About myself, a solution algorithm for the Navier …...About myself My Trayectory Born in Mainz....

About myself, a solution algorithm for the Navier-Stokes

Equations and the Stokes Resolvent Problem

M.Sc. Fabian Gabel

Institut für MathematikTechnische Universität Hamburg

M.Sc. Fabian Gabel Inaugural Talk November 1, 2018 1 / 14

Contents

About myself


Contents

About myself

Masterthesis Computational Engineering


Contents

About myself


Masterthesis Mathematics


Contents

About myself



At TUHH


About myself

My Trayectory

Born in Mainz.


About myself

My Trayectory

Born in Mainz.

High school (5th - 9th grade) in

Riedlingen.


About myself

My Trayectory

Born in Mainz.


Riedlingen.

10th - 12th grade in Puebla

(Abitur, Bachillerato).


About myself

My Trayectory

Born in Mainz.


Riedlingen.



Study of B.Sc. MPE, TU

Darmstadt.


About myself

My Trayectory

Born in Mainz.


Riedlingen.




Darmstadt.

Study of M.Sc. CE, TU

Darmstadt.


About myself

My Trayectory

Born in Mainz.


Riedlingen.




Darmstadt.

Study of M.Sc. CE, TU

Darmstadt.

Study of B.Sc, M.Sc. Math, TU

Darmstadt.


About myself

Research at TU Darmstadt

Student assistant at the chairs of

Production Management and Technology (experimental research, tool wear-out, Prof. Abele)


About myself




Technical Thermodynamics (Shear-stress induced flow, numerical evaporation research, Prof.

Stephan)


About myself





Stephan)

Numerical Methods (parallelisation of solution algorithm, CFD, HPC, Prof. Schäfer)


About myself





Stephan)

Numerical Methods (parallelisation of solution algorithm, CFD, HPC, Prof. Schäfer)

Math. Modelling and Analysis (numerical investigation of contact line movement, Prof. Bothe)



Implementation and Performance Analyses of a Highly Efficient

Algorithm for Pressure Velocity Coupling





Challenges for applications for Computational Fluid Dynamics (CFD):

Results with little waiting time

High accuracy

Complex geometries

Multiphysical problems







High accuracy

Complex geometries


Figure: Jet turbine (GE)







High accuracy

Complex geometries


Figure: Jet turbine (GE)

Requirements

Adaptivity

Coupling of variables

Scalability (parallel computations)



Physical Model - Navier-Stokes Equations + Scalar Transport

auiP uP,i +

∑

F∈NB(P)

auiF uF,i + a

ui,p

P pP +∑

F∈NB(P)

aui,p

F pF

︸︷︷︸

Pressure-velocity coupling

+ aui,T

P TP

︸︷︷︸

Boussinesq approximation

= bP,uii = 1, ..., 3

ap

P pP +∑

F∈NB(P)

ap

F pF +

3∑

j=1

ap,uj

P uP,j +∑

F∈NB(P)

ap,uj

F uF,j

︸︷︷︸

Pressure-velocity coupling

= bP,p

aTPTP +

∑

F∈NB(P)

aTFTF +

3∑

j=1

aT,uj

P uP,j +∑

F∈NB(P)

aT,uj

F uF,j

+ aT,p

P pP +∑

F∈NB(P)

aT,p

F pF

︸︷︷︸

Newton-Raphson linearization

= bP,T



The PETSc Framework

A freely available and supported research

code for the parallel solution of nonlinear

algebraic equations

Usable from C, C++, FORTRAN

77/90, Python and MATLAB

Portable to any parallel system

supporting MPI

1012 unknowns, full-machine

scalability on Top-10 systems (see

top500.org)

Figure: Numerical libraries of PETSc



Adaptivity and Block Structured Grids

Figure: Domain of channel flow

Cubes as obstacles

laminar flow

nontrivial block transitions

Figure: Blocking around obstacles





Cubes as obstacles

laminar flow


Figure: Numerical grid on west- and east boundary.





Cubes as obstacles

laminar flow


Ll , Ll+1

Rl

Rl+1

Sl

Sl+1

left grid block right grid block

block boundary

3 6 9

2 5 8

1 4 7

2 4

1 3

1

2

3

4

5

7

8

8

9

1

2

VecGhostUpdate( )Vec on Proc 1

Ghost values from Proc 2

1

2

3

4

Vec on Proc 2

Block 1 on Proc 1 Block 2 on Proc 2



Matrix Structure and Variable Coupling

Figure: block structured

grid and corresponding

matrix structure (no

variable coupling)

0

0

1

1

2

2

3

3

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aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aP

aN aE aT

aS aE aT

aNaW aT

aSaW aT

aN aEaB

aS aEaB

aNaWaB

aSaWaB

aN aE aT

aS aN aE aT

aS aE aT

aW aN aE aT

aW aS aN aE aT

aW aS aE aT

aW aN aT

aW aS aN aT

aW aS aT

aN aE aTaB

aS aN aE aTaB

aS aE aTaB

aW aN aE aTaB

aW aS aN aE aTaB

aW aS aE aTaB

aW aN aTaB

aW aS aN aTaB

aW aS aTaB

aN aEaB

aS aN aEaB

aS aEaB

aW aN aEaB

aW aS aN aEaB

aW aS aEaB

aW aNaB

aW aS aNaB

aW aSaB

aR aR aR aR

aR aR aR aR

aR aR aR aR

aR aR aR aR

aL

aL

aL

aL

aL

aL

aL

aL

aL

aL

aL

aL

aL

aL

aL

aL



Matrix Structure and Variable Coupling

au1

P

au2

P

au3

P

apP

aTP

au1,pP

au2,pP

au3,pP

ap,u1

P ap,u2

P ap,u3

P

au1,TP

au2,TP

au3,TP

aT,u1

P aT,u2

P aT,u3

P aT,pP

unterschiedliche Kopplungsterme:

Pressure-Velocity

Velocity

Temperature-Velocity/Pressure

instead of an scalar entry: dense 5x5 matrix



MIT Benchmark - Velocity-Pressure-Temperature Coupling



MIT Benchmark - Velocity-Pressure-Temperature Coupling

Resolution Solver configuration Time s No. of nonlinear its.

SEG 0.3719E+02 203

CPLD 0.6861E+02 62

TCPLD 0.1012E+03 3132x32x32

NRCPLD 0.2153E+02 22

SEG 0.1997E+04 804

CPLD 0.7687E+03 63

TCPLD 0.1278E+04 5964x64x64

NRCPLD 0.4240E+03 17

SEG 0.5197E+05 3060

CPLD 0.1860E+05 74

TCPLD 0.1950E+05 50128x128x128

NRCPLD 0.6155E+04 18



Existence of Solutions to the Navier-Stokes Equations - Agenda

Navier-Stokes equations for homogeneous, incompressible fluids:

∂tu − ν∆u + (u · ∇)u +∇π = f t ∈ (0, T), x ∈ Ω

∇ · u = 0 t ∈ (0, T), x ∈ Ω

u(0) = a x ∈ Ω

u|∂Ω = 0 t ∈ (0, T).




1 Construct operator A : D(A) → X, X suitable Banach space such that

u ∈ D(A) ⇐⇒ ∇ · u = 0, u|∂Ω = 0 and ∃π with Au = −∆u +∇π.






2 Give meaning to etA (semigroup theory).







3 Non-linearity in NSE as right-hand side. Variation of constants formula:

u(t) = e−tAa −

∫ t

0

e−(t−s)A

P(u(s) · ∇)u(s)dx.







3 Non-linearity in NSE as right-hand side. Variation of constants formula:

u(t) = e−tAa −

∫ t

0

e−(t−s)A

P(u(s) · ∇)u(s)dx.

4 Kato iteration:

uj+1(t) := u0(t)−

∫ t

0

e−(t−s)A

P(uj(s) · ∇)uj(s)ds.



On Resolvent Estimates in Lp for the Stokes Operator

in Lipschitz Domains

Let Ω ⊂ Rd, d ≥ 3, a bounded Lipschitz domain.






Theorem (Fabes, Mendez, Mitrea, 1998)

There is ε = ε(Ω, d) > 0, s.t. for all 3

2− ε < p < 3 + ε the Helmholtz projection exists on L

p(Ω;Cd).Furthermore, the projection

P : Lp(Ω;Cd) → L

p(Ω;Cd)

is continuous.






Theorem (Fabes, Mendez, Mitrea, 1998)

There is ε = ε(Ω, d) > 0, s.t. for all 3

2− ε < p < 3 + ε the Helmholtz projection exists on L

p(Ω;Cd).Furthermore, the projection

P : Lp(Ω;Cd) → L

p(Ω;Cd)

is continuous.

Theorem (Shen, 2012)

Let θ ∈ [0, π). There exists ε(θ, d,Ω) > 0, s.t. for all

2d

d + 1− ε < p <

2d

d − 1+ ε

the Stokes operator Ap on Lpσ(Ω) is sectorial with angle θ. Ap is closed, densely defined, 0 ∈ ρ(Ap) and −Ap

generates an analytic semigroup.





Main tools:

Bessel and Hankel functions (fundamental solutions)





Main tools:


Harmonic analysis (Cauchy integral formula for Lipschitz domains, Hardy-Littlewood maximal

operators)





Main tools:



operators)

Method of boundary layer potentials





Main tools:



operators)


Rellich estimates on solutions to the Stokes problem





Main tools:



operators)



Fredholm theory





Main tools:



operators)



Fredholm theory

Interpolation theory (Refined version of the Calderón-Zygmund Lemma)





Main tools:



operators)



Fredholm theory

Interpolation theory (Refined version of the Calderón-Zygmund Lemma)

Help of my supervisor: Dr. Patrick Tolksdorf



At TUHH

Research

Limit Operator Method

Fibonacci Hamiltonian

. . .

Teaching

Linear Algebra (Marko Lindner)

Mathematical Image Processing (Marko Lindner)

ZLL

. . .


About myself, a solution algorithm for the Navier …...About myself My Trayectory Born in Mainz....

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