Topology optimization of unsaturated flows in multi-material porous media: Application to a simple diaper model

Abstract

We present a mathematical approach to optimize the material distribution for fluid transport in unsaturated porous media. Our benchmark problem is a simplified diaper model as an exemplary liquid absorber. Our model has up to three materials with vastly different properties, which in the reference configuration are arranged in parallel layers. Here, swelling is neglected and the geometry of a swollen diaper is assumed and treated as a porous medium of high porosity. The imbibition process is then simulated by solving an unsaturated flow problem based on Richards' equation. Our aim is to maximize the amount of absorbed liquid by redistributing the materials. To this end, a density based multi-material topology optimization (based on the well known SIMP model) is performed. A sensitivity analysis for the nonlinear transient problem is provided, which enables the application of first order optimization solvers. We perform two-and three-material optimization and discuss several variants of the problem setting. We present designs with up to 45% more absorbed liquid compared to the reference configuration.

A Review of Topology Optimisation for Fluid-Based Problems

“…Ha et al [187] used the Darcy-Stokes interpolation method [175] to maximise the permeability of three-dimensional woven materials. A multimaterial approach is taken by Wein et al [188], where a highly nonlinear saturated porous model with multiple materials is applied to design diapers that quickly transports fluid away from the surface to capture it in the interior. Takezawa et al [189] used a Brinkman-Forchheimer macro model to optimise material microstructures for minimum flow resistance considering the trade-off between permeability and form-drag.…”

“…PM is the least used method with only a single paper [79]. Surprisingly, FVM is the second least used method with only 12 papers [21,47,52,80,82,115,129,130,137,144,159,188] or 7%, despite the fact that FVM for many years has been the preferred discretisation method for computational fluid dynamics. This can probably be explained by several factors: topology optimisation originates from solid mechanics where FEM is the preferred method; discrete adjoint approaches are easier using FEM than FVM; stabilised FEM has grown to be a mature and accurate method [193,194].…”

“…Analysing all papers, the vast majority use a steady-state laminar flow model with 158 papers or 85% as seen in Figure 8a. Only 13 papers or 7% consider a transient laminar flow model [72][73][74][75][76][77][78][79]145,150,152,158,188], with a meager six papers or 3% treating turbulent flow [21,[80][81][82][83]129]. In the case of time-dependent problems, this is most likely due to the vast increase in computational cost related to simulation of transient flow problems, where all temporal details must be resolved sufficiently and all temporal solutions saved in memory (or recomputed) for the adjoint solve.…”

“…The fluid-structure interaction category counts four papers [164,168,172,173], but it must be noted that, common for all, the three-dimensional design freedom is severely limited, as the design domain is restricted in the third dimension. Finally, a very large share of the three-dimensional results belong in the category of microstructure and porous media problems with 14 papers [175][176][177][179][180][181][182][183][184][187][188][189]191,192]. Since, in material design, it is natural to consider both stiffness and permeability, the need for three-dimensional design freedom is obvious, as two-dimensional models would have either the fluid or the solid phase disconnected.…”

“…Only 13 papers in total treat time-dependent problems [72][73][74][75][76][77][78][79]145,150,152,158,188], eight of which are for fluid flow only. Since most realistic flow applications exhibit some form of time-dependent behaviour through either time-dependent boundary conditions or flow instabilities, it is strongly recommended that more community effort is dedicated to expanding the research on applying topology optimisation to time-dependent fluid-based problems.…”