تجزیه و تحلیل حساسیت از پوسته های آلیاژی حافظه شکل

This paper presents procedures for efficient design sensitivity analysis for shape memory alloy (SMA) structures modeled with shell elements. Availability of sensitivity information at low computational cost can dramatically improve the efficiency of the optimization process, as it enables use of efficient gradient-based optimization algorithms. The formulation and computation of design sensitivities of SMA shell structures using the direct differentiation method is considered, in a steady state electro-thermo-mechanical finite element context. Finite difference, semi-analytical and refined semi-analytical sensitivity analysis approaches are considered and compared in terms of efficiency, accuracy and implementation effort, based on a representative finite element model of a miniature SMA gripper.

Shape memory alloys (SMAs) are active materials with a high power density, capable of producing comparatively large actuation strains and stresses [1]. Their actuation properties originate from a solid-state phase transformation, which is affected by changes in temperature or stress, and strains associated with this transformation can be used for actuation. SMA actuators are widely used in wire or spring configurations, but upcoming applications in, e.g., medical instrumentation or microsystems also demand more complex shapes. 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, interest in the application of systematic computational design approaches, such as design optimization techniques, to the design of SMA structures is increasing. Design optimization has been applied to SMA wire-based configurations [2] and [3] and to SMA structures modeled by analytical models [4]. However, the models used in these studies cannot be extended to more general SMA structures. In addition, others have applied heuristic peak stress reduction algorithms [5] to more complex SMA designs [6] and [7]. However, their approach is less versatile than the more general and systematic design optimization techniques developed in the structural optimization community, based on a formal mathematical problem formulation combined with optimization algorithms (see, e.g., [8] and [9]). Recently, Dumont and Kuhl [10] have demonstrated design optimization of SMA structures modeled by finite elements, using a genetic algorithm. Genetic algorithms are however known to be rather inefficient, which makes that this optimization approach is limited to relatively small problems. To realize efficient SMA design optimization suited for a wide range of problems of realistic complexity, the availability of sensitivity information is crucial. Various approaches exist to perform sensitivity analysis, and the available techniques and their characteristics are discussed extensively in dedicated books and review papers [9], [11], [12], [13] and [14]. An essential aspect is that with the appropriate techniques, design sensitivities can often be obtained at low computational cost, compared to the response evaluation itself. This advantage is particularly evident in the case of history-independent non-linear models [11]. In that case the analysis itself is quite expensive, since the non-linearity usually requires an incremental–iterative solution strategy. In comparison to this significant computational effort, the sensitivity analysis for path-independent models is far less demanding. The sensitivity analysis presented in this paper is based on a simple constitutive model for SMA behavior based on the R-phase transformation in NiTi [15]. In contrast to the majority of existing SMA models, this model is history-independent and therefore well suited for use in sensitivity analysis and design optimization. However, the considered model is specifically aimed at the R-phase/austenite transformation in NiTi in a selected temperature range, and is not directly applicable for describing more general SMA behavior. The formulation of SMA constitutive models for general SMA behavior, which include hysteresis and dynamic effects, but which at the same time are sufficiently simple to allow sensitivity analysis and design optimization, still remains a challenge. For a discussion of possible alternatives to the presented approach, such as, e.g., the reduction of complicated models using centre-manifold theory, see Ref. [15] and the references therein. The focus of the present paper is however on the outlined SMA behavior based on the R-phase transformation in particular, which offers attractive properties for actuator applications [1]. For this situation, we develop and evaluate effective sensitivity analysis approaches, which allow gradient-based design optimization. This paper starts with a brief overview of various sensitivity analysis approaches in Section 2. The present work is aimed particularly at SMA shell structures, as these can generate large actuator displacements through bending deformation. The most general case of actuation by means of resistive heating is considered, which requires a sequentially coupled electrical, thermal and mechanical finite element analysis. Simpler situations, e.g., a given temperature distribution, are also covered by this general formulation. Section 3 discusses the derivation and computation of design sensitivities for SMA shell structures in this setting. Numerical results based on finite difference, semi-analytical and refined semi-analytical sensitivity analysis approaches are subsequently presented and discussed in Section 4, using a representative case study of a miniature SMA gripper, followed by conclusions. The application of the developed sensitivity analysis routines to a gradient-based design optimization procedure of this SMA gripper is outside the scope of the present paper, but is presented in a forthcoming article [16]. 2. Sensitivity analysis approaches Several approaches exist to perform sensitivity analysis, and the relevant techniques and their characteristics are briefly reviewed here. Detailed discussions can be found in dedicated books and reviews [9], [11], [12], [13] and [14]. In the following subsections, the system response of interest will be denoted by f and the state variables by u. For simplicity, only a single design variable s is considered, without loss of generality. The response is considered to be a function of both u and s, where the state variables also implicitly depend on the design variable, i.e., f = f(u(s), s). Adjoint formulations are not considered, since for the intended shape optimization problems they are not expected to offer significant advantages over the direct differentiation method.

In this paper, the sensitivity analysis of SMA structures actuated by Joule heating has been considered. The behavior of these structures is described by a non-linear path-independent model, which allows for sensitivity analysis procedures that require significantly less computational effort than the analysis itself. A restarted global finite difference (GFD) approach as well as direct semi-analytical (SA) and refined semi-analytical (RSA) procedures have been studied in this context. As the number of design variables in the intended SMA shape optimization applications is generally modest, an adjoint formulation has not been considered in this paper. Numerical testing on an SMA miniature gripper model has revealed that particularly for shape design sensitivities of mechanical response quantities, the selection of a proper relative design perturbation is critical. By testing a range of perturbations, it was found that in certain cases only a small interval exists for which accurate sensitivities are obtained. The semi-analytical approaches appeared to be slightly more sensitive to the perturbation, compared to the GFD method. The reported improved accuracy of the RSA method over the SA formulation was not observed in all cases, probably because the dominant errors in the present problem are not related to large rigid body motions, but rather to the non-linearity of the SMA model. Generally, one or two iterations proved sufficient to obtain a perturbed solution in the GFD case, starting from the nominal final configuration. For the linear electrical and thermal analyses, the superior efficiency of the semi-analytical approach is undisputed. However, considering the entire SMA analysis as a whole, the GFD method turns out to be quite competitive for the considered class of problems, due to the relatively high cost of the incremental–iterative scheme used to solve the non-linear mechanical problem. For the representative miniature gripper example, the GFD approach proved to be only 16% slower than the SA case, for 12 design variables. This makes it an attractive option for sensitivity analysis, particularly when its comparatively straightforward and generic implementation is taken into account. In contrast, the complexity of the computation of the pseudo-load in the semi-analytical approaches was found to increase significantly due to the fact that the problem involves a coupled electrical, thermal and mechanical analysis. Particularly the term associated with thermo-mechanical coupling required additional implementation effort. A fully analytical as well as a finite-difference-based formulation has been implemented, and it was found that both procedures perform well. In spite of the fact that an iterative process is used inside the material model, inaccuracies due to numerical noise have not been detected in case of the finite-difference approach, and results obtained with both methods were in good agreement. An additional consideration is the sensitivity analysis of derived quantities. An important response quantity in the used SMA model is the maximum effective strain. In case of the finite difference method, evaluation of its sensitivity is straightforward, as the finite difference formula can simply be applied, using the computed nominal and perturbed responses. In contrast, because of the complex SMA material model, in the semi-analytical cases a rather lengthy and complex procedure is required to compute the sensitivity of the effective strain from the state variable sensitivities. In conclusion, it can be stated that for the present class of SMA problems, given a suitable ratio of design variables versus response quantities, the restarted finite difference approach is a viable option for sensitivity analysis, particularly considering the significantly smaller implementation effort. However, for larger problems with considerably more degrees of freedom, the advantage of semi-analytical methods will increase, as the cost of system matrix decomposition will increase sharply. Also, when the number of design variables increases, the GFD approach will become increasingly unattractive, as the computational cost per design variable is higher than in the (R)SA methods. For large numbers of design variables, an adjoint semi-analytical approach is likely to be the most efficient. The developed sensitivity analysis methods are expected to be of great use in further work on design optimization of SMA structures.