Evolutionary Topology Optimization of Structures ... - Semantic Scholar

Evolutionary Topology Optimization of Structures with Multiple Displacement and Frequency Constraints Zhihao Zuo, Yimin Xie* and Xiaodong Huang School of Civil, Environmental and Chemical Engineering, RMIT University GPO Box 2476, Melbourne 3001, Australia (Received: 28 January 2011; Received revised form: 31 May 2011; Accepted: 5 July 2011)

Abstract: Topological design with multiple constraints is of great importance in practical engineering design problems. The present work extends the bi-directional evolutionary structural optimization (BESO) method to multiple constraints of displacement and frequency in addition to the amount of material usage. Besides the binary design variables, the Lagrange multipliers for constraints are considered as additional continuous variables and determined by a search scheme. The enhanced approach can include a number of constraints besides the simple volume constraint. To demonstrate the effectiveness of the proposed BESO approach, several examples are presented for the maximization of structural overall stiffness subject to the material volume, displacement and frequency constraints.

Key words: topology optimization, bi-directional evolutionary structural optimization (BESO), multiple constraints, optimal design, displacement constraint, frequency constraint. 1. INTRODUCTION Topology optimization is a considerably practical aid in structural design as it produces design solutions of high performance by finding the optimal structural layout. It is capable of speeding up the structural design process and producing reliable solutions to various engineering problems. This area has been extensively investigated in the past three decades since the modern formulation of optimal layout theories by Prager and Rozvany (1977). Several methods have been developed for topology optimization of continuum structures, one of the most popular techniques is the solid isotropic microstructure with penalization (SIMP) method (Bendsøe 1989; Rozvany, Zhou et al. 1992). The SIMP method assumes isotropic materials and takes the element relative density as the continuous design variable. The design variable determines the Young’s modulus through a power-law interpolation scheme

(also known as the SIMP material model). The method of moving asymptotes (MMA) (Svanberg 1987) or the optimality criteria (OC) method (Zhou and Rozvany 1991) is often incorporated in the SIMP method for seeking optima. Another group of popular topology optimization techniques are the evolutionary structural optimization (ESO) method and its descendent versions that use binary design variables to represent the structural design. The ESO method was first proposed by Xie and Steven (1993, 1997) in the early 1990s. The original ESO follows a straightforward algorithm of iteratively removing inefficient material to make the structures evolve towards possible optima. The concept of bidirectional ESO (BESO) method was proposed as an extension (Querin et al. 1998) that additionally allows material to be added to the most demanding places. The ESO/BESO methods are simple approaches (Tanskanen

*Corresponding author. Email address: [email protected], Fax: +61-3-9639-0138; Tel: +61-3-9925-3655.

Advances in Structural Engineering Vol. 15 No. 2 2012

385

Evolutionary Topology Optimization of Structures with Multiple Displacement and Frequency Constraints

2002, 2006) and have been applied to various optimization problems (Zhao et al. 1997; Manicharajah and Xie 1998; Li et al. 1999a, b; Rong et al. 2000). However the early ESO/BESO methods treated material removal/addition in a rather empirical way and failed to present the mathematical justification for producing optima. Therefore they were reviewed and criticized (Sigmund and Peterson 1998; Zhou and Rozvany 2001; Edwards et al. 2007; Rozvany 2008) and given suggestions for possible improvements, e.g. in (Rozvany and Querin 2002). Later Huang and Xie (Huang and Xie 2007; 2009; Huang et al. 2010) proposed convergent and meshindependent BESO algorithms to solve the stiffness optimization problem using a “soft-kill” technique. The advanced algorithms realize the determination of void elements’ sensitivities rather than heuristic estimation. With the optimality criterion described for problems with binary design variables, this formulation claims the justification for optima output from a convergent and mesh-independent procedure. The advanced developments of evolutionary topology optimization are presented and summarized in (2010). However, almost all ESO methods have been dealing with only one simple constraint, most likely the volume constraint, which is actually easy to control. Unlike multi-objective problems, the weighting factors for objectives/constraints must be determined as Lagrange multipliers. Since in most traditional ESO methods element sensitivities of void elements remain unsolved or heuristically designated, the constraint values cannot be precisely controlled and the previous ESO methods have been criticized due to the incapability of easily solving multiconstrained problems (Tanskanen 2002, 2006; Rozvany 2008). Recently Huang and Xie (2010) successfully included an additional local displacement constraint in a mean compliance minimization problem with the volume constraint using the soft-kill BESO technique. The use of soft material enables the calculation of void element sensitivities for both constraints and objectives. In every iterative step, the only Lagrange multiplier for the displacement constraint is determined asymptotically by estimating the displacement with trial addition/removal of elements. The displacement constraint therefore can be made satisfied so long it is possible. However, this approach becomes difficult when more than one Lagrange multipliers are considered. So far multiple complex constraints still remain unsolved in ESO methods and worth investigating due to the practical potential. In real structural engineering design, various criteria could be taken into account as

386

multiple constraints, e.g. the material usage, deflection and frequency etc. In this paper, an approach to topology optimization of continuum structures is presented with constraints of displacement and frequency in addition to the common volume constraint. The displacement constraint reflects certain functional requirements that the deformation at some points (not only the load point) needs to be under a prescribed limit. The frequency constraint comes from the design need for a structure to avoid large deformation caused by resonance effects due to external excitations such as wind, machine vibration or earthquake etc. Not losing generality, the optimization problem for BESO is first generally formulated with any number of general equality and inequality constraints. Formulating the Lagrange function, the Lagrange multipliers for the constraints are treated as additional continuous design variables that are sought steadily using their derivatives. As the Lagrange multipliers take values theoretically in an infinite domain, replacement factors are used to represent them through a simple scaling function. The traditional stiffness optimization with the common volume constraint is then extended with additional constraints of displacement and frequency. Several numerical examples are demonstrated in order to examine the proposed approach. The subsequent sections are outlined as follows. The second section formulates the general multi-constrained optimization problem and derives the search algorithms for the Lagrange multipliers. The third section presents the numerical implementation of applying the algorithms to stiffness optimization with volume, displacement and frequency constraints. The fourth section shows a series of numerical examples of the selected optimization problem. The conclusions are drawn in the final section. 2. SOLVING MULTI-CONSTRAINED OPTIMIZATION PROBLEMS 2.1. General Optimization Formulation for BESO The present paper considers the optimization problems with multiple constraints. As the basis of BESO, the binary nature of the design variables is also taken into account as an intrinsic condition. The structure volume can be easily controlled with binary design variables of the elements. Later it is demonstrated that determination of the Lagrange multiplier of the volume constraint is unnecessary for calculating element sensitivities when the structure is meshed into equally sized elements. Therefore, the common volume constraint is simply treated separately. Then the general topology optimization problem for BESO can be described in the following minimization formulation:


Zhihao Zuo, Yimin Xie and Xiaodong Huang

Minimize

L ( x, λ , µ , S 2 ) = F ( x) + λ v

F ( x) = ∫ f ( x)dΩ, x = {x1 , x 2 , L , xi ,L , x n} Ω

(1)

+∑

( xi = x min or 1) Subject to n

∑ 1 xi vi − V * = 0 h j ( x) = 0, j = 1, 2, L , l

(2)

(3)

2.2. Lagrange Relaxation of General BESO Optimization To solve the general multi-constrained optimization problem, the afore-mentioned problem formulation needs to be transformed into an equivalent formulation using the Lagrange relaxation. First a slack variable S 2k is used to transform an inequality constraint into an equality constraint. gk ( x) + Sk2 = 0

(5)

By introducing Lagrange multipliers λv for the volume constraint, λj and µk for the general equality and inequality constraints respectively, the following Lagrange function, i.e. the relaxed objective function is obtained.


l λ h ( x) + 1 j j

∑

m µ 1 k

*

(6)

 gk ( x) + Sk2   

2 Minimize L( x, λ , µ , S ) ,

x = {x1 , L , x n}, nλ = {λ v , λ1 , L , λl },

(7)

µ = {µ1 , L , µm}, S = {S1 , L , Sm}

(4)

In the above general formulation, x is the vector of design variables in the design domain Ω and F is the objective function. The binary design variable nature is described in Eqn 1, where 1 is the upper bound of the design variable and xmin the lower bound. The design variable xi indicates the corresponding element’s status, namely presence (solid) or absence (void). In the soft-kill formulation, elements are not really eliminated from the structure by “removal” but replaced with very weak material. In this sense, the design variable represents the element relative density with two candidate values: 1 representing element presence, or xmin which is a very small value such as 10−4 representing element absence. The volume constraint is formulated in Eqn 2 with vi the element volume and V* the objective volume. The general equality and inequality constraints are represented in Eqns 3 and 4 respectively. Note that all the inequality constraints can be formatted into the “less than” form of Eqn 4.

n 1 i i

Towards optimality, the above Lagrange function is equivalent to the original objective function F(x) with the all the constraints being satisfied. Therefore the minimization problem of F(x) is transformed to the 2 minimization of L( x, λ , µ , S ) .

where gk ( x) < 0, k = 1, 2, L , m

(∑ x v − V )

Then the overall element sensitivity of the Lagrange function due to the design variables xi is derived in the following way.

αi =

∂h j ∂g ∂L ∂F l m + ∑ 1 µk k + λ v vi + ∑ 1 λ j = ∂xi ∂xi ∂xi ∂xi

(8)

Comparing this element sensitivity with the one of the original objective function, one can see that the Lagrange multipliers are actually utilized to compromise the original objective function and the constraints. Notice that the second term in the above sensitivity is identical for all the elements when they have the same volume (vi = v). In the BESO methods, element removal/admission is based on the relative ranking of all elements. Therefore we can construct an equavalent sensitivity number that does not affect the sensitivity ranking.

α%i = α i − λ v vi = α i − λ v v =

∂h j ∂F l + ∑1λ j ∂xi ∂xi

+∑

m µ 1 k

(9)

∂gk ∂xi

This way, the Lagrange multiplier λv for the volume constraint becomes trivial and does not need to be solved. In the BESO formulation, the volume constraint is actually easily satisfied as the element update is under the control of fixed parameters, e.g. the objective volume fraction and the evolutionary volume ratio that will be introduced later. As the other constraints generally cannot be directly made satisfied, they remain to be dealt with in a more sophisticated manner.

387


2.3. Searching for the Lagrange Multipliers Lagrange multipliers and the slack variables are introduced as additional variables of the Lagrange function. Like the design variables x, they also need to be determined in order for a solution to the optimization problem to be obtained. The sensitivities of the Lagrange function due to these additional variables are expressed as the following additional sensitivities.

practice, λj and µk are further defined through a scaling function of replacement factors γj and ϕk that range in a narrow domain [0, 1).

λj =

µk = ∂L = h j ( x) ∂λ j

(10)

∂L = gk ( x) + Sk2 ∂µ k

(11)

∂L = 2 µk Sk ∂Sk

(12)

Employing the Kuhn-Tucker necessary condition (KTC) of optimality, the above additional sensitivities need to vanish. A preliminary update trend for the Lagrange multipliers can be determined with the help of the above additional sensitivities and the KTC. For the equality conditions, if the sensitivity of the Lagrange function due to λj is positive, i.e. constraint value hj(x) is greater than zero, λ j needs to be decreased in order to reduce (i.e. minimize) the Lagrange function, and vice versa. On the other hand, if the constraint value hj(x) is equal to zero, the KTC implies a stationary point on λj with the current value as a candidate solution and λ j should remain unchanged. In the same way for µk of the inequality constraints, if the constraint value gk(x) is positive, the additional sensitivity gk(x) + S 2k must also be positive and µk needs to be decreased for the Lagrange function minimization. On the other hand, the vanishing of Eqn 12 implies that µk needs to be zero with Sk ≠ 0, while Sk ≠ 0 leads to gk(x) ≤ 0 according to Eqn 5. This means that if an inequality constraint is satisfied, the corresponding Lagrange multiplier is zero and the constraint is not considered for minimizing the Lagrange function. Unlike the design variables x, the additional variables are continuous in the infinite domain. It is practically infeasible to search such a domain for a solution of the additional variables with a direct method. Therefore in

388

γj

, γ j ∈[0, 1)

(13)

ϕk , ϕ k ∈[0, 1) 1 − ϕk

(14)

1− γ j

In the above definition, the Lagrange multipliers are represented in the whole range with the replacement factors γj and ϕk, e.g. zero ϕk leads to zero µk and ϕk approaching 1 pushes µk to infinity. Increase or decrease of the Lagrange multipliers can be realized by increasing or decreasing the corresponding replacement factors. This way, the determination of Lagrange multipliers is realized by the searching for replacement factors within [0, 1) with appropriate increments in programming implementation. Note that the slack variables are not required to be determined with a searching scheme. Keeping the vanishing of Eqn 12, the slack variables serve as indicators whether the corresponding constraints are active. 3. STIFFNESS OPTIMIZATION WITH DISPLACEMENT AND FREQUENCY CONSTRAINTS 3.1. Problem Formulation Drawn from the general problem, we now deal with a specific optimization problem as an example for the proposed approach. Demonstrated below is the formulation of stiffness maximization with three constraints, namely the volume, displacement and frequency constraints. As the common inverse indicator of the structural overall stiffness, the mean compliance is taken as the objective function. Minimize

C = C ( x) =

1 T u Ku 2

(15)

Subject to Eqn 2 and d j ( x) − d * < 0

(16)

ω * − ω n ( x) < 0

(17)



where u is the displacement vector, K is the global stiffness matrix, dj is the displacement magnitude of the jth node, ωn is the nth natural frequency, with ω* and d* being the imposed constraint values. The displacement and the frequency constraints are demonstrated in Eqns 16 and 17 respectively. In some cases, the natural frequencies may need to be confined under a certain limit in order to escape from the dangerous frequency range. In such scenarios, the frequency constraint is changed to ωn − ω*< 0 and the later Lagrange function varies accordingly. 3.2. Lagrange Relaxation The optimization problem is relaxed by constructing the Lagrange function with the inequality constraints of the local displacement and natural frequency. L ( x, λ v , µ , S 2 ) = C ( x) + λ v

(∑ x v − V ) n 1 i i

α istf

pxip −1 T 0 ∂C =− = ui K i ui ∂xi 2

(21)

where ui is the element displacement vector, K0i is the element stiffness matrix calculated with solid material, i.e. using the Young’s modulus E 0. With the same material model, the element sensitivity for the displacement (Huang and Xie 2010) is obtained as

α idis =

∂d j ∂xi

= pxip −1u Tij K i0 u i

(22)

*

+ µ1  d j − d * + S12  + µ2 ω * − ω n + S22  (18) Then the overall element sensitivity number is calculated in the following.

α%i =

material model, the element sensitivity for the mean compliance takes the following form (Huang and Xie 2009).

∂d j ∂ω ∂L ∂C − λv v = + µ1 − µ2 n ∂xi ∂xi ∂xi ∂xi

It is seen that the element sensitivity number of the overall Lagrange function is a combination of the element sensitivities of the mean compliance − αistf, the local displacement on the jth node − αidis, and the nth natural frequency − α ifrq . Thus the overall sensitivity calculation is realized with these three parts. 3.3. Sensitivity Calculation The element sensitivity for stiffness optimization is commonly calculated as the element strain energy density (such as in Bendsøe and Sigmund 2003). The calculation takes advantage of the SIMP material model (Bendsøe and Sigmund 1999) that defines the element Young’s modulus through the following power law penalization. (20)

where p is the penalty exponent and E 0 is the Young’s modulus of solid material, i.e. when xi = 1. Based on this


x − xp  E ( xi ) =  min p min (1 − xip ) + xip  E 0  1 − x min 

(23)

(19)

= α istf + µ1α idis − µ2α ifrq

E ( xi ) = E 0 xip

where uij is the element displacement vector under the virtual load. In order to avoid the artificial and localized modes, Huang et al. (2010) proposed a modified SIMP material model that is described as

With the help of the above modified SIMP material model, the element sensitivity for the natural frequency is calculated as the following. More details of this material model and frequency sensitivity calculation are seen in (Huang et al. 2010). Alternative material models can be also found in (Pedersen 2000).

α ifrq =

∂ω n 1 T ui, n = ∂xi 2ω n  1 − x min p−1 0 2 0 × p pxi K i − ω n Mi  u i , n  1 − x min 

(24)

where ui, n denotes the element eigenvector of the nth mode and M0i is the element mass matrix with solid material. Sometimes the adjacent eigenmodes may become multiple by having the same or very close frequencies. In this case, BESO methods simply takes the average of the sensitivities from the relevant modes to solve this problem (see Yang et al. 1999 or section 2.2 of Zuo et al. 2010).

389


Note that a common penalty exponent p = 3 is used for the examples in this paper. 3.4. Mesh Independence and Convergence Additional measurements on the overall element sensitivities need to be taken in order to produce meshindependence and convergent solutions. To circumvent numerical instabilities (Sigmund and Peterson 1998) such as checkerboards, a mesh-independence filter scheme (Huang and Xie 2007) is employed to blur the element sensitivities using a low-pass filter of radius rmin. This scheme is briefly summarized into two steps: First the raw element sensitivity α i is equally distributed to its nodes as α nj; then the modified element sensitivity αˆ i is obtained by summing up weighted α nj using a weighting function w of the distance rij between the element centre and the jth node.

αˆ i =

∑ w(rij )α nj , ∑ w(rij )

where w(rij ) = max(rmin − rij , 0) (25)

Details of this filter scheme can be found in (Huang and Xie 2007) and more information about the previous filter techniques can be found in (Sigmund and Peterson 1998) that summarized the numerical instabilities. In order to relax the solution convergence difficulty due to the binary nature of design variables, the element sensitivities are further history-averaged (Huang and Xie 2007), i.e. the above modified element sensitivity is averaged between the current (kth) and last (k-1th) iteration to yield the final modified sensitivity α˜ i. In this way, the variation of element sensitivities through the history is smooth. 1 α%i = (αˆ i , k + αˆ i , k −1 ) 2

(26)

3.5. Updating Variables in the Iterative Procedure The variables to be determined in the proposed approach include the additional variables, i.e. Lagrange multipliers, and the design variables that indicate the element status. The Lagrange multipliers for the constraints need to be determined first in order to construct the overall element sensitivity. As the overall sensitivity calculation is sensitive to the contributions from the element sensitivities of the original objective function and the additional constraints, it is beneficial to scale

390

the element sensitivities of different kinds into the same magnitude order. The Lagrange multipliers usually start from zero and are gradually updated according to the corresponding additional sensitivities. For instance, the replacement factor ϕ k for the Lagrange multiplier is decreased if the corresponding additional sensitivity is positive, as described in the previous section. The increment for the replacement factors in each iterative step is usually set to a small value, e.g. 1%. On the other hand, the increment gets smaller when the constraint value approaches close to the imposed value, e.g. from 10−3 to 10−5 according to the difference between the constraint value and the imposed value. After the Lagrange multipliers are updated, the element sensitivity of the Lagrange function is constructed with the element sensitivities of the original objective function and additional constraints. The update of element design variables follows the routine BESO method, which can be briefly described in these steps: first determine a target volume for a new design according to the current volume and the objective volume; sort all the element sensitivity number in a descent sequence; then find a threshold of the sensitivity numbers and update the element status, so that by switching on all solid elements with sensitivity numbers below this threshold, and vice versa, the volume of the new design reaches the current target volume. The details of this update scheme can be found in reference (Huang and Xie 2007). The volume constraint and the solution convergence are checked for a decision whether to terminate the optimization procedure at the end of each iterative step. Moreover for the present problem, the satisfaction of the additional conditions is checked with the average through the final iterative steps. 1 N − d* < 0 d ∑ i =1 j , q −i +1 N

(27)

1 N ω * − ω n , q−i +1 < 0 ∑ i =1 N

(28)

where q denotes the current iterative step, and N in this paper is selected to be 10 so that the additional conditions are satisfied in the final ten designs. The iterative procedure for the proposed evolutionary topology optimization approach is summarized into the steps shown in the flowchart of Figure 1 with the description of the update scheme of the Lagrange multipliers.



Update the Lagrange multipliers∗

Start

Construct sensitivities of the Lagrange function

Define the design domain and FE discretization

∗ In

d −d ∗< 0? / ω ∗−ωn < 0?

Yes

No Filter and history-average the element sensitivities

Determine imposed values for constraints and BESO parameters

Vf∗, Er, rmin etc.

(d −d ∗)/d ∗> t ? / (ω ∗−ωn)/ ω ∗< t ?

No

Yes Determine the target volume for next iterative step and update design variables

Carry out FEA

ϕd = ϕd + e /ϕω = ϕω +e ϕd = ϕd + (d −d ∗)/d ∗ / ϕω = ϕω + (ω ∗− ω n)/ω∗

Calculate sensitivities for objective function and constraints

Constraints satisfied and objective converged?

No

Yes End

Out Note: the tolerance t and increment e can be prescribed as small numbers, e.g. 1%

Figure 1. Flowchart of the overall BESO optimization procedure and the determination of the Lagrange multipliers

4. NUMERICAL EXAMPLES Numerical examples are demonstrated in this section using the proposed approach described in last section. ABAQUS/Standard is used as the FEA engine and a computer program is utilized as ABAQUS’s postprocessor to perform topology optimization. 4.1. Example 1 The beam structure shown in Figure 2 is simply supported at the lower corners and vertically loaded at the middle point of the upper edge. The roller-supported corner is denoted point A where a horizontal local displacement constraint is to be applied. The properties of the material used are: Young’s modulus E = 1 GPa, mass density ρ = 8000 kg/m3 and Poisson’s ratio υ = 0.3. The whole design domain is discretized into 200 × 100 quadrilateral plane stress elements. The structure is optimized for the overall stiffness and the volume constraint is V* = 30%, i.e. 30% material of the design 100 N

50 mm

A 100 mm

Figure 2. Design domain of the beam structure in Example 1: geometry and boundary conditions


domain. The optimization parameters are: the evolutionary volume ratio (Huang and Xie 2007) ER = 2.0%, rmin = 1.5 mm. Several additional constraints on the horizontal displacement dA at point A and the fundamental frequency ω1 are included in the optimization of the beam structure. The additional constraints are combined into the following sets: (a) no constraint on dA or ω1; (b) dA < 1.2 mm, ω1 > 1.9 × 10−3 rad/s; (c) dA < 1.0 mm, ω1 > 1.9 × 10−3 rad/s; (d) dA < 1.0 mm, ω1 < 1.1 × 10−3 rad/s. Note that due to the complexity of a multiconstrained problem, some constraints may not be satisfiable if the imposed value is set inappropriate. Therefore most of the constraints in this paper are defined based on realistic estimations, e.g. the frequency constraint value is set with 30% increase from that of the initial full design. The optimal designs are displayed in Figure 3 while the corresponding resulted performances are listed in Table 1. The design with no additional constraints of the local displacement and fundamental frequency possesses the lowest resulted mean compliance since all the other designs are extra constrained. In the same way, design (b) is stiffer than design (c) as the displacement constraint is less strict for (b) than (c). Because the mode shape of the simply supported beam is basically a combination of horizontal and vertical vibrations, unsymmetrical final designs are obtained when the fundamental frequency constraints are present. Figure 4 shows the evolutionary histories for the mean compliance, local displacement and fundamental frequency respectively. One sees the convergence in the mean compliance during the final iterative steps for all the

391


(a)

Mean compliance (Nmm)

(a)

(b)

120.000 115.000 110.000 105.000 100.000 95.000 90.000 85.000 80.000 75.000 70.000 65.000 60.000 55.000 50.000

d < 1.2 mm, ω > 0.0019 rad/s d < 1.0 mm, ω > 0.0019 rad/s d < 1.0 mm, ω > 0.0011 rad/s No d or ω constraint 1

11

21

31

41

51

61

71

81

91 101 111 121

81

91 101 111 121

81

91 101 111 121

Iteration

Displacement (mm)

(b)

(c)

1.500 1.450 1.400 1.350 1.300 1.250 1.200 1.150 1.100 1.050 1.000 0.950 0.900 0.850 0.800

d < 1.2 mm, ω > 0.0019 rad/s d < 1.0 mm, ω > 0.0019 rad/s d < 1.0 mm, ω > 0.0011 rad/s No d or ω constraint

1

11

21

31

41

51

61

71

Iteration

(d)

(c)

cases with different constraints. The convergence of the local displacement and the fundamental frequency is also obtained in the final stage as long as the relevant constraint is active. This is due to the stable development of the Lagrange multipliers and history-averaging of the element sensitivities. Since the convergence criteria (for both the

Fundamental frequency (rad/s)

Figure 3. Optimal stiffness designs of the beam structure in Example 1 considering various displacement and frequency constraints: (a) no constraints on dA or ω1; (b) dA < 1.2 mm, ω1 > 1.9 × 10−3 rad/s; (c) dA < 1.0 mm, ω1 > 1.9 × 10−3 rad/s; (d) dA < 1.0 mm, ω1 < 1.1 × 10−3 rad/s

0.0020 0.0019 0.0018 0.0017 0.0016 0.0015 0.0014 0.0013

d < 1.2 mm, ω > 0.0019 rad/s d < 1.0 mm, ω > 0.0019 rad/s d < 1.0 mm, ω > 0.0011 rad/s No d or ω constraint

0.0012 0.0011 0.0010 1

11

21

31

41

51

61

71

Iteration

Figure 4. Evolutionary optimization histories for the optimal designs of Example 1: (a) mean compliance; (b) local displacement; (c) fundamental frequency

objective and the additional constraints) and the constraint satisfaction criteria need to be checked, the evolution with additional constraints needs more iterative steps to insure an optimum.

Table 1. Summary of the optimal designs under different constraints in Example 1 Constraints

Performances

(a) No constraints on dA and ω1 (b) dA < 1.2 mm, ω1 > 1.9 × 10−3 rad/s (c) dA < 1.0 mm, ω1 > 1.9 × 10−3 rad/s (d) dA < 1.0 mm, ω1 < 1.1 × 10−3 rad/s

392

C (Nmm) 95.533 98.845 104.181 102.800

d A (mm) 1.414 1.198 0.999 0.999

ω 1 (rad/s) 1.627 × 10−3 1.912 × 10−3 1.902 × 10−3 1.099 × 10−3



4.2. Example 2 In this example, we consider only the frequency constraints in the optimization of a bridge-type structure with non-structural lump masses shown in Figure 5. The bridge is pinned at both lower corners and subject to a concentrate force P = 1 × 107N at the centre of the lower edge. Three non-structural masses are placed along the lower edge, with M1 = 32000 kg and M2 = 16000 kg. The structure material properties are: Young’s modulus E = 200 GPa, mass density ρ = 8000 kg/m3 and Poisson’s ratio υ = 0.3. The bridge design domain is divided into 160 × 50 quadrilateral plane stress elements. The bridge is optimized for stiffness and the objective volume fraction is 40%. The optimization parameters are: ER = 2%, rmin = 0.12 m. The optimal designs for the bridge are shown in Figure 6 and the solutions are summarized in Table 2. First the bridge is optimized under no additional frequency constraints and the optimal design is shown in Figure 6(a). As a standard Michel-type solution, the non-structural masses M2 are excluded. The low natural frequencies are due to the hanging masses on soft material. Further the natural frequency constraints are considered. The optimal design with ω1 > 480 rad/s is shown in Figure 6(b) and the other one with ω2 > 540 rad/s in Figure 6(c). When constraining only the first frequency, the first two eigenmodes are detected to be nearly multiple. Finally the optimal design under both ω1 > 480 rad/s and ω2 > 540 rad/s are obtained in Figure 6(d), in which case both resulted frequencies are above the separate prescribed limits. 6m

(a)

(b)

(c)

(d)

Figure 6. Optimal stiffness designs of the bridge in Example 2 considering various frequency constraints: (a) no frequency constraints; (b) ω1 > 480 rad/s; (c) ω2 > 540 rad/s; (d) ω1 > 480 rad/s, ω2 > 540 rad/s

Figure 7 shows the evolutionary history for the optimization under both ω1 > 480 rad/s and ω2 > 540 rad/s. Not only the mean compliance converges to around 5410 Nm, but also the both two frequencies produce good convergence to the imposed constraint values in the final stage.

2m M2

M1

M2

P

Figure 5. Design domain of the bridge in Example 2: geometry and boundary conditions

4.3. Example 3 Optimization of three dimensional structures is very useful in some engineering practice. This is easy to implement in the BESO approach as one only needs to employ the 3D structural analysis in FEA software packages, e.g. ABAQUS and ANSYS.

Table 2. Summary of the optimal designs under different constraints in Example 2 Constraints

Performances

(a) No constraints on ω1 and ω2 (b) ω1 > 480 rad/s (c) ω2 > 540 rad/s (d) ω1 > 480 rad/s, ω2 > 540 rad/s


C (Nmm) 4122.61 5080.92 5079.46 5411.77

ω 1 (rad/s)

ω 2 (rad/s)

26.75 480.21 394.60 480.79

26.76 490.31 541.15 540.04

393


580

6000

570

5750

560

5500 5250

550 Frequency (rad/s)

4750

530

4500

520

4250

510

4000

500

3750

490

3500 3250

480

Mean compliance (Nm)

5000

540

3000

470

2750 1st frequency 2nd frequency Mean compliance

460 450 440

2500 2250 2000

1

11

21

31

41

51

61

71

81

91

Iteration

Figure 7. Evolutionary optimization histories for the optimal design with both constraints on the first and second frequencies in Example 2: ω1 > 480 rad/s, ω2 > 540 rad/s

Figure 8 shows a solid cube in three dimensions. The cube is pin-supported at the four lower corners. A concentrate force of 1 × 107N is applied at the centre of the upper face and a non-structural mass of 5000 kg is placed at the centre of the lower face. The properties of the material used are: Young’s modulus E = 200 GPa, mass density ρ = 7800 kg/m3 and Poisson’s ratio υ = 0.3. The whole design domain is meshed into 40 × 40 × 20 eight-node cubic elements. The overall stiffness is to be maximized and the objective volume fraction is predefined as 15% as the volume constraint. The common optimization parameters are: ER = 2.0%, rmin = 0.075 m. Figure 9 shows a summary of the optimal solutions. First a conventional stiffness optimal design of the cube

is shown in Figure 9(a) that considers no additional constraints. Since the natural frequency is not taken into (a)

FE model

CAD model

FE model

CAD model

FE model

CAD model

(b)

Force (c)

0.5 m Mass 1m 1m

Figure 8. Design domain of the cube in Example 3: Geometry and boundary conditions: pinned at four lower corners, concentrate load at the centre of the upper surface and non-structural mass at the centre of the lower surface

394

Figure 9. Optimal stiffness designs of the cube in Example 3: (a) no constraints on ω1 and dmass (resulted in C = 46736.10 Nm, ω1 = 17.89 rad/s, dmass = 5.04 × 10−7m); (b) ω1 > 450 rad/s (resulted in C = 47072.30 Nm, ω1 = 451.26 rad/s, dmass = 5.30 × 10−3 m); (c) ω1 > 450 rad/s, dmass < 4.5 × 10−3m (resuled in C = 48883.79 Nm, ω1 = 450.95 rad/s, dmass = 4.49 × 10−3 m)



account of optimization, the non-structural mass is excluded from the final design, which is not expected. Both the fundamental frequency and the vertical displacement of the mass point are very low because of the soft connection with the non-structural mass. Considering the fundamental frequency, Figure 9(b) shows the final optimal stiffness design of the cube with the addition constraint ω1 > 450 rad/s. Bars are obtained to hold the non-structural mass in order to confine the free vibration. Further a vertical displacement constraint of the mass point is added besides the frequency constraint, namely both additional constraints ω1 > 450 rad/s and dmass > 4.50 × 10−3 m are taken into account. Figure 9(c) shows the final optimal design which has a much lower local displacement than the frequency-only-constrained one (4.49 × 10−3m vs 5.30 × 10−3 m). Comparing the three final designs, one sees that the mean compliance increases as additional constraints are added into the problem. Due to the zigzag shaped finite elements, the FE models of the final designs need to be converted into processed CAD models that allow for further detail design and preparation for manufacture. Figure 9 shows the CAD models of the final optimal designs beside the corresponding FE models. Based on the prototypical FE models, the boundary curves and surfaces are fitted so that the structural layouts can be better interpreted. The same model was optimized for the fundamental frequency with the same objective volume fraction in the previous work (Huang et al. 2010) where no static load was considered. 4.4. Example 4 The final example deals with a design problem of reinforcing a hybrid-steel-concrete frame building shown in Figure 10. The original design of the frame work consists of two opposite steel shear walls on the shorter sides of the building. Horizontal loadings on the building are supported by six exterior steel columns and two interior reinforced concrete lift cores, while vertical loads such as the roof snow load and other live loads are applied on the reinforced concrete roof and floors. The geometries of the main structural members are demonstrated in Figure 10. Due to the plan aspect ratio of 6:20 and the configuration of the lift cores and columns, the deformation in the direction of the shorter side (identified as the weak direction) is much more critical than that of the longer side (the strong direction). Therefore the loadings in the weak direction are considered and the shear walls are optimized to improve the resistance against the deformation in the weak direction. The loading and boundary conditions are shown in Figure 11. Two load cases are taken into account for wind loads from opposite directions, both


Glass wall t = 0.02 m 4m 0.5 m × 0.5 m t = 0.03 m

2m

4m

2m

10 m

1 m × 0.5 m 2 m t = 0.0 3 m 2m t = 0.15 m Glass wall t = 0.02 m Plan view 2.1 m 1.8 m 2.1 m

5m

t = 0.15 m 5m

5m

5m

5m

5m

5m

4m

2m

4m Front view

10 m Side view

Figure 10. The geometries of the eight-storey frame building considered in Example 4 for reinforcement: plan view, front view and side view

load cases will be considered in sensitivity calculation (see Chapter 6 of Huang and Xie 2010). Since the ground floor is designed with large open space, the glass wall on the longer sides covers until the top of the ground floor. Therefore the wind pressure in the weak direction of the building is linearly distributed in a trapeziform from the first floor to the roof, assuming the wind speed at the ground level is zero. Besides the self-weight of the main structural members, non-structural masses are applied to the floors and roof simulating non-structural attachments such as the interior partition walls. The properties of the construction materials are listed as following: Esteel = 200 GPa, ρsteel = 8000 kg/m3, υsteel = 0.3 for steel, Econcrete = 32 GPa, ρconcrete = 2500 kg/m3, υconcrete = 0.3 for reinforced concrete, Eglass = 100 GPa, ρglass = 2500 kg/m3, υglass = 0.3 for toughened glass. The whole frame is simulated using sheel elements with various thicknesses for different parts. The original frame with shear walls has the roof

395


pwind

proof

proof Mroof

pwind

Mroof

pfloor

pfloor

original shear walls, i.e. 30% of the design domain. The optimization parameters are: ER = 2% and rmin = 0.4 m. Figure 12 shows the final solution for reinforcing the frame building. A bracing system is obtained to replace

Mfloor Mfloor pfloor

pfloor Mfloor Mfloor

pfloor


pfloor


pfloor


pfloor

pfloor

Mroof = 7000 Kg Mfloor Mfloor Mfloor = 10000 Kg proor = 5000 N/m2 pfloor = 10000 N/m2 Load case 2 pwind = 12000 N/m2 Load case 1

Figure 11. Two load cases considered for the frame building with the wind pressures from opposite directions, pressures on the floors and roof, and non-structural masses in the floors and roof

drifting of 416.28 mm in the weak direction under the wind load, and the fundamental frequency 5.20 rad/s of which mode the vibration happens in the weak direction. The reinforcement problem aims to stiffen the frame building with additional constraints on the roof drifting and the fundamental frequency. Various building codes set a limit for building drifting under wind load, which usually varies from height/500 to height/600. Taking the height of this building being 40 m, the roof drifting droof < 68 mm is added in the optimization problem. The control points for the local displacement constraints are placed at the roof corners on the wind-loaded side and in the direction of the wind load. Since two load cases are considered, four roof corners (two for each load case) are constrained for drifting. However the displacements of each corner under the relevant load case are with the same magnitude in opposite/same directions due to symmetries of geometry and loading. Therefore the displacement constraints on the four corners can be actually formulated into a single one. On the other hand, a low fundamental frequency may easily lead to structure resonance due to external excitations. Concerning this, a second constraint for the fundamental frequency ω1 > 12 rad/s is also included. The design domain is the whole walls of the shorter sides and is discretized using 10680 four node shell elements on each side. The objective volume fraction is set to match the material usage of the

396

Side

Perspective (a) Shear wall design

Side

Perspective

(b) Optimal design with bracing

Figure 12. Comparison of designs for the building: (a) design with uniform shear walls (C = 369044.4 Nm, droof = 416.28 mm, ω1 = 5.20 rad/s); (b) optimal design with resulted bracing system (C = 108907.30 Nm, droof = 67.94 mm, ω1 = 12.01 rad/s)



the original shear walls, resulting in the great performance improvements compared with the frame/shear wall system: mean compliance reduced by 70.5%, roof drifting decreased by 83.7%, fundamental frequency increased by 130.96%, while the material usage remains the same. The final solution satisfies both additional constraints of roof drifting and fundamental frequency. 5. CONCLUSIONS This paper presents an extended BESO method to deal with multiple additional constraints besides the volume constraint in topology optimization of continuum structures. General algorithms are derived using the mathematical formulation of the KuhnTucker condition for optimality. The Lagrange function is constructed as a relaxed objective function that may include any number of additional constraints. The Lagrange multipliers are determined by a search strategy using the sensitivities. To avoid searching in an infinite range, replacement factors within a finite range are proposed to represent the Lagrange multipliers in the whole range through a scaling function. To evolve the topological design, an update scheme for both the element design variables and the additional Lagrange multipliers is employed through optimality criteria based on the relaxed objective function. Following the proposed general algorithms for multiple constrained BESO, an optimization problem is formulated for stiffness maximization under additional constraints of local displacements and natural frequencies besides the simple volume constraint. Softkill technique is used to accurately calculate element sensitivities of both the objective function and additional constraints. Lagrange multiplier update is determined following a search strategy based on the element sensitivities. Several numerical examples of 2D and 3D structures for this type of multiple constraint optimization problems are examined. Convergent solutions are obtained for stiffness maximization that satisfies all the constraints. These examples clearly demonstrate the effectiveness of the proposed approach. Multiple load cases for optimal structural designs have also been considered by the BESO method, see e.g. Example 4 of this paper or Chapter 6 of (Huang and Xie 2010). ACKNOWLEDGEMENTS This work is supported by the Australian Research Council under its Discovery Projects funding scheme (Project No. DP1094401).


REFERENCES Bendsøe, M.P. (1989). “Optimal shape design as a material distribution problem”, Structural and Multidisciplinary Optimizition, Vol. 1, No. 4, pp. 193–202. Bendsøe, M.P. and Sigmund, O. (2003). Topology Optimization: Theory, Methods and Applications, Springer, Berlin, Heidelberg, German. Bendsøe, M.P. and Sigmund, O. (1999). “Material interpolation schemes in topology optimization”, Archive of Applied Mechanics, Vol. 69, No. 9–10, pp. 635–654. Edwards, C.S., Kim, H.A. and Budd, C.J. (2007). “An evaluative study on ESO and SIMP for optimizing a cantilever tie-beam”, Structural and Multidisciplinary Optimizition, Vol. 34, No. 5, pp. 403–414. Huang, X. and Xie, Y.M. (2007). “Convergent and meshindependent solutions for the bi-directional evolutionary structural optimization method”, Finite Element in Analysis and Design, Vol 43, No. 14, pp. 1039–1049. Huang, X. and Xie, Y.M. (2009). “Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials”, Computational Mechanics, Vol. 43, No. 3, pp. 393–401. Huang, X. and Xie, Y.M. (2010). “Evolutionary topology optimization of continuum structures with an additional displacement constraint”, Structural and Multidisciplinary Optimizition, Vol. 40, No. 1–6, pp. 409–416. Huang, X. and Xie, Y.M. (2010). Evolutionary Topology Optimization of Continuum Structures: Methods and Applications, John Wiley & Sons, Ltd, Chichester, England. Huang, X., Zuo, Z.H. and Xie, Y.M. (2010). “Evolutionary topological optimization of vibrating continuum structures for natural frequencies”, Computers and Structures, Vol. 88, No. 5–6, pp. 357–364. Li, Q., Steven, G.P., Querin, O.M. and Xie, Y.M. (1999a). “Evolutionary shape optimization for stress minimization”, Mechanics Research Communications, Vol. 26, No. 6, pp. 657–664. Li, Q., Steven, G.P. and Xie, Y.M. (1999b). “Shape and topology design for heat conduction by evolutionary structural optimization”, International Journal of Heat and Mass Transfer, Vol. 42, No. 17, pp. 3361–3371. Manicharajah, D. and Xie, Y.M. (1998). “An evolutionary method for optimization of plate buckling resistance”, Finite Element in Analysis and Design, Vol. 29, No. 3–4, pp. 205–230. Pedersen, N.L. (2000). “Maximization of eigenvalues using topology optimization”, Structural and Multidisciplinary Optimization, Vol. 20, No. 1, pp. 2–11. Prager, W. and Rozvany, G.I.N. (1977). “Optimization of structural geometry”, Dynamical Systems, Academic Press, New York, USA, pp. 265–293. Querin, O.M., Steven, G.P. and Xie, Y.M. (1998). “Evolutionary structural optimization (ESO) using a bi-directional algorithm”, Engineering Computations, Vol. 15, No. 8, pp. 1034–1048. Rong, J.H., Xie, Y.M., Yang, X.Y. and Liang, Q.Q. (2000). “Toplogy optimization of structures under dynamic response constraints”, Journal of Sound and Vibration, Vol. 234, No. 2, pp. 177–189.

397


Rozvany, G.I.N. (2008). “A critical review of established methods of structural topology optimisation”, Structural and Multidisciplinary Optimizition, Vol. 37, No. 3, pp. 217–237. Rozvany, G.I.N. and Querin, O.M. (2002). “Combining ESO with rigorous optimality criteria”, International Journal of Vehicle Design, Vol. 28, No. 4, pp. 294–299. Rozvany, G.I.N., Zhou, M. and Birker, T. (1992). “Generalized shape optimization without homogenization”, Structural and Multidisciplinary Optimization, Vol. 4, No. 3–4, pp. 250–254. Sigmund, O. and Peterson, J. (1998). “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima”, Structural and Multidisciplinary Optimization, Vol. 16, No. 1, pp. 68–75. Svanberg, K. (1987). “The method of moving asymtotes - a new method for structural optimization”, International Journal of Numerical Methods in Engineering, Vol. 24, No. 2, pp. 359–373. Tanskanen, P. (2002). “The evolutionary structural optimization method: theoretical aspects”, Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 47–48, pp. 5485–5498. Tanskanen, P. (2006). “A multiobjective and fixed elements based modification of the evolutionary structural optimization method”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 1–3, pp. 76–90. Xie, Y.M. and Steven, G.P. (1993). “A simple evolutionary procedure for structural optimization”, Computers and Structures, Vol. 49, No. 5, pp. 885–886. Xie, Y.M. and Steven, G.P. (1997). Evolutionary Structural Optimization, Springer, London, UK. Yang, X.Y., Xie, Y.M., Steven, G.P. and Querin, O.M. (1999). “Topology optimization for frequencies using an evolutionary method”, Journal of Structural Engineering, ASCE, Vol. 125, No. 12, pp. 1432–1438. Zhao, C.B., Steven, G.P. and Xie, Y.M. (1997). “Evolutionary natural frequency optimization of two-dimensional structures

398

with additional non-structural lumped masses”, Engineering Computations, Vol. 14, No. 2, pp. 233–251. Zhou, M. and Rozvany, G.I.N. (1991). “The COC algorithm, Part II: topological geometry and generalized shape optimization”, Computational Methods in Applied Mechanics and Engineering, Vol. 89, No. 1–3, pp. 197–224. Zhou, M. and Rozvany, G.I.N. (2001). “On the validity of ESO type methods in topology optimization”, Structural and Multidisciplinary Optimization, Vol. 21, No. 1, pp. 80–83. Zuo, Z.H., Xie, Y.M. and Huang, X.D. (2010). “An improved bidirectional evolutionary topology optimization method for frequencies”, International Journal of Structural Stability and Dynamics, Vol. 10, No. 1, pp. 55–75.

NOTATIONS α element sensitivity number C mean compliance d displacement E Young’s modulus F objective function g inequality constraint h equality constraint K stiffness matrix L Lagrange function M mass matrix p penalty exponent in material interpolation scheme rmin filter radius S2 slack variable u element displacement vector; eigenvector V* objective volume x design variable ω frequency λ, µ Lagrange multipliers γ, ϕ replacement factors ρ mass density
