HYBRID NUMERICAL METHOD FOR NON-STATIONARY

CONTINUUM MECHANICS

N.G. Burago1,3, I.S. Nikitin2,3

1IPMech RAS, 2ICAD RAS, 3Bauman MSTU

Volodarka - 2015

Under consideration:

Problems of Continuum Mechanics in Moving Regions of Complex Geometry

Advantage of three method components:

1. Moving Adaptive Grids2. Equilibrated Viscosity Scheme3. Overlapping Grids

Types of Grid Adaption1) Region boundaries of complex shape

2) Minimization of approximation errors ( min |hdy/dx| )

min ( ,T)V

dV ε

( , ) :tx x x

0.5( ) Tε F ×F - I

*( , )x V

t

x x x

T || || yTF x

0 0 ε 0 0 ε

Basement: Non-linear thermo-elasticity equations for adaptive grid generation

N.G.Bourago and S.A.Ivanenko, Application of nonlinear elasticity to the problem of adaptive grid generation // Proc. Russian Conf. on Applied Geometry, Grid Generation and High Performance Computing, Computing

Center of RAS, Moscow, 2004, June 28 - July 1. P. 107-118.

Grid is treated as isotropic thermo-elastic medium:

V~

*213 V~

dJ]I~2]T~

I2/)1I[(K~

[min

)F~

det(J

5.0),~(*~~ txxxVx

)10,max( 4*

JJ' ( : I)I / 3

I:~I1 '~:'~I2 2T

3 J)F~

:F~

det(I

Governing variational equation

Cell shape control parameter

Cell compressibility parameter

Thermal expantion (monitor function, “anti-temperature”)

Method for grid equation - stabilization method

[ ( : ) : ] 0t

V

x x L x x f J dM V

)t,x~(xx *V~x~

Algorithm: two-layered explicit scheme of stabilization which

provides equilibrium (by norm) of members with L and f

(...)Jf )1( 5.0

Details: ipmnet.ru/~burago

2. Equilibrated Viscosity Scheme (Variant of stabilized Petrov-Galerkin scheme)

Уравнения для задач механики жидкости и газа

( ) 0art dVt

V

u

dS)

S

(dV

V

:)p(t vv unuugIuuu

TdS

S

dV

V

T:Tr))pE((t

E

nqqu

2 ( ( : ) / 3) ( )v v art σ e e I I u U)1(p

))((5.0 Tuue

( )T artk T E q 2/UE uu

Уравнения для твердых деформируемых сред аналогичны

Variational Equations for Fluid Dynamics

Equations for Solid Mechanics are analogical

Simplified explicit scheme SUPG FEM1 (variant of stabilized Petrov-Galerkin method)

n( ) 1k

n2k

d

dk n

art k

2 0 21

1

( ) [ ( ) ]M

n n nk k kl

l

d d

u

n 2 22k

1

(d ) [ ]M

nk kl

l

| | / 1 0u c u 0.5nk

1.0nk иначе

тоесли

, , , ,x y zu u u E 0( , )

1

M

k kl J k ll

d

( , )1

M

k kl J k ll

1Brooks A.N., Hughes T.J.R. Streamline Upwind Petrov-Galerkin formulations for convection dominated flows // Computer Methods in Applied Mechanics and Engineering. 32. (1982) pp. 199-259.

Artifitial diffusion “equilibrates” by norm inviscous fluxes

Inviscous fluxes = convective terms + conservative parts of fluxes

Simplification: used FEM scheme is analogical to central difference scheme

)3/):((~2~vv IIeeσ Tk

~~Tq

art

art

v

vv /

1

//~

art

art

T

TT /k

1

/k/k

~

Коррекция физической вязкости по А.А.Самарскому(“экспоненциальная подгонка”)

n

knk

nk

nk

nk

N,1k

n

h/)!1D(||c(

hmint

2 u

Условие устойчивости (Курант-Фридрихс-Леви)

Physical viscosity is decreasing with growth of artificial viscosity

Corrected (by A.A.Samarsky) physical viscosity (“exponential correction scheme”)

Courant–Friedrichs–Lewy stability condition

Doolan E.P., Miller J.J.H., Schilders W.H.A. Uniform numerical methods for

problems with initial and boudary layers. BOOLE PRESS. Dublin. 1980.

Brief resume of numerical method.

Galerkin formulation. Simplex finite elements. Adaptive moving grid. All major unknowns in nodes.

FEM analogy of central difference schemes in space.

Explicit scheme for compressible media:

Variant of stabilized Petrov-Galerkin scheme

Explicit (convection fluxes) - implicit (concervative and dissipative fluxes) scheme for incompressible or low Mach flows

Exponential correction of physical viscosity.

Adaption: separated stage at the end of each time step

Ideal gas flow in channel with a step: М=3; =1.4; t = 0; 0.5; Adapted grid

Ideal gas flow in channel with a step: М=3; =1.4; t = 1.0; 2.0; Adapted grid

Ideal gas flow in channel with a step: М=3; =1.4; t = 3.0; 4.0; Adapted grid

Ideal gas flow in channel with a step: М=3; =1.4; t = 4.0;

Isolines of density; Adaption using as monitor functionu

Adaptive grids for supersonic flows in channels with obstacles

maintenance of uniform distribution of nodes

K T 0

Adaptive grid for turbine blade forming process.

3. Overlapping grids for calculation problems with complex geometry

Overlapping grids – what for? For example: Supersonic flows over several obstacles

Grid with triangular hole Main and overlapping grids

Overlapping or Chimera grids

Main bordering grid + Additional overlapping grids

Calculation is carried out by steps according to the explicit scheme or by iterations according to the implicit scheme separately on the main grid and on the overlapping grids, thus after each step (iteration) by means of interpolation the exchange of data between grids in an overlapping zone is carried out.

Simpified overlapping grid method

Main bordering grid + Additional overlapping grids

The overlapping grids are used only for complex boundary description and for boundary conditions on the main grid ==========================

Instead of overlapping grids here the overlapping areas defined by the set of conditions may be used

Target: simple solution of complex geometry problem

Part of solution region near overlapping grid (an obstacle)(velocities at t=0)

Part of solution region near overlapping grid (an obstacle). Main grid is adaptive and moving (velocities at t=0.1)

Part of solution region near overlapping grid (an obstacle). Main grid is adaptive and moving (velocities at t=0.2)

Part of solution region near overlapping grid (an obstacle). Main grid is adaptive and moving (velocities at t=0.3)

Part of solution region near overlapping grid (an obstacle). Isolines of vertical velocity at t=0.

Part of solution region near overlapping grid (an obstacle). Isolines of vertical velocity at t=0.1

Part of solution region near overlapping grid (an obstacle). Isolines of vertical velocity at t=0.2

Part of solution region near overlapping grid (an obstacle). Isolines of vertical velocity at t=0.3

Part of solution region near overlapping grid (an obstacle). Isolines of local Mach number at t=0.45

Whole solution region with two overlapping grids (an obstacles). Isolines of local Mach number at t=0.45

Whole solution region with two overlapping grids (obstacles). Isolines of local Mach number at t=7.0

Whole solution region with two overlapping grids (obstacles). Isolines of major unknown functions at t=7.0

Whole solution region. Main adaptive grid at t=7.0

Conclusions.

Implementation is rather easy possible as an addition to any normal gas dynamic code, capable to solve problems in rectangular or brick like solution region. It just needs to add separated blocks for setting overlapping areas and at the end of each step (iteration) it needs to provide correction of solution on overlapping parts of major grid in order to satisfy boundary conditions.

Advantages. Simplified overlapping grid method allows easily take into account complex geometry and boundary conditions. Equilibrated viscosity scheme and elastic grid adaptation technique provide together high accuracy of solution and robustness. There is no dependence of governing parameters of scheme on type of problem, the number of parameters is small and each has clear meaning.

Drawbacks. To provide necessary accuracy of solution rather high performance computers are required. Therefore personal computers are effective and give good accuracy only for 2D unsteady problems.

For мore detailed information see WEB: http://www.ipmnet.ru/~burago

The end