## Abstract

We present a novel optimization framework for optimal design of structures exhibiting memory characteristics by incorporating shape memory polymers (SMPs). SMPs are a class of memory materials capable of undergoing and recovering applied deformations. A finite-element analysis incorporating the additive decomposition of small strain is implemented to analyze and predict temperature-dependent memory characteristics of SMPs. The finite element method consists of a viscoelastic material modelling combined with a temperature-dependent strain storage mechanism, giving SMPs their characteristic property. The thermo-mechanical characteristics of SMPs are exploited to actuate structural deflection to enable morphing toward a target shape. A time-dependent adjoint sensitivity formulation implemented through a recursive algorithm is used to calculate the gradients required for the topology optimization algorithm. Multimaterial topology optimization combined with the thermo-mechanical programming cycle is used to optimally distribute the active and passive SMP materials within the design domain. This allows us to tailor the response of the structures to design them with specific target displacements, by exploiting the difference in the glass-transition temperatures of the two SMP materials. Forward analysis and sensitivity calculations are combined in a PETSc-based optimization framework to enable efficient multi-functional, multimaterial structural design with controlled deformations.

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## Notes

- 1.
Note that implicit derivatives, \(\frac {d*}{d\boldsymbol {\rho }}\), capture implicit dependence of a function or state variable with respect to

*ρ*due to the solution of the residual, whereas explicit derivatives capture only direct dependence. Consequently, implicit derivatives are more expensive to evaluate, and therefore we seek to eliminate them from the sensitivity calculation

## References

Aage N, Andreassen E, Lazarov BS (2015) Topology optimization using PETSc: an easy-to-use, fully parallel, open source topology optimization framework. Struct Multidiscip Optim 51(3):565–572

Baghani M, Naghdabadi R, Arghavani J, Sohrabpour S (2012) A thermodynamically-consistent 3D constitutive model for shape memory polymers. Int J Plast 35:13–30. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0749641912000083

Behl M, Lendlein A (2011) Shape-memory polymers, pp. 1–16. American Cancer Society, [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/0471238961.1908011612051404.a01.pub2

Bowen CR, Kim HA, Weaver PM, Dunn S (2014) Piezoelectric and ferroelectric materials and structures for energy harvesting applications. Energy Environ Sci 7:25–44. [Online]. Available: https://doi.org/10.1039/C3EE42454E

Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654. [Online]. Available: https://doi.org/10.1007/s004190050248

Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26):3443–3459. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0045782500002784

Carbonari RC, Silva ECN, Paulino GH (2008) Topology optimization applied to the design of functionally graded piezoelectric bimorph. AIP Conf Proc 1:291–296. [Online]. Available: https://aip.scitation.org/doi/abs/10.1063/1.2896793

Carrell J, Tate D, Wang S, Zhang H-C (2011) Shape memory polymer snap-fits for active disassembly. J Clean Prod 19(17):2066–2074. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0959652611002393

Chen Y-C, Lagoudas DC (2008a) A constitutive theory for shape memory polymers. Part I: large deformations. J Mech Phys Solids 56(5):1752–1765. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0022509607002323

Chen Y-C, Lagoudas DC (2008b) A constitutive theory for shape memory polymers. Part II: a linearized model for small deformations. J Mech Phys Solids 56(5):1766–1778. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0022509607002311

Cho J-H, Azam A, Gracias DH (2010) Three dimensional nanofabrication using surface forces. Langmuir 26(21):16534–16539. pMID: 20507147. [Online]. Available: https://doi.org/10.1021/la1013889

Choi M-J, Cho S (2018) Isogeometric configuration design optimization of shape memory polymer curved beam structures for extremal negative Poisson’s ratio. Struct Multidiscip Optim 58(5):1861–1883. [Online]. Available: https://doi.org/10.1007/s00158-018-2088-y

Frecker MI (2003) Recent advances in optimization of smart structures and actuators. J Intell Mater Syst Struct 14(4–5):207–216. [Online]. Available: https://doi.org/10.1177/1045389X03031062

Gaynor AT, Meisel NA, Williams CB, Guest JK (2014) Multiple-material topology optimization of compliant mechanisms created via polyjet three-dimensional printing. J Manuf Sci Eng 136(6):061015–061015–10. [Online]. Available: https://doi.org/10.1115/1.4028439

Ge Q, Dunn CK, Qi HJ, Dunn ML (2014) Active origami by 4D printing. Smart Mater Struct 23(9):094007

Ge Q, Sakhaei AH, Lee H, Dunn CK, Fang NX, Dunn ML (2016) Multimaterial 4D printing with tailorable shape memory polymers. Sci Rep 6:31110 EP. [Online]. Available: https://doi.org/10.1038/srep31110

Geiss MJ, Maute K (2018) Topology optimization of active structures using a higher-order level-set-XFEM-density approach. In: 2018 Multidisciplinary analysis and optimization conference, p 4053

Geiss MJ, Boddeti N, Weeger O, Maute K, Dunn ML (2019) Combined level-set-XFEM-density topology optimization of four-dimensional printed structures undergoing large deformation. J Mech Des 5:141

James KA, Waisman H (2015) Topology optimization of viscoelastic structures using a time-dependent adjoint method. Comput Methods Appl Mech Eng 285:166–187. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S004578251400437X

Langelaar M, van Keulen F (2008) Modeling of shape memory alloy shells for design optimization. Comput Struct 86(9):955–963

Langelaar M, Yoon G H, Kim Y, Van Keulen F (2011) Topology optimization of planar shape memory alloy thermal actuators using element connectivity parameterization. Int J Numer Methods Eng 88 (9):817–840

Lendlein A, Jiang H, Jünger O, Langer R (2005) Light-induced shape-memory polymers. Nature 434(7035):879–882. [Online]. Available: https://doi.org/10.1038/nature03496

Leng J, Yu K, Liu Y (2010) Recent progress of smart composite material in HIT. [Online]. Available: https://doi.org/10.1117/12.854803

Li G, Nettles D (2010) Thermomechanical characterization of a shape memory polymer based self-repairing syntactic foam. Polymer 51(3):755–762. http://www.sciencedirect.com/science/article/pii/S0032386109010672

Liu Y, Gall K, Dunn ML, Greenberg AR, Diani J (2006) Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. Int J Plast 22(2):279–313. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0749641905000604

Liu Y, Du H, Liu L, Leng J (2014) Shape memory polymers and their composites in aerospace applications: a review. Smart Mater Struct 23(2):023001. [Online]. Available: https://doi.org/10.1088/0964-1726/23/2/023001

Maute K, Tkachuk A, Wu J, Jerry Qi H, Ding Z, Dunn ML (1402) Level set topology optimization of printed active composites. J Mech Des 2015(11):10. [Online]. Available: https://doi.org/10.1115/1.4030994

Oliver K, Seddon A, Trask RS (2016) Morphing in nature and beyond: a review of natural and synthetic shape-changing materials and mechanisms. J Mater Sci 51(24):10663–10689. [Online]. Available: https://doi.org/10.1007/s10853-016-0295-8

Qi HJ, Nguyen TD, Castro F, Yakacki CM, Shandas R (2008) Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers. J Mech Phys Solids 56(5):1730–1751. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S002250960700230X

Reed JL Jr, Hemmelgarn CD, Pelley BM, Havens E (2005) Adaptive wing structures. [Online]. Available: https://doi.org/10.1117/12.599922https://doi.org/10.1117/12.599922

Reese S, Böl M, Christ D (2010) Finite element-based multi-phase modelling of shape memory polymer stents. Comput Methods Appl Mech Eng 199(21):1276–1286. Multiscale models and mathematical aspects in solid and fluid mechanics. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S004578250900262X

Rupp CJ, Evgrafov A, Maute K, Dunn ML (2009) Design of piezoelectric energy harvesting systems: a topology optimization approach based on multilayer plates and shells. J Intell Mater Syst Struct 20 (16):1923–1939. [Online]. Available: https://doi.org/10.1177/1045389X09341200

Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.1620240207

Siéfert E, Reyssat E, Bico J, Roman B (2019) Bio-inspired pneumatic shape-morphing elastomers. Nat Mater 18(1):24–28. [Online]. Available: https://doi.org/10.1038/s41563-018-0219-x

Sigmund O, Torquato S (1999) Design of smart composite materials using topology optimization. Smart Mater Struct 8(3):365–379. [Online]. Available: https://doi.org/10.1088/0964-1726/8/3/308

Silva ECN, Kikuchi N (1999) Design of piezoelectric transducers using topology optimization. Smart Mater Struct 8(3):350–364. [Online]. Available: https://doi.org/10.1088/0964-1726/8/3/307

Tibbits S (2014) 4D printing: multi-material shape change. Archit Des 84(1):116–121. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/ad.1710

Volk BL, Lagoudas DC, Chen Y-C, Whitley KS (2010) Analysis of the finite deformation response of shape memory polymers: I. Thermomechanical characterization. Smart Mater Struct 19(7):075005

Volk BL, Lagoudas DC, Maitland DJ (2011) Characterizing and modeling the free recovery and constrained recovery behavior of a polyurethane shape memory polymer. Smart Mater Struct 20(9):094004. [Online]. Available: https://doi.org/10.1088/0964-1726/20/9/094004

Wache HM, Tartakowska DJ, Hentrich A, Wagner MH (2003) Development of a polymer stent with shape memory effect as a drug delivery system. J Mater Sci: Mater Med 14(2):109–112. [Online]. Available: https://doi.org/10.1023/A:1022007510352

Yin L, Ananthasuresh G (2002) A novel topology design scheme for the multi-physics problems of electro-thermally actuated compliant micromechanisms. Sens Actuat A: Phys 97–98:599–609. Selected papers from Eurosenors XV. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0924424701008536

Yu K, Yin W, Sun S, Liu Y, Leng J (2009) Design and analysis of morphing wing based on SMP composite. [Online]. Available: https://doi.org/10.1117/12.815712

## Funding

The authors received financial support of the National Science Foundation under grant # CMMI-1663566.

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Detailed descriptions of the algorithms used to generate all results are provided throughout the paper. Additionally, we have included all relevant material properties, and all algorithm parameters. Copies of the code used to generate the results will be made available upon request.

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## Appendices

### Appendix 1:: Finite element derivations

The subscript, *n*, represents the time step.

The terms \(\mathbb {A}_r\), \(\mathbb {A}_{g}\), \(\mathbb {H}_r\), \(\mathbb {H}_{g}\), \(\mathbb {B}_r\), \(\mathbb {B}_{g}\), and \({\Delta }\phi ^{g}_{n+1}\) required in (A.2) are computed as:

The terms \(\mathbb {W}_r\), \(\mathbb {W}_{g}\), \(\mathbb {M}\), \(\mathbb {N}\), \(\mathbb {P}\), \(\mathbb {O}\), \(\mathbb {E}\), and \(\mathbb {F}\) for (9) are defined as:

Here, \(\mathbb {I}\) is the fourth-order identity tensor given by:

Here, *δ*_{ij} is the Kronecker delta. Isotropic linear elastic constitutive law is utilized to compute the fourth-order elasticity tensors \(\mathbb {K}^r_{eq}\) and \(\mathbb {K}^r_{neq}\) corresponding to the rubbery-phase and \(\mathbb {K}^{g}_{eq}\) and \(\mathbb {K}^{g}_{neq}\) for the glassy-phase material.

### Appendix 2: Derivation of sensitivity analysis

Having discussed the generalized formulation for time-dependent adjoint sensitivity analysis in Section 5, we focus on deriving the sensitivity formulation specifically for shape memory polymers. To avoid confusion in the notation representing inelastic strain components and time steps, from here on the current time step will be denoted by subscript {*n* + 1}, the previous time step will be denoted by subscript {*n*}, and so on.

The sensitivity of the objective function is calculated via (25). This equation has two components: the first is the *adjoint* vectors (** λ**) and the other is the component capturing the explicit dependence of the residual term on the design variable. The

*adjoint*vectors are computed via (24). Evaluation of both of these components requires the residual term (

**). The residual equation for the SMP can be stated as:**

*R*where the terms \(\mathbb {X}^{(r)}_{n+1} \), \(\mathbb {X}^{(g)}_{n+1} \), \(\mathbb {Y}^{(r)}_{n+1} \), \(\mathbb {V}^{(r)}_{n+1} \), \(\mathbb {Z}^{(r)}_{n+1} \) are given by:

The differentiation of the residual equation, *R*_{n+ 1}, with respect to the design variables can be computed by:

To evaluate the *adjoint* vectors, it is required to capture the explicit dependence of the residual for the *k*^{th} time step on the displacement of the *i*^{th} time step, i.e., \(\frac {\partial \boldsymbol {R}_{k}}{\partial \boldsymbol {u}_{i}}\). These terms are referred to as the “coupling” terms. Finding the \(\frac {\partial \boldsymbol {R}_{k}}{\partial \boldsymbol {u}_{i}}\) terms are more involved since at each time step there is an exponential growth of terms from the previous time step. For example, let us evaluate the term \(\frac {\partial \boldsymbol {R}_{n+1}}{\partial \boldsymbol {u}_{n-1}}\). The coupling term \(\frac {\partial \boldsymbol {R}_{n+1}}{\partial \boldsymbol {u}_{n-1}}\) is proportional to \(\frac {\partial \boldsymbol {R}_{n+1}}{\partial \boldsymbol {\varepsilon }_{n-1}}\), since strain is a linear function of displacement (*u*). We can use the chain rule to write:

Equation (B.4) gets contributions from *term I* and *term II*. The parameter *R*_{n+ 1} which represents the residual, obtained during the forward analysis, is given by (B.1) which has seven terms. Each of the terms, at a particular time step, is dependent not only on the current time step of the evaluation but also on the previous time step as shown in (9). For example, if we calculate the coupling coefficients from the second term, \({\int \limits }_{\Omega }^{} \boldsymbol {B}^T\mathbb {X}^{(r)}_{n+1}\boldsymbol {\varepsilon }^{(ir)}_{n} dv\), of the residual equation, and track the evolution of the term in time, we will get the map as shown in Fig. 24. The coefficient *C*_{f} is defined as:

The terms \(\mathbb {A}_n\) and \(\mathbb {B}_n\) are given by:

If we collect the terms to evaluate \(\frac {\partial \boldsymbol {\varepsilon }^{(ir)}_{n}}{\partial \boldsymbol {\varepsilon }^{(r)}_{n-1}}\), we get:

Equation (B.5) represents *term I* in terms of \(\varepsilon ^{(ir)}_n\). A similar procedure is adopted for all the other six terms present in the (B.1) to make a total of twenty-three terms for the coupling term \(\frac {\partial \boldsymbol {R}_{n+1}}{\partial \boldsymbol {u}_{n-1}}\). The computation of *term II* is straightforward and is given by:

Capturing the evolution of all the components required to accurately calculate the sensitivities makes this process computationally expensive and a highly time-consuming procedure. The time taken increases exponentially with the total number of time steps required to simulate the thermo-mechanical cycle of the SMP increases. The function and the recursive algorithm used to compute the {\(\frac {\partial R_k}{\partial u_i}\)} terms for the total sensitivity analysis are shown in Algorithms 2 and 3. Note that for the recursive algorithm shown in Algorithm 3, parameters *k* and *i* represent the time steps. Here, the functions func_eir, func_eig, func_is, and func_i are programmable versions of *ε*^{(ir)},*ε*^{(ig)},*ε*^{(is)}, and *ε*^{(i)}, shown in (9), implemented for the *k*^{th} step. The variable [*M*] is a collection of parameters representing the intrinsic material properties. The function *f* represents a general function manipulating its inputs and giving a desired output.

The individual functions have similar structures and one such function func_eir has been shown in details in Algorithm 3.

To verify the implementation of the sensitivity analysis, the design domain shown in Fig. 5 is discretized with a coarse mesh of 45 elements. The structure is initialized with a uniform distribution of design variable ** ρ** = 0.3. It was then subjected to an axial stretching load

**= 0.025**

*F**N*during the cooling phase of the thermo-mechanical cycle. The load was removed during the relaxation and heating phases of the thermo-mechanical programming cycle. The function of interest is the tip displacement \(\boldsymbol {u}^M_a\) as shown in (19). In this case, the parameter

*a*is the

*y*− degree-of-freedom of the node shown in Fig. 5 and

*M*is the time step at the end of the step III of the thermo-mechanical programming cycle. The material parameters used for this analysis are same as listed in Table 1. The adjoint method and the forward difference method were used to evaluate the derivative of the tip displacement with respect to the mixing ratio of each element. Figure 25 shows the normalized error of the sensitivity values obtained by the finite-difference approach and the adjoint sensitivity analysis. The normalized error (NE) for each element is evaluated as:

Note that for elements where the sensitivity is at or near zero, we have omitted the normalized error to avoid the indication of an artificially high error due to an extremely small denominator. The displacement obtained at the end of step III was − 0.0130 mm. The sensitivity values obtained through the adjoint formulation and the finite-difference method are tabulated in Table 2. The maximum error between these values was found to be 2.6 × 10^{− 7}. This established that the framework developed can successfully compute the sensitivities for SMP materials with a high degree of accuracy.

Figure 26 shows the time required to calculate \(\frac {\partial \boldsymbol {R}_{n+1}}{\partial \boldsymbol {u}_{n-7}}\), the contribution of a total of 8 simulation steps, for a finite-element mesh of 50 elements by a single processor. As we can see, just using eight steps to simulate the entire SMP thermo-mechanical programming cycle even for a coarse mesh can incur high computational costs. This result motivated the development of PETSc-based parallel implementation of the FEA and sensitivity evaluation framework using CPUs on the *Golub Cluster* at the University of Illinois. Since the bottleneck for the entire algorithm is the sensitivity evaluation and particularly the time-dependent algorithm, the parallelization is done with the objective of distributing the elements onto the processors such that each processor has the optimum number of elements for efficient computations. A total of 144 processors (6 nodes with 24 processors each) were utilized for generating the 2D results. For the 3D optimization implementation, a total of 250 processors (10 nodes with 25 processors each) were utilized. The structural optimization problem is solved using the Method of Moving Asymptotes (MMA) (Svanberg 1987). The PETSc implementation of the MMA algorithm is based on the paper by Aage et al. (2015).

### Appendix 3: Validation of the finite element model

After implementing the constitutive model using the finite element framework for a single shape memory polymer material with the material properties as tabulated in Table 1 for *S**M**P*^{1}, the accuracy of the implementation was verified against existing experimental and computational results from the literature. The results and the comparisons here are for the *full* thermomechanical programming cycle, not the modified cycle described in Fig. 3. Two broad cases, time-independent SMP behavior and time-dependent SMP behavior, were analyzed and their results were compared.

### 3.1 Time-independent SMP behavior

To verify the current finite element implementation, the results obtained for a time-independent *stress free strain recovery* cycle were compared with the experimental results obtained by Liu et al. (2006). Figure 27b shows that the internal stress in a SMP sample increases as the temperature is reduced. This increase in the internal stress is due to an increase in the thermal stresses since the sample cannot contract with the decrease of temperature. We can observe that near the vicinity of the glass-transition temperature, the internal stress is negligible. This can be attributed to the low thermal stresses in this region. In the regions away from the glass-transition temperature, the internal stress increases sharply due to the presence of the glassy phase. The nature of evolution of the internal stresses, as observed experimentally in Fig. 27a for different amounts of pre-strains, is captured successfully by the current implementation. The discrepancies in the magnitude of the stresses can be attributed to the different materials used in the experimental studies and the numerical simulations. The difference in the material properties arises mainly due to the fact that the current analysis is geared toward application in the topology optimization algorithm and is based on the rheological model as shown in Fig. 2, where the material properties used for the different components are as per the values tabulated in Table 4 of ref. Baghani et al. (2012), which are different from the material properties reported in the experiments. This analysis was carried out to demonstrate that the current formulation can identify and mimic the nature of stress-strain evolution as observed in experimental results.

Figure 27d shows the free strain recovery for different amounts of fixed pre-strains with increase in temperature. As the temperature is increased, the amount of strains stored in the SMP sample decreases and the structure comes back to its initial configuration. It can be observed that even for different types of deformations, the paths followed during the recovery process are similar. The results obtained by the current implementation closely resemble the experimental results as shown in Fig. 27c. These results show that the current finite-element implementation can correctly capture the time-independent nature of the stress and strain evolution for a SMP material.

### 3.2 Time-dependent SMP behavior

The experiment performed by Li and Nettles (2010) was computationally simulated to validate the time-dependent aspect of the current implementation. Here, an SMP-based foam was compressed under a constant stress, held for 30 min and was subjected to the thermo-mechanical cycle. The main objective is to analyze the nature and form of the strain-time behavior. A comparison of the results of the current implementation with the numerical studies reported by Baghani et al. (2012) is shown in Fig. 28. Note that the deviations observed in Fig. 28a from those in Fig. 28b are mainly due to the use of different thermal strain function. We have used the function of thermal strain as given by (8) to maintain a continuity in our implementation. Also, the material parameters used differ in the two cases since we have not included any hard phase. Since our objective was to show that the current implementation sufficiently captures the SMP mechanics fit for moving forward with the topology optimization design, we can conclude that the overall correlation between the experimental results and the current implementation agrees to a level sufficient for our implementation.

Figure 29 compares the current implementation with the experiments reported by Volk et al. (2010) regarding the time-dependent uniaxial loading of SMPs followed by the thermo-mechanical cycle. We also compare our results with the implementation of Baghani et al. (2012). The results show that the current implementation can successfully capture the time-dependent effects with a moderate level of irreversible strains. From Fig. 29, we can observe that the strain at *T* = *T*_{H} is not 0, i.e., we do not recover all the strain that is put into the structure while it is deformed. This is due to the fact that while applying deformation a part of the total strain, irreversible strain component(*ε*^{i}), is permanently lost and cannot be recovered.

Figure 30 contains results from simulation of the multiaxial loading of an SMP material and compares the temperature vs. strain and time vs. strain plots obtained from the current implementation with the results reported by Baghani et al. (2012). The similarity of the results indicates that the current implementation can successfully capture the multiaxial loading of SMPs.

Having verified that the current finite element implementation of the constitutive model proposed by Baghani et al. (2012) can capture the essential characteristics of SMPs to an acceptable level of accuracy, we move forward with the computational design aspect using topology optimization.

### Appendix 4: The symmetry assumption

The results in Section 6.2 assume a symmetric design due to the symmetry of the loading and boundary conditions. To verify the assumption, we have also solved the problem using the full domain. Figure 32 shows the optimized material distribution for the self-actuating gripper corresponding to the full-design domain as shown in Fig. 31a without the assumption of symmetry.

The optimization problem statement for the full-domain case is written as:

The optimized value of *U*_{y}^{N} for the same node and under the same loading conditions is 2.2758 mm, in the negative *y*-direction. If we compare the optimized material distribution for the half-domain case as shown in Fig. 14 and the full-domain case as shown in Fig. 32, we observe that the two results have very similar topologies, with minor differences in the material distributions. The differences between the two solutions can be explained by the nonconvex nature of the optimization problem, which makes the optimization solutions dependent on both the starting point (initial guess) of the optimization, the search path followed to arrive at the final solution.

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Bhattacharyya, A., James, K.A. Topology optimization of shape memory polymer structures with programmable morphology.
*Struct Multidisc Optim* **63, **1863–1887 (2021). https://doi.org/10.1007/s00158-020-02784-0

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### Keywords

- Multimaterial topology optimization
- Shape memory polymers
- Multi-physics design
- Adjoint sensitivity analysis
- High-performance computing