Weighted Residual Method. By choosing different weight functions, we obtained different numerical approximation methods. Point-Collocation Method.
MCEN 5023/ASEN 5012 Chapter 8
Energy Theorem and Variational Method
Fall, 2006
1
Linear Elasticity Linear Elasticity
What is linear elasticity about?
P
P deformed
undeformed
X2
Question: If we apply a force on a material, what are the stresses, strains and displacements at EACH material point?
X2
X1
X1
Input Boundary conditions (Applied force; Applied displacement …)
Output Stresses, strains, displacements, at each material point (x1,x2,x3)
??? 2
Linear Elasticity Summary of Linear Elasticity Kinematics:
Constitutive:
eij = eij =
1 (ui, j + u j ,i ) 2
ν 1 +ν σ ij − σ kk δ ij E E
6 equations
σ ij = 2Geij + λekk δ ij 6 equations
σ ij , j + f i = 0
Equilibrium:
3 equations
Totally: 15 equations Unknowns: Stresses:
σ 11 σ 22 σ 33 σ 12 σ 21 σ 23
Strains:
ε 11 ε 22 ε 33 ε 12 ε 21 ε 23
Displacements:
u1 u 2 u3
Totally: 15 unknowns 3
Linear Elasticity Good news: We have 15 unknowns, but we have 15 equations. So with proper boundary conditions, we have a unique (one, and only one) solution to this problem.
Bad news: How to get the solution?
4
Linear Elasticity P deformed
eij =
1 (ui, j + u j ,i ) 2
σ ij = 2Geij + λekk δ ij σ ij , j + f i = 0 X2
Boundary conditions X1
What we mean by saying a solution to a linear elasticity problem? A solution that satisfies above equations and boundary conditions at EACH material point. This is a very strong requirement!! 5
Linear Elasticity Since finding a precise solution is too strong, we step back a little.
σ (u1 , u2 , u3 )ij , j + f i = 0
P deformed
In stead of asking above equations are satisfied at each material point, we ask the above equations to be satisfied in the sense of integral, i.e.
X2
Ω
[∫ σ (u~ , u~ , u~ ) 1
X1
2
3 ij , j
]
+ f i dV = 0
V
u~k
is an approximate solution
This is a big release, because
σ (u~1 , u~2 , u~3 )ij , j + f i = Ri ≠ 0 Ri is residual.
but
∫ R dV = 0 i
V
6
Linear Elasticity Weighted Residual Method
[∫ σ (u~ , u~ , u~ ) 1
2
3 ij , j
]
+ f i dV = 0
V
[∫ σ (u~ , u~ , u~ ) 1
2
3 ij , j
]
+ f i W dV = 0
V
Here, we punish the residuals. W is a weight function. 7
Linear Elasticity Weighted Residual Method By choosing different weight functions, we obtained different numerical approximation methods.
Point-Collocation Method Subdomain Collocation Method Method of Moment Least Squares Methods Method Galerkin Method
8
Linear Elasticity Weighted Residual Method Example:
d 2u +u + x = 0 2 dx
0 ≤ x ≤1
Boundary conditions
x=0
u=0
Accurate solution:
x=1
u=0
sin x u= −x sin 1
Assuming an approximate solution is:
u = x(1 − x)(a1 + a2 x + a3 x 2 + L) 1
Weighted Residual is:
∫ W Rdx = 0 0
i
9
Linear Elasticity
d 2u +u + x = 0 2 dx
Weighted Residual Method: example
0 ≤ x ≤1
First order approximation:
u = a1 x(1 − x)
n=1
(
R1 = x + a1 − 2 + x − x 2
)
Second order approximation
u = x(1 − x)(a1 + a2 x)
n=2
(
)
(
R2 = x + a1 − 2 + x − x 2 + a2 2 − 6 x + x 2 − x 3
)
10
Linear Elasticity Weighted Residual Method: example Point collocation method: Allocate several points within the domain
(0 ≤ x ≤ 1)
R( xi ) = 0 n=1, we choose the mid-point
x = 1/ 2
(
)
R1 ( x = 1 / 2 ) = x + a1 − 2 + x − x 2 = 0 2 a1 = 7
1 7 − a1 = 0 2 4 n=2, we choose
x = 1/ 3
2 u1 = x(1 − x) 7
x = 2/3
1 16 2 R2 ( x = 1 / 3) = − a1 + a2 = 0 3 9 27 2 16 50 R2 ( x = 2 / 3) = − a1 − a2 = 0 3 9 27
a1 = 0.1948
a2 = 0.1731
u 2 = x(1 − x)(0.1948 + 0.1731x ) 11
Linear Elasticity Weighted Residual Method: example n
Galerkin method
u = ∑ N i ai
Wi = N i
i =1
First order approximation:
W1 = N1 = x(1 − x )
u = a1 x(1 − x)
n=1
(
R1 = x + a1 − 2 + x − x 2
(
(
)
))
2 ( ) W Rdx = 0 = x 1 − x x + a − 2 + x − x dx 1 ∫ 1 ∫ 1
1
0
0
a1 =
5 = 0.2727 18
u = 0.2727 x(1 − x) 12
Linear Elasticity Weighted Residual Method: example n
Galerkin method
u = ∑ N i ai
Wi = N i
i =1
Second order approximation:
W1 = N1 = x(1 − x )
u = x(1 − x)(a1 + a2 x)
W2 = N 2 = x 2 (1 − x )
(
)
(
R = x + a1 − 2 + x − x 2 + a2 2 − 6 x + x 2 − x 3
)
[ ( ) ( ∫ W Rdx = ∫ x (1 − x )[x + a (− 2 + x − x ) + a (2 − 6 x + x
)] − x )]dx = 0
2 2 3 ( ) W Rdx = x 1 − x x + a − 2 + x − x + a 2 − 6 x + x − x dx = 0 1 2 ∫ 1 ∫ 1
1
0
0
1
0
1
2
0
2
2
1
2
2
3
u = x(1 − x)(0.1924 + 1707 x) 13
Linear Elasticity Weighted Residual Method: example Errors Solution
u=
x=0.25
sin x −x sin 1
Point coll. n=1 2 u1 = x(1 − x) 7 Galerkin, n=1
u1 = 0.2727 x(1 − x)
Error %
0.04401
x=0.5
Error %
0.06975
x=0.75
Error %
0.06006
0.05355
21.7
0.07143
2.4
0.05357
-10.8
0.05208
18.3
0.06944
-0.4
0.05208
-13.3
0.04464
1.4
0.07034
0.8
0.06087
1.3
0.04408
0.2
0.06944
-0.4
0.06008
0.03
Point coll. n=2 u 2 = x(1 − x)(0.1948 + 0.1731x )
Galerkin, n=1 u = x(1 − x)(0.1924 + 1707 x)
14
Linear Elasticity Principle of Virtual Displacements Concepts Admissible
Admissible displacements
P
A set of displacements that satisfy the geometric constraints.
S P1
Virtual displacements
S P2
A set of small displacement variations upon which the geometric constraints are satisfied.
δui
dui
t
u Su
X2
Β
X1
Not Admissible 15
Linear Elasticity Principle of Virtual Displacements
σ ij , j + f i = 0 B.C.:
σ ij n j − Ti = 0
P
in B
S P1
on ∂B
T Now, we want to use weighted residual method to obtain the integral form of equilibrium equations. We use the virtual displacements δ ui as weight function
∫ (σ B
ij , j
u Su
X2
Β
X1
+ f i )δui dV − ∫ (σ ij n j − Ti )δui dS = 0 ∂B
16
Linear Elasticity Principle of Virtual Displacements
∫ (σ B
ij , j
+ f i )δui dV − ∫ (σ ij n j − Ti )δui dS = 0 ∂B
(ab ), j = a, j b + ab, j
∫σ B
ij , j
[
a, j b = (ab ), j − ab, j
]
δui dV = ∫ (σ ijδui ), j − σ ijδui , j dV B
= ∫ (σ ijδui ), j dV − ∫ σ ijδui , j dV B
∫ a dB =∫ B
,i
∂B
a ni dS
∫ (σ B
B
ij
δui ), j dV = ∫ (σ ijδui )n j dS ∂B
17
Linear Elasticity Principle of Virtual Displacements
1 ∫B σ ijδui , j dV = ∫B 2 (σ ij + σ ij )δui, j dV
1 = ∫ (σ ij + σ ji )δui , j dV B 2
1 1 = ∫ (σ ijδui , j + σ jiδui , j )dV = ∫ (σ ijδui , j + σ ijδu j ,i )dV B 2 B 2 1 = ∫ σ ij (δui , j + δu j ,i )dV B 2
⎡1 ⎤ ( ) = ∫ σ ijδ ⎢ ui , j + u j ,i ⎥ dV B ⎣2 ⎦
= ∫ σ ijδε ij dV B
18
Linear Elasticity Principle of Virtual Displacements
∫σ B
∫ (σ B
ij , j
δui dV = ∫ (σ ijδui )n j dS − ∫ σ ijδε ij dV
ij , j
∂B
B
+ f i )δui dV − ∫ (σ ij n j − Ti )δui dS = 0 ∂B
∫ (σ B
ij
δε ij − f iδui )dV = ∫ Tiδui dS
Internal work on virtual displacements
∂B
External work on virtual displacements
Principle of virtual displacement: The work done by external force on virtual displacements is equal to the work done by the internal force on virtual displacements. 19
Energy Theorem Principle of Minimum Potential Energy General form of linear elastic constitutive equations
σ ij = 2Gε ij + λε kk δ ij
σ ij = Dijkl ε kl
Dijkl = Dklij
From principle of virtual displacements
∫ (σ B
ij
δε ij − f iδui )dV − ∫ Tiδui dS = 0 ∂B
∫ (D B
ε δε ij − f iδui )dV − ∫ Tiδui dS = 0
ijkl kl
∂B
Consider
1 ⎞ 1 ⎛1 δ ⎜ Dijkl ε kl ε ij ⎟ = Dijkl (δε kl )ε ij + Dijkl ε kl (δε ij ) = Dijkl ε kl (δε ij ) 2 ⎝2 ⎠ 2
20
Energy Theorem Principle of Minimum Potential Energy
⎡ ⎛1 ⎤ ⎞ ∫B ⎢⎣δ ⎜⎝ 2 Dijklε klε ij ⎟⎠ − f iδui ⎥⎦ dV − ∫∂B Tiδui dS = 0 1 Dijkl ε kl ε ij = U (ε ij ) = U (ui ) 2
Strain energy density Stored energy density
Stress
Strain
21
Energy Theorem Principle of Minimum Potential Energy
∫ [δU (u ) − f δu ]dV − ∫ i
B
We say both fi and
i
i
∂B
Tiδui dS = 0
Ti are conservative forces.
− Tiδui = δΨ (ui )
− f iδui = δΦ (ui )
∫ [δU (u ) + δ Φ(u )]dV + ∫ i
B
δ
i
∂B
δ Ψ (ui )dS = 0
{∫ [U (u ) + Φ(u )]dV + ∫ B
i
i
∂B
}
Ψ (ui )dS = 0 22
Energy Theorem Principle of Minimum Potential Energy
System potential energy
Π p = ∫ [U (ui ) + Φ(ui )]dV + ∫ Ψ (ui )dS ∂B
B
δΠ p = 0 Among all the possible displacements, the true solution (real displacements) results in the smallest system potential energy. In other words, any approximate solutions give larger system potential energy than the real solution does. 23
Energy Theorem Principle of Minimum Potential Energy
Πp
*
( )
Π p ui
*
( )
Π p ui > Π p (ui ) *
*
Π p (ui )
Approximate solution
Real solution
Possible displacements 24
Energy Theorem Principle of Minimum Potential Energy Application: 1. Formulating finite element 2. Judge the properties of approximate solutions Properties of approximate solutions
From energy balance, strain energy = work done by external force
1 1 ∫B U (ui )dV = ∫B 2 fiui dV + ∫∂B 2 Tiui dS work done by external force 25
Energy Theorem Properties of approximate solutions
Π p = ∫ [U (ui ) + Φ(ui )]dV + ∫ Ψ (ui )dS ∂B
B
Π p = ∫ [U (ui ) − f i ui ]dV − ∫ Ti ui dS ∂B
B
∫
B
− ∫ 2U (ui )dV
1 1 f i ui dV + ∫ Ti ui dS B 2 ∂B 2
U (ui )dV = ∫
B
Π p = − ∫ U (ui )dV B
26
Energy Theorem Properties of approximate solutions
Since,
( ) − ∫ U (u )dV > − ∫ U (u )dV ∫ U (u )dV < ∫ U (u )dV Π p ui > Π p (ui ) *
Π p = − ∫ U (ui )dV B
*
*
B
i
i
B
*
B
i
B
i
*
ui < ui The approximate solutions obtained from principle of minimum potential energy are smaller than the real solution. Displacement FEA makes the system stiffer!! 27
Variational Theorem How did we obtain the Principle of Minimum Potential Energy? Equilibrium equations and BC
σ ij , j + f i = 0
in B
σ ij n j − Ti = 0
on ∂B
ij , j
i
ij
∂B
i
j
i
i
Principle of V.D.
Π p = ∫ [U (ui ) + Φ (ui )]dV + ∫ Ψ (ui )dS ∂B
δΠ p = 0
∫ (σ + f )δu dV − ∫ (σ n − T )δu dS = 0 B
Principle of M.P.E. B
Weak form of EE and BC
∫ (σ δε − f δu )dV = ∫ T δu dS ij
B
∂B
ij
i
i
i
i
28
Variational Theorem If the functional Пp is given, then we can obtained the E.E. and B.C. by taking the variation of Пp. Principle of M.P.E.
Principle of V.D.
Π p = ∫ [U (ui ) + Φ (ui )]dV + ∫ Ψ (ui )dS ∂B
B
δΠ p = 0
ij
B
∂B
in B
i
i
i
i
∫ (σ + f )δu dV − ∫ (σ n − T )δu dS = 0 ij , j
B
σ ij n j − Ti = 0
ij
Weak form of E.E. and B.C.
E.E. and B.C.
σ ij , j + f i = 0
∫ (σ δε − f δu )dV = ∫ T δu dS
on ∂B
∂B
i
ij
j
i
i
i
29
Variational Theorem Functional is a function of functions. ( For example, Πp is a function of ui, which are functions of xi ) For PDEs, there might exist a functional, whose variation is equivalent to the PDEs and B.C.
Π
δΠ
A(u)=0, in B C(u)=0, on ∂ B
Some PDEs may not have a functional. The variational theorem is a powerful tool in developing numerical techniques, such as FEM. 30
Variational Theorem Variational theorem in linear elasticity ¾ Hu-Washizu Variational Theorem ¾ Hellinger-Reissner Variational Theorem ¾ Principle of Minimum Complementary Energy
31
Variational Theorem Hu-Washizu Variational Theorem In formulating the principle of minimum potential energy, we demand that
1 ε ij = (ui , j + u j ,i ) 2
and
ui = ui
on
∂B u
In Hu-Washizu variational theorem, we introduce these requirements into the functional
Π H −W
1 ⎡ ⎤ = Π p + ∫ α ij ⎢ε ij − (ui , j + u j ,i )⎥ dV + ∫ u bi (ui − ui )dS B ∂B 2 ⎣ ⎦
αij and bi are weight functions. Note, by introducing this new functional, εij becomes independent of ui.
32
Variational Theorem Hu-Washizu Variational Theorem
δ Π H −W = 0
33
Variational Theorem Hu-Washizu Variational Theorem
34
Variational Theorem Hu-Washizu Variational Theorem
35
Variational Theorem Hu-Washizu Variational Theorem
⎧1 1 ⎡ ⎤⎫ Π H −W (ui , ε ij , σ ij ) = ∫ ⎨ Dijkl ε ij ε kl − f i ui − σ ij ⎢ε ij − (ui , j + u j ,i )⎥ ⎬dV B 2 2 ⎣ ⎦⎭ ⎩ − ∫ p Ti ui dS − ∫ u σ ij n j (ui − ui )dS ∂B
∂B
36
Variational Theorem Hellinger-Reissner Variational Theorem In H-W variational theorem, εij and σij are independent functions. In H-R variational theorem, εij is related to σij through constitutive equations.
ε ij = Cijklσ kl In linear elasticity,
1 1 Dijkl ε ij ε kl = Cijklσ ijσ kl 2 2
37
Variational Theorem Hellinger-Reissner Variational Theorem
⎧1 1 ⎡ ⎤⎫ Π H −W (ui , ε ij , σ ij ) = ∫ ⎨ Dijkl ε ij ε kl − f i ui − σ ij ⎢ε ij − (ui , j + u j ,i )⎥ ⎬dV B 2 2 ⎣ ⎦⎭ ⎩ − ∫ p Ti ui dS − ∫ u σ ij n j (ui − ui )dS ∂B
∂B
38
Variational Theorem Hellinger-Reissner Variational Theorem
1 ⎧1 ⎫ Π H − R (ui , σ ij ) = ∫ ⎨ σ ij (ui , j + u j ,i ) − Cijklσ ijσ kl − f i ui ⎬dV B 2 2 ⎩ ⎭ − ∫ p Ti ui dS − ∫ u σ ij n j (ui − ui )dS ∂B
∂B
39
Variational Theorem Principle of Minimum Complementary Energy In H-R variational theorem, σij does not have to satisfy the equilibrium equations and BC. In principle of minimum complementary energy, we require that σij satisfies the equilibrium equations and BC.
40
Variational Theorem Principle of Minimum Complementary Energy
1 ⎧1 ⎫ Π H − R (ui , σ ij ) = ∫ ⎨ σ ij (ui , j + u j ,i ) − Cijklσ ijσ kl − f i ui ⎬dV B 2 2 ⎩ ⎭ − ∫ p Ti ui dS − ∫ u σ ij n j (ui − ui )dS ∂B
∂B
41
Variational Theorem Principle of Minimum Complementary Energy
1 Π C (σ ij ) = ∫ Cijklσ ijσ kl dV − ∫ p Ti ui dS B 2 ∂B
42
Body forces and external forces are conservative
Variational Theorem Weak form
Summary Equilibrium equations and BC
Principle of M.P.E.
Principle of V.D.
Π p (ui )
Release straindisplacement relationship and displacement BC, constitutive eq. Principle of M.C.E.
Π c (σ ij )
H-R theorem
Π H − R (ui , σ ij ) Enforce equilibrium equations
H-W theorem
Π H −W (ui , ε ij , σ ij )
Enforce constitutive equations
43
