Thermomechanical Topology Optimization of Shape-Memory AlloyStructures Using a Transient Bi-Level Adjoint Method

Ziliang Kang, Kai A. James

University of Illinois at Urbana-Champaign, Department of Aerospace Engineering

Urbana, Illinois, United States

Abstract

We present a novel method for computational design of adaptive shape-memory alloy (SMA) struc-tures via topology optimization. By optimally distributing a shape-memory alloy within the pre-scribed 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 mate-rial model, the thermomechanical response of the SMA structure is solved through inelastic finiteelement analysis, while assuming a transient but spatially uniform temperature distribution. Thematerial distribution is parameterized via a SIMP formulation, with gradient-based optimizationused to perform the optimization search. We derive a transient, bi-level adjoint formulation foranalytically computing the design sensitivities. We demonstrate the proposed design frameworkusing a series of two-dimensional thermomechanical benchmark problems. These examples includedesign for optimal displacement due to the two-way shape memory effect, and design for maximummechanical advantage while accounting for pseudoelasticity.

Keywords: shape-memory alloys, computational inelasticity, two-way shape memory effects,pseudoelasticity, transient adjoint sensitivity analysis, topology optimization

1. Introduction

Active materials are capable of sensing and self-actuation while maintaining their mechanicaland thermal characteristics, thus they are ideal for multifunctional tasks. Such characteristicsmake active materials useful in a wide range of applications from biomedical devices to aerospacesystems [1]. However, due to the complex properties of these materials, for decades, scientists andengineers have endeavored to develop computational design techniques that enable accurate andefficient modeling of these materials so that their capabilities can be fully leveraged through thedesign of smart structures [2].

Among the various active materials, piezoelectric materials, dielectric elastomers, ionic polymers-metal compsites, shape memory polymers and shape memory alloys are the most common withinthe research literature [3]. Piezoelectric materials can be easily controlled using electricity [4], yetthey are largely limited to vibration control applications due to their high actuation frequency andlow actuating force[5]. Dielectric elastomers are capable of providing large deformation and actu-ating force, with a low actuating frequency. However, a pre-stretched membrane shape is requiredto trigger the actuating behaviors [6, 7]. Ionic polymer–metal composites are synthetic compositeswith actuating behavior similar to that of dielectric elastomers[8]. Yet researchers have struggledto achieve large deformations and actuation forces simultaneously [9]. Shape memory polymers areanother group of polymers which are capable of large deformation. Although efforts to create such

Preprint submitted to International Journal for Numerical Methods in Engineering July 27, 2020

polymers with two-way shape memory effects have been reported [10, 11], the more typical one-wayshape memory effect of shape memory polymers has limited their usage in actuators [12]. Shape-memory alloys (SMAs) contain high actuating density and medium actuating frequency [13]. StableTwo-way Shape Memory Effects (TWSMEs) and pseudoelasticity (superelasticity) are unique fea-tures of SMAs. These two effects, triggered by phase transformation, enable shape memory alloys tohave recoverable deformations under a prescribed temperature cycle or loading cycle. Nevertheless,the complicated training process and manufacturing cost are weaknesses of these materials.

With recent advances in additive manufacturing, it has become easier than ever to fabricatestructures that have complex geometries [14], thus opening up new possibilities for manufacturingthe intricate designs often produced by topology optimization. This duality between additive manu-facturing and topology optimization has generated a renewed focus the design of smart materials viatopology optimization. A number of researchers have investigated topology optimization of piezo-electric materials, with applications to mechanical amplifiers [15], flextensional actuators [16], plateand shell actuators [17], piezoelectric fans [18], piezoelectric sensors/actuators for vibration control[19], energy harvesting systems [20], bimorph-type actuators[21] and transducers[22]. Efforts havealso been directed to other material candidates, such as dielectric materials [23], magnetic material[24, 25], elastomers [26] and hyperelastic polymers[27]. Researches have also investigated the useof functionally graded representations [28] and level-set methods [29] for the design of smart struc-tures. In addition, non-gradient based optimization methods, such as genetic algorithms, have alsobeen studied to improve the robustness of design algorithms dealing with complicated constitutivemodels [24, 30].

Although substantial work has been done to improve these design techniques, there remainsrelatively little research on topology optimization of shape memory alloys — the most widelyused active material. One previous study proposed an optimization approach for SMA structuresfocusing on pseudoelasticity [31], however there remains no similar study on design for the two-wayshape memory effect. Since the phase transition behavior of SMAs varies significantly from onematerial to the other, SMA-based designs are the result of repeated testing and experimentation.Hence the ability to generate an algorithm that can apply to many different SMA materials andcan be used for different applications would be highly beneficial. However, there are three majorchallenges that must be addressed in order to achieve this capability. First, there is a need tointroduce more accurate constitutive models to capture the latent heat exchange physics of thephase transformation of SMAs. Second, transient sensitivity analysis is necessary to enable efficientnumerical optimization, since SMAs are a highly nonlinear materials and their thermomechanicalresponse must be modeled using computationally intensive finite element analysis. Third, it isimportant to develop an appropriate optimization strategy and corresponding optimization problemstatement that mathematically represents the desired peusdoelastic and two-way shape memorybehavior.

In this paper, we propose a novel framework for computational design of shape-memory al-loys, via topology optimization. We address the aforementioned problems through the followingcontributions: we implement an accurate finite element model based on the phenomenological con-stitutive relationship originally proposed by Lagoudas [32] to simulate the nonlinear physical andmechanical properties of SMAs. We also derive a transient adjoint sensitivity formulation to ana-lytically differentiate the inelastic material response of SMAs. Furthermore we derive a consistentanalytical formula for evaluating the tangent matrices used in the sensitivity analysis via a nestedSchur complement procedure. The resulting formula allows users to circumvent the issue of anill-conditioned tangent matrix, to obtain accurate sensitivities. Finally, we propose an original op-timization problem formulation and material interpolation scheme in which the elastic and thermalproperties of the material are controlled via a single design variable. We demonstrate the proposed

2

method using a set of benchmark problems designed specifically to leverage the unique characteris-tics of SMAs. The optimization results show that our framework is able to tailor the thermoelasticresponse, including both the TWSME and pseudoelasticity to achieve the desired performance.

2. Methodology

2.1. Constitutive Relationship of SMAs

SMAs have two stable phases corresponding to different molecular lattice structures, martensite(M) and austenite (A). The TWSME and pseudoelasticity properties of SMAs are the result of phasetransformation triggered by latent heat exchange, under a temperature cycle or a loading cycle. Inthe presence of this phase transformation, SMAs show inelastic characteristics; otherwise they canbe assumed to behave elastically.

Two classes of constitutive relationships of SMAs have been proposed, one using micromechanical-based models and the other using phenomenological models. Both theories provide plausible physi-cal depictions of the diffusionless, solid-solid phase transformation process. However, micromechanical-based models provide more accurate results at the microscale since they capture the directionalbehavior of the SMA microstructure [33], while phenomenological models are more computationallyefficient yet maintain accuracy at the continuum scale. In this study, the phenomenological modelproposed by Lagoudas [32] is used, since the process of optimization is computationally intensive.To further ensure computational tractability, we assume SMA materials are isotropic, homogeneousand polycrystallized. We also assume small deformations and uniform temperature distributions,and we neglect the body forces.

The phenomenological constitutive models of SMAs are derived from the kinematics of thematerial, conservation (balancing) laws, and the entropy inequality principle, which describes thethermodynamics of SMAs. Based on the assumption of small deformations, we define the totalstrain tensor ε in terms of the gradient of the displacement field u as below

ε =1

2

[(5u) + (5u)T

](1)

where the superscript “T” (without italics) refers to the transpose operator and 5 is the gradientoperator, calculated as

5(f) =∂(f)

∂xx+

∂(f)

∂yy +

∂(f)

∂zz (2)

with the symbols x, y, z representing unit basis vectors in three-dimensional space. The total straintensor is composed of three parts, the elastic strain εe, pure thermal expansion εth and inelasticstrain, of which we only consider the transformation strain εt

The conservation laws are composed of conservation of mass, linear momentum, angular mo-mentum and energy. The local form of the conservation laws are are follows.

Conservation of Mass : ρ+ div(ρv) = 0

Conservation of Linear Momentum : div(σ) = ρv

Conservation of Angular Momentum : σ = σT

Conservation of Energy : ρu = σ : ε− div(q) + ρQ

(3)

where ρ is the local material density, v is the velocity of the material point, σ is cauchy stresstensor, u is the specific internal energy per unit mass, q is the heat flux and Q represents internalheat sources. The operators “div” and (·) refer to the divergence and time derivative operator,respectively.

3

To model the latent heat exchange process, the entropy inequality principle, also known as theClausius-Duhem inequality (4), is needed to define the thermodynamic state of the material. Inthe below inequality, s refers to entropy of the system, and T is the local temperature.

ρs+1

Tdiv(q)− ρQ

T≥ 0 (4)

To model the evolutionary phase transformation, we track the martensite volume fraction, ξ.Choosing σ, T , ξ and εt as internal state variables, we define the thermodynamic potential of SMAsvia the Gibbs-free energy as

G(σ, T, ξ, εt) =− 1

2ρσ : S : σ − 1

ρ:[α(T − T0) + εt

]+ c

[(T − T0)− T ln

T

T0

]− s0T + u0 +

1

ρf(ξ)

(5)

where f(ξ) is the hardening function of the SMA obtained from experiments. The parameters S,α, c, s0 and u0 are the effective compliance tensor of the material, effective thermal expansionparameter, effective specific heat, effective specific entropy and effective specific internal energyrespectively. The effective values of these parameters can be expressed in terms of their referencevalues in the pure martensite (M) and austenite (A) phases as well as the martinsite volume fractionξ. Here, “:” refers to double dot product of tensors. In the uniform temperature distribution case,we assume the effective thermal expansion parameter and specific heat are constant.

S = SA + ξ(SM − SA) = SA + ξ∆S

α = αA = αM

c = cA = cM

s0 = sA0 + ξ(sM0 − sA0 ) = sA0 + ξ∆s0

u0 = uA0 + ξ(uM0 − uA0 ) = uA0 + ξ∆u0

(6)

To further simplify the model, an evolution law (flow rule) [32] is proposed to establish arelationship between the transformation strain εt and the martensite volume fraction ξ. Thusthe transformation strain εt is no longer an independent variable and we only have three internalvariables, σ, T and ξ, to consider. The definition of the transformation tensor Λ is based onexperimental results. Here we use the definition proposed by Boyd and Lagoudas [34].

εt = Λξ

Λ =

32H

σsσeffs

ξ > 0

H εt−rεefft−r

ξ < 0

(7)

whereH is the maximum transformation strain of the SMA material. In the forward transformation,σs is the deviatoric stress tensor and σeffs is its associated effective (von Mises) stress. In the

reverse transformation, εt−r is the transformation strain tensor at the reversal point, and εefft−r isits associated effective strain.

With the three sets of thermomechanical relationships for SMA materials stated in equation 1,equation 3 and the definition of Gibbs free energy in equation 5, we can redefine the Clausius-Duheminequality in equation 4 as πξ ≥ 0, where π is given by

π(σ, T, ξ) =σ : Λ +1

2σ : ∆S : σ + σ : ∆α(T − T0)

− ρ∆c

[(T − T0)− T ln

T

T0

]+ ρ∆s0T − ρ∆u0 −

∂f(ξ)

∂ξ

(8)

4

Note that the Clausius-Duhem inequality needs to be satisfied for all possible thermomechanicalloading paths. For the forward transformation, ξ is a positive value and hence we assume π tohave a positive threshold value Y . Similarly, we assume π to have a negative threshold value −Yduring the inverse transformation, when ξ < 0. The aforementioned threshold can be expressed asa transformation function Φ, defined in equation 9.

Φ =

π − Y ξ > 0 (A→ M)

−π − Y ξ < 0 (M→ A)(9)

From here on, we can use Kuhn-Tucker conditions to obtain the threshold of the phase trans-formation process, shown in equation 10. This condition assumes that transformation will notbegin if the transformation function Φ does not reach the threshold value (< 0), in which casethe materials behave elastically (ξ = 0). When the material exhibits inelastic behavior (ξ 6= 0),the transformation region should always satisfy the condition that Φ = 0. In other words, for allpossible thermomechanical loading paths, either Φ or ξ needs to be zero.

A→ M : Φ ≤ 0, ξ ≥ 0; Φξ = 0

M→ A : Φ ≤ 0, ξ ≤ 0; Φξ = 0(10)

Furthermore, a consistency condition is needed to guarantee that the stress and temperaturefields remain continuous at the transformation surface.

Φ =∂Φ

∂σ: σ +

∂Φ

∂T: T +

∂Φ

∂ξ: ξ (11)

Using the Kuhn-Tucker condition and the consistency condition, the inelastic constitutive relationfor SMAs can be expressed in an incremental form as below. In equation 12, “⊗” refers to thetensor product operator and “d” refers to the difference operator. The tensor L is referred to asthe tangent stiffness modulus.

dσ = L : dε

L =

S−1 − S−1:∂σΦ⊗S−1:∂σΦ∂σΦ:S−1:∂σΦ−∂ξΦ

ξ > 0

S−1 − S−1:∂σΦ⊗S−1:∂σΦ∂σΦ:S−1:∂σΦ+∂ξΦ

ξ < 0

(12)

Note that we assume that the phase transformation is rate-independent in equation 11. This isbecause the phase transformation between martensite and austenite is a diffusionless process thatoccurs by shear distortion of the lattice structure (movement of atoms from their original position),and thus can be viewed as a rate-independent process. [32]

2.2. Finite Element Analysis of SMAs

Based on the constitutive relationship discussed in the previous section, the equilibrium condi-tion of the material domain, Ω, can be described in residual form as

Global level : force equilibrium∫

Ω ε(d)σ(d, T, ξ)dΩ−∫

ΓP dPdΓp = 0Local level : KKT− condition Φ ≤ 0 in Ω

ξΦ = 0 in Ω

consistency Φ = ∂Φ∂σ : σ + ∂Φ

∂T : T + ∂Φ∂ξ : ξ in Ω

flow rule εt = Λξ

(13)

5

On the boundary of the design domain Γ, there are two subsets of boundary conditions in whichwe prescribe portions of the displacement field d on the boundary Γd or the surface traction F onΓp.

d = dc on Γd

F = P on Γp(14)

We discretize the system via a Backward-Euler time integration scheme in which the progressionof time is implcitly represented by changes in temperature. These temperature increments serve asa proxy for the time step in the time-marching procedure. The discrete model is defined throughtwo sets of residuals. The global residual contains nodal information at global degrees of freedom,while the local residual captures local information at the Gauss points within a given element. Thecombined discretized residual equations are shown below.

Global level : Rn+1 =∧el

(∑Gd

(wBTGdσGd,n+1)−

∑Gf

($NGfpGf ,n+1)) = 0

Local level :

Hn+1 =

Hξ = Φ(σn+1, Tn+1, ξn+1)

Hεt = εtn + Λ(ξn+1 − ξn)− εtn+1

HS = Sn + ∆S(ξn+1 − ξn)− Sn+1

Hσ = σn + S−1n+1 : [Bdn+1 −α(Tn+1 − T0)− εtn+1]

− S−1n : [Bdn −α(Tn − T0)− εtn]− σn+1

= 0 ξ 6= 0

Hn+1 = σn + S−1 : [(Bdn+1 −α(Tn+1 − T0)− εt)− (Bdn −α(Tn − T0)− εt)]− σn+1 = 0 ξ = 0

(15)Here Gd and Gf are Gauss points, of which the superscripts d and f refer to 2D gauss points

of displacement field and 1D gauss points of surface traction p, respectively. w and $ are thecorresponding Gauss weights. The matrix B contains the spacial derivatives of the shape functions,N , which are used to interpolate the stress and displacement fields. The symbol

∧denotes the

assembly operator for assembling global matrices. Note that the subscripts Gd and Gf in theabove equation refer to the evaluation of the parameter at the indexed Gauss point (e.g. BGd =B(xGd , yGd , zGd), σGd = σ(xGd , yGd , zGd). At a fully thermal-elastic stage of the material (ξ = 0),εt equals either zero or a nonzero constant, depending on whether the SMA is in the austenite ormartensite phase. Since the Young’s modulus is the only variant in the evolution of the compliancetensor, we define the local residual HS in terms of the compliance modulus, which is the reciprocalof the Young’s modulus (S = 1/E). The relationship between the compliance tensor S and thecompliance modulus S is defined through a normalized constitutive matrix as shown in equation16, where υ is the Poisson’s ratio.

S = S : C

C =

1 −υ −υ 0 0 0−υ 1 −υ 0 0 0−υ −υ 1 0 0 00 0 0 2(1 + υ) 0 00 0 0 0 2(1 + υ) 00 0 0 0 0 2(1 + υ)

(16)

In addition, we assume the thermal expansion is purely hydrostatic, and the relationship between

6

the thermal expansion tensor, α, and thermal expansion parameter, α, is stated in equation 17

α = α[

1 1 1 0 0 0]T

(17)

To solve the transient coupled nonlinear system shown in equation 15, we implement a global-localapproach and uncouple the differential equation system as shown in equation 18. The uncoupledsystem (18) is then linearized numerically by implementing the Newton-Raphson process in twonested iterative loops, in accordance with the return mapping algorithm [35].

R(u,ν(u)) = 0

H(u,ν(u)) = 0(18)

In the equation above, we treat the local state variables ν = [ξ, εt, S,σ]T, each one associated witha unique Gauss point, as a function of the global state variable u = [F c,df ]T. The local residualsare initially solved based on a trial state of u, and then we return to solve the global residual. Here,df refers to the displacements at free degrees of freedom, and F c stands for the unknown appliedforce at constrained degrees of freedom.

In the case where the SMA behaves elastically, ν has a much simpler form, given by ν = [σ]T.From here we can deduce the following expression for a nested Newton-Raphson procedure givenby

outer loop :

[∂R

∂u(u(k),ν(u(k))) +

∧el

∑G

∂R

∂ν(u(k),ν(u(k))

dν

du(u(k))

]δu = −R(u,ν(u(k)))

u(k+1) = u(k) + δu

inner loop :

[∂H

∂ν(u(k),ν(l)(u(k)))

]δν = −H(u(k),ν(l)(u(k)))

ν(l+1) = ν(l) + δν

(19)where k and l are the iteration counters of the Newton-Raphson loops for each residual. A flow chartof the return mapping procedure is shown in figure 1. Note that in the linear system described inthe outer loop above, we can concisely express the quantity in brackets using the symbol K (shownin equation 20), which is generally referred as the consistent tangent stiffness matrix1. In equation20, we express the tangent stiffness matrix in terms of the consistent tangent stiffness modulus L

(defined in equation 12). The symbol J refers to the element Jacobian matrix that describes themapping from local to global coordinates (as before, the subscript G indicates that the parameteris evaluated at the Gauss point G within the isoparametric element. The subscript “el” indicatesthe portion of a vector (u for example) that contains only the degrees of freedom associated withthe element referenced by the index “el”. The result shown in equation 21 was first derived byLagoudas in [32], which we invoke here without loss of generality. This relation appears in boththe finite element analysis and the sensitivity analysis, and we provide an original derivation for itin the appendix.

1Note that in equation 19, the partial derivative ∂R∂u

is nonzero only due to the partitioning of the global residual,in which case the vector u contains variables corresponding to the unknown applied forces at degrees of freedomwhere the displacement is prescribed. In the general case, the global residual has no direct dependence on the nodaldisplacements, therefore this term vanishes

7

K =∧el

∑G

(∂R

∂νG

)(dνG

duel

)= −

∧el

∑G

(∂R

∂νG

)(∂HG

∂νG

)−1(∂HG

∂uel

)(20)

=∧el

∑G

wBTGLGBGdetJG (21)

where we have made use of the following relation, which is obtained by differentiating the localresidual in equation 18.

∂H

∂u(u,ν(u)) +

∂H

∂ν(u,ν(u))

dν

du(u) = 0 (22)

Figure 1: A flow chart of return mapping algorithm

To better illustrate the implementation procedure, we also provide pseudo-code for the returnmapping algorithm in table 1. Since −(∂H/∂ν)−1H is expensive to calculate, we use the closest-point return mapping algorithm [35], where we force the local residuals Hσ and HS to be zero whenthe local iteration is implemented.

Out Loop: Implement global iteration to update external state variables

I. Let k = 0, Tn+1 = Tn + n · dT,F n+1 = F n + n · dF ,d(0)n+1 = dn

II. Loop over Gauss points, calculate total strain ε(k)n+1 = Bd

(k)n+1 and go to inner loop

Inner Loop: Implement global iteration to update internal state variables

1. Let l = 0, ξ(0)n+1 = ξn, ε

t(0)n+1 = εtn, S

(0)n+1 = Sn

2. Calculate trial stress and evaluate the local residual, H

σ(l)n+1 = S

(l)n+1

−1: [ε

(k)n+1 − α(Tn+1 − T0)− εt(l)n+1]

HΦ = Φ[σ(l)n+1, Tn+1, ξ

(l)n+1]

8

Hεt = εtn + Λ(l)n+1[ξ

(l)n+1 − ξn]− εt(l)n+1

If |HΦ| ≤ eH1 and ||Hεt || ≤ eH2Update consistent tangent stiffness modulus L and internal variables. Go to outer loop

ElseProceed to transformation

3. Compute increment of inner state variables −(∂H/∂ν)−1H

∆ξ(k)n+1 =

Φ(k)n+1−∂σΦ

(k)n+1:S

(k)n+1

−1:Hεt

±∂σΦ(k)n+1:S

(k)n+1

−1:∂σΦ

(k)n+1+∂ξΦ

(k)n+1

(+ : ξ > 0,− : ξ < 0)

∆σ = S(l)n+1

−1: [−Hεt ∓∆ξ

(k)n+1∂σΦ

(l)n+1] (− : ξ > 0,+ : ξ < 0)

∆εt(l)n+1 = −S(k)

n+1 : ∆σ − [∆S : σ + ∆α(Tn+1 − T0)]∆ξ(k)n+1

4. Update martensite volume fraction, transformation strain and compliance modulus

ξ(l+1)n+1 = ξ

(l)n+1 + ∆ξ

(l)n+1

εt(l+1)n+1 = ε

t(l)n+1 + ∆ε

t(l)n+1

S(l+1)n+1 = Sn + ∆S∆ξ

(l)n+1

5. Let l = l + 1 and return to step 2III. Assemble global stiffness tangent matrix K and internal force vector F int

K =∧el

∑G

wBTGLGBGdetJG,F

(k)int =

∧el

∑G

wBTGσGdetJG

IV. Update displacement field

d(k+1)n+1 = d

(k)n+1 −K

−1(F(k)int − F n+1)

V. Evaluate the convergence condition

If ||d(k+1)n+1 − d

(k)n+1|| ≤ eR

Finalize the external variables u and internal variables ν, n = n + 1.Else

Let k = k + 1, return to step II

Table 1: Implementation procedure for finite element analysis of SMAs

Generally, the hardening function, f(ξ), is the result of curve fitting of experimental data [34].Hence the hardening function plays an important role in the constitutive relationship, as it definesthe shape of the hysteresis curve in the TWSME and pseudoelasticity. Before embarking on thedesign optimization, we simulate both two-way shape memory effects and pseudoelasticity of ourchosen design material, half-half NiTi (table 2) with a quadratic hardening model.

Austenite (A) Martensite (M)

Young’s modulus E (Pa) 32.5× 109 23.0× 109

Thermal coefficient α (K−1) 22.0× 10−6 22.0× 10−6

Specific heat c (J/kgK) 400.0 400.0Transformation start temperature (K) 241 226Transformation finish temperature (K) 290 194Highest transformation strain H 0.033Material density ρ (kg/m3) 6500Reference temperature T0 (K) 300Entropy difference ρ∆s0 (J/m3K) −11.55× 104

Table 2: Material properties of NiTi50

The quadratic hardening function we have selected (see equation 23), is the most widely used

9

option [32]. The parameters bM , bA, µ1 and µ2 are constants that depend on the intrinsic materialproperties. The formulas for these parameters can be found in [32]. In figure 2, we presentillustrative 1D results for the evolution of strain and martensite volume fraction under a temperaturecycle, and the evolution of stress and martensite volume fraction under a loading cycle.

Quadratic : f(ξ) =

12ρb

Mξ2 + (µ1 + µ2)ξ (ξ > 0)12ρb

Aξ2 + (µ1 − µ2)ξ (ξ < 0)(23)

160 180 200 220 240 260 280 300

Temperature(K)

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Str

ain

Total

Transformation

160 180 200 220 240 260 280 300

Temperature(K)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Mar

tin

site

vo

lum

e fr

acti

on

(a) Two-way shape memory effect

0 0.01 0.02 0.03 0.04 0.05 0.060

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

8

Strain

Str

ess

(Pa)

Total

Transformation

0 1 2 3 4 5

x 108

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Stress(Pa)

Mart

insi

te v

olu

me f

racti

on

(b) Pseudoelasticity

Figure 2: Simulation of the TWSME and pseudoelasticity in NiTi50

2.3. Optimization Problem Definition

We seek an optimization framework that enables us to tailor and maximally leverage the uniquebehaviors of SMAs. As illustrated in figure 3, the transformation temperature of SMA materialsis a result of both thermal and stress conditions. Here, we use topology optimization to optimizethe material distribution of the design domain, in order to tailor the thermomechanical responseof the structure to optimally leverage both the TWSME and superelasiticity properties of SMAstructures. Hence, a series single-material benchmark problems are proposed to demonstrate theability of optimization to achieve the desired TWSME and pseudoelasticity effects.

10

Figure 3: Influence of thermomechanical loading on the transformation temperature of SMAs

2.3.1. Design Objectives

In this study, to be able to optimize the characteristics of both TWSMEs and pseudoelasticity,two kinds of objective functions are presented. For the case of optimizing the TWSME, we pursue anobjective, shown in equation 24, to maximize the deformation of a structure triggered by TWSMEsunder a prescribed temperature cycle.

fobj = LTd, (24)

where L is a basis vector which selects for the displacement of interest. By optimizing this function,we are able to harness to TWSME to tailor the displacement of the structural at our desired degreeof freedom. In the case of pseudoelasticity, we seek to increase the mechanical advantage of the SMAstructure undergoing superelastic transformation. In this case, we seek to maximize the output forceof a force inverter or displacement amplifier while considering the phase transformation of SMAsunder a loading cycle. The definition of the objective function (i.e. mechanical advantage) is shownin equation 25.

fobj =F out

|F in|=LToutF

|F in|(25)

In the above equation, |Fin| refers to the magnitude of the input force, and Lout is a vector thatselects for the force magnitude at the output degree of freedom.

2.3.2. Interpolation of Effective Material Properties

We solve the optimal design problem by determining which elements of the design domainshould contain material, and which should be void. In accordance with the SIMP formulation [36],we parameterize the state of each element (solid or void) via a relative material density variable,r ∈ [0, 1]. Then the effective physical properties of material within each element can be presentedusing the standard SIMP interpolation function.

Eeff = rpE∗

dT eff = rpdT ∗

V eff = rV ∗

(26)

11

where p is the penalization constant, E∗ is the actual Young’s modulus of the design material, anddT ∗ is the actual temperature increment when we implement a temperature cycle on the designdomain. Note that the effective volume, V eff is a fraction of the actual element volume V ∗, with nopenalization applied. Note also that the effective temperature increment, dT eff, is also controlledby the local design parameter, r. This allows us to implicitly model the thermal conductivity andphase change throughout the design domain, since void regions contain no temperature change orphase transformation.

Note that in order to prevent numerical instabilities, such as checkerboarding and mesh-dependency[37], we implement the density filtering technique introduced by Bruns and Tortorelli [38]. Underthis formulation, the effective (filtered) density variables, r, are computed as a weighted sum of aset of unfiltered independent variables, r, within a prescribed neighborhood of each element. Inthe examples presented in Section 3, the radius of the filter neighborhood is equal to the width oftwo elements within the uniform finite element mesh. For details of the filtering procedure, readersare directed to the reference paper [38].

2.4. Sensitivity Analysis

2.4.1. The Transient Adjoint Sensitivity Formulation

To evaluate the design sensitivities of the system, we use a transient, path-dependent adjointmethod. This work is an extension of the formulation, which was proposed by Michaleris et al.[39] for sensitivity analysis of elastoplastic materials. We first express all functions of interest, fint(whether objective of constraint functions), in Lagrangian lagrangian form as follows.

Π =fint(uNt(r),νNt(r), r) +

Nt∑n=1

λTnRn(un(r),un−1(r),νn(r),νn−1(r), r)

+

Nt∑n=1

(∧el

∑G

γTG,nHG,n(un(r),un−1(r),νG,n(r),νG,n−1(r), r)

) (27)

where the vectors γ and λ are free parameters whose values can be chosen to maximize computa-tional expediency. Note that since the residuals, R and H are all zero vectors, the Langrangianfunction satisfies Π = fint, for all values of the vectors γn and λG,n.

Taking the first derivative of Π with respect to the design variable vector r and applying thechain rule, we obtain

dfintdr

=dΠ

dr=∂fint∂r

+

Nt∑n+1

(∂fint∂un

dundr

+∑G

∂fint∂νG,n

dνG,n

dr

)

+

Nt∑n=1

λTn

[∂Rn

∂r+∂Rn

∂un

dundr

+∂Rn

∂un−1

dun−1

dr+∧el

∑G

(∂Rn

∂νG,n

dνG,n

dr+

∂Rn

∂νG,n−1

dνG,n−1

dr

)](28)

+

Nt∑n=1

[∧el

∑G

γTG,n

(∂HG,n

∂r+∂HG,n

∂un

dundr

+∂HG,n

∂un−1

dun−1

dr+∂HG,n

∂νG,n

dνG,n

dr+

∂HG,n

∂νG,n−1

dνG,n−1

dr

)]

Note that the above equation contains both implicit ( ddr ) and explicit derivatives ( ∂

∂r ). Implicitderivatives capture implicit dependence of the state variable, u, with respect to the design vari-ables. Consequently, these terms are expensive to compute, therefore the objective of the adjointformulation is to find an adjoint solution for the free parameters, γn and λG,n, which will cause all

12

implicit terms to vanish. To avoid having to evaluate implicit terms, we set the sum of all termscontaining implicit derivatives to zero. This results in the following system of equations.

n = Nt :

( ∂fint∂uNt+ λT

Nt

∂RNt∂uNt

+∧el

∑G

γTG,Nt

∂HG,Nt∂uNt

)duNt

dr = 0

( ∂fint∂νG,Nt

+ λTNt

∂RNt∂νG,Nt

+ γTG,Nt

∂HG,Nt∂νG,Nt

)dνG,Nt

dr = 0, ∀ G

n < Nt :

[λTn∂Rn∂un

+ λTn+1

∂Rn+1

∂un+∧el

∑G

(γTG,n

∂HG,n

∂un+ γT

G,n+1∂HG,n+1

∂un)

]dundr = 0

(λTn∂Rn∂νG,n

+ λTn+1

∂Rn+1

∂νG,n+ γT

G,n∂HG,n

∂νG,n+ γT

G,n+1∂HG,n+1

∂νG,n)

dνG,n

dr = 0, ∀ G

(29)

The solution of the above adjoint equation is given in equation 30. Note that the final adjoint vector,λNt , depends on the tangent matrices and the explicit derivatives ∂fint

∂uNtand ∂fint

∂νG,Nt. Therefore, we

begin by solving for the last adjoint vector in the time sequence. Every other adjoint vector, λnwhere n < Nt, depends on the adjoint vector that follows it in the time sequence, λn+1. Therefore,we solve for the remaining adjoint vectors in reverse chronological order until all adjoint responsesare computed.

n = Nt :

λNt =

[∂RNt

∂uNt−∧el

∑G

∂RNt

∂νG,Nt

(∂HG,Nt

∂νG,Nt

)−1 ∂HG,Nt

∂uNt

]−T

·

[∧el

∑G

∂fint∂νG,Nt

(∂HG,Nt

∂νG,Nt

)−1 ∂HG,Nt

∂uNt− ∂fint∂uNt

]T

γG,Nt =−(∂HG,Nt

∂νG,Nt

)−T( ∂fint∂νG,Nt

+ λTNt

∂RNt

∂νG,Nt

)T

, ∀ G

(30)

n < Nt :

λn =

[∧el

∑G

∂Rn

∂νG,n

(∂HG,n

∂νG,n

)−1 ∂HG,n

∂un− ∂Rn

∂un

]−T

·

λTn+1

[∂Rn+1

∂un−∧el

∑G

∂Rn+1

∂νG,n

(∂HG,n

∂νG,n

)−1 ∂HG,n

∂un

]

+∧el

∑G

γTG,n+1

[∂HG,n+1

∂un− ∂Hn+1

∂νG,n

(∂HG,n

∂νG,n

)−1 ∂HG,n

∂un

]T

γG,n =−(∂HG,n

∂νG,n

)−T(λTn

∂Rn

∂νG,n+ λT

n+1

∂Rn+1

∂νG,n+ γT

G,n+1

∂HG,n+1

∂νG,n

)T

, ∀ G

Once all implicit terms have been eliminated using the above adjoint solution, the total sensitivityexpression reduces to the following.

dΠ

dr=∂fint∂r

+

Nt∑n=1

(λTn

∂Rn

∂r+∧el

∑G

γTG,n

∂HG,n

∂r

)

+ λT1

(∂R1

∂u0

du0

dr+∧el

∑G

∂R1

∂νG,0

dνG,0

dr

)+∧el

∑G

γTG,1

(∂H1

∂u0

du0

dr+

∂H1

∂νG,0

dνG,0

dr

) (31)

2.4.2. Analytical Solution of the Adjoint Vectors

To calculate the adjoint vectors λn and γn, eight tangent matrices, ∂Rn/∂un, ∂Rn/∂νn,∂Hn/∂un, ∂Hn/∂νn, ∂Rn+1/∂un, ∂Rn+1/∂νn, ∂Hn+1/∂un, and ∂Hn+1/∂νn in equation 30

13

need to be specified. In the context of the phase transformation, the aforementioned matricesshould be considered carefully to account for the different cases since temperature and loadingconditions will determine whether the SMAs behave elastically or inelastically. If the materialexhibits inelastic behavior in the current step, the local residual H should refer to the case whereξ 6= 0 in equation 15, and the form of the corresponding matrices should be as follows.

∂Rn

∂un=

∂Rcn∂F cn

∂Rcn∂dcn

∂Rfn∂F fn

∂Rfn∂dfn

=

[−Icc 0

0 0

]∂Rel,n

∂νn=[

0 0 0 wBTdetJ]

∂Hn

∂uel,n=

0 00 00 0

06×c S−1n : Bf

(32)

∂Hn

∂νn=

∂Hn

Φ∂ξn

∂HnΦ

∂εtn

∂HnΦ

∂Sn

∂HnΦ

∂σn∂Hn

εt

∂ξn

∂Hnεt

∂εtn

∂Hnεt

∂Sn

∂Hnεt

∂σn∂Hn

S∂ξn

∂HnS

∂εtn

∂HnS

∂Sn

∂HnS

∂σn∂Hn

σ∂ξn

∂Hnσ

∂εtn

∂Hnσ

∂Sn

∂Hnσ

∂σn

=

∂ξΦn 0 0 ∂σnΦT

n

Λn −I6×6 0 0∆S 0 −I1×1 00 −S−1

n −S−1n : σn −I6×6

Here, the subscripts f and c refers to entries within the vector that correspond to free and con-strained degrees of freedom, respectively. If the local material is still in an elastic state (ξ = 0),the derivative of residual for the current step with respect to the current state variables will havea much simpler form.

∂Rn

∂un=

[−Icc 0

0 0

]∂Rel,n

∂νn=[wBTdetJ

]∂Hn

∂uel,n=[

06×c S−1n : Bf

](33)

∂Hn

∂νn=[

∂Hn∂σn

]=[−I6×6

]Next, we discuss the tangent matrices of the form ∂Rn+1

∂un. There are four possible cases, each of

which is shown below.

∂Rn+1

∂un= 0

∂Rn+1

∂νn= 0

14

∂Hn+1

∂uel,n=

[

08×c 08×f06×c −S−1

n : Bf

]ξn = 0, ξn+1 = 0 or ξn 6= 0, ξn+1 = 0[

014×c −S−1n : Bf

]ξn = 0, ξn+1 6= 0 or ξn 6= 0, ξn+1 6= 0

(34)

∂Hn+1

∂νn=

[I6×6

]ξn = 0, ξn+1 = 0[

08×6

I6×6

]ξn = 0, ξn+1 6= 0[

0 I6×6

]ξn 6= 0, ξn+1 = 0

0 0 0 0−Λn+1 I6×6 0 0−∆S 0 I1×1 0

0 S−1n S−1

n : σn I6×6

ξn 6= 0, ξn+1 6= 0

Computing the various matrices used to solve for the adjoint vectors is computationally ex-pensive. In addition, computing the inverse of the matrix ∂Hn/∂νn can be problematic, sincethe compliance matrix S and the compliance modulus S are dependent on the Martensite volumefraction, which varies during phase transformation. Furthermore, the large magnitude of the off-diagonal entries in ∂Hn/∂νn will make the matrix ill-conditioned. Thus the accuracy of the inverseoperation is not guaranteed. To resolve this issue, we introduce a nested Schur complement proce-dure that provides an alternate path to analytically evaluating the matrix quantities presented inequation 35. Below, we present the analytical solution of the adjoint vectors for the case when thematerial behaves purely inelastically during the current and following pseudo-time step. A detaileddescription of the derivation is provided in the appendix. Beginning with the adjoint equationsgiven in 30, we can express the matrices that define the linear systems being solved in terms ofknown element properties as follows.

∂Rn

∂un−∧el

∑G

∂Rn

∂νG,n

(∂HG,n

∂νG,n

)−1 ∂HG,n

∂uel,n=

[−Icc Kcf

0cf Kff

]∂Rel,n

∂νn

(∂Hn

∂νn

)−1

= wBT[−S

−1n :∂σΦnQ Ln Ln : C−1 : σn −Ln : Sn

]detJ

(∂Hn

∂νn

)−1 ∂Hn

∂uel,n=

∂σΦn

T:S−1n

Q Λn

∆S

:∂σΦnT:S−1

n

Q

−Ln

B (35)

∂Hn+1

∂νn

(∂Hn

∂νn

)−1

=

[A8×8 B8×6

06×8 −I6×6

]

∂Hn+1

∂νn

(∂Hn

∂νn

)−1 ∂Hn

∂uel,n=

01×6(

Λn−Λn+1

Q : ∂σΦnT : S−1

n

)01×6

−S−1n

B

15

where

A =

01×1 01×7Λn−Λn+1

Q

01×1−I7×7 − Λn−Λn+1

Q : ∂σΦnT :

[S−1n Sn

−1 : σn01×6 01×1

] B =

01×6Λn−Λn+1

Q : ∂σΦnT

01×6

(36)

Again, we provide a detailed description of the implementation procedure for calculating tran-sient adjoint vectors and evaluating the resulting sensitivities in table 3.

I. Calculate adjoint vectors

Case A. n = Nt

1. Let n = Nt

2. Global Level

1) Let −∧el

∑G

∂RNt∂νG,Nt

(∂HG,Nt∂νG,Nt

)−1 ∂HG,Nt∂uel,Nt

= KNt , calculate ∂fint∂uNt

2) Loop over elements, calculate ∂fint∂νNt

(∂HNt∂νNt

)−1 ∂HNt∂uNt

, ∂fint∂νNt

(∂HNt∂νNt

)−1,∂RNt∂νNt

(∂HNt∂νNt

)−1

3) Assemble to get global ∂fint∂νNt

(∂HNt∂νNt

)−1 ∂HNt∂uNt

4) Calculate λNt3. Local Level

1) Loop over elements, calculate γNt for each Gauss point2) Let n = n− 1, go to Case B.

Case B. n < Nt

1. Global Level

1) Let −∧el

∑G

∂Rn∂νG,n

(∂HG,n

∂νG,n

)−1 ∂HG,n

∂uel,n= Kn, calculate ∂Rn+1

∂un

2) Loop over elements, calculate ∂Rn+1

∂νn

(∂Hn∂νn

)−1∂Hn∂un

, ∂Hn+1

∂un, ∂Hn+1

∂νn

(∂Hn∂νn

)−1∂Hn∂un

∂Rn∂νn

(∂Hn∂νn

)−1, ∂Rn+1

∂νn

(∂Hn∂νn

)−1, ∂Hn+1

∂νn

(∂Hn∂νn

)−1

3) Assemble ∂Rn+1

∂νn

(∂Hn∂νn

)−1∂Hn∂un

, ∂Hn+1

∂un,∂Hn+1

∂νn

(∂Hn∂νn

)−1∂Hn∂un

4) Calculate λn2. Local Level

1) Loop over elements, calculate γn for each Gauss point2) Check termination condition

If n > 0Let n = n− 1, go back to Case B.

elsego to step II.

II. Calculate sensitivity

Calculate sensitivities according to equation 31.

Table 3: Implementation procedure of adjoint sensitivity analysis of SMAs

Based on our numerical results, finite element analysis for one psuedotime step of a 2D mesh with2000 quadrilateral elements takes approximately 4 seconds to converge, when running Matlab on aplatform that uses an intel i5 cpu and contains 8GB memory. This computation time scales linearly

16

with the number of pseudo time steps. The computational cost of the adjoint sensitivity analysistakes approximately half of the time required by the finite element analysis. The lower computationtime is due to the fact that we store the state variables for each time step, and therefore, there isno need to perform an iterative Newton-Raphson procedure during the sensitivity analysis.

3. Example Problems

3.1. Optimal Design for Two-way Shape Memory Effects

The topology optimization problem described in figure 4 is used to investigate optimal tailoringof the TWSME. We maximize the tip displacement in the x-direction at the center of the right edgeof the design domain, where a small input force is applied. Note that the displacement is triggeredby a combination of elastic deflection and phase transformation due to cooling the structure. Thegeometry and boundary conditions are shown in figure 4, and the problem statement is given below.

min fobj = −LTd

s.t LTd ≤ 3× 10−4m

V ≤ 0.3

Eeffn = rpEn, dT eff

n = rpdT

(37)

Here, the vector L contains a value of unity at the degree of freedom where the displacementresponse is to be measured, and zeros at all other entries. To avoid an overly compliant design, weseparately enforce a constraint on the instantaneous displacement, d, which is computed at timet = 0 when the material exhibits a purely thermoelastic response.

Figure 4: Boundary Condition for TWSMEs optimization

An additional constraint is implemented to limit the volume of the optimized structure to 30%of the design domain. The start temperature of simulation is 270K and the end temperature is setto 240K, while the reference temperature T0 is chosen to be 290K. Note that a nominal constantforce, F = 5× 106N, is applied in order to ensure the presence of stress within the material. Sinceour constitutive model relies on the relationship between the transformation function and the stressstate (see Eqn. 7 ) we require some requisite stress level in order to accurately model the TWSME.The penalization factor p is chosen to be 3, and the temperature increment is dT = −0.1K per step.The optimized result of a mesh of 4096 quadrilateral elements and the corresponding convergenceplot are shown in figure 5.

17

(a) Optimized structure

0 20 40 60 80 100

Iteration number

10-6

10-4

10-2

100

102

KK

T-n

orm

-10

-9

-8

-7

-6

-5

-4

-3

f obj

10-3

(b) Convergence history

Figure 5: Optimization result of TWSMEs case

From the combined results presented in figure 5b and figure 7, we see that the optimizationis convergent and that it successfully produces the desired displacement response. Figure 6 showsthe shape of the optimized structure at different points in the temperature cycle. We also presenta comparison of displacement contributions caused by phase transformation along with the dis-placement due to thermoelastic effects. In figure 7, we simulate the optimized structure for anextended thermomechanical cycle, that goes beyond the temperature range used in the optimiza-tion algorithm. From the results, we see that cooling the structure leads to a negative thermoelasticdisplacement, while the expansion of the structure is due to the phase transformation.

0 0.2 0.4 0.6 0.8 1

x(m)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y(m

)

Base Line

220K

200K

Figure 6: Deformation of the structure at differ-ent temperature states

190 200 210 220 230 240 250 260 270 280

Temperature(K)

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Tip

Dis

pla

cem

ent(

m)

TotalThermal-elasticTransformation

Figure 7: Tip displacement due to thermoelas-tic and phase transformation effects

In this case, since the extent of the phase transformation is proportional to the temperaturechange, we deduce that maximizing displacement shortly after the onset of the phase transformation(at T ≈ 240K) will lead to an increased displacement after the transformation is complete. Thisapproach improves computational efficiency since modeling thermal conduction is not required, nordo we need to simulate the SMA structure for a whole transformation cycle.

3.2. Optimal Design for Combined TWSMEs and Pseudoelasticity

To investigate the ways in which the interaction between TWSMEs and pseudoelasticity canbe optimally leveraged in mechanical design, we present an example problem involving design

18

optimization of a force inverter. The geometry and boundary conditions for the problem are shownin figure 8. The mechanism is subjected to a cooling cycle from T = 290K to T =250K, withdT = −0.1K. During each pseudotime step, the applied load F in is also increased incrementallyfrom 0N to a maximum of 20 × 106N, using a constant increment. The optimization problemstatement is shown below.

min fobj = −F out/F in,

s.t din < 2× 10−3m

V ≤ 0.3

Eeffn = rpEn, dT eff

n = rpdT

(38)

Figure 8: Boundary Condition for pseudoelasticity optimization

In the optimization problem statement, the penalization constant is chosen as p = 3 in thesimulation. To avoid an overly compliant design, we enforce a displacement constraint on din atthe degree of freedom corresponding to the input force location at time t = 0, as well as a volumeconstraint. Again, we use a mesh of 4096 quadrilateral elements. The optimization results areshown in figure 9.

(a) Optimized structure

0 50 100 150 200

Iteration number

10-4

10-3

10-2

10-1

100

KK

T-n

orm

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2f o

bj

(b) Convergence rate

Figure 9: Optimization result of pseudoelasticity case

19

0 0.5 1 1.5 2 2.5 3 3.5 4

Fin

(N) 107

-0.5

0

0.5

1

1.5

2

2.5

Fout(N

)

107

Total

Thermal-elastic

Transformation

Figure 10: Contribution of different thermomechanical behaviors to output force

As can be seen from the figures, the optimization produces fully converged designs that exhibitthe desired mechanical response. We also present a portion of output force in regards to differentthermomechanical behaviors in figure 10. To calculate the output force due to transformation, weconsider the following relationship

σe = S−1 : εtotal − S−1 : α(T − T0)− S−1 : εt (39)

σe = σtotal − σthermal − σt

= σthermoelastic − σt(40)

where σe and σt refer to the elastic stress tensor and transformation stress tensor, respec-tively. Since the calculation of the global output force follows Gauss quadrature rules with constantweights, the output force due to thermoelastic effects can be calculated according to equation 39using the history of the effective Young’s modulus. We then calculate the output force due totransformation using the superposition relationship shown in equation 40.

In figure 11, we provide an illustrative sketch of the stress-strain relation of a typical SMA (NiTi)versus that of steel. Generally, SMAs have a higher yielding strain than other metals. For example,the yield strain of steel is around 0.001 to 0.002 with a yield strength from 100MPa∼400MPa [40],while the yield point of NiTi is beyond its maximum transformation strain (0.03 to 0.08) with ayield strength in the range 150MPa∼700MPa [41]. Hence, the SMA mechanism will be able towithstand larger input forces while accommodating larger deformations than similar metals thatdo not exhibit pseudoelasticity.

It can be observed that the output force is significantly augmented by the superelastic behaviorof the SMA material. This is indicative of the fact that the TWSME and superelastic effectcontribute to a significant increase in internal stress, and therefore they can enhance mechanicaladvantage. Furthermore, due to the unique hysteretic stress-strain behavior of SMAs, the forceinverter made from SMAs will also be able to function within a wider range of input forces.

4. Conclusions

We present a novel framework for topology optimization of structures and mechanisms con-taining shape memory alloys. The proposed approach uses a phenomenological constitutive model,which is able to accurately model the intrinsic latent heat exchange process and its accompanying

20

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Strain

0

1

2

3

4

5

6

7

Str

ess

(Pa)

108

NiTi

Steel

Figure 11: Schematic illustration of yield point of SMA and steel

phase transformation. Based on the model, we implement nonlinear finite element analysis and de-rive a path-dependent adjoint sensitivity formulation for gradient-based optimization. Analyticalsolutions of the tangent matrices used in the sensitivity calculation are also provided. The solutionsare shown to overcome the ill-conditioned nature of the various linear systems encountered duringthe numerical calculation of sensitivities. Finally, we present several examples of topology optimiza-tion problems in which we tailor the nonlinear thermomechanical response of the SMA materialto achieve enhanced mechanical performance. The results show that the optimized structures areconverged, and they exhibit the desired mechanical response. These results indicate the power ofthe proposed framework to address the various design challenges encountered in optimization ofSMA-based structures.

5. Acknowledgements

This research was supported by the National Science Foundation through grant number CMMI-1663566.

Appendix

Analytical Derivation of Adjoint Vector

In this appendix, we present the derivation of an analytical formulation for computing thetangent matrices described in equation 35. The motivation behind deriving these formuli is toavoid directly factorizing the ill-conditioned matrix ∂Hn/∂νn. For conciseness, we focus on thecase where the behavior of the SMA in the current pseudo-time step and next step is inelastic. Tostart, we first represent the inverse of ∂Hn/∂νn using the Schur complement [42].(

∂Hn

∂νn

)−1

=

[(A + BC)−1 (A + BC)−1BC(A + BC)−1 −I + C(A + BC)−1B

](41)

with

∂Hn

∂νn=

∂ξΦn 0 0Λn −I6×6 0∆S 0 −I1×1

∂σnΦ′n

00

0 − S−1n − S−1

n : σn −I6×6

=

[A BC D

](42)

21

Note that the analytical solution in which we directly factorize ∂Hn/∂νn is not recommended forcalculating the sensitivities. This analytical solution will not change the ill-conditioned character-istics of the matrix, since (A + BC) is nearly singular due to the fact that the large number S−1

still appears on the lower triangle. Similarly, ∂Hn+1/∂νn can be defined in block matrix form as

∂Hn+1

∂νn=

0 0 0

−Λn+1 I6×6 0−∆S 0 I1×1

000

0 S−1n S−1

n : σn I6×6

=

[A BC D

](43)

The five aforementioned matrices then can be represented in a new form shown below.

−∧el

∑G

∂Rn

∂νG,n

(∂Hn

∂νG,n

)−1 ∂HG,n

∂uel,n=∧el

∑G

wBTG[I − C(A + BC)−1B]GS

−1G,nBGdetJG

∂Rel,n

∂νn

(∂Hn

∂νn

)−1

= wBT[C(A + BC)−1 −I + C(A + BC)−1B

]detJ(

∂Hn

∂νn

)−1 ∂Hn

∂uel,n=

[(A + BC)−1B

−I + C(A + BC)−1B

]B (44)

∂Hn+1

∂νn

(∂Hn

∂νn

)−1

=

[A(A + BC)−1 A(A + BC)−1B

(C + C)(A + BC)−1 −I + (C + C)(A + BC)−1B

]∂Hn+1

∂νn

(∂Hn

∂νn

)−1 ∂Hn

∂uel,n=[A(A + BC)−1B− I

]S−1n B

It can be observed that the key to obtaining an accurate solution of the tangent matrices lies insolving the inverse of (A + BC). Here we use the Schur formulation again to calculate the inverseof the matrix.

(A + BC)−1 =

[(A′ + B′C′)−1 (A′ + B′C′)−1B′

C′(A′ + B′C′)−1 −I + C′(A + B′C′)−1B′

](45)

with

A + BC =

∂ξΦn −∂σΦn : S−1n −∂σΦn : S−1

n : σnΛn

∆S−I6×6 06×1

01×6 −I1×1

=

[A′ B′C′ D′

](46)

This time, A′ + B′C′ turns out to be a scalar, with a value of Q = −∂σΦn : S−1n : ∂σΦn + ∂ξΦn.

Hence the issue of the ill-conditioned matrix has been solved, and we are able to accurately evaluatethe five matrices as follows.

−∧el

∑G

∂Rn

∂νG,n

(∂HG,n

∂νG,n

)−1 ∂HG,n

∂uel,n=∧el

∑G

wBTG(S−1

G,n −S−1

G,n : ∂σΦG,n ⊗ S−1G,n : ∂σΦG,n

∂σΦG,n : S−1G,n : ∂σΦG,n − ∂ξΦG,n

)BGdetJG

=∧el

∑G

wBTGLG,nBGdetJG

∂Rel,n

∂νn

(∂Hn

∂νn

)−1

= wBT[S−1n :∂σΦnQ Ln Ln : C−1 : σn −Ln : Sn

]detJ

(∂Hn

∂νn

)−1 ∂Hn

∂uel,n=

∂σΦn

T:S−1n

Q Λn

∆S

:∂σΦnT:S−1

n

Q

−Ln

B (47)

22

∂Hn+1

∂νn

(∂Hn

∂νn

)−1

=

[A8×8 B8×6

06×8 −I6×6

]

∂Hn+1

∂νn

(∂Hn

∂νn

)−1 ∂Hn

∂uel,n=

01×6(

Λn−Λn+1

Q : ∂σΦnT : S−1

n

)01×6

−S−1n

Bwhere

A =

01×1 01×7Λn−Λn+1

Q

01×1−I7×7 − Λn−Λn+1

Q : ∂σΦnT :

[S−1n S−1

n : σn01×6 01×1

] B =

01×6Λn−Λn+1

Q : ∂σΦnT

01×6

(48)

The first equation in 47 can be recognized as the formula for the tangent stiffness tensor asderived by Lagoudas [32], and referred to in equation 20. Therefore, in addition to improvingthe accuracy of the sensitivity evaluation, we are able to conserve computational resources by re-using the tangent stiffness matrix already computed during the final step of the Newton-Raphsonprocedure. Note that the above derivation applies to cases where the transformation tensor Λ doesnot explicitly depend on the stress tensor σ (e.g. the case where ξ < 0 in equation 7), and thereforethe partial derivative ∂σΛ vanishes. For the cases where ∂σΛ is non-zero, one can obtain the similarresult using a similar Schur complement-based procedure, which we omit here for conciseness.

10-2

10-1

100

101

E/E*

10-4

10-3

10-2

10-1

100

101

max

|K-K

dir

|/m

ax|

K|

Figure 12: Accuracy of the tangent stiffness matrix obtained by directly factorizing ∂Hn/∂νn

Using the aforementioned Schur complement-based solution, we plot the error in the tan-gent stiffness matrix obtained via the naıve method of direct factorization (i.e. Kdir =−∧

el

∑G(∂Rn/∂νG,n)(∂HG,n/∂νG,n)−1(∂HG,n/∂uel,n)). The direct factorization is implemented

using the built-in Matlab function “\”. The reference value in the error calculation is taken as thetangent stiffness matrix computed using the analytical formula given in equation 20. The resultingerror is plotted as a function of varying Young’s modulus values in a 2D SMA problem (see figure12). Note that the function “\” uses LU decomposition to solve the linear system via double-precision floating-point arithmetic. The plotted error values are based on the psuedo-time step

23

with the largest error. The plot shown in Figure 12 reveals that the error in the tangent stiffnessmatrix K grows as the Young’s modulus increases, which in turn will introduce inaccuracies intothe sensitivity calculation.

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