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A new stabilisation approach for level-set based topology optimisation of hyperelastic materials

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Abstract

This paper introduces a novel computational approach for level-set based topology optimisation of hyperelastic materials at large strains. This, to date, is considered an unresolved open problem in topology optimisation due to its extremely challenging nature. Two computational strategies have been proposed to address this problem. The first strategy resorts to an arc-length in the pre-buckling region of intermediate topology optimisation (TO) iterations where numerical difficulties arise (associated with nucleation, disconnected elements, etc.), and is then continued by a novel regularisation technique in the post-buckling region. In the second strategy, the regularisation technique is used for the entire loading process at each TO iteration. The success of both rests on the combination of three distinct key ingredients. First, the nonlinear equilibrium equations of motion are solved in a consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. Third, and with the purpose of avoiding excessive numerical stabilisation, a scalar degradation function is applied on the regularised linearised elasticity tensor, based on a novel regularisation indicator field. The robustness and applicability of this new methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton-Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10− 12.

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Notes

  1. Solid isotropic material with penalisation.

  2. The well-known augmented Lagrangian method (Bonet et al. 2016b) is a prototypical example of these approaches.

  3. Lowercase (or uppercase) indices are used to refer to the spatial (material) configuration.

  4. Note how the regularisation parameter featuring in (40) is frozen at incremental step n. This removes the nonlinearity altogether and avoids the need to linearise in a Newton-Raphson–type manner.

  5. λcr can be approximated by the point in the equilibrium path where λ decreases.

  6. The hydrostatic pressure is obtained as \(p = \frac {1}{3}\text {tr}\boldsymbol {\sigma }\) where σ is the Cauchy stress tensor defined as σ = J− 1PFT in 3D problems and σ = (λ33J2D)− 1PFT for 2D problems.

References

  • Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550(84):84–86

    Article  Google Scholar 

  • Allaire G (2006) Conception optimale de structures. Springer

  • Allaire G, Jouve F, Toader A (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

    Article  MathSciNet  Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202

    Article  Google Scholar 

  • Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224

    Article  MathSciNet  Google Scholar 

  • Bonet J, Gil AJ, Lee CH, Aguirre M, Ortigosa R (2015a) A first order hyperbolic framework for large strain computational solid dynamics - Part I: total Lagrangian isothermal elasticity. Comput Methods Appl Mech Eng 283(0):689–732

    Article  MathSciNet  Google Scholar 

  • Bonet J, Gil AJ, Ortigosa R (2015b) A computational framework for polyconvex large strain elasticity. Comput Methods Appl Mech Eng 283:1061–1094

    Article  MathSciNet  Google Scholar 

  • Bonet J, Gil AJ, Ortigosa R (2016a) On a tensor cross product based formulation of large strain solid mechanics. Int J Solids Struct 84:49–63

    Article  Google Scholar 

  • Bonet J, Gil AJ, Wood RD (2016b) Nonlinear continuum mechanics for finite element analysis: statics. Cambridge University Press

  • Bruns TE, Tortorelli DA (2003) An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. Int J Numer Methods Eng 57:1413–1430

    Article  Google Scholar 

  • Burden RL, Faires JD, Burden AM (2015) Numerical analysis. Cengage learning

  • Burger M, Stainko R (2006) Phase-field relaxation of topology optimization with local stress constraints. SIAM J Control Optim 45(4):1447–1466

    Article  MathSciNet  Google Scholar 

  • Chen F, Wang Y, Wang M, Zhang Y (2017) Topology optimization of hyperelastic structures using a level set method. J Comput Phys 351:437–454

    Article  MathSciNet  Google Scholar 

  • de Boer R (1982) Vektor- und Tensorrechnung für Ingenieure. Springer

  • Geiss M, Boddeti N, Weeger O, Maute K, Dunn M (2018) Combined level-set-xfem-density topology optimization of 4d printed structures undergoing large deformation. ASME J Mech Des

  • Gursel A (2018) Softer is harder: what differentiates soft robotics from hard robotics? MRS Advances 3 (28):1557–1568. https://doi.org/10.1557/adv.2018.159

    Article  Google Scholar 

  • Ha S, Cho S (2008) Level set based topological shape optimization of geometrically nonlinear structures using unstructured mesh. Comput Struct 86(13):1447–1455

    Article  Google Scholar 

  • Hesch C, Gil A, Ortigosa R, Dittmann M, Bilgen C, Betsch P, Franke M, Janz A, Weinberg K (2017) A framework for polyconvex large strain phase-field methods to fracture. Comput Methods Appl Mech Eng 317:649–683

    Article  MathSciNet  Google Scholar 

  • Jensen JS, Sigmund O (2011) Topology optimization for nanophotonics. Laser Photonics Rev 5(2):308–321

    Article  Google Scholar 

  • Jung D, Gea H (2004) Topology optimization of nonlinear structures. Finite Elem Anal Des 40(11):1417–1427

    Article  Google Scholar 

  • Lahuerta R, Simões E, Campello E, Pimenta P, Silva E (2013) Towards the stabilization of the low density elements in topology optimization with large deformation. Comput Mech 52(4):779–797

    Article  MathSciNet  Google Scholar 

  • Laurain A (2018) A level set-based structural optimization code using FEniCS. Struct Multidiscipl Optim 58 (3):1311–1334

    Article  MathSciNet  Google Scholar 

  • Liu L, Xing J, Yang Q, Luo Y (2017) Design of large-displacement compliant mechanisms by topology optimization incorporating modified additive hyperelasticity technique. Mathematical Problems in Engineering

  • Munk DJ, Vio GA, Steven GP (2015) Topology and shape optimization methods using evolutionary algorithms: a review. Struct Multidiscip Optim 52(3):613–631

    Article  MathSciNet  Google Scholar 

  • Norato JA, Bendsøe MP, Haber RB, Tortorelli DA (2007) A topological derivative method for topology optimization. Struct Multidiscip Optim 33(4–5):375–386

    Article  MathSciNet  Google Scholar 

  • Ortigosa R, Gil AJ (2017) A computational framework for incompressible electromechanics based on convex multi-variable strain energies for geometrically exact shell theory. Comput Methods Appl Mech Eng 317:792–816

    Article  MathSciNet  Google Scholar 

  • Ortigosa R, Gil AJ, Bonet J, Hesch C (2015) A computational framework for polyconvex large strain elasticity for geometrically exact beam theory. Comput Mech 57:277–303

    Article  MathSciNet  Google Scholar 

  • Osher S, Sethian JA (1988) Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J Comp Phys 78:12–49

    Article  Google Scholar 

  • Poya R, Sevilla R, Gil AJ (2016) A unified approach for a posteriori high-order curved mesh generation using solid mechanics. Comput Mech 58(3):457–490

    Article  MathSciNet  Google Scholar 

  • Rus D, Tolley M (2015) Design, fabrication and control of soft robots. Nature 521:467–475

    Article  Google Scholar 

  • Sethian J, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(1):489– 528

    Article  MathSciNet  Google Scholar 

  • Sigmund O, Maute K (2013) Topology optimization approaches a comparative review. Struct Multidiscip Optim 48(6):1031– 1055

    Article  MathSciNet  Google Scholar 

  • Sokolowski J, Zochowski A (1999) On the topological derivative in shape optimization. SIAM J Control Optim 37(4):1251– 1272

    Article  MathSciNet  Google Scholar 

  • Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229(7):2697–2718

    Article  MathSciNet  Google Scholar 

  • Wallin T, Pikul J, Shepherd R (2018) 3d printing of soft robotic systems. Nat Rev Mater 3:84–100

    Article  Google Scholar 

  • Wang M, Wang X, Guo D (2003) A level-set method for structural topology optimization. Comput Meth Appl Mech Eng 192(1–2):227–246

    Article  MathSciNet  Google Scholar 

  • Wang F, Lazarov BS, Sigmund O, Jensen JS (2014) Towards the stabilization of the low density elements in topology optimization with large deformation. Comput Methods Appl Mech Eng 276:453–472

    Article  Google Scholar 

  • Wehner M, Truby R, Fitzgerald D, Mosadegh B, Whitesides G, Lewis J, Wood R (2016) An integrated design and fabrication strategy for entirely soft, autonomous robots. Nature 536:451–455

    Article  Google Scholar 

  • Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336

    Article  Google Scholar 

  • Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Meth Eng 23(4):595–622

    Article  MathSciNet  Google Scholar 

Download references

Funding

The first and third authors acknowledge the support provided by the S\(\hat {e}\)r Cymru National Research Network under the Ser Cymru II Fellowship “Virtual engineering of the new generation of biomimetic artificial muscles”, funded by the European Regional Development Fund. The first author also acknowledges the support provided by Ministerio de Ciencia, Innovación y Universidades, for the award of a Juan de la Cierva Formación Fellowship. Additionally, this research was partially supported by the AEI/FEDER and UE under the contracts DPI2016-77538-R and by the Fundación Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia (Spain)) under the contract 20911/PI/18.

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Correspondence to Jesús Martínez-Frutos.

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Appendix: Particularisation to plane stress

Appendix: Particularisation to plane stress

Plane stress is suitable for many industrial applications. In this context, the three-dimensional deformation gradient tensor F is expressed in terms of its in-plane component F2D and the out-of-plane thickness stretch λ33 component as:

$$ \boldsymbol{F} = \left[\begin{array}{llll} \boldsymbol{F}_{2D} & \boldsymbol{0}_{2\times1}\\ \boldsymbol{0}_{1\times 2} & \lambda_{33} \end{array}\right]. $$
(52)

Making use of (2), the Jacobian J and the co-factor H of F can be defined as:

$$ \begin{array}{@{}rcl@{}} J& =& J_{2D}\lambda_{33};\\ \boldsymbol{H}& =& J\boldsymbol{F}^{-T} = \left[\begin{array}{llll} \lambda_{33}\boldsymbol{H}_{2D} & \boldsymbol{0}_{2\times1}\\ \boldsymbol{0}_{1\times2} & J_{2D} \end{array}\right], \end{array} $$
(53)

with H2D and J2D defined as:

$$ \boldsymbol{H}_{2D}=\left( \boldsymbol{F}_{2D}:\boldsymbol{I}\right)\boldsymbol{I} - \boldsymbol{F}_{2D}^{T};\qquad J_{2D}=\frac{1}{2}\boldsymbol{H}_{2D}:\boldsymbol{F}_{2D}. $$
(54)

Equations (52) and (53) enable particularising the strain energy e (0ϕ) and its extended representation \(W\left (\mathcal {V}\right )\) in (5) to plane stress as:

$$ e\left( \boldsymbol{\nabla}_{0}\boldsymbol{\phi}\right) = \widetilde{e}\left( \boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D},\lambda_{33}\right) = \widetilde{W}\left( {\mathcal{V}_{2D}},\lambda_{33}\right), $$
(55)

with \({\mathcal {V}_{2D}} = \{\boldsymbol {F}_{2D},J_{2D}\}\). Making use of (52) and (53), the invariants IIF and IIH featuring in the constitutive model in (6) can be written in terms of the set \(\{{\mathcal {V}_{2D}},\lambda _{33}\}\) as:

$$ \begin{array}{llll} II_{\boldsymbol{F}} = II_{\boldsymbol{F}_{2D}} + \lambda_{33}^{2}; \qquad II_{\boldsymbol{H}} = \lambda_{33}^{2}II_{\boldsymbol{F}_{2D}} + J_{2D}^{2}. \end{array} $$
(56)

Making use of (56), the Mooney-Rivlin model in (6) can be particularised to plane stress as:

$$ \begin{array}{@{}rcl@{}} \widetilde{W} & =& \left( \frac{\mu_{1}}{2} + \lambda_{33}^{2}\frac{\mu_{2}}{2}\right)II_{\boldsymbol{F}_{2D}} + \frac{\mu_{1}}{2}\lambda_{33}^{2} + \frac{\mu_{2}}{2}J_{2D}^{2}\\ &&- \left( \mu_{1} + 2\mu_{2}\right)\left( \ln J_{2D} + \ln \lambda_{33}\right) + \frac{\kappa}{2}\left( J_{2D}\lambda_{33} - 1\right)^{2} \\ &&- \frac{3}{2}\left( \mu_{1} + \mu_{2}\right). \end{array} $$
(57)

Let δϕ2D and Δϕ2D denote virtual and incremental variations of the in-plane mapping ϕ2D. The directional derivative of \(\widetilde {e}\left (\boldsymbol {\nabla }_{0}\boldsymbol {\phi }_{2D},\lambda _{33}\right )\) with respect to possible virtual variations of the in-plane mapping is:

$$ D\widetilde{e}[\delta\boldsymbol{\phi}_{2D}] = \partial_{\boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D}}\widetilde{e}:\boldsymbol{\nabla}_{0}\delta\boldsymbol{\phi}_{2D} + \left( \partial_{\lambda_{33}}\widetilde{e}\right)D\lambda_{33}[\delta\boldsymbol{\phi}_{2D}], $$
(58)

where the second term on the right-hand side of (58) vanishes due to the plane stress assumption (refer to Remark 2 below (60)). Making use of (58) and of the left-hand side of (8), it can be concluded that the first Piola-Kirchhoff stress tensor in plane stress is defined as:

$$ \boldsymbol{P}_{2D} = \partial_{\boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D}}\widetilde{e}\left( \boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D},\lambda_{33}\right). $$
(59)

Remark 2

The plane stress implies the out-of-plane component of the stress to vanish, i.e.:

$$ \partial_{\lambda_{33}}\widetilde{e}\left( \boldsymbol{\nabla}_{0}\boldsymbol{\phi},\lambda_{33}\right) = 0. $$
(60)

Generally, above (60) is nonlinear and needs to be solved iteratively. In addition, the directional derivative of λ33 namely Dλ33[δϕ2D] can be computed from the condition:

$$ D\left( \partial_{\lambda_{33}}\widetilde{e}\left( \boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D},\lambda_{33}\right)\right)[\delta\boldsymbol{\phi}_{2D}] = 0. $$
(61)

The directional derivative of the plane stress condition (61) can be expanded as:

$$ \begin{array}{@{}rcl@{}} &&D\left( \partial_{\lambda_{33}}\widetilde{e}\left( \boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D},\lambda_{33}\right)\right)[\delta\boldsymbol{\phi}_{2D}]\\ &=&\partial_{\lambda_{33}}\boldsymbol{P}_{2D}:\boldsymbol{\nabla}_{0}\delta\boldsymbol{\phi}_{2D}\\ && + \left( \partial^{2}_{\lambda_{33}\lambda_{33}}\widetilde{e}\right)D\lambda_{33}[\delta\boldsymbol{\phi}_{2D}] = 0. \end{array} $$
(62)

Finally, Dλ33[δϕ2D] can be obtained from (62) as:

$$ D\lambda_{33}[\delta\boldsymbol{\phi}_{2D}] = -{\left( \partial^{2}_{\lambda_{33}\lambda_{33}}\widetilde{e}\right)}^{-1}\left( \partial_{\lambda_{33}}\boldsymbol{P}_{2D}:\boldsymbol{\nabla}_{0}\delta\boldsymbol{\phi}_{2D}\right). $$
(63)

In the case of plane stress, the fourth-order elasticity tensor \(\boldsymbol {\mathcal {C}}\) emerges from the second directional derivative of the strain energy as:

$$ \begin{array}{@{}rcl@{}} D^{2}\widetilde{e}[\delta\boldsymbol{\phi}_{2D};\!{\Delta}\boldsymbol{\phi}_{2D}]\!&=&\!\boldsymbol{\nabla}_{0}\delta\boldsymbol{\phi}_{2D} \!:\partial^{2}_{\boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D}\boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D}}\widetilde{e}: \boldsymbol{\nabla}_{0}{\Delta}\boldsymbol{\phi}_{2D}\\ &&\!+\left( \boldsymbol{\nabla}_{0}\delta\boldsymbol{\phi}_{2D}\!:\partial_{\lambda_{33}}\boldsymbol{P}_{2D}\right)D\lambda_{33}[{\Delta}\boldsymbol{\phi}_{2D}].\\ \end{array} $$
(64)

Making use of (63) in (64) and comparison against (9) enables to obtain the following expression for the in-plane constitutive tensor \(\boldsymbol {\mathcal {C}}_{2D}\):

$$ \begin{array}{llll} \boldsymbol{\mathcal{C}}_{2D} = \partial^{2}_{\boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D}\boldsymbol{\nabla}_{0}\boldsymbol{\phi}_{2D}}\widetilde{e} - {\left( \partial^{2}_{\lambda_{33}\lambda_{33}}\widetilde{e}\right)}^{-1}\partial_{\lambda_{33}}\boldsymbol{P}_{2D}\otimes \partial_{\lambda_{33}}\boldsymbol{P}_{2D}. \end{array} $$
(65)

Remark 3

Proceeding similarly as in Remark 1, the following expression for the two-dimensional first Piola-Kirchhoff stress tensor P2D can be obtained when considering the extended representation \(\widetilde {W}\) (55):

$$ \boldsymbol{P}_{2D}=\partial_{\boldsymbol{F}_{2D}}\widetilde{W} + \partial_{J_{2D}}\widetilde{W}\boldsymbol{H}_{2D}. $$
(66)

Furthermore, the following expression (equivalent to that in (65)) can be obtained for \(\boldsymbol {\mathcal {C}}_{2D}\) in terms of the derivatives of \(\widetilde {{W}}\):

$$ \begin{array}{@{}rcl@{}} \boldsymbol{\mathcal{C}}_{2D}& =& \partial^{2}_{{\boldsymbol{F}_{2D}}{\boldsymbol{F}}_{2D}}\widetilde{W} + \partial^{2}_{{J_{2D}}{J_{2D}}}\widetilde{W}\boldsymbol{H}_{2D}\otimes\boldsymbol{H}_{2D}\\ && +\partial^{2}_{\boldsymbol{F}_{2D}{J_{2D}}}\widetilde{W}\otimes\boldsymbol{H}_{2D} + \boldsymbol{H}\otimes\partial^{2}_{J_{2D}\boldsymbol{F}_{2D}}\widetilde{W}\\ &&+\partial_{J}\widetilde{W}\left( \boldsymbol{I}\otimes\boldsymbol{I} - \boldsymbol{\mathcal{I}}^{T}\right) - {\left( \partial^{2}_{\lambda_{33}\lambda_{33}}\widetilde{W}\right)}^{-1}\partial_{\lambda_{33}}\boldsymbol{P}_{2D}\\ &&\otimes \partial_{\lambda_{33}}\boldsymbol{P}_{2D}, \end{array} $$
(67)

with

$$ \partial_{\lambda_{33}}\boldsymbol{P}_{2D} = \partial^{2}_{{\boldsymbol{F}_{2D}}{\lambda_{33}}}\widetilde{W} + \partial^{2}_{J_{2D}\lambda_{33}}\widetilde{W}\boldsymbol{H}_{2D}. $$
(68)

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Ortigosa, R., Martínez-Frutos, J., Gil, A.J. et al. A new stabilisation approach for level-set based topology optimisation of hyperelastic materials. Struct Multidisc Optim 60, 2343–2371 (2019). https://doi.org/10.1007/s00158-019-02324-5

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