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Structural and Multidisciplinary Optimization

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01-02-2012 | Research Paper

Modeling and optimal design of Neuro-Mechanical Shape Memory Devices

Authors: Carl-Johan Thore, Anders Klarbring

Published in: Structural and Multidisciplinary Optimization | Issue 2/2012

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Abstract

In this paper we describe the modeling and optimization of what we refer to as Neuro-Mechanical Shape Memory Devices (NMSMDs). These are active mechanical structures which are designed to take on specific shapes in response to certain external stimuli. An NMSMD is a particular example of a Neuro-Mechanical Network (NMN), a mechanical structure that consists of a network of simple but multifunctional elements. In the present work, each element contains an actuator and an artificial neuron, and when assembled into a structure the elements form an actuated truss with a superimposed recurrent neural network. The task of designing an NMSMD is cast as an optimization problem in which a measure of the error between the actual and desired shape for a number of given stimuli is minimized. The optimization problems are solved using a gradient based solver, and some numerical examples are provided to illustrate the results from the design process and some aspects of the proposed model.

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In this paper, the term passive structure denotes a structure where any deformation is solely due to applied mechanical forces. The response, deformation and/or reaction forces, of an active structure may also be affected by internal mechanical forces, active forces, which require supply of non-mechanical energy (e.g. thermal or electric), referred to as auxiliary energy.

The term “neural network” can refer to either a network of actual biological neurons (or a model thereof); or to a, biologically inspired, device for solving computational problems; see e.g. Haykin (2009). Herein, the latter interpretation of the term is more appropriate.

The fact that the two subsystems are coupled through ideal amplifiers implies that, in order to do so, we could simply add terms relating to the neural network on both sides of (3). After noting that the two subsystems become decoupled, one could then proceed by treating them separately.

An equilibrium point \({\boldmath {U}}^{*}\) is stable if for each ϵ > 0 there exists δ(ϵ) > 0 such that \(||{\boldmath {U}}^{*}-{\boldmath {U}}_{0}|| < \delta \Rightarrow ||{\boldmath {U}}^{*}-{\boldmath {U}}|| < \epsilon\) for all t ≥0.

By choosing E _max large enough, (21) can always be satisfied, so it is clear that (for reasonable choices of

https://static-content.springer.com/image/art%3A10.1007%2Fs00158-011-0684-1/MediaObjects/158_2011_684_Figa_HTML.gif

) the set of feasible points to (23) is nonempty. Then since z ≥ 0 by the displacement constraints, assuming φ is continuously differentiable, existence of optimal solutions follows from fairly non-restrictive coercivity conditions on φ on the set

; cf. e.g. Andreasson et al. (2005, Chapter 4).

More formally, one wants to enforce smoothness of the input-output mapping (Haykin 2009, Chapter 7) in the sense that similar inputs should result in similar outputs. Removal of excess weights might be seen as a heuristic to accomplish this.

Setting the active force in (13) to + / − this value and solving for u when F _i = 0 yields u/l _i ≈ − 0.4/0.3, i.e. a maximum contraction/elongation of 40/30% of the elements original length, (deviating only slightly from the values predicted by the linear model (12)).

Another consequence of the existence of many local minima is that it can make it difficult to draw conclusions about the effect of varying parameters in the problem. This is because unless the global optimum is found for all parameter values of interest, there may be no easy way to distinguish between what is a result from changing the parameter and simply ending up in a different local minima.

Strictly speaking, all equilibrium points should be isolated in order to guarantee convergence; see e.g. Marco et al. (2003).

For the case when

https://static-content.springer.com/image/art%3A10.1007%2Fs00158-011-0684-1/MediaObjects/158_2011_684_Figc_HTML.gif

for some i, we note that

https://static-content.springer.com/image/art%3A10.1007%2Fs00158-011-0684-1/MediaObjects/158_2011_684_Figac_HTML.gif

as discussed in Section 3, while the sum of integrals in (26) can be split into two parts: one for all i such that

< 0 and one for the rest of the form \(\sum_{i} \frac{1}{\tau}\int_{0}^{\tanh{v}_{i}}\tanh^{-1}(s)\,\mathrm{d}s\).

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Title: Modeling and optimal design of Neuro-Mechanical Shape Memory Devices
Authors: Carl-Johan Thore
Anders Klarbring
Publication date: 01-02-2012
Publisher: Springer-Verlag
Published in: Structural and Multidisciplinary Optimization / Issue 2/2012
Print ISSN: 1615-147X
Electronic ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-011-0684-1

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