A design sensitivity analysis principle and its implementation into ADIANA

Compums & Smcrures Vol. 32, No. 314,pp. 691-705.1989 Printedin Great Britain.

0

1989 Maxwll

0045.7949/8953.00+0.00 PcrSrmmn Macmillan plc

A DESIGN SENSITIVITY ANALYSIS PRINCIPLE AND ITS IMPLEMENTATION INTO ADINA J. S. ARORA and J. E. B. C~~~cxso Optimal Design Laboratory, College of Engineering, The University of Iowa, Iowa City, IA 52242-1593, U.S.A. Ahatract-This paper describes a continuum principle of design sensitivity analysis of linear and nonlinear structures for shape and nonshape optimization problems. The principle has been derived previously using the virtual work equation, and the concepts of reference volume and adjoint structure. It is stated in a form that is more convenient to interpret and discretixe for numerical calculations. Physical interpretation for each implicit design variation term in the principle is given relating it to the virtual work of internal or external forces, or their explicit design variations. A discretixation of the principle using usual shape functions of finite element analysis is developed for implementation into computer programs, such as

ADINA. Some numerical examples are solved to show practical use of the principle.

2. NONLINEAR

1. INTRODUCTION The

problem of design sensitivity analysis and optimization of nonlinear structures has become important in many applications. As the structure is optimized for its performance, it may become more flexible requiring use of large displacement formulation. In many applications, the structural material behaves nonlinearly and may also go into the inelastic range. Therefore, it is important to develop a theory and numerical methods for design sensitivity analysis before optimum design of nonlinear structures can be attempted. Considerable work has already been done for design sensitivity analysis of linear structures with static and dynamic response. constraints since the mid 1960s. For a detailed review of the subject, Refs [l-S] should be consulted. Design sensitivity analysis of nonlinear structures is just beginning to be investigated and the literature on the subject is beginning to grow. The discrete form of design sensitivity analysis of nonlinear structures (modeled by finite elements) has been developed [6-l 11. The continuum form of the sensitivity analysis has also been investigated [ 12-181. However, more work needs to be done to fully develop the methods and apply them to practical problems. The present paper re-examines a recently developed unified theory of design sensitivity analysis of nonlinear structures for shape, nonshape and material selection problems [ 12, 131.The theory is stated as a design sensitivity analysis principle and physical interpretation is given to each implicit design variation term in the principle. Several analytical problems have been solved in the literature to show operations of the principle [12]. The paper also presents discretization of the principle for implementation into a finite element analysis program, such as ADINA. A few sample problems modeled by the finite elements are optimized and solutions are discussed. 691

STRUCTURAL

ANALYSIS

We first summarize the notation used, the nonlinear analysis equations and the reference volume concept before stating the design sensitivity analysis principle. We shall use the Total Lagrangian formulation. To identify the quantities in the deformed configuration, we use a left superscript to indicate the configuration in which the quantity occurs and left subscript to indicate the reference conliguration. For more details of the notation and analysis procedures, Ref. [19] should be consulted. We start with the principle of virtual work for equilibrium in the deformed configuration:

where &?Iis 2nd Piola-Kirchhoff stress tensor, &cis the Green-Lagrange strain tensor, $ is the body force, $” is the surface traction, ‘IIis the displacement field at time t, ‘V is the initial volume with Or as its boundary, Tr is the traction specified boundary, r, is the displacement specified boundary, 6 refers to arbitrary variation of the state field, and * refers to standard tensor product. The 2nd Piola-Kirchhoff stress can be calculated from the Cauchy stress tensor *e as $4 = I&Xl(tJ)-’ * (‘a) * (6X)-‘; where ;X is Green-Lagrange

&=

c?‘x,/Px,

the deformation gradient. strain tensor is given as

& = ;[oV’Ur+ (ov’ur)r + (oV’u’)(,V’u~~

(2) The

(3)

where oV = @/aox, a/aOy, a/aOz)r. The nonlinear constitutive law is taken as $S = d&e, b); where b is a design variable.

692

J. S. Aao~a and J. E. B. CARDOSO

To solve for the nonlinear response, we use the incremental decompositions as ‘fA’u=‘u+u,

‘+“&=;e+&,

‘+AOT=;T+oT,

‘+*;f=;f+d

‘+AS=*++,

with 0 = [0 0 O]r. The strain variation and incremental strains are given as

(4)

,S=dr**&

oe = ,e + o’l; oB= f[(oVu3(oVu3rl

S$ = ,f@‘u) + ;A@‘;

(5)

oe = oat + &u’,

(6)

oij = f o&u‘

,e = $oVur + (oVu3’ + (oVu3(oV’u3r + (oV’u3(oVurY-l (7) where u, $3, ,,e, ,,f and ,T are the increments in the corresponding fields. The arbitrary variation of strains becomes ,‘+A& = Bee. Now the virtual work equation can be written in the incremental form at the load level (t + At) after having determined the equilibrium state at the load level t. After neglecting the higher order term &3- 6,~ the incremental equilibrium equation becomes

(14)

where & and ou’ are increments in ;A and ku’; ,S *6,e = ,$r~08 and $3 * 6,q = @S,f) A constraint functional representing the point constraints, average constraints, reaction force constraints and surface constraints is taken as G@, ;s, ‘II, b) ‘d l’ +

Y=

7i(‘u,b) Odl-‘, s 00

s

&T, b)‘dT,.

+

(15)

Or.

3. REFERENCE VOLUME CONCEPT

(8)

The incremental stress-strain law given in eqn (5) is linearized for use in eqn (8) by using 6,~ = 6,e, i.e. 0s = & * ,e. It is more convenient to use the matrix notation for discretization of the continuum equations for implementation on the computer. Therefore, we introduce the matrix notation here and define the 2nd Piola-Kirchhoff stress and Green-Lagrange strain vectors $3 and $Z as (^ is used to indicate vectors) $ = [XX &

X, &

& = kx &, b%:&,

oxo”x(‘x);

ZA, &Jr

‘dl’ = J dE J=]X];

J=J]X’rr]

g = ;qx);

The constraint

= ,iY’u+ f 3;u

OdT = Jdr; (16)

X, = aoxi/arxj; 'U= ‘u(‘x);

x = x-1;

(9)

&X

3,:

To treat both shape and nonshape design sensitivity analysis problems, we use the concept of a reference volume [ 121. In this approach, the configuration at time 0 is mapped to a reference volume v with boundary F. As the design is changed, the mapping to the reference volume changes but the reference volume and its shape never change. Using a left superscript r to refer to the reference configuration, we define various transformations and equations as

8

= fs(‘x).

functional

(17)

becomes

(10)

where ;u’ and linear and bilinear operators oa and ;A

G($, &,‘u, b)J dV +

Y=

s

s

h(‘u, b)Jdr,

are given as + ;u’ = [$I,~;u,, &]r, Or = Ox, “y,

,a’=

alaox 0 0 [

0 a/soy 0

0 0 a/a%

Au,,= Wa”r, or

Oz

a/soy alaox 0

(11)

a/aoz 0 alaox

Au,,

0

&I,,

0

&I,,

0

0

;u,2

0

bu,,

bu,,

1 (12)

-

The Green-Lagrange

(13)

& = #zr(lv’ul) S$ = oa(6’ur)

i&T,b)Jdr,.

virtual work equation

0 a/aoz a/soy

bb, 0 ;u,, 00;u,, 1 0

The incremental

s

(18)

becomes

,T.GufdTr=O.

(19)

strain and its variation

are

+ (,V’urYx + Xr(,v’u~(ov’u~~; (20)

A design sensitivity analysis principle and its implementaton into ADINA P = i{mJ01+ + mYo1

won

specified displacement

+ ~T,v'"7Lv~1~

(21)

LV’“Tv .

The reference volume concept gets translated quite naturally into isoparameteric elements when finite element discretization of the continuum is used for numerical calculations. This is discussed later.

& = G,s; P

ANALYSIS

(%IJ+ h 3.7)dr,

(22)

-

(~~.EaJ+~.‘&“J+~..p_gj)dP

(23)

where the superscript a refers to the fields for an adjoint structure which is defined during the derivation of the design sensitivity analysis expression as follows. Equilibrium equation for the adjoint structure:

0.

P = G,,; on Or,.

e’ = oa(uO).

(24)

s Initial conditions, body force, surface traction, and

(25)

(26)

law for the adjoint structure: S” = $,, . (E”- CO’)+ s”‘.

+

T”“-6’uJdr,=

S”’ = -G,,;

Strain field for the adjoint structure:

PRINCIPLE

The procedure for derivation of a design sensitivity analysis expression that is valid for shape and sizing design problems is to take design variation of the constraint functional given in eqn (18). This will contain explicit as well as implicit variations of the state fields. The implicit design variations are then obtained in terms of certain adjoint fields by using the design variations of the equilibrium equation and the constitutive law for the primary structure (the original structure will be called the primary structure). They are then substituted into the design variation of the functional to obtain the final sensitivity expression. Details of the derivation are given in [12-141. Here Ee#ve only the final expressions. Let 6, S and srepresent the total, the explicit, and the implicit design variations. Thzn total design variation of Y is given as JY = S$ + &. In the following, we first give expressions for & and & and then define the quantities used in them.

-

for the adjoint structure:

= h,, on OTT; I@ = -g,,

Stress-strain 4. DESIGN SENSITWITY

693

(27)

The variation Sri’ is obtained from eqn (26) as 6”” = f~r(,V”‘r)(,V6’u~T8 + Xf(,VG’ur)(,Vu”‘)~.

(28)

In the foregoing expressions, the superscript i refers to the initial conditions and the superscript 0 refers to the boundary conditions. Note that the equilibrium equation for the adjoint structure, eqn (24), is the same as the incremental equilibrium equation, eqn (8), for the primary structure at the final load level r. The adjoint loads in eqn (24) are delined in eqn (25) according to the form of the constraint functional. It is interesting to note that the explicit design variation term

in eqn (22) accounts for design variation of the displacement specified boundary in shape optimization problems. Note also that the principle given in eqns (22) and (23) is written in the most general form where integrals over the volume and surface of the continuum are used. It is important to realize that it can be specialized to one and two dimensional structures as well as fixed domain design problems. For such applications, simplified expressions can be readily obtained from eqns (22) and (23). Use of the principle on analytical examples has been demonstrated in [l2-141. 5. PHYSICAL

INTERPRETATION

OF THE PRINCIPLE

The explicit design variations of the functional !f’ given in eqn (22) are quite straightforward to calculate. The implicit variations given in eqn (23) require definition of an adjoint structure. Each term of the implicit variation can be given the following physical interpretation which can be useful in numerical evaluation of design gradients as well as in optimum design of structures.

virtual work done by the design variation of the body force in going through the displacements of the adjoint structure.

694

J. S. AROMA and J. E. B. Cm (ii)

&f*@$Jdl? s virtual work done by the body force in going through the displacements of the adjoint structure over only the changed part of the structure.

(iii) virtual work done by the explicit design variation of the specified boundary traction in going through displacements of the adjoint structure. (iv) virtual work done by the specified surface traction in going through displacements of the adjoint structure for the changed part of the structure. (v)

s

&S.c”JdP:

virtual strain energy stored in the structure due to explicit design variations of the stresses in going through strains of the adjoint structure. (vi) virtual strain energy due to stresses in the primary structure in going through explicit design variations of the strains in the adjoint structure. (vii) virtual strain energy stored in the changed part of the structure due to stresses in the primary structure in going through strains of the adjoint structure. With these interpretations, it can be seen from eqn (23) that the implicit design variation 8Y is given as the difference of the explicit design variation of virtual work of the external loads in going through displacements of the adjoint structure and the explicit design variation of the virtual strain energy of internal stresses in going through strains of the adjoint structure. In [3] it is shown that for bar structures the design ~~iti~ty coefficients for certain stress and displacement constraint functions are related to virtual strain energy densities produced by the internal forces of the primary structure in going through displacements due to a virtual loading system (defined by the form of the constraint function). The foregoing interpretations of the principle generalize that notion for a #ntinuum.

The idea of virtual loads and virtual displacements gets generalized to adjoint loads and adjoint structures, respectively. These inte~retations can also be useful in gaining insights into the behavior of optimum structures. For example, the optimality conditions for a structure under single loading condition may lead to the conclusion that the strain energy density at the optimum must be uniform throughout the continuum as in the case of bar st~ctures[20]. 6.

FINITE ELEMENT DLWRETIZATION OF THE PRINCIPLE

Usual finite element procedures can be used to discretize the principle and to develop ~mpu~tio~l algorithms for numerical evaluation of the design sensitivity coefficients. We shall first summarize the discretization of the analysis equations for the primary structure and then the design sensitivity analysis equations. The reference volume is identified as isoparametric volume having natural coordinates for its description. Therefore we shall use the isoparametric finite element formulation and the notations defined in [ 191for the most part. To simplify the presentation, it is assumed that the system is modeled by a single finite element, It is, however, understood that most systems will be modeled by a number of finite elements which can be of different types. Usual assembly procedures are used to combine contributions from individual finite elements to calculate various quantities for the entire model. It is understood that this will be done during the analysis and design ~nsitivity analysis phases. It is also important to note that equations presented in the following are symbolic in nature. Further development of the equations is usually necessary for each type of finite element to convert them into forms suitable for numerical implementations [ 13,191. The displacement, incremental displacement, and the strain from eqn (10) at load level t, are discretized using the shape function matrix ,,H as ‘u=oH’U;

u=&KJ;

$ = (bB, +- 4 ;B,, >‘U = $tZr

(29)

where ‘U and U are the displacement and incremental displacement vectors, and the strain-displacement matrices are

bB, = clw;

;A =

&I = l4 I;s,,

(30)

to%VT

OT

OT

0=

&H,,VT

OT

OT

0'

(OK.” v T w,x’U)T

tfn.! VT

OT

(OR,VT

OT

cPw w

OT

(OK,*VT

to%, VT

A design sensitivity analysis principle and its irnplementaton into ADINA

analysis into an existing structural analysis code. This point is further explained in the next section. The equilibrium equation for the adjoint structure, eqn (24), after substituting discretizations of eqn (38) is given as

Variation of strains from eqn (14): S$~Y = ($m + ;BLI )6 ‘U = &S ‘U. The equilibrium eqn discretization, becomes

(1)

at

time

(32) t,

;Q(‘u. b) = ;R(‘U, b) - $(W, b) = 0

after

(33)

where the internal and external force vectors are, respectively, given as

Substituting the adjoint constitutive law from eqn (27), using eqns (25), (36) and (38), and the fact that 6’U are arbitrary, eqn (39) becomes ;KU” =

(G,a)TJ

dP +

I + ;R = I&I’EIJ

695

dI’+ J,,Hr;pld&.

(34)

i

(h,, ,$DTJdrT I

(GsQ,, WTJ

dp +

(G,,

s

&YJ dl? (40)

Define: The incremental

equilibrium

sUe’+A;R-8;

eqn (8) becomes

with

$=&+&.

(35)

Thus, the adjoint equilibrium equation, eqn (40), in a discretized form is given as

& = $U” = ( ‘p,u)T

(42)

where $CKL is the linear strain incremental stiffness matrix and & is the nonlinear (geometric, or initial stresses) incremental stiffness matrix. In eqn (36), gS is a matrix of stresses (9 x 9 in the general threedimensional case) derived using eqn (14) as

= S$P~S,ll’ For discretization u”= OHIP;

= 6 V&

$&&.

(37)

of the adjoint structure, define i? = pBLU”;

Sq” = 6&,Ua

6&”= &,GU”;

= &A ;BNLUa.

(38)

An important point to be noted here is that the discretization for the primary and adjoint structures can be different; i.e. the shape functions, and the number and type of finite elements can be different. Although in most implementations the two discretizations will be the same to effect efficiency, the continuum formulation opens up the possibility of having different models for adjoint and primary structures. This flexibility is not obvious if a discretized model is used in the derivation of sensitivity expression [6-91. This observation has some implications for implementation of design sensitivity

+

(G,, s

WJ d?‘+ (G, e,,)J dv. (43) s

It is important to note that the stiffness matrix for the adjoint structure is the same as the tangent stiffness matrix for the primary structure at the load level t

where design sensitivity analysis is being performed. This matrix is usually available in the decomposed form from the structural analysis phase, so it can be directly utilized in solving the adjoint equilibrium equation. The first two integrals on the right-hand side of eqn (43) that defines loads for the adjoint structure correspond to a displacement constraint and the last two integrals correspond respectively to stress and strain constraints. The design sensitivity eqns (22) and (23) after discretization become

$!P=

s

[(?G + E”’ - &I

-

S"'

l

&)J + GZJ] d I’

696

J. S. ARORAand J. E. B. CAIWOSO

If we identify the first three integrals as BY-the explicit design variation of Y-and combine the remaining terms, eqn (44) can be written as

.

Finite Element Node

o Design Element Node

4 design elements,16 finite elements

Fig. 1. Design element definition.

where ;Q = 0 is the equilibrium equation defined in eqn (33). Thus to calculate the design sensitivity coefficient gorn eqn (45) we need to calculate design variations SY and %OQ,and solve the adjoint equilibrium eqn (42) for U”. It can be seen again that the implicit part of the design variation of Y is the difference between virtual work of the explicit design variations of external and internal forces in going through displacements of the adjoint structure. Equations (45) and (42) coincide with the corresponding sensitivity and adjoint equations derived with a discrete model [6-91. Thus the two approaches for derivation of computer implementable expressions for design sensitivity analysis lead to the same result. However, the continuum formulation offers more flexibilities and generalizes the notion of adjoint variables to adjoint structure as noted before. 7. CONCEPT OF DESIGN ELEMENTS

In the previous sections, the concepts of reference volume and adjoint structure were the tools to perform design sensitivity analysis with continuum and discrete models. Using the finite element approach, equivalence between continuum and discrete models was shown. For the analysis problem, a finite element is defined by its stiffness matrix, right-hand side load vector and the response quantities such as displacements and stresses. This idea can be extended to define the concept of a sensitivity element by defining additional quantities as derivatives of internal forces, stresses and load vector with respect to design and state variables. Since the constraints are equations containing the response quantities, we can assemble the sensitivity elements and get the design sensitivity analysis model for the entire structure. The sensitivity elements may use the same interpolation functions as for the original finite elements, or different ones. Also, for convenience of design, we can group several finite and sensitivity elements in only one macroelement. This macroelement has no other function

than to model the structure for design. We call it a design element which can also be modeled using the isoparametric concept. Key nodes of the design element can be treated as design variables for shape optimization problems. At the beginning of each iteration of the optimum design process, an analysis model has to be generated from the design mode1 as a result of design changes in the last iteration. Figure 1 illustrates this process. Two levels of discretization are defined; the first level corresponds to the finite element model for analysis, and the second level corresponds to the design element mode1 for optimization. The finite element mesh at the beginning of each analysis is generated by dividing the design elements into smaller elements. Each design element is defined by the number and topology of its nodes relative to the finite element mesh, and shape functions. The design element concept is also valid for cross-sectional dimensions. We can have a macroelement as a group of finite elements whose crosssectional dimensions are related to the cross-sectional dimensions of the macroelement or design element. Each time the design geometry is updated we use such relations to update the geometry of the finite elements. 8. IMPLEMENTATION OF THE PRINCIPLE INTO ADINA

It is clear that in order to treat realistic design a sophisticated nonlinear structural problems, analysis capability is needed. The computer program ADINA provides such a capability for static and dynamic analysis with several material models. Also, a computer program IDESIGN (Interactive DESIGN Optimization of Engineering Systems) is a sophisticated, state-of-the-art software that provides nonlinear programming capability (211. Cost and constraint functions as well as their gradients with respect to the design variables have to be supplied to the optimization program. A design sensitivity analysis capability has to be added between analysis and

A design sensitivity analysis principle and its impletnentaton into ADINA optimization phases. During the optimization phase, changes in the design are made which affect the analysis model. This process of analysis/sensitivity/ design update must be executed several times before an optimum is reached. Since structural analysis and optimization capabilities are available, we concentrate here on implementation of design sensitivity analysis capability. The quantities needed to _ implement design sensitivity analysis are: ‘U, 3, &, ‘G,, the von Mises equivalent stress; m, the element mass; 6% X, -6 $3 fiKt’ee*. the derivative of von Mises stress with respect to Cauchy stress; ‘8,s. the derivative of Cauchy stresses with respect to 2nd piolaKirchhoff stress vector; &&/ab and b, the explicit design derivatives of element straindisplacement matrices; aX/ab and aJ/ab, the explicit design derivatives of the Jacobian and its determinant; dm/ab, the explicit design derivative of the element mass; and &R/a b, the equivalent nodal force design derivative. Steps needed beyond the analysis phase to implement design sensitivity analysis are as follows. 1. Calculation of explicit design variations ZY’ for each constraint functional from eqn (44). 2. Calculation of loads for the adjoint structure for each constraint function from eqn (43). 3. Solution of the adjoint equilibrium equation (42). 4. Calculation of explicit design variations of the external and internal forces for the entire structure as indicated in eqn (45). 5. Calculation of total design variations of a constraint functional using eqn (45). Here zOQ may be assembled for the entire structure before completing calculations of eqn (45), or element-by-element calculations may be done first and used later to assemble IJ”QOQ. Two strategies can be used to implement design sensitivity analysis capability with existing finite element analysis packages, such as ADINA. These strategies have recently been discussed in great detail [I. There shape optimization problems were not included and actual implementation was limited to only axial force elements. In this paper, we include shape optimization problems and extend the implementation to other finite elements. Therefore, we shall briefly review the two strategies, compare them and present details of impl~entation of one of them into ADINA. 8.1. First strategy: design sensitivity extending analysis program

analysis hy

In this strategy, the existing analysis program is extended slightly to calculate and save additional data needed for design sensitivity analysis. Some of the data needed for gradient calculations are routinely calculated during response calculations but

697

they are not saved. So, they are not available at the end of analysis. However they can be saved on external files by inserting WRITE statements at appropriate places. A separate module is then developed to use the saved data along with the response quantities and several ‘lower level’ subroutines already available in the existing analysis package to calculate final gradients. 8.2. Second strategy : design outside analysis program

sensitivity

analysis

This strategy uses the response output already saved from the analysis and calculates the additional information needed for design sensitivity analysis in a separate module. No modifications are made (or possible) to the analysis program. All calculations in eqn (45) must be performed completely outside the analysis program. Since many of the data required in the design phase may not be saved, we need to calculate them again. The data to be recalculated are the element level matrices, such as Jacobian of the transformation, deformation gradient, incremental constitutive matrix, Piola-Kirchhoff stresses, straindisplacment matrices, etc. These data can be G&Ulated using the shape functions and the original input data for the analysis program. This is the part where the flexibility of using shape functions that are different from the analysis phase offered by the ~ntinuum formulation can be extremely useful. Many times the shape functions used in the analysis program are not known. Therefore different shape functions have to be used for design sensitivity analysis. Once the loads for the adjoint structure have been assembled for each constraint using eqn (43), the analysis program is used again to calculate the adjoint displacements, strains and stresses. Once all the data needed for eqn (45) have been calculated, the design variations of the constraint functionals can be calculated. 8.3. Comparison of the two strategies Advantages of the first strategy are its computational efficiency and the simplicity of programming. One needs to save the calculated element level data and develop small subroutines to calculate and save partial derivatives of element level quantities. A separate small module is then developed to assemble loads for the adjoint structure. Existing subrou~n~ are called to solve for adjoint fields. Calculations of eqn (45) are programmed to complete gradient evaluations. A major disadvantage of the strategy is that it requires access to the source program for analysis and familiarity with it. Also, care has to be taken to avoid the sensitivity calculations when the program is used just for analysis. The second strategy is very attractive because it does not require knowledge of the analysis program or its source code, at least conceptually. In reality the computational as well as programming effort can be substantial depending on what data are available at the end of analysis phase. If the program does not

698

J. S. ARORAand J. E. B. C,+anoso

have a re-start capability to calculate adjoint displacements, the programming effort to implement design sensitivity analysis with this option can essentially amount to ‘re-write’ of the analysis code. This can be very time consuming as well as costly. Even if re-start capability is available, substantial programming effort is needed to calculate element level quantities and their partial derivatives with respect to design. This must be done outside the analysis program for all finite element and other options available in the program. In this strategy, the shape functions used during design sensitivity analysis can be different from those used in the analysis phase. This may have to be the case because shape functions used in the analysis program may not be known or some implicit shape functions may have been used. In such a case accuracy of design sensitivity coefficients may not be good. While comparing the two strategies one should also not forget that sensitivity analysis is not, by itself, an ultimate goal, but a tool for design. The implementation is intended to achieve design improvements and not just to produce sensitivities. Such a design model at the end of each optimization iteration has to be transformed to the corresponding analysis model, the input data needed for the analysis program have to be changed for new analysis. Therefore, a module is needed in the second strategy to create new input data for analysis after each optimization iteration. The module should read the entire input data file, modify the data and save the file for further processing. This can be quite cumbersome. In the first strategy this is not needed. The input module for the analysis program can be modified to read new data from a different file. In addition to the preceding difficulties, the second strategy will be quite cumbersome to implement for path-dependent problems. It has been shown in [14] that design sensitivity analysis of such problems is also path dependent, and a direct differentiation method is more appropriate than the adjoint structure method. The first strategy is relatively simple to implement for such problems, but not the second one. As a matter of fact it may be almost impossible to implement the second strategy without some extensions of the analysis code. 8.4. Implementation of design sensitivity analysis with ADINA Based on the preceding discussion, the tirst strategy is selected for implementation of design sensitivity analysis into ADINA. This option is adopted in the present work to advocate the philosophy that design of systems is the ultimate goal and not just their analysis. Therefore, objective of structural analysis capability should be broadened to calculate sensitivities of the structural response to changes in the initial design. This capability can then be combined with optimization programs to optimize designs, if desired. All the gradient calculations are implemented

and saved at the element level in ADINA for use in the design sensitivity analysis and optimization phases. The final sensitivities of the cost function and response quantities are then calculated outside ADINA by assembling the element sensitivities and solving the adjoint structure. Some of ADINA’s subroutines are recalled to accomplish these objectives. Considerable implementation details for design sensitivity analysis with ADINA have been presented in [7]. Details of the current implementation are also discussed in [13]. Here we shall present these only in summary form. Figure 2 gives an overview of design sensitivity analysis and optimization with ADINA and IDESIGN programs. Most of the figure is selfexplanatory. Some of the terminology is defined as follows. IDESIGNIN is the input file for the program IDESIGN containing initial design variables, lower and upper bounds for design variables, constraint information, and parameters for convergence, constraint violation, etc. OPTADIIN is an input file defining the number and topology of the design elements. It contains such data as number of design elements, relationships between the design elements and the finite elements, nodes for the design element, etc. Both the input files define the initial design that is to be updated at each design iteration as a temporary design. ADINAIN is the input file for the analysis program ADINA. Data for design variables in this file (nodal coordinates and cross-section properties) have to be updated accordingly to the temporary design. This is done by slightly modifying ADINA’s subroutines. New coordinates of the design element nodes (actual design variable values) and the design element shape functions are used in the subroutines to calculate the coordinates of the new finite element mesh. At the element level, the cross-sectional properties are updated. OPTADIINZ is another input file for design which contains information about stress and displacement constraints, points of constraint, and their allowable values. ADINA.OUT is the analysis output file. The response to the analysis corresponding to the final design is available at the end of the design process. Figure 2 also shows that appropriate data are saved from ADINA on external files to calculate cost and constraint functions and their derivatives. Some of the data are [13]: element mass, number of displacement constraints, number of stress constraints, allowable displacement, allowable stress, number of degrees of freedom (d.o.f.) of the element, global degree of freedom, number of the local ith degree of freedom in the element N, displacement corresponding to the local d.o.f. i, Gaussian point number (IPT), determinant of the Jacobian matrix, derivatives of the Jacobian determinant with respect to design variables at IPT, weight factor at IPT, equivalent stress (von

A

designsensitivityanalysisprincipleand its implementatoninto ADINA

699

DATANEEDEDPORDESIGN SENSITIVITY ANALYSIS AND Ol’lWIZAmN SAVED ENFILES. IT IS USED BY ~~~~~~~DF OFI’IMlZATlON.

DATA BASE

Fig.

2. Design sensitivity analysis and optimization with ADINA and IDESIGN.

Mises) at IFT, number of design elements, number of cross-sectional design variables of the element, number of element nodes, element mass derivatives with respect to design variables, equivalent stress derivative with respect to the ith degree of freedom, derivative with respect to the .&h design variable of the element force component corresponding to the ith d,o.f., and information about the use of the factorized tangent stiffness matrix at the final load level. Subroutines are inserted into the program at the element level to calculate element sensitivities, such as for calculation of the adjoint load of the displacement response, derivatives of the strain-displacement matrices with respect to design variables, explicit element force design derivatives, and explicit design derivatives and adjoint load for the element stress response. Derivatives of the determinant of the Jacobian are calculated by modifiying ADINA’s subroutines where the determinant is calculated.

framed structural design problem. Although these problems are academic in nature, they show the wide range of applicability of the nonlinear design sensitivity analysis principle and the optimum design procedure. For all the examples, the gradients calculated by implementing the design sensitivity analysis principle into ADINA are compared with finite difference calculations. This is done to verify aauracy and computer implementation of the principle. Three methods for verifying numerical evaluation of the gradients suggested in f24 are used. These procedures use the following three parameters. Relative value: First variation:

FV=(ISY’l/lA!f’)

x 100

Normalized inner product: 9. EXAMPLE PROBLEMS

The system that combines ADINA and IDESIGN described in Sec. 8 has been used to solve several design problems 1131. These include structures modeled by plane elements, straight beams, curved beams, and shell elements. Three example problems are presented here: a perforated tension strip with a hole, a planar fillet design problem, and a planar CA.& 32/34-N

(N’/8b)r(A!?‘/Ab) Nrp = fW/cYb 11\IAljAb

11

where H/a b and A’PJAbam the analytical and &rite difference gradients of a constraint Y; S!P is the tirst variation of !P due. to a design change Ab, and AYr= !I’@+ Sb) - F(b). For Snite difference gradient evaluation, an option in IDESIGN is used to

700

J. S. AROM and J. E. B. Cumoso

Fiite Element Model

Fig. 3. Perforated tension strip. automatically calculate them. When the analytical and finite difference gradient calculations are close to each other, RV and FV should be close to 100 and (1 - NIP) should be close to zero. These parameters will be reported for each problem. The mass or volume of the structure is taken as the cost function. Constraints are imposed on nodal displacements written in the normalized form as 1.O.O, < wh ere Vi is the generalized disIU,l/lU,-placement along the ith global degree of freedom and U, is its allowable value. Cauchy stresses-recovered from the 2nd Piola-Kirchhoff stresses-are used in imposing the stress constraints. The stress constraint is expressed in a normalized form as ~~/a, - 1.O < 0, where efi is either the von Mises stress at the Gaussian point j of element k, the mean von Mises stress in element k, or the maximum von Mises stress in element k; and 6, is an allowable stress value. Since the sensitivity data are enormous, gradients of only some of the constraints are reported. Note also that the sequential quadratic programming (SQP) algorithm in IDESIGN is used to obtain optimum solutions. A very precise convergence criterion of 1.OE - 03 and constraint feasibility requirement of within 0.1% of allowable value are used. 9.1. Perforated tension strip As the first numerical example, a perforated strip given in [23] and shown in Fig. 3 is optimized with respect to its thickness. Three cases are considered: linearly elastic material with small displacements (LE), linearly elastic material with geometric nonlinearity (GN), and material-nonlinear-only case (MNO). For the latter case, an isotropic hardening elastic-plastic model is used. The data for the

problem are L = 56 mm, w = 20 mm, d = 10 mm, E = 70,000 N/mm2, v = 0.25, mass density = 2.7E - 06 kg/mm3 and p = 150 N/mm. Because of the symmetry, only one quarter of the strip is discretized with 11 elements and 50 nodal points. The displacement compatibility is imposed along the boundary between nodes 46 to 50 (zero displacement in the Z direction) and nodes 5 to 34 (zero displacement in the Y direction). The element thicknesses are selected as design variables (11 design variables). As constraints, the displacement in the Z direction along the edge between nodes 1 to 5 should not exceed 0.1 mm, and the maximum equivalent von Mises stress at the integration points in every element must not exceed 400 N/mm*. The finite element and design models are shown in Fig. 3. A grid of 2 x 2 Gaussian points is used in numerical integration. An increment of 0.01% in the design variables was found to give convergence to the finite difference calculations of gradients for the first two cases (LE and GN) while 0.001% increment was needed for the MN0 case. The initial thickness is selected as 1.5 mm. The smallest and largest allowed value for thickness are 0.1 mm and 10 mm respectively, except in the MN0 case where the lower limit is changed to 0.5 mm to prevent the strip from becoming unstable during equilibrium iterations. Initial mass of the structure is 1.0544E - 3 kg. Table 1 shows comparison of the analytical and finite difference gradient evaluations for each constraint at the initial design for the GN and MN0 cases (with the hardening module E, = 1400 N/mm2). There is no large discrepancy between the analytical and finite difference gradient calculations. In fact, comparison of each gradient component shows al-

701

A design sensitivity analysis principle and its implementaton into ADINA Table 1. Verification of design sensitivity coeti&nts for displacement and stress constraints (maximum stress) for perforated tension strip Cieometricagy nonlinear analysis

Material-nonlinear-only analysis

RV

Fv

(1 - NIP)

RY

FY

cost

100.00

100.00

0.ooooE + 00

100.00

100.00

0.ooooE + 00

Disp.2 Disp.4 Disp.8 Disp.9

100.01 100.01 100.01 100.01 100.01

100.00 100.00 100.00 100.00 100.00

1.192lE - 07 0.ooooE + 00 5.96ME - 08 1.192lE -07 1.192lE -07

100.69 100.74 loo.74 100.77 100.80

101.08 101.08 101.08 101.08

1.5438E - 04 1.729lE-CM 1.8972E - 04 1.8620B - 04 1.86206 - 04

Stress 1 Stress 2 Stress 3 Stress 4 Stress 5 stress 6 Stress 7 Stress 8 Stress 9 Stress 10 Stress 11

100.01 100.01 100.01 100.01 100.01 100.01 100.00 100.00 100.01 100.01 100.01

100.00 100.00 100.00 100.00 100.00 100.00 99.99 100.00 100.00 100.01 100.01

5.9605E - 08

100.53 100.32 100.68 101.18 100.36 100.06 100.01 100.25 100.06 100.00 100.00

100.31 101.69 101.81 92.80 101.69 99.19 98.35 101.19 99.25 100.12

Fun~ion

DiSP.6

O.OOOOE + 00 0.OOOOE+ 00 0.OOOOE+ 00

2.38428 - 07 59605E -OS

1.1921E-07 5.96052 - 08 0.OOOOE + 00 0.OOOOE+ 00 1.192lE -07

(1 -NIP)

101.08

99.95

1.5926E -04 5.2088E - 04 1.3292E - 04 3.457lE - 04 l.l146E-04 3.3319E - 04 1.0312E -04 5.9128E - 04 2.7001E - 05 5.36446 - 07 1.192lE -07

most 100% agreement between the two calculations for each constraint gradient. For the LE case the agreement was also 100%. The analytical gradients of some of the constraints are as follows. Case LE: Z-displacement

at Node 5 -7.7545E-03 -2.0255E-02 -2.3%8E-02 -2.8072E-02-3.637414:-02 -1.936OE-02 -7.2379E-03 -7.6075E-02 -3.206OE-02 -1.7615E -02 -7.47766 -02

Case LE: stress in Element 2 -3.2486E -02 -4.578lE -01 -1.2012E -01 1.172lE -01 -7.3035E -02 -94029E - 04 -7.2580E - 02 2.2048143- 02 -3.715OE - 03 4.5281E - 03 Case GN: Z-displacement at Node 3 - 7.726614:- 03 - 1.9908E - 2 -2.3775E

-02

-2.7668E

-02

-3.539lE

-02

4.5698E -02

-1.9035E-02

-7.2821E-03-7,9342E-02 -2.6647E-02-4.9344E-02-4.4136E-02 Case GN: Stress in Element 4 -4.26438 -02 8.932lE -02 -5.74258 -02 -3.8093E -4.2856E - 04 -6.4296E - 02 1.9333E - 02 -3.3216E

-01 -03

Case MNG: Z-displacement - 1.0002E - 02 -4.5365E -6.2342E - 03 -9.32008

- 02 -4.07668 - 02 -4.4839E

at Node 1 - 02 -3.2262E - 02 -2.1123E - 02 - 1.5846E - 02 -9.1438E

-7.133915 -02 3.9308E - 03

4.6825E -02

- 02 - 1.6246E - 02 - 03

Case MNG: Stress in Element 6

-1.2534E-03 2.8098E -02 -2.166lE -02 8.3656E-03 -8.03715-02-2.0042E-01 4.4030E -03 -2.1597E-02 1.1581E -02 -3.3293E-03 3.1024E -03 The optima

designs are as follows.

Case LE: thicknesses = (5.18903E - 01, 2.21926, 6.93418E - 01, 1.74739, 9.43258 - 01, 9.14277E - 01, 2.78538 - 01,. 8.2262E - 01,4.37994E - 01, 5.91066E - 01, 6.54126E - 01); all displacement constraints are nearly active but constraints at Nodes 2 and 5 are tight with Lagrange multiplies as 2.130493 - 04, 2.73549B - 06; optimum mass is 4.856256 - 4 kg; and the program took 17 iterations to converge. Case GN: thicknesses = (5.12058E - 01, 2.17103, 6.88648E - 01, 1.72723, 9.3632115 - 01, 9.1746E - 01, 2.75619E -01, 8.17268E - 01,4.36693E - 01, 5.88836E-Ol,6.52179E - 01); all displacement ~ns~ints are nearly active but constraints at Nodes 2 and 5 are tight with Lagrange multipliers as 2.13009E - 04

702

J. S. AROIU and J. E. B. Cm Table 2. Verification of design sensitivity co&icients for displacement and stress constraints (maximum stress) for uerforated tension strin with lame strains Strains = 3%

Strains = 7%

Function

RV

FV

RV

FV

cost

100.00

100.00

O.OOOOE + 00

100.00

100.00

O.OOOOE + 00

Disp.2 Disp.4 Disp.6 Disp.8 Disp.9

101.48 101.47 101.45 101.44 101.44

101.58 101.58 101.56 101.55 101.55

9.29838 - 06 1.0073E - 05 1.09678 - 05 1.1265E -05 l.l265E-05

102.97 102.95 102.92 102.90 102.91

103.18 103.18 103.16 103.14 103.13

3.7253E 4.0472E 4.45256 4.4405E 4.512lE

-

05 05 05 05 05

99.88 99.45 100.08 99.41 99.78 99.98 99.81 99.93 99.92 99.99 100.01

100.63 98.51 100.16 98.83 100.27 99.70 100.24 99.88 99.92 99.79 100.07

1.2815E - 05 2.37236 - 05 1.90736 - 06 8.7619E - 06 5.4836E - 06 3.9339E - 06 3.2306E - 05 2.3842E - 06 1.1384E -05 89407E - 07 2.9802E - 07

99.69 98.75 100.11 98.71 99.51 99.70 99.59 99.83 99.82 99.97 99.99

101.17 96.92 100.22 97.55 100.47 99.33 100.49 99.71 99.85 99.57 100.13

5.14396 9.1672E 7.0333E 3.43326 2.1696E 1.5676E 1.2022E 9.7156E 4.4227E 3.0994E 7.74866 -

05 05 06 05 05 05 04 06 05 06 07

Stress Stress Stress Stress Stress Stress Stress Stress Stress Stress Stress

1 2 3 4 5 6 7 8 9 10 1I

(1 -NIP)

(1 -NIP)

and 2.733758 - 04, optimum mass is 4.82936E - 4 kg; and the program took 17 iterations to converge. This case gives a slightly better design than Case LE. Case JUNO: thicknesses = (1.15875, 1.90637, 9.5441E - 01, 1.58117, 1.0537, 7.433B - 01, 5.OE - 01, 7.97812E - 01, 5.OE - 01, 5.58933E - 01,5.9475E - 01); all displacements constraints are nearly active but displacements at Nodes 2 and 5 are tight with Lagrange multipliers as 2.09927E - 4 and 2.2423E - 04; lower bound constraints for design variables 7 and 9 are tight with Lagrange multipliers as 1.02363E - 05 and 1.69341 E - 05; optimum mass is 4.933E - 04 kg; and the program took 16 analyses to converge. 9.2. Perforated tension strip with large strain The structure of the previous example is now considered for large strains under the incompressible Mooney-Rivlin material law. The stress-strain relation is given as

a==~,

[{/}-D2{

_;,j]

1 +2c,

D [

1 (10

+{I--D2(C,,+C22))

CZ2 C,, 1 -G2

11

where C,, = (1 + 2c,,), C,, = (1 + 2&,

C,, = t12, and

D = (C,,C,, - Cf2)-‘. The material properties c, and

c2 are 21.605 N/mm2 and 15.747 N/mm*, respectively, and density = 2.7E - 6 kg/mm’. Two load cases are used to have strains of approximately 3% and 7% (Case I with a uniform load of 1.875 N/mm and Case II with a load of 3.75 N/mm). Table 2 shows comparison of analytical and finite difference (increment of 0.01% at the initial design) gradient evaluations. Both cases show a good agreement between the two gradient calculations. Some of the gradients are as follows.

Case I: Z-displacement at Node 5 -3.1276E -02 -8.0321E -02 -9.8846E -02 -1.1325E-01 -1.5323E-01 -8.5974E -02 -3.11728 -02 -3.0980E- 01 - 1.2997E -01 -7.0427E-02 -2.8883E-01 Case I: stress in Element 2 -8.27356 - 02 - 1.5170E + 00 -4.3365E - 01 3.6621E - 01 -24036E - 01 4.79148E - 03 - 2.2739E - 01 7.95176 - 02 -9.8943E - 03 1.9848B - 02 Case II: Z-displacement at Node 4 -6.7532E - 02 - 1.6927E - 01 -2.0681E -6.5504E - 02 -6.5913E - 01 -2.4055E

- 01 -2.34918 - 01 -3.1292E - 01 - 1.93896 - 01 -5.5041E

1.50996 - 01

- 01 - 1.73026 - 01 - 01

Case II: stress in Element 4 -3.04998 - 01 4.89896 - 01 -3.6708E - 01 -2.4780E +00 -4.95258 - 01 1.3834E - 01 - 1.7232E - 02 3.31846 - 02 1.4094E - 02 -4.0883E - 01

3.1691E - 01

A design sensitivity analysis principle and its implementaton into ADINA

703

The optimum design with allowable stress of 1.5 N/mm’ and density of 2.7E - 06 kg/mm3 are: Case I: thickness = (2.20084,8.56832,

2.93923,6.53118,3.85726, 3.60370, 1.23314, 3.25202, 1.76414,2.32369, 2.54699); all displacement constraints are nearly active but constraints at Nodes 2 and 5 are tight with Lagrange multipliers as 8.1013E - 04 and 1.0904E - 03; optimum mass is 1.9256E - 3 kg; and the program took 12 iterations to converge. =(6.51152, 10.0, 7.59865, 10.0, 8.64508, 6.27787, 2.29004, 6.95987, 3.30468, 4.68431, 5.23824); all displacement constraints are nearly active but constraints at Nodes 2 and 5 are tight with Lagrange multipliers as 1.56032E - 3 and 2.40293E - 3; design variable numbers 2 and 4 are at their upper limit with Lagrange multipliers as 7.872lE - 05 and 4.598E - 05; optimum mass is 3.882E - 3 kg; and the program took 12 iterations to converge. Case II: thickness

9.3. Fillet shape design problem Figure 4 shows the finite element model for a quarter of a flat piece. Appropriate boundary conditions are used along edges l-5 and 14 to impose symmetry in the problem. The model has 12 eightnode isoparametric finite elements and 50 nodes. Only one eight-node isoparametric design element (quadratic interpolation) defined by the nodes 17,19, 21, 25, 29, 33, 35 and 37 is considered. The only design variable is the vertical coordinate b of Node 29 having initial value of 6.75 in. The problem is to design the shape of the edge 21-37 to minimize the volume of the piece. The edge 46-50 is subjected to a load of 27 psi. Constraints are imposed on the mean von Mises stress in elements 4, 6, 8 and 10. The allowable stress is a, = 8 psi. The material model is the same as for the example given in Sec. 9.2. The material is quite soft and large strains are allowed. The thickness of the piece is 1 in. and the initial volume is 145.125 in’. The program IDESIGN takes 13 iterations to converge to the optimum point given as 9.0, 5.7539, 3.9218, 3.5039 and 4.5 in. as locations

Fig. 4. Model for design of the fillet.

for the nodes 21,24,29,32 and 37. The final volume is 132.808 in3 which is a reduction of 8.5% from the initial volume. At the optimum stress constraint in element 8 is tight. Optimum shape of edge 21-37 is shown as a dotted curve in Fig. 4. 9.4. Six beam frame In this section, the frame shown in Fig. 5 is optimized with respect to its cross-sectional dimensions. Linear and geometrically nonlinear responses are considered for the purpose of optimum design. Results obtained with linear and nonlinear analysis are compared. Also, results are compared with the ones obtained for the same structure modeled as a six bar truss [8, lo]. Dimensions and loads are shown in Fig. 5. The Young modulus is 3.OE + 7 psi and Poisson’s ratio is 0.3 for al1 the members. Six isoparametric three-node beam elements and 11 nodes are used to model the structure. All the elements have rectangular cross-section. Depth and width of each element are chosen as design variables (total of 12 design variables) and the cost function is the volume of the structure. The starting cost is 1062.3 in3. Starting design, lower and upper bounds for the design variables are: 1.0, 0.5 and 2.0 in. for all the depths; and 2.0, 1.0 and 5.0 in. for all the widths. Note that the starting design has the same areas (2 in’) as for the truss structure [lo]. Also the same lower and upper bounds for the areas are used in both models. The cross-section orientations have been selected in an unfavorable position for the beam model so that this is not a factor for comparison with the truss structure. Displacement constraints of 20 and 40 in. in the Y and Z directions are imposed at the tip of the structure where the loads are applied.

k 1.SE+S lbs

Fig. 5. Cantilever framed structure.

704

J. S. AROIU and J. E. B. Clurwso Table 3. Verification of design sensitivity analysis for six beam frame Analytical variation

Function

Finite difference variation

RV

FV

cost Disp. 1 Disp.2

Linear analysis 0.15936+04 0.1593E + 04 -0.16756 + 00 -0.1683 + 00 -0.9919E + 00 - 0.9967E + 00

100.0 99.9 99.8

100.0 99.5 99.5

cost Disp. 1 Diso.2

Geometrically nonlinear analysis 0.2125E + 04 0.21256 + 04 -0.38758 + 00 -0.3911E +00 -0.9071E + 00 -0.9165E + 00

100.0 99.4 97.0

100.0 99.1 99.0

Table 4. Optimum designs for six beam frame Element

1 2 3 4 5 6 Frame volume, in’ Truss volume, in’

Linear case Width Depth

Nonlinear case Width Depth

2.4729 1.0000 1.8804 1.4178 1.6305 1.3271

1.7485 1.7650 1.7467 1.7863 1.7539 1.7565

0.67343 0.50000 0.80928 1.06670 0.57467 0.84322

617.25 714.39

Stress constraints are not imposed in this example so that the results can be compared with those given in [8] and [lo]. Before optimizing the design, analytical sensitivities of the constraints with respect to the design variables at the initial design are calculated and compared with those calculated by the finite differences. An increment of 0.01% in the design variables is used for the finite difference calculation and a 3 x 4 x 4 grid of Gaussian points (respectively for axial and transverse directions) is used for numerical integrations. Table 3 shows the results for the analytical and finite difference calculations for both linear and geometrically nonlinear cases. The discrepancy between analytical and finite difference variation is less than 0.5% for the linear case, and less than 1% for the nonlinear case. The optimal designs obtained for the linear and nonlinear cases are given in Table 4. The optimal volumes are obtained as 617.25 in’ for the linear case and 474.23 in’ for the nonlinear case in 17 and 3 iterations, respectively. These two values of the cost function show that, when there is stiffness hardening of the structure (caused by the axial force), the design obtained with the linear model is conservative compared to the nonlinear method. The optimal volumes obtained with the truss model of the structure are V = 714.39 in3 and V = 509.87 in3 for the linear and nonlinear cases, respectively. This shows that the beam model for the structure allows reductions of 13.6% (linear case) and 7% (nonlinear case) in the cost function with respect to the truss model. The truss model is conservative.

0.51584 0.50000 0.51481 0.50000 0.51481 0.50000 474.23 509.87

10. DISCUSSION AND CONCLUSIONS

A principle for design sensitivity analysis of linear and nonlinear response of a continuum is described. The principle can be. used to calculate gradients of stress, strain, and displacement constraints. Shape, nonshape and material selection design problems can be treated. A physical interpretation for the implicit design variation terms is given relating them to virtual strain energy or virtual work of external loads, This interpretation of the sensitivity coefficients can lead to insights into behavior of optimum structures. The concept of reference volume used in the derivation of the principle gets translated into the isoparametric finite element formulation for discretization of the principle. It is found that discretization of the primary structure for response calculations and discretization used during design sensitivity analysis can be different. This offers certain flexibilities in numerical implementation of design sensitivity analysis into existing structural analysis programs that are not obvious when a discrete model is used for their derivation. It is shown in the paper, however, that continuum design sensitivity analysis expression when discretized matches identically with the one obtained with the discretized model (when the same discretizations are used in both cases). A concept of design elements is introduced to discretize the design problem. Two procedures to implement design sensitivity analysis into existing structural analysis codes are studied and discussed. Comparison of the procedures-their advantages and

A design sensitivity analysis prineipie and its implementaton

disadvantages-is presented, A detailed plan of implementation into ADINA and integration into a design optimization software is presented and discussed. Several design examples are used to verify the principle and its implementation. Optimum solutions for linear, geometrically nonlinear and materially nonlinear problems are obtained and discussed. It is shown that inclusion of geometric nonlinearities in the problem can give substantially different optimum designs compared to the linear case. In conclusion, a unified theoretical basis for design sensitivity analysis of linear and nonlinear structures has been established. The theory has already been extended to problems that have path-dependent response 1141. More work needs to be done to extend

the theory to other nonlinear problems dynamics and buckling problems.

such as

Acknowledgement-This work is a part of the research carried out under a grant from the National Science Foundation, No. MSM 86- 13314. REFERENCES 1. H. M. Adelman and R. T. Haftka, Sensitivity analysis

2.

3.

4.

5.

6. 7.

8.

of discrete structural svstems. AI&i Jni 24. 823-832 (1986). H. M. Adelman and R. T. Haftka (Editors), Sensitivity analysis in engineering, NASA-CP-2457, Proceedings of the NASA Symposium, NASA Langley Research Center, Hampton, VA, 25-26 September (1986). J. S. Arora and E. J. Haug, Methods of design sensitivity analysis in structural optimi~tion. AfAA Jnl 17, 970-974 (1979). R. T. Haftka and H. M. Adelman, Recent developments in structural sensitivity analysis. Third International Conference on CAD ICAM. poetics Ed Factoriesofthe Future, Southfield, Michigan, 14-17 August (1988). R. T. Haftka and R. V. Grandhi, Structural shape optimization-a survey. Comput. Meth. appt. Mech. Engng 57, 91-106 (1986). Y. S. Ryu, M. Haririan, C. C. Wu and J. S. Arora, Structural design sensitivity analysis of nonlinear response. Comput. Struct. 21, 245-255 (1985). M. Haririan, J. 3. Cardoso and J. S. Arora, Use of ADINA for design optimization of nonlinear structures. Compuf. Struct. 26, 123-134 (1987). J. S. Arora and C. C. Wu, Design sensitivity analysis

into ADINA

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and optimization of nonlinear structures. In Proceedings of NATO Advanced Study htstitute on ComputerAided Opts Design (Edited by C. A. Mota Soares), pp. 589-604. Springer, New York (1987). 9. C. C. Wu and J. S. Arora, Design sensitivity analysis of nonlinear response using incremental procedure. AZAA Jni 25, 1118-l 125 (1987). 10. C. C. Wu and J. S. Arora, Simultaneous analysis and design optimization of nonlinear response. Ettgmr Cornnut. 2, 53-63 (1987). Il. C.$.Wu and J. S. Arora,‘De&n sensitivity analysis of nonlinear buckling load. Comput. Mech. 3, 129-140 (1988). 12. J. E. B. Cardoso and J. S. Arora, A variational method for design sensitivity analysis in nonlinear structural mechanics. AIAA Jnl26. 595-603 0988). 13. J. E. B. Cardoso and J: S. Arora; De&u sensitivity analysis and optimi~tion of nonlinear structures-a unified approach. Technical Report No. ODL-88.3, (available at NTIS; No. PB88-235171/Ail) Optimal Design Laboratory, College of Engineering, The Uni~rsity of Iowa, IA (1988). 14. J. J. Tsay and J. S. Arora, Design sensitivity analysis of nonlinear structures with history dependent effects. Technical Report No. ODL-88.4, Optimal Design Laboratory, College of Engineering, The University of Iowa, IA (1988). 15. R. T. Haftka and 2. Mroz, First and second order sensitivity analysis of linear and nonlinear structures. AZAA Jnl24, 1187-l 192 (1986). 16. Z. Mroz, M. P. Kamat and R. H. Plant, Sensitivity analysis and optimal design of nonlinear beams and plates. J. Struct. Mech. 13, 245-266 (1985). 17. Z. M&, Sensitivity analysis and optimal design with account for varying shape and support conditions. In Proceedings of NATO Study Institute on ComputerAided Op~imai Design (Edited by C. A. Mota Soares), DO. 407-438. Sorineer. New York (1987). 18. K. K. Choi and i L. T. Santos; Des@ sensitivity analysis of nonlinear structural systems-Part I: theory. Int. J. Numer. Mefh. Engng 24, 20392055. 19. K. J. Bathe, Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). 20. V. B. Venkayya, Design of optimum structures. Compur. Strwt. 1, 265-310 (1971). 21. J. S. Arora and C. H. Tseng, Interactive design optimization. Engng Optimization 13, 173-188 (1988). 22. C. H. Tseng and J. S. Arora, Numerical verification of design sensitivity analysis. AIAA Jnl27, 117-l 19 (1988). 23. ADINA Verification Manual-Problem Solution. Report ARD 84-5, ADINA R&D, Inc., Watertown, MA (1984).