Numerical Analysis Interpolation and fitting

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Numerical AnalysisInterpolation and fitting

• Adam Fic

2
Interpolation and fitting
Kind of approximations 1) expanding functions
in Taylor series 2) expanding functions in
series of orthogonal functions 3) interpolation
(extrapolation) 4) fitting (least square
approximation)
Problem - function f(x) is given in analytical
form - values of function f(x) are given at
selected points (nodes) x1, y1, x2, y2,
....... xn, yn,
How to find function Pn(x) which approximates
function f(x) ???
3
Interpolation and fitting
Interpolation
Fitting, least square approximation
nm
ngtm
4
Fitting, Least square approximation
Necessary conditions
Coefficient of correlation
5
Interpolation
nm
Extrapolation ?
6
Lagrange interpolation
Lagrange functions, Lagrange coefficients
properties of Lagrange functions

power ?

lni(xk) ?
General Lagrange polynomial
Examples
Linear interpolation
Quadratic interpolation
-
x
x

1
)
3
(
)
(
2
x
l
L
2
-
x
x
-
-
)
)(
(
x
x
x
x
2
1
1

)
6
(
)
(
3
2
x
l
L
-
3
-
-
x
x
)
)(
(
x
x
x
x

2
)
4
(
)
(
1
x
l
3
1
2
1
L
2
-
x
x
-
-
)
)(
(
x
x
x
x
1
2
2

)
7
(
)
(
3
1
x
l
L
3
-
-


2
1
)
)(
(
x
x
x
x
)
5
...(
)
(
)
(
)
(
x
l
y
x
l
y
x
L
3
2
1
2
2
2
2
1
2
-
-
)
)(
(
x
x
x
x

2
1
3
)
8
(
)
(
x
l
L
3
-
-
)
)(
(
x
x
x
x
2
3
1
3



3
2
1
)
9
...(
)
(
)
(
)
(
)
(
x
l
y
x
l
y
x
l
y
x
L
3
3
3
2
3
1
3
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Selecting power of the interpolation polynomials
Runges problem
3
4
2
1
5
How to avoid oscillations ?
Use piecewise polynomial, i.e. piecewise linear
Spline (thick elastic rod) a function, which
has several number of derivatives everywhere in
the considered interval of interpolation a, b,
i.e. also at nodes of interpolation.
Bezjer cubic splines - perhaps the most
important they have continuous first and second
derivatives.
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Cubic Bezier splines
Application for the least square fitting
Definition
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Radial basis functions
Radial functionInverse multiquadraticMultiquadra
ticGaussThin splines (cienkiej plytki)
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Shape functions
Basic properties of shape functions basic
applications (interpolation, element
transformation)
Lagrange shape functions in 1D
Lagrange shape functions satisfies eq. (1) and
(2) !
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Shape functions for 2D elements

• Elements used in 2D
• triangles
• square elements used as basic elements to
  construct
• quadrilateral elements and elements
• with curvilinear edges

• Shape functions for quadratic elements
• Lagrange family
• Serendipity family

Shape functions for triangles (first order
triangle element)
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Serendipity family in 2D
4 nodes basic elements
3
4
Definition of shape functions
1
1
-1
-1
2
1
N3
N1
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Serendipity family in 2D
8 nodes basic elements
Definition of shape functions
3
4
7
1
1
-1
8
6
-1
1
2
5
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Serendipity family in 3D
8 nodes element (first order element)
7
8
5
6
4
3
1
2
20 nodes element (second order element)
19
7
8
18
20
17
5
6
15
16
13
4
11
14
3
12
10
1
2
9
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Transformation of straight element into a curve
Selected nodes
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Transformation of 2D elements
Examples of element transformation
Elements proper for modelling boundary problems