Huilin
Ye
^{a},
Ying
Li
^{ab} and
Teng
Zhang
*^{cd}
^{a}Department of Mechanical Engineering, University of Connecticut, 191 Auditorium Road, Unit 3139, Storrs, Connecticut 06269, USA. E-mail: yingli@engr.uconn.edu; Fax: +1-860-4865088; Tel: +1-860-4867110
^{b}Polymer Program, Institute of Materials Science, University of Connecticut, 97 North Eagleville Road, Unit 3136, Storrs, Connecticut 06269, USA
^{c}Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, NY 13244, USA. E-mail: tzhang48@syr.edu; Tel: +1-315-443-2969
^{d}BioInspired Syracuse, Syracuse University, Syracuse, NY 13244, USA

Received
15th September 2020
, Accepted 29th November 2020

First published on 7th December 2020

Magnetic actuation has emerged as a powerful and versatile mechanism for diverse applications, ranging from soft robotics, biomedical devices to functional metamaterials. This highly interdisciplinary research calls for an easy to use and efficient modeling/simulation platform that can be leveraged by researchers with different backgrounds. Here we present a lattice model for hard-magnetic soft materials by partitioning the elastic deformation energy into lattice stretching and volumetric change, so-called ‘magttice’. Magnetic actuation is realized through prescribed nodal forces in magttice. We further implement the model into the framework of a large-scale atomic/molecular massively parallel simulator (LAMMPS) for highly efficient parallel simulations. The magttice is first validated by examining the deformation of ferromagnetic beam structures, and then applied to various smart structures, such as origami plates and magnetic robots. After investigating the static deformation and dynamic motion of a soft robot, the swimming of the magnetic robot in water, like jellyfish's locomotion, is further studied by coupling the magttice and lattice Boltzmann method (LBM). These examples indicate that the proposed magttice model can enable more efficient mechanical modeling and simulation for the rational design of magnetically driven smart structures.

Among various numerical techniques,^{32–39} finite element methods (FEMs) are widely used to simulate the nonlinear and active deformation of ferromagnetic materials.^{32,39} For example, Zhao et al.^{32} derived a finite element simulation scheme for the hard-magnetic soft materials and implemented it into the FEM software ABAQUS through a user-defined element (UEL). This robust and powerful platform has greatly promoted the computational contributions to the design of magnetic smart structures. However, it is still difficult to include the multiphysics interactions into the framework of UEL in ABAQUS, such as fluid–structure interactions. Besides, the commercial software's parallel simulation capabilities are also constrained by the setup of the computational environments and the availability of the research licenses. Lattice models are another kind of numerical techniques for simulating elastic solids that are efficient and easy to integrate with other methods for multiphysics problems.^{40–48} For example, lattice spring gel models have been applied to study the fluid-driven motion of microcapsules on compliant surfaces.^{44,49} Recently, the authors derived irregular lattice models directly from the FEM framework,^{46} and further coupled the lattice model with the lattice Boltzmann method (LBM) for complicated fluid–structure interactions (e.g., the platform of OpenFSI^{50}). Although the simplicity and powerfulness of the lattice model make it very promising, there is still a lack of a rigorous derivation to include the magnetic forces.

In this work, we present a modeling and simulation platform of hard-magnetic soft materials that combines the advantages of the recently developed FEM schemes and lattice models, named ‘magttice’. We show that the magnetic actuation can be incorporated into the existing lattice model as nodal forces, which can be pre-computed if the external magnetic field is uniform. We implement the magttice model into the open-source molecular dynamics package, LAMMPS, and leverage its highly parallelized simulation capability and the versatile simulation techniques for multiphysics problems, such as fluid–structure interactions, by coupling the magttice model with the LBM. We then apply the magttice model to investigate the folded deformation of an origami plate, nonlinear deformation, and swimming of small-scale robots made by using a magnetic strip. These examples indicate that the proposed magttice model can enable more efficient mechanical modeling and simulation for the rational design of magnetically driven smart structures.

(1) |

The lattice model is derived based on energy equivalence through comparing with the finite element model. In the framework of the FEM, the deformation gradient tensor F in an irregular element can be calculated as:^{56}

(2) |

(3) |

By rearranging the summations in eqn (3), the energy can be obtained in the lattice model (28 lattice springs in a general case Fig. 1(a)) with the form^{46}

(4) |

(5) |

Fig. 1 Computational model and benchmark of the proposed magttice model. (a) Illustration of the proposed magttice model for ferromagnetic materials. (b) Diminished magnetic forces for the interior part of a beam with uniformly residual and external magnetic fields. (c) Deformed configurations of a representative beam (length L = 17.5 mm, width W = 5 mm, and height H = 1 mm) at different external magnetic fields. The dimensionless variable characterize the effective strength of magnetic forces, in comparison to the elasticity. (d) The normalized maximum displacement as a function of γ. The finite element solution is taken from previous work done by Zhao et al.^{32} |

For the magnetic energy density, it can also be pre-computed as

U_{magnetic} = V_{0}^{−1}(−f_{m})^{a}_{i}x^{a}_{i}, | (6) |

Viewing the magnetic actuation as forces can also help simplify the modeling systems. Taking a ferromagnetic beam structure as an example with ^{r} = [1, 0, 0]^{T} and B^{applied} = [0, 0, 1]^{T} (Fig. 1(a) and (b)), the magnetic forces applied to the interior nodes will be cancelled out by summing the forces of the adjacent elements. This will lead to zero internal forces and non-zero nodal forces at the two ends, which is consistent with the recent observations in the modeling of the magnetic actuation of ferromagnetic beams with the elastic model.^{60} We further simulate the deformed configurations of the representative ferromagnetic beam (length L = 17.5 mm, width W = 5 mm, and height H = 1 mm) using the proposed magttice model (Fig. 1(c)), where we have G = 330 kPa, K = 20 G and as a tunable parameter. As shown in Fig. 1(d), our simulation results are in excellent agreement with FEM results reported by Zhao et al.^{32}

Hu et al.^{9} already showed that the robot's final configurations depended on the direction of the external magnetic field and had “C” and “V” stable configurations. To demonstrate these aspects, we run eight simulations by varying the direction θ from 0° to 315°. As shown in Fig. 3, a “V” shape is observed when θ is closer to 45° and a “C” shape can be obtained when θ is closer to 215°. Furthermore, rigid body rotation can be noticed for θ = 90°, 180°, 270°, and 315°. These findings are all in excellent agreement with the analytical solutions and experiments in the previous study.^{9} However, the configurations for θ = 135° and θ = 315° are more complicated, which is neither a “V” nor a “C” shape. In addition, there is no rigid body rotation and two contact points between the robot and the platform.

(7) |

(8) |

(f_{m}′)_{i}^{a} = R_{ij}(f_{m})_{i}^{a}, i, j = 1, 2, 3, a = 1,2,…,8. | (9) |

We demonstrate this function by using the robot turning-deformation as an example. The robot is first deformed into a “C” shape by setting θ = 225° and |B^{applied}| = 16.9 mT. Then we perform three sequential rotations along the clockwise direction with Δθ = −45°. The robot is rotating with the external magnetic field and maintains the “C” shape (Fig. 5) (see Movie 1 of ESI†). Following this deformation sequence, it is very interesting to note that a “C” shape can be achieved for θ = 90°. This is different from the “C” shape starting from the fixed direction of the external magnetic field (Fig. 3), indicating that the structure is bi-stable.

Fig. 5 (a) Time variation of the direction of the external magnetic field. (b) Structures with spatially varied residual magnetic flux for robotic-turn. Here we set Δθ = −45°. |

(10) |

∇·u = 0, | (11) |

(12) |

Fig. 6 Swimming of the magnetic robot. (a) Fluid–structure interaction scheme for the swimming of the magnetic robot. (b) Sequence of the applied magnetic field B^{applied}, α is equal to 360°–θ. (c) Comparison of the gaits of the magnetic robot in one period: from recovery stroke to power stroke. The experimental results are adapted from Hu et al.^{9} |

The magnetic robot is represented by a flat sheet and accounted for by the magttice model introduced in Section 2. The mechanical properties of the magnetic robot are the same as those in the experimental study (G = 33 kPa, K = 20 G, density ρ = 1.86 g cm^{−3} and ). The coupling between the magnetic robot and fluid is fulfilled by the immersed boundary method (IBM), which has been extensively used to study the fluid–structure interaction problems.^{48,50,65,66,67} In IBM, the FSI force F^{f} is calculated as

F^{f} = β(u^{f}(t) − u^{s}(t)), | (13) |

(14) |

(15) |

The sequence of B^{applied} is given in Fig. 6(b), where the maximum B^{applied}, B^{applied}_{max} = 17 mT and frequency f = 25 Hz. Accordingly, the robot can perform locomotion when B is oscillating along the direction α = 105° and α = 285°. In one period, the prescribed B^{applied} sequence can make the robot demonstrate a slow recovery stroke, in which the robot changes from the “C”-shape to “V”-shape (0–19.5 ms as shown in Fig. 6(c)). Following is a fast power stroke (19.5–32 ms as shown in Fig. 6(c)), which takes the robot back to the “C”-shape. From the comparison of the configuration evolution of the magnetic robot between experiment and simulation results (cf.Fig. 6(c)), we can see that the current fluid–structure interaction platform can capture the whole swimming gaits of the robot underwater. Furthermore, the asymmetry of the “C”-shape and “V”-shape makes the robot experiences more propulsion in power stroke than the reduction in recovery stroke. This net propulsion will propel the robot to move upwards along the z-direction. It resembles that of an actual jellyfish swimming,^{61} and it is also observed in our simulation (see Movie 2 of ESI†).

- J. Ginder, S. Clark, W. Schlotter and M. Nichols, Int. J. Mod. Phys. B, 2002, 16, 2412–2418 CrossRef CAS .
- Y. Li, J. Li, W. Li and H. Du, Smart Mater. Struct., 2014, 23, 123001 CrossRef .
- W. Li, K. Kostidis, X. Zhang and Y. Zhou, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2009, pp. 233–238.
- H. Böse, R. Rabindranath and J. Ehrlich, J. Intell. Mater. Syst. Struct., 2012, 23, 989–994 CrossRef .
- S. Hong, Y. Jung, R. Yen, H. F. Chan, K. W. Leong, G. A. Truskey and X. Zhao, Lab. Chip, 2014, 14, 514–521 RSC .
- S. Kashima, F. Miyasaka and K. Hirata, IEEE Trans. Magn., 2012, 48, 1649–1652 Search PubMed .
- S. Opie and W. Yim, J. Intell. Mater. Syst. Struct., 2011, 22, 113–125 CrossRef .
- S. Wu, Q. Ze, R. Zhang, N. Hu, Y. Cheng, F. Yang and R. Zhao, ACS Appl. Mater. Interfaces, 2019, 11, 41649–41658 CrossRef CAS PubMed .
- W. Hu, G. Z. Lum, M. Mastrangeli and M. Sitti, Nature, 2018, 554, 81–85 CrossRef CAS PubMed .
- Y. Kim, G. A. Parada, S. Liu and X. Zhao, Sci. Robot, 2019, 4, 7329 CrossRef PubMed .
- T. Xu, J. Zhang, M. Salehizadeh, O. Onaizah and E. Diller, Sci. Robot., 2019, 4, 4494 CrossRef PubMed .
- S. Wu, C. M. Hamel, Q. Ze, F. Yang, H. J. Qi and R. Zhao, Adv. Intell. Syst., 2020, 2000060 CrossRef .
- K. E. Peyer, L. Zhang and B. J. Nelson, Nanoscale, 2013, 5, 1259–1272 RSC .
- Z. Wu, Y. Chen, D. Mukasa, O. S. Pak and W. Gao, Chem. Soc. Rev., 2020, 49, 8088–8112 RSC .
- Z. Ren, T. Wang, W. Hu and M. Sitti, Robotics: Science and Systems, 2019.
- G. Mao, M. Drack, M. Karami-Mosammam, D. Wirthl, T. Stockinger, R. Schwödiauer and M. Kaltenbrunner, Sci. Adv., 2020, 6, eabc0251 CrossRef CAS PubMed .
- H.-W. Huang, M. S. Sakar, A. J. Petruska, S. Pané and B. J. Nelson, Nat. Commun., 2016, 7, 1–10 Search PubMed .
- H. Song, H. Lee, J. Lee, J. K. Choe, S. Lee, J. Y. Yi, S. Park, J.-W. Yoo, M. S. Kwon and J. Kim, Nano Lett., 2020, 20(7), 5185–5192 CrossRef CAS PubMed .
- J. Cui, T.-Y. Huang, Z. Luo, P. Testa, H. Gu, X.-Z. Chen, B. J. Nelson and L. J. Heyderman, Nature, 2019, 575, 164–168 CrossRef CAS PubMed .
- Y. Kim, H. Yuk, R. Zhao, S. A. Chester and X. Zhao, Nature, 2018, 558, 274–279 CrossRef CAS PubMed .
- F. Fahrni, M. W. Prins and L. J. van IJzendoorn, Lab Chip, 2009, 9, 3413–3421 RSC .
- S. Zhang, Z. Cui, Y. Wang and J. den Toonder, Lab Chip, 2020, 20, 3569–3581 RSC .
- S. Khaderi, C. Craus, J. Hussong, N. Schorr, J. Belardi, J. Westerweel, O. Prucker, J. Rühe, J. D. Toonder and P. Onck, Lab Chip, 2011, 11, 2002–2010 RSC .
- J. Belardi, N. Schorr, O. Prucker and J. Rühe, Adv. Funct. Mater., 2011, 21, 3314–3320 CrossRef CAS .
- H. Gu, Q. Boehler, H. Cui, E. Secchi, G. Savorana, C. De Marco, S. Gervasoni, Q. Peyron, T.-Y. Huang and S. Pane, et al. , Nat. Commun., 2020, 11, 1–10 Search PubMed .
- X. Zhao, J. Kim, C. A. Cezar, N. Huebsch, K. Lee, K. Bouhadir and D. J. Mooney, Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 67–72 CrossRef CAS PubMed .
- H. K. Yap, J. H. Lim, F. Nasrallah, F.-Z. Low, J. C. Goh and R. C. Yeow, IEEE International Conference on Rehabilitation Robotics (ICORR), 2015, pp. 735–740.
- P. R. Buckley, G. H. McKinley, T. S. Wilson, W. Small, W. J. Benett, J. P. Bearinger, M. W. McElfresh and D. J. Maitland, IEEE Trans. Biomed. Eng., 2006, 53, 2075–2083 Search PubMed .
- H. Gu, S. W. Lee, J. Carnicelli, T. Zhang and D. Ren, Nat. Commun., 2020, 11, 1–11 Search PubMed .
- S. Zhang, P. Zuo, Y. Wang, P. R. Onck and J. M. den Toonder, ACS Appl. Mater. Interfaces, 2020, 12(24), 27726–27736 CrossRef CAS PubMed .
- K. Yu, N. X. Fang, G. Huang and Q. Wang, Adv. Mater., 2018, 30, 1706348 CrossRef PubMed .
- R. Zhao, Y. Kim, S. A. Chester, P. Sharma and X. Zhao, J. Mech. Phys. Solids, 2019, 124, 244–263 CrossRef CAS .
- G. Z. Lum, Z. Ye, X. Dong, H. Marvi, O. Erin, W. Hu and M. Sitti, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, E6007–E6015 CrossRef CAS PubMed .
- J. Li, T. D. Pallicity, V. Slesarenko, A. Goshkoderia and S. Rudykh, Adv. Mater., 2019, 31, 1807309 CrossRef PubMed .
- A. Goshkoderia, V. Chen, J. Li, A. Juhl, P. Buskohl and S. Rudykh, Phys. Rev. Lett., 2020, 124, 158002 CrossRef CAS .
- S. M. Montgomery, S. Wu, X. Kuang, C. D. Armstrong, C. Zemelka, Q. Ze, R. Zhang, R. Zhao and H. J. Qi, 2020, arXiv preprint arXiv:2006.12721.
- K. H. Lee, K. Yu, H. Al Ba’ba’a, A. Xin, Z. Feng and Q. Wang, Research, 2020, 2020, 4825185 CrossRef PubMed .
- W. Gao, D. Kagan, O. S. Pak, C. Clawson, S. Campuzano, E. Chuluun-Erdene, E. Shipton, E. E. Fullerton, L. Zhang and E. Lauga, et al. , Small, 2012, 8, 460–467 CrossRef CAS PubMed .
- K. Danas, S. Kankanala and N. Triantafyllidis, J. Mech. Phys. Solids, 2012, 60, 120–138 CrossRef CAS .
- M. Ostoja-Starzewski, Appl. Mech. Rev., 2002, 55, 35–60 CrossRef .
- X. He, M. Aizenberg, O. Kuksenok, L. D. Zarzar, A. Shastri, A. C. Balazs and J. Aizenberg, Nature, 2012, 487, 214–218 CrossRef CAS PubMed .
- O. Kuksenok and A. C. Balazs, Sci. Rep., 2015, 5, 1–7 Search PubMed .
- O. Kuksenok and A. C. Balazs, Mater. Horiz., 2016, 3, 53–62 RSC .
- A. Alexeev, R. Verberg and A. C. Balazs, Macromolecules, 2005, 38, 10244–10260 CrossRef CAS .
- V. V. Yashin and A. C. Balazs, J. Chem. Phys., 2007, 126, 124707 CrossRef PubMed .
- T. Zhang, Extreme Mech. Lett., 2019, 26, 40–45 CrossRef .
- P. Dayal, O. Kuksenok and A. C. Balazs, Langmuir, 2009, 25, 4298–4301 CrossRef CAS PubMed .
- H. Ye, Z. Shen and Y. Li, Comput. Mech., 2018, 62, 457–476 CrossRef .
- G. A. Buxton, R. Verberg, D. Jasnow and A. C. Balazs, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2005, 71, 056707 CrossRef PubMed .
- H. Ye, Z. Shen, W. Xian, T. Zhang, S. Tang and Y. Li, Comput. Phys. Commun., 2020, 256, 107463 CrossRef CAS .
- A. Dorfmann and R. Ogden, Eur. J. Mech. A Solids, 2003, 22, 497–507 CrossRef .
- S. Kankanala and N. Triantafyllidis, J. Mech. Phys. Solids, 2004, 52, 2869–2908 CrossRef CAS .
- T. K. Kim, J. K. Kim and O. C. Jeong, Microelectron. Eng., 2011, 88, 1982–1985 CrossRef CAS .
- P. Saxena, M. Hossain and P. Steinmann, Int. J. Solids Struct., 2013, 50, 3886–3897 CrossRef .
- D. Garcia-Gonzalez, Smart Mater. Struct., 2019, 28, 085020 CrossRef CAS .
- T. Belytschko, W. K. Liu, B. Moran and K. Elkhodary, Nonlinear finite elements for continua and structures, John wiley & sons, 2013 Search PubMed .
- S. Tang, Y. Li, Y. Yang and Z. Guo, Soft Matter, 2015, 11, 7911–7919 RSC .
- Z. Zhou, Y. Li, W. Wong, T. Guo, S. Tang and J. Luo, Soft Matter, 2017, 13, 6011–6020 RSC .
- L. Dorfmann and R. W. Ogden, Nonlinear theory of electroelastic and magnetoelastic interactions, Springer, 2014, vol. 1 Search PubMed .
- L. Wang, Y. Kim, C. F. Guo and X. Zhao, J. Mech. Phys. Solids, 2020, 104045 CrossRef .
- B. J. Gemmell, J. H. Costello, S. P. Colin, C. J. Stewart, J. O. Dabiri, D. Tafti and S. Priya, Proc. Natl. Acad. Sci. U. S. A., 2013, 110, 17904–17909 CrossRef CAS PubMed .
- Z. Ren, W. Hu, X. Dong and M. Sitti, Nat. Commun., 2019, 10, 1–12 CrossRef CAS PubMed .
- S. Succi, The lattice Boltzmann equation: for fluid dynamics and beyond, Oxford University Press, 2001 Search PubMed .
- F. Mackay, S. T. Ollila and C. Denniston, Comput. Phys. Commun., 2013, 184, 2021–2031 CrossRef CAS .
- C. S. Peskin, Acta Numer., 2002, 11, 479–517 CrossRef .
- F.-B. Tian, H. Luo, L. Zhu, J. C. Liao and X.-Y. Lu, J. Comput. Phys., 2011, 230, 7266–7283 CrossRef CAS .
- H. Ye, H. Wei, H. Huang and X.-Y. Lu, Phys. Fluids, 2017, 29, 021902 CrossRef .

## Footnote |

† Electronic supplementary information (ESI) available: Simulation movies for turning and swimming of the magnetic robot. See DOI: 10.1039/d0sm01662d |

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