1. Error Minimization of Diffusion Operator
黃耀新 (Yao-Hsin Hwang)
國立高雄海洋科技大學 輪機工程系
National Kaohsiung Marine University
Department of Marine Engineering

2. Outlines Derivation of isotropic distribution
One-dimensional formulation
Two-dimensional formulation
Validations

3. Particle smoothing procedure
Original formulation (PS) or
with

4. Accuracy analysis Taylor-series expansion Original formulation
therefore

5. Isotropic distribution(1)
Invariant after rotation
Geometrical relation
n=1
n=2

6. Isotropic distribution(2)
others

7. Isotropic distribution(3)
Consider
We have
and

8. One-dimensional formulation
First derivative
Second derivative
Elimination of artificial velocity

9. Two-dimensional formulation(1)
First derivative
Second derivative

10. Two-dimensional formulation(2)
Elimination of artificial velocity
Effective diffusivity

11. Error measure
Difference between numerical and physical diffusivity tensor

12. One-parameter Based on isotropic formulation Minimization of error
which leads to
and

13. Two-parameter Add cross derivative term Minimization of error
which leads to

14. Three-parameter Complete second derivative terms Consistent expression
leads to

15. 1a-smooth field Solution field Resulting terms
leads to exact solution:

16. 1b-operator test
Solution field and particle distribution
Results

17. 1c-conduction
Governing equation and exact solution
Results

19. 1d-flow
Governing equation and exact solution

21. 2a-operator test
Solution field and particle distribution

24. 2b-conduction
Governing equation and exact solution

27. 2c-Elliptic duct flow Governing equation and elliptical duct
Steady-state solution
with

32. 2-d Fully developed temperature
Governing equation and elliptical duct
Fully developed solution
with

35. 2-e Flow over cylinder
Governing equation
Steady-state solution
with

40. 2-e Flow across corner
Governing equation
Steady-state solution