequate versatility in the formulations of

constramts and design variables required for problem applications.

Operating system best series the purpose of connecting different programs whose data structures happen to be dlffirent and distinct. such as NASTRAN and SPAR. for example, rather than connecting a functional routine to a data-based system such as E.r\L.

Acknoir,/rdyenrenr-The author would like to express sincere thanks to Dr. W. D. Whetstone for his effons in implementing PARS on E.-IL, and to >fr. J. F. Emerson for his help during its developmental phase.

I.

2.

C. L. Giles. C. B. Blackburn and S. C. Dixon. Auto- mated procedures for sizing aerospace vehicle structures (SAVES). J. .-Iircrqfi 19. XI?-819 (1972). J. Sobieszczanski. Sizing of complex structure by the integration of several different optimal design afgo- rithms. AGARD-LS-70. Hampton, Virginia (IO-1 i Oct. 1974).

3.

4.

J. Sobieszczanski-Sobieski and R. B. Bhat, Adaptable structural synthesis usmg advanced analysis and optimization coupled by a computer operating system. A Col!ection of Techniccd Pffpers on Strucrures- .~IA/~IASI~/E!‘ASCE,‘,~HS 20th SDM Co& April 1979. pp. 20-71, AIA.4 Paper No. 79-0723. Also in J. .-lircrirji 18, 142-139 (1981). J. L. Rogers. Jr.. J. Sobieszczanski-Sobieski and R. B. Bhat, An impIementation of the programming structural synthesis system (PROSSS). NASA TM 83180 (Dec. 1981).

5.

6.

7.

s.

9.

Xl. L. Adelberg r! (11.. OPUS-a programming system approach to structural optimization. 4th fnt. CorzjY Vehick S~rueturai Mech. pp, 151-160 (18-20, Nov. 1981). R. T. Haftka. Automated procedure for design of wing structures to satisfy strength and flutter requirements. NASA TN D-7264 (July 1973). A. R. Do&, ISSYS-an integrated synergistic system. NASA CR-159221 (Feb. 1980). R. T. Haftka and B. Prasad. Programs for analysis and r&zing of complex structures (PARS). Tren& in Corn- puterized Structur~ii Analysis and Synthesis, pp. 323-330. Pergamon Press, New York (1975). Also in Conrpur. SIructurrs IO. 323-330 (1979). B. Prasnd and R. T. Haftka. Oreanization of PARS-a

”

IO.

II.

12.

13.

14.

15.

structural resizing system. /Idwnces in Compufer Terlmol~gv-1980. (Edited by A. Seireg), pp. X-273, 2, ASME Century 2 Publication, (1980). Also in A&. in Eegn\e .So/iwure J. 1, 9-19 (1982). W. D. Whetstone, SPAR Structura/ Analysis System ReJerence Manual, System Level I?, Vol. I, NASA CR-145098-i (Feb. 1977), and System LZZ& 13A. Vol. I. NASA CR-1%970-I (Dec. 1978). W. D. Whetstone, EISI-EAL: Engineering Analysis Lan- guage. Proc. 2nd Conf. on Computing in Civil Engng, ASCE. pp. 7-76-28.i (1980). G. L. Gifes and R. T. Haftka. SPAR data handiing utilities. NASA TM 78701 (Sept. 1978). J. Sobieszczanski-Sobieski and J. L. Rogers. A programming system for research and applications in structural optimization. Proc. Inr. Symp. on Optimum Struc- tural Desigrr, 19-X Ocr. 198 1. Tuscan, Arizona, pp. 11-9, 11-E Control Data Comparison. Z;OS Version I Refirence ,tlanuul, NOS 1.2. CDC ?Jo. 60445300 (March 1978). B. Prasad, Novel concepts for constraint treatments and approximations in eff;cient structural synthesis. Af.U J. (to appear) 1985.

16. B. Prassd and J. F. Emerson, A general capability of

REFERENCES

An integrated system for optimal structural synthesis and remodelling

Table 3. PARS data showing input sequences for 2-bar truss example

839

PRuN, s rRs2,,,30,1000 CAD0 ISo*S.EAL @ADO J?E*OATA.TRS2/GENIRIC CAD0 ~ORD~OPTIRIZE.OP1INI/BUIU) @ADO ?ORD*OPTIMIZE.SENS/BUIID PXQT II1 *(PARS 0esV1 t E23: l,l,l 1. 123: 2,2,1 2. DESIGN: 1,l 0.

l (PARS DCON) 6

OISCON 3 1 1000.

*(PARS SCOW) 6 STRESS -~E23

1 1 10000. 10000. 2 2 10000. 10000.

*(PARS OPTIONS) s ILIRG-29 lGRNl=GP.NERIC lGRN2=TRS2 lORNl=REGISTERS lORN2=TRS2 ISROTEST-1

*(DESIGN VARIABLES) 6 'EEIGRT

*(VARIABLE ORSCRIPTIONI I 'ECIGRT OF TRUSS

l (STRRSS CONSTRAINTS) s E23 1 E23 2

*(STRESS ALLOYABLES) I 10000. 10000.

l (SENSITIVI~ oPTIoNs) 5 ILIBG-29 IGMNl-GENERIC lGllN2=TRS2 IORNl-REGISTERS IORN2=TRS2 ISEOTEST-1

'(LOMS) S

PARS

PARS

PARS

PARS

DESIGN VARIABLE DESCRIPTION

OISPLACENENT CONSTRAINTS

STRESS CONSTRAINTS

ANALYSIS OPTIONS

CASE 1 I-2: J-3: 1.0+5

*(CASE TITLES) $

1'VLRTICAL WAD ON VERTEX *(OPT VARIABLES) s

' .A1 'BLIGHT

*(VARIABLE SELECTION) 5 11 21

'(OPT LINITS) 5 3.0 100.0 1.5 2.5

*(CONSTRAINT se~EcTx0~) 5

1,1,2 2,1,1 1,1,3 2,1,2

*(WENTON OPTIONS) S IICNT-7 IR-.l IGlC-.l IOEPP-3 IKET--1

tXOT 01

DESIGN-SEWS

DESIGN-SEWS

DESIGN-SEWS

VARIABLES

VARIABLE TITLES

STRESS CONSTRAINTS

DESIGN-SINS STRESS ALLOUABLES

DESIGN-SENS OPTIONS

LOAD DEPINITION POR OPTIRIZATION

LOM CASE TITLES POR OPTIRIZATION

VARIARLRS POR USE IN OPTIUIZATION

SELECT VARIABLES POR OPTIMZATION

OPTIMIZATION SIDE CONSTRAINTS

MATRIX ROAD MAP POR COMBINATION

RESETS POR l CUBE= EXECGTION

LUMP-10 l PERFORM(29 OPT ANALYSIS) @PIN

design sensitivity for finite element systems. A Col!ecfion 19. B. Prasad, Explicit constraint approximation forms in of Technical Papers-AIAAIASMEIASCEIAHS. 23rd structural optimization-Part 1. Analyses and projec- SDM Conf., iMay 1982, Part 2. AIAA Paper No. tions. Comput. Meth. Appl. Mech. Engng 40(l). 1-26 824680. pp. 175-186. (1983).

17. 9. Prasad, A generalized class of variable penait) 20. methods for nonlinear programming. J. Optimixrion Theory Applic. 35, 159-182 (1981).

18. G. N. Vanderplaats, CONMIN-A FORTRAN Ko- gram for Constrained Funcrion Minimizarion User’s Manual. NASA TM X-62282 (Aug. 1973).

B. Prasad and J. F. Emerson, Optimal structural remodelling of multi-objective systems. Cornput. Structures, 18, 619-628 (1984).

an integrated system for optimal structural synthesis and remodelling

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