In the previous section, we investigated the application of reanalysis using ROM in stress-constrained topology optimization through a 2D numerical example. In this section, we apply the previously described ROM methods to the 3D compliance minimization problem of the L-shaped beam. To better understand the influence of the stress constraint on the optimization results, we then consider making an in depth analysis of some of the parameters associated with the stress constraint based on the AdapBi-ROM method. The L-shaped beam is shown in Fig. 8, where the structure’s top surface is fixed and a vertically downward load (f=1) is applied along the top right side.

We use standard hexahedral elements to discretize the design domain with each element having unit volume and thickness. The maximum number of iterations is set to 300.

### ROM methods on 3D L-shaped beam

Here, we set (L=80) and (W=50), thus the design domain is discretized by a total of 32768 elements. 5 ROM methods were tested for this 3D case, The corresponding design results are given in Table 6.

From Table 6 we see that the combined AdapBi-ROM method behaves well both as far as computational time saving as well as final objective accuracy are concerned, as expected, in the 3D large-scale test case. Next, we will use this method as the ROM approach of choice to conduct a parameter study.

### Parametric study

For this, we set (L=50) and (W=6) in Fig. 8, thus the design domain is discretized by a total of 9600 elements. Some parameters are predefined as follows: initial limit of objective relative error (hat{e}^{0}_{c}=5times 10^{-3}) and its decay factor (varsigma =10^{-2}), for equilibrium equations (N_{b}=4):,initial force residual threshold (hat{e}^{0}_{c}=0.04) and its scale factor (theta =20), for adjoint equations (N_{b}=6):, initial force residual threshold (hat{e}^{0}_{c}=0.01) and its scale factor (theta =50).

#### Influence of p norm factor p

Here, we vary the *p* norm factor from 2 to 30, and while fixing the stress penalty factor at (q=0.8) and the allowable maximum stress at (hat{sigma }). Table 7 gives the design results.

It can be seen from Table 7 that the CPU runtime trend for both the reference method as well as the combined AdapBi-ROM method is not monotonic. More computation time is spent when using a smaller *p* value (like 2 and 4) while less time is spent when using a larger *p* value (like 20 and 30) for these two methods. However, both methods still have some differences in terms of the changing trend of the CPU runtime, where the CPU time of the reference method goes down while that of the AdapBi-ROM method goes up, when we vary the *p* value from 6 to 8.

Interestingly, we find that the optimal compliance obtained using both methods has the same trend of first dropping and then increasing as the value of *p* increases, which means that we can likely obtain the smallest optimal compliance (meaning the “stiffest” possible structure) by using a mid-range value for *p*. In general, to better balance CPU time and structural performance, the *p* value should neither be too large nor too small.

Figures 9, 10 and 11 show the optimal topology and stress iterative history as well as the volume fraction iteration history for different values of the P-norm p norm factor with the AdapBi-ROM method.

We see from Fig. 9 that the optimal topologies obtained using different values of *p* based on AdapBi-ROM method are visually different when using (p=4) and (p=6) (or (p=12) and (p=20)).

From Fig. 10, we can see that the stress value during the entire procedure satisfy its constraint limit progressively faster with a larger value of p. Meanwhile, it seems that the larger the value of *p*, the smaller the value of the final maximum von Mises stress (sigma _{max }^{VM}). For (p=20) and (p=30), the curves are below the stress limit within 50 iterations.

Fig. 11 shows that the volume fraction during the iterations can rapidly, and in a stable manner, satisfy its respective constraint limiting value, provided (p > 2).

Next, we give a comparison of the compliance (objective function) iteration history using different values of p based on AdapBi-ROM method in Fig. 12 where it appears that the larger the p value, the faster the rate of convergence of the optimization.

#### Investigation of varying stress penalty factor q

We vary the stress penalty factor *q* while fixing the p norm factor (p= 8) and the allowable maximum stress at. Table 8 gives the design results.

It can be seen from Table 8 that the trends for CPU time and optimal compliance for both the reference method and the AdapBi-ROM method are exactly the same and not monotonic as the *q* value decreases. Therefore, the ROM retains the same impact of the stress penalty factor parameter as for the reference case. Interestingly, the CPU time has a general tendency to first increase and then decrease while the optimal compliance has the opposite tendency, implying a potential conflict between CPU time and optimal compliance for several *q* values. This means that *q* needs to be properly determined to better balance the computational efforts and structural performance.

Just like in “Influence of p norm factor p”, we give the optimal topology and stress iteration history as well as the volume fraction iteration history for different values of the stress penalty factor, using the AdapBi-ROM method, as shown in Figs. 13, 14 and 15, respectively.

From Fig. 13 we can see that the optimal topologies using different values of q based on AdapBi-ROM method are visually distinguishable from each other, which can be seen from the test case with (q=0.5) and (q=0.8).

From Fig. 14 we note that the stress constraint is not satisfied when using relatively smaller values of (q (0.1 – 0.3)) which means the stress penalization is not sufficient. In addition, using a larger value of *q* can cause the maximum von Mises stress to satisfy its constraint limiting value as soon as possible during the procedure. In the case of (q=0.9), the curves end up below the stress limit after about 25 iterations.

Figure 15 shows that the volume fraction during the iterations can rapidly and in a stable manner satisfy the constraint for all 5 different values of *q* tested here.

Next, we compare the compliance iteration histories using different values of *q* based on the AdapBi-ROM method in Fig. 16 where different convergence rates of optimization can be obtained by varying *q*, but no monotonicity is observed as *q* increases. It seems plausible that a midrange value of *q* would lead to a relative lower convergence rate as shown below.

#### Varying the allowable maximum stress

In this case, we vary the allowable maximum stress (hat{sigma })while simultaneously fixing the p norm factor *p* at 8 and the stress penalty factor *q* at 0.8. Table 9 gives the design results.

From Table 9 we see that the trends for the CPU time are the same for both the reference as well as the AdapBi-ROM methods, first increasing and then decreasing as the (hat{sigma }) value increases. A similar trend is seen for the optimal compliance for both methods.

Just like in the previous investigations, we give the optimal topology and stress iteration history as well as the volume fraction iteration history for different values of the allowable maximum stress based on the AdapBi-ROM method, as shown in Figs. 17, 18 and 19, respectively.

From Fig. 17 we can see that the optimal topologies using different values of (hat{sigma }) based on the AdapBi-ROM method are clearly very different from each other. When using smaller values of (hat{sigma }) ((hat{sigma }=0.15) ,(hat{sigma }=0.3) ), the final design structural branches are discontinuous and in a “fractured” state which means that the optimized design has not yet converged sufficiently despite having completed maximum iterations. Interestingly, we find a clear curvature forming at the corner of the L-shaped beam by using these two smaller values of (hat{sigma }) . As for other values of (hat{sigma }) , the optimal topologies are also different from each other (see the case of (hat{sigma }=0.45) and (hat{sigma }=0.65) ).

From Fig. 18 we can see that the stress constraint is not satisfied in the case of the smallest value of (hat{sigma }) ((hat{sigma }=0.15) ). Also the maximum von Mises stress constraint is increasingly easier to satisfy during the procedure as (hat{sigma }) increases, but we note that an overtly large value of (hat{sigma }) could potentially render the stress constraint inactive (as seen in the case of (hat{sigma }=0.65) and (hat{sigma }=0.75) ).

From Fig. 19 we can see that the volume constraint is not satisfied for smaller values of (hat{sigma }=0.15) or (hat{sigma }=0.3) despite having been satisfied for larger values of (hat{sigma }).

Finally, we compare the compliance iteration history using different values of (hat{sigma }) and the AdapBi-ROM method in Fig. 20, where the compliance clearly has different convergence rate depending on the values of (hat{sigma }) and a relatively large value of (hat{sigma }) could potentially accelerate convergence.

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