Proposed approximate single-loop chaos control (ASLCC) methodmethod

f_FE g_FE g₁ g₂ PMA-AMV 9.5253 (2.4538,3.8819) 77 13930 3.0025,3.0320 10 PMA-CGA 9.5253 (2.4531,3.8829) 77 26936 3.0357,3.0045 10

SLSV 9.5253 (2.4538,3.8819) 175 370 3.0025,3.0320 3

SLSV-CG 9.5253 (2.4538,3.8819) 182 384 3.0025,3.0320 3 SORA 9.2840 (2.5802,3.5981) 325 2498 2.5696,3.0000 3 ASORA 9.5253 (2.4538,3.8819) 260 570 3.0022,3.0357 5 SLShV-CG (Proposed) 9.5252 (2.4538,3.8819) 443 1122 3.0185,3.0433 13

MPTP was updated using the conjugate gradient search direction in each iteration. Four mathematical and four engineering RBDO examples were solved for the verification of SLShV- CG. Examples 2 and 3 demonstrated that SLShV-CG converged to the desired reliable solution irrespective of the convex or concave nature of the performance functions, whereas other promi- nent methods failed to converge. Also, SLShV-CG is found to be computationally efficient than those methods, which are able to generate the reliable optimal solution.

As the proposed method utilizes CG for obtaining MPTP, there is always a chance of oscillation during the convergence of MPTP in case of highly non-linear performance function.

Example 2 demonstrated such oscillation while converging to the reliable optimal solution.

This leads to the second objective of this thesis in which these oscillations are traced and appropriate theory is applied. In the next section, an approximate single-loop chaos control method is discussed in detail.

3.5 Proposed approximate single-loop chaos control (ASLCC) method

The solution obtained by SLA may get stuck to periodic oscillation due to the use of steepest descent search or conjugate gradient search directions while estimating MPTPs for highly nonlinear performance functions. Therefore, the chaos control theory is used to obtain MPTP when it starts oscillating in the search space. In order to have proper understanding of the method a brief discussion of chaos control theory is presented.

3.5.1 Chaos control theory (CC)

Pingel et al. (2004) introduced an appropriate linear transformation to modify the eigenvalues of Jacobian matrix of dynamical system and stabilize the unstable fixed points of original system without changing their value and location. The iterative point x^(k+1) is updated according to equation (3.23).

x^(k+1)=x^(k)+λC[f(x^(k))−x^(k)], (3.23)

where λ is the chaos control factor which lies between 0 and 1. C is a n×n dimensional involutory matrix, i.e., a orthogonal matrix in which each row and each column will comprise of only one element 1 or−1 and other elements are 0. The total number of involutory matrix can be 2ⁿn!. To enhance the efficiency of chaos control method, matrixC must be the minimum set of involutory matrix (i.e., minimum number of 2ⁿn! matrix). The selection of C also depends on the property of saddle point and spiral point of unstable fixed points. Generally, selecting the unit matrix of C (Pingel et al., 2004) can stabilize the unstable fixed points.

Therefore, for a two dimensional dynamical system the matrix C can be taken as an identity matrix of size 2×2. The factor λ should be small enough to make the unstable fixed point stable. The selection of factor λalso depends on the original Jacobian matrix. The larger the maximum of the absolute eigenvalues of the Jacobian matrix, the smaller λ value should be taken for stabilization. This leads to more number of iteration in order to achieve stabilization.

Chaos control factor also acts as an important parameter which controls the efficiency as well as accuracy of any RBDO method in which chaos control method performs the reliability analysis.

It has also been observed that this chaos control factor is a sensitive parameter for convergence.

Therefore, the efficiency of the RBDO methods becomes sensitive to the initial value of chaos control factor.

As mentioned earlier, the iterative points to update MPTP in PMA can be obtained by steepest descent direction or by conjugate gradient direction. As these directions can yields to periodic oscillation, chaos control method (Yang and Yi, 2009) is used to evaluate the iterative points of the MPTP search, which is calculated as

u^(k+1)=u^(k)+λC[f(u^(k))−u^(k)], (3.24)

where

f(u^(k)) =u^(k)_{AM V} =−β^t ∇_uG(u^(k)_{AM V})

∥∇_uG(u^(k)_{AM V})∥, (3.25) In equation (3.24), the involutory matrix C is taken as Identity matrix (I) and the value of chaos control factorλ is taken between 0 and 1. Equation (3.24) is used in place of equation (2.10) to update the MPTP in PMA.

The use of chaos control theory in the double-loop method makes it inefficient as it is implemented at every iteration. To have a better implementation of chaos control theory, a proper oscillation criterion for tracking oscillation during the convergence of MPTPs is needed.

Therefore, a new RBDO method using the single-loop method with the chaos control theory is proposed to address these issues. In the following section, the proposed method is discussed in detail.

3.5.2 The proposed ASLCC method

As discussed earlier, SLA (Liang et al., 2008) employs KKT optimality conditions to approxi- mate MPTP from the standard normal space to the normal space. This enhances the efficiency of RBDO method by eliminating the reliability analysis loop. The deterministic optimization that is used in RBDO formulation is as follows.

min:f(µX),

s.t.: G_i(d,X^(k)_{i,M P T P})≤0, i= 1, . . . , M,

(3.26) wheref(µ_X) is the objective function,G_i is thei−th deterministic performance function,d is the deterministic design variable andµ_Xis the mean values of random variable vector X. The approximated MPTP is expressed as

X^(k)_{i,M P T P} = T⁻¹(U) =µ^(k)_X +σXU^(k)_i,ASLCC, (3.27) where U is the vector of random variables in the standard normal space, σ_X is the standard deviation of the random variables,X^(k)_{i,M P T P} is the approximated MPTP fori−th constraint in the normal space atk−th iteration. Since the accuracy of the reliable optimal solution depends onX^(k)_{i,M P T P}, this thesis thus focuses on evaluating accurateU^(k)_i,ASLCC given in equation (3.27) using the proposed method. An oscillation criterion is also proposed to track MPTPs. When these MPTPs start oscillating, the current MPTP of equation (3.27) is updated using chaos control theory. The oscillation criterion is given in equation (3.28).

γ_i^(k+1) = (U^(k+1)_i,ASLCC −U^(k)_i,ASLCC)·(U^(k)_i,ASLCC −U^(k−1)_i,ASLCC) γ_i^(k+1) >0 : no oscillation or converging of MPTPs

≤0 : oscillation or diverging of MPTPs

(3.28)

where the value of γ_i^(k+1) decides the oscillation among consecutive MPTPs of i−th perfor- mance function. If the dot product of equation (3.28) is positive, the two vectors (U^(k+1)_i,ASLCC− U^(k)_i,ASLCC) and (U^(k)_i,ASLCC −U^(k−1)_i,ASLCC) are almost in the same direction, signifying there is convergence as shown in Figure 3.18. In this case, the conjugate gradient search direction is used to estimate MPTP because it is found to be robust (Jeong and Park, 2017). On the other hand, if the dot product is negative, the two vectors (U^(k+1)_i,ASLCC −U^(k)_i,ASLCC) and (U^(k)_i,ASLCC −U^(k−1)_i,ASLCC) are moving away from each other. The angle between these vectors thus increases, signifying oscillation among MPTPs as shown in Figure 3.19. In this case, the chaos control theory is utilized to estimate the new MPTP, which helps to eliminate oscilla- tion of MPTPs for highly nonlinear performance functions. Moreover, the control factor λis determined from the angle updating criterion, which is given in equation (3.29).

λ=







0.2λ, 0.2θ^(k)> θ^(k−1), λ^θ(k−1)_θ(k) , θ^(k)> θ^(k−1) ≥0.2θ^(k),

λ, θ^(k)≤θ^(k−1),

(3.29)

where θ^(k) and θ^(k−1) are the angles between (U^(k+1),U^(k)) and (U^(k),U^(k−1)), respectively.

µ^(k)

µ^(k+1)

||D||

D

(k)

X α =

MPTP (k)

α^(k)

XMPTP (k+1)

Figure 3.17: Schematic diagram for MPTP estimation using conjugate gradient method

θ^(k−1)

(k+1) (k)

u

u

u u

1 2

u(0)

θ^(k) u

(k−1)

Figure 3.18: converging MPTP

(k+1)

u(k) u

u2

u₁ u(0)

(k−1)

θ θ

(k)

u

(k−1)

Figure 3.19: diverging MPTP

3.5.2.1 Estimation of MPTP using conjugate gradient search direction

The formulation to approximate MPTP is as follows

X^(k)_{i,M P T P} =µ^(k)_X +β_i^tσ_Xα^(k−1)_i , (3.30)

α^(k)_i =











σ_X∇G_i

∥σ_X∇G_i∥

µ⁽⁰⁾_X , ifk= 0, D_i

∥D_i∥

X^(k)_{i,M P T P}, otherwise,

(3.31)

whereα^(k)_i is the normalized direction at thek−th iteration that is estimated by the conjugate gradient search direction,D^(k)_i . It is given as

D^(k)_i =







σ_X∇G_i, k≤1 σX∇G^(k)_i + ^(σX∇G

(k)

i )^T(σX∇G^(k)_i )

(σX∇G^(k−1)_i )^T(σX∇G^(k−1)_i ) ·D^(k−1)_i , otherwise, (3.32) Figure 3.17 shows the iterative updating of MPTP using conjugate gradient search directions.

3.5.2.2 Estimation of MPTP using chaos control theory

After calculating X^(k)_{i,M P T P} using equation (3.30), it is transformed to the standard normal space to find the corresponding value ofU^(k)_{i,M P T P} using equation (3.33). Thereafter,U^(k)_i,ASLCC

is updated using equation (3.34).

U^(k)_{i,M P T P} = T(X_i) = (X^(k)_{i,M P T P} −µ_X)/σ_X. (3.33)

U^(k)_i,ASLCC =







U^(k)_{i,M P T P}, ifk <2 or γi >0, β_i^{t U}

(k−1)

i +λ^(k)_i C[U^(k)_{i,M P T P}−U^(k−1)_i ]

∥U^(k−1)_i +λ^(k)_i C[U^(k)_{i,M P T P}−U^(k−1)_i ]∥, otherwise. (3.34) It can be seen that U^(k)_i,ASLCC is the same asU^(k)_{i,M P T P} whenk <2 orγ_i >0 of equation (3.28).

Otherwise, U^(k)_i,ASLCC is evaluated separately using chaos control theory. The chaos control factor λ^(k)_i is changed in each iteration using the angle condition given in equation (3.29). If the angle (θ^(k)) as shown in Figure 3.18 is increasing from the previous angle (θ^(k−1)), λ^(k)_i is decreased to 0.2 ×λ^(k)_i , otherwise λ^(k)_i is kept constant. The converging and diverging scenarios of MPTP in the standard normal space are shown in the Figure 3.18 and Figure 3.19, respectively. It can also be seen in Figure 3.18 that the consecutive MPTPs are in the same direction and therefore, achieve stable convergence. On the other hand, if the direction of vectors changes, it signifies divergence or oscillation as shown in Figure 3.19.

The proposed method is referred to as approximate single-loop chaos control method (ASLCC) because the chaos control method is used when MPTPs start oscillating. The fol- lowing are the steps for ASLCC method and the flowchart is shown in Figure 3.20, in which all steps are shown.

Step 1. Initialize with mean values of random variables,µ⁽⁰⁾_X , standard deviation of the random variables, σX, target reliability index of the performance function, β^T, chaos control factor, λ^(k)_i , and the involutory matrix,C, that is considered as identity matrix I.

Step 2. Set k = 0, initial design variables d⁽⁰⁾ = µ⁽⁰⁾_X , initial MPTP as X⁽⁰⁾_{M P T P} = µ⁽⁰⁾_X and U⁽⁰⁾_{M P T P} =T(X⁽⁰⁾_{M P T P}).

Step 3. Perform deterministic optimization as min f(d,µ_X),

s.t. G_i(d,X^(k)_i )≤0 i= 1, . . . , M, where X^(k)_i =

( X⁽⁰⁾_{M P T P}, k= 0 µ^(k)_X +σ^T_XU^(k)_i,ASLCC, otherwise,

(3.35)

and generate the mean value of random variablesµ^(k+1)_X .

Step 4. Calculate D^(k)_i using equation (3.36)

D^(k)_i =







σ_X∇G_i, k≤1

σ_X∇G^(k)_i + ^(σX∇G

(k)

i )^T(σX∇G^(k)_i )

(σX∇G^(k−1)_i )^T(σX∇G^(k−1)_i ) ·D^(k−1)_i , otherwise.

α^(k)_i = _∥D^Di

i∥

X^(k)_{i,M P T P},

(3.36)

where, α^(k)_i is estimated with conjugate gradient search direction D^(k)_i ifk ≥2. Other- wise,α^(k)_i is estimated with steepest descent search direction,

α^(k)_i = σ_X ∇G_i(X)(d^(k),X^(k)_i )

∥σ_X ∇G_i(X)(d^(k),X^(k)_i )∥. (3.37)

Step 5. Update X^(k+1)_{i,M P T P} in the normal space as

X^(k+1)_{i,M P T P} =µ^(k+1)_X +β_i^tσXα^(k)_i , (3.38) and calculateU^(k+1)_{i,M P T P} =T(X^(k+1)_{i,M P T P}) and proceed.

Step 6. Check the value of oscillation criterionγ^(k) when k >1. If the value is positive ork≤1, estimateU^(k+1)_i,ASLCC asU^(k+1)_{i,M P T P} and goto step 8. Else, proceed to step 7.

Step 7. Assign U^(k+1)_i,ASLCC as given in equation (3.39) and proceed to Step 8.

U^(k+1)_i,ASLCC =β^T U^(k)_i +λ^(k)_i C[U^(k+1)_{i,M P T P} −U^(k)_i ]

∥U^(k)_i +λ^(k)_i C[U^(k+1)_{i,M P T P} −U^(k)_i ]∥ (3.39)

Step 8. If the convergence criterion ∥f(d^(k+1),µ^(k+1)_x )−f(d^(k),µ^(k)_x )∥/∥f(d^(k),µ^(k)_x )∥ ≤0.001 or

∥µ^(k+1)_x −µ^(k)_x ∥ ≤0.001 is satisfied, terminate. Otherwise, go to Step 9.

Step 9. Evaluate the angle between two vectors as θ^(k)=cos⁻¹

U^(k+1)_i,ASLCC·U^(k)_{i,M P T P}

∥U^(k+1)_i,ASLCC∥∥U^(k)_{i,M P T P}∥

, (3.40)

and calculate the iterative chaos control factor for k≥1 as

λ^(k+1)_i =







0.2λ^(k)_i , 0.2θ^(k)> θ^(k−1), λ^(k)_i ^θ(k−1)_θ(k) , θ^(k)> θ^(k−1) ≥0.2θ^(k), λ^(k)_i , θ^(k)≤θ^(k−1)

(3.41)

Step 10. Check the iteration counterk. Ifk >2, calculate the oscillation criterion using equation (3.42). Else, goto step 11.

γ_i^(k+1)= (U^(k+1)_i,ASLCC−U^(k)_i,ASLCC)·(U^(k)_i,ASLCC−U^(k−1)_i,ASLCC) (3.42) Step 11. Update MPTP,X^(k+1)_{i,M P T P}, usingU^(k+1)_i,ASLCCas given in equation (3.43) and updatek=k+1

and goto Step 3.

X^(k+1)_{i,M P T P} =µ^(k+1)_X +σ_XU^(k+1)_i,ASLCC. (3.43)