Keywords: sensitivity analysis, semi-analytical design sensitivities, shape memory alloys, shape optimization, NiTi, R-phase.

Shape memory alloys (SMAs) are active materials with a high power density capable of producing comparatively large actuation strains and stresses. These materials have found application in a wide variety of fields, ranging from civil engineering to microsystems and from medicine to aerospace. However, designing effective multi-dimensional SMA actuators is a challenging task, due to the complex behavior of the material and the fact that often electrical, thermal and mechanical aspects have to be considered simultaneously. For this reason, the application of systematic computational design approaches, such as design optimization techniques, to the design of SMA structures is of great interest. To enable efficient SMA design optimization procedures, the availability of sensitivity information that allows the use of gradient-based optimization algorithms is crucial.

This paper presents the formulation and computation of design sensitivities of SMA shell structures, and their first use in a gradient-based design optimization procedure of a miniature SMA gripper. The sensitivity analysis is carried out using the direct differentiation method, in a steady state electro-thermo-mechanical finite element context. Finite difference, semi-analytical and refined semi-analytical approaches are considered and compared. The SMA constitutive model used in this study is specifically intended for design optimization of SMA structures and actuators. In contrast to the majority of SMA models, the formulation of this model is history-independent, which simplifies the sensitivity analysis considerably. The model is specifically aimed at the description of the pseudo-elastic behavior of NiTi alloys, based on the austenite/R-phase transformation. This behavior is characterized by its negligible hysteresis, which is very attractive for actuator applications.

This research is aimed at SMA shell structures, which can generate large actuator displacements through bending. The most general case of actuation is considered, where the actuation effects are initiated by temperature changes induced by Joule heating. This implies that a coupled electro-thermo-mechanical finite element analysis is required. As a consequence, the sensitivity analysis includes coupling terms between three different physical fields. A challenging aspect of this work lies in the fact that the constitutive model in the considered plane stress setting contains implicit equations, which lead to complications in the sensitivity analysis. This problem manifests itself in the thermo-mechanical sensitivity coupling terms and in sensitivities of derived mechanical responses such as stresses or equivalent strains. Solutions for this difficulty based on finite difference and analytical approaches are discussed and evaluated.

As a numerical example of the proposed model-based design procedure for SMA structures, the shape optimization of a miniature gripper is considered. An electro-thermo-mechanical shell finite element model of this gripper is used in this optimization process. The objective of the study is to maximize the displacements of the gripper tips, with constraints on the maximum local temperature and equivalent strain values. These local constraints are treated in an aggregated manner using the Kreisselmeier-Steinhauser constraint aggregation technique, in order to reduce the number of responses.

Suitable relative perturbations of the design variables are determined by comparing the results obtained by finite-difference, semi-analytical and refined semi-analytical methods. The accuracy improvement obtained by the refined approach in this particular case is also evaluated. Using the developed sensitivity analysis, shape optimization of the SMA miniature gripper structure is carried out using two gradient-based optimization algorithms, Sequential Quadratic Programming (SQP) and the Method of Moving Asymptotes (MMA). Subsequently, a comparison is made between these gradient-based design optimization procedures and results of a response-surface-based technique, regarding the quality of the optimal solution and the computational efficiency of the optimization process. On average, the number of response evaluations was reduced by nearly a factor of 25, through the use of gradient-based optimization techniques.

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