Publications

This paper brings to the attention of the AES community our experience with developing and trying out a new course that focuses on the generation of system configuration (geometry, architecture, and drawing) during the optimization of performance. The configuration is free to morph. Real systems are destined to remain imperfect because of finiteness constraints. They are plagued by resistances to the flow of fluid, heat, and electricity. Resistances are always finite because of constraints. The balancing and distributing of resistances (irreversibility) through the available volume is the mechanism that generates the architecture. Our approach to this key topic is evolutionary, and from simple to complex. In the 2004 format, the course covers several main topics: the relationship between optimization and the generation of configuration, multi-scale hierarchical structures for fluid flow and heat flow, and tree-shaped networks for collection and distribution. The paper relies on classroom-level examples to illustrate these topics. The paper also discusses the teaching method, and the applicability of this course concept to other domains (cost minimization, urban design, etc.).

No abstract is provided for this article.

This paper outlines the fundamentals of the methods of exergy analysis and entropy generation minimization (or thermodynamic optimization—the minimization of exergy destruction). The paper begins with a review of the concept of irreversibility, entropy generation, or exergy destruction. Examples illustrate the accounting for exergy flows and accumulation in closed systems, open systems, heat transfer processes, and power and refrigeration plants. The proportionality between exergy destruction and entropy generation sends the designer in search of improved thermodynamic performance subject to finite-size constraints and specified environmental conditions. Examples are drawn from energy storage systems for sensible heat and latent heat, solar energy, and the generation of maximum power in a power plant model with finite heat transfer surface inventory. It is shown that the physical structure (geometric configuration, topology) of the system springs out of the process of global thermodynamic optimization subject to global constraints. This principle generates structure not only in engineering but also in physics and biology (constructal theory). Copyright © 2002 John Wiley & Sons, Ltd.

This paper documents the fundamental problem of designing a porous flow architecture that meets the requirements of facilitating flow access while storing and releasing heat to a flowing fluid. Examples of such designs are regenerators that operate cyclically in various types of heating or reheating furnaces. The main geometrical scales are determined for parallel flow channels in a fixed regenerator volume with a fixed porosity, by matching the time scales of convection along the channels and thermal diffusion. In accord with the constructal law, the route to better architectures for maximum heat transfer and minimum pressure losses is the morphing of the regenerator architecture from parallel channels to dendritic channels. Copyright © 2012 John Wiley & Sons, Ltd.

Ephraim Maurice SparrowOn Thursday Aug. 1, 2019, we were deeply saddened to hear that Professor Ephraim M. Sparrow passed away at the age of 91 years. Professor Ephraim M. Sparrow was born on May 27, 1928, and through his extensive professional career he became one of the best-known and respected researchers in the field of heat transfer. He received his B.Sc. and M.S. in 1948 and 1949, respectively, from the Massachusetts Institute of Technology (MIT). Then, he moved to Harvard University to receive his M.A. and Ph.D. in 1950 and 1952, respectively. His Ph.D. thesis is titled “Free Convection With Variable Properties and Variable Wall Temperature” under the supervision of Professor Howard Wilson Emmons.After finishing his Ph.D. in 1952, Professor Ephraim M. Sparrow joined Raytheon, where he worked on different heat transfer problems such as radar, electronics cooling, and the first microwave oven. Later, he joined the National Advisory Committee for Aeronautics (NACA) (now National Aeronautics and Space Administration (NASA)), Lewis Flight Propulsion Laboratory, Cleveland, OH where he worked on fluid mechanics and heat transfer. He had many publications from his work in NASA such as two papers [1,2] in the first issue of ASME Journal of Heat Transfer (JHT), which was published in February 1959 with 15 papers, two papers [3,4] in the second issue of JHT (May 1959), one paper [5] in the third issue of JHT (August 1959), and two papers [6,7] in the fourth issue of JHT (November 1959).In 1959, he started his career at the Thermodynamics and Heat Transfer Laboratory (THTL), Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN. THTL was established by Professor Ernst R. G. Eckert in 1950. Within the period 1968–1980, Professor Ephraim M. Sparrow served as the fluid mechanics chairman program. Within the period 1986–1988, Professor Ephraim M. Sparrow worked at the National Science Foundation (NSF), first as Program Manager of Heat Transfer and posteriorly as the CBET division director in the engineering directorate. At NSF, he stressed the need to focus on practical problems and important fundamental issues. At the time of his death, he was both the longest serving faculty member and the oldest in the Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN.During his academic life, Professor Sparrow supervised more than 250 Master's degree students and more than 100 doctoral degree students. Many of his previous Ph.D. graduate students became professors at well-known universities across the globe. Professor Sparrow has published many papers, reports, and books on fluid mechanics, and heat transfer such as Refs. [8] and [9]. His paper in ASME Journal of Heat Transfer (JHT), which was published in May 1977 about “Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area” [10] was his most cited paper. He pioneered research in many important areas of heat transfer and his extensive contributions provided tremendous incentive to researchers and engineers around the world.In May 1988, Eph's 60th Birthday was celebrated with an article published in the International Journal of Heat and Mass Transfer [11]. Following his 60th birthday in 1988 until his death in 2019, Eph did not show any evidence of slowing down and was, in his typical manner, very active in pursuing his research interests. Professor Ephraim M. Sparrow was very active until the last year of his life. For example, he had these publications in 2019 [12–18].During his academic career, following Professor W. H. Giedt in 1972, Professor Ephraim M. Sparrow served as the Editor of the ASME Journal of Heat Transfer until 1980 when this position was succeeded by Professor K. T. Yang.In 1979, he cofounded with Professor W. J. Minkowycz the journal titled Numerical Heat Transfer in order to serve as a forum for the dissemination of ideas and research in the field of numerical heat transfer and computational fluid dynamics (CFD). He was the guest editor on many occasions. For example, he was the guest editor of a Biomedical-Related Special Issue in the International Journal of Heat and Mass Transfer with Professor John P. Abraham, University of St. Thomas, in November 2008 [19]. He was co-author with Professors J. P. Abraham, and R. D. Lovik in a paper about “Unsteady, three-dimensional fluid mechanic analysis of blood flow in plaque-narrowed and plaque-freed arteries” in the Biomedical-Related Special Issue [20]. Also, volume 50, Advances in Heat Transfer was edited by Ephraim M. Sparrow, John P. Abraham, and John M. Gorman.1 This volume had six chapters. Professor Ephraim M. Sparrow was the co-author of the first chapter about “Enhancement of Jet Impingement Heat Transfer by Means of Jet Axis Switching” [21].Furthermore, the contributions of Professor Ephraim M. Sparrow have spanned many areas and have been recognized by numerous honors and awards. They include: Max Jakob Memorial Award in 1976, Member of the National Academy of Engineering in 1986, Monie A. Ferst Award in 1993, Fellow of the American Society of Mechanical Engineers (ASME), and the ASME Ralph Coates Roe Medal.In July 2020, there will be a Memorial Track of Professor Ephraim Sparrow for celebrating his life and research contributions. This special symposium will be organized by the Heat Transfer Division (HTD) in his memory. Contributions for this symposium are solicited from all pertinent technical topic areas involving personal recollections from the three programs: ICNMM 2020, SHTC 2020, and FEDSM 2020 at Rosen Shingle Creek, Orlando, FL, July 12–15, 2020.2It is hard to find proper words to express our feelings at this sad moment, but without any doubt the professional community will remember Professor Ephraim M. Sparrow as an excellent and prolific scientist, teacher, mentor and above all a pleasant and honest person, and a highly respectful colleague with an outstanding character and ethics. We will all miss him dearly.

Detailed studies of some material in this chapter have been made in the books by Straughan (2008, 2015a, b, c, d).

This paper reviews the recent progress made on the study of melting in the presence of natural convection in a porous medium saturated with a phase-change material. The basic geometries that have been considered are the confined porous medium that is heated from the...

No abstract is provided for this article.

Urban heat dome flow, which is also referred to as urban heat island circulation, is important for urban ventilation and pollutant transport between adjacent cities when the background wind is weak or absent. A “dome-shaped” profile can form at the upper boundary of the urban heat island circulation. The horizontal extent of the heat dome is an important parameter for estimating the size of the area it influences. This study reviews the existing data on the horizontal extent of the urban heat dome flow, as determined by using either field measurements or numerical simulations. A simple energy balance model is applied to obtain the maximum horizontal extent of a single heat dome over the urban area, which is found to be approximately 1.5 to 3.5 times the diameter of the city’s urban area at night. A linearized model is also re-analysed to calculate the horizontal extent of the urban heat dome flow. This analysis supports the results from the energy balance model. During daytime, the horizontal extent of the urban heat dome flow is found to be about 2.0 to 3.3 times the urban area’s diameter, as influenced by the convective turbulent plumes in the rural area.

Mohamed Gad-el-HakProfessor Mohamed Gad-el-Hak is an engineering scientist, globally well-known in the field of classical physics and the subfields of mechanics, biomechanics, fluid mechanics, turbulence, flow control, microelectromechanical systems (MEMS), microfluidics, and nanotechnology. He was born on Feb. 11, 1945 in Tanta, Egypt, a city in the heart of the Nile Delta, 94 km north of Cairo. Gad-el-Hak's elementary, secondary, and tertiary public education took place in Cairo. He received a B.Sc. in mechanical engineering from Ain Shams University in 1966, where he graduated summa cum laude and ranked first in his class. Gad-el-Hak moved to U.S. in 1968 to start his graduate studies. He received a Ph.D. Degree in fluid mechanics in 1973 from the Johns Hopkins University under the tutelage of Professor Stanley Corrsin. (Corrsin's graduate advisors at Caltech were Hans W. Liepmann and Theodore von Kárrmán. The latter's doctoral advisor at Göttingen was Ludwig Prandtl.) Gad-el-Hak's doctoral thesis is entitled “Experiments on the Nearly Isotropic Turbulence Behind a Jet-Grid.”After obtaining his Ph.D., Mohamed Gad-el-Hak joined the department of aerospace engineering at the University of Southern California, and afterward the department of engineering science and systems at the University of Virginia. After that, he was a senior research scientist and program manager at Flow Research Company in Seattle, Washington, where he managed different hydrodynamic and aerodynamic research projects. After this 10-year industrial experience, Gad-el-Hak returned to academia, joining the department of aerospace and mechanical engineering at the University of Notre Dame.In 2002, Gad-el-Hak accepted a position at Virginia Commonwealth University as chair of mechanical engineering and the Inez Caudill Eminent Professor of Biomedical Engineering. During his tenure as department chair, Gad-el-Hak initiated a unique graduate and undergraduate program in nuclear engineering. The unit is now known as the department of mechanical and nuclear engineering.Other academic appointments include visiting professorship at Institut de Mécanique de Grenoble, Université de-Poitiers, Friedrich-Alexander-Universität Erlangen–Nürnberg, Technische Universität München, Technische Universität Berlin, Brandenburgische Technische Universität Cottbus, Université de Valenciennes, and Peking University. Nonacademic appointments include the Naval Undersea Warfare Center in Newport, RI, and Forschungszentrum Rossendorf in Dresden, Germany.The present authors and numerous others describe Gad-el-Hak as original, pioneering, visionary, creative, indefatigable, nonconformist, having breadth and depth, and ahead of the curve. (The subject passionately opposed to the label “stable genius”.) Dr. Gad-el-Hak's teaching is innovative, rigorous, and does not appeal to the lowest common denominator, and his research is always leading edge. One of Gad-el-Hak's traits is his ability to invent novel measuring techniques where none existed. He has done that with the laser-induced fluorescence, compliant coating deformation, wind-waves characterizations, micropumping, and superhydrophobic surface's longevity, amount of entrapped air, and microscale thickness, among others.Dr. Gad-el-Hak holds two patents: one for a drag-reducing method for airplanes and underwater vehicles, and the other for a lift control device for delta wings. He has additionally disclosed seven patents. Gad-el-Hak has authored/edited 20 books [1–11], and authored 140 journal articles, 52 essays in magazines and newspapers, 63 book chapters, 223 conference papers, and 28 book reviews. He presented over 310 invited lectures in every continent except Antarctica.As of April 2020, Gad-el-Hak's papers have been cited more than 14,600 times in the technical literature, and his h-index is 53, i10-index is 142, and i100-index is 28 (source Google Scholar). Two of Gad-el-Hak's books have been translated into Chinese [9,10], and several of his articles/essays have been translated into Arabic, Chinese, the Czech language, French, German, Japanese, Spanish, and Turkish. Gad-el-Hak is the author of the book Flow Control: Passive, Active, and Reactive Flow Management, and editor of the books Frontiers in Experimental Fluid Mechanics, Advances in Fluid Mechanics Measurements, Flow Control: Fundamentals and Practices, The MEMS Handbook (first and second editions), Transition and Turbulence Control, and Large-Scale Disasters: Prediction, Control and Mitigation.Recent work on large-scale disasters by Gad-el-Hak [11] resulted in the establishment of a universal metric by which the severity of all natural and manmade disasters is measured. That book was the first in the U.S. and second in the world to view large-scale disasters from the physical point of view, in contrast to the social, logistical, or medical viewpoint.Professor Gad-el-Hak's scholarship has been featured in NPR, PBS, Nature magazine, Newsweek, and The New York Times. Additional to working in the broad field of mechanics, he penned essays and op/ed's on global warming [12]; energy crisis [13]; proliferation of scholarly publications [14]; massive open online courses (MOOC) [15]; university governance [16]; STEM and the humanities [17]; engineering education [18]; and societal values of basic research [19].In 1981, Dr. Gad-el-Hak authored the first archival paper to describe the laser-induced fluorescence flow visualization technique [20]. The novelty lies in the ability to generate a very thin sheet of laser light as to be able to see one plane at a time, and the use of extremely small amounts of fluorescent dye as not to make the fluid's interior opaque. Among the technique's advantages are its high signal-to-noise ratio and its ability to dissect the flow field, as a CAT scan would to solid objects. Laser-induced fluorescence is now routinely used in numerous laboratories around the world, for both gas and liquid flows. He also identified the mechanism by which a turbulent region grows into a laminar, vortical flow [20]. The efficient growth by destabilization mechanism is an order of magnitude more effective than the conventional entrainment process in which a turbulent region incorporates/engulfs the surrounding irrotational flow.The article “Reynolds Number Effects in Wall-Bounded Turbulent Flows” by Gad-el-Hak, published in Applied Mechanics Reviews, marked a paradigm shift in the approach to this subject [21]. Funding programs by the U.S. Defense Advanced Research Projects Agency (DARPA), Office of Naval Research (ONR), and Air Force Office of Scientific Research (AFOSR) were inspired by this seminal paper. He introduced the concept of selective/ targeted/opposition/reactive control [22] to achieve drag reduction, lift enhancement, mixing augmentation, and noise suppression in wall-bounded flows. This patented closed-loop control is now researched intensively around the world. Entire scientific conferences and funding programs are dedicated to reactive (in contrast to active) control.Dr. Gad-el-Hak conducted the groundbreaking experiments that detailed the fluid–compliant surface interactions in turbulent boundary layers [23–27]. He also introduced a noninvasive technique to probe the coating's instability waves. The laser-based probe has a spatial resolution of 1 μm and temporal resolution of several kHz. He was among the first group of aerodynamicists in the U.S. to work on the “supermaneuverability” research program [28], a word coined by the German aerodynamicist Wolfgang Herbst. The DARPA/AFOSR unsteady aerodynamics program formed the foundation of the millions of unmanned aerial vehicles flying today.Gad-el-Hak's paper in ASME Journal of Fluids Engineering, “A Novel Pump for MEMS Applications” [29], presented a new method for pumping fluids in microelectromechanical systems (MEMS) applications at very low Reynolds numbers. Inertial pumps do not work at low Reynolds numbers. The only type that worked for MEMS, prior to introducing Gad-el-Hak's rotary pump, was of the reciprocating variety. His most cited paper was published in the ASME Journal of Fluids Engineering, “The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture” [30]. He is the first scientist to place the fledgling field of microfluidics on firm physical ground. Whole books, courses, and funding programs sprang worldwide as a result of this single paper. As of April 2020, that article has been cited over 1480 times (source Google Scholar).Dr. Gad-el-Hak's analytical, numerical, and experimental research on the characterization of superhydrophobic coatings resulted in better understanding of the fledgling field. Twenty journal publications, including two invited review papers [31,32], resulted from his three-year effort. Two measurement techniques were introduced for the first time: (i) an in situ, noninvasive probe to assess the longevity of such coatings and (ii) a method to measure the coating's thickness down to the micron scale as well as the amount of entrapped air in the coating's micropockets [31–34]. Additionally, using electrospinning, Gad-el-Hak advanced a cost-effective fabrication technique that can produce superhydrophobic surfaces with both random and aligned fibrous micro/nanostructures. He also developed an alternative method for engineering affordable superhydrophobic surfaces by randomly depositing size-controlled hydrophobic aerogel particles on a substrate. Dr. Gad-el-Hak advanced theoretical and experimental strategies to characterize the micro/nanostructure of the surface morphology. His research showed that the laboratory-scale surfaces could be scaled-up for marine applications.To commemorate the 90th Anniversary of the ASME Fluids Engineering Division, Professor Mohamed Gad-el-Hak delivered the plenary talk “Nine Decades of Fluid Mechanics” at the ASME FEDSM 2016 meeting, July 10–14, 2016, Washington, DC. His talk was published in October 2016 in a special issue of the ASME Journal of Fluids Engineering. In this review paper [35], Gad-el-Hak covered the progress in fluid mechanics during the period 1926–2016. He described selected experimental, theoretical, and numerical advances, and elucidated the grand benefits of inventing the computer and the laser to the field of fluid dynamics.Dr. Gad-el-Hak's research covers a remarkable range of Reynolds, Mach, and Knudsen numbers. His recent work on hypersonic flows identified a new principle for aerodynamic heating [36,37]. The resulting five journal papers were independently highlighted in 2018 by two prestigious publications: American Institute of Physics' SciLight, and Oxford's National Science Review.In his role as an engineering professor, Gad-el-Hak developed a unique writing course for advanced undergraduate and beginning graduate students in science and engineering [38]. The class is writing intensive and, through assessments, has proven to be more effective than similar classes taught by English and communications faculty. The semester-long course has also been offered as a short course in other universities. His superbly penned essay for The Chronicle of Higher Education, “We Must Stop the Avalanche of Low-Quality Research” [39], was chosen in 2011 by the British Science Council to be part of a standardized English examination.Professor Gad-el-Hak has served as editor of eight international journals including Applied Mechanics Reviews, AIAA Journal, and Bulletin of the Polish Academy of Sciences. Gad-el-Hak is also a contributing editor for McGraw-Hill's Year Book of Science and Technology, for Springer-Verlag's Lecture Notes in Engineering and Lecture Notes in Physics, and for CRC Press's Mechanical Engineering Series. He serve as a consultant to the United Nations, the governments of 12 countries, and many industrial and academic concerns. Professor Gad-el-Hak has been a member of many advisory panels for the U.S. Department of Defense (DOD), Department of Energy (DOE), National Aeronautics and Space Administration (NASA), and National Science Foundation (NSF).In testament to Dr. Gad-el-Hak's successes, he has been the elected Fellow of the American Academy of Mechanics (AAM) [40], American Association for the Advancement of Science (AAAS), American Institute of Physics (AIP), American Physical Society (APS), and American Society of Mechanical Engineers (ASME). In addition, Gad-el-Hak is the recipient of numerous international awards, for example, the 14th Freeman Scholar Award by the American Society of Mechanical Engineers [41], the Japanese Government Research Award for Foreign Scholars, and the German Alexander von Humboldt Prize. He was designated an ASME Distinguished Lecturer as well as inducted into the Johns Hopkins Society of Scholars in 2002. In 2016, Dr. Gad-el-Hak was awarded an ASME Medal for seminal contributions to the discipline of fluids engineering. In the same year, he also received an ASME certificate of appreciation in testimony of the high regard of his associates and the deep appreciation of the society for his valued services in advancing the engineering profession.On the occasion of his 75th birthday, and on behalf of his friends, colleagues, and students all over the globe, the present authors wish Professor Mohamed Gad-el-Hak a continuous active life in happiness and good health, and a very happy birthday!

Realistic models of energy systems demand the treatment of installations and their flowing surroundings together, more so when the installations are large and their spheres of impact greater. The interface between energy systems and the environment is formed by...

Freedom, safety, and ease of movement are innate human urges attributed to conscience along with many other preferences such as attractiveness (beauty), economy, and life. This article addresses the physics basis of the innate urge to have freedom. It unveils the connection between animal freedom and the universal (constructal) tendency toward easier movement and greater access in all evolutionary systems throughout nature (animate & inanimate). The demonstration is made with a model of lack of freedom in animal movement: a man who walks his dog on a leash. When two animals are coerced to move at the same speed, their combined effort (the spent power) is greater than when they move freely, and independently. When the speed of the couple is dictated by the big body (man), the big one walks freely, and the small one must run. Participants in organized movement (life, society) are not equal. All participants move with less effort when they are not coerced to move the same way. The implications of this part of physics (nature) are numerous and help unify the animal realm with the design and evolution of human society. If you want diversity, give the population freedom, not prescriptions.

WhySocial organization and innovation does social organizationOrganization happen by itself? Why does it evolve as the movement of the members of the societySociety increases? Why does the organizationOrganization become more hierarchical, with greater...

The fundamental question in heat transfer engineering is to determine the relationship between the heat transfer rate and the driving temperature difference. In nature, many saturated porous media interact thermally with one another and with solid surfaces that...

This article is adapted from the Introduction of Design in Nature: How the Constructal Law Governs Evolution in Biology, Physics, Technology, and Social Organization, written by Adrian Bejan and J. Peder Zane. This law sweeps the entire mosaic of nature from inanimate rivers to animate designs, such as vascular tissues, locomotion, and social organization. The constructal law tears down the walls that have separated the disciplines of science by providing a new understanding of what it means to be alive. The constructal law defines life in physics terms, and it covers all live-system phenomena. The author believes that the constructal law also challenges another idea that has become dogma since Darwin—that there is no overarching direction to evolution. The constructal law, by contrast, predicts that evolution should occur because of the tendency of all flow systems to generate better and better designs for the currents that flow through them.

“Inequality” is a common observation about us, as members of society. In this article, we unify physics with economics by showing that the distribution of wealth is related proportionally to the movement of all the streams of a live society. The hierarchical distribution of wealth on the earth happens naturally. Hierarchy is unavoidable, with staying power, and difficult to efface. We illustrate this with two architectures, river basins and the movement of freight. The physical flow architecture that emerges is hierarchical on the surface of the earth and in everything that flows inside the live human bodies, the movement of humans and their belongings, and the engines that drive the movement. The nonuniform distribution of wealth becomes more accentuated as the economy becomes more developed, i.e., as its flow architecture becomes more complex for the purpose of covering smaller and smaller interstices of the overall (fixed) territory. It takes a relatively modest complexity for the nonuniformity in the distribution of wealth to be evident. This theory also predicts the Lorenz-type distribution of income inequality, which was adopted empirically for a century.

“Design in nature” is a topic of growing interest throughout science. The constructal law is the physics law of design generation and evolution in nature: “for a flow system to persist in time (to live), it must evolve such that it provides easier and easier access to its currents.” In this paper, we show how the constructal law accounts for the main features of the design of the biosphere: global movement of mass as the action of constructal engines (geophysical, animal, and human made) that dissipate their power into brakes, animal locomotion, vision, cognition, and hierarchy. The architecture and hierarchy of vegetation results from the constructal tendency to generate designs that facilitate the flow of water and “the flow of stresses” (i.e., mechanical strength per unit volume). Natural porous media have multiple scales because their flows are also configured as trees. The paper concludes with the oneness of design in nature, global design, and science and technology evolution—all as manifestations of the natural tendency captured by the constructal law and unified constructal theory of evolution.

This paper shows that the geometry of the heat flow path between a volume and one point can be optimized in two fundamentally different ways. In the “growth” method of the original constructal theory the structure is optimized starting from the smallest volume element of fixed size. Growth, or optimal numbers of constituents assembled into larger volumes, is one route to resistance minimization. In the “design” method the overall volume is fixed, and the designer works “inward” by optimizing the internal features of the heat flow path. The design method is new. It is shown analytically that the two methods produce comparable geometric results in which the high-conductivity channels form constructal tree networks, and where the low-conductivity material fills the interstices. For simplicity, it is assumed that the high-conductivity channels and their tributaries make 90-deg angles. In both approaches, the overall resistance decreases as the internal complexity of the conductive composite increases. In the growth method the number of constituents in each assembly can be optimized. In the design method, some of the constituent numbers cannot be optimized: these numbers assume the roles of weak parameters. The growth method is the simplest, and provides a useful approximation of the design and performance that can be achieved using the design method. Numerical solutions of the volume-to-point optimization problem confirm the results obtained analytically, and show that the geometric features of the optimal design are robust.