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Robust Cassie–Baxter wettability of a rough solid surface with micro-textures is a key factor for stable hydrophobicity. Overlayed micro-textures are potentially more effective in ensuring the robustness of the surface properties, because of the layer-by-layer increase of the duty ratio and their effective approximation of the full hierarchy. However, a design methodology that includes considering manufacturability is lacking. In this article, we address this deficiency and present a monolithic inverse design approach, composed of a series of topology optimizations, to derive micro-textures with hierarchy approximated by overlayed geometries. The optimization are implemented in a dimensionless manner using a periodic regular-polygon tiling of the plane, in which the corresponding dimensionless Young-Laplace equation is used to describe the physics at the liquid/vapor interface. Two sequential and neighboring optimization tasks are linked through the design domain of the downward layer, determined by a conformal extension of the physical density representing the pattern of the upward layer. This ensures the manufacturability e.g. for an overlayed lithography process. Layer-by-layer robustness enhancement is thereby achieved, and the capability to anchor the three-phase contact line after the collapse of the liquid/vapor interface supported by the upward layer. In generating the overlayed micro-textures, a rigorous scaling factor for the patterns was determined, leading to a recursion inequality based on the depth of the liquid/vapor interfaces at the critical static pressures that determines the extrusion distance of the patterns. The trace height and minimal aspect ratio of the micro-textures are specified by the scaling factor and extrusion distance for a layer. This allows a compromise between performance and manufacturability, and thereby avoid instabilities caused by elasto-capillary collapse of the micro-/nano-structures. We computationally confirm the optimality by comparing the derived micro-textures with previously reported designs.

In confined microfluidic spaces such as microchannels, electroosmosis is a convenient Coulomb-force mechanism used to electrically actuate charged particles and ions presented in the fluid and pump the electrolytic fluid itself through drag forces. The shape and position of electrode pairs, whose induced charges are in contact with the fluid, determine the electric field and hence the resulting fluid-dynamic velocity distribution. In this paper, we address the inverse design of the electrode-pair patterns in such actuation mechanisms. Our approach is to use topology optimization to inversely determine the patterns of an electrode pair. The optimization procedure requires a mathematical description of the desired fluid behaviour, and then drives the patterns of the electrode pairs to achieve the goal performance. We demonstrate the behaviour of the procedure, which couples the Navier-Stokes equations with charge transportation, to implement an efficient electroosmotic micromixer for laminar microflow. We show that the procedure allows to investigate such microflows under the influence of selected parameter variations, thereby exploring the design space towards optimal device performance. This developed method is novel on the topology optimization of a surface structure to control bulk performance and its implementation over a lower-dimensional surface of an otherwise volumetric domain, where the material interpolation is implemented between Dirichlet and Newmann types of boundary conditions.

The robust Cassie–Baxter mode of the wetting behaviour on a micro-textured solid surface, is a key topography element yielding stable super-hydrophobicity. To meet this purpose, we propose an inverse computational design procedure for the discovery of suitable periodic micro-textures, based on three different tilings of the plane. The symmetric tiles of the lattice are regular triangles, quadrangles, and hexagons. The goal of the inverse design procedure is to achieve the robust Cassie–Baxter state, in which the liquid/vapour interface is mathematically described using the Young–Laplace equation on the lattice, and a topology optimisation approach is utilised to construct a variational problem for the inverse design procedure. Based on numerical calculations of the constructed variational problem, underlying effects are revealed for several factors, including the Bond number, duty ratio, feature size, and lattice constant. The effects of feature size and lattice constant provide approaches for compromisingly considering the robustness of the Cassie–Baxter mode and manufacturability of the inversely designed micro-textures; the effect of the lattice constant permits the scaling properties of the derived patterns, and this in turn provides an approach to avoid the elasto-capillary instability driven collapse of the micro/nanostructures in the derived micro-textures. Further, a monolithic inverse design procedure for the periodic micro-textures is proposed in the conclusions, with synthetically considering the manufacturability as well as contact angle and surface-volume ratio of the liquid bulge held by the supported liquid/vapour interface.

In topology optimization of electromagnetic waves, the Gâteaux differentiability of the conjugate operator to the complex field variable results in the complexity of the adjoint sensitivity, which evolves the original real-valued design variable to be complex during the iterative solution procedure. Therefore, the self-inconsistency of the adjoint sensitivity is presented. To enforce the self-consistency, the real part operator has been used to extract the real part of the sensitivity to keep the real-value property of the design variable. However, this enforced self-consistency can cause the problem that the derived structural topology has unreasonable dependence on the phase of the incident wave. To solve this problem, this article focuses on the self-consistent adjoint analysis of the topology optimization problems for electromagnetic waves. This self-consistent adjoint analysis is implemented by splitting the complex variables of the wave equations into the corresponding real parts and imaginary parts, sequentially substituting the split complex variables into the wave equations with deriving the coupled equations equivalent to the original wave equations, where the infinite free space is truncated by the perfectly matched layers. Then, the topology optimization problems of electromagnetic waves are transformed into the forms defined on real functional spaces instead of complex functional spaces; the adjoint analysis of the topology optimization problems is implemented on real functional spaces with removing the variational of the conjugate operator; the self-consistent adjoint sensitivity is derived, and the phase-dependence problem is avoided for the derived structural topology. Several numerical examples are implemented to demonstrate the robustness of the derived self-consistent adjoint analysis.

This paper presents the topology optimization of steady Navier–Stokes flows with body forces that influence the optimal shape and topology of fluid flows. Based on the implicit expression of the fluid flow with the level set method, an optimization problem is formulated and analyzed using the continuous adjoint method. The shape and topological sensitivities are computed based on the adjoint and asymptotic analysis of the optimization problem. In the optimization procedure, the level set surface is evolved based on the shape sensitivity and nucleated based on the topological sensitivity simultaneously. Three kinds of body forces that are commonly used in the design of fluid devices, i.e. constant, nonuniform, and solution-dependent body forces, are considered in the two-dimensional and three-dimensional numerical examples. Numerical results demonstrate that this method can effectively achieve the topology optimization of the Navier–Stokes flows with body forces.

This paper discusses the topology optimization of unsteady incompressible Navier-Stokes flows. An optimization problem is formulated by adding the artificial Darcy frictional force into the incompressible Navier-Stokes equations. The optimization procedure is implemented using the continuous adjoint method and the finite element method. The effects of dynamic inflow, Reynolds number and target flux on specified boundaries for the optimal topology of unsteady Navier-Stokes flows are presented. Numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady Navier-Stokes flows.