List of Figures

• 1.1. Top: internal orientation. Bottom: external orientation for geometric objects in three dimensions.
• 1.2. Left: Integers form a chain, totally ordered by $\leq$ . Middle: Incomparable items forming an anti-chain. Right: The power-set of $\{a,b,c\}$ ordered by $\leq$ as a partially ordered set.
• 1.3. Cell topology of a simplex cell in two dimensions.
• 1.4. Cell topology of a cuboid cell in two dimensions.
• 1.5. Cell topology of a 3-simplex cell.
• 1.6. Complex topology of a simplex cell complex.
• 1.7. Complex topology of a cuboid cell complex.
• 1.8. Representation of a 1-chain with boundary (left) and a 2-chain with boundary (right).
• . Cochain complex with the corresponding coboundary operator: ${\ensuremath{\mathfrak{K}}}^1 \xrightarrow{\delta} \delta {\ensuremath{\mathfrak{K}}}^1 \xrightarrow{\delta} \delta \delta {\ensuremath{\mathfrak{K}}}^1 = 0$
• 1.10. A graphical representation of (co)homology for a three-dimensional cell complex.
• 1.11. Illustration of cycles $A,B,C$ and a boundary $C$ . $A,B$ are not boundaries.
• 1.12. Left: a fiber bundle with the homeomorphism $f$ . Right: A homeomorphism into $U_b \times F$ , which does not preserve the projection, thus not revealing a fiber bundle [1].
• 1.13. Zero section of a vector bundle [2].
• 1.14. A hierarchy of concepts with partial specialization. The most general form is represented by a sheaf concept. The concept of fiber bundles is obtained by using fibers with a certain dimension. If the fiber space satisfies linear vector space properties, the concept of a vector bundle is derived. Finally, by confining the dimension of the base and fiber space, a tangent bundle is obtained.
• 1.15. Illustration of the correlation of tangent space and cotangent space.
• 1.16. The identification of the concept of fiber bundles and the chain and cochain concept as dual spaces.
• 1.17. A representation of the intrinsic fiber bundles of the respective skeleton base spaces.
• 1.18. Overview of the mathematical concepts which have been introduced for the separation of a physical field domain into a base space and fiber space related to the concept of fiber bundles. The topological toolkit provides concepts to describe the internal structure of the cells of the cell complex to enable a generic data structure specification as well as generic traversal mechanisms. Concepts from computational topology such as chain and cochain complexes to obtain vector space structures within the tangent and cotangent space, are presented. Generic data access mechanisms are based on discrete field representations by cochains with their corresponding geometrical orientation and dimension.
• 2.1. Discretization concept for the primary and secondary cell complex with consistent orientation.
• 2.2. Time stepping for the discretization concepts.
• 2.3. Space-time stepping for the discretization concepts.
• 2.4. Finite volume requirements for a primary and secondary cell complex.
• 2.5. Primary control volumes used in the finite volume method. Left: cell-centered. Right: vertex-centered.
• 2.6. Finite volume and the rendered cell average value within each cell for a one-dimensional cell complex.
• 2.7. Geometrical interpretation of the basis of the expanded solution variable $u_h$ for finite elements for a one-dimensional cell complex.
• 2.8. Finite difference requirements for the mesh.
• 2.9. Different geometric interpretations of the first-order finite difference approximation related to forward, backward, and central difference approximation.
• 2.10. Approximation of two-dimensional mixed derivatives.
• 2.11. Boundary treatment for the finite difference scheme. The necessary grid points are not available for all different types of approximation schemes.
• 4.1. A simplex representation of a data model and the corresponding interfaces.
• . Application design based on the object-oriented programming paradigm. Each of the inner blocks has to be implemented, resulting in an implementation effort of $\mathcal{O}(D \cdot A)$ , where $D$ stands for the number of data structures and $A$ for the number of algorithms.
• . Application design based on the generic programming paradigm. Only the topmost and leftmost parts have to be implemented, which reduces the implementation effort to $\mathcal{O}(D + A)$ .
• 5.1. Complex topology of a singly linked list.
• 5.2. Complex topology of a doubly linked list.
• 5.3. Complex topology of a tree.
• 5.4. Complex topology of an array.
• 5.5. Complex topology of a graph.
• 5.6. Local cell complex (left), normally called mesh, and a 2-cell representation (right). Vertices are marked with filled circles.
• 5.7. Global cell complex (left), normally called grid, and a 2-cell representation (right).
• 5.8. Boundary operator applied onto a 3-simplex poset.
• 5.9. Connection matrix $C_j^i$ for a cell complex.
• 5.10. Left: a fiber bundle over a 0-cell complex. Right: a fiber bundle over a 2-cell complex.
• 5.11. Illustration of an index space depth of zero (left) and one (right). The base space is modeled by a 0-cell complex.
• 7.1. Building blocks of the GSSE.
• 7.2. The generic topology library of the GSSE.
• 7.3. Example of the connection matrix $C_j^i$ for a 3-simplex cell complex.
• 7.4. Representation of a 0-cell complex with a topological structure equivalent to a standard container.
• 7.5. Representation of a 1-cell complex with cells (edges, C) and vertices (V).
• 7.6. Incidence relation and traversal operation.
• 7.7. The cell poset of a 4-simplex as implemented by the GTL.
• 7.8. Rendering of a 4-simplex.
• 7.9. Renderings of a 4-cube.
• 7.10. The generic functor library of the GSSE.
• 7.11. Traversal methods induced by the incidence relation. The rows illustrate traversal schemes of the same base element, whereas columns depict traversal schemes of the same traversal element.
• 7.12. Topological traversal of vertices.
• 7.13. 0-cell traversal on the P4; the units are iterations per second.
• 7.14. 0-cell traversal on the AMD64 (right); the units are iterations per second.
• 7.15. 0-cell traversal on the G5; the units are iterations per second
• 7.16. 0-cell traversal on the IBM; the units are iterations per second
• 7.17. Topological traversal of cells for a one-dimensional cell complex.
• 7.18. Incidence traversal for the BGL and the GTL approach on the P4; the units are iterations per second
• 7.19. Incidence traversal for the BGL and the GTL approach on the AMD64; the units are iterations per second
• 7.20. Incidence traversal for the BGL and the GTL approach on the G5; the units are iterations per second
• 7.21. Incidence traversal for the BGL and the GTL approach on the IBM; the units are iterations per second
• 7.22. Evolution of compiler enhancements on the P4 for the DAXPY benchmark using the GCC 4.0.4
• 7.23. Evolution of compiler enhancements on the P4 for the DAXPY benchmark using the GCC 4.1.2
• 7.24. Evolution of compiler enhancements on the P4 for the DAXPY benchmark using the GCC 4.2.1
• 7.25. Evolution of compiler enhancements on the AMD X2 for the DAXPY benchmark using the GCC 4.0.4
• 7.26. Evolution of compiler enhancements on the AMD X2 for the DAXPY benchmark using the GCC 4.1.2
• 7.27. Evolution of compiler enhancements on the AMD X2 for the DAXPY benchmark using the GCC 4.2.1
• 7.28. Best compiler performance for the Intel Core using the GCC 4.2.1.
• 7.29. Best compiler performance for the AMD X2 using the GCC 4.2.1.
• 7.30. Best compiler performance for the G5 using the system compiler.
• 7.31. Best compiler performance for the IBM using the system compiler.
• 8.1. Potential of a PN diode during different stages of the Newton iteration. From initial (left) to the final result (right).
• 8.2. Visualization of non-converging process. Here the potential is illustrated where small oscillations can be observed from left to right.
• 8.3. Illustration of a complete breakdown of the solution procedure (from left to right).
• 8.4. The left figure depicts Faraday's law by the corresponding projection onto a finite cell, whereas the right figure illustrates Amperé's law.
• 8.5. Illustration of the $x$ -component of $\ensuremath{\mathbf{E}}$ with a harmonic oscillating source in the x-y plane at $t=t_1$ .
• 8.6. Illustration of the $x$ -component of $\ensuremath{\mathbf{E}}$ with a harmonic oscillating source in the x-y plane at $t=t_2$ .
• 8.7. Wave equation with a harmonic oscillating source in the x-y plane where the source is switched of.
• 8.8. The $y$ -component of $\ensuremath{\mathbf{E}}$ is depicted on the right side.
• 8.9. Two-dimensional structured grid with an initial doping profile (concentration in $\textnormal {cm}^{-1}$ ).
• 8.10. Two-dimensional diffusion simulation for a structured grid after $200$ time steps.
• 8.11. Two-dimensional diffusion simulation for a structured grid after $2000$ time steps.
• 8.12. Three-dimensional device structure with an initial phosphorus doping profile (concentration in $\textnormal {cm}^{-1}$ ).
• 8.13. Three-dimensional diffusion simulation for a device structure with an initial doping profile. Two subsequent simulation steps are depicted.
• 8.14. Domain for a two-dimensional PN-junction diode.
• 8.15. Netto doping concentration of the two-dimensional PN diode. Donors are given in red, acceptors are blue; units are given in parts per cubic meter.
• 8.16. Two-dimensional diode with potential distribution for equilibrium mode; units given in Volt.
• 8.17. Two-dimensional diode with potential distribution for reverse mode; units given in Volt.
• 8.18. Two-dimensional diode with potential distribution for forward mode; units given in Volt.
• 8.19. Two-dimensional diode with charge distribution for reverse mode; units given in parts per cubic meter.
• 8.20. Two-dimensional diode with charge distribution for forward mode; units given in parts per cubic meter.
• 9.1. Resulting structure from the given input specification.
• 9.2. A structure suitable for capacitance simulation, created by VGM.
• 9.3. The offset of the second layer of contact 1. Right: the corresponding mesh.
• 9.4. The offset of the second layer of contact 1. Right: the corresponding mesh.
• 9.5. Illustration of isosurfaces of the potential distribution.
• 9.6. Interconnect structure for resistance analysis.
• 9.7. Interconnect structure for resistance analysis.
• 9.8. Structures created by LAYGRID.
• 9.9. Structure created by VGM. With this approach different rations between the interconnect line (red) and the covering layer (blue) can be easily modeled. Here a ratio between the thickness of the interconnect line and the covering layer is given by $1/10000$ . As can be seen, the thin layer does not impose additional points for the interconnect line.
• 9.10. Resistance progression related to the spatial expansion of the via part.
• 9.11. Temperature distribution due to self-heating in a tapered interconnect line with cylindrical vias.
• 9.12. Comparison of the finite element assembly times in equations per millisecond, GSSE and SAP.
• . A relation $R$ of a set $A$ and a set $B$ .
• . The image of $a$ under $f$ .
• . The kernel of $f$ .
• C.1. Geometrical interpretation of the basis functions for the (left) finite volume method and (right) for the finite element method. The finite volume method describes consistent crisp cells which can be interpreted as a cell complex, whereas the finite element method uses spread cells which do not conform with the properties of a consistent cell complex.