This paper demonstrates the application of the topology optimization method as a general and systematic approach for microfluidic mixer design. The mixing process is modeled as convection dominated transport in low Reynolds number incompressible flow. The mixer performance is maximized by altering the layout of flow/non-flow regions subject to a constraint on the pressure drop between inlet and outlet. For a square cross-sectioned pipe the mixing is increased by 70% compared with a straight pipe at the cost of a 2.5 fold increase in pressure drop. Another example where only the bottom profile of the channel is a design domain results in intricate herring bone patterns that confirm findings from the literature. TOPOLOGY OPTIMIZATION OF MICROFLUIDIC MIXERS 499 diffusive properties, one can alter the geometry such that the flow distributes the matter more evenly in the solvent. By doing so, convection of matter is used as a mechanism to enhance mixing which; however, comes at the cost of an increased pressure drop between inlet and outlet. Stroock et al. [2] presented a mixer that induces chaotic advection by sequencing asymmetric microchannel sections containing staggered herringbones. For the systematic design of such microchannel mixers, topology optimization could be useful since no prerequisites are taken with respect to the geometry, only a design domain and boundary conditions need to be specified a priori.The material distribution method for topology optimization was first presented by Bendsøe and Kikuchi [3] for solid mechanics problems. Since then, topology optimization has been introduced in several other branches of physics such as optics, acoustics and flows (see e.g. Bendsøe and Sigmund [4] for an overview of the subject).Optimal design in fluid mechanics has been studied long before topology optimization was invented and optimal shapes minimizing the dissipated power for different profiles subjected to Stokes flow were already determined analytically in the 1970s by Pironneau [5] using shape optimization. As opposed to shape optimization, however, topology optimization allows introduction of new boundaries as the optimization progresses. This allows the topology to change several times during the optimization, which is impossible in shape optimization where the topology (i.e. the number of boundaries and connectivity) is predetermined.In topology optimization the geometry is represented as a gray-scale image. The color in each pixel (finite element) represents a value of a physical parameter, e.g. permeability, such that black pixels represent small permeability (no-flow regions with 'solid-like' material) and white pixels represent large permeability (fluid regions). Computationally, the gray-scale in each element is a design variable. Based on repeated finite element analyses the design variables are updated using gradient driven math programming tools as described in e.g. [4].Topology optimization in fluid mechanics was introduced by Borrvall and Petersson [6] modeling 2D flow in a Brinkman med...

a b s t r a c tWe consider a model problem of isogeometric shape optimization of vibrating membranes whose shapes are allowed to vary freely. The main obstacle we face is the need for robust and inexpensive extension of a B-spline parametrization from the boundary of a domain onto its interior, a task which has to be performed in every optimization iteration. We experiment with two numerical methods (one is based on the idea of constructing a quasi-conformal mapping, whereas the other is based on a spring-based mesh model) for carrying out this task, which turn out to work sufficiently well in the present situation. We perform a number of numerical experiments with our isogeometric shape optimization algorithm and present smooth, optimized membrane shapes. Our conclusion is that isogeometric analysis fits well with shape optimization.

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This paper deals with isogeometric analysis of the 2-dimensional, steady state, incompressible Navier-Stokes equation subjected to Dirichlet boundary conditions. We present a detailed description of the numerical method used to solve the boundary value problem. Numerical inf-sup stability tests for the simplified Stokes problem confirm the existence of many stable discretizations of the velocity and pressure spaces, and in particular show that stability may be achieved by means of knot refinement of the velocity space. Error convergence studies for the full Navier-Stokes problem show optimal convergence rates for this type of discretizations. Finally, a comparison of the results of the method to data from the literature for the the lid-driven square cavity for Reynolds numbers up to 10,000 serves as benchmarking of the discretizations and confirms the robustness of the method.

SUMMARYThis paper studies topology optimization of a coupled opto-mechanical problem with the goal of finding the material layout which maximizes the optical modulation, i.e. the difference between the optical response for the mechanically deformed and undeformed configuration. The optimization is performed on a periodic cell and the periodic modeling of the optical and mechanical fields have been carried out using transverse electric Bloch waves and homogenization theory in a plane stress setting, respectively. Two coupling effects are included being the photoelastic effect and the geometric effect caused by the mechanical deformation.For the studied objective and material choice it is concluded that the photoelastic effect and the geometric effect counteract each other, which yields designs which are fundamentally different if the optimization takes only one effect into account. When both effects are active a compromise is found; however, a strong regularization is needed in order to achieve reasonable 0-1 designs with a clear physical interpretation.