Design of microvascular flow networks using multi-objective genetic algorithms Alejandro M. Aragón a , Jessica K. Wayer b , Philippe H. Geubelle c, * , David E. Goldberg d , Scott R. White c a Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801, USA b Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 South Wright Street, Urbana, IL 61801, USA c Beckman Institute of Advanced Science and Technology, University of Illinois at Urbana-Champaign, 405 North Mathews Avenue, Urbana, IL 61801, USA d Illinois Genetic Algorithms Laboratory, University of Illinois at Urbana-Champaign, 104 South Mathews Avenue, Urbana, IL 61801, USA article info Article history: Received 30 January 2008 Accepted 13 May 2008 Available online 5 June 2008 Keywords: Microvascular network Genetic algorithms Multi-objective optimization Bio-mimetic material abstract A multi-objective genetic algorithm is used to design 2D and 3D microvascular networks embedded in bio-mimetic self-healing/self-cooling polymeric materials. Various objective functions and constraints are considered, ranging from flow efficiency and homogeneity to network redundancy and void volume fraction. The design variables include the network topology defined over a template and the microchan- nel diameters chosen among a finite set of values. The effect of network redundancy, template geometry and microchannel diameters on the Pareto-optimal fronts generated by the genetic algorithm is investigated. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Inspired by vascular networks in living organisms, materials consisting of a network of microchannels embedded in a polymeric matrix offer great potential in various autonomic healing, cooling and sensing applications. One example of a bio-mimetic material uses hollow fibers embedded in a polymeric matrix and filled with an uncured healing agent, which is released when the fibers expe- rience damage [13,23]. In a recent publication [35], Toohey and co- workers have demonstrated repeated healing of a polymer coating with the aid of a subsurface microvascular network containing a healing agent in monomeric form. As cracks form in the coating, the healing agent is wicked to the crack surfaces through capillar- ity and encounters solid catalyst particles contained in the coating. The healing process is thus initiated and can be repeated as long as the three-dimensional microvascular network contained in the polymeric substrate provides enough healing agent to the coating. The circulation of a liquid in the microvascular network is also being considered for thermal management of a structural compo- nent subjected to external thermal loading. An example of such an application can be found in [30] for the case of a microvascular network embedded in an epoxy matrix. Interest in this class of bio-mimetic materials has also been dri- ven by recent advances in manufacturing techniques such as the robotic deposition process, which allows for the creation of com- plex two- and three-dimensional microvascular network struc- tures [34,38]. In this process, the microvascular network is drawn by extruding a fugitive ink on a polymer substrate through needle tips so that the resulting microchannels have diameters that can be as small as 10 lm. After the drawing process has taken place, the resulting structure is embedded in a liquid polymer that is subsequently cured. The ink contained within the polymer is then evacuated by heating the material, leaving the embedded microvascular network. The use of needle tips with different sizes and a fully automated nozzle with three-dimensional motion re- sults in a powerful manufacturing methodology for the creation of complex patterns. Various methods have been proposed in the literature for the design of flow networks, which presents a set of unique challenges in terms of the complexity of the objective functions, design vari- ables and constraints. In the constructal theory [3], optimal flow structures are obtained by applying the constructal law, which states that the optimal flow structures should provide easier access to the flow than those that are non-optimal. This method has been used to optimize very simple geometries [4,19,39]. Another ap- proach relies on topology optimization, which has been used pri- marily in structural design, but has been recently extended to the design of flow networks considered either as a continuum or as a discrete system [5]. A comprehensive study of flow networks using a discrete topology is given in [20], where the diameter vari- ables are chosen from the positive real number set. Flow networks can also be designed by evolutionary algorithms, a family of biol- ogy-inspired methods that are increasingly gaining popularity be- cause of their simplicity and applicability to the optimization of a large set of problems in numerous fields. Starting from a popula- tion of candidate solutions, evolutionary algorithms search for bet- ter candidates by applying a set of genetic operators and using a 0045-7825/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2008.05.025 * Corresponding author. Tel.: +1 217 244 7648; fax: +1 217 244 0720. E-mail address: geubelle@uiuc.edu (P.H. Geubelle). Comput. Methods Appl. Mech. Engrg. 197 (2008) 4399–4410 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma