K. Brakke, Some new values of Sylvester's function for n noncollinear points, J. Undergrad. Math.

K. Brakke, The Motion of a Surface by its Mean Curvature, Princeton University Press, 1978. The expanded book version of my Ph.D. thesis. Now out of print, so this is a scan of the book in PDF format (only 8MB!). And a LaTeX version, with a table of notation and some misprints corrected.

J. Mantock, K. Fukunage, and K. Brakke, Systematic feature extraction, IEEE Transactions on Pattern Analysis and Machine Intelligence

Voronoi tessellation manuscripts. Five unpublished papers on simulating Voronoi tessellations that I did in the mid-1980's.

K. Brakke, Minimal cones on hypercubes, J. Geom. Anal. vol. 1 (1991) 329-338.

PDF file, 130KB.

Abstract: It is shown that in dimension greater than 4, the minimal area hypersurface separating the faces of a hypercube is the cone over the edges of the hypercube. This constrasts with the cases of two and three dimensions, where the cone is not minimal. For example, a soap film on a cubical frame has a small rounded square in the center. In dimensions over 6, the cone is minimal even if the area separating opposite faces is given zero weight. The proof uses the maximal flow problem that is dual to the minimal surface problem.

K. Brakke, The Surface Evolver, Experimental Mathematics vol. 1 no. 2 (1992), 141-165.

PDF file, 400KB.

Abstract: The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity, and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface.

K. Brakke, The Opaque Cube Problem, Am. Math. Monthly vol. 99 (Nov. 1992), 866-871.

PDF file, 240KB.

Abstract: It is a classic puzzle to find the shortest set of curves that intersect all straight lines through a square, and the conjectured solution is still unproven. This paper asks the analogous question for a cube, and comes up with the best known solution.

K. Brakke, Minimal Surfaces, Corners, and Wires, J. Geom. Anal. vol. 2 no. 1 (1992), 11-36.

PDF file, 940KB.

Abstract: Weierstrass representations are given for minimal surfaces that have free boundaries on two planes that meet at an arbitrary dihedral angle. The contact angles of a surface on the planes may be different. These surfaces illustrate the behavior of soapfilms in convex and nonconvex corners. They can also be used to show how a boundary wire can penetrate a soapfilm with a free end, as in the overhand knot surface. They should also cast light on the behavior of capillary surfaces.

K. Brakke, Soap films and covering spaces, J. Geom. Anal. vol. 5 no. 4 (1995) 445-514.

PDF file, 3MB.

Abstract: A new mathematical model of soap films is proposed, called the "covering space model." The two sides of a film are modelled as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities, films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have proved troublesome for previous soap film models.

K. Brakke, Numerical Solution of Soap Film Dual Problems, Experimental Mathematics vol. 4 no. 4 (1995) 269-287.

PDF file, 1MB (and cover illustration)

Abstract: The soap film problem is to minimize area, and its dual is to maximize the flux of a divergenceless bounded vectorfield. This paper discretizes the continuous problem and solves it numerically. This gives upper and lower bounds on the area of the globally minimizing film. In favorable cases, the method can be used to discover previously unknown films. No initial assumptions about the topology of the film are needed. The paired calibration or covering space model of soap films is used to enable representation of films with singularities.

K. Brakke, R. Phelan, and D. Weaire, Computation of equilibrium foam structure using the Surface Evolver, Experimental Mathematics

PDF file, 5.5MB. (and cover illustration)

Abstract: The Surface Evolver has been used to minimise the surface area of various ordered structures for monodisperse foam. Additional features have enabled its application to foams of arbitrary liquid fraction. Early results for the case of dry foam (negligible liquid fraction) produced a structure haveing lower surface area, or energy, than Kelvin's 1887 minimal tetrakaidecahedron. The calculations reported here show that this remains the case when the liquid fraction is finite, up to about 11%, at which point an f.c.c arrangement of the cells becomes preferable.

K. Brakke and T. Singler, Computer simulation of solder bridging phenomena, Transactions of the ASME

Abstract: Solder bridging is investigated under the assumption that liquid solder bridges are equilibrium capillary surfaces and that the principal factor that determines whether a bridge will freeze to form a permanent short is its configurational stability. A computational paramemtric bridge stability study is conducted to determine the response of bridging to the system volume, the distance between pads, the contact angle between the liquid metal ant resist surface and the relevant physiochemical properties of the liquid metal.

K. Brakke, The Surface Evolver and the stability of liquid surfaces , Phil. Trans. R. Soc. A vol. 354 (1996) 2143-2157.

PDF file, 1MB

Abstract: The Surface Evolver is an interactive program for studying the shapes of liquid surfaces. Recently added features permit the calculation of the Hessian matrix of second derivatives of the energy. The Hessian can be used for fast convergence to an equilibrium, and eigenvalue analysis of the stability of that equilibrium. This paper describes the use of the Hessian by the Surface Evolver, presents some sample stability analyses, and gives some numerical results on the accuracy and convergence of the methods. It is also shown how one can evolve unstable surfaces.

G. Francis, J. M. Sullivan, R. Kusner, K. Brakke, C. Hartman, and G. Chapell, The minimax sphere eversion, in Mathematics and Visualization, ed. K. Polthier and H. Hege, Springer-Verlag, Berlin, 1997, 3-20.

PDF file, 2MB

Abstract: We consider an eversion of a sphere driven by a gradient flow for
elastic bending energy. We start with a halfway model which is an unstable
Willmore sphere with 4-fold orientation-reversing rotational symmetry. The
regular homotopy is automatically generated by flowing down the gradient of
the energy from the halfway model to a round sphere, using the Surface
Evolver. This flow is not yet fully understood; however, our numerical
simulations give evidence that the resulting eversion is isotopic to one of
Morin's classical sphere eversions. These simulations were presented as
real-time interactive animations in the CAVE automatic virtual environment
at Supercomputing'95, as part of an experiment in distributed, parallel
computing and broad-band, asynchronous networking.

Video available.

K. Brakke and J. Sullivan, Using symmetry features of the Surface Evolver to study foams, in

PDF file, 1.3MB

Abstract: This paper describes the use of various symmetry features, including periodic boundary conditions, mirror boundaries, and rotational symmetry, in the Evolver. As a test case, we use these features to study foams, in particular the equal-volume foams of Kelvin and Weaire-Phelan. To compute the shape and energy of one of these compound surfaces, it is most efficient to work with only the smallest possible fundamental domain.

F. Baginski and K. Brakke, Modeling ascent configurations of strained high-altitude balloons, AIAA Journal

PDF file, 2MB.

Abstract: We consider the problem of estimating stresses in the ascent shape of an elastic high-altitude scientific balloon. The balloon envelope consists of a number of long, flat, tapered sheets of polyethylene called gores that are sealed edge-to-edge to form a complete shape. Because the film is so thin, it has zero bending stiffness and cannot support compressions. In particular, the balloon film forms internal folds of excess material when the volume is not sufficiently large. Because of these factors, a standard finite element approach will have difficulty computing partially inflated balloon shapes. In our approach, we develop a variational principle for computing strained balloon shapes that incorporates regions of folded material as a part of the geometric model. We can apply our model to fully inflated or partially inflated configurations. The equilibrium shape is the solution of minimum energy satisfying a given volume constraint. We apply our model to a design shape representative of those used in scientific ballooning and compute a family of ascent configurations with regions of external contact for a volume as low as 22% of its float value.

K. Brakke and F. Morgan, Instability of the wet X soap film, J. Geom. Anal. 8, no. 5 (1998), 749-768.

PDF file, 680KB.

Abstract: For idealized, infinitely thin ("dry") soap films, an X is unstable, while for very thick ("wet") soap films it is minimizing. We show that for soap films of relatively small but positive wetness, the X is unstable. Full stability diagrams for the constant liquid fraction case and the constant pressure case are generated. Analogous questions about other singularities remain controversial.

K. Brakke and F. Morgan, Instabilities of cylindrical bubble clusters, Eur. Phys. J. E

PDF file, 650 KB

Abstract: Small bubbles in an experimental two-dimensional foam between glass plates regularly undergo a three-dimensional instability as the small bubbles shrink under diffusion or equivalently as the plate separation increases, and end up on one of the plates. The most recent experiments of Cox, Weaire, and Vaz are accompanied by Surface Evolver computer simulations and rough theoretical calculations. We show how a recent second variation formula may be used to perform exact theoretical calculations for infinitesimal perturbations for such a system, and verify results with Surface Evolver simulations.

K. Brakke, Instability of the wet cube cone soap film, Colloids and Surfaces A: Physiochem Eng. Aspects 263 (2004) 4-10.

PDF file, 389 KB.

Abstract: A "dry" conical soap film on a cubical frame is well known not to be stable. Recent experimental evidence seems to indicate that adding liquid to form "Plateau borders" stabilizes the conical film, perhaps to arbitrarily low liquid volumes. This paper presents numerical simulation evidence that the wet cone is unstable for low enough liquid volume, with the critical volume fraction being about 0.000274.

F. Baginski, K. Brakke, and W. Schur, Stability, clefting, and other issues related to undesirable equilibria in large pumpkin balloons, 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference; Austin, TX; USA; 18-21 Apr. 2005. pp. 1-13.

PDF file, 430 KB.

Abstract: NASA's effort to develop a large payload, high altitude, long duration balloon, the Ultra Long Duration Balloon, focuses on a pumpkin shape super-pressure design. It has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. Hoop stress considerations in the pumpkin design leads to choosing the lowest possible bulge radius, while robust deployment is favored by a large bulge radius . Some qualitative understanding of design aspects on undesired equilibria in pumpkin balloons has been obtained via small-scale balloon testing. Poorly deploying balloons have clefts, but most gores away from the cleft deploy uniformly. Mechanical locking may be a contributing factor in the formation of such undesired equilibria. Long term success of the pumpkin balloon for NASA requires a thorough understanding of the phenomenon of multiple stable equilibria. This paper uses the notion of stability to cl assify balloon designs. When we applied our model to a balloon based on the NASA Phase IV-A pumpkin design, we found the fully inated/fully deployed strained equilibrium float configuration to be unstable. To explore the sensitivity of this particular design and to demonstrate our general approach, we carry out a number of parametric studies that are variations on the Phase IV-A design. In this paper, we will focus on analytical studies, but we also compare our results with experimental and flight data wh enever possible. We will discuss the connection between stability and the generic deployment problem.

D. Weaire, S. Cox, and K. Brakke, Liquid Foams, book chapter in Cellular Ceramics, ed. P. Colombo and M. Scheffler, Wiley-VCH, Weinheim, 2005.

PDF file, 460 KB

Abstract: The elegant structure of a liquid foam and its constituent parts have fascinated scientists for centuries. A combination of experiments, theory and simulations has elucidated most of its static and quasi-static properties. However, this is only part of a wider subject: dynamic effects remain as a considerable challenge, particularly for wet foams.

M. Anderson, C. Egger, J. Casci, G. Tiddy, and K. Brakke, A new minimal surface and the structure of mesoporous silicas, Angew. Chem. Int. Ed.

PDF files, 670 KB and 315 KB

Abstract: We report our studies of the structure of the surfactant-templated, cubic, mesoporous silica superstructure SBA-1 and provide a formulation in terms of curvature that has important repercussions for both surfactant structures and the mechanism of formation of inorganic replicas. We establish that the crucial interface that determines the inorganic structure is between the silica and water adsorbed at the micelle surface, not between silica and surfactant, thus challenging the present synthesis me chanisms. We adopt a general protocol for understanding the surface curvature and energy which could be applied widely to the growth of inorganic structures in biology, including nonperiodic and disordered structures.

F. Baginski, K. Brakke, and W. Schur, Stability of cyclically symmetric strained pumpkin balloon configurations and the formation of undesired equilibria, Journal of Aircraft

PDF file, 3MB

Abstract: NASA'S effort to develop a large payload, high altitude, long-duration balloon, the ultralong duration balloon, focuses on a pumpkin shape superpressure design. It has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. Hoop stress considerations in the pumpkin design lead to choosing the lowest possible bulge radius, whereas robust deployment is favored by a large bul ge radius. Mechanical locking may be a contributing factor in the formation of undesired equilibria. Long term success of the pumpkin balloons for NASA requires a thorough understanding of the phenomenon of multiple stable equilibria. This paper uses the notion of stability to classify balloon designs. When we applied our finite element model to a balloon based on the NASA Phase IV-A pumpkin design, we found the fully inflated/fully deployed strained equilibrium float configuration was unstable. To demonst rate our approach for exploring the stability of constant bulge radius designs and their sensitivity to parameter changes we carry out a number of parametric studies. We focus on analytical studies, but we also compare our resuts with flight data whenever possible.

F. Baginski, K. Brakke, and W. Schur, Unstable, cyclically symmetric and stable, asymmetric pumpkin-balloon configurations, Journal of Aircraft

PDF file, 4MB

Abstract: By design, a pumpkin balloon is intended to assume a cyclically symmetric "pumpkin-like" shape once it reaches float altitude and is fully inflated. Recent work by the authors showed that under certain circumstances, a strained cyclically symmetric pumpckin balloon configuration can be unstable. This means the balloon must assume an alternate non-cyclically symmetric stable equilibrium shape. Julian Nott's round-the-world balloon Endeavoru was on of the first pumpkin-type balloons to encounter this instability. In this paper, we will explore the phenomena of unstable cyclically symmetric and stable asymmetric balloon configurations.

F. Baginski, K. Brakke, and W. Schur, Cleft formation in pumpkin balloons, Advances in Space Research

PDF file, 3MB

Abstract: NASA's development of a large payload, high altitude, long duration balloon, the Ultra Long Duration Balloon, centers on a pumpkin shape super-pressure design. Under certain circumstances, it has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. In this paper, we will use th concept of stability to classify equilibrium shapes of fully pressurized/fully deployed strained ball oons.

G. C. Shearman, B. J. Khoo, M. Motherwell, K. A. Brakke, O. Ces, C. E. Conn, J. M. Seddon, and R. H. Templer, Calculations of and evidence for chain packing stress in inverse lyotropic bicontinuous cubic phases, Langmuir

PDF file, 470 KB

Abstract: Inverse bicontinuous cubic lyotropic phases are a complex solution to the dilemma faced by all self-assembled water-amphiphile systems: how to satisfy the incompatible requirements for uniform interfacial curvature and uniform molecular packing. The solution reached in this case is for the water-amphiphile interfaces to deform hyperbolically onto triply periodic minimal surfaces. We have previously suggested that although the molecular packing in these structures is rather uniform the relative phase behavi or of the gyroid, double diamond, and primitive inverse bicontinuous cubic phases can be understood in terms of subtle differences in packing frustration. In this work, we have calculated the packing frustration for these cubics under the constraint that their interfaces have constant mean curvature. We find that the relative packing stress does indeed differ between phases. The gyroid cubic has the least packing stress, and at low water volume fraction, the primitive cubic has the greatest packing stress. However, at very high water volume fraction, the double diamond cubic becomes the structure with the greatest packing stress. We have tested the model in two ways. For a system with a double diamond cubic phase in excess water, the addition of a hydrophobe may release packing frustration and preferentially stabilize the primitive cubic, since this has previously been shown to have lower curvature elastic energy. We have confirmed this prediction by adding the long chain alkane tricosane to 1-monoolein in excess water. The model also predicts that if one were able to hydrate the double diamond cubic to high water volume fractions, one should destabilize the phase with respect to the primitive cubic. We have found that such highly swollen metastable bicontinuous cubic phases can be formed within onion vesicles. Data from monoelaidin in excess water display a well-defined transition, with the primitive cubic appearing above a water volume fraction of 0.75. Both of these results lend support to the proposition that differences in the packing frustration between inverse bicontinuous cubic phases play a pivotal role in their relative phase stability.

D-Y. Shin, K. A. Brakke, Theoretical analysis of jetting and ink filling processes for TFT LCD colour filters, Proceedings of the 10th Asian Symposium on Information Display, August, 2007, Singapore.

PDF file, 217 KB

The fabrication of TFT LCD colour filters with the piezo Drop-On-Demand (DOD) inkjet printing technology has gained attention from industries. However, this technology differs from previous processes such as spin and slit coating technologies in terms of the degree of complexity. Different from spin and slit coating processes, the piezo DOD inkjet printing technology has the capability to selectively deposit ink droplets on the positions, which greatly saves the waste of materials in producing TFT LCD colo ur filters. This feature, however, draws two engineering difficulties. First, the ink droplet volume should be carefully controlled to avoid the total ink volume variation among subpixels, which, otherwise, could cause visible swathe marks. Second, ink droplets must be confined without the introduction of unfilled regions in a subpixel and spilling over into the adjacent subpixels. In this study, two fundamental theoretical analyses are performed to investigate one possible cause of visible swathe marks an d suggest a concise way to derive the optimum surface conditions which eventually confine ink in a subpixel.

D-Y. Shin, K. A. Brakke, Theoretical Investigation of Jetting and Wetting Phenomena for the Fabrication of TFT LCD Color Filters, August, 2007, Daegu, South Korea, 376-379.

PDF file, 217 KB

Abstract: Although years of trials for the fabrication of TFT LCD color filters with the piezo Drop-On-Demand (DOD) inkjet printing technology have been made, the underlying physics of jetting and wetting has not been fully understood. In this study, the key engineering issues, jetting and wetting, are investigated with mathematical models.

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