Lecture # 04 Grid Generation

Grid Generation – An Overview

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What a Grid is?

• A mesh/grid is an artificial geometric construction that facilitates the spatial discretization of the governing equations to be solved.

• The mesh determines the locations in the field where the variables will be evaluated and the stencil of the discrete equations.

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The Importance

• The final accuracy and efficiency of any numerical solution are highly dependent on the particular meshing strategy and mesh density distribution employed.

• A good matching of the strengths and weaknesses of the grid generation and flow solution techniques and a strong and favourable interplay between the two is the key to an efficient overall numerical solution.

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Classification of Meshing Strategies

• Structured Mesh – Physical location of any mesh point and the identity of its neighbours are known implicitly. Physical locations may have to be stored.

• Unstructured Mesh – Physical location of a mesh point and the identity of its neighbours, i.e., the connectivity of the mesh are to be determined explicitly.

• Hybrid Mesh – A combination of the two above.

• Gridless Mesh– A set of disconnected points distributed throughout the field.

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Structured Mesh

• Cartesian Mesh– Mesh generation is trivial. The grid points and their connectivity are known implicitly. Methods can be extended to complex geometries using cut-cell approach.

• Body-Fitted Mesh– Grid lines/surfaces conform to the boundary lines/surfaces. A warped or mapped Cartesian-type mesh where the boundaries of the mesh coincide exactly with the the boundaries of the physical domain. Physical location of the mesh points must be stored but the identity of the neighbours known implicitly.

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Structured Mesh – Contd. • Overset Mesh– Multiple overlapping grids

to discretize the domain, the solver interpolates values between the various grids in the regions of overlap.

• Block-Structured Mesh – The domain is decomposed into a number of topologically simpler domains and each domain is meshed independently with a structured grid.

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Single block structured mesh about a wing configuration

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An overset grid for a complex geometry

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A multi-block structured grid

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Grid Generation Methods

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Structured Grid Generation

• Algebraic Methods – Geometric data of the Cartesian coordinates in the interior of a domain are generated from specified values at the boundaries through interpolations or specific functions of the curvilinear coordinates.

• PDE Mapping Methods– Mapping by solving PDEs with the dependent and independent variables being the physical domain coordinates and transformed computational domain coordinates respectively.

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Algebraic Methods• Domain Vertex Method– utilize tensor

products of unidirectional FEM interpolation functions (Lagrangian, Hermite or Spline) for two or three dimensions.

( ) ( ) ( ) ( )

( ) ( )

ˆ ˆ ˆ , 1,2,3, , , 1,2

, , , 1,2,3, 1,....,8

i L M N iLMN

i N iN

x x i LMN

x x i N

ξ η ζ

ξηζ

=Φ Φ Φ = =

=Φ = =

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Algebraic Methods – Contd.

• Transfinite Interpolation – tensor products of unidirectional interpolation but with all sides of the boundaries interpolated and matched. The corner nodes are also matched.

• Steps (2-dimensions)1. Pick four points on Gwhich are

identified as the images of the four corners of

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TFI – Contd.2. The resulting four curve segments are identified as

the graphs of the four vector valued functions F(0,h), F(1,h), F(x,0) and F(x,1) – the 4 segments of the physical boundary are images of the 4 sides of the computational domain.

3. A bilinearly blended transfinite function U(x, h) is constructed using (Boolean sum projection) the four F functions that maps the boundary of the computational domain to that of the physical domain.

4. Check for univalency criteria – nonsingular Jacobian

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TFI – Contd.• The univalent function matches F on

the boundary of and interpolates to F at a finite set of points.

:U Ω → ΩΓ

( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )[ ] ( ) ( )

1 1 2 2

N M NM

U F F F F

ξ η ξ η η ξ ξ η

ξ η ξ η ξ ηη ξ η ξ η ξξ η ξ η

= ℘ ⊕℘ =℘ +℘ −℘ ℘

℘ =Φ +Φ

℘ ℘ =Φ Φ

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Physical domain

Transformed computational domain

Transfinite interpolation

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FNM match the function at four corners but not on all boundaries

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Parameterization for 2D C-type structured grid

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PDE Mapping Methods

• Elliptic Grid Generator – Solution of a set of elliptic PDE, (Laplace or Poisson equations)

Iterative solution in the computational domain to determine the grid coordinates (x,y).

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

2 2 2 2 2

x y x x x y y x x y x J Px Qx

x y y x x y y y x y y J Py Qy

η η ξξ ξ η ξ η ξη ξ ξ ηη ξ η

+ − + + + =− +

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1. Smooth grid point distribution

2. Orthogonality at boundaries

3. Desired clustering using appropriate control functions P and Q

4. Construction of the control functions is often difficult

5. Larger computational time

6. Most widely used

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Hyperbolic Grid Generator

• Applicable to open domain problem

• Computationally efficient and less expensive marching type solution

• Inability to match prescribed point distribution on all boundaries

• Hyperbolic PDE for constraints of orthogonality and cell volume/arc length

( )22 2 2 2

0x x y y and

either x y y x V

or x y x y s

ξ η ξ η

ξ ξ η η

− = ∆

+ + + = ∆

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Treatment of doubly and multiply connected domain for O-type grid

Treatment of doubly and multiply connected domain for O type grid

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O type elliptic grid with control

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Geometry Definition – Surface Modeling & Surface Grid

• Point Sets– Union of ordered point sets that define multiple cross-sections of the geometry. Inaccurate and ambiguous form of surface discretization. Geometry details like small gaps, slope and curvature continuity not preserved.

• B-Rep– Geometry definition by a set of 3 or 4 sided curved surface patches and trimmed surfaces.

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Approximation of a surface with hole by two patches and by a single trimmed surface

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• Surface Repair – Removal of unrealistic gaps, discontinuities and small overlaps created by the CAD packages – modified input geometry.

• Projection Surface – The surface grid is constructed on a projection surface which is then placed over the collection of surface patches that defines the actual geometry.

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Mesh generation on the surface patches

• Physical space approach – grid points must coincide with the actual surface and need to be determined from the actual surface geometry.

• Parametric space approach – 2D meshing problem. To be mapped back to physical space. Possibility of invalid physical surface mesh for highly warped surface or irregular parameterization. Global or quilted patches solely for meshing.

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Elliptic Surface Grid• The governing equations are

( ) ( )

1 222 ,11 11 ,22 12 ,12 1 ,1 2 ,2 1 2

2 2 2 2 2 211 11

12 ,1 ,2

,11 ,22 ,12

1 21 2

, , , , , ,

, , , , , , , ,

a r a r a r Pr Pr b b na

a x y x y a x y z a x y z

a x x y y z z r x y z r x y z

r x y z r x y z r x y z

b b ar

ξ η η ξ ξ ξ ξ η η η

ξ η ξ η ξ η ξ ξ ξ η η η

ξξ ξξ ξξ ηη ηη ηη ξη ξη ξη

+ − − − = +

= − = + + = + +

= + + = =

e principal curvatures

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Algebraic Surface Grid• Construction of curves on the surfaces and

surface patches using appropriate basis polynomials and control vectors – NURBS are most widely used.

• Union of the patches is the global surface.

• For valid mesh the curves bordering each patch are to be meshed the same way in all patches containing them.

• Mesh each patch, parametric space preferred.

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Structured surface grid on the top surface of a generic hyperplane

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Structured surface grid on the bottom surface of the hyperplane

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Surface patches created on a hypersonic vehicle for unstructured grid generation

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Adaptive Meshing• Mesh point movement or mesh redistribution –

structure and connectivity preserved.1. Spring analogy – each edge a spring, stiffness depends

on the quantity to be minimized.

2. Variational principle – minimization functional containing various solution based criteria as well as grid quality criteria simultaneously.

3. Control functions – modified to produce clustering based on solution gradients or truncation errors.

• Mesh enrichment – addition of extra vertices, structure and connectivity lost.

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Unstructured Grid

• Requirement of structure in the mesh removed offering increased flexibility.

• Nodes numbered in any order, and have arbitrary number of neighbours.

• Arbitrary but homogeneous connectivity ⇒Single data structure for the entire mesh unlike block structured mesh.

• Adaptive meshing is easy to implement• Algorithms closely tied to computational

geometry.

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Unstructured Grid

• Elements are generally triangles and tetrahedrons – but need not be.

• Two most prevalent mesh generation approaches.

1. Advancing Front Method

2. Delaunay Triangulation Method

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Advancing Front Method• Initial Front – union of the edges that discretize the

geometry boundary. This front advances out into the field. A stack or priority queue.

• Selecting an edge from this list, a new point is created based on specified criteria so that an optimal triangle is formed.

• Updating the front – by removing the current edge and adding the two newly created edges depending on their visibility.

• Process terminates when the stack (front) is empty.

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Advancing Front Concept. Dotted line is the initial front

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Point Selection

Field points are created to produce triangles of optimal shape and size.

1. Specification of parameters

2. Field function or distribution function

3. Background grid

4. Interpolation

5. Cross-over (intersection) with other edges

6. Smaller edge or angle later

7. Smooth variation of triangle sizes

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For 3D

1. Initial front is the surface grid (2D triangular mesh on the boundary surfaces.

2. New points ahead of the front to form tetrahedra.

3. Both edge-edge and edge-face intersection check.

Local transformation (edge and/or face swapping) for quality improvement.

Boundary integrity guaranteed

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Delaunay Triangulation• Triangulation of a set of points using Delaunay

criterion – “No triangle can contain a point other than its forming vertices within its circumcircle”

Unique triangulation (in 2D) More efficient than AF Boundary integrity lost, boundary to be

recovered Max-min property – maximizes the

smallest angle in the triangulation.

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Incremental Delaunay Triangulation

Predetermined mesh points put in a list.

Initial triangulation of just a few triangles to completely cover the domain to be gridded.

Mesh points inserted sequentially into the existing triangulation

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Incremental Delaunay Triangulation

Insertion of a new pointinto an existing triangulation is locating and deleting all existing triangles whose circumcircle contain the inserted point. A new triangulation is then constructed by joining the new point to all boundary

vertices of the cavity created. (Bowyer-Watson Algorithm)

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Point insertion technique

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Automatic Point Placement

• An initial coarse triangulation• A priority queue based on some triangle

parameter.• Field distribution of the parameter as desired.• Triangles in the queue are sequentially examined

and if required a point is inserted at the circumcentre

• New triangles are put in the queue if not acceptable.

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Boundary Integrity

• Not guaranteed if the domain is concave. Edges or faces that define the boundary do not form a subset of the triangulation.

• Boundary to be recovered by local transformation (edge and face swapping) and modifying the boundary point resolution.

• Constrained triangulation.

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Edge swapping process

Edge-face swapping

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Breakthrough of boundary in Delaunay triangulation

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Example of a 2D unstructured grid

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A tetrahedral unstructured grid for a 3D geometry

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Lecture # 04 Grid Generation

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