Surface Evolver Manual Version 2.26 August 25, 2005 Kenneth A. Brakke Mathematics Department Susquehanna University Selinsgrove, PA 17870 [email protected] http://www.susqu.edu/brakke Contents 1 Introduction. 9 1.1 General description ............................................ 9 1.2 Portability ................................................. 9 1.3 Bug reports ................................................ 10 1.4 Web home page .............................................. 10 1.5 Newsletter ................................................. 11 1.6 Acknowledgements ............................................ 11 2 Installation. 12 2.1 Microsoft Windows ............................................ 12 2.2 Macintosh ................................................. 13 2.3 Unix/Linux ................................................ 13 2.3.1 Compiling ............................................ 14 2.4 Geomview graphics ............................................ 14 2.5 X-Windows graphics ........................................... 15 3 Tutorial 16 3.1 Basic Concepts .............................................. 16 3.2 Example 1. Cube evolving into a sphere ................................. 17 3.3 Example 2. Mound with gravity ..................................... 20 3.4 Example 3. Catenoid ........................................... 22 3.5 Example 4. Torus partitioned into two cells ............................... 24 3.6 Example 5. Ring around rotating rod ................................... 26 3.7 Example 6. Column of liquid solder ................................... 31 3.8 Example 7. Rocket fuel tank ....................................... 34 3.8.1 Surface energy .......................................... 34 3.8.2 Volume .............................................. 35 3.8.3 Gravity .............................................. 36 3.8.4 Running .............................................. 37 3.9 Example 8. Spherical tank ........................................ 39 3.9.1 Surface energy .......................................... 40 3.9.2 Volume .............................................. 41 3.9.3 Gravity .............................................. 42 3.9.4 Running .............................................. 43 3.10 Example 9. Crystalline integrand ..................................... 45 3.11 Tutorial in Advanced Calculus ...................................... 46 2 Surface Evolver Manual 3 4 The Model 50 4.1 Dimension of surface ........................................... 50 4.2 Geometric elements ............................................ 50 4.2.1 Vertices .............................................. 50 4.2.2 Edges ............................................... 51 4.2.3 Facets ............................................... 52 4.2.4 Bodies. .............................................. 53 4.2.5 Facetedges ............................................ 53 4.3 Quadratic model .............................................. 53 4.4 Lagrange model .............................................. 54 4.5 Simplex model .............................................. 54 4.6 Dimension of ambient space ....................................... 54 4.7 Riemannian metric ............................................ 54 4.8 Torus domain. .............................................. 55 4.9 Quotient spaces and general symmetry .................................. 55 4.9.1 TORUS symmetry group ..................................... 56 4.9.2 ROTATE symmetry group .................................... 56 4.9.3 FLIP ROTATE symmetry group ................................. 56 4.9.4 CUBOCTA symmetry group ................................... 57 4.9.5 XYZ symmetry group ...................................... 57 4.9.6 GENUS2 symmetry group .................................... 57 4.9.7 DODECAHEDRON symmetry group .............................. 57 4.9.8 CENTRAL SYMMETRY symmetry group ........................... 58 4.9.9 SCREW SYMMETRY symmetry group ............................. 58 4.10 Symmetric surfaces ............................................ 58 4.11 Level set constraints ............................................ 58 4.12 Boundaries ................................................ 59 4.13 Energy. .................................................. 59 4.14 Named quantities and methods ...................................... 61 4.15 Pressure .................................................. 62 4.16 Volume or content ............................................. 62 4.17 Diffusion ................................................. 62 4.18 Motion ................................................... 63 4.19 Hessian .................................................. 63 4.20 Eigenvalues and eigenvectors ....................................... 64 4.21 Mobility .................................................. 65 4.21.1 Vertex mobility .......................................... 65 4.21.2 Area normalization ........................................ 65 4.21.3 Area normalization with effective area .............................. 66 4.21.4 Approximate polyhedral curvature ................................ 66 4.21.5 Approximate polyhedral curvature with effective area ...................... 66 4.21.6 User-defined mobility ...................................... 66 4.22 Stability .................................................. 66 4.22.1 Zigzag string ........................................... 66 4.22.2 Perturbed sheet with equilateral triangulation .......................... 67 4.23 Topology changes ............................................. 67 4.24 Refinement ................................................ 67 4.25 Adjustable parameters and variables ................................... 67 4.26 The String Model ............................................. 68 3 Surface Evolver Manual 4 5 The Datafile 69 5.1 Datafile organization ........................................... 69 5.2 Lexical format ............................................... 69 5.2.1 Comments ............................................ 69 5.2.2 Lines and line splicing ...................................... 69 5.2.3 Including files .......................................... 69 5.2.4 Macros .............................................. 70 5.2.5 Case ................................................ 70 5.2.6 Whitespace ............................................ 70 5.2.7 Identifiers ............................................. 70 5.2.8 Strings .............................................. 70 5.2.9 Numbers ............................................. 70 5.2.10 Keywords ............................................. 70 5.2.11 Colors ............................................... 71 5.2.12 Expressions ............................................ 71 5.3 Datafile top section: definitions and options ............................... 72 5.3.1 Macros .............................................. 72 5.3.2 Version check ........................................... 72 5.3.3 Element id numbers ....................................... 72 5.3.4 Variables ............................................. 73 5.3.5 Arrays ............................................... 73 5.3.6 Dimensionality .......................................... 73 5.3.7 Domain .............................................. 74 5.3.8 Length method .......................................... 74 5.3.9 Area method ........................................... 75 5.3.10 Volume method .......................................... 75 5.3.11 Representation .......................................... 75 5.3.12 Hessian special normal vector .................................. 76 5.3.13 Dynamic load libraries ...................................... 76 5.3.14 Extra attributes .......................................... 76 5.3.15 Surface tension energy ...................................... 77 5.3.16 Squared mean curvature ..................................... 77 5.3.17 Integrated mean curvature .................................... 77 5.3.18 Gaussian curvature ........................................ 77 5.3.19 Squared Gaussian curvature ................................... 78 5.3.20 Ideal gas model .......................................... 78 5.3.21 Gravity .............................................. 78 5.3.22 Gap energy ............................................ 78 5.3.23 Knot energy ............................................ 78 5.3.24 Mobility and motion by mean curvature ............................. 78 5.3.25 Annealing ............................................. 79 5.3.26 Diffusion ............................................. 79 5.3.27 Named method instances ..................................... 79 5.3.28 Named quantities ......................................... 80 5.3.29 Level set constraints ....................................... 81 5.3.30 Constraint tolerance ....................................... 82 5.3.31 Boundaries ............................................ 83 5.3.32 Numerical integration precision ................................. 83 5.3.33 Scale factor ............................................ 83 5.3.34 Mobility ............................................. 83 5.3.35 Metric ............................................... 84 4 Surface Evolver Manual 5 5.3.36 Autochopping ........................................... 84 5.3.37 Autopopping ........................................... 84 5.3.38 Total time ............................................. 84 5.3.39 Runge-Kutta ........................................... 84 5.3.40 Homothety