Vector and Tensor Analysis - Structural Analysis Laboratorystrana.snu.ac.kr/lecture/fem_2008/Note/Vector-Tensor... · · 2007-09-04탄성체2,황찬규 Fall, 2007 Changyu Hwang

탄성체2,황찬규

Fall, 2007

Changyu Hwang

Vector and Tensor Analysis


Vector Calculus 연습



(from Maze’s book)









두개의 막대기를 직각으로 놓고는 반시계 방향으로돌리면, 평면을 뚫고 올라가는 힘.

임의의 평면의 Normal 방향 찾을 때 유용

Reminder !


SIMPLY


Permutation symbol을 이용한 cross product 표시


연습: 다음 식을 증명하시오.

Permutation symbol sign rule


SolutionSolution


?



m kjqi q

(first)(second)-(outer)(inner)(첫번째*두번째)- (밖에서*안으로)

증명은어떻게 하~나?


(from Maze’s book)증명


(from Maze’s book)연습문제




다음 식을 Indicial notation으로 풀어쓰시오.

Ak


Ak

SOLUTIONS











각자 연습


각자 연습


Contravariant vectors

HOW ?


Example:


Covariant vectors


Example:



The gradient g = ∇y is an example of a covariant tensor.The differential position d = dx is an example of a contravariant tensor.

Contravariant/covariant tensors

1 21 2

nn

x

y y ydy dx dx .... dxx x xˆ ˆˆy d g d

∂ ∂ ∂= + + +∂ ∂ ∂

=∇ =

1 2

1 2

x n

n

y y yg y , ,.....,x x x

d dx ,dx ,.....,dx

∂ ∂ ∂⎢ ⎥= ∇ = ⎢ ⎥∂ ∂ ∂⎣ ⎦⎢ ⎥= ⎣ ⎦

여기서

1 21 2

nn

X

y y ydy dX dX .... dXX X X

ˆˆ ˆy D G D

∂ ∂ ∂= + + +∂ ∂ ∂

=∇ =

.ˆˆ ˆg G, d D↔ ↔ 사이관계를구하는것이목표이다

Scalar

Scalar


1st rank Contravariant tensors (vector)

새 좌표계의 미소길이를 구 좌표계의 미소길이로 표시하면,

1 21 2

i i ii n

n

ij j

j

X X XdX dx dx .... dxx x x

X NEWdx dxOLDx

∂ ∂ ∂= + + +∂ ∂ ∂

∂= =∂

iNEW

jOLD

Xx

∂∂

1X

2X3X

NEW

1dX

2dX3dX

1x

2x

3x OLD

1dx

2dx3dx

변환대상이VECTOR


1 21 2

i i ii n

nX X XdX dx dx .... dxx x x

∂ ∂ ∂= + + +∂ ∂ ∂

1 1 11 1 2

1 2

2 2 22 1 2

1 2

1 21 2

nn

nn

n n nn n

n

X X XdX dx dx .... dxx x xX X XdX dx dx .... dxx x x

..........

X X XdX dx dx .... dxx x x

∂ ∂ ∂= + + +∂ ∂ ∂∂ ∂ ∂

= + + +∂ ∂ ∂

∂ ∂ ∂= + + +∂ ∂ ∂

In a matrix form1 1 1

1 21 12 2 2

2 21 2

1 2

n

n

n nn n n

n

X X X.....x x xdX dxX X XdX dx.....x x x

..... .......... ..... ..... .....dX dxX X X.....

x x xˆD J d

⎡ ⎤∂ ∂ ∂⎢ ⎥∂ ∂ ∂⎧ ⎫ ⎧ ⎫⎢ ⎥

⎪ ⎪ ⎪ ⎪⎢ ⎥∂ ∂ ∂⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬∂ ∂ ∂⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎩ ⎭∂ ∂ ∂⎢ ⎥

⎢ ⎥∂ ∂ ∂⎣ ⎦

=

Jacobian matrix

Contravariant Tensors (vector)

ii NEW j OLDNEW

jOLD

XdX dxx

− −∂=∂


구 좌표계의 길이를 새 좌표계의 길이로 표시하면,

1 21 2

i i ii n

nx x xdx dX dX .... dXX X X∂ ∂ ∂

= + + +∂ ∂ ∂

풀어서 써보면,1 1 1

1 1 21 2

2 2 22 1 2

1 2

1 21 2

nn

nn

n n nn n

n

x x xdx dX dX .... dXX X Xx x xdx dX dX .... dXX X X

.......

x x xdx dX dX .... dXX X X

∂ ∂ ∂= + + +∂ ∂ ∂∂ ∂ ∂

= + + +∂ ∂ ∂

∂ ∂ ∂= + + +∂ ∂ ∂

1st rank Covariant tensors (vector)


1 21 2

i i ii n

nx x xdx dX dX .... dXX X X∂ ∂ ∂

= + + +∂ ∂ ∂

를 원식에 대입하면,

1 21 2

nn

y y ydy dx dx .... dxx x x∂ ∂ ∂

= + + +∂ ∂ ∂

1 1 11 2

1 1 2

2 2 21 2

2 1 2

1 21 2

nn

nn

n n nn

n n

y x x xdy dX dX dXx X X X

y x x x dX dX dXx X X X

......

y x x x dX dX dXx X X X

⎛ ⎞∂ ∂ ∂ ∂= + + ⋅⋅+⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂+ + + ⋅⋅+⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂+ + + ⋅⋅+⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

1 1 11 1 2

1 2

2 2 22 1 2

1 2

1 21 2

nn

nn

n n nn n

n

x x xdx dX dX .... dXX X Xx x xdx dX dX .... dXX X X

.......

x x xdx dX dX .... dXX X X

∂ ∂ ∂= + + +∂ ∂ ∂∂ ∂ ∂

= + + +∂ ∂ ∂

∂ ∂ ∂= + + +∂ ∂ ∂

1 2 ndX ,dX ,...,dX 에대하여정리하면


1 21

1 1 1 2 1

1 22

2 1 2 2 2

1 2

1 2

1 2

1 21

n

n

n

n

nn

n n n n

n

k k k nk

x y x y x ydy dXX x X x X x

x y x y x y dXX x X x X x

......

x y x y x y dXX x X x X x

x y x y x yX x X x X x=

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂= + +⋅⋅+⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

+ + +⋅⋅+⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ + +⋅⋅+⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

= + +⋅⋅+⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

nk

X

dX

ˆˆ ˆy D G D=∇ ⋅ = ⋅

∑

1 2

1 2

1 2

1 2

n

i i i i n

n

ni i i

x y x y x yGX x X x X x

x x xˆ ˆ ˆg g gX X X

∂ ∂ ∂ ∂ ∂ ∂= + +⋅⋅+∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂= + +⋅⋅+∂ ∂ ∂

Cartesian Coordinate에서는 둘이 같다.

iOLD

i NEW j OLDjNEW

OLDX x

NEW

xG gX

xy yX

− −∂

=∂

∂∇ = ∇

∂



2nd rank Covariant tensors

예: 4차원=시간 + 3차원공간



Notice that each component of the new metric array is a linear combination of the old metric components, and the coefficients are the partials of the old coordinates. with respect to the new. Arrays that transform in this way are called covariant tensors

( ) ( )y x

x xg gy y

μ ν

αβ μνα β∂ ∂

=∂ ∂


if we define an array Aμν with the components (dxμ/ds)(dxν/ds) where s denotes a path parameter along some particular curve in space, then equation (2) tells us that this array transforms according to the rule

This is very similar to the previous formula, except that the partial derivatives are of the new coordinates with respect to the old. Arrays whose components transform according to this rule are called contra-variant tensors.

( ) ( )y x

y yA Ax x

α βαβ μν

μ ν∂ ∂

=∂ ∂

Documents

Vector and Tensor Analysis - Structural Analysis Laboratorystrana.snu.ac.kr/lecture/fem_2008/Note/Vector-Tensor... · · 2007-09-04탄성체2,황찬규 Fall, 2007 Changyu Hwang