Alexander Dbert

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Transformation Techniques for block structured
grids

• Alexander Döbert
• Seminar on Grid Generation
• Summer Term 2005

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Motivation

• What is a transformation ?
• Why do we need transformation techniques ?

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What is a transformation ?
Physical domain
Computational domain
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Why do we need transformation techniques I.
Problem A normal Cartesian grid with constant
parameter lines does not approximate the contour
of an object good enough.
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Why do we need transformation techniques II.
Example of an well adapted grid
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Overview

• Coordinate Transformation
• Modeling of curves and surfaces
• Distribution of points
• Generating structured grids

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Coordinate Transformation

• The basis of an object adapted grid relates to
  the parameters , and . These parameter
  lines continue curvilinear in the Cartesian
  framework. Therefore they are called curvilinear
  coordinates.

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Mapping of coordinates

• physical domain

computational domain
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Modeling of curves and surfaces
The boundaries of the calculation domain are
shaped as curves in 2D or surfaces in 3D. These
are generally curvilinear. The grid points inside
can be distinguished as intersections of curves
or surfaces.
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Lagrange Interpolation

• Lagrangian interpolation formula

Lagrangian polynomial
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Lagrange Interpolation

• Pro
• The polynomial itself and its derivatives are
  continuous.
• Contra
• In application of many points it leads to heavy
  oscillations.

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Hermite Interpolation

• Hermitian interpolation formula

Hermitian polynomials
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Hermite Interpolation

• Pro
• The polynomial itself and its derivatives are
  continuous.
• Additionally to the discrete curve points the
  gradients of these points are included.
• Contra
• In application of many points it leads to heavy
  oscillations.

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Spline function
Because of the oscillating character of
polynomials they are not convenient to
interpolate curves through multiple points. A
corrective is the so-called Spline
function. The interpolation polynomial is
constructed in intervals. It is demanded that the
polynomial is continuous differentiable on the
interval.
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Types of Spline functions

• Cubical spline function
• Parameter spline function
• Bézier spline function

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Cubical splines I

• Cubical spline functions are polynomials of third
  order on an interval .

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Cubical splines II

• Include four constants to get enough flexibility
  to ensure two continuous derivatives on the
  interval boundaries.

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Parameter splines

• Both of the visualized methods are unsuitable to
  generate closed curves because they require a
  monotone gradient.
• Parameter splines are more general and
  interpolate the individual components of the
  curve-coordinates each with spline functions.

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Cubical Bézier splines I

• Bézier splines apply polynomials of third order.
• They approximate the curve progression through
  four so-called Bézier-Points in each interval.

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Cubical Bézier splines II
Example Four points
on the interval are given. The
progression of the curve from is
approximated as follows
With Lagrangian Polynom
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Cubical Bézier splines II
The result is the curve through
and
and the tangents at the positions
and
point to the same direction as
and
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Distribution of points
The sense of an exact description of curves and
surfaces is, that a point for an arbitrary curve
or surface parameter lies exactly on the defined
contour. The chosen points serve as boundary grid
points and therefore their distribution is very
important for the shaping of the grid.
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Distribution on curves

• Distribution of the curve Parameter in a way that
  the points on the curve satisfy certain demands.
  (e.g. aggregation in a part of the domain)
• The relation between every point and the
  appropriate curve parameter can be defined with
  linear interpolation.
• Determination of the distribution function
  dependents on the demands.

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Distribution on curvesExample
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Distribution on surfaces I

• The determination of surface parameters is mostly
  done in two steps.
• First Lines in one direction

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Distribution on surfaces II

• Second Lines in the other direction

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Generating structured grids

• Shear transformation
• Tensor product transformation
• Transfinite interpolation

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Shear transformation I

• Lagrangian interpolation through two points.
• Two shearings between two boundary points

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Shear transformation II
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Tensor product transformation

• successive application of the shear
  transformation.
• Combines the two equations of the shear
  transformation (bilinear transformation).

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Transfinite Interpolation I

• Combination of shear-transformation and
  tensor-product-transformation.

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Transfinite Interpolation II
Derivation

• Sum of A and B from Shear-Transformation contains
  all four boundary curves and connections lines
  between adjacent vertices.
• Tensor-product-transformation also generates
  linear coordinate progressions.
• Therefore subtract T from A B.
• The result is called Transfinite-interpolation.

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Transfinite Interpolation 3D I

• Needs three curvilinear coordinates
• This leads to

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Transfinite Interpolation 3D II
Example
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Thank you for your attention!