Page 1 JASS 2004 Tobias Weinzierl Sophisticated construction ideas of ansatz- spaces How to...

JASS 2004 Tobias Weinzierl

Sophisticated construction ideas of ansatz-spaces

How to construct Ritz-Galerkin ansatz-spaces for the Navier-Stokes equation that preserve the mass continuity.

JASS 2004 – Tobias Weinzierl

On some ideas of-Cornelia Blanke

-Prof. Dr. Christoph Zenger-Dr. Miriam Mehl


Principles of numerical simulations

consistency

convergence

stability

grid independence

mass conservation

laws of nature


Importance of conservation laws

• mass / energy explosion

• incorrect

• formalization:

XXh,τt

τ ttXtXh )()(

Some examples for X:•energy•mass•momentum


The simulated phenomenon

),,(:),(

),,(

),,(),,(:

:

:

21

212

21121

2

txxutxu

txxu

txxutxxu

p

u

Incompressible viscous fluid:•three degrees of freedom:

•kinematic pressure p•kinematic velocity u (vector field)

•constant density


Mathematical stuff

p

x

pxp

2

1:

2

1:)(

uu

uuuu

2,1

)(:i

iiu

xudivu

gradient

divergence operator

2,1

2

2

2,1

1

2

2

1

2

2

:

i i

i i

ux

ux

u

uu

Laplace operator

nonlinear term


Navier-Stokes-equation

0)(Re

1

0Re

1)(

122

2

2

121

2

12

11

2

1

px

ux

ux

ux

ux

u

uut

puuuut

i

velocitypressure

0

0

22

11

ux

ux

u

momentum equation:•nonlinear second order PDE•konvection term is nonlinear•diffusion is linear (Laplace operator)•built in: energy and momentum conservation•high Reynolds number: turbulent flow


Agenda

• Navier-Stokes Problem

• Recapitulation of FEM ideas

• Grid types and evaluation of (bi-)linear ansatz functions

• Construction of a more sophisticated ansatz-space with respect to mass conservation

• Some further properties of our ansatz-functions

• What we work on now


Things I will not talk about

• Handling nonlinearity

• Stability and consistence evaluations

• Handling sideconditions

• Hanging points

• Multilevel

• Time discretization


The properties of FEM by Ciarlet – part I: The grid

FEM approximates the solution function using linear combination of some basis functions (ansatz-functions). The solution functionitself is approximated.

•dofs are no hanging points•uniformity (Zlamal condition)

CHu 1

geometric element


The properties of FEM by Ciarlet – part II: The ansatz-functions

•Polyonomial Ansatz-functions•Local support•Affine Families

hh VVV dim

Conformity:


Grids for Navier-Stokes problems

colocated grid fully staggered grid partially staggered grid

u,p u,p

u,p u,p

u

u

u

u

u u

u u

p p

•nonregular quadratic domain approximation•finite element approximation: continous solutions•Navier-Stokes-equations are ‚solved‘•use other proofs: „checkerboard instability“

01PQ


The discrete mass conservation

02222

)()()()()(

0)(

0)(

43213142

2412

hbb

hbb

haa

haa

ndSundSudxudiv

Udxudiv

udiv

U

U

12Tba ),( 11

Tba ),( 22

Tba ),( 44

Tba ),( 33

24

4331

•control volume (Gauss)•quadratic cells are control volumes•should be valid for every fem solution•set of linear equations (constraints on solution)


Agenda



• Grid types and evaluation of (bi-)linear ansatz-functions





Bilinear ansatz-functions for velocity

0)(

),(:|

1)1)(1(),(:|

122211

22122212

1211211121

21212121

xcaxcaux

udiv

dxxcxbxa

dxxcxbxaxxu

xxxxxxxx

i

ii

T

T

without proof

•linear on the axes of the coordinate system•nonlinear on the support (see Maple sheet)•discrete mass conservation doesn‘t imply pointwise mass conservation


Lagrange ansatz-functions on triangles

21

22212

1211121

2121

)(

),(:|

1),(:|

baux

udiv

cxbxa

cxbxaxxu

xxxx

i

ii

T

T

•evaluate other grid types •on a triangle the div is constant•if two dofs are given, the side condition determines the value of the third ‚unknown‘


Handling the side condition

1

1

b

a

2

2

b

a

3

3

b

a

4

4

b

a

5

5

b

ah

h

)(4

1

)(4

1

432143215

432143215

bbbbaaaav

bbbbaaaau

02222

0|)(

43213142

hbb

hbb

haa

haa

udiv

•divide square into four triangles•assumption: discrete conservation formula is valid•new point isn‘t a real dof for its value is determined by other four points•result is a piecewise linear function


The sophisticated ansatz-function

•piecewise linear (see Maple sheet)•if continuity is preservedon border, continuity ispreserved in every innerpoint•there are some other niceproperties


Agenda








Energy conservation

0),(Re

2

),(),()),(),,((

0)),(),,((

0)(

||||2

1)( 2

uu

uuuutxutxut

txutxut

tEt

dxutE

tt

0Re

1)(

puuuut

•energy of system modelled by Navier-Stokes-equations is constant iff diffusion isn‘t modelled•if there‘s no friction, energy equality holds•otherwise energy decreases•roadmap:

•insert momentum equation•apply continuity condition•some technical work


Discrete energy conservation

0))()(( hT

hT

hTh uDDuCuCu

00)(

0)(0Re

1)(

hhT

hhhh MupMDuuuCu

udivpuuuut

convection diffusionmassmatrix

),(),()),(),,(()),(),,((

0)(

tt uuuutxutxut

txutxut

tEt


Navier-Stokes approximations

No friction means D is zero and energy should be conserverd. Therefore:

-C has to be antisymmetric-D has to be symmetric positiv definit

Some calculations show: Constructed ansatz-space produces matrices with properties needed.

For mass conservation is given pointwise adaptive grids and refinement processes are no problem for this ansatz-space.

0)(

0)()(

hTT

h

hT

h

uDDu

uCuC

0))()(( hT

hT

hTh uDDuCuCu


Agenda








What we work on now

• implement ideas within space-filling-curves framework

• use multigrid algorithms

• extract application domain independ techniques


Lessons learned

• validate every approximation against laws of nature. This shows drawbacks / possible problems of solution

• use freedom of choosing ansatz-space properly

• try to use problem-specific ansatz-spaces: If dofs are coupled (here x1 and x2), perhaps the ansatz-functions have to be coupled, too

• ask application domain experts, too - not only mathematicians

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