List of Figures

• 1.1. Overview of algorithms for unstructured mesh generation with emphasis on four widely used mesh element types in two and three spatial dimensions (picture partly adapted from [3]).
• 1.2. Typical computational analysis cycle which can be divided into the pre-analysis and the analysis phase. The incorporation of error estimation and mesh adaptation improves the quality of the numerical solution.
• 2.1. Local three-dimensional simplex partitioning. One new vertex can be inserted on an edge, a face, or in the interior of an initial tetrahedron. The original tetrahedron is then cut into two, three, or four tetrahedrons, respectively.
• 2.2. Inserting a new vertex on a particular edge - the so-called refinement edge - yields to bisection of all tetrahedra which are connected to the refinement edge. This h-refinement-method (see Section 2.2) is therefore called tetrahedral bisection.
• 2.3. Figure 2.3(a) and Figure 2.3(c) jointly show two tetrahedra with different refinement edges, colored red and blue. The constellation is said to be non-compatibly divisible because refinement edges of neighboring tetrahedra are not identical. Figure 2.3(b) and Figure 2.3(d) show the resulting configuration where the red refinement edge was chosen for the bisection.
• 2.4. Compared to Figure 2.3 in this case the blue refinement edge is not ignored and chosen for the first refinement step. This additional step solves the non-compatibly divisible problem and the red refinement edge can be bisected afterwards. This recursive process yields more regular mesh elements compared to obstinate tetrahedral bisection.
• 2.5. Anisotropic property of a tetrahedron can be understood as variation of the coordinate system related distances between the vertices. Principal Component Analysis (PCA) applied to a tetrahedron forms a symmetric $3 \times 3$ covariance matrix. The eigensystem of the covariance matrix constitutes a quadratic surface which is used for visualization. Figure 2.5(a) and Figure 2.5(b) give the orthogonal eigenvectors scaled by the eigenvalues (red arrows) and the according quadratic surface (transparent blue hull) for a geometrically distorted and a regular tetrahedron, respectively.
• 2.6. Metric representation for an isotropic Euclidian space, where the metric tensor function is set to the identity matrix $\mathbf{M}=\mathbf{M}(x,y,z) = \mathbf{I}=\operatorname{const}$ .
• 2.7. Metric representation for an anisotropic distorted space, where for exemplification the metric tensor function $\mathbf{M} = \mathbf{M}(x,y,z) = (3,2,1)^T \cdot \mathbf{I} = \operatorname{const}$ was chosen. The sphere depicted in Figure 2.6 is transformed to an ellipsoid.
• 3.1. Possible configurations of scalar, vector, and dyad data values related to 0 -faces of a $3$ -simplex.
• 3.2. Possible relations between scalar data values and faces of a $3$ -simplex.
• 3.3. Affine map between an arbitrary tetrahedron (left) and the standard 3-simplex (right).
• 3.4. A linear interpolation scheme is used to obtain a continuous data distribution over the $3$ -simplex from 0 -face related scattered scalar data. Iso-level contours form always a flat planes.
• 3.5. Figure 3.5(a) shows a typical plate-capacitor structure. Two coplanar metal planes are connected to a voltage supply. The lower plane is riddled with a bead. The resulting electrostatic field is shown in 3.5(b). The electrical field is a vector quantity which is perpendicular to the iso-surfaces of the electrostatic potential $\psi$ (picture adapted from [60]).
• 3.6. Dilation function $f(\psi )$ for stretching parameter $\lambda$ according to Equation (3.11).
• 3.7. Figure 3.7(a) shows the initial mesh of an eighth of a sphere. Appropriate boundary conditions are applied, where the spike is set to unity and the outer rounded hull is set to zero. On all other boundaries Neumann boundary conditions are given. Laplace's equation solution is carried out and iso-surfaces and the gradient field are depicted in Figure 3.7(b).
• 3.8. Refined sphere part with $441$ points and $1801$ tetrahedra. The upper part is fractionalized that means, tetrahedra are cut away to see the interior of the solid. The red iso-line marks the refinement region which is limited to a ``potential'' of approximately $70 \%$ of the maximum.
• 3.9. Rotation evolution for the transformation of the dilation function matrix $\mathbf{S}$ demonstrated on the standard $3$ -simplex. The first rotation is around the $z$ -axis and the second is about the rotated $x$ -axis so that the former $y$ -axis is parallel to the direction of the gradient $\vec{\nabla}f$ of the scalar data function $f(x,y,z)$ .
• 3.10. Example structure of the gradient refinement method, where a cubic body is covered by an L-shaped mask. For the approximation of the surface distance field the solution of the Laplace equation is chosen with Dirichlet boundary conditions. On all other surfaces Neumann boundary conditions are applied. Figure 3.10(c) shows the iso-levels of the Laplace equation solution and Figure 3.10(d) the corresponding gradient field.
• 3.11. The gradient refinement method was chosen to obtain an anisotropic refinement in regions of scalar data values above $70 \%$ of the maximum. For the anisotropic primary stretching direction the gradient field of the Laplace equation solution $f(x,y,z)$ was used. The refinement is limited to the cubic body, the L-shaped mask on top is influenced by mesh conformity reasons only.
• 3.12. For the Hessian refinement strategy the metric tensor function $M(x,y,z)$ is defined pointwise. For the anisotropic edge length calculation the edge is split into two parts each with constant metric.
• 3.13. Figure 3.13(a) shows a plot of the two-dimensional scalar data distribution which is applied to the three-dimensional cylindric tetrahedral mesh structure, depicted in Figure 3.13(b).
• 3.14. Metric function evaluation of the propaedeutic example structure. The ellipsoidal glyphs are scaled for a proper visualization. The blue ellipsoid at $p=(0,0,2)^T$ is almost a sphere, since the entries of the Hessian given in Equation (3.23) are nearly zero for this point, but for the metric definition, the Hessian is offset by the identity matrix. Therefore the semiaxis of the ellipsoidal glyph are approximately unity.
• 3.15. The Hessian refinement method was used to obtain an anisotropic refinement in regions of high second derivatives of the initial data profile. The mesh is untouched in regions with smooth iso-levels and the anisotropic orientation is aligned according to the curvature of the iso-level surfaces.
• 3.16. One quarter of the refined structure in an orthographic view. The iso-surfaces are drawn end-to-end. One can clearly see that the refinement is restricted to regions where the iso-surfaces have a high curvature.
• 3.17. Initial, mostly regular mesh. $3.884$ points and $20.534$ tetrahedra. The coloration gives the solution of the Laplace equation.
• 3.18. Non-planar surface example for anisotropic refinement. The refinement takes place only in a well-defined layer beneath the surface. The contours of the structure show iso-surfaces of the Laplace equation solution and the according gradient field.
• 3.19. Aerial image simulation result of the floating gate mask of a $3 \times 3$ EEPROM cell array. The black rectangle shows the smallest region of the symmetric constellation, which was used for three-dimensional manufacturing process simulation.
• 3.20. Comparison between the simulated shape of the floating gate, Figure 3.20(a) and a scanning electron microscope (SEM) picture, Figure 3.20(b). Figure 3.20(c) shows a quarter of the entire memory cell.
• 3.21. The Laplace equation is used to obtain an approximation of the surface distance field which is used later on for the layer refinement method. To introduce anisotropic mesh elements a combination of the layer and the gradient refinement method is applied. A detailed view of the gradient field of the Laplace equation solution is depicted in Figure 3.21(b).
• 3.22. One quarter of an EEPROM memory cell. Additional mesh points have been introduced by an anisotropic refinement procedure which processes the top part of the structure, other regions are untouched. The white rectangles in the upper pictures show the zooming regions for the detailed views in the second row.
• 4.1. The vertex gradient $\vec{G}_{P}$ , see Equation (4.2), is constructed as volume weighted for $\Bbb{R}^3$ or area weighted for $\Bbb{R}^2$ arithmetic average for the two- and three-dimensional case, respectively. All element gradients ( $\vec{G}_{E_1} \ldots \vec{G}_{E_n}$ ) attached to the vertex $P$ are evaluated according to Equation (3.16).
• 4.2. One-dimensional scalar data test profile and norm of the second derivative (scaled to unity). This profile is used in $x$ direction in the mesh structure shown in Figure 4.3(a). The mesh adaptation according to the second derivative can be seen in Figure 4.4.
• 4.3. Figure 4.3(a) shows the initial coarse mesh constellation with the one-dimensional test profile depicted in Figure 4.2. Due to the exceptional case that for this illustration also an analytical solution exists, the modified error estimator given by Equation (4.12) can be used to obtain error values for the initial mesh structure depicted in Figure 4.3(b).
• 4.4. The Hessian refinement method was used to obtain an anisotropic refinement in regions of high second derivatives of the initial attribute, indicated through a modified gradient recovery based error estimator.
• 4.5. Histogram of the error distribution before (gray scale) and after refinement (red). The error was normalized to the maximum error and divided into ten error classes from $0.1$ (low error) to $1.0$ (high error). Since the amount of tetrahedrons changes after the refinement, the amount regarding the error class is given in $\%$ . The error estimation is performed with Equation (4.5) for each element over the domain and a clear shift from higher to lower error classes has been achieved by the refinement.
• 4.6. Initial diffusion simulation domain, a silicon block is partially covered by a silicon nitride mask. For the diffusion simulation a non-vanishing normal derivative of the dopant concentration was applied to the upturn part of the silicon block, marked with red arrows.
• 4.7. Iso-surfaces and gradient field of the diffused quantity. The gradient field shows strong variation along the $x$ and the $y$ -direction. One can expect a finer mesh along these directions after the Hessian refinement. Along the $z$ -direction the gradient field shows a very smooth behavior and the mesh density along this directions should be kept.
• 4.8. Due to the combination of error estimation and anisotropic mesh refinement a good balance between the accuracy and the number of mesh elements was found. The refinement region is located in regions of high gradient field variations.
• 5.1. Figure 5.1(a) gives a schematic overview of mass transport of metal atoms along different diffusion paths in a typical Cu interconnect line. Figure 5.1(b) shows an failure of a copper conductive strip due to electromigration, viewed with a scanning electron microscope (SEM). The passivation layer was removed using reactive ion etching and hydrofluoric acid [86].
• 5.2. Illustration of an intersection between a sphere (blue) and a cube (red). The intersection plane (yellow) of these solids forms an interface which is part of further discussions.
• 5.3. Illustration of an intersection between one quarter of a sphere (blue) and a cube (red). The intersection plane (yellow) of these solids forms an interface which is part of the tetragonal tessellation and therefore called sharp interface or explicit interface.
• 5.4. Implicit interface representation between the sphere and the cube. Due to the usage of linear interpolation between the mesh points, the interface defined by Equation (5.3) smears out. A cross cut through the mesh structure and level surfaces are plotted for iso-contour values $-0.98$ (blue), $0.00$ (green) , and $0.98$ (red) in the right picture.
• 5.5. Tunable one-dimensional interface definition function.
• 5.6. Diffuse refined interface representation between the sphere and the cube. A refined region surrounds the interface with higher mesh density and, therefore, a higher numerical accuracy is reached. A cross cut through the mesh structure and level surfaces are plotted for iso-contour values $-0.98$  (blue), $0.00$  (green) , and $0.98$  (red) in the right picture.
• 5.7. Data structure of a hierarchical refinement-coarsement scheme.
• 5.8. Three-dimensional domain for the interconnect electromigration simulation with trapezoid shaped tantalum (Ta) covered copper (Cu) lines, round conical via, and horizontal silicon carbide (SiC) etch stop layers embedded in some low-$\kappa$ material.
• 5.9. Three-dimensional interconnect electromigration simulation domain with trapezoidal shaped tantalum ($Ta$ ) covered copper ($Cu$ ) lines, round conical via, and horizontal silicon carbide ($SiC$ ) etch stop layers embedded in some low-$\kappa$ material.
• 5.10. During void formation and movement a dynamic mesh adaptation scheme is used to guarantee a good spatial resolution of the void copper interface. Copper grain boundaries are taken into account and an appropriately fine mesh was computed.
• 6.1. Silicon lattice, known as diamond structure is adopted by solids with four symmetrically placed covalent bonds. The translational symmetry is a FCC lattice with a basis of two atoms, one at $a_0(0,0,0)^T$ an the other at $a_0($¼$,$¼$,$¼$)^T$ , where $a_0$ is the lattice constant.
• 6.2. The reciprocal lattice structure of a face-centered cubic (FCC) basis forms a body-centered cubic (BCC) lattice. Figure 6.2(a) shows a primitive reciprocal lattice part and the Wigner-Seitz cell which is referred as the first Brillouin zone. Figure 6.2(b) shows the periodicity of the Brillouin zone cells.
• 6.3. First Brillouin zone of the reciprocal lattice with emphasis on the first octant which carries the first irreducible wedge.
• 6.4. $E(\mathbf{k})$ for the conduction and valence bands of silicon.
• 6.5. Inconsistent disambiguation. Different possible iso-surface representation for an ambiguous scalar data vertex relation shown in the left and in the middle. Inconsistent disambiguation between adjacent cubes causes a surface rupture, shown right.
• 6.6. Decompositions of a cube into five tetrahedra (first row) and six tetrahedra (second row). The decomposition of the cube into six tetrahedra can be reached by an intermediate step, where the cube is split into two prisms. Each of the prisms is afterwards divided into three tetrahedra.
• 6.7. First octant of the silicon Brillouin zone. For the spatial discretization a cubic grid based tessellation scheme was used with constant grid spacing. The right part of the picture shows surfaces of constant energy for the first conduction band of silicon, the left part additionally holds mesh information. The wave vector is plotted in units of $2\pi /a_0$ .
• 6.8. First octant of the silicon Brillouin zone. For the spatial discretization a pure unstructured tessellation scheme was used. The right part of the picture shows surfaces of constant energy for the first conduction band, the left part additionally holds mesh information. The wave vector is plotted in units of $2\pi /a_0$ .
• 6.9. One quarter of $\{100\}$ -planes; Contours of constant energy for the first three conduction bands and the heavy hole band in silicon with additional mesh information are shown. The wave vector is plotted in units of $2\pi /a_0$ and the energy in $\MR{eV}$ with a band gap offset for silicon of $1.12 \MR{eV}$ for the valence band.
• 6.10. In the left part of the figure the density of states for the first three conduction bands and the sum of them is plotted versus energy. Note that the energy axes have an offset according to the band gap energy of silicon $E_g \approx 1.12 \MR{eV}$ . The right part shows a direct comparison between two analytical models and the more accurate full band approach.
• 6.11. One quarter of the $\{100\}$ -planes of the first octant of the silicon Brillouin zone. For the spatial discretization an cubic grid based tessellation scheme was used with constant grid spacing of $40$ (Figure 6.11(a)), and $80$ (Figure 6.11(b)) ticks along the coordinate axes were used. The coloration gives the energy values of the first conduction band of silicon. The wave vector is plotted in units of $2\pi /a_0$ and the energy in $\MR{eV}$ with a band gap offset for silicon of $1.12 \MR{eV}$ .
• 6.12. Average kinetic energy calculations in units of $\mathrm{k_B}{}T$ over the temperature. Structured and unstructured mesh strategies have been used as discretization scheme.
• 6.13. Electron velocity versus field along $[100]$ direction at $77K$ and $300K$ .
• A.1. System of four points form a tetrahedron.
• A.2. A Principal Component Analysis (PCA) scheme was used to extract the ellipsoidal parameters for a glyph visualization of the anisotropic property of a tetrahedron.
• B.1. Definition of Euler angles $(\phi ,\theta ,\psi )$ in the so-called $x$ -convention rotation scheme according to the rotation components given in Equation (B.2), picture adapted from [133].