#
Multi-Objective Design Optimization of a Shape Memory Alloy Flexural Actuator^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{90}Y) which delivers high-energy, low-penetrating radiation that destroys tumor tissue while limiting adverse effects to surrounding healthy liver tissue [4]. Currently, SIRT uses a single-lumen microcatheter (1.0 mm diameter) and a manually operated syringe to deliver the

^{90}Y microspheres into the hepatic artery. Recent studies have documented significant increases in patient survival (29.4 months with SIRT and systemic chemotherapy vs. 12.8 months with chemotherapy alone [5]). Despite the documented advantages of SIRT, the inability to directly target tumor sites has limited its widespread adoption. The tortuous and patient-variable arterial anatomy limits the direct targeting of tumors, causing damage to healthy liver tissue and other organs via ischemia or radiation [6].

## 2. System Model

_{0}, where

_{d}(Figure 3c). As the tendon is heated to temperatures above the austenitic finish temperature A

_{f}, the material undergoes a phase transformation to austenite and recovers some of this strain, establishing a larger activated equilibrium curvature κ

_{a}, such that κ

_{d}< κ

_{a}< κ

_{0}.

_{d}must be minimized while the activated displacement y

_{a}must be maximized (Figure 4).

#### 2.1. Homogenized Energy Model of SMA Flexural Actuators

_{A}, martensite plus x

_{M+}, and martensite minus x

_{M-}. The relative stress σ

_{R}and interaction stress σ

_{I}are assumed to be manifestations of underlying densities that affect the equilibrium phase fractions. In Equation (4), E

_{A}is the austenitic elastic modulus, E

_{M}is the martensitic elastic modulus, and ε

_{T}is the maximum recoverable strain.

#### 2.2. Equilibrium Equations

_{0}, the SMA has zero strain over its cross-sectional area.The moment integral (6) is discretized using rectangular elements [15], as shown in Figure 6. Assuming the stress distribution is symmetric about the neutral axis, this discretization yields

_{i}is the midpoint of the ${i}^{th}$ rectangle, and $2\sqrt{{a}^{2}-{y}_{i}^{2}}$ is the width of the rectangle.

_{∞}is the ambient temperature. Equation (9) can be solved using a variety of nonlinear optimization algorithms; here, we employ the golden section search method [16]. After finding the deactivated curvature (and associated equilibrium phase fractions), the equilibrium activated curvature minimizes

_{ss}is the SMA’s activated steady-state temperature.

_{0}and sleeve flexural rigidity EI. The nonlinear relationship between the initial and final curvature in Figure 7 is due to the nonlinear stress–strain characteristics of SMA, which are included in our HEM model. At very low initial strains, the SMA is in its austenite phase, and the stress–strain relationship is relatively linear. This characteristic is evident in Figure 7a for initial curvatures up to approximately 100 m

^{−1}. At larger strains, the austenite to martensite transformation causes a plateauing of the stress–strain relationship. This characteristic is evident in Figure 7a for initial curvatures between approximately 100 and 700 m

^{−1}. When the phase transformation is complete (to fully detwinned martensite), a linear stress–strain relationship resumes. This characteristic is evident in Figure 7a for initial curvatures above approximately 700 m

^{−1}. As shown in Figure 7, the accuracy of computed actuator curvature improves with the number of discretization layers (N). However, because computational time is directly proportional to N, there exists a tradeoff between model accuracy and computational time. Based on the simulation results, N = 8 was found to adequately balance this tradeoff, as it produced curvature estimates within 2% of the full-order model (N = 128) with only 6.25% of the computational burden.

## 3. Experimental Setup

^{−1}), with bolts and washers used to secure the SMA tendon during shape setting (Figure 9). This range of initial curvatures was determined from simulations to provide the largest difference in the activated and deactivated curvature (Figure 10b). To standardize the electrical resistance of each 0.31 mm diameter SMA tendon (Dynalloy, Inc., Irvine, CA, USA), each curvature profile was machined to provide the same arc length and straight length. Constant tendon tension was maintained by suspending a 400 g mass vertically from each SMA tendon during the fixture tightening process.

_{0}, a programmable power supply (Agilent E3615A) was used to activate the specimen (maintaining 1.4 amps of DC current for 5.0 s, resulting in thermally-induced austenitic phase transformation). The curvature of the actuated specimen was measured by comparison to a printed scale.

^{−6}to 3 × 10

^{−4}Nm

^{2}) was determined from simulations to provide the largest change in activated and deactivated curvature (Figure 10). The molds were fabricated using rapid prototyping in ABSplus material (Dimension Elite, Stratasys, Eden Prairie, MN, USA). Nylon monofilament (0.31 mm diameter) was tensioned along the neutral axis of each rectangular mold to provide space for SMA tendon insertion. Each mold was cast with polydimethylsiloxane (PDMS, Dow Corning SYLGARD 184) and allowed to cure for two days.

_{a}and κ

_{d}. The process of activating, deactivating and image acquisition was automated using a custom LabView program (National Instruments, Austin, TX, USA).

_{a}and κ

_{d}, an image processing code was developed using MATLAB’s Image Processing toolbox (Mathworks, Inc., Natick, MA, USA). The code applied golden-section optimization to find the bending radius R (Figure 12) that minimizes the difference between predicted (L

_{pred}) and actual bending length (L

_{act}) according to

## 4. Results

#### 4.1. HEM Parameter Optimization

_{k,m}and β

_{k}) and seven SMA constitutive model parameters (E

_{A}, E

_{M}, σ

_{L}, Δσ

_{T}, ε

_{T}, h and T

_{max}) were chosen as design variables. The design variables and their associated bounds are listed in Table 1.

^{−5}Nm

^{2}) and six different initial curvatures (κ

_{0}= 100, 125, 167, 200, 250, and 333 m

^{−1}). Additional data was obtained for a fixed initial curvature (κ

_{0}of 200 m

^{−1}) and four different values of flexural rigidity (EI = 6 × 10

^{−6}, 1.5 × 10

^{−5}, 3 × 10

^{−5}, 6 × 10

^{−5}Nm

^{2}). The GA function in the MATLAB optimization toolbox was employed for single-objective design optimization. The design variables were real value encoded and 36 individuals were systematically populated within initial design variable bounds. The bounds were chosen from knowledge of the design space to speed convergence. Heuristic crossover was utilized with a crossover fraction of 1.2. The convergence criteria consisted of a minimum objective function gradient (1 × 10

^{−6}) and a generation limit (200). The initial and optimized model parameters are shown in Table 2 and in Figure 13. The initial parameters were taken from [10].

#### 4.2. Multi-Objective Optimization

_{0}and EI as the design variables. The optimized HEM and equilibrium curvature model are used to evaluate κ

_{a}and κ

_{d}, which are related to y

_{a}and y

_{d}via Equation (2). The actuator length (L = 0.011 m) was chosen to achieve 0.007 m displacement; the actuator is $\theta ={90}^{\circ}$(Figure 12). To improve numerical conditioning, log

_{10}(EI) was used to normalize the individuals for reproduction. The gamultiobj function in MATLAB was employed for the MOGA. Most of the GA settings were identical to the single-objective optimization case, with the exceptions of population (50) and minimum objective function gradient (1 × 10

^{−4}). Upper and lower parameter bounds were utilized for the population initialization (UB

_{i}and LB

_{i}, respectively), and values for subsequent populations (UB and LB) are shown in Table 4. The MOGA converged based on the minimum objective function gradient.

_{d}and y

_{a}: no design combination of EI and κ

_{0}perfectly minimizes y

_{d}(y

_{d}= 0 m) while simultaneously maximizing y

_{a}(y

_{a}≥ 0.007 m). The results of Figure 17a indicate that the Pareto-optimal displacements y

_{d}and y

_{a}have strong non-linear dependencies on EI (as was the case with κ

_{a}and κ

_{d}in Figure 15a). Furthermore, the correlation between y

_{d}, y

_{a}and κ

_{0}in Figure 17b resembles the quasi-linear dependence of κ

_{a}and κ

_{d}on κ

_{0}in Figure 15b. It is also clear from Figure 17 that the change in activated displacement occurs predominantly over initial curvatures ranging from 50 to 375 (m

^{−1}) and flexural rigidities, EI, ranging from 1 × 10

^{−5}to 5 × 10

^{−4}Nm

^{2}.

_{0}in Figure 18 illustrates how these design variables relate to actuation range: generally, this range is proportional to the initial curvature and inversely proportional to flexural rigidity. An additional performance metric, the actuation ratio (y

_{a}/y

_{d}), helps illustrate the design tradeoff between y

_{a}and y

_{d}, as shown in Figure 19. Arguably, the “best” design balances both performance metrics, such as the design (y

_{a}= 0.004 m, y

_{d}= 0.0019 m) indicated in Figure 19.

#### 4.3. Experimental Validation of Pareto Frontier

_{d}at low κ

_{0}(100, 125 m

^{−1}) were higher than predicted and high κ

_{0}(200, 250, 333 m

^{−1}) were less than predicted.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Computational fluid dynamic (CFD) simulation results: microsphere trajectories in the common hepatic artery (CHA) are dependent on release location [7].

**Figure 2.**Shape memory alloy (SMA) flexural actuators could enable precise positioning of microcatheter tips within the common hepatic artery.

**Figure 7.**Dependence of the simulated actuator curvature on discretization number (N): (

**a**) as a function of the initial tendon curvature ${\kappa}_{0}$, (

**b**) as a function of the sleeve flexural rigidity EI.

**Figure 8.**Custom fixture used for shape-setting SMA specimens: top—dimensioned drawing; bottom—photograph of machined fixture.

**Figure 10.**Simulation results used to determine the parametric range of (

**a**) EI and (

**b**) ${\kappa}_{0}$.

**Figure 11.**(

**a**) Rectangular polydimethylsiloxane (PDMS) sleeves (edges highlighted in black for image contrast) with (

**b**) molds used for fabrication.

**Figure 13.**Comparison between initial and optimized (

**a**) relative stress density and (

**b**) interaction stress density.

**Figure 14.**Predicted vs. experimentally measured actuator curvatures for the initial (non-optimized) model: (

**a**) as a function of the sleeve flexural rigidity EI, (

**b**) as a function of the initial curvature ${\kappa}_{0}$.

**Figure 15.**Predicted vs. experimentally measured actuator curvatures for the optimized model: (

**a**) as a function of the sleeve flexural rigidity EI, (

**b**) as a function of the initial curvature κ

_{0}.

**Figure 16.**Comparison of Pareto-optimal designs’ activated and deactivated displacement, ${y}_{a}$ and ${y}_{d}$.

**Figure 17.**Effects of Pareto-optimal design parameters on activated (${y}_{a}$) and deactivated (${y}_{d}$) displacement: (

**a**) flexural rigidity, EI, (

**b**) initial curvature, κ

_{0}.

**Figure 18.**Interaction of Pareto-optimal design variables (initial curvature, κ

_{0}, and flexural rigidity, EI) as a function of actuation range $\left({y}_{a}-{y}_{d}\right)$.

**Figure 19.**Actuator performance trade-off: displacement ratio $\left({y}_{a}/{y}_{d}\right)$ and change in displacement $\left({y}_{a}-{y}_{d}\right)$ versus activated displacement ${y}_{a}$.

**Figure 20.**Pareto frontier: Comparing selected Pareto-optimal designs to their experimental evaluation.

Variable | Description | Lower Bound | Upper Bound | Units |
---|---|---|---|---|

${\alpha}_{k,m}$ | Relative stress density coefficients | 0 | 3.0 | - |

${\beta}_{k}$ | Interaction stress density coefficients | 0 | 3.0 | - |

${E}_{A}$ | Austenitic elastic modulus | 10 | 100 | GPa |

${E}_{M}$ | Martensitic elastic modulus | 10 | 100 | GPa |

${\sigma}_{L}$ | Martensitic transition stress at 348 K | 100 | 400 | MPa |

$\Delta {\sigma}_{T}$ | Hysteresis loop’s temperature dependence | 1.0 | 15.0 | MPa/K |

${\epsilon}_{T}$ | Maximum recoverable strain | 1.0 | 7.0 | % |

h | Convection Coefficient | 0.01 | 1.6 | W/(m^{2}K) |

T_{max} | Maximum SMA Temperature | 353 | 403 | K |

Variable | Initial Value | Optimized Value | Units |
---|---|---|---|

${E}_{A}$ | 30.7 | 59.02 | GPa |

${E}_{M}$ | 26.0 | 16.83 | GPa |

${\sigma}_{L}$ | 295 | 204.17 | MPa |

$\Delta {\sigma}_{T}$ | 9.2 | 9.96 | MPa |

${\epsilon}_{T}$ | 4.4 | 7 | % |

h | 0.8 | 0.4279 | W/(m^{2}K) |

T_{max} | 380 | 381.4 | K |

SSE Deactivated | SSE Activated | SSE Sum | |
---|---|---|---|

Initial Model | 4792 | 2483 | 7263 |

Optimized Model | 478.1 | 36.7 | 514.2 |

Percent Improvement | 90% | 99% | 93% |

**Table 4.**Upper and lower parameter bounds for the initial population (UB

_{i}and LB

_{i}, respectively) and subsequent populations (UB and LB).

${\mathit{\kappa}}_{0}$ | EI | |
---|---|---|

UB_{i} | 600 | 1 × 10^{−5} |

LB_{i} | 0 | 1 × 10^{−2} |

UB | 600 | 1 |

LB | 0 | 0 |

Design | Initial Curvature (m ^{−1}) | Sleeve EI (Nm ^{2}) | Deactivated Displacement (m) | Activated Displacement (m) |
---|---|---|---|---|

1 | 100 | 3.05 × 10^{−5} | 0.0010 | 0.0025 |

2 | 125 | 2.23 × 10^{−5} | 0.0016 | 0.0036 |

3 | 167 | 1.76 × 10^{−5} | 0.0025 | 0.0049 |

4 | 200 | 1.58 × 10^{−5} | 0.0031 | 0.0057 |

5 | 250 | 1.47 × 10^{−5} | 0.0037 | 0.0065 |

6 | 333 | 1.16 × 10^{−5} | 0.0049 | 0.0075 |

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**MDPI and ACS Style**

Haigh, C.D.; Crews, J.H.; Wang, S.; Buckner, G.D. Multi-Objective Design Optimization of a Shape Memory Alloy Flexural Actuator. *Actuators* **2019**, *8*, 13.
https://doi.org/10.3390/act8010013

**AMA Style**

Haigh CD, Crews JH, Wang S, Buckner GD. Multi-Objective Design Optimization of a Shape Memory Alloy Flexural Actuator. *Actuators*. 2019; 8(1):13.
https://doi.org/10.3390/act8010013

**Chicago/Turabian Style**

Haigh, Casey D., John H. Crews, Shiquan Wang, and Gregory D. Buckner. 2019. "Multi-Objective Design Optimization of a Shape Memory Alloy Flexural Actuator" *Actuators* 8, no. 1: 13.
https://doi.org/10.3390/act8010013