- 2.1. Medial axis and medial object, M. Price et al. [124].
- 2.2. Thin layers in two and three dimensions with the local feature size (radius of the circles/spheres) at example locations (stars).
- 3.1. Various types of not so well shaped elements and some parameters.
- 3.2. The relation between the edge length and its opposite angle in a triangle follows from and therefore .
- 3.3. With constant edge length and circumsphere radius the opposite dihedral angle in a tetrahedron can have arbitrary values.
- 3.4. A Voronoi box which intersects the boundary and an outside Voronoi point . The Voronoi regions for each point are shaded differently.
- 3.5. Finite element mesh criterion for two dimensions.
- 3.6. tessellation and Crit. 3.3.
- 3.7. tessellation, no obtuse dihedral angles.
- 3.8. Global stiffness matrix for a tessellation and local matrices of those four elements which are adjacent to edge .
- 3.9. type tessellation with a shifted point.
- 3.10. Element matrices which contribute to the entry in the global stiffness matrix for the edge . Due to the symmetry of the mesh the three matrices on the left and on the right side possess the same entries.
- 3.11. Delaunay mesh (), 3072 tetrahedra.
- 3.12. Delaunay mesh (), 2560 tetrahedra.
- 3.13. Non-Delaunay mesh, 2560 tetrahedra.
- 3.14. Red-Green refinement using mixed elements in three dimensions.
- 3.15. 3-2 or 2-3 local transformation. The internal facet which is being swapped is drawn shaded.
- 3.16. Splitting an edge to ensure a well connected surface topology. Correctly splitting a polygon requires an expensive calculation of all intersections.
- 3.17. Staircase effects approximate slopes and result in unnecessary large meshes.
- 3.18. Marching cubes algorithm applied to discrete data which describes the distribution of a finite number of materials.
- 3.19. Pathological cases and alternative templates.
- 3.20. Detail of a trench consisting of 2912 triangles before and 288 triangles after data reduction by locally discarding points.
- 4.1. Unstructured quadrilateral surface mesh, MENTAT II [98].
- 4.2. LOCOS: (a) structured mesh, 2000 tetrahedra (b) unstructured mesh, 957 tetrahedra.
- 4.3. Overlaying two layer descriptions and lateral two-dimensional unstructured mesh.
- 4.4. Layout structure description.
- 4.5. Interconnect simulation of a part of the layout using a layer-based product mesh.
- 4.6. Intersection of the cartesian cells with the boundary, M. Berger et al. [2].
- 4.7. Intersection based octree mesh of Flash EEPROM, ISE ETH [52].
- 4.8. Various situations in two dimensions and different patterns depending on the angle .
- 4.9. Three necessary tests to avoid collisions in three dimensions. The arrows show the direction of the advancing front and the tetrahedron which is tested and built.
- 4.10. Boundary mesh of the floating gate structure, 36 tetrahedra.
- 5.1. Each Voronoi box associated with a point is differently shaded. Two triangles with their circumcenters which are the vertices of the Voronoi boxes are depicted for the correct Delaunay case and for the non-Delaunay case. Incorrect Voronoi boxes which are derived from non-Delaunay triangles overlap.
- 5.2. The Delaunay edge (a) and Delaunay triangle (b) criteria.
- 5.3. (a) Boundaries which are not conform with the Delaunay Triangulation (b) A constrained Delaunay Triangulation (c) A conforming Delaunay Triangulation
- 5.4. A constrained Delaunay Triangulation with a non-Delaunay edge . The point does not affect edge . The half of the smallest sphere which lies inside the mesh is highlighted.
- 5.5. An untetrahedralizable twisted prism where the diagonals of the three side facets almost intersect.
- 5.6. Steiner point insertion at the circumcenter, removal of non-Delaunay elements, and triangulation of the resulting cavity.
- 5.7. Delaunay Triangulation vs. quality improved Steiner Triangulation. The original 130 triangles (94 points) were refined with 128 Steiner points resulting in 376 triangles.
- 5.8. The worst case element with a angle and minimum edge length . The largest circumcircle has radius .
- 5.9. A naive approach where the bisection of boundary edges and the insertion of circumcenters runs into an endless loop. The small angle which causes the insertion of a Steiner point at the center of the dotted circumcircle is shaded. A better solution can be obtained and is shown in the bottom left corner.
- 5.10. (a) Non-Delaunay sliver with circumsphere and two adjacent tetrahedra in the back (b) Strict sense Delaunay sliver with an empty circumsphere (c) Delaunay sliver with a cospherical point set.
- 6.1. Overall concept
- 6.2. Finite octree point generation.
- 6.3. Mesh for the octree point distribution, 25253 tetrahedra.
- 6.4. A triangle which is at first not flip-able and the state of flip saturation.
- 6.5. Multiple connected edges.
- 6.6. Non-convex coplanar triangles. The common edge is by definition flip-able, while the triangles are not.
- 6.7. Refinement types for structural edges.
- 6.8. Refining structural edges for the trivial case of a planar polygon.
- 6.9. Double sphere criterion.
- 6.10. Locating the triangle which contains the projection of the circumcenter and flipping of the non-structural edge.
- 6.11. Polygonal boundary description of a MOS Transistor with a spacer.
- 6.12. Structural edges of the MOS transistor example.
- 6.13. Delaunay surface triangulation.
- 6.14. Adapted surface mesh after Steiner point insertions.
- 6.15. Modified advancing front algorithm.
- 6.16. The triangle to which the next tetrahedron is attached is shaded for each step.
- 6.17. A snapshot of the growing mesh generated by the modified advancing front algorithm.
- 6.18. A non-uniform point bucketing scheme and a rectangular search region which is aligned with the bounding box and which is associated with a circle and a given value (two-dimensional analogy).
- 6.19. Every found point in the search region defines a .
- 6.20. The scope of for the two-dimensional case. Depending on the location of a point its value can reach a critical value. Singular regions of are indicated.
- 6.21. An open surface description and the growing mesh.
- 6.22. Complete boundary representation after tetrahedralization.
- 6.23. Runtime on an HP 9000-735/100Mhz
- 6.24. (a) Overlapping triangles which share two points (b) Only one
common point (c) No points are shared (d) The convex local sphere
boundary,
*LSPB* - 6.25. Possible error in a three-dimensional tetrahedralization of a cospherical point set.
- 6.26. Approximately cospherical points can form unexpected constellations.
- 6.27. The advancing front of the global queue does not pass through a subset of cospherical points.
- 6.28. (a) One adjacent triangle to merge (b) Of two adjacent triangles only one can be merged correctly (c),(d) No adjacent triangles exist
- 6.29. A twisted prism with cospherical vertices and three cocircular point subsets. It can be converted into a convex polyhedron or into an untetrahedralizable polyhedron while keeping the vertices cospherical. Thus, in all cases all triangles satisfy the Delaunay criterion.
- 7.1. The advancing front at several moments during the meshing process. It separates the meshed region from the empty parts of the volume.
- 7.2. Surface mesh of Beethoven's bust.
- 7.3. Structural edges form a contour plot.
- 7.4. The model of a cow.
- 7.5. The mesh of a cow with 11608 elements.
- 7.6. Edges and points of the final mesh.
- 7.7. Final mesh of Beethoven's bust with 17665 elements.
- 7.8. Human skull, mesh with 28512 elements.
- 7.9. Crossections of the human skull mesh.
- 7.10. Structure of the discretized conductors in a DRAM ( x ).
- 7.11. Mesh generated with the layer-based method, 6480 tetrahedra. The vertical propagation of refinement through all layers can be observed.
- 7.12. Fully unstructured Delaunay mesh of the DRAM, 5290 tetrahedra.
- 7.13. Silicon bulk (5139 elements) and four metal lines (51682 elements).
- 7.14. Oxide layer, 47203 elements.
- 7.15. Flow diagram for the high pressure
*CVD*model. - 7.16. A cylindrical via, mesh with 7324 elements
- 7.17. A cross-section of a cylindrical via with a uniform mesh point distribution, 7324 elements.
- 7.18. The cross-section of the same cylindrical via meshed differently shows the non-uniform distribution of mesh points, 75720 elements.
- 7.19. The volume meshes used for the continuum transport model and the
corresponding three-dimensional film profiles for a sequence of time steps of
a Ti/TiN/W plug fill process. The profiles show from bottom to
top the initial circular via, the
*PVD*TiN layer formed by sputter deposition and the*CVD*tungsten layer. - 7.20. Iso surfaces of the concentration in a damascene structure, mesh with 18148 elements.
- 7.21. Structural edges.
- 7.22. NMOS Transistor with a thin oxide layer.
- 7.23. Boron implantation profile, mesh with 134374 elements.
- 7.24. Typical CMOS inverter structure with two transistors.
- 7.25. Initial coarse mesh of the CMOS device.

2000-01-20