Weighted residual method (метод взвешенных невязок)
Initial equation: |
2U (x, y) |
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2U (x, y) |
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2U (x, y) 0 |
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(x, y) |
- arbitrary function (произвольная функция) |
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Evidently: 2U (x, y) (x, y) 0
2U (x, y) (x, y)dS 0 S - is the problem domain area
S
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Galerkin method (метод Бубнова-Галеркина)
2U (x, y) (x, y)dS 0 |
This relation should be valid for all |
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possible functions (x, y) |
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Let us replace the unknown function by its approximation:
2U~(x, y) (x, y)dS R 0
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R is called ‘weighted residual’
(x, y) is called ‘weighting function’
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Galerkin method
Main ideas of the method are: |
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to use the weighted residual method; |
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to use approximation functions for weighting: |
j (x, y) j (x, y) |
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to set residuals to zero; |
Rj 0 |
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to apply integration-by-parts procedure to the integral (week formulation)
integration-by-parts = интегрирование по частям week formulation = ослабленная формулировка
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Week formulation
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Initial equation is: |
2U~(x, y) (x, y)dS R |
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2 ~ |
(x, y) j (x, y)dS 0 |
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For Galerkin formulation: |
U |
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S |
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~ |
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Remember: |
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U (x, y) Ui i (x, y) |
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i 1 |
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Ui (x, y) (x, y)dS 0 |
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So: |
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i 1 S |
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2-nd order derivative from the linear polynomial is = 0:
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Week formulation
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Ui 2 i (x, |
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i 1 |
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Let us transform the integral
For this purpose we shall use a relation from vector algebra
y) j (x, y)dS 0
2 i (x, y) j (x, y)dS
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F G F G F G
F i (x, y) |
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G j (x, y) |
i (x, y) j (x, y) i (x, y) j (x, y) i (x, y) j (x, y) |
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So we can |
2 i (x, y) j (x, y) i (x, y) j (x, y) i (x, y) j (x, y) |
express: |
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Week formulation
Integrating the last relation over the problem domain:
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2 i (x, y) j (x, y)dS i (x, y) j (x, y) dS |
i (x, y) j (x, y)dS |
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S |
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The Gauss theorem: |
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FdS F d F d |
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S |
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j (x, y) |
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2 i (x, y) j (x, y)dS |
i (x, y) |
d i (x, y) j (x, y)dS |
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S |
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S |
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is the border of the problem domain |
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For the elements inside the problem domain the function: |
i (x, y) 0 |
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So |
2 i (x, y) j (x, y)dS i (x, y) j (x, y)dS |
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S |
S |
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Week formulation
So the equation takes form of:
N |
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Ui i (x, y) j (x, |
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i 1 |
S |
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Finally we have got a system of equations:
N
aij Ui 0 i 1
With the coefficients
y)dS 0
aij i (x, y) j (x, y)dS
S
If the boundary potentials are known in advance, several equations in the system will have non-zero right hand sides
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1-st type boundary conditions
1-st type boundary conditions: |
U F1 |
Un U
If the potential at the boundary is equal to zero:
Un 0
The 1-st type boundary conditions keep a symmetry of the main problem matrix
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The potential and field intensity
After a system of equations is solved we can express a potential as:
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N |
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U |
(r) Ui i (x, y) |
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i 1 |
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Inside a triangle the same expression: |
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U |
(r) Ui i (x, y) |
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i 1 |
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N |
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The field intensity: |
E(r) U Ui |
i (x, y) |
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Inside a triangle: |
E(r) Ui i (x, y) |
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i 1 |
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2-nd type boundary conditions
2-nd type boundary conditions: |
U |
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For the elements near the boundary:
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2 i (x, y) j (x, y)dS |
i (x, y) j |
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S |
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Now this term can not be neglected
Equation for the potential:
N |
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(x, y)dS |
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(x, y) d i (x, y) j (x, y)dS
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(x, y)dS |
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ij |
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(x, y) |
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i n |
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j (x, y)d |
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