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.
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Acknowledgements
This research was conducted within the DCAMM Research School through a grant from the Danish Agency for Science, Technology and Innovation. The authors would like to thank the TopOpt research group (www.topopt.dtu.dk) for fruitful discussions.
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Appendix A: Adjoint sensitivity analysis
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:
The derivative of the Lagrangian is obtained using the chain rule
rearranging and splitting yields: Find \({\boldsymbol \lambda}\) such that
and the sensitivities can be evaluated by computing
This leads to the following adjoint problem for the objective function:
Find for all such that
where the solution is used for the evaluation of the sensitivities
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
and the sensitivities of the entries can be derived using the adjoint method.
which yields the following adjoint problem
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
1.2 A.2 Permeability constraints
For each constraint based on the normal permeability (i = k),
the sensitivity needs to be computed. The derivation is as follows. Forming the residual
Then forming the derivative of the Lagrangian
which leads to the following adjoint problem
Find and \(p^k\in\mathcal{Q}\) such that and \(\hat{p}\in \mathcal{Q}\)
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
but only for i = k.
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Andreasen, C.S., Sigmund, O. Saturated poroelastic actuators generated by topology optimization. Struct Multidisc Optim 43, 693–706 (2011). https://doi.org/10.1007/s00158-010-0597-4
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DOI: https://doi.org/10.1007/s00158-010-0597-4