June 5, 2019 1 / 26
10.1 ( ) 2 / 26
10.2 8 2 3 4 3 4 6 10.1 4 2 3 4 3 (a) 4 (b) 2 3 (c) 2 4 10.1: 3 / 26
8.3 3 5.1 4 10.4 Gauss 10.1 Ω i 2 3 4 Ξ 3 4 6 Ξ ( ) Ξ 5.1 Gauss ˆx : Ξ Ω i ˆx h u 4 / 26
10.2.1 ( ) i E Ω i R d Ξ R d i N i = {1,..., N i } ˆφ = ( ˆφ (1),..., ˆφ ( Ni )) ξ Ξ u Ωi û h (ξ) = ˆφ (ξ) ū i, ˆv h (ξ) = ˆφ (ξ) v i, ˆx h1 (ξ) = ˆφ (ξ) x i1,. ˆx hd (ξ) = ˆφ (ξ) x id ū i, v i R Ni u v x i1,..., x id R Ni Ω i 5 / 26
ξ Ξ u x Ω i Jacobi 6 / 26
10.3 2 4 ξ 2 x i(4) x i(3) ξ (4) ξ (3) x Ω i x 2 x Ξ=(0,1) 2 i(2) ξ 1 ξ x i(1) ξ (1) ξ (2) x 1 10.1: 4 7 / 26
10.1 4 i E Ω i 4 Ξ = (0, 1) 2 ξ Ξ u i(1) û h (ξ) = ( ˆφ (1) (ξ) ˆφ (2) (ξ) ˆφ (3) (ξ) ˆφ (4) (ξ) ) u i(2) u i(3) = ˆφ (ξ) ū i, u i(4) ˆv h (ξ) = ˆφ (ξ) v i, ˆx h1 (ξ) = ˆφ (ξ) x i1, ˆx h2 (ξ) = ˆφ (ξ) x i2 ˆφ (1) (1 ξ 1 )(1 ξ 2 ) ˆφ = ˆφ (2) ˆφ (3) = ξ 1 (1 ξ 2 ) ξ 1 ξ 2 ˆφ (4) (1 ξ 1 )ξ 2 8 / 26
ˆφ x 1 x 2 ˆφ ξ α {1, 2, 3, 4} ( ) ( ) ( ) ˆφ(α) / ξ ξ ˆφ (α) (ξ) = 1 ˆx1 / ξ = 1 ˆx 2 / ξ 1 ˆφ(α) / x 1 ˆφ (α) / ξ 2 ˆx 1 / ξ 2 ˆx 2 / ξ 2 ˆφ (α) / x 2 = ( ξ ˆx T) x ˆφ (α) (ξ) ( ) ˆφ(α) / x x ˆφ (α) (ξ) = 1 ˆφ (α) / x 2 = 1 ( ) ( ) ˆx2 / ξ 2 ˆx 2 / ξ 1 ˆφ(α) / ξ 1 ω i (ξ) ˆx 1 / ξ 2 ˆx 1 / ξ 1 ˆφ (α) / ξ 2 = ( ξ ˆx T) ξ ˆφ (α) (ξ) (10.3.1) 9 / 26
( ξ ˆx T) T ω i (ξ) = det ( ξ ˆx T) (10.3.2) ˆx : Ξ Ω i Jacobi Jacobi 10 / 26
( a i ( φi(α), φ i(β) ))α,β R4 4 ( ( ) φi(α) φ i(β) a i φi(α), φ i(β) = + φ ) i(α) φ i(β) dx Ω i x 1 x 1 x 2 x 2 = x φ i(α) (x) x φ i(β) (x) dx Ω i = x ˆφ (α) (ξ) x ˆφ (β) (ξ) ω i (ξ) dξ (10.3.3) (0,1) 2 (10.3.3) x ˆφ (α) (ξ) ω i (ξ) (10.3.1) (10.3.2) Gauss 11 / 26
10.4 Gauss Gauss n N f n : (, 1) R n n {1, 3, 5,...} f 1 (y) dy = 2f 1 (0), ( f 3 (y) dy = f 3 1 ) ( ) 1 + f 3 3, 3 ( f 5 (y) dy = 5 ) 3 9 f 5 + 8 5 9 f 5 (0) + 5 9 f 5 ( ) 3, 5 Gauss i {1, 2,..., (n + 1) /2} w i f n (η i ) η i Gauss 10.1 f n Gauss 12 / 26
f 1 f 3 f 5 1 η 1 0 y η 1 η 2 y η 1 η 2 η 3 y 1 1 0 1 1 0 1 n = 1 n = 3 n = 5 10.1: 1 Gauss 13 / 26
Gauss Legendre n m 10.4.1 (Legendre ) l n : (, 1) R Legendre { d (1 x 2 ) } d dx dx l n + n (n + 1) l n = 0 (10.4.1) l n Legendre 14 / 26
l 0 1 l 2 l 4 l 5 1 x l 3 l 1 10.2: Legendre l n 15 / 26
Legendre Rodrigues d n l n (x) = 1 { (x 2 2 n n! dx n 1 ) } n (10.4.2) l 0 (x) = 1, l 1 (x) = x, l 2 (x) = x 2 1 3, l 3 (x) = x 3 3 5 x, 10.2 l 0 l 5 l n n f : (, 1) R n f (x) l n (x) dx = 0 (10.4.1) (10.4.2) [ { 1 f (x) l n (x) dx = f n (n + 1) ( 1 x 2 ) }] 1 dl n dx 16 / 26
1 2 n n! = ()n 2 n n! df dx d n dx n { (x 2 1 ) n } dx d n f dx n ( x 2 1 ) n dx = 0 {l n } n l n (x) l m (x) dx = 2 2n + 1 δ nm x 0 < x 1 < < x n ϕ i (x) = j {1,...,n}, j i x x j x i x j = (x x 0) (x x 1 ) (x x i ) (x x i+1 ) (x x n ) (x i x 0 ) (x i x 1 ) (x i x i ) (x i x i+1 ) (x i x n ) 17 / 26
ϕ i (x j ) = δ ij f : R R ˆf (x) = i {1,...,n} ϕ i (x) f (x i ) ˆf (x i ) = f (x i ) ϕ i (x) Lagrange ˆf (x) Lagrange Legendre l n η 1,..., η n Lagrange φ i (x) = j {1,...,n}, j i x η j η i η j (10.4.3) 10.3 φ 1 φ 5 φ i (x) Lagrange Gauss 18 / 26
ϕ 1 ϕ 2 1 ϕ 3 ϕ 4 ϕ 5 1 x 10.3: 5 Legendre φ i (x) 19 / 26
10.4.2 (Gauss ) η 1,..., η n n Legendre l n f : (, 1) R 2n f (x) dx = i {1,...,n} w i f (η i ) (10.4.3) φ i (x) w i = φ i (x) dx 20 / 26
f (x) n 1 f (x) = φ i (x) f (η i ) i {1,...,n} n f n f (x) dx = φ i (x) f (η i ) dx = w i f (η i ) i {1,...,n} i {1,...,n} f n 2n f (x) = l n (x) g (x) + r (x) 21 / 26
g (x) r (x) n Legendre l n (x) g (x) dx = 0 l n (η i ) = 0 f (η i ) = r (η i ) r (x) n r (x) dx = f (x) dx = i {1,...,n} w i r (η i ) r (x) dx = i {1,...,n} w i r (η i ) = i {1,...,n} w i f (η i ) 22 / 26
23 / 26
10.4.2 (0, 1) f 2n (y) dy = 1 ( ) ηi 1 w i f 2 2 0 i {1,...,n} 2 (, 1) 2 n 10.4 Gauss f 2n (ξ) dξ = w i w j f (η ij ), (,1) 2 (i,j) {1,...,n} 2 f 2n (ξ) dξ = 1 w i w j f (0,1) 4 2 (i,j) {1,...,n} 2 (( ( ) 1 η ij 2 1))/ (10.4.4) (10.3.3) (10.4.4) 24 / 26
n ( 0 ) y 2 y 2 y 2 η11 y 1 η12 η22 y 1 y 1 (,1) 2 η11 η21 n = 1 n = 3 n = 5 10.4: 2 Gauss 25 / 26
10.5 1 2 Gauss 26 / 26