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Dr. Noemi Friedman, 24. 01. 2018.
Introduction to PDEs and Numerical Methods
Lecture 13.
The finite element method: assembling the matrices,
isoparametric mapping, FEM in higher dimension
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture| Seite 2
RECAP: How to solve PDE with
FEM with nodal basis, piecewise linear shape functions
Finite Element method with piecewise linear functions in 1D, hom DBC
1) Weak formulation of the PDE, definition of the โenergyโ inner product (the bilinear
functional, ๐) and and the linear functional (๐น)
๐ ๐ข, ๐ฃ = ๐น ๐ฃ2) Define approximating subspace by definition of a mesh (nodes 0,1,..N, with coordinates,
elements) and setup the hat functions on them
3) Compute the elements of the stiffness matrix (Grammian) โ evaluation of integrals
๐พ๐๐ = ๐ ๐๐(๐ฅ), ๐๐(๐ฅ) = ๐๐(๐ฅ), ๐๐(๐ฅ) ๐ธ๐, ๐ = 1. . ๐ โ 1
4) Compute the elements of the vector of the right hand side โ evaluation of integrals
๐๐ = ๐น ๐๐ , ๐ = 1. . ๐ โ 15) Solve the system of equations:
for ๐ฎ, which gives the solution at the nodes.
The solution in between the nodes can be calculated from:
ฮฆ๐ ๐ฅ = Ni x =
๐ฅ โ ๐ฅ๐โ1
๐๐ฅ โ [๐ฅ๐โ1, ๐ฅ๐]
๐ฅ๐+1 + ๐ฅ
๐๐ฅ โ [๐ฅ๐ , ๐ฅ๐+1]
0 else
๐๐ฎ = ๐
๐ข x โ
๐=1
๐
๐ข๐ ๐๐(x)
๐ = 1. . ๐ โ 1
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 3
Recap:
1D Example with linear nodal basis
๐(๐ฅ)
๐
๐/5 ๐/5 ๐/5๐/5 ๐/5
instead:
Compute stiffness matrix elementwisely and
then assemble
๐ ๐๐ฎ
Global stiffness matrix
1 2 3 4 5 6
๐พ4๐ =
1 2 3 4 5 6
1
2
3
4
5
6
๐ข1
๐ข2
๐ข3
๐ข4
๐ข5
๐ข6
0
๐2
๐3
๐4
๐5
0
=
4 5
4
5
๐พ4๐ 1,1 = ๐ธ๐ด
ฮฉ4
๐๐4(๐ฅ)
๐๐ฅ
๐๐4(๐ฅ)
๐๐ฅ๐๐ฅ
๐พ4๐(1,1) ๐พ4
๐(1,2)
๐พ4๐(2,1) ๐พ4
๐(2,2)
๐4 ๐5๐3๐2
๐พ4๐ 1,2 = ๐ธ๐ด
ฮฉ4
๐๐4(๐ฅ)
๐๐ฅ
๐๐5(๐ฅ)
๐๐ฅ๐๐ฅ
๐พ4๐ 2,1 = ๐ธ๐ด
ฮฉ4
๐๐5(๐ฅ)
๐๐ฅ
๐๐4(๐ฅ)
๐๐ฅ๐๐ฅ
๐พ4๐ 2,2 = ๐ธ๐ด
ฮฉ4
๐๐5(๐ฅ)
๐๐ฅ
๐๐5(๐ฅ)
๐๐ฅ๐๐ฅ
๐พ4๐(1,2)๐พ4
๐(1,1)
๐พ4๐(2,1) ๐พ4
๐(2,2)๐พ5
๐(1,1)
๐พ3๐(2,2)
๐พ3๐ 1,1
๐พ2๐(2,2)
๐พ2๐ 1,1
๐พ1๐(2,2)
1
1
๐พ3๐(1,2)
๐พ2๐(1,2)
๐พ3๐(2,1)
๐พ2๐(2,1)
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 4
Recap:
1D Example with linear nodal basis
๐(๐ฅ)
๐
๐/5 ๐/5 ๐/5๐/5 ๐/5
instead:
Compute stiffness matrix elementwisely and
then assemble
๐ ๐๐ฎ
1 2 3 4 5 6
๐4๐ =
1 2 3 4 5 6
1
2
3
4
5
6
๐ข1
๐ข2
๐ข3
๐ข4
๐ข5
๐ข6
0
0
=
4
5
๐4๐ 1 =
ฮฉ4
๐(๐ฅ)๐4(๐ฅ)๐๐ฅ
๐4๐ 1
๐4๐ 2
๐4 ๐5๐3๐2
๐4๐ 2 =
ฮฉ4
๐(๐ฅ)๐5(๐ฅ)๐๐ฅ
๐พ4๐(1,2)๐พ4
๐(1,1)
๐พ4๐(2,1) ๐พ4
๐(2,2)๐พ5
๐(1,1)
๐พ3๐(2,2)
๐พ3๐ 1,1
๐พ2๐(2,2)
๐พ2๐ 1,1
๐พ1๐(2,2)
1
1
๐พ3๐(1,2)
๐พ2๐(1,2)
๐พ3๐(2,1)
๐พ2๐(2,1)
๐4๐ 1
๐4๐ 2
๐2๐ 1
๐1๐ 2
๐3๐ 1
๐2๐ 2
๐3๐ 2
๐5๐ 1
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture| Seite 5
The same but elementwisely: How to solve PDE with
FEM with nodal basis, piecewise linear shape functions
Finite Element method with piecewise linear functions in 1D, hom DBC
1) Weak formulation of the PDE, definition of the โenergyโ inner product (the bilinear
functional, ๐) and and the linear functional (๐)๐ ๐ข, ๐ฃ = ๐ ๐ฃ
2) a.) Define reference element, define maping between global and local coordinate systems
ฮพ ๐ฅ ๐ฅ(๐)
b.) Define reference linear shape functions
3) Compute the โelement stiffnessโ matrix โ evaluation of integrals
๐พ๐๐ = ๐ ๐๐(๐ฅ), ๐๐(๐ฅ) = ๐๐(๐ฅ), ๐๐(๐ฅ) ๐ธ๐, ๐ = 1. . 2
4) Compute the right hand side elementwisely ๐๐๐ = ๐ ๐๐ , ๐ = 1,2
5) Compileโ global stiffnessโ matrix
6) Solve the system of equations:
for ๐ฎ, which gives the solution at the nodes.
The solution in between the nodes can be calculated from:
๐๐ฎ = ๐
๐ข x โ
๐=1
๐
๐ข๐ ๐๐(x)
N1 ฮพ = 1 โ ฮพ N2 ฮพ = ฮพ
๐พ ๐ = ๐พ4๐(1,1) ๐พ4
๐(1,2)
๐พ4๐(2,1) ๐พ4
๐(2,2)
๐4๐ = ๐4
๐ 1
๐4๐ 2
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 6
Local/global coordinate system 1D
๐พ4๐ ๐, ๐ = ๐ธ๐ด
ฮฉ4
๐๐๐(๐)
๐๐
๐๐
๐๐ฅ
๐๐๐(๐)
๐๐
๐๐
๐๐ฅ๐๐ฅ =
๐ธ๐ด
๐4๐ 2
ฮฉ4
๐๐๐(๐)
๐๐
๐๐๐(๐)
๐๐๐๐ฅ
๐ = [0,1]
1
๐๐
1
๐๐
๐พ4๐ ๐, ๐ =
๐ธ๐ด
๐ ๐ 2 0
1 ๐๐๐(๐)
๐๐
๐๐๐(๐)
๐๐
๐๐ฅ(๐)
๐๐๐๐ =
๐ธ๐ด
๐ ๐ 0
1 ๐๐๐(๐)
๐๐
๐๐๐(๐)
๐๐๐๐
๐ ๐
Idea:
coordinate transformation to have unit length elements element stiffnes matrix is the same for each element
๐๐
๐๐ฅ=
1
๐ ๐
๐พ4๐ ๐, ๐ = ๐ธ๐ด
ฮฉ4
๐๐4(๐ฅ)
๐ฅ
๐๐5(๐ฅ)
๐๐ฅ๐๐ฅ
๐, ๐ โ [1,2]
๐, ๐ โ [4,5]
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 7
Local/global coordinate system 1D
๐4๐ ๐ =
ฮฉ4
๐(๐ฅ)๐4(๐ฅ)๐๐ฅ
ฮฉ4
๐(๐ฅ)๐5(๐ฅ)๐๐ฅ=
0
1
๐(๐)๐๐(๐)๐๐ฅ(๐)
๐๐๐๐ = ๐ ๐
0
1
๐(๐)๐๐(๐)๐๐
๐ โ [1,2]
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 8
Local/ coordinate system, isoparametric mapping 1D
coordinate transformation
using the ansatzfunctions isoparametric mapping
functions of lower order: subparametric
functions of higher order: superparametric
๐ฅ ๐ = ๐ฅ๐๐1 ๐ + ๐ฅ๐+1๐2 ๐ = ๐1 ๐ ๐2 ๐๐ฅ๐
๐ฅ๐+1
local coordinate
global coordinate
๐ = 0 ๐ = 1
๐ฅ๐ ๐ฅ2
Shape functions:
Transformation from local to global coordinates:
Stiffness matrix with isoparametric elements:
๐ = [0,1]๐ฅ
๐1 ๐ = 1 โ ๐
๐2 ๐ = ๐
๐พ4๐ ๐, ๐ = ๐ธ๐ด
ฮฉ4
๐๐๐(๐ฅ)
๐ฅ
๐๐๐(๐ฅ)
๐๐ฅ๐๐ฅ = ๐ธ๐ด
ฮฉ4
๐๐๐(๐)
๐๐
๐๐
๐๐ฅ
๐๐๐(๐)
๐๐
๐๐
๐๐ฅ๐๐ฅ
๐, ๐ โ [1,2]
๐, ๐ โ [4,5]
๐พ4๐ ๐, ๐ = ๐ธ๐ด
0
1 ๐๐๐(๐)
๐๐
๐๐ฅ
๐๐
โ1๐๐๐(๐)
๐๐
๐๐ฅ
๐๐
โ1๐๐ฅ(๐)
๐๐๐๐
๐๐ฅ
๐๐= ๐ฅ๐
๐๐1 ๐
๐๐+ ๐ฅ๐+1
๐๐2 ๐
๐๐
๐๐ฅ
๐๐=
๐๐1 ๐
๐๐
๐๐2 ๐
๐๐
๐ฅ๐
๐ฅ๐+1
๐๐ฅ
๐๐
โ1 ๐๐ฅ
๐๐
โ1
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 9
The same but differently:
Local/ coordinate system, isoparametric mapping 1D
coordinate transformation
using the ansatzfunctions isoparametric mapping
functions of lower order: subparametric
functions of higher order: superparametric
๐ฅ๐๐๐๐ ๐ = ๐ฅ๐๐1 ๐ + ๐ฅ๐+1๐2 ๐ = ๐1 ๐ ๐2 ๐๐ฅ๐
๐ฅ๐+1
local coordinate
global coordinate
๐ = โ1 ๐ = 0 ๐ = 1
๐ฅ๐ ๐ฅ2
Basis functions:
Transformation from local to global coordinates:
Stiffness matrix with isoparametric elements:
ยฑ1/2 ยฑ1/2
๐
๐๐๐ฅ๐๐๐๐ ๐ =
๐
๐๐๐1 ๐
๐
๐๐๐2 ๐
๐ฅ๐
๐ฅ๐+1
=1
2๐๐โ1/2 +1/2
2๐๐ 2๐๐ 1/2๐๐
๐พ๐๐ = ๐ธ๐ด
๐พ4๐ =
๐ธ๐ด
๐ ๐1 โ1
โ1 1
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture| Seite 10
FROM STRONG FORM TO WEAK FORM in higher dimension
Steps of formulating the weak form (recipe)
๐ฟ๐ข, ๐ฃ โ ๐, ๐ฃ = 0 โ๐ฃ โ ๐
1.) Multiply by test function ๐ and integrate
๐ฟ๐ข(๐ฑ) = ๐(๐ฑ)
๐ฟ๐ข ๐ฑ ๐ฃ ๐ฑ ๐๐ฑ โ ๐ ๐ฑ ๐ฃ ๐ฑ ๐๐ฑ = 0
2.) Reduce order of ๐ฟ๐ข, ๐ by using Greenโs theoreem (generalized integration by
parts)
3.) Apply boundary conditions
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture| Seite 11
FROM STRONG FORM TO WEAK FORM
Recap: differential operators (grad, div, curl), Greenโs theorem
Greenโs identity
1) rewrite equation with the product rule in multiple dimensions
๐ป โ ๐ฃ ๐ป๐ข = ๐ป๐ฃ โ ๐ป๐ข + ๐ฃ โ๐ข
2) integrate both sides over the domain ฮฉ (bounded by ๐ฮฉ)
ฮฉ
๐ป โ ๐ฃ ๐ป๐ข ๐ฮฉ = ฮฉ
๐ป๐ฃ โ ๐ป๐ข ๐ฮฉ + ฮฉ
๐ฃ โ๐ข ๐ฮฉ
3) apply divergence theorem
ฮฉ
๐ป โ ๐ฃ ๐ป๐ข ๐ฮฉ = ฮฉ
๐๐๐ฃ ๐ฃ ๐ป๐ข ๐ฮฉ = ๐ฮฉ
๐ฃ ๐ป๐ข โ ๐ ๐๐ฮฉ
โ ฮฉ
๐ฃ โ๐ข ๐ฮฉ = ฮฉ
๐ป๐ฃ โ ๐ป๐ข ๐ฮฉ โ ๐ฮฉ
๐ฃ ๐ป๐ข โ ๐ ๐๐ฮฉ
similar to integration by part in multiple dimensions
Recap: Multidimensional stationary heat equation
with inhomogeneous Dirichlet and Neumann BC.
๐ฟ๐ข(๐ฑ) = ๐(๐ฑ)Strong form:
Example:
1.) Multiply by test function ๐ฃ and integrate
โฮ๐ข ๐ฑ ๐ฃ ๐ฑ ๐ฮฉ โ ๐ ๐ฑ ๐ฃ ๐ฑ ๐ฮฉ = 0
โฮ๐ข ๐ฑ ๐ฃ ๐ฑ ๐ฮฉ = ฮฉ
๐ป๐ข ๐ฑ โ ๐ป๐ฃ ๐ฑ ๐ฮฉ โ ๐ฮฉ
๐๐ข
๐๐๐ฃ ๐ฑ ๐ฮ
2.) Reduce order of ๐ฟ๐ข, ๐ฃ by using divergence theorem
โฮ๐ข ๐ฑ = ๐๐ข = ๐๐๐ข
๐๐= โ
convert to homogeneous problem:
๐ข = ๐ + ๐ข๐:known function,๐ = ๐ on ฮ๐ท ๐ข:new function that we look for
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture| Seite 12
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture| Seite 13
Recap: Multidimensional stationary heat equation
with inhomogeneous Dirichlet and Neumann BC.
ฮฉ
๐ป๐ข ๐ฑ โ ๐ป๐ฃ ๐ฑ ๐ฮฉ = ฮฉ
๐ ๐ฑ ๐ฃ ๐ฑ ๐ฮฉ + ฮ๐
โ๐ฃ ๐ฑ ๐ฮ
โฮ๐ข ๐ฑ ๐ฃ ๐ฑ ๐ฮฉ = ฮฉ
๐ป๐ข ๐ฑ โ ๐ป๐ฃ ๐ฑ ๐ฮฉ โ ๐ฮฉ
๐๐ข
๐๐๐ฃ ๐ฑ ๐ฮ
3.) Apply boundary conditions
๐ฮฉ
๐๐ข
๐๐๐ฃ ๐ฑ ๐ฮ =
ฮ๐
๐๐ข
๐๐๐ฃ ๐ฑ ๐ฮ +
ฮ๐ท
๐๐ข
๐๐๐ฃ ๐ฑ ๐ฮ =
ฮ๐
โ๐ฃ ๐ฑ ๐ฮ
โ 0
ฮฉ
๐ป ๐ ๐ฑ + ๐ข ๐ฑ โ ๐ป๐ฃ ๐ฑ ๐ฮฉ = ฮฉ
๐ ๐ฑ ๐ฃ ๐ฑ ๐ฮฉ + ฮ๐
โ๐ฃ ๐ฑ ๐ฮ
ฮฉ๐ป ๐ข ๐ฑ โ ๐ป๐ฃ ๐ฑ ๐ฮฉ = ฮฉ๐ ๐ฑ ๐ฃ ๐ฑ ๐ฮฉ + ฮ๐โ๐ฃ ๐ฑ ๐ฮ โ ฮฉ๐ป๐ ๐ฑ โ ๐ป๐ฃ ๐ฑ ๐ฮฉ
from natural/Neumann BC from essential/Dirichlet BC
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 14
Isoparametric linear mapping 2D
triangular elements
Basis functions:
Transformation from local to global coordinates:
Stiffness matrix:
1
๐, ๐ โ [1,2,3]
๐ฅ๐๐๐๐ ๐, ๐
๐ฆ๐๐๐๐ ๐, ๐=
๐1 ๐, ๐ ๐2 ๐, ๐
๐1 ๐, ๐
๐3 ๐, ๐
๐2 ๐, ๐ ๐3 ๐, ๐
๐ฅ1๐ฆ1
๐ฅ2๐ฆ2
๐ฅ3
๐ฆ3
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 15
Isoparametric linear mapping 2D
triangular elements
Stiffness matrix:
Stiffness matrix with local coordinates:
where:
substitution rule
determinant should not be negative or zero!
๐ =
๐=1
3๐๐๐ ๐, ๐
๐๐๐ฅ๐
๐=1
3๐๐๐ ๐, ๐
๐๐๐ฅ๐
๐=1
3๐๐๐ ๐, ๐
๐๐๐ฆ๐
๐=1
3๐๐๐ ๐, ๐
๐๐๐ฆ๐
๐, ๐ โ [1,2,3]
๐, ๐ โ [1,2,3]๐ฒ๐๐ =
0
1
0
1โ๐
๐ฑโ๐ป
๐๐๐
๐๐๐๐๐
๐๐
โ ๐ฑโ๐ป
๐๐๐
๐๐๐๐๐
๐๐
๐ฑ ๐๐๐๐
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 16
Isoparametric linear mapping 2D
triangular elements, example
๐ฅ ๐, ๐
๐ฆ ๐, ๐=
1 โ ๐ โ ๐ ๐
1 โ ๐ โ ๐
๐๐ ๐
237197
Transformation from local to global coordinates (isoparametric mapping):
๐ฅ๐๐๐๐ ๐, ๐
๐ฆ๐๐๐๐ ๐, ๐=
๐1 ๐, ๐ ๐2 ๐, ๐
๐1 ๐, ๐
๐3 ๐, ๐
๐2 ๐, ๐ ๐3 ๐, ๐
๐ฅ1๐ฆ1
๐ฅ2๐ฆ2
๐ฅ3
๐ฆ3
1 2
3
4
5 6
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 17
Local/ coordinate system, isoparametric mapping 2D
triangular elements, example
Stiffness matrix with local coordinates:
๐, ๐ โ [1,2,3]๐ฒ๐๐ = 0
1
0
1โ๐
๐ฑโ๐ป
๐๐๐
๐๐๐๐๐
๐๐
โ ๐ฑโ๐ป
๐๐๐
๐๐๐๐๐
๐๐
๐ฑ ๐๐๐๐
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 18
Local/ coordinate system, isoparametric mapping 2D
triangular elements, example
๐ฑ๐ป = ๐ฑ๐ป = ๐ฑ = 34๐ฑ =๐ ๐โ๐ ๐
๐ฑโ๐ป =๐
๐ฑ๐ป๐ ๐โ๐ ๐
๐ฒ๐๐
๐=
0
1
0
1โ๐ 1
344 2โ7 5
๐๐๐
๐๐๐๐๐
๐๐
โ1
344 2โ7 5
๐๐๐
๐๐๐๐๐
๐๐
34๐๐๐๐
๐ฒ21
๐= 0
1 01โ๐ 1
34
4 2โ7 5
โ1โ1
โ1
34
4 2โ7 5
10
34๐๐๐๐ == 01 โ1
1โ๐ 1
34
โ62
โ4โ7
๐๐๐๐
๐ฒ21
๐==โ1.118 0
1 01โ๐
๐๐๐๐ = โ1.118 โ 1
2= โ0.559
๐ฒ๐๐
๐=
0
1
0
1โ๐
๐ฑโ๐ป
๐๐๐
๐๐๐๐๐
๐๐
โ ๐ฑโ๐ป
๐๐๐
๐๐๐๐๐
๐๐
๐ฑ ๐๐๐๐
๐ฒ11
๐๐ฒ12
๐๐ฒ13
๐
๐ฒ21
๐๐ฒ22
๐๐ฒ23
๐
๐ฒ13
๐๐ฒ23
๐๐ฒ33
๐
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 19
Local/ coordinate system, isoparametric mapping 2D
triangular elements, example
๐ฑ๐ป = ๐ฑ = 34๐ฒ11
๐๐ฒ12
๐๐ฒ13
๐
๐ฒ21
๐๐ฒ22
๐๐ฒ23
๐
๐ฒ13
๐๐ฒ23
๐๐ฒ33
๐
๐๐ =
ฮฉ๐
๐(๐ฅ)๐1(๐ฅ, ๐ฆ)๐๐ฅ
ฮฉ๐
๐(๐ฅ)๐2(๐ฅ, ๐ฆ)๐๐ฅ
ฮฉ๐
๐(๐ฅ)๐3(๐ฅ, ๐ฆ)๐๐ฅ
=
0
1
0
1โ๐
๐(๐)๐1(๐, ๐) ๐ฑ ๐๐๐๐
0
1
0
1โ๐
๐(๐)๐2(๐, ๐) ๐ฑ ๐๐๐๐
0
1
0
1โ๐
๐(๐)๐3(๐, ๐) ๐ฑ ๐๐๐๐
๐ข1๐
๐ข2๐
๐ข3๐
๐1๐
๐2๐
๐3๐
=
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 20
Local/ coordinate system, isoparametric mapping 2D
triangular elements, example
๐ฒ11
๐๐ฒ12
๐๐ฒ13
๐
๐ฒ21
๐๐ฒ22
๐๐ฒ23
๐
๐ฒ31
๐๐ฒ32
๐๐ฒ33
๐
๐ข1๐
๐ข2๐
๐ข3๐
๐1๐
๐2๐
๐3๐
=
1 2
3
4
5
local 1 2 3global 4 3 6
6
๐ฒ22
๐๐ฒ21
๐๐ฒ23
๐
๐ฒ12
๐๐ฒ11
๐๐ฒ13
๐
๐ฒ32
๐๐ฒ31
๐๐ฒ33
๐
๐ข1
๐ข2
๐ข3
๐ข4
๐ข5
๐ข6
๐2๐
๐1๐
๐3๐
4
3
6
1
2
3
4
5
6
1
2
3
1 2 3 4 5 6
=
4
3
6
24. 01. 2018. | Dr. Noemi Friedman | PDE lecture | Seite 21
Local/ coordinate system, isoparametric mapping 2D
quadrilateral elements
๐
๐(โ1,1)
(1, โ1)(โ1, โ1)
(1,1)
1
32
4
Basis functions:
Transformation from local to global coordinates:
Stiffness matrix:
1
๐ฅ๐๐๐๐ ๐, ๐
๐ฆ๐๐๐๐ ๐, ๐=
๐1 ๐, ๐ ๐2 ๐, ๐
๐1 ๐, ๐
๐3 ๐, ๐
๐2 ๐, ๐ ๐3 ๐, ๐
๐4 ๐, ๐
๐4 ๐, ๐
๐ฅ1๐ฆ1
๐ฅ2๐ฆ2
๐ฅ3๐ฆ3
๐ฅ4
๐ฆ4
๐พ๐๐ = ฮฉ
๐ป๐๐(๐ฑ) โ ๐ป๐๐ ๐ฑ ๐ฮฉ