Saturated poroelastic actuators generated by topology optimization

Abstract

In this paper the fluid-structure interaction problem of a saturated porous media is considered. The pressure coupling properties of porous saturated materials change with the microstructure and this is utilized in the design of an actuator using a topology optimized porous material. By maximizing the coupling of internal fluid pressure and elastic shear stresses a slab of the optimized porous material deflects/deforms when a pressure is imposed and an actuator is created. Several phenomenologically based constraints are imposed in order to get a stable force transmitting actuator.

Appendix A: Adjoint sensitivity analysis

The Lagrangian $\mathcal{L}$ is formed by adding the adjoint variable multiplied by the residual (= 0) to the objective function, Φ:

$$ \mathcal{L}=\Phi(\mathbf{u},\rho)+{\boldsymbol \lambda}^TR(\mathbf{u},\rho) $$

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The derivative of the Lagrangian is obtained using the chain rule

$$ \begin{array}{rll} \frac{{\rm d} \mathcal{L}}{{\rm d} \rho}&=&\frac{\partial \mathcal{L}}{\partial \rho}+\frac{\partial \mathcal{L}}{\partial \mathbf{u}}\frac{{\rm d} \mathbf{u}}{{\rm d} \rho} \\ &=&\frac{\partial \Phi(\mathbf{u}, \rho)}{\partial \rho} +\frac{\partial \Phi(\mathbf{u}, \rho)}{\partial \mathbf{u}}\frac{{\rm d} \mathbf{u}}{{\rm d} \rho} \\ &&+{\boldsymbol \lambda}^T\frac{\partial R(\mathbf{u},\rho)}{\partial \rho} +{\boldsymbol \lambda}^T\frac{\partial R(\mathbf{u},\rho)}{\partial \mathbf{u}}\frac{{\rm d} \mathbf{u}}{{\rm d} \rho} \end{array} $$

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rearranging and splitting yields: Find ${\boldsymbol \lambda}$ such that

$${\boldsymbol \lambda}^T\frac{\partial R(\mathbf{u}, \rho)}{\partial \mathbf{u}}\frac{{\rm d} \mathbf{u}}{{\rm d} \rho} +\frac{\partial \Phi(\mathbf{u}, \rho)}{\partial \mathbf{u}}\frac{{\rm d} \mathbf{u}}{{\rm d} \rho}=0 $$

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and the sensitivities can be evaluated by computing

$$ \frac{{\rm d} \mathcal{L}}{{\rm d} \rho}=\frac{\partial \Phi(\mathbf{u}, \rho)}{\partial \rho}+{\boldsymbol \lambda}^T\frac{\partial R(\mathbf{u},\rho)}{\partial \rho} $$

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This leads to the following adjoint problem for the objective function:

Find for all such that

$$ \int\! E_{ ijlm}\varepsilon_{ij}({\boldsymbol \lambda})\varepsilon_{lm}(\hat{{\boldsymbol \lambda}}){\rm d} \Omega \!=\!-\!\int\!\frac{1}{2}(\!-\!\delta_{ij}) E_{ijlm}\varepsilon_{lm} (\hat{{\boldsymbol \lambda}}){\rm d} \Omega $$

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where the solution is used for the evaluation of the sensitivities

$$ \begin{array}{rll} \frac{{\rm d} \Phi}{{\rm d} \rho}&=& \int \bigg[\frac{1}{2}(1-\delta_{ij})E_{ijlm}'\varepsilon_{lm}(\mathbf{u})+\delta_{ij}\varepsilon_{ij}({\boldsymbol \lambda})\frac{E'}{E_0} \\ && \qquad -\,\varepsilon_{ij}({\boldsymbol \lambda})E_{ijlm}\varepsilon_{lm}(\mathbf{u})\bigg]{\rm d} \Omega \end{array} $$

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1.1 A.1 Stiffness constraints

The stiffness constraints are related to the problem above as the equation system is the same except for the loading that changes according to the direction considered. The stiffness tensor entries are computed as

$$ E_{ijkh}=E_{ijkh}-E_{ijlm}\varepsilon_{lm}(\mathbf{u}^{kh})$$

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and the sensitivities of the entries can be derived using the adjoint method.

$$ \begin{array}{rll} R_{ijkh}&=&\int_\Omega \varepsilon_{pq}({\boldsymbol \lambda}^{ij}) E_{pqkh}{\rm d} \Omega \\ &&-\int_\Omega \varepsilon_{pq}( {\boldsymbol \lambda}^{ij}) E_{pqlm} \varepsilon_{lm}(\mathbf{u}^{kh}) {\rm d} \Omega \\ \frac{\partial R_{ijkh}}{\partial \rho}&=&\int_\Omega \varepsilon_{pq}({\boldsymbol \lambda}^{ij}) E_{pqkh}'{\rm d} \Omega \\ && -\int_\Omega \varepsilon_{pq}( {\boldsymbol \lambda}^{ij}) E_{pqlm}' \varepsilon_{lm}(\mathbf{u}^{kh}) {\rm d} \Omega \\ \frac{\partial E_{ijkh}}{\partial \rho}&=&\int_\Omega E_{ijkh}'-E_{ijlm}'\varepsilon_{lm}(\mathbf{u}^{kh}) {\rm d} \Omega \\ \frac{\partial R_{ijkh}}{\partial \mathbf{u}}\frac{{\rm d} \mathbf{u}}{{\rm d} \rho} &=&-\int_\Omega \varepsilon_{pq}({\boldsymbol \lambda}^{ij}) E_{pqlm} \varepsilon_{lm}'(\mathbf{u}^{kh}) {\rm d} \Omega \\ \frac{\partial E_{ijkh}}{\partial \mathbf{u}}\frac{{\rm d} \mathbf{u}}{{\rm d} \rho} &=&- \int_\Omega E_{ijlm} \varepsilon_{lm}'(\mathbf{u}^{kh}) {\rm d} \Omega \end{array} $$

which yields the following adjoint problem

$$ -\!\int_\Omega \varepsilon_{pq}({\boldsymbol \lambda}^{ij}) E_{pqlm} \varepsilon_{lm}'(\mathbf{u}^{kh}) {\rm d} \Omega \!-\! \int_\Omega E_{ijlm} \varepsilon_{lm}'(\mathbf{u}^{kh}) {\rm d} \Omega\! =\! 0 $$

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$$ \Rightarrow\! \int_\Omega \varepsilon_{pq}({\boldsymbol \lambda}^{ij}) E_{pqlm} \varepsilon_{lm}(\hat{{\boldsymbol \lambda}}) {\rm d} \Omega \!=\!-\!\int_\Omega E_{ijlm}\varepsilon_{lm}(\hat{{\boldsymbol \lambda}}) {\rm d} \Omega $$

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which due to the symmetry of E is equivalent to the original problem but with opposing sign and hence the sensitivities can be computed from the original solution as

$$ \begin{array}{rll} \frac{{\rm d} E_{ijkh}}{{\rm d} \rho}&=&\int_{\Omega}\biggl[E_{ijkh}'-E_{ijlm}'\varepsilon_{lm}(\mathbf{u}^{kh}) \\ && \qquad +\,\varepsilon_{pq}(\mathbf{u}^{ij})E_{pqlm}'\varepsilon_{lm}(\mathbf{u}^{kh}) \\ && \qquad -\,E_{pqkh}'\varepsilon_{pq}(\mathbf{u}^{ij})\biggl]{\rm d} \Omega \end{array} $$

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1.2 A.2 Permeability constraints

For each constraint based on the normal permeability (i = k),

$$ K_{ik}=\frac{1}{|\Omega|}\int_\Omega \emph{v}_i^k {\rm d} \Omega $$

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the sensitivity needs to be computed. The derivation is as follows. Forming the residual

$$ \begin{array}{rll} R(\hat{\mathbf{v}},\mathbf{v}^k,\rho) &= &-\int_\Omega \hat{\emph{v}}_{i,j}{\emph{v}}_{i,j}^k {\rm d}\Omega - \int_\Omega\zeta(\rho){\emph{v}}_i^k\hat{\emph{v}}_{i} {\rm d}\Omega \\ && +\, \int_\Omega\hat{\emph{v}}_{i,i}p {\rm d}\Omega+ \int_\Omega \hat{\emph{v}}_i\delta_{ik}{\rm d}\Omega - \int_\Omega \hat{p}{\emph{v}}_{i,i}^k{\rm d}\Omega \end{array} $$

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Then forming the derivative of the Lagrangian

$$ \begin{array}{rll} \frac{{\rm d} \mathcal{L}}{{\rm d} \rho}&=&\frac{\partial K(\mathbf{v})}{\partial \mathbf{v}}\frac{{\rm d} \mathbf{v}}{{\rm d} \rho}+\frac{\partial R({\boldsymbol \lambda},\mathbf{v},\rho)}{\partial \rho} \\ &&+\frac{\partial R({\boldsymbol \lambda},\mathbf{v},\rho)}{\partial \mathbf{v}}\frac{{\rm d} \mathbf{v}}{{\rm d} \rho} +\frac{\partial R({\boldsymbol \lambda},\mathbf{v},\rho)}{\partial p}\frac{{\rm d} p}{{\rm d} \rho} \end{array} $$

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which leads to the following adjoint problem

Find and $p^k\in\mathcal{Q}$ such that and $\hat{p}\in \mathcal{Q}$

$$ \begin{array}{rll} -\int_\Omega \lambda^k_{i,j}\hat{\lambda}_{i,j} {\rm d}\Omega &-& \int_\Omega\zeta(\rho)\hat{\lambda}_i\lambda^k_{i} {\rm d}\Omega - \int_\Omega\lambda^k_{i,i}p {\rm d}\Omega \\ &+& \,\int_\Omega \lambda^k_i{\rm d}\Omega + \int_\Omega p\hat{\lambda}_{i,i}{\rm d}\Omega = 0 \end{array} $$

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which is seen to be equivalent to the state problem when i = k and $\lambda_i = \emph{v}_i$. Hence the sensitivities can be computed by evaluating

$$ \frac{{\rm d} K_{ik}(\mathbf{v})}{{\rm d} \rho}=-\frac{{\rm d} \zeta(\rho)}{{\rm d} \rho}\emph{v}_i^k\emph{v}_i^k $$

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but only for i = k.