Circular domain with Dirchlet BC Compared with analytical solution

Solve the field solution of the origional problem by optimal points and optimal parameter

Case 1-1: Circular domain with Dirchlet BC Compared with analytical solution

Novel estimation technique(M=10) Novel estimation technique(M=20) Novel estimation technique(M=30)

Fig. 2-5. The optimal parameter versus the number of source points for case 1-1 in example1.

0.01 0.1 1 10 100

CASE 1-2: 80 points

Compared with analytical solution Novel estimation technique(M=12) Novel estimation technique(M=20) Novel estimation technique(M=28)

0.01 0.1 1 10 100

CASE 1-2: 40 points

Compared with analytical solution Novel estimation technique(M=12) Novel estimation technique(M=20) Novel estimation technique(M=28)

0.01 0.1 1 10 100

CASE 1-2: 120 points

Compared with analytical solution Novel estimation technique(M=12) Novel estimation technique(M=20) Novel estimation technique(M=28)

Fig. 2-6. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 1-2 of example1. (a) 40 points (b) 80 points (c) 120 points.

40 80 120

Number of points

1E-015 1E-014 1E-013 1E-012 1E-011 1E-010 1E-009

R.M.S Error

Case1-2 :

Compared with analytical solution Novel estimation technique(M=10) Novel estimation technique(M=20) Novel estimation technique(M=30)

Fig. 2-7. R.M.S error versus the number of source points for case 1-2 in example 1.

40 80 120

Number of points

0 10 20 30 40 50

dopt

Case 1-2 :

Compared with analytical solution Novel estimation technique(M=12) Novel estimation technique(M=20) Novel estimation technique(M=28)

Fig. 2-8. The optimal parameter versus the number of source points for case 1-2 in example1.

x y

= 0

∇ u

u

x y

= 0

∇ u

u

Fig.2-9. Problem sketch and source points distribution of the case 2-1 in example 1.

0.01 0.1 1

CASE 2-1: 30 points

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

0.01 0.1 1

CASE 2-1: 10 points

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

0.01 0.1 1

CASE 2-1: 80 points

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

Fig. 2-10. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 2-1 of example 1. (a) 10 points (b) 30 points (c) 80 points.

20 40 60 80 100

Number of points

1E-008 1E-007 1E-006 1E-005 0.0001 0.001 0.01 0.1

R.M.S Error

Case 2-1:

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

Fig. 2-11. R.M.S error versus the number of source points for case 2-1 in example 1.

20 40 60 80 100

Number of points

0 0.2 0.4 0.6 0.8 1

dopt

Case 2-1:

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

Fig. 2-12. The optimal parameter versus the number of source points for case 2-1 in example 1.

x y

= 0

∇ u

=1 u

−1

= u

x y

= 0

∇ u

=1 u

−1

= u

x y

= 0

∇ u

=1 u

−1

= u

Fig.2-13. Problem sketch and source points distribution of the case 2-2 in example 1.

0.01 0.1 1

CASE 2-2: 60 points

Compared with analytical solution Novel estimation technique(M=40) Novel estimation technique(M=60) Novel estimation technique(M=80)

0.01 0.1 1

CASE 2-2: 30 points

Compared with analytical solution Novel estimation technique(M=40) Novel estimation technique(M=60) Novel estimation technique(M=80)

0.01 0.1 1

CASE 2-2: 100 points

Compared with analytical solution Novel estimation technique(M=40) Novel estimation technique(M=60) Novel estimation technique(M=80)

Fig. 2-14. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 2-2 of example 1. (a) 30 points (b) 60 points (c) 100 points.

0 40 80 120 160 200

Number of points

0 0.05 0.1 0.15 0.2 0.25

R.M.S Error

Case 2-2

Compared with analytical solution Novel estimation technique(M=40) Novel estimation technique(M=60) Novel estimation technique(M=80)

Fig. 2-15. R.M.S error versus the number of source points for case 2-2 in example 1.

0 40 80 120 160 200

Number of points

0 0.4 0.8 1.2 1.6 2

d opt

Case2-2:

Compared with analytical solution Novel estimation technique(M=40) Novel estimation technique(M=60) Novel estimation technique(M=80)

Fig. 2-16. The optimal parameter versus the number of source points for case 2-2 in example 1.

= 0

∇ u

u

x

y

d

= 0

∇ u

u

x

y

d

Fig.2-17. Problem sketch and source points distribution of the case 1 in example 2.

0.1 1 10 100 1000

CASE 1: 160 points

Compared with analytical solution Novel estimation technique(M=5) Novel estimation technique(M=10) Novel estimation technique(M=15)

0.1 1 10 100 1000

CASE 1: 100 points

Compared with analytical solution Novel estimation technique(M=5) Novel estimation technique(M=10) Novel estimation technique(M=15)

0.1 1 10 100 1000

CASE 1: 200 points

Compared with analytical solution Novel estimation technique(M=5) Novel estimation technique(M=10) Novel estimation technique(M=15)

Fig. 2-18. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 1 of example 2. (a) 100 points (b) 160 points (c) 200 points.

40 80 120 160 200 240

Number of points

1E-015 1E-014 1E-013 1E-012 1E-011 1E-010 1E-009

R.M.S Error

Case 1:

Compared with analytical solution Novel estimation technique(M=5) Novel estimation technique(M=10) Novel estimation technique(M=15)

Fig. 2-19. R.M.S error versus the number of source points for case 1 in example 2.

80 120 160 200 240

Number of points

0 2 4 6 8 10

dopt

Case 1:

Compared with analytical solution Novel estimation technique(M=5) Novel estimation technique(M=10) Novel estimation technique(M=15)

Fig. 2-20. The optimal parameter versus the number of source points for case 1 in example 2.

x y

= 0

∇u

u

x y

= 0

∇u

u

x y

u t

= 0

∇u

x y

u t

= 0

∇u

(a) Dirchlet problem

(b) Mixed type problem

Fig.2-21. Problem sketch and source points distribution of the case 2-1 and case 2-2 in example 2 (a) Dirchlet problem (b) Mixed type problem.

0.1 1 10

CASE 2-1: 100 points

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

0.1 1 10

CASE 2-1: 40 points

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

0.1 1 10

CASE 2-1: 160 points

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

Fig. 2-22. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 2-1 of example 2. (a) 40 points (b) 100 points (c) 160 points.

40 80 120 160

Number of points

1E-013 1E-012 1E-011 1E-010 1E-009 1E-008 1E-007

R.M.S Error

Case 2-1:

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

Fig. 2-23. R.M.S error versus the number of source points for case 2-1 in example 2.

40 80 120 160

Number of points

4 8 12 16 20

dopt

Case 2-1:

Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=30) Novel estimation technique(M=40)

Fig. 2-24. The optimal parameter versus the number of source points for case 2-1 in example 2.

0.1 1 10

CASE 2-2: 100 points

Compared with analytical solution Novel estimation technique(M=30) Novel estimation technique(M=40) Novel estimation technique(M=50)

0.1 1 10

CASE 2-2: 60 points

Compared with analytical solution Novel estimation technique(M=30) Novel estimation technique(M=40) Novel estimation technique(M=50)

0.1 1 10

CASE 2-2: 260 points

Compared with analytical solution Novel estimation technique(M=30) Novel estimation technique(M=40) Novel estimation technique(M=50)

Fig. 2-25. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 2-2 of example 2. (a) 60 points (b) 100 points (c) 260 points.

100 200 300

Number of points

1E-015 1E-014 1E-013 1E-012 1E-011 1E-010 1E-009 1E-008 1E-007 1E-006 1E-005 0.0001 0.001 0.01

R.M.S Error

Case 2-2:

Compared with analytical solution Novel estimation technique(M=30) Novel estimation technique(M=40) Novel estimation technique(M=50)

Fig. 2-26. R.M.S error versus the number of source points for case 2-2 in example 2.

100 200 300

Number of points

0 4 8 12 16 20

dopt

Case 2-2:

Compared with analytical solution Novel estimation technique(M=30) Novel estimation technique(M=40) Novel estimation technique(M=50)

Fig. 2-27. The optimal parameter versus the number of source points for case 2-2 in example 2.

x

y

= 0

∇u

u

d

x

y

= 0

∇u

x

y

x

y

= 0

∇u

u

d

Fig.2-28. Problem sketch and source points distribution of the case 3 in example 3.

0.1 1 10

CASE 3: 120 points

Compared with analytical solution Novel estimation technique(M=10) Novel estimation technique(M=20) Novel estimation technique(M=30)

0.1 1 10

CASE 3: 40 points

Compared with analytical solution Novel estimation technique(M=10) Novel estimation technique(M=20) Novel estimation technique(M=30)

0.1 1 10

CASE 3: 200 points

Compared with analytical solution Novel estimation technique(M=10) Novel estimation technique(M=20) Novel estimation technique(M=30)

Fig. 2-29. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 3 of example 2. (a) 40 points (b) 120 points (c) 200 points.

0 100 200 300

Number of points

1E-015 1E-014 1E-013 1E-012 1E-011 1E-010 1E-009 1E-008 1E-007 1E-006 1E-005

R.M.S Error

Case 3:

Compared with analytical solution Novel estimation technique(M=10) Novel estimation technique(M=20) Novel estimation technique(M=30)

Fig. 30. R.M.S error versus the number of source points for case 3 in example 2.

0 100 200 300

Number of points

0 40 80 120

d opt

Case 3:

Compared with analytical solution Novel estimation technique(M=10) Novel estimation technique(M=20) Novel estimation technique(M=30)

Fig. 31. The optimal parameter versus the number of source points for case 3 in example 2.

2 = 0

∇ u

r

x y

2 = 0

∇ u

r

x y

(a) Problem sketch

d d’

max

1 /

' r d d

d = −

d d’

max

1 /

' r d d

d = −

Fig.2-32. Problem sketch and source points distribution of the case 2 in example 3 (a) Problem sketch (b) source point distribution.

(b) Source point distribution

1 10

CASE 1: 100 points

Compared with analyticalsolution Novel estimation technique(M=32)

CASE 1: 160 points

Compared with analytical solution

CASE 1: 220 points

Compared with analytical solution Novel estimation technique(M=32) Novel estimation technique(M=35) Novel estimation technique(M=40)

Fig. 2-33. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 1 of example 3. (a) 100 points (b) 160 points (c) 220 points.

50 100 150 200 250 300

Number of points

1E-014 1E-013 1E-012 1E-011 1E-010 1E-009 1E-008 1E-007 1E-006 1E-005 0.0001

R.M.S Error

Case 1:

Compared with analytical solution Novel estimation technique(M=100) Novel estimation technique(M=160) Novel estimation technique(M=220)

Fig. 2-34. R.M.S error versus the number of source points for case 1 in example 3.

40 80 120

Number of points

0 4 8 12 16 20

d opt

Case 1:

Compared with analytical solution Novel estimation technique(M=100) Novel estimation technique(M=160) Novel estimation technique(M=220)

Fig. 2-35. The optimal parameter versus the number of source points for case 1 in example 3.

(a) Problem sketch

Fig.2-36. Problem sketch and source points distribution of the case 3 in example 3 (a) Problem sketch (b) source point distribution.

(b) Source point distribution

0 50 100 150 2 00 C ompared wit h anal ytical solut ion Novel esti mation t echnique(M= 16) Novel esti mation t echnique(M= 24) Novel esti mation t echnique(M= 32)

0 50 1 00 15 0

Compared with analytical solution Novel estimation technique(M=16) Novel estimation technique(M=24) Novel estimation technique(M=32)

0 50 100 1 50

Compared with analytical solution Novel estimation technique(M=16) Novel estimation technique(M=24) Novel estimation technique(M=32)

Fig. 2-37. The error analysis versus the paramenters, d, with the different terms of Trefftz basis for case 2 of example 3. (a) 80 points (b) 160 points (c) 200 points.

50 100 150 200 250

Number of points

1E-010 1E-009 1E-008 1E-007 1E-006 1E-005 0.0001 0.001 0.01 0.1 1 10

R.M.S Error

Case 2:

Compared with analytical solution Novel estimation technique(m=16) Novel estimation technique(m=24) Novel estimation technique(m=32)

Fig. 2-38. R.M.S error versus the number of source points for case 2 in example 3.

50 100 150 200 250

Number of points

0 40 80 120 160 200

dopt

Case 3:

Compared with analytical solution Novel estimation technique(m=16) Novel estimation technique(m=24) Novel estimation technique(m=32)

Fig. 2-39. The optimal parameter versus the number of source points for case 2 in example 3.

Define contour boundary condition type of new problem

Create quasi-analytical solution of new problem by Trefftz set

Solve the new problem by BEM

Obtain B.C. of new problem by quasi-analytical solution

Estimate R.M.S error comparing with quasi-analytical solution

Obtain optimal number of elements Start

Solve the field solution the adopting the optimal Number of elements

End

Define contour boundary condition type of new problem

Create quasi-analytical solution of new problem by Trefftz set

Solve the new problem by BEM

Obtain B.C. of new problem by quasi-analytical solution

Estimate R.M.S error comparing with quasi-analytical solution

Obtain optimal number of elements Start

Solve the field solution the adopting the optimal Number of elements

End

Fig. 3-1. Flowchart of the systematic error estimation scheme.

= 0

∇ u

x u =

= 0 u

x

y

= 0

∇ u

x u =

= 0 u

x

y

Fig. 3-2. Problem sketch for the case 1.

0 400 800 1200 1600 Number of elements

0 0.004 0.008 0.012

R.M.S Error

Case 1: Square domain

Compared with analytical solution Novel estimation technique(M=120) Novel estimation technique(M=136) Novel estimation technique(M=160)

Optimal element (200)

Fig. 3-3. The error analysis for the field solution with the different terms of Trefftz basis for case 1.

10 100 1000 Number of elements

0 20 40

Time( second )

Case 1: Square domain

Fig. 3-4. Convergence rate of computational time with the different number of elements.

= 0 t y

= 0

∇ u

= 0 t

= 0 u

= 100 u

R

₁

=1 x

R

₂

=4

= 0 t y

= 0

∇ u

= 0 t

= 0 u

= 100 u

R

₁

=1 x

R

₂

=4

Fig. 3-5. Problem sketch for the case 2.

0 400 800 1200 1600

Number of elements

0 0.04 0.08 0.12 0.16

R.M.S Error

Case 2: Quarter tube cross-section domain Compared with analytical solution Novel estimation technique(M=30) Novel estimation technique(M=35) Novel estimation technique(M=39)

Optimal element (80)

Fig. 3-6. The error analysis for the field solution with the different terms of Trefftz basis for case 2.

0 0.4 0.8 1.2 θ

68.2 68.22 68.24 68.26

u(r=2.43,q)

Case 2: Quarter tube cross-section domain Analytical solution

80 elements 120 elements 400 elements 800 elements 1600 elements

Fig. 3-7. The error analysis for the field solution along the radius ,r=2.43, with the different number of elements.

0 200 400 600 800

Number of elements

68.16 68.2 68.24 68.28

u(x=0.248,y=0.391)

Case 2: Quarter tube cross-section domain Analytical solution

BEM's result(solving Original problem) BEM's result(solving new problem,M=30) BEM's result(solving new problem,M=35) BEM's result(solving new problem,M=39)

Fig. 3-8. u (0.248, 0.391) versus number of elements and with the different terms of Trefftz basis.

2 =0

∇ u

u

x y

2 =0

∇ u

u

x y

2 =0

∇ u

t u

x y

2 =0

∇ u

t u

x y

(a)

Fig. 3-9. Problem sketch for the case 3. (a) case 3-1 (b) case 3-2.

(b)

0 200 400 600 800 1000

Number of elements

0 0.4 0.8 1.2 1.6

R.M.S Error

Case 3-1: Arbitrary domain(Dirchlet B.C.) Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=25) Novel estimation technique(M=30)

Optimal element (100)

Fig. 3-10. The error analysis for the field solution with the different terms of Trefftz basis for case 3-1.

0 200 400 600 800 1000 Number of elements

0 0.4 0.8 1.2 1.6

R.M.S Error

Case 3-2: Arbitrary domain(Mixed type B.C.) Compared with analytical solution Novel estimation technique(M=20) Novel estimation technique(M=25) Novel estimation technique(M=30)

Optimal element (100)

Fig. 3-11. The error analysis for the field solution with the different terms of Trefftz basis for case 3-2.

在文檔中國立宜蘭大學土木工程學系（研究所）碩士論文 Department of Civil Engineering National Ilan University Master Thesis (頁 49-94)