Simultaneous optimization of topology and geometry of a regular plane truss

Pergamon Computers & Svuctures Vol. 66, No. I, pp. 69-71, 1998

0 1997 Elsevier Science Ltd. All rights reserved

PII: soo45-7949(97)ooo5o-3 Printed in Great Britain 0045.7949/98 Sl9.00 + 0.00

SIMULTANEOUS OPTIMIZATION OF TOPOLOGY AND (GEOMETRY OF A REGULAR PLANE TRUSS

Makoto Ohsaki Department of Architecture and Architectural Systems, Kyoto University, Sakyo, Kyoto 606-01, Japan

(Received 21 December 1996; accepted 1 May 1997)

Abstract-An algorithm is presented for simultaneous optimization of geometry and topology of a regular plane truss with uniform cross-sectional area. Difficulties arising from singularity and discontinuity in eliminating unnecessary members and nodes from the ground structure are discussed in detail by using a c;mtilever-type plane truss. Note that the optimization problem becomes more complicated by limiting the feasible designs to regular trusses with uniform cross-sectional areas. In the proposed method, a pin-jointed truss is first modeled as a rigidly jointed frame. A sigmoid function is used for modeling continuous transition between frames with different topologies. In this manner, coalescent nodes and members are successfully removed from the initial ground structure. In the examples, optimal solutions are found for a rectangular plane truss which consists of rectangular units, and the effect of nodal cosl on the optimal topology is discussed. 0 1997 Elsevier Science Ltd

Key wor&-Displacement constraint, Geometry optimization, Regular truss, Sigmoid function, Top- ology optimization

INTRODUCTION

Shape optimization of trusses is categorized into topology optimizal:ion and geometry optimization.

In the widely used approach of topology optimization, unnecessary members are removed from highly connected ground structures while the nodal locations are fixed [l, 21. In this approach, optimal geometry or configuration cannot be found without

introducing infinite number of nodes and members in the initial ground structure, because no efficient method or criteria has been developed for adding new nodes or members to the truss [3]. Most of the topology optimization methods developed so far are for plane trusses under stress and/or displacement constraints [4-81. It is well known that the optimal topology under stress constraints of a truss subject to multiple loading conditions often turns out to be a singular solution, because the stress constraint need not be satisfied in nonexistent members

[5,6,% 101. Nakamura and Ohsaki [l l] presented a method

for finding optimal topologies of trusses for specified fundamental natural frequency. It has been shown in Ref. [l l] that their method is successfully applied for the carse of multiple fundamental frequency. They presl:nted a practical method of topology optimization, where the unstable nodes between colinear members are fixed after generating unstable optimal topology. There is no such instability for the case of rigidly jointed frames [12].

In most of the papers concerning topology optimization of trusses, the effect of nodal cost on the op-

timal topology seems to be overlooked. Ohsaki [13] discussed importance of consideration of nodal cost and presented an optimization algorithm based on the genetic algorithm. It has been shown in Ref. [13] that an optimal topology with smaller numbers of nodes and members can be found by adding nodal cost in the objective function. A technique has been presented for accelerating the process of convergence to the optimal topology starting from a ground structure with sufficiently many members and nodes.

In geometry or configuration optimization, the nodal coordinates of the truss with fixed topology are chosen as design variables [1418]. In this approach, side constraints should be given for nodal coordinates so that a node does not coincide with another, thereby preventing unfavorable numerical instability due to the existence of extremely short members or closely spaced parallel members [14,15]. Therefore it is very difficult to change the topology of the truss, in the process of geometrical optimization, by continuously moving the nodes and by removing the coalescent nodes. For these reasons stated above, it is widely recognized that simultaneous optimization of topology and geometry of a truss is extremely difficult.

It is important to note that topology optimization problem is a discrete optimization problem, whereas configuration optimization is a continuous problem. There have been several optimization algorithms such as branch-and-bound method [8] and genetic algorithm [ 19,201 which are successfully applied to

69

70 M. Ohsaki

discrete optimization problems. In those algorithms, however, a hybrid approach should be introduced in which configuration optimization problems are to be solved successively, if the nodal locations, which are continuous variables, are also considered as design variables. Therefore computational effort seems to be very large if those algorithms are applied for optimizing large trusses. In addition to this difficulty, it is not realistic to provide the data of connectivity for constructing the stiffness matrix of all the possible topologies.

Optimal topology and configuration may also be found simultaneously by the homogenization method [21]. Using this method, approximate solutions are easily found. There is an intrinsic difference, however, between the solution modeled by plate elements and the optimal pin-jointed truss which consists of straight bars. In the case where uniform distribution is assumed for the member cross-sectional areas of the truss, an optimal solution cannot be found by the homogenization method.

In this paper, an algorithm is presented for simultaneously optimizing topology and geometry of a plane regular truss with uniform cross-sectional area, where displacement constraints are considered. Since the trusses in practical use for building structures are usually regular, and since the cross-sectional areas of the members are not allowed to have arbitrary different positive values for those structures, it is desirable from a practical point of view to develop an optimization method for such a class of trusses. It should be noted here that an optimum design problem is not made simpler by limiting the feasible designs to the trusses with uniform cross- sectional areas. Additional difficulties arise such as discontinuous property between the trusses with different topologies.

A pin-jointed truss is first modeled as an equivalent rigidly jointed frame with moderately small flexural stiffness to prevent instability due to removal of some members. A sigmoid function is used for simulating continuous transition between frames with different topologies. Nodal cost as well as member cost is included in the objective function to be minimized. The difficulties in finding optimal trusses and the key concept of the proposed algorithm are illustrated by using a cantilever-type regular plane truss under displacement constraint. In the examples, optimal solutions are found for a rectangular plane truss which consists of rectangular units, and the effect of the nodal cost on the optimal topology is discussed.

OPTIMIZATION PROBLEM

Consider a pin-jointed regular plane truss which consists of units with fixed topology. The shape of a unit is allowed to vary under the condition that the regularity of the truss is preserved. The numbers

of units in the X- and Y-directions are denoted, respectively, by N: and N;. The coordinates (x,y) do not have to be Cartesian coordinates; e.g. X may represent a curved coordinate for the case of arch trusses. The cross-sectional area is assumed to be same for all the members. The problem considered here is to find the optimal topology, optimal nodal locations and the optimal value of the cross-sectional area of a regular plane truss.

The truss is subjected to a set of nodal loads P = {F'j}. Note that the algorithm proposed in this paper is easily extended to the case of multiple loading conditions. The locations of the supports as well as the geometrical properties such as the span length for defining the total dimension of the truss are fixed in order to prevent the existence of infeas- ible or impractical optimal solution. The size of the vector of nodal loads P depends on NY and NY. The loads are applied at the limited number of nodes and the locations of those nodes are fixed.

The displacement vector U = { Uj} is calculated from

KU=P, (1)

where K is the elastic stiffness matrix. The effect of geometrical nonlinearity is not considered here. Displacement constraints are given as

IujlCfl,t (2)

where Vj is the upper bound for the absolute value of the jth displacement component.

Design variables are Nt, N;, the vector of nodal coordinates X and the cross-sectional area A which is the same for all the members. Note that the sizes of X, U and K are functions of Nj: and Ny”. The set of the vectors of nodal coordinates corresponding to feasible regular geometry is denoted by x. The elastic modulus is denoted by E. Then the axial stiffness is given as AE/Li, where Li is the length of the ith member.

In practical design, the trusses have some regularity properties, and it is not practically admissible to allow the member cross-sectional areas to have arbitrary different positive values. In this paper, regular trusses with uniform cross-sectional area are considered as feasible designs. Note that the optimization problem is not made simpler by limiting the feasible designs. Difficulties due to the discreteness of Nz and NY” are discussed in detail in the next section.

In designing trusses, the cost of nodes is usually equivalent to that of the members. Therefore, from a practical point of view, it is critical to consider the nodal cost in topology optimization problem. The cost of each node is denoted by B which is same for all the nodes. Let N” and N” denote the total numbers of members and nodes of the truss, respectively, which are functions of NY and NY. Then the simultaneous topology and geometry op-

Topology and geometry of a regular plane truss

timization problem is stated as follows:

find

N$ N;, X E x and A

which minimize m

C = c”‘5AiL.i + N n B i=l

under displacement constraints

IUjllUj,

where cm is the cost coefficient for the unit volume of the members.

Note again thai. Nj: and NJ are discrete variables, and A and X are contmuous variables. A hybrid optimization problem can be solved by combining a conventional gradient-based optimization algorithm and a dllscrete optimization method such as the branch-and-bound method and genetic algorithm. In this case, however, configuration optimization problems sh’ould be solved successively, and computational effort appears to be very large if those algorithms are applied for optimizing large trusses. In addition to this difficulty, it is not realistic to provide the data of connectivity for constructing the stiffness matrix of all the possible topologies.

In this paper, the hybrid problem of discrete and continuous optimization stated above is solved by varying the nodal locations, and by removing the unnecessary nodes and members from the highly connected ground structure. It is shown in the following sections th.at there exists serious difficulty due to the discretlmess in the design variable and the assumption of uniform cross-sectional areas.

ILLUSTRATIVE EXAMPLE

To illustrate the difficulties in simultaneous optimization of topology and geometry of a truss with uniform cross-sectional area, optimum designs are found for cantilever-type plane trusses with fixed total length W and the height H. Nodal cost is not considered h.ere for a simple presentation of the method. The truss corresponding to (N$ Ni) = (3,l) is shown in Fig. 1. The vertical member is assumed to remain vertical during design modification, i.e. the trusses which consist of rectangular units are considered as feasible designs. The X-coordinate Xi(O I is NY) of the boundary of the units is defined as shown in Fig. 1. The coordinates Xi(l I is Nj: - 1) and the member cross-sectional area A are chosen as design variables, i.e. XI, X, and A are varied for the truss with N: = 3. An upper-bound for the

1, t, (1

Fig. 1. A plane truss (N : = 3, N; = 1).

71

I

H

X

absolute value of the displacements is 1.0 cm. The load of 49.0 kN is applied in the Y-direction of node a. The elastic modulus of the members is 205.8 GPa, the cost coefficient cm is equal to 1.0 x 10V3 cme3, and the values of W and H are 400.0 cm and 200.0 cm, respectively. Optimal trusses are found for Nj: = 1, 2, 3 and 4 by using the optimization package IDESIGN [22] which utilizes recursive quadratic programming algorithm.

The values of Xi, A and C of the optimum designs with fixed topologies are listed in Table 1. It is interesting to note that equally divided configurations are optimum for these statically determinate trusses. Note that the same optimal configurations are found from several randomly generated initial configurations for each cases. It is observed from Table 1 that the optimal truss for Nj: = 2 has the lowest value of the objective function. The important point here is that no node or member is allowed to be removed in the conventional optimization algorithm. The objective of this paper is to present an algorithm which leads to the optimal truss with N]: = 2 starting from the initial trusses corresponding to NY 23, by removing the coalescent nodes and members.

In the following, examples are shown for illus- trating each specific difficulty.

Singularity due to existence of extremely short members

Response displacements have been calculated for trusses with A = 1.0 cm* and NY = 3, in order to

Table 1. Optimal cross-sectional area (cm2), nodal location (cm) and total cost for N 1: = 1,2,3 and 4

x=1 N,” = 2 N, = 3 N, = 4

A 9.1335 6.0271 6.7095 6.1132 x0 0.00 0.0 0.0 0.0 Xl 400.0 200.0 133.33 100.00 x2

- 400.0 266.67 200.0 x3

- 400.0 300.0 x4

- - 400.0

c 13.218 10.642 12.967 16.736

M. Ohsaki

z 2 $ .9 P 6.965 --... ..-.I-..-. -.L..- --.........-....--.. .-..-

6.96 I I I 1 I I 200 200.2 200.4 200.6 200.8 201

Nodal Coordinate (cm)

Fig. 2. Variation of displacement U, with respect to the nodal coordinate Xx.

address the difficulty in removing the nodes and members by continuously varying the nodal coordinates. The coordinate X2 is varied from 201.0 to 200.0 cm while Xi is fixed at 200.0 cm. The displacement in the Y-direction of node a is denoted by U,. Note that U, has the maximum value, for all the cases, among the absolute values of the displacement com- ponents. Variations of U, and C with respect to X2 are plotted in Figs 2 and 3, respectively. As X2 is decreased to Xi (= 200.0 cm), the stiffness matrix K becomes singular when X,-Xi reaches 1 .O x lo-‘* cm, because the length of members 5 and 7 are extremely small and the extensional stiffness AE/Li of those members has a very large value.

ger than that for the case of Nz = 2. The displacement U, for (NY, X2) = (2200.0 cm) and (N;, X2, X3) = (3200.0 cm, 200.1 cm) are 6.9793 and 6.0271 cm, respectively. The difference observed here is due to the fact that the members 4, 6 and 8 for NI = 3 almost overlap with each other and the cross-sectional area of member 4 in Fig. 4 for Ni = 2 should be three times as large as that of other members, if the response quantities are expected to be same as those for the truss with Nt = 3. Note again that there is no such discontinuity if the cross-sectional area of each member is allowed to have different values. In that case, the cross-sectional areas of the co-linear members are simply to be added.

Discontinuity in total cost and displacement Discontinuity in deformation

The values of U, and C are calculated for the trusses with NY = 2 and 3 to investigate other difficulties for finding optimal topologies by removing unnecessary nodes and members. The member cross-sectional areas are 1.0 cm* for both trusses. The total cost for (N’$ X2) = (2200.0 cm) is 1.7657, whereas that for (N!$ X2, X3) = (3200.0 cm, 200.1 cm) is 2.1657. It is obvious that C for N: = 3 is larger than that for N: = 2, because A is same for both cases, and the sum of the length of the members of the truss with Nz = 3 is lar-

Finally, the difficulty in optimizing pin-jointed trusses without rotational stiffness at the joints is discussed in the following. Let Ai denote the cross-sectional area of the ith member. The values of u, and C for N: = 2, where A4= 3.0 cm* and Aj= 1.0 cm2 for the remaining members, are 5.7096 cm and 2.1657, respectively. Although the value of C agrees with that for (N$ X2, X3) = (3200.0 cm, 200.1 cm) up to five digits, the values of U, are still very different. Figure 5 shows the mode of deformation of the truss with (NY, X2, X3) = (3200.0 cm, 200.1 cm). Note that there exists discontinuity in the displacement in the Y-direction between X = Xi and X2. This disconti-

2.166 I I I I

2.164 ’ I I I I I 200 200.2 200.4 200.6 200.8 201

Nodal Coordinate (cm)

Fig. 3. Variation of total cost C with respect to the nodal coordinate X,.

Fig. 4. A plane truss (N : = 2).

----

Fig. 5. Mode of deformation of a plane truss with N i = 3 and XI = 200.1 cm.

Topology and geometry of a regular plane truss 73

nuity is due to the .fact that the stiffness of member 6 in Fig. 1 is not infinite and the rectangular unit defined by the nodIes b, c, e and d deforms into a parallelogram. To prevent this discontinuity, an additional constraint should be given such that the relative displacement between nodes b and c as well as d and e should be sufficiently small when the distance between Xi and Xz almost vanishes in the process of optimization. However, it is very difficult to impose such an additional constraint in the conventional optimization process. There is no such discontinuity for a rigidly jointed frame, because relative rotation between the members is restricted at each node.

Summary of d@culr’ies.

For these reasons, there exist many difficulties for finding the optimal number of units of a truss by removing coalescent nodes and members. The difficulties stated above are summarized as follows:

Existence of extremely short members or elimination of some members leads to singularity of the stiffness matrix. A member which has a larger cross-sectional area is generated if coalescent members are simply combined into a single member. In con- trast, the global stiffness changes discontinuously if some among the coalescent members are removed, leaving only one member. Additional side constraints are required to restrict the relative displacement of the closely spaced nodes. Any node or mlember cannot be removed com- pletely during the optimization process, because once removed node or member may sub- sequently become necessary to exist in the final optimal solution.

Continuous modeling by frame and sigmoid function

The first difficulty stated above is successfully avoided by modeling the truss as a frame which has flexural stiffness and is connected rigidly at the joints. The existen’ce of extremely small members can also be avoided by introducing side constraints for limiting the lower bound of the member length. It is shown in the examples that the effect of flexural stiffness of the frame on response displacements is very small if the radius of gyration y of the members is modera.tely small. In the following, y is assumed to be independent of A. Note that the deformation of truss is very different from that of an equivalent frame. However, only the extensional deformation is to be considered here, and local bend- ing deformation ca:n be ignored without any loss of generality.

A sigmoid function is used to define the cross-sectional areas of the closely spaced members. For instance, A*( = A6 =: As) of the truss with Nl = 3 is

1

0.8

2 0.6

3 %y 0.4

0 20 40 60 80 100

Fig. 6. Sigmoid function for various values of parameter.

defined as a function of AXi = X2 - X, as

,

where x* is a parameter for defining the nondimen- sional value of the distance, and s is the parameter for the shape of the sigmoid function. Variations of A* with respect to AX, are illustrated in Fig. 6 for s = 0.1, 0.3, 0.5 and 0.9, where A = l.Ocm* and x* is fixed at lOO.Ocm. By using this function, A6 and A8 are successfully decreased to null as AXi is decreased, while Ai for members 4, 5 and 7 are equal to A. This way, the configurations corresponding to NY = 2 and 3 are continuously connected. Since the existence of a member with null cross-sectional area or with null length may lead to unfavorable singularity of the stiffness matrix, side constraints

AXi>D

are given and D = 0.1 cm is adopted here.

(4)

Optimum designs have been found for a frame with NY = 1, 2, 3 and 4, where y = lO.Ocm*, P = lOO.Ocm and s = 0.6. The values of Xi, A and C at optimum are listed in Table 2. It is observed from Table 2 that the optimal topology of Ni = 2 and Xi=201 .O has been successfully found starting from the initial topologies with Ni = 2, 3 and 4. Note that the differences between the optimal total costs for Nj: = 2, 3 and 4 are sufficiently small. The cross-sectional areas at optimum of the truss with Nz = 4 is listed in Table 3. The members 10, 12, 14 and 16 with small cross-sectional areas

Table 2. Optimal cross-sectional area (cm*), nodal location (cm) and total cost of a plane frame

Y=l N.; = 2 N.; = 3 N.; = 4

A 9.0456 5.9840 5.9826 5.9835 x0 0.0 0.0 0.0 0.0

XI 400.0 201.08 201.05 201.00 x2 - 400.0 201.15 399.80 x3 - - 400.0 399.90 x4 - - - 400.0 c 13.091 10.551 10.552 10.557

14 M. Ohsaki

Table 3. Optimal cross-sectional areas (cm2) for N i = 4

Member number Ai

1, 5, 9, 13 5.9835 3, 7, 11, 15 5.9835 2, 4 5.9688 6, 8 5.9676 10, 12, 14, 16 0.0099724

are to be removed. A slight difference exists between the cross-sectional areas of the remaining members, because a moderately large value has been given for the parameter s defining the sigmoid function. If s is smaller, the possibility of conver- ging to local optima increases, and short members may exist in the optimum solution. Although the possibility of global optimal solutions with extremely short members is excluded if s is moderately large, those optimal solutions have not been found as a result of optimization of a truss with fixed topology. Those designs are also not optimum from a practical point of view.

It is observed from Tables 1 and2 for N: = 2 that the differences in the nodal coordinates and total cost between pin-jointed truss and the rigidly jointed frame are moderately small. The displacement U, of the pin-jointed truss with NY = 2, A,&.9840 cm* and X1 = 201.08 cm is 1.0072 cm, which is slightly larger than that of the equivalent frame. After optimal topology of a frame is found, additional geometrical and cross-sectional optimization may be carried out for the pin-jointed truss with the same topology for the purpose of verification. Such verification, however, is unnecessary if y is sufficiently small and if only an approximate optimal solution needs to be found.

OPTIMIZATION ALGORITHM

A general algorithm is presented in this section based on the results of the small cantilever truss in the previous section. Nodal cost together with member cost is included in the objective function. Unnecessary nodes are removed in a similar way as illustrated in the previous section for the elimination of members. Only regular plane trusses are considered here, where the configuration of the truss is defined in terms of the design variables {Xi} and { Yi} which define the coordinates of the nodes located at the intersection of the lines in X- and Y- directions. Since the algorithm is a natural extension of a conventional geometry optimization pro- cedure, detailed formulation of sensitivity analysis is not presented here.

Let Bk denote the nodal cost of the kth node of the initial ground structure. Dependence of Aj and Bk on AXi and AYi ( = Yi + I-YJ should be expli- citly defined before starting the optimization process. Sigmoid functions s” and P with respect to

‘i+l

Fig. 7. A unit of a rectangular truss.

AXi and AYi are defined as

,

, (5b)

'i+l

'i

64

where rC is a parameter for defining the nondimen- sional value of the distance in the Y-direction. For a rectangular unit as shown in Fig. 7, Bk and Ai are calculated from

BI = BS”(AXi_l)S’(AYi),

B2 = BS”(AXi)S’(AYi),

B3 = BS”(AXi_I)S’(AYi_I),

B4 = B,S”(AXi)Sy(AYi_,),

A, = ASx(AXi_,),

A2 = ASX(AXi),

A3 = ASY(AYi),

A4 = ASY(A Yi-,),

A5 = ASX(AXi)Sy(A Yi),

A6 = ASY(A Yi),

A7 = ASX(AXi). (6)

If AXi approaches 0, for example, then A2 and A7 have extremely small values, and the members 3 and 6 degenerate to a single member as a result of removal of node two. This process cannot be successfully carried out if the structure is modeled as a pin-jointed truss, because node two becomes unstable after members 3 and 6 are removed. Similar relations are easily formulated for other types of plane units. Differentiation of s” with respect to Xi + 1 and Xi leads to the following equations:


asx m=- co,h*(~~i~; xi>. (7b)

Similar formulations are derived for partial differentiation of Sy with respect to Yi and Yi + 1.

For a large truss which consists of large number of units, the optimllzation process may converge to a local optimal solu.tion which has larger number of units than that of the global optimal solution. In this case, an additional optimization should be carried out starting from the truss of which the topology is same as that of the local optimal solution, after randomly assigning the nodal coordinates. The optimization algoril.hm is summarized as follows:

Simultaneous optimization of topology and geometry is carried out for a regular plane truss as shown in Fig. 8, where E = 205.8 GPa, y = 10.0 cm*, P, = 98.0 MN, W = 300.0 cm, H = lOO.Ocm, x* = r* = 5O.Ocm, D = 0.1 cm, cm = 1.0 x lop5 cmp3, and Vj = 1.0 cm. The cost of the supports is not included because the number of supports are fixed during optimization and consideration of the cost of the supports does not lead to any difference in the optimal topology.

Step 1. Define dependence of Aj and Bk on AXi and AYi, and assign the parameters s, X* and p.

Step 2. Set parameters cm, rrj, B and D as well as mechanical properties E and y.

Step 3. Carry out sensitivity analysis of displacements with respect to Aj, Xi and Ye Then find the optimal solution of the frame by using any optimization algorithm.

Note that the solution converges to a local optimal topology with large numbers of nodes and members, if a small value is assigned for the parameter s. Conversely, if s has a large value, the distribution of Ai and Bk are far from uniform and the global optimal topology might be different from that found by the proposed algorithm. Therefore s is reduced linearly from 0.6 to 0.3 in each optimization process in step 3 in the previous section. This way, the possibility of convergence to a local optimal solution is reduced, and the accuracy of the optimal solution is improved.

Step 4. Update the topology by combining closely spaced nodes and members. Then randomly generate the initial values of nodal coordinates, and carry out another optimization.

Step 5. If imtial and final topology as the result of optimiz,ation in step 4 are different, then return to step 4.

Consider a case where the large nodal cost of B = 1.0 is given. Optimization results starting from the frame with (NF, N ;) = (6, 3) are as listed in Table 4. It is observed from Table 4 that the side constraint XI-&,>D( =O.l cm) is active and the nodes at X = X1 should be removed. Similarly, the nodes at X = X2, X3, &, X5, X6 and Y = Y2, Y3 are also to be removed. Therefore a frame with (N$Nl) = (1, 1) has been found. In this case it is clear that no further optimization is required.

Step 6. After (optimal topology is found, carry Optimal cross-sectional area has been found for out conventiona:l geometrical and cross-sectional the pin-jointed truss with (N$N 1,) = (1, 1) for the optimization for an equivalent truss without con- purpose of verification. The values of A and C at sidering elimination of any member or node. optimum are 137.00 cm* and 3.6925, respectively,

Note that step 6 is required only for the purpose of verification, or for the case where the strict optimum design is needed. Therefore step 6 may be omitted in practical application.

EXAMPLES

W

Fig. 8. A plane truss (N 1 = 6, N; = 3).

76

Table 4. Optimal cross-sectional area (cm*), nodal location (cm) and total cost of a plane rectangular frames

(B = 1.0)

N.: > NJ (6 3)

A 135.24 x0 0.0 XI 0.1 x2 0.2 x3 0.3 x4 0.4 X5 299.9 x6 300.0 yo 0.0 Yl 149.8 y2 149.9 y3 150.0 c 3.7881

M. Ohsaki

where the total cost of 2.0 of the two nodes except the two supports is included. The value of the cross-sectional area of the pin-jointed truss is larger than that of the frame, because there is no flexural stiffness at the joints of a truss. However, the existence of unnecessary members with small cross-sectional areas in the frame leads to a larger total cost

value. Next, optimization has been carried out for the

case of B = 0.1, which is 10% of that for the previous example. The optimization results are listed in the second column of Table 5. In this case, the optimal topology of (NU,,N ;) = (2, 1) has been found as a result of the first optimization starting from the initial topology of (N!$Ni) = (6, 3). Since the global optimal truss might have smaller number of units, another optimization has been carried out, starting from (Ni,NJ) = (2,l). The results are listed in the third column of Table 5. In this case the optimal values of Nl and N i are 2 and 1, respectively.

Next geometrical and cross-sectional optimization has been carried out for the pin-jointed truss with (N;, N,;) = (2, 1) for verification of the validity of approximation in the optimization process of frame. The optimal values of A, X1 and C are

Table 5. Optimal nodal location, cross-sectional area and total volume of a plane rec-

tangular frames (B = 0.1)

(G> NJ) (6, 3) (2, 1)

A 88.960 89.377 x0 0.0 0.0 XI 149.81 150.0 x2 149.91 300.0 x3 150.10 x4 150.11 X5 299.9 X6 300.0 yo 0.0 0.0 YI 149.80 150.0 y2 149.90 y3 150.0 c 1.7346 1.7177

90.400 cm2, 150.0 cm and 1.7327, respectively. Note that C includes the cost of four nodes which is equal to 0.4. Since a sufficiently small value has been given for the radius of gyration of the members of the frame, the differences in the optimal values between the truss and the frame are moderately small.

It may be concluded from the examples of

B = 1.0 and B = 0.1 that larger value of nodal cost leads to an optimal topology with a smaller number of nodes. Note again that the solution converges to a local optimal topology with large numbers of nodes and members, if a small value is assigned for the parameter s. On the contrary, if s has a large value, the distribution of Ai and Bk are far from uniform and the optimal topology might be different from that found by the proposed algorithm. Development of more efficient algorithm of continuous update of s during optimization process remains as a subject of future research.

DISCUSSION AND CONCLUSIONS

An algorithm has been presented for simul-

taneous optimization of topology and geometry of a regular plane truss which consists of rectangular units and uniform member cross-sectional area. Nodal cost together with member cost is included in the objective function to be minimized, and displacement constraints are considered. The locations of supports and the total dimension of the truss such as span length are fixed, and the cross-sectional area and the nodal coordinates are chosen as design variables.

Difficulties in the process of eliminating unnecess-

ary nodes and members of a truss with uniform cross-sectional areas have been discussed using an example of cantilever-type plane truss. Note that the optimization problem has not been made simpler by limiting the feasible design to regular trusses with uniform cross-sectional areas. Additional difficulties arise such as discontinuity between trusses with different topologies. In the proposed method, the truss is first modeled as an equivalent rigidly jointed frame with sufficiently small radius of gyration. Using this method, unfavorable instability due to the removal of nodes and members are successfully avoided. Transition between trusses with different topologies is simulated continuously by using a sigmoid function. By using the proposed technique, optimal topology and geometry are found simultaneously starting from a frame with moderately small numbers of nodes and members. If the ground structure approach for topology optimization with fixed nodal locations is used for finding the optimal geometry, infinite numbers of nodes and members should exist in the initial truss. In this method, configuration optimization only needs to be carried out a few times at most, and the


data of connectivity between nodes and members are required only for the initial ground structure.

In the examples given, optimal trusses have been found for a plane truss which consists of rectangular units. Note that the proposed method may be easily applied for regular trusses with other types of units. It has been shown that the optimal topology strongly depends on the value of the nodal cost. An additional elimination process needs to be carried out if the solution does not appear to converge towards the global optimal topology, though such a process has not been necessary in the examples. It should be noted that the convergence property depends on the parameter defining the sigmoid function. The development of criteria for defining the parameter or developing an efficient update algorithm during the optimization process remains as a subject of future research.

8.

9.

10.

11.

Ringertz, U.T. A branch and bound algorithm for topology optimization of truss structures. Engineering Optimizations, 1986, 10, 11 l-124. Cheng, G. and Jiang, Z. Study on topology optimization with stress constraints. Engineering Optimizations, 1992, 20, 129-148. Save, M.A. Remarks on minimum-volume designs of a three-bar truss. Journal of Structural Mechanics, 1983, 11, 101-l 10. Nakamura, Tsuneyoshi and Ohsaki, M. A natural generator of optimum topology of plane trusses for specified fundamental frequency. Computer Methods and Applications in Mechanical Engineering, 1992, 94, 113-129.

12.

13. Extension to the case of multiple loading con-

ditions also appears to be straightforward. However, consideration of stress constraints will be rather difficult, because the stress constraints do not need to be satisfied in nonexistent members.

Ohsaki, M. and Nakamura, T. A natural generator of optimum topology of a space frame for specified fundamental frequency. In Space Structures 4, Proc. 4th Int. Co@ on Space Structures, Surrey, UK, ed. G. Parke and C. Howard, Thomas Telford, London, 1993, pp. 1221-1229. Ohsaki, M. Genetic algorithm for topology optimization of trusses. Computers and Structures, 1995, 57, 219-225.

14. Sadek, E. Dynamic optimization of framed structures with variable layout. International Journal of Numerical Methods in Engineering., 1986, 23, 12733 1294.

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Imai, K. and Schmit, L.A. Configuration optimization of trusses. Proceedings of the ASCE, 1982, 107(ST5), 745-756.

1. Kirsch, U. Optimal topologies of truss structures. Applied Mechanics Review, 1989, 42, 223-239.

2. Topping, B. H. V., Mathematical programming tech- niques for shape optimization of skeletal structures, In Shape and Layout Optimization of Structural Systems and Optimality Criteria Methods, ed. G. I. N. Rozvany. Springer, Wien, 1992, pp. 349-375.

3. Kirsch, U. Integration of reduction and expansion process in layout optimization. Structural Optimization, 1996, 11, 13-18.

4. Bendsae, M.P., Ben-Tal, A. and Zowe, J. Optimization method for truss geometry and topology design. Structural Optimization, 1994, 7, 141-159.

5. Rozvany, G.I.N. and Zhou, M. A note on truss design for stress and dnplacement constraints by optimality criteria methods. Structural Optimization, 1991, 3, 45 50.

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Dems, K. and Gatkowski, W. Optimal design of a truss configuration under multiloading conditions. Structural Optimization, 1995, 9(ST5), 262-265. Lin, J.H., Che, W.Y. and Yu, Y.S. Structural optimization on geometrical configuration and element sizing with statical and dynamical constraints. Computers and Structures, 1982, lS(STS), 507-515. Saka, M.P. Shape optimization of trusses. Proceedings of the ASCE, 1980, 106(ST5), 1155-l 174. Jenkins, W.M. Towards structural optimization via the genetic algorithm. Computers and Structures, 1991, 40(ST5), 1321-1327. Hajela, P. and Lin, C-Y. Genetic search strategies in multicriterion optimal design. Structural Optimization, 1992, 4(ST5), 99-107. Bendee, M.P. and Kikuchi, N. Generating optimal topologies in structural design using a homogenization method. Computers and Methods of Application in Mechanical Engineering, 1988, 71(ST5), 197-224. Arora, J. S. and Tseng, C. H., IDESIGN User’s Manual, Ver. 3.5, Optimal Design Laboratory, The University of Iowa, 1987.

Documents

Simultaneous optimization of topology and geometry of a regular plane truss