# Mesh Generation

Math 228B Numerical Solutions of Differential Equations

Mesh Generation

Motivation: Most numerical methods for
PDEs require a mesh for non-trivial domains

Various methods might use different components of the mesh:

• Nodes (vertices)

• Edges (faces in 3D)

• Elements

Structured vs. Unstructured Meshes

Natural classification of meshes based on connectivity of nodes:

• In structured meshes, all nodes have the same connections to their neighbors (at least away from the boundaries)

• Unstructured meshes allow for arbitrary connectivities (as long as the mesh remains conforming)

• Hybrid meshes combine the two, e.g. by having structured parts in certain areas of the domain

Structured Mesh Generation

Why Structured Meshes?

• Lead to very efficient numerical methods

• High quality for sufficiently simple geometries

• Larger grid control when high anisotropy is required

• Multi-block approach allows for realistic geometries

Single-Block Grid Generation

• Construct a one-to-one mapping between a rectangular computational domain and a physical domain

• Ideally, grid size in physical space should be dictated by solver/solution requirements

• Ensure grid quality e.g. smoothness, orthogonality

Single Block Grid Generation - Creating the Mapping

• Transfinite Interpolation (TFI)

• Conformal Mapping

• Solving PDE’s

• Elliptic

• Parabolic/Hyperbolic

Algebraic Mappings

• Construct a mapping between the boundaries of the unit square (cube) and the boundaries of an "arbitrary" region which is topologically equivalent

• Combine 1D interpolants using Boolean sums to construct mapping - Transfinite Interpolation (TFI)

• Not guaranteed to be one-to-one

• Orthogonality not guaranteed

• Very Fast

• Quite General

• Grid quality not always assured

Algebraic Mappings - 1D Interpolants

• General 1D interpolant of $$f(x)$$ for $$x \in (0,1)$$ $\hat{f}(x) \equiv \Pi_x f = \sum_{i=0}^L \sum_{n=0}^P \alpha^n_i(x) \left. \frac{d^n f}{d x^n} \right|_{x=x_i}$

• $$\alpha^n_i(x)$$ are the blending functions

• Examples

• Linear Lagrange interpolation - $$P=0, L=1$$ $\Pi_x f = (1-x) f(0) + x f(1)$

• Quadratic Lagrange interpolation - $$P=0, L=2$$ $\Pi_x f = (2x^2 - 3x + 1) f(0)+ (4x-4x^2)f(0.5) + (2x^2-x) f(1)$

• Hermite interpolation - $$P=1, L=1$$ \begin{aligned} \Pi_x f & = & (2x^3-3x^2+1)f(0) + (3x^2-2x^3) f(1) + \\ & & (x^3-2x^2+x) f'(0) + (x^3-x^2) f'(1) \end{aligned}

Algebraic Mappings - Transfinite Interpolation

• Start from 1D boundary mappings of $${\bf R}\equiv (x,y)$$, e.g.

$${\bf R}(\xi,0), {\bf R}(\xi,1), {\bf R}(0,\eta), {\bf R}(1,\eta)$$

• Construct 1D interpolants in the $$\xi$$ and $$\eta$$ directions (e.g. linear) \begin{aligned} \Pi_\xi {\bf R} & = & (1-\xi){\bf R}(0,\eta) + \xi {\bf R}(1,\eta) \\ \Pi_\eta {\bf R} & = & (1-\eta){\bf R}(\xi,0) + \eta {\bf R}(\xi,1) \end{aligned}

Algebraic Mappings - Transfinite Interpolation

• Construct two-dimensional interpolant by doing the Boolean sum $\hat{{\bf R}}(\xi,\eta) = (\Pi_\xi \oplus \Pi_\eta) {\bf R} = (\Pi_{\xi} + \Pi_\eta - \Pi_\xi \Pi_\eta) {\bf R}$ Expanding: $\hat{{\bf R}}(\xi,\eta) = (1-\xi,\xi)\left(\begin{array}{c} {\bf R}(0,\eta) \\ {\bf R}(1,\eta) \end{array}\right) + ({\bf R}(\xi,0),{\bf R}(\xi,1))\left(\begin{array}{c} 1-\eta \\ \eta \end{array}\right)$ $- (1-\xi,\xi)\left(\begin{array}{cc} {\bf R}(0,0) & {\bf R}(0,1)\\ {\bf R}(1,0) & {\bf R}(1,1) \end{array}\right)\left(\begin{array}{c} 1-\eta \\ \eta \end{array}\right)$ $=(1-\xi){\bf R}(0,\eta) + \xi{\bf R}(1,\eta) + (1-\eta){\bf R}(\xi,0) + \eta{\bf R}(\xi,1)$ $-(1-\xi)(1-\eta){\bf R}(0,0) - (1-\xi)\eta{\bf R}(0,1) - \xi(1-\eta){\bf R}(1,0) -\xi\eta{\bf R}(1,1)$

• Important property: Preserves $${\bf R}$$ at the domain boundary

• Extends to general 1D interpolants and any dimension

Algebraic Mappings - Example

Algebraic Mappings - Example

$$\Pi_\xi {\bf R}$$

$$\Pi_\eta {\bf R}$$

$$(\Pi_\xi \oplus \Pi_\eta) {\bf R}$$

Algebraic Mappings - Grid Control

• Use non-regular subdivisions in $$(\xi, \eta)$$ (e.g. exponential functions) to obtain desired element sizes in $$(x,y)$$

• Use derivative boundary conditions to enforce boundary orthogonality $\frac{\partial{\bf R}}{\partial \xi} \cdot \frac{\partial{\bf R}}{\partial \eta} = 0$

Conformal Mapping

• An analytic function $$\alpha = f (z)$$ such that $$\displaystyle \frac{df}{dz} \ne 0$$ defines a one-to-one (conformal) mapping between $$z = x + i y$$ and $$\alpha = \xi + i \eta$$, or between $$(x,y)$$ and $$(\xi, \eta)$$.

• The functions $$\xi(x,y)$$ and $$\eta(x,y)$$ satisfy the Cauchy- Riemann equations (e.g. $$\xi_x=\eta_y$$, and $$\eta_x= - \xi_y$$) and as a consequence, they are harmonic $\nabla^2 \xi = 0, \qquad \nabla^2 \eta = 0 \qquad \mbox{(smoothness)}$

• Preserve angles (grid orthogonality)

• Preserve ratios

• Lead to high quality grids

• Limited to 2D

Conformal Mapping Transformations

• Joukowski (maps circle of radius $$c$$ to segment $$[-2c, 2c]$$) $\alpha = z + \frac{c^2}{z}, \quad \mbox{or} \qquad \frac{\alpha + 2c}{\alpha-2c} = \left(\frac{z+c}{z-c}\right)^2$

• Karman-Trefftz $\frac{\alpha + 2c}{\alpha-2c} = \left(\frac{z+c}{z-c}\right)^n$

• Schwarz-Christoffel (maps polygon into half plane) $\frac{d \alpha}{dz} = K\prod_{k=1}^n\left( 1-\frac{z}{z_k}\right)^{\beta_k}$

Conformal Mapping - Schwarz-Christoffel

Ref. “Schwarz-Christofell Mapping”, Driscoll and Trefethen, Cambridge Univeristy Press, 2002.

PDE Grid Generation

• Construct mapping by solving a PDE

• Elliptic Equations (smooth grids) $\nabla^2 \xi (x,y) = P(x,y), \quad \nabla^2\eta(x,y) = Q(x,y)$

• Hyperbolic equations (orthogonal grids) \begin{aligned} x_\xi y_\eta - x_\eta y_\xi & =& J \qquad \mbox{(size control)}\\ x_\xi x_\eta + y_\xi y_\eta & = & 0 \qquad \mbox{(orthogonality)} \end{aligned}

• Most widely used approach

• Grids usually have high quality

Elliptic Grid Generation

We are interested in solving $\begin{array}{rcll} - \nabla^2 \xi & = & P & \qquad \mbox{in} \quad \Omega \\[2ex] \xi & = & g & \qquad \mbox{on} \quad \Gamma_D \\[2ex] \displaystyle \frac{\partial \xi}{\partial n} & = & h & \qquad \mbox{on} \quad \Gamma_N = \Gamma \backslash \Gamma_D \end{array}$

where $$P$$, $$g$$, and $$h$$ are given.

Similarly for $$\eta(x,y)$$

Elliptic Grid Generation

?  ?

$\begin{array}{c} x = x (\xi, \eta)\\ y = y (\xi, \eta)\\ \longrightarrow \end{array}$

$$\nabla^2 \xi = P$$

Can we determine an equivalent problem to be solved on $$\hat{\Omega}$$?

Elliptic Grid Generation

$$\displaystyle \begin{array}{ccc} \xi = \xi (x,y) & & x = x (\xi, \eta)\\ \eta = \eta (x,y) & & y = y (\xi, \eta)\\ & & \ \ \\ \left( \begin{array}{c} d \xi \1.5ex] d \eta \end{array} \right) = \left( \begin{array}{ccc} \xi_x & \ & \xi_y \\[1.5ex] \eta_x & & \eta_y \end{array} \right) \left( \begin{array}{c} d x \\[1.5ex] d y \end{array} \right) & & \left( \begin{array}{c} d x \\[1.5ex] d y \end{array} \right) = \left( \begin{array}{ccc} x_{\xi} & \ & x_{\eta} \\[1.5ex] y_{\xi} & & y_{\eta} \end{array} \right) \left( \begin{array}{c} d \xi \\[1.5ex] d \eta \end{array} \right) \end{array}$$ $$\Rightarrow \displaystyle \left( \begin{array}{ccc} \xi_x & \ & \xi_y \\[1.5ex] \eta_x & & \eta_y \end{array} \right) = \left( \begin{array}{ccc} x_{\xi} & \ & x_{\eta} \\[1.5ex] y_{\xi} & & y_{\eta} \end{array} \right)^{-1} = \frac{1}{J} \left( \begin{array}{ccc} y_{\eta} & \ & - x_{\eta} \\[1.5ex] - y_{\xi} & & x_{\xi} \end{array} \right)$$ $$J = x_{\xi} y_{\eta} - x_{\eta} y_{\xi}$$ Elliptic Grid Generation $$\begin{array}{rclcrcl} \xi_x & = & \frac{y_\eta}{J} & \quad & \xi_y & = & -\frac{x_\eta}{J}\\[1ex] \eta_x & = & -\frac{y_\xi}{J} & \quad & \eta_y & = & \frac{x_\xi}{J} \end{array}$$ and $$\begin{array}[t]{rcl} \xi_{xx} & = & \displaystyle \frac{\partial}{\partial x} \: (\xi_x) \ = \ \left(\xi_x \: \frac{\partial}{\partial \xi} + \eta_x \: \frac{\partial}{\partial \eta} \right) \left(\frac{y_\eta}{J}\right) \\[1ex] & = & \displaystyle \frac{1}{J} \left( y_{\eta} \: \frac{\partial}{\partial \xi} - y_{\xi} \frac{\partial}{\partial \eta} \right) \left(\frac{y_\eta}{J}\right) \\[1ex] & = & \ldots\\[2ex] \xi_{yy} & = & \ldots \end{array}$$ Elliptic Grid Generation - Thompson’s Equations Finally, $$\xi_{xx} + \xi_{yy} = 0$$ and $$\eta_{xx} + \eta_{yy} = 0$$, become $$\fbox{ \displaystyle \begin{array}{rcl} a x_{\xi\xi} - 2 b x_{\xi \eta} + cx_{\eta\eta} & = & 0 \\ a y_{\xi\xi} - 2 b y_{\xi\eta} + cy_{\eta\eta} & = & 0 \end{array} }$$ $$a$$, $$b$$, $$c$$ depend on the mapping. $$\begin{array}{ccc} a = x^2_{\eta} + y^2_{\eta} & b = x_{\xi} x_{\eta} + y_{\xi} y_{\eta} & c = x^2_{\xi} + y^2_{\xi} \end{array}$$ • These equations can be solved using central finite differences on a regular grid in the $$(\xi, \eta)$$ domain to determine the $$(x,y)$$ coordinates of each grid point. • Picard iteration: Start from initial grid coordinates $$x,y$$. Compute $$a,b,c$$, solve the PDE, and repeat until convergence. Elliptic Grid Generation - Grid Control • Modify grid by e.g. adding source terms to the PDE: $$\xi_{xx} + \xi_{yy} = P(x,y)$$ and $$\eta_{xx} + \eta_{yy} = Q(x,y)$$ $$\fbox{ \displaystyle \begin{array}{rcl} a x_{\xi\xi} - 2 b x_{\xi \eta} + cx_{\eta\eta} & = & -J^2 (x_\xi P + x_\eta Q) \\ a y_{\xi\xi} - 2 b y_{\xi\eta} + cy_{\eta\eta} & = & -J^2 (y_\xi P + y_\eta Q) \end{array} }$$ • The functions $$P(\xi,\eta)$$ and $$Q(\xi,\eta)$$ can be used to obtain grid control • Derivative boundary conditons can be used to enforce grid orthogonality at the boundary Ref: “Numerical Generation of Two-Dimensional Grids by Use of Poisson Equations with Grid Control”, Sorenson and Steger, in Numerical Grid Generation Techniques, Smith, R.E. (Ed.), NASA-CP-2166, pp. 449-461, 1980 Single-Block Grid Common Topologies  O-Grid C-Grid …plus combinations Examples: Single-Block O-Grids Examples: Single-Block C,H-Grids Examples: H-Grids  H-Grid H-Grid/I-Grid Multi-Block Grid Generation • Subdivide domain into an unstructured assembly of quadrilaterals/hexahedra • Obtaining block topology automatically is hard • Obtaining block geometry automatically (e.g. point coordinates) once topology is known is tractable Examples: Multi-Block Grids Examples: Multi-Block Grids Block Topology Generators (from ICEM CFD) Automatic $$H \Rightarrow O$$ conversion Unstructured Mesh Generation Unstructured Mesh Generation • Approximate a domain in $$\mathbb{R}^d$$ by simple geometric shapes • Determine node points and element connectivity • Goal: Resolve the domain accurately with well-shaped elements, but use as few elements as possible • Applications: Numerical solution of PDEs (FEM, FVM, DGM, BEM), interpolation, computer graphics, visualization Geometry Representations Explicit Geometry • Parameterized boundaries Implicit Geometry • Boundaries from contour Unstructured Meshing Algorithms • Delaunay refinement • Refine an initial triangulation by inserting centroid points and updating connectivities • Efficient and robust, provably good in 2-D • Advancing front • Propagate a layer of elements from boundaries into domain, stitch together at intersection • High quality meshes, good for boundary layers, but somewhat unreliable in 3-D Unstructured Meshing Algorithms • Octree mesh • Create an octree, refine until geometry well resolved, form elements between cell intersections • Guaranteed quality even in 3-D, but poor element qualities • DistMesh • Improve initial triangulation by node movements and connectivity updates • Easy to understand and use, handles implicit geometries, high element qualities, but non-robust and low performance Delaunay Triangulation • Find non-overlapping triangles that fill the convex hull of a set of points • Properties: • Every edge is shared by at most two triangles • The circumcircle of a triangle contains no other input points • Maximizes the minimum angle of all the triangles Constrained Delaunay Triangulation • The Delaunay triangulation might not respect given input edges • Use local edge swaps to recover the input edges Delaunay Refinement Method • Algorithm: • Form initial triangulation using boundary points and outer box • Replace an undesired element (bad or large) by inserting its circumcenter, retriangulate and repeat until mesh is good • Will converge with high element qualities in 2-D • Very fast – time almost linear in number of nodes The Advancing Front Method • Discretise the boundary as initial front • Add elements into the domain and update the front • When front is empty the mesh is complete Grid Based and Octree Meshing • Overlay domain with regular grid, crop and warp edge points to boundary • Octree instead of regular grid gives graded mesh with fewer elements Mesh Size Functions • Function $$h(\boldsymbol{x})$$ specifying desired mesh element size • Many mesh generators need a priori mesh size functions • Physically-based methods such as DistMesh • Advancing front and Paving methods • Discretize mesh size function $$h(\boldsymbol{x})$$ on a background grid Mesh Size Functions • Based on several factors: • Curvature of geometry boundary • Local feature size of geometry • Numerical error estimates (adaptive solvers) • Any user-specified size constraints • Also: $$|\nabla h(\boldsymbol{x})|\le g$$ to limit ratio $$G=g+1$$ of neighboring element sizes Explicit Mesh Size Functions • A point-source \[\begin{aligned} h(\boldsymbol{x}) = h_\mathrm{pnt} + g |\boldsymbol{x}-\boldsymbol{x}_0| \end{aligned}

• Any shape, with distance function $$\phi(\boldsymbol{x})$$ \begin{aligned} h(\boldsymbol{x}) = h_\mathrm{shape} + g \phi(\boldsymbol{x}) \end{aligned}

• Combine mesh size functions by $$\min$$ operator: \begin{aligned} h(\boldsymbol{x}) = \min_i h_i(\boldsymbol{x}) \end{aligned}

• For more general $$h(\boldsymbol{x})$$, solve the gradient limiting equation [Persson’05] \begin{aligned} \frac{\partial h}{\partial t} + |\nabla h| &= \min (|\nabla h|,g), \\ h(t=0, \boldsymbol{x}) &= h_0(\boldsymbol{x}). \end{aligned}

Mesh Size Functions – 2-D Examples

Mesh Size Function $$h(\boldsymbol{x})$$

Mesh Based on $$h(\boldsymbol{x})$$

Laplacian Smoothing

• Improve node locations by iteratively moving nodes to average of neighbors: \begin{aligned} \boldsymbol{x}_i \leftarrow \frac{1}{n_i}\sum_{j=1}^{n_i} \boldsymbol{x}_j \end{aligned}

• Usually a good postprocessing step for Delaunay refinement

• However, element quality can get worse and elements might even invert:

Face and Edge Swapping

• In 3-D there are several swappings between neighboring elements

• Face and edge swapping important postprocessing of Delaunay meshes

Boundary Layer Meshes

• Unstructured mesh for offset curve $$\psi(\boldsymbol{x})-\delta$$

• The structured grid is easily created with the distance function

The DistMesh Mesh Generator

The DistMesh Mesh Generator

1. Start with any topologically correct initial mesh, for example random node distribution and Delaunay triangulation

2. Move nodes to find force equilibrium in edges

• Project boundary nodes using implicit function $$\phi$$

• Update element connectivities

Internal Forces

For each interior node: \begin{aligned} \sum_i \boldsymbol{F}_i = 0 \end{aligned} Repulsive forces depending on edge length $$\ell$$ and equilibrium length $$\ell_0$$: \begin{aligned} |\boldsymbol{F}| = \begin{cases} k(\ell_0-\ell) & \text{if }\ell<\ell_0, \\ 0 & \text{if }\ell\ge\ell_0. \end{cases} \end{aligned} Get expanding mesh by choosing $$\ell_0$$ larger than desired length $$h$$

Reactions at Boundaries

For each boundary node: \begin{aligned} \sum_i \boldsymbol{F}_i + \boldsymbol{R} = 0 \end{aligned} Reaction force $$\boldsymbol{R}$$:

• Orthogonal to boundary

• Keeps node along boundary

• Move nodes $$\boldsymbol{p}$$ to find force equilibrium: \begin{aligned} \boldsymbol{p}_{n+1}=\boldsymbol{p}_n+\Delta t \boldsymbol{F}(\boldsymbol{p}_n) \end{aligned}
• Project boundary nodes to $$\phi(\boldsymbol{p})=0$$