CMn.1 Computational Continuum Mechanics

GWSh /TWSh FE implementation widely applicable n-D computational mechanics forms illustrate

structural mechanics heat/mass transfer electromagnetics fluid mechanics mechanical vibrations

coupled PDE systems matrix PDEs constraints non-linearity

CMn.2 Structural Mechanics, Conservation Forms

Newton’s law, statics

σ ij ≡ stress tensor = σ x τ xy τ xz σ y τ yz(sym) σ z

Principal of virtual work l inear Hooke’s law, matr ix vector form

DΕ: Π = 1 2 ε T E ε - u T B d vol - u

T T d surf ∂Ω

Ω

T = surface traction, ε = strain matrix, u = displacement vector

surfiTiuvoljBjuijjivoleE ddd0 ,d2/1d:D ∫Ω∂

−∫Ω

−−∫∫Ω

=∫Ω

≡∏ φεε σ

, Bi = body force

DP:

0ij ii

Bxσ∂

+ =∂

CMn.3 Structural Mechanics, Formulations

F i r s t F E a p p l i c a t i o n i n s t r u c t u r e s , v i a Ω ⇒ Ω h = ∑ e Ω e

w h e n e x t r e m i z e d , p r o d u c e s a l g e b r a i c s y s t e m G W S N d e v e l o p m e n t , p l a n e s t r e s s / s t r a i n , n = 2

D P :

∂σ x

∂ x + ∂τ xy

∂ y + B x = 0

∂σ y

∂ y + ∂τ xy

∂ x + B y = 0

form ulation ingredients:

∑ ∫ ∫Ω Ω∂⎟⎟⎠⎞

⎜⎜

⎝

⎛−−≡ΠΠ=Π≈Π

e e e

TTTee

h dσ}{}{d}{}{}]{[}{21, TuBuεEε τ

y

x

T T

y

x

undeflected geom etry exaggerated deflected state

{ } { }xyyxTi

j

j

iij x

uuu

yxvyxuyx

γ,ε,εε:formmatrix

2/1ε:kinematics

ˆ),(ˆ),(),(:ntdisplaceme

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=

+= jiu

CMn.4 Linear Elasticity, Constitutive Model

Constitutive model relates stress and strain tensor form: σij = λεkk +2Gεij matrix form: {σ} = [E] {ε} Hooke’s law, l inear, homogeneous media

{ } { }

{ } { } [ ]

)σσ(νσ2/)1(00

0101

)21()1(E,,,

,,:strainplane

yxz

Txyyx

Txyyx

+=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−

−−≡≡

≡

ννν

νν

ννγεεε

τσσσ

E

{ } { }

{ } { } [ ]

)ν1/()εε(νε2/)ν1(00

01ν0ν1

ν1E,γ,ε,εε

τ,σ,σσ:

2

stressplane

−+−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

≡≡

≡

yxz

Txyyx

Txyyx

E

CMn.5 DP for Plane Elasticity

Substitute definitions into DP

plane stress: L u = ∇

2u + 1 + v 1 - v

∂

∂ x ∂ u

∂ x + ∂

v ∂ y

+ B xG

= 0

L v = ∇ 2v + 1 + v

1 - v ∂

∂ y ∂ u

∂ x + ∂

v ∂ y

+ B yG

= 0

plane strain: L u = ∇ 2u + 1

1 - 2 v ∂

∂ x ∂ u

∂ x + ∂

v ∂ y

+ B xG

= 0

L v = ∇ 2v + 1

1 - 2 v ∂

∂ y ∂ u

∂ x + ∂

v ∂ y

+ B yG

= 0

material propert ies : DP + Hooke produces coupled laplacian PDE system

∴ BCs required on total i ty of ∂Ω

for displacement field: u(x, y) ≡ u(x, y)i + v(x, y)j

ν ≡ Poisson rat io E = Youngs modulus G ≡ shear modulus = E / (2(1 + ν )

CMn.6 Vector DP for Plane Elasticity

Contributions to laplacian embedded in second terms

directional diffusion: k v = 2/ 1 - ν , plane stress2 1 - ν / 1 - 2 ν , plane strain

plane stress/strain directional diffusion form: b ≡ B/G

L u = k v ∂

2u ∂ x 2

+ ∂ 2u

∂ y 2 + k v - 1

∂ 2v

∂ x ∂ y + b x = 0

L v = k v ∂

2v ∂ y

2 + ∂

2v ∂ x

2 + k v - 1 ∂

2u ∂ x ∂ y

+ b y = 0

Vector form identifies divergence constraint for g(ν) = ( 1 + ν)/(1 - ν) or 1/(1 - 2ν) DP+ Hooke : L (u) = - ∇2u - g(ν) ∇ (∇•u) - b = 0

CMn.7 Matrix DP For Plane Elasticity

How to handle ∇(∇⋅u) ⇒ matrix DP

0σ

:D =+∂

∂i

j

ij Bx

P

2

0

0

][

=⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

=

nxy

y

xD

DP+Hooke matrix PDE

DP+Hooke : [D]T [E][D]{u} + {B} = {0}

{ } [ ]{ } { } [ ]{ }uDE == εandεσfor

{ }0BDP =+ }{}σ{][:D T ⇒

CMn.8 GWSN, Plane Elasticity Matrix DP

For any approximation u N ≡ ∑ Ψ α (x)U α

)(,0d}{}]{][[][)(GWSββ

xBuDEDx ΨΩ

≡⎟⎠⎞⎜

⎝⎛ −−Ψ≡ ∫ allforNTN τ

Green-Gauss theorem symmetr izes

∫∫ ∫

Ω∂≡Ψ−

Ω ΩΨ−Ψ=

0d σ}]{][[][β

d}{βd}]{][[]β[GWS

NT

NTN

uDEN

BuDED ττ

Neumann BC surface integra l TT ][ˆ][ DnN •≡

t e r m in t r o d u c e s s u r f a c e t r a c t io n s T D E e x t r e m u m wi l l p r o vid e t h e m a t r ix f o r m

CMn.9 DE Virtual Work Extremum, Plane Elasticity

A u g m e n t i n g Π f o r i n i t i a l s t r e s s / s t r a i n , p o i n t l o a d s

Π = 1 2 ε T E ε - ε T E ε 0 + ε T σ 0 d v o l

Ω

- u T B dvo l -Ω

u T T s dsurf - U T P∂Ω

E v a l u a t e i n t e g r a l a s s u m o v e r F E d o m a i n s Ω e E x t r e m i z e w . r . t n o n - c o n s t r a i n e d F E n o d a l D O F

}{}]{[}0{DOF

RUK =⇒≡⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂Π∂ h

( ) ( ) ( )∑ ∫∫∑ ⎥⎦⎤

⎢⎣⎡ ⋅−⋅−⋅=Π=Π≈Π

Ω∂∩Ω∂ΩΩ e

eh surfvol

eee

dd

w h ere : [K ] = g lo b a l “s tiffn ess” m atrix {R } = g lo b a l “ lo ad v ec to r”

D E :

CMn.10 DE Virtual Work, FE Stiffness Matrix

I m p l e m e n t i n g D E e x t r e m u m v i a F E b a s i s o n Ω e

approximation: ue(x,y) ≡ {Nk}T{U}e

matrix form: u e ≡ u v e = U V e

{ } { }{ } { } ⎥⎦

⎤⎢⎣

⎡T

k

Tk

NN

00

kinematics: {ε} ≡ [D] {u} S t i f f n e s s m a t r i x c o n t r i b u t i o n t o Π h

eeV

UT

kN

TkN

xy

yx

exy

yx

}]{[}{}0{

}0{}{0

0}]{[

γ

εε

UBUD ≡⎭⎬⎫

⎩⎨⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

≡≡

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

{ } [ ]{ }

{ } [ ] [ ][ ] { }

{ } [ ] { } [ ] matrixstiffnesselement,21

dd21

ddεε21

≡⋅⋅⋅+=

⋅⋅⋅+=

⋅⋅⋅+≡Π

∫

∫

Ω

Ω

eeeTe

e

TTe

Te

yx

yx

e

e

kUkU

UBEBU

E

CMn.11 DE Virtual Work, Plane Elasticity FE Extremum, BCs

E x t r e m iz a t io n o f Π e f o r D O F n o t B C - c o n s t r a in e d L o a d m a t r ix e x t r e m u m

C o m p a r in g D P a n d D E e x p o s e s N e u m a n n B C fo r G W S h a s

∫ Ω∂ ⎭⎬⎫

⎩⎨⎧

⇒⎭⎬⎫

⎩⎨⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

e eeT

kN

TkN

kNkN

}TY]{200A[}TX]{200A[

dσTYTX

}{0

0}{}{0

0}{

k e U e{ } =∂

Π∂

e

e

U

( ) eSβ =Ψ∫ ⋅

{ } { } { } [ ] [ ]{ } [ ] { }( ){ }[ ] { }[ ]{ }

{ }[ ] { }[ ]{ } { }PTg

BEBrrR

++

+

−==

∫∫

∫

Ω∂∩Ω∂

Ω

Ω

surfNN

volNN

vol

Tkk

Tkk

Te

Teee

e

e

e

d

dρ

dσε,S 00

w h e r e : { P } = p o i n t l o a d s a t D O F { T } = s u r f a c e t r a c t i o n n o d a l f o r c e

CMn.12 Structures GWSh FE Template Essence

With traction BC, FE implementation GWSh yields

GWS h = S e {WS} e ≡ {0} {WS} e = [STIFF] e {U} e – {b} e

[STIFF] e template essence

∫Ω≡ eyxe

Tee dd]][[][]STIFF[ BEB

{WS} e = CONST e DET e - 1 [ETLI ] e T [B2LKE ] [ETKI ] e

Load vector template essence

∫Ω ⎟⎠⎞⎜⎝⎛ +−≡ eyxe

TNNeTee

Tee dd}ρ]{}}][{[{}0ε{][}0ε]{[][}b{ gBEB

∫ Ω∂∩Ω∂ ++ e eeTNN }δ{dσ}]{}}][{[{ PT

{WS}e = CONSTe [ETLI]eT [B2L0E] {EPS0}e + CONSTe [ETLI]eT {SIGL0}e + P{δ}e + DETe [B200]{RHOG}e + DETe [A200]{T}e

1 2

3

Ωe

(X1, Y1)e (X2, Y2)e

ζ3

ζ1

ζ2

x

y (X3, Y3)e

For {N1(ζ)} spanning triangle Ωe , index notation

e

i

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−

=

1221

3113

2332

α

XXYYXXYYXXYY

ζ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∂∂∂∂∂∂

∂∂≡ TNT

TTN

xxx

x

e }1{}0{

}0{}1{

1/2/2/0

01/][B

via chain rule, ∂/∂xi =∂/∂ζα(∂ζα/∂xi)e,

[ ]

0α1 {δ } {0}1 α[ ] 0 α2det {0} {δ }αα2 α1

0 0 011 21 311 10 0 0 det ET12 22 32det

12 22 32 11 21 31 3 6

T T

e T Tee

KI eee

e

ζ

ζ

ζ ζ

ζ ζ ζ

ζ ζ ζ

ζ ζ ζ ζ ζ ζ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

⎡ ⎤⎢ ⎥

−⎢ ⎥≡ ≡⎢ ⎥⎢ ⎥⎣ ⎦ ×

B

and recalling ∂{N1}/∂ζα = {δα}

CMn.13 GWSh k = 1 FE Basis Implementation

CMn.14 GWSh k = 1 Implementation, Stiffness Matrix

For [B]e ⇒ [ETKI]e , k = 1 element “stiffness matrix”, plane stress

e

v

e

e

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−=

21

31ζ21ζ11ζ32ζ22ζ12ζ

32ζ22ζ12ζ31νζ21νζ11νζ32νζ22νζ12νζ31ζ21ζ11ζ

31ζ32ζ021ζ22ζ011ζ12ζ032ζ031ζ22ζ021ζ12ζ011ζ

)2ν1(2

E1det

)(

[ ] [ ] [ ][ ] [ ] [ ][ ]

[ ]e

e

eTeeee

Te

KIv

vv

v

yx

KILIyx

e

e

ET2/)1(00

0101

ζζ0ζζ0ζζ0ζ0ζζ0ζζ0ζ

)1(det

ddE

ETB2LKETDET CONSTddSTIFF

3132

2122

1112

3231

2221

1211

22

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−=

=≡

∫

∫

Ω

−

ΩEBEB

CMn.15 GWSh k = 1, Stiffness Matrix Resolution

R e s o l u t i o n e

ee

ee ]SHEAR[)1(4

E1det]NORML[

)21(2

E1det]STIFF[

νν +

−+

−

−=

e

sym

e

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

232ζ32ζ22ζ32ζ12ζ31ζ32νζ21ζ32νζ32ζ11νζ

222ζ22ζ12ζ31ζ22νζ21ζ22νζ22ζ11νζ

212ζ31ζ12νζ21ζ12νζ12ζ11νζ

231ζ31ζ21ζ31ζ11ζ

)(221ζ21ζ11ζ

211ζ

]NORML[

e

sym

e

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

231ζ31ζ21ζ31ζ11ζ32ζ31νζ22ζ31νζ12ζ31νζ

221ζ21ζ11ζ32ζ21νζ22ζ21νζ12ζ21νζ

211ζ32ζ11νζ22ζ11νζ12ζ11νζ

232ζ22ζ32ζ32ζ12ζ

)(222ζ22ζ12ζ

212ζ

]SHEAR[

CMn.16 GWSh, [DIFF]e ⇔ [STIFF]e Comparison

The {N1(ζ )} [STIFF] e is ordered for {Q}e ≡ {U1, U2, U3, V1, V2, V3}e T

extract ing common mult ipl ier

[ ] [ ][ ] [ ] ⎥⎦

⎤⎢⎣

⎡+

≡

⎟⎠⎞

⎜⎝⎛ +

−+=

eeee

e

eee

e

STIFFVVSTIFFVUSTIFFUVSTIFFUU

det)ν1(2E

]SHEAR[21]NORML[

ν11

det)ν1(2E]STIFF[

Comparing to scalar heat transfer construction

ee

ee QQ

k]DIFF[

det2]DIFF[ =

common divisor : 2det e

diffusion coeff ic ient :⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+⇔

21,

11

ν1E

vke

ν < 1 , E ≈ O (10 6) ⇒ e las t ic i ty is very dif fus ive!

CMn.17 GWSh, [DIFF]e ⇔ [STIFF]e Comparison

The metric data [ETLI] e play the fundamental role

for heat t ransfer

e = k e

2 det e

ζ 112 + ζ 12

2 , ζ 11 ζ 21

+ ζ 12ζ 22 , ζ 11 ζ 31 + ζ 12

ζ 32

ζ 21 ζ 11

+ ζ 22 ζ 12

, ζ 21 2 + ζ 22

2 , ζ 21 ζ 31 + ζ 22

ζ 32

ζ 31 ζ 11

+ ζ 32 ζ 12

, ζ 31ζ 21 + ζ 32ζ 22 , ζ 312 + ζ 32

2 e

[DIFFQQ]e

for plane s t ress , representat ive se l f coupl ing term

ee

e

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−+−

+−

+=

........................................,2ζζ)ν1(ζζ

........................................,2ζζ)ν1(ζζ

........................................,2ζ)1(ζ

det)ν1(2E]STIFFUU[

3212

22122111

2

12

2

11

3111

ν

c ross-coupling terms possess s imilar regular i ty

CMn.18 GWSh k = 1 Implementation, Load Vector

Body force, surface traction and point load contributions

[ ] yxee

TNNe dd}ρ{}1{}1{}b{ g∫Ω ⎥⎦⎤⎢⎣⎡=

ee

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

RHOGYRHOGX

]200B[]0[]0[]200B[

6

det

{ }δPYPX

TYTX

]200A[]0[]0[]200A[

3

det

eee

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

element-independent matrices [B200], [A200] same as defined in HTn

∫Ω ++ e eeeT

kNkN∂}δ{dσ}]{}}][{[{ PT

CMn.19 Plane Stress: Plate with a Hole

GWSh {N1} basis implementation

meshing, Ωh von Mises stress

x displacement, uh y displacement, vh

F lu id -th erm a l-stru c tu ra l sy stem d es ig n u ses “ C F D ”

fo r R ey n o ld s-a v e ra g ed , tu rb u len t, u n s te ad y in c o m p re ss ib le m e an f lo w

D M : ∇ ⋅u = 0

D P : ( ) 0guuuu =Θ+∇+⋅∇−∇+∇⋅+∂∂ − ˆ

ReGrReEu 2

1 tvpt

D E : ( ) 0u =+Θ∇+⋅∇−Θ∇⋅+∂Θ∂

Θ−− sv

tt

t1PrPe 1

D E ′ :

D E m ′ :

( ) 0uTu =−∇+∇+⋅∇−∇⋅+∂∂ −− εPrPe 11 kvk

tk t

t

( ) 0uTu =−∇+∇⋅∇−∇⋅+∂∂ − k

kv

tt

t /εCεCεPrCεε 22ε

1ε

1ε

w h ere :

n o n -D g ro u p s a re d e fin e d in tex t C h ap te r C M n

a lg e b ra ic R ey n o ld s s tre ss m o d e l

( )εC

...)(32

2kv

vk

t

Tt

μ≡

+∇+∇+−=− uuδT

CMn.20 Navier-Stokes, Computational Fluid Dynamics

NS turbulent flow conservation law PDEs

initial-value explicitly non-linear! multiple DOF/node! as given, are ill-posed! intrinsically unstable for Re >> 1

Commercial CFD codes contain required capabilities

have 15-20 year development history are based on FD, FV and/or FE discrete methods learning curve is steep! chances for error are numerous output interpretation requires color graphics & animations

CMn.21 Navier-Stokes, CFD, Algorithm Issues

For n = 2: kuku ˆωˆψ ⋅×∇≡×∇= and

DM: yidenticall0ˆψ =×∇⋅∇=⋅∇ ku

:Dˆ Pk ×∇⋅ 0ωReωˆψω 21 =∇−∇⋅×∇+ −kt

kinematics: ψˆˆψˆω 2−∇=⋅×∇×∇=⋅×∇= kkku

Steady-state N-S PDEs + BCs:1 2

2

ˆ(ω) Re ω ψ ω 0(ψ) ψ ω 0

−= − ∇ + ∇ × ⋅∇ =

= −∇ − =

kLL

)ω,ψ(ωˆconstantψψ:

0)ψω,(ˆ:calculusviaψ,ω),(:

wall

out

inininin

w

w

f

xy

=∇⋅==Ω∂

=∇⋅Ω∂⇒Ω∂

n

nu

CMn.23 Streamfuction-Vorticity Navier-Stokes

Consider unsteady isothermal laminar NS

DM: ∇⋅u = 0

DP: !onnone,onPDEs4

0ReEu 21

ppt

uuuuu

⇒=∇−∇+∇⋅+ −

Mathematically, DM is a constraint on solutions to DP

theories to enforce the constraint include

pseudo-compressibility: 0βD 1 =⋅∇+⇒ − utpM

pressure projection: ε 0, itera tive lyh h∇ ⋅ ⇒ >u

vector field theory: Ψu ×∇=guaranteesD M

∇×DP eliminates pressure appearance

⇒ for n = 2, produces streamfunction-vorticity formulation

CMn.22 Incompressible N-S, Well-Posedness

Galerkin weak statements:

for

thus:

α αα

( , ) ψ ( ) , {ω,ψ} , {OMG,PSI}N T Tq x y Q q Qα≡ = =∑ x

N N 1 2 N N Nβ βΩ Ω

1β α α β γ γ α αΩ

N Nβ β α α β α αΩ Ω

ˆGWS (ω) ψ (x)L(ω )dτ ψ Re ω ψ k ω dτ

Re ψ ψ OMG ψ ψ PSI ψ OMG dτ BC 0, β,α

GWS ( ) ψ L(ψ )dτ [ ψ ψ PSI ψ ψ OMG ] dτ BC 0, β,αψ

−

−

⎡ ⎤= = − ∇ + ∇× ⋅∇⎣ ⎦⎡ ⎤= ∇ ⋅∇ + ∇× ⋅∇ + = ∀⎣ ⎦

= = ∇ ⋅∇ − + = ∀

∫ ∫∫∫ ∫

GWSN ⇒ GWSh = Se{WS}e ≡ {0}

1

1w

w

{WS (ω )} Re [B2 ] {OMG} {PSI} [B3 0 ] {OMG}

Re [A200] { (ψ ,ω , U }

{WS(ψ )} [B2 ]{PSI} [B200] {OMG} [A200] {U }

h Te e e e e e

h he w e

he e e e e e

KK K K

f

KK

−

−

= +

+

= − +

CMn.24 GWSh, Streamfunction-Vorticity NS, n = 2

s

n

Ωh

∂Ωwall

y

x

Ωe

^n

wψ

ωkinvolvesωforGWS ∇⋅×∇ ˆψhh

Vorticity Robin BC generated via TS on Ωh:

yxxy ∂∂

∂∂

−∂∂

∂∂

=∇⋅×∇ωψωψωˆψk

[ ]

ee

TT

e yxxNN

xN

xNN

yNKK

e

]B3X0Y[]B3Y0X[

dd}{}{}{}{}{}{0B3

−⇒

⎥⎦

⎤⎢⎣

⎡∂

∂∂

∂−

∂∂

∂∂

=∴ ∫Ω

wall

ωdψd0ωψψωψ 2

2

2

2

2

22

Ω∂

−=⇒=−∂∂

−∂∂−

=−∇−nns

TS:

6/)d/dω(2/ωUψ

)(6d

ψd2d

ψdddψψ)(ψ

3w

2www

43

w3

32

w2

2

ww

nnnn

nOnn

nn

nn

n

Δ−Δ−Δ+=

Δ+Δ

+Δ

+Δ+=Δ

BC: ( ) 0)ψψ)(/6()U)(/6(ω)/3(-ωˆω w1w3w2 =−Δ−Δ+Δ∇⋅= +nnnn

V(y)

CMn.25 GWSh Details, Streamfunction-Vorticity NS

nΔ

GWSh ⇒ global Newton statement

( ) ( )eepeepp QQQQ }F{S}δ{]JAC[S}F{}δ]{JAC[ 11 −=⇔−= ++

Template pseudo code for {FQ}e

{WS}e ≡ (const) (avg)e{dist}e(metric;det)e[matrix]{Q or data}e

}ψ,(UOMG]{200A)[1}(){)(Re,3(}OMG]{0B3-03B)[1;2E1E}(PSI){)((

}OMG]{2B)[1;EE}(){)((Re}{FOMG

ww1

-1

Δ+Δ−+

−+−=

− fnKJJKKJ

JKKIJIe

}U]{200A)[1}(){)((}OMG]{200B)[1}(){)((

}PSI]{2B)[1;EE}(){)((}{FPSI

w+−+

−= JKKIJIe

CMn.26 Newton {FQ}e Template, (ωh, ψh) GWSh

Newton jacobian formed via differentiation

ee

ee Q

Q⎥⎦

⎤⎢⎣

⎡Ω

ΩΩΩ=

∂∂

≡]Jψψ[]Jψ[]ψJ[]J[

}{}F{]JAC[

Jacobian template pseudo-code

]][2B)[1;EE}(){)((}PSI{}FPS{]Jψψ[

]][200B)[1}(){)(1(}OMG{}FPSI{]Jψ[

]][200A)[1}(){)(Re,6(]][0B3-03B)[1;2E1E}(OMG){)((}PSI{

}FOMG{]ψJ[

]][200A)[1}(){)(Re,3(

]][03B03B)[1;2E1E}(PSI){)((]][2B)[1;EE}(){)((Re}OMG{}FOMG{]J[

31

1

1

JKKIJII

nJKKJKJ

n

KJJKKJJKKIJI

e

ee

e

ee

e

ee

e

ee

−=∂

∂≡

−=∂∂

≡Ω

Δ−−=∂

∂≡Ω

Δ−

−−+−=∂

∂≡ΩΩ

−

−

−

CMn.27 Newton Jacobian Template, (ωh, ψh) GWSh

G W S h + N ew ton ⇒ com p u tab le m atrix sta tem en t

S tep -w ell d iffu ser b en ch m ark

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎭⎬⎫

⎩⎨⎧

−=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡Ω

ΩΩΩ + p

e

p

eee FPSI

FOMGδPSIδOMG

JψψJψψJJ

S1

0

2

4

6

8

10

12

14

0 100 200 300 400 500 600 700

R e

L/s

C om pu tedE xperim en tal, A rm aly, et. a l (19 83 )

R e = 100

R e = 600

V alidation

CMn.28 Incompressible N-S, Step Wall Diffuser

CMn.29 Mechanical Vibrations, Continuum Systems

Lagrangian viewpoint, mechanical continuum

E-L:

22 E21,ρ21 xuVuT ==

for Lagrangian L ≡ T – V = data) ,,( uu ∇f , and dM = 0 = dE

0)(

:equationLEd =∇∂

∂⋅∇−⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

−⇔uu

P LLt

2( ) 0, E ρxxu u c u c= − = =L

Longitudinal oscillations of a bar, n = 1

Lagrangian

L = T – V = data) ,,( uu ∇f

0:dL-E =⎟⎠⎞

⎜⎝⎛

∂∇∂

⋅∇−⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

uuP LL

t

Transverse vibrations of a plate, n = 2

0)υE,(:d 22

=∇⋅∇−∂∂ yf

tyP

Sound propagation in free air, n = 3

Tcpctp γR,0:d 222

2

==∇−∂∂P

Normal mode solution tietq ω)(),( xQx =

eigenmodes 0QQ =−∇⋅∇− 2ωf

homogenous BCs ii ni QQ ⇒≤≤⇒ ,1,ωω

22

CMn.30 Mechanical Vibrations, n-D Continuum Systems

T ransverse vibrations of a plate

0),(:BCs

0)υ,(:d 22

=

=∇⋅∇−∂∂

ty

yEfty

bx

P

normal mode solution tiety ω)(Q),( xx =

GWSh for eigenmodes [ ] }0{}{]MASS[ω]STIFF[ 2 =− Qhomogeneous BCs det([MASS]-1 [STIFF] - ω i2[I]) = {0}

G W S h normal mode so lutions , ω ih = 45 , 71 , 99 for i = 7 , 12 ,

CMn.31 Transverse Vibrations, L-Shaped Membrane