EVALUATION OF LEVEL SET TOPOLOGY OPTIMIZATION FORMULATIONS FOR DESIGN OF MINIMUM-DISPERSION MICROFLUIDIC DEVICES

Applying topology optimization to minimize the sample dispersion of a microfluidic device is challenging in part because the fictitious permeability used to classify topological regions tends to a smooth non-Boolean field, in contradiction to the requirement that a valid solution be Boolean. Imposing the Boolean condition must be done carefully in order to avoid spurious local minimizers or an excessively smooth solution. Here we investigate the behavior of two di! ere nt formulations of level-set based topology optimization on problems where we must find the Boolean field that best approximates a specified non-Boolean field. We introduce a formulation based on inequality constraints that can e! ect ively satisfy the Boolean condition. 1. Introduction. The goal in shape optimization is to identify the shape of a domain I such that when a physical problem is solved on that domain, an objective function is minimized. The most general form of shape optimization is topology opti- mization (2), in which the topology of the domain is allowed to vary; i.e., topological holes and/or islands can appear or disappear. By its nature, topology optimization is a large-scale boolean programming problem, because each point in a larger, embed- ding domain is either in I or not. However, in practice it is rarely approached directly in that form; rather, each point in the domain is characterized by a real number to be driven, somehow, to the boolean extremes of 0 or 1. Much of the art in topology optimization is choosing this reparametrization in such a way closely approximates a boolean field without introducing spurious local minimizers. In many problems, there is a natural reparametrization that is simple and works well. In structural topology optimization, for example, if the relationship between material density and sti! n ess is taken to be nonlinear and monotonic, then for certain objective functions there will be no payo! to intermediate material densities. In other words, in such problems the nature of the physical problem, objective function, and constraints drives the material density naturally to a boolean solution (2). In this case, nothing special has to be added to the method to produce a boolean solution. Here, though, we are interested in problems where no such lucky accident hap- pens. The motivating problem for this study is the design of minimum-dispersion microfluidic sensors; by minimum-dispersion we mean that a sample of particles that enters the device together will not be dispersed by the flow. We represent the shape of the channel by introducing an approximately-boolean permeability. The challenging feature of this problem is that for the physics and objective function under consid- eration, the solution to this problem is a permeability that varies smoothly between the boolean values of zero and one, with most of the domain being at intermediate permeabilities. Such a device cannot be manufactured. In this problem, then, we must somehow externally - by means of a penalty or constraint - enforce the boolean condition. The obvious way to do this is with a penalization on any non-boolean values, but that is easily seen to be non-convex, and indeed, in practice it is found that such a penalization introduces numerous artificial local minimizers. An improved formulation was introduced by Cunha in his thesis (4). Cunha penalizes the slope of a level set function, which, as will be discussed below, allows