Topology optimization and 3D printing of large deformation compliant mechanisms for straining biological tissues

Abstract

This paper presents a synthesis approach in a density-based topology optimization setting to design large deformation compliant mechanisms for inducing desired strains in biological tissues. The modelling is based on geometrical nonlinearity together with a suitably chosen hypereleastic material model, wherein the mechanical equilibrium equations are solved using the total Lagrangian finite element formulation. An objective based on least-square error with respect to target strains is formulated and minimized with the given set of constraints and the appropriate surroundings of the tissues. To circumvent numerical instabilities arising due to large deformation in low stiffness design regions during topology optimization, a strain-energy-based interpolation scheme is employed. The approach uses an extended robust formulation, i.e., the eroded, intermediate, and dilated projections for the design description as well as variation in tissue stiffness. Efficacy of the synthesis approach is demonstrated by designing various compliant mechanisms for providing different target strains in biological tissue constructs. Optimized compliant mechanisms are 3D printed and their performances are recorded in a simplified experiment and compared with simulation results obtained by a commercial software.

Appendix: Evaluating the derivative \(\frac {{\partial }f_{k}}{{\partial }\mathbf {u}}\)

In view of (14), one finds the derivative $\frac {{\partial }f_{k}}{{\partial }\mathbf {u}_{e}}$ as⁸

$$ \begin{array}{@{}rcl@{}} \frac{{\partial}f}{{\partial}\mathbf{u}_{e}} &=&\frac{2}{N_{be}}\\&&\times\displaystyle \sum\limits_{e= 1}^{N_{be}}\left( \frac{w_{1}(\epsilon_{\text{xx}}^{e} \!-\epsilon_{\text{xx}}^{*})\frac{{\partial}\epsilon_{\text{xx}}^{e}}{{\partial}\mathbf{u}_{e}} \!+w_{2}(\epsilon_{\text{yy}}^{e} \!-\epsilon_{\text{yy}}^{*})\frac{{\partial}\epsilon_{\text{yy}}^{e}}{{\partial}\mathbf{u}_{e}} \!+ \!+w_{3}(\epsilon_{\text{xy}}^{e} -\epsilon_{\text{xy}}^{*})\frac{{\partial}\epsilon_{\text{xy}}^{e}}{{\partial}\mathbf{u}_{e}}} {w_{1}(\epsilon_{\text{xx}}^{*})^{2} + w_{2}(\epsilon_{\text{yy}}^{*})^{2} + w_{3}(\epsilon_{\text{xy}}^{*})^{2}} \right)\\ \end{array} $$

(A.1)

One needs $\frac {{\partial }\epsilon _{\text {xx}}^{e}}{{\partial }\mathbf {u}_{e}}$, $\frac {{\partial }\epsilon _{\text {yy}}^{e}}{{\partial }\mathbf {u}_{e}}$ and $\frac {{\partial }\epsilon _{\text {xy}}^{e}}{{\partial }\mathbf {u}_{e}}$ and they can be extracted from the derivative $\frac {{\partial }\mathbf {E}_{e}}{{\partial }\mathbf {u}_{e}}$. Now, using (7) and (6), we have⁹

$$ \mathbf{E} = \frac{1}{2}\left( \nabla_{0}\mathbf{u} + {(\nabla_{0}\mathbf{u})^{\top}} + \nabla_{0}\mathbf{u}{(\nabla_{0}\mathbf{u})}^{\top}\right), $$

(A.2)

In view of FE setting, the displacement vector u of an element in terms of its nodal displacements ${u_{I}^{A}}$ and bi-linear shape functions N_A can be written as¹⁰:

$$ \mathbf{u} = \sum\limits_{A}N_{A}(\boldsymbol{\zeta}){u_{I}^{A}} = N_{A}(\boldsymbol{\zeta}){u_{I}^{A}}. $$

(A.3)

Now, Eq. A.2 yields using Eq. A.3 as:

$$ E_{IJ} = \frac{1}{2}\left( \frac{{\partial}N_{A}}{{\partial}X_{J}}{u_{I}^{A}} + \frac{{\partial}N_{A}}{{\partial}X_{I}}{u_{J}^{A}} + \frac{{\partial}N_{A}}{{\partial}X_{I}}\frac{{\partial}N_{B}}{{\partial}X_{J}}{u_{K}^{A}} {u_{K}^{B}}\right). $$

(A.4)

One finds derivative of E_IJ with respect to ${u_{I}^{A}}$ as:

$$ \frac{{\partial}E_{IJ}}{{\partial}{u_{K}^{A}}} = \frac{1}{2}\frac{{\partial}N_{A}}{{\partial}X_{J}}\left( 2\delta_{IK} + 2\frac{{\partial}N_{B}}{{\partial}X_{I}}{u_{K}^{B}}\right), $$

(A.5)

and hence, $\frac {{\partial }\epsilon _{\text {xx}}^{e}}{{\partial }\mathbf {u}_{e}}$, $\frac {{\partial }\epsilon _{\text {yy}}^{e}}{{\partial }\mathbf {u}_{e}}$ and $\frac {{\partial }\epsilon _{\text {xy}}^{e}}{{\partial }\mathbf {u}_{e}}$.