Thermomechanical topology optimization of shape‐memory alloy structures using a transient bilevel adjoint method

Abstract

We present a novel method for computational design of adaptive shape-memory alloy (SMA) structures via topology optimization. By optimally distributing a SMA within the prescribed design domain, the proposed algorithm seeks to tailor the two-way shape-memory effect (TWSME) and pseudoelasticity response of the SMA materials. Using a phenomenological material model, the thermomechanical response of the SMA structure is solved through inelastic finite element analysis, while assuming a transient but spatially uniform temperature distribution. The material distribution is parameterized via a SIMP formulation, with gradient-based optimization used to perform the optimization search. We derive a transient, bilevel adjoint formulation for analytically computing the design sensitivities. We demonstrate the proposed design framework using a series of two-dimensional thermomechanical benchmark problems. These examples include design for optimal displacement due to the TWSME, and design for maximum mechanical advantage while accounting for pseudoelasticity. K E Y W O R D Scomputational inelasticity, pseudoelasticity, shape-memory alloys, topology optimization, transient adjoint sensitivity analysis, two-way shape-memory effects 1 2558

Multiphysics design of programmable shape-memory alloy-based smart structures via topology optimization

“…Hence the terms ∂Gn ∂un , ∂G n+1 ∂un , ∂Gn ∂νn and ∂G n+1 ∂νn are intrinsically zero. Consequently, the adjoint vectors λ n and γ G,n only rely on the mechanical residuals R and H. Hence, the two adjoint vectors have the same formulas as those obtained for sensitivity analysis of SMAs with uniform temperature distribution [38]. Meanwhile, the adjoint vector ψ n , which is related to transient thermal conduction, is influenced by all of the residuals R, H and G. Note also that ∂Rn ∂ωn and ∂R n+1 ∂ωn are zero for SMAs, since the coupling between the mechanical and thermal behaviors of SMAs is determined solely via the local residual H, i.e.…”

“…Sensitivity Analysis 2.6.1. The Transient Adjoint Sensitivity Formulation Building upon on the authors' previous work on sensitivity analysis of SMA-based structures with a uniform temperature distribution [38], we again use a transient, path-dependent adjoint method. Considering the coupled transient thermal conduction problem, the global thermal residual G must be included into Lagrangian functional in addition to the mechanical residuals R and H,…”

“…Moreover, computing the inverse of the matrix ∂H n /∂ν n may result in ill-conditioned matrices, due to the large magnitude of the off-diagonal entries of the compliance matrix S and the compliance modulus S in ∂H n /∂ν n . Our previous work [38] has provided the analytical solution for evaluating the matrix, in the case where we adopt a flow rule that is independent of the stress tensor σ. Herein, we discuss another case in which the flow rule is a function of stress and ∂Λ is nonzero, as shown in Equation 35. Note that when SMAs behave elastically, i.e.…”

“…For SMAs, Langelaar proposed a topology optimization algorithm considering a piecewise linear constitutive material model [37]. In addition, the authors' previous work studied topology optimization of singlematerial SMA structures, using an inelastic model that accounts for the phase transformation of SMAs and assumes uniform temperature distribution [38]. However, multi-material design of SMAbased smart structures that enables complex motion with less active materials, as well as nonlinear thermal-responsive behaviors of smart materials coupled with transient heat conduction have yet to be rigorously explored.…”

Topology optimization structure design of shape memory alloy with multiple constraints

“…Several studies in recent years deserve attention. Kang and James (2020) proposed a density-based topology optimization framework to unitize the two-way shape memory effect and superelastic. Subsequently, they extended their research to the multimaterial topology optimization design of SMA innovative structures that considered transient heat conduction (Kang and James, 2022).…”