mesh generation concept. structured grids v. selmin

Mesh Generation Concept.Structured Grids

V. Selmin

Multidisciplinary Computation and Numerical Simulation

Mesh Generation Concept


The computational grid on which a discretisation of an equation is a fundamental component of the numerical solution procedure. The general requirement for a grid are:

1- The computational grid must discretise the domain and all the boundaries . This closely links the grid to the geometry of the domain. The distribution of points in the domain must adequately represent the geometry of the region.

2- The nodes and elements of the grid should be inter-related or connected, so that appropriate approximations to mathematical operators, i.e. the discrete approximation, can be computed in a efficient manner. This determines the inherent structure within the grid.

3- The spatial distribution of nodes should provide a well-shaped elements with no large discontinuities in nodal an element spacing. The accuracy of a numerical discretisation is dependent of the properties of the underlying spatial discretisation. For example, the classical second order accurate finite difference representation of a second spatial derivative is only second order if the grid point spacing is uniform. Furthermore, it can be shown that other spatial derivatives only achieve second order accuracy if the grid lines are approximately orthogonal. Hence the spatial discretisation should be achieved without discontinuous grid point spacing and without the introduction of higly skewed cells or elements.

4- The distribution of nodes in the grid should allow for an appropriate resolution of features in the solution. For accurate simulations, it is necessary that there are adequate points in regions where there is high activity, i.e. where the unknown variables change rapidtly.

5- Ideally, the minimum number of nodes should be used to achieve a given level of accuracy of the solution of the governing equations.



Structured GridsMesh structure:• Domain divided into a structured assembly of quadrilateral cells • Each interior nodal points is surrounded by exactly the same number of mesh cells (or elements)• Directions within the mesh can be immediately identify by associating a curvilinear co-ordinates system• it is possible to immediately identify the nearest neigthbours of any node j on the mesh

Advantages:• Large number of algorithms for discretisation are available• The algorithms can be normally implemented in a computationallly efficient manner

Disadvantages:• Difficulty to generate grids of regions of general shapes multi-block grids• Very high elapsed time necessary to produce a grid for domains of extremely complex shape

Finite difference, finite volume and finite element discretisations



Unstructured GridsMesh structure:• Computational domain divided into an unstructured assembly of computational cells• The number of cells surrounding a typical interior node is not necessarily constant• The nodes and the elements has to be numbered• To get the necessary information on the neightbours the numerotation of the nodes wich belong to each element has to be stored• The concept of directionality does not exist anymore

Advantages:• Powerful tool for discretising domains of complex shapes• Unstructured mesh methods naturally offer the possibility of incorporating adaptivity

Disadvantages:• Alternative solution algorithms are more limited• Computational implementation places large demands on both computer memory and CPU

Finite volume and finite element discretisations

Geometry modelling

Geometry Modelling

The boundary of the domain to be discretised needs to be represented in a suitable manner before the generationcan start. If the automatic discretisation of an arbitrary domain is to be achieved, the mathematical description of the domain topology has to possess the greatest possible generality.The computer implementation of this description must provide means for automatically computing any geometrical quantity relevant to the generation procedure.

Planar two-dimensional caseIn the planar two dimensional case, the boundary is represented by closed loops of orientated composite curves. For simple connected domains these curves are orientated in a counter-clockwise sense while for multiple connected regions the exterior boundary curves are given a counter-clockwise orientation and all the interiorboundary curves are orientated in a clockwise sense

Mathematical description of the geometry → CAD model

Geometry modellingt

Geometry ModellingThree dimensional caseIn three dimensions, the domain to be discretised is viewed as a region bounded by surfaces which intersect along curves. The portions of these curves and surfaces needed to define the three dimensional domain of interest are called curve and surface components, respectively.

Decomposition of the boundary of a three dimensional domain into its surfaces and curve components

Geometry modelling

Geometry ModellingIn addition, boundary curves and surfaces are oriented. This is important in the generating process as it is used to define the location of the region to be discretised.The orientation of a boundary surface is defined by thedirection of the inward normal. The orientation of the boundary curves is defined with respect to the boundary surfaces that contain them. Each boundary curve will be common to two boundary surfaces and will have opposite orientations with respect to each of them.

An example of the approximated geometry of the surfaces component and its curve components is herepresented.

Geometry modelling

Curve RepresentationA support curve may be described by a piecewise parametric representation. In those representation, the curve is subdivided into arcs, and the position vector r of a generic point on each arc is expressed as a function of a single real parameter u, which by convention varies into the interval [0,1].

In general, this function is represented by a polynomial whose rank may change from arc to arc and which can beexpressed as

where each is a vector formed by three coefficients

which represents the components of with respect to a cartesian referencesystem (x,y,z) , whereas n is the rank of the polynomial.The position vector can be rewritten as follows

where

and

A point on the curve can then be identified by the number of the arc on which it is lying and the value of the parametric coordinates u, which is usually called local parametric coordinate.

an

zyx ,,r

nnuuuu aaaarr 2

210)(

ka

zk

yk

xkk aaa ,,a

ka

AUr )(u

naaaaA ,,,, 210

Tnuuu ,,,,1 2 U

Geometry modelling

Surface RepresentationThe surfaces are represented by using a similar scheme. They are subdivided into patches which forms a regular grid on the so-called parametric plane. On this plane, we can define with respect to each patch the local parametric coordinates u and v with varies on the interval [0,1].The position r of a node on the surface can then be expressed as a polynomial expansion in u and v on each patch :

The total rank of the polynomial is nxm, whereas n and m represent the rank of the polynomial with respect to the cartesian reference system (x,y,z):

The position vector can be rewritten in the following form

where

The quantities U and V are expressed as

mnmn vuuvuvuvu ,1,1

20,21,00,10,0),( aaaaaar

zlk

ylk

xlklk aaa ,,,, ,,a

UAVr ),( vu

),,0(

),,0(,

ml

nkk,l

aA

TmTn vvvuuu ,,,,1,,,,,1 22 VU

Structured grid generation

Structured Grid Generation and Mapping ConceptConsider in a one-dimensional domain , with boundaries and .To generate a set of n+1 equally spaced nodes within the domain, it is convenient to introduce a new variable such that

This representation may be generalised to give, for an arbitrary distribution of nodes,

where is the domain within which the nodes are uniformly spaced, x is the domain of and the function

defines the nodal distribution. In this sense, the generation of a grid is equivalent to finding a function which mapsnodes distributed evently in one space (or domain) to the required distribution of nodes in another space.More specifically, the equally spaced nodes in the logical (computational) space are mapped through the grid generation function, to the physical space.

The basic concept also applies in two and three dimensions. An arbitrary shape domain is mapped by

into a square in two dimensions and by

into a cube in three dimensions.The grid in the logical space is uniform and when mapped to the physical space provides the required grid.

L,0 01 L2

Lx

xLx

nini

nii

i

0

,

,0,/

0

ninixx iii ,0,/,)(

),(,),( yyxx

),,(,),,(,),,( zzyyxx

0 L

0 1

x


Mappings and JacobiansConsider the continuous mapping of the form . If a function f exists for which

then

For any differential function f(x) , it is clear that the change of f with respect to is equal to the change of f with

respect to x multiplied by a scaling parameter J which is the change of x with respect to the new variable .

Using this equation to solve for gives

The term is called the jacobian of the mapping and can be seen to be an effective scaling introducedthrough the change of variables. The quantity is the ratio of the length in the physical space to the lengthin the logical space. If the Jacobian vanishes within the domain, then the transformation is invalid

In two dimensions, let consider a continuous transformation of the form

If we consider a function f with

then

)(xx

)(xff

x

Jx

fJ

x

x

ff,

xf /

f

Jx

f 1

/xJ /x

),(,),( yyxx

),( yxff

y

y

fx

x

ff

y

y

fx

x

ff


Mappings and Jacobiansthat may be rewritten in the more concise form

Solving this equation for the derivative of f with respect to the vector x gives

If vanishes the the transformation is invalid. Thus, if the Jacobian is never zero, the inverse function guarantees the existence of the local inverse map:

In a similar manner, in three dimensions, we consider a continuous transformation of the form

Again consider a function f with

then

),(,),(;//

//yx

ff

yx

yxf

xξx

Jxξ

ξJ

x1- ff

)det(J

),(,),( yxyx

),,(,),,(,),,( zzyyxx

),,( zyxff

),,(,),,(; zyxfff

xξx

Jxξ

x

ξ


Mappings and JacobiansThere is clearly an infinite number of maps between the logical and physical space. The issue in grid generation is to find a map which has the appropriate properties. The properties which are required to a mapping include• Positive Jacobian determinant, • Smoothness• Orthogonality or near othogonality• Uniformity of cells

The following table provides some useful quantities used in grid generation

0J

Dimensions

One dimension: x=(x)

Two dimensions: x=(x,y)

Three dimensions: x=(x,y,z)

Geometrical Property

Ratio of lengths

Ratio of lengths

Ratio of areas

Ratio of lengths

Ratio of areas

Ratio of volumes

/x

J

xx ,/,/

J

xx

xxxx

xxx

//

,//,//

,/,/,/


Algebraic and Transfinite InterpolationInterpolationThe root of algebraic methods for grid generation are in simple interpolation.In one dimension, the boundaries of a domain are the two end points and .It is natural and intuitive that a grid is formed by explicit interpolation between theseboundaries.Linear interpolation in one dimension can be written

This represents a linear interpolation between the end points with and providingthe linear variation.The more general formulation is

with

The functions and are called blending functions since they blend the boundary data and into the domain .This interpolation may be generalised by choosing other blending functions, for example functions that allows to control slopes of the grid lines at the boundaries

It is clear that such simple techniques cannot be in general be applied to domains in two dimensions which have four general boundary curves. In general, interpolation between curves would lead to grid cross-over.

0s 1s

10,)1()( 10 ssR

10,)()()( 10 ssR

1)1(,0)0(;0)1(,1)0(

)( )(10

0s 1s

)1( 0s

1s

0 1


Algebraic and Transfinite InterpolationMulti-directional Interpolation – Two DimensionsA two-dimensional domain can be represented by four bounding curves, two for which and twowith . The boundaries are defined as

and the four curves intersect at the four corners of the domain:

constantconstant

)1,(:Curve)0,(:Curve

),1(:Curve),0(:Curve

32

31

RCRC

RCRC

),1()1,(;)1,(),1(;),1()0,(;)0,(),0( 4321 RRSRRSRRSRRS


Algebraic and Transfinite InterpolationAs a starting position, the one-dimensional interpolation scheme can be written for a two-dimensional domain in the form

where represents the interpolation derived between the boundaries . Similarly, for theboundary curve , the interpolant is

Finally, the following operator is formed

The operator is called the Boolean sum of and , and can be written as

or

This type of interpolation is called transfinite interpolations and the functions are called blending functions.This interpolation may be generalised by choosing other blending functions, for example functions that allows to control slopes of the grid lines at the boundaries

10,),1(),0()1(),( RRRa

10,)1,()0,()1(),( RRRb

),( aR constantconstant

bababa RRRRRR *

ba RR aR bR

)1,0()1()1,1()0,1()1()0,0()1)(1(

)1,()0,()1(),1(),0()1(),(

RRRR

RRRRRRR ba

1

)1,1()0,1(

)1,0()0,0(1

)1,(

)0,(1

),1(

),0(1),(

RR

RR

R

R

R

RRRR ba

1,,1,


Algebraic and Transfinite Interpolation2D Example

Without boundary slope control With boundary slope control


Algebraic and Transfinite InterpolationMulti-directional Interpolation – Three DimensionsTransfinite interpolation schemes can be extended into three dimensions where a domain is assume to be

boundedby six faces and naturaly mapped into a cube with local coordinates . The resulting grid will consist of astructured set of hexahedra.To form the interpolant , three unidirectional interpolation schemes are formed between opposite faces of the domain, namely:

Finally the interpolant is formed from

),,(

),,( R

,)1,,()0,,()1(),,(

,),1,(),0,()1(),,(

,),,1(),,0()1(),,(

RRRc

RRR

RRR

b

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),,( R

cbabccbbacba

bac

acb

cba

RRRRRRRRRRRR

RRR

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RRRR

*****

)(

)(

)(),,(


Grid Generation by PDEPDE: Partial Differential Equation The mathematical formulation of may problems in science and engineering involve partial differential equations.Special cases of the two-dimensional second order equation:

where a, b, c, d, e, f and g may be functions of the independent variables x and y and of the dependent variable , occur frequently because they often take the mathematical form of a conservation principles in physics.

This general equation can be classified according to the coefficients. The equation is

Elliptic equations are generally associated with equilibrium or steady-state problems. For example, steady flow of incompressible irrotational fluid, or heat conduction in a domain. The solution of an elliptic equation is a function of x and y which satisfies the partial differential equation at every point in a domain.Problems involving changes of variables with time usually lead to parabolic or hyperbolic equations. A common example of a parabolic equation derives from the theory of heat conduction, when the temperature is a function of time.Hyperbolic problems generally originate from vibration problems, or from problems where discontinuities can

persist in time, such as shock waves. Perhaps the simplest hyperbolic equation is the one-dimensional wave equation.

Although these different types of equations exhibit quiet different properties, they have a role to play in grid generation.

02

22

2

2

gfy

ex

dy

cyx

bx

a

04whenHyperbolic

04whenParabolic

04whenElliptic

2

2

2

acb

acb

acb


Grid Generation by PDEGrid Generation using Laplace and Poisson equationsAn appropriate form of grid generation equation in one dimension can be expressed as

subject to the generalised boundary conditions

where P is an arbitrary function used to control the grid point distribution.Thus, the grid generation problem involves1- the specification of the two end nodes, and ,2- the generation of the one-dimensional grid by solving the equation

In two dimensions, the equivalent equations are

where (x,y) are the coordinates in the physical space, the coordinates in the logical space, P and Q are arbitrary functions of the independent variables . The grid generation problem then involves1- the specification of the four corner nodes,2- the generation of one-dimensional grids on the four bounding curves, 3- the generation of the two-dimensional grid by solving the elliptic system of equations

)(2

2

Px

21 )1(,)0( sxsx

1s 2s

),(,),(2

2

2

2

2

2

2

2

Qyy

Pxx

),( ),(


Grid Generation by PDEThe elliptic operator naturally extend to three dimensions where the form is

The equations are solved for the grid coordinates (x,y,z) as a function of the independent coordinates of the logical space . The functions P, Q and R can be used to control the grid node spacing.The grid generation problem then involves1- the specification of the eight corner nodes,2- the generation of one-dimensional grids on the twelve bounding curves, 3- the generation of the two-dimensional grids on the six bounding faces,4- the generation of the three-dimensional grid by solving the elliptic system of equations

For the numerical solution procedure, second order accurate discretrisation of the operators are usually used combined with iterative solution procedure and over-relaxing techniques in order to improve convergence.The iterative solution process requires that initial conditions be prescribed to enable the initial solution to becomputed. It is possible to begin the iteration with any suitable conditions. However, an effective procedure is to use transfinite interpolation to form the initial guess of the solution.

),,(

),,(

,),,(

,),,(

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Rzzz

Qyyy

Pxxx


Grid Generation by PDEA function that satisfies Laplace’s equation is said to be harmonic which has the following properties:

1- Smoothness It is expected that the solution at any given point to be an average of the solution in surrounding region and, as such, be smoothly varying over the domain. In formals terms, it is possible to prove the mean value theorem for harmonic functions. This states that the value of a harmonic function at a point, is equal to the average of its values over the area of any circle centred on those point. In this case, the average would be defined as

2- Extremum principle It is possible to demonstrate to prove by using the mean value theorem, that if is harmonic, then the

maximum or minimum of must occur on the boundary. An immedfiate consequence of the maximum principle is that the solution to the problem

with specidied on every boundary, is unique and is indefinitely differentiable, i.e. .

For these properties, it is possible to infer the behaviour to be expected of grids generated using the system of Lapacian equations:1- The nodal distribution in the grids will be smooth and any discontinuities in the boundary data will not be propagated into the domain. 2- The solution will be unique3- The extremum principle may guarantee a one-to-one mapping between spaces. This is equivalent to ensuring a mapping exists for which the Jacobian J of the mapping is positive everywhere. However, this is not true for the grid generation equations written in the previous Laplacian forms: the solution admits negative Jacobians, which means the grid lines fold, or overlap, creating cells that have a negative area or volume.

dxdyarea

1

02

2

2

2

yx C


Grid Generation by PDETo overcome this practical limitation of the use of the Laplacian operator for the solution of the grid coordinates, it is necessary to consider the inverse mapping,using x and y as the independent variables and and as the independentvariables.

In two dimensions, the Laplacians for and with control functions and are

where

Inverting these equations leads

with

It is noted that the previous equations are nonlinear, elliptic and coupled in x and y.Unlike the original formulation, which could be solved for x and y , these equations must be solved together.

Grid cross-over

),( P),( Q

),(,),(2

2

2

2

2

2

2

2

Q

yxP

yx

),(,),( yxyx

)(2

,)(2

22

22

2

2

22

22

2

2

Qy

Pyyyy

Qx

Pxxxx

J

J

2222

,,

yxyyxxyx


Grid Generation by PDEThe major motivation of the use of the inverse equations is to obtain a mapping which possesses the property of the Jacobian being positive everywhere. Rado’s theorem shows that by requiring the inverse map to be harmonic instead of the map itself, then the mapping is one-to-one and onto and hence exhibits the property of a positive Jacobian everywhere.Certainly, in the exact differential form, the inverse equations provide a mean of generating a grid which is smooth,and does not produce a folding of the grid lines. It is emphatised that this property is exhibited by the exact differential form of the equations. This property does not necessarily hold for the numerical solution.Moreover, the advantageous property of no grid folding exhibited by the inverse equations does not in general hold true for the inhomogeneous form of the equations. Rados’ theorem no longer applies. The property of non-folding of grid lines does hold true provided the source term do not change sign within the domain.

Grid control functionsThe inherent smoothing properties of Laplace equations ensures, that, in absence of boundary curvature, the mesh points are evently spaced. However, near convex boundaries the grid points will become more closely spaced, whilst near concave boundaries the mesh spacing will be more sparse.


Grid Generation by PDEThe properties are not always desirable for mesh generation where it is essential that the mesh near a boundary reflects the shape of that boundary. Control of mesh point spacing can be achieved by introducing source terms into Laplace equation thus converting the system of equations into Poisson type. The source terms, often call grid control functions, can be so designed to provide the flexibility required to control mesh points spacing.

Negative values of Q tend to cause the -coordinates lines to move in the direction of decreasing , while negative values of P cause the -lines to move in the direction of decreasing . Thomson at al have reviewed the many forms of functions which can be used to attract or repel the mesh to lines or points. A typical function is

and similar for Q. Here the effect of the amplitude of the is to attract -lines towards the lines while theeffect of the amplitude of is to attract -lines toward the single point .

ia iib ),( ii

22exp)sign()exp()sign(),( iiii

iiiii

ii dbcaP


Grid Generation by PDEGrid Generation using Hyperbolic equationsComputational grids can also be generated by solving hyperbolic partial differential equations which are marchedoutwards from a specified distribution of points on an inner boundary. Constraints on mesh size and angle are generally used to determine the governing grid generation equations.Because the outer boundary of a grid can not specified in hyperbolic mesh generation, the method is best suited to applications which have asymptotic like outer boundaries. In external aerodynamics, for example, the outer boundary is in general a far-field condition of undisturbed flow.However, domain decomposition methods that use overset composite meshes for complex field simulations can also effectively employ hyperbolic grid generation.In two dimensions, the idea is to march outward from a user specified distribution of nodes on an inner boundary.The marching is performed by using a hyperbolic systems of equations, in which one of the independent variablescorresponds to the marching direction. It is assumed that the boundary is defined at

Following Steger, two equations are chosen which represent constraints on mesh orthogonality and mesh cell area.

The orthogonality constraint is

or

and the mesh area

or

where A is the area of any one mesh cell and must be prescribed

0),( yx

0

yyxx

),(,0 yx

xxx

Axyyx

AJ


Grid Generation by PDEThese two equations form a system of non-linear partial differential equations with initial data specified on the boundary .Steger proposed to solve these equations using a linearisation procedure, leading to

with

If is the marching direction, then

Using the fact that the inverse of a matrix H is

where is the determinant of H , and G is the transpose of the matrix of co-factors of H , then it can be shown that

The matrix exists if , and the matrix is symmetric. A symmetric matrix has real eigenvalues,

so the system is hyperbolic with used as the marching direction.

constant

fx

Bx

A

00

000

00

000

00

0

0,

//

//,

//

//

AAxy

yx

xy

yxfBA

x

ABfBx 1-1-

000 -

TGH

H11

H

2020

00

0

00

0

1000000000

00000000

00

10 ,

)

)1

,1

yx

A(Ax

A(Ay

yyxxxyyx

xyyxyyxx

BB

fBB

AB

10B 00 B 0

10 AB


Grid Generation by PDEThe numerical solution is achieved by using backward differences in the -direction and central differences in the -direction, i.e.

This gives the marching equations

These equations can be solved by either an explicit or implicit finite difference schemes. The implicit scheme can lead to marching the equations for any arbitrary step interval in the -direction.The grid is controlled by the initial nodal spacing on the boundary and by the cell area A , which is clearly a function of x and y, A=A(x,y). The cell areas, which provide the grid control, must be specified at every node in the mesh and as such it is necessary to devise automatic techniques to specify this information.Two approaches may be identified:1- Use a reference (background) grid where the area distribution is specified2- Determine the cell area as the grid is generated (extrapolation).

An example of a mesh around an aerofoil is here shown. The discontinuity propagated from the trailing edge is clearly visible

The advantage of a hyperbolic mesh generation procedure are that orthogonalgrids which can offer good user control can be generated very rapidly.The disadvantages are that boundaries in the field, including outer boundarycan not be specified and discontinuities in the boundary data can be propagatedinto the field.

2, 11

,

1

,

,kj,kj

kj

j,kj,k

kj

-- xxxxxx

1,)(2

1,1,10001,1, kjkjkjkj - xxABfBxx 1-1-

0


Grid Generation by PDEGrid Generation using Parabolic equationsGrid Generation based upon partial differential equations can also be performed.In general, such equations are constructed by modifying elliptic generation systems so that second derivatives, in one co-ordinate direction, do not appear. The solution of the parabolic system can then be marched away from a boundary in much the same way as that for hyperbolic grid generation. Here, however, some influence of the other boundaries is retained in the equations. Although some development of such schemes has been made, it has proved to be limited in its flexibility and applicability.


Multiblock Mesh Generation

It is clear that structured meshes result in a set of curvilinear coordinates.One way to seeing this is that, in two dimensional space, the physical domain is mapped to a rectangle in the logical (computational) domain. That means that a cartesian mesh in this logical space is mapped through a transformation to the physical space to provide a mesh consisting of quadrilaterals which are not necessary regular. The global property of curvilinear coordinates makes structured mesh generation somewhat limited in its application to complex geometries.

The idea behind multiblock mesh generation is that, instead of using one global curvilinear coordinate system,several local curvilinear systems are constructed and connected together.The domain is subdivided into blocks and within each block a curvilinear system is derived.The block subdivision provides the necessary flexibility to construct structured meshes for complex geometrical shapes.The multiblock concept is not restricted to any particular mesh point generation technique. The generation of points can be performed using the algebraic or the elliptic partial differential approach.



Topology Definition:Central to the multiblock concept is the block subdivision of a domain.Here a block is defined as a geometrical domain which is topologically equivalent to a rectangle in two dimensions or a cube in three dimensions.

A trivial example of a block subdivision of a domain around an airfoil is given in the following figure.Here the mesh topology is a so-called C-mesh. The block subdivision for the mesh topology is not unique.Two equally valid subdivisions are denoted by a and b. In the case a, the aerofoil boundary is restricted to one block side whilst in the second case the aerofoilboundary extends over one complete side and partially over two others. Either are possible but for discussion purposes it will be assumed that only one type of boundary can exist on any block side. This assumption does not in practice restrict the applicability of the multiblock approach, but does provide some simplification in the implementation.

a

b

Block structure for C-mesh topology



Another two examples of block subdivisions are shown for a O and a H type grid.

The block structure for a polar or O-topology can be reduced to a single block in which two opposite sides map onto each other.

The cartesian or H-topology mesh around an aerofoil is given by six blocks suitable connected.

Block structure for O-mesh topology

Block structure for H-mesh topology



Let consider now the more complex geometry of two aerofoils, with rounded leading edges, arranged in a tandem configuration. It is possible in this case to generate a mesh which effectively has one global coordinate system. A cartesian mesh could be constructed local to each airfoil or alternatively the forward aerofoil could be favouredwith a C-mesh but the aft-element would then have a cartesian mesh.

On the grounds of mesh quality and flow sensitivity in the vicinity of the leading edges a cartesian mesh local to leading edge may be found to be unsuitable. Ideally, it can be argued that a C-mesh or a polar mesh is required to provide the necessary grid quality for accurate flow simulations. If this is the case, it is no longer possible to build such mesh topologies and conserve a global curvilinear coordinate system. However, it is topologically possible to construct such meshes and the concept of a multiblock subdivision of the domain provides the appropriate mechanism.

Cartesian mesh around both airfoils C-mesh around the leading airfoil and H-mesh around the second airfoil


Mesh Generation within a BlockEach block has its own curvilinear coordinate system. This system is independent of those in the adjacent blocks.To generate a mesh within a block, it is necessary to construct an algorithm which operates on a rectangular computational domain.With a suitable treatment of boundary points, the generation of the global mesh then involves a loop over all

blocks.

The mesh within each block is in general build after discretisation of its boundaries.Thus, assuming that the boundary points are known, the interior points may be determined by using a system of elliptic partial differential equations - in this case, the boundary points location corrisponds to a Dirichlet condition –or by using transfinite interpolations.The basic idea is that the boundary points are either known or compute by some other techniques.


Point and Slope Continuity at InterfacesClearly, a computational mesh can not contain holes or void regions. For this condition to be satisfied, it is necessary that given two blocks which are adjacent and have a common

edge,the grid points in physical space, which correspond to points on the edge, must be continuous.

With fixed data points on block boundaries, it is possible to form a well defined problem for the generation of points within a block. But, unless special treatment to be applied, the grid lines extending from such boundaries will not possess special angular relationships with boundary. If a block boundary is adjacent to another block boundary,

andthe boundary points are fixed, the resulting mesh will in general be discontinuous in gradients (and higher derivatives) accross the boundary.

It is possible to achieve slope continuity at block interfaces, by providing to the internal nodes grid generation technique, grid lines discretisation and grid lines intersection angles on both block edges.


Singular Points Treatment

Flexibility for structured multi-block grid generation introduces singular points in the mesh.

Usually in two dimensions two families of lines pass through a given point.

Singular points are those points in the field which more or less than two families of curvilinear lines intersect.

Classical discretisations, like finite difference formulationscannot be defined or have to be adapted at singular points.

Alternative means of expressive derivatives at those points must be found.

→ Impact on the solver


Example of Multiblock Structured Grid


SummaryIntroduction of the mesh generation concept.

Mathematical representation of the geometry.


• Algebraic and Transfinite Interpolations• Grid generation by using PDE (elliptic, hyperbolic, parabolic)• Multi-block structured grid generation concept

Structured grid generation is efficient for simple geometry, but lack of efficiency for complex geometry.

Strong impact on the solver that has to be adapted at particular features of the grid like singular points.

New developments (Chimera techniques, ….) transfer the complexity from the grid generation process to the solver: easier(?) grid generation process, but more complex solvers.

The technologies based on the unstructured grid generation approach seems to be more adapted for application of CFD (Computational Fluid Dynamics) around complex shapes