Nonclassified activity of CML RFNC-VNIITF
Oleg V. Diyankov
Head of Computational MHD Laboratory
Russian Federal Nuclear Center-All-Russian Institute for Technical Physics
Snezhinsk, Chelyabinsk region (the Urals), Russia
About 9000 employees, among which there are scientists: physicists, chemists, mathematicians; designers, engineers, etc.
RFNC has many divisions. We’re representing the division of theoretical physics and applied mathematics.
Division of Theoretical Physics and applied mathematics
260 scientists in computational mathematics and theoretical physics.
Computational MHD lab was created on the 1st of April 1996.
14 scientists are working in it.
The picture of the laboratory
The main directions of the work 2.5D MHD code 2D Irregular Grid for Mathematical Modeling Linear Solvers for Flows in Porous Media Modeling Difference Schemes for Hyperbolic Systems Treatment 3D Elastic-Plastic Modeling of Processes of Ceramics
Formation 3D Gas Dynamic Code for Instability Investigation Development of Special Software Tools for CERN
2.5D MHD Code
The Physical Model Realized in the Code The Application of the Code to Laser Beam
Interaction with the Matter Modeling The Application of the Code to Liner
Magnetic Compression Modeling The Application of the Code to the Plasma
Channel Formation Modeling
2½D MHD MAG Code 2½D MHD MAG Code
O.V.Diyankov, I.V.Glazyrin,
S.V.Koshelev, I.V.Krasnogorov, A.N.Slesareva, O.G.Kotova
Russian Federal Nuclear Center - VNIITFP.O.Box 245, Snezhinsk, Chelyabinsk Region, Russia
The main goal of MAG code creation was the necessity of modeling of hot dense plasmas in magnetic field.
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The system of equations used in MAG code is determined by Braginskii model for one-temperature case:
The MAG Code Model:The MAG Code Model:
Let us assign indices x and y in the case of the axial symmetry to the r- and z- components of vector variables and r- and z- components of independent spatial variables correspondently, and index z to the - component of vector variables.
Then we receive a basic system of equations, which depends on the parameter of symmetry ( = 0 - the plane symmetry, = 1 - the symmetry is axial).
The equations have been written in arbitrary moving coordinate system, and then splitting into two systems has been performed. The diffusive terms have been splitted to a separate system of equations, the remained terms produce a quasilinear hyperbolic system.
Two equations for x- and y- components of magnetic field was written in form of an z-component vector potential A:
The first system is a hyperbolic one and it describes the ideal MHD flows in arbitrary moving coordinate system.
The second one is a diffusive system of equations. It includes the equations for energy, z (or ) – component of magnetic field and z (or ) component of vector potential.
Details of Numeric:Details of Numeric:
x
AxB
y
AxB yx
;
1. Gas dynamics conditions:
• applied pressure: P|b= f(t), where P|b means the pressure at the corresponding boundary, f(t) is a given time dependent function;
• rigid wall: un|b = 0, where un is a normal to boundary
component of mass velocity;
• piston: un|b = f(t);
Boundary conditions:Boundary conditions:
2. Conditions for heat conductivity:
),( tTfTTbbbn
They are: given temperature, given heat flux, heat flux as a function of temperature. These conditions may be written in the form:
Here, T|b is the temperature at the corresponding boundary, , are numerical parameters (=0 and =1 for given temperature and =1 and =0 for heat flux), f(T|b, t) is a given function. nT|b is a normal to the boundary component of the temperature gradient.
Boundary conditions:Boundary conditions:
3. Conditions for magnetic field
Boundary conditions:Boundary conditions:
• symmetry: Bz /n=0, B /n=0, Bn =0 ;
• conducting wall: Bz=0, B=0, Bn =0 ;
• axis: Bz=0, B=0, Bn =0, where B is used for the axial symmetry and Bz for the plane one;
• given current: Bz=2j /c – plane symmetry, B= 2I /cr – axial symmetry,
where Bn is the normal to the boundary component of the magnetic field, j - current density, I is the whole current in z - direction, r - upper radius.
Algorithms of mesh reconstruction:Algorithms of mesh reconstruction:
Lagrange (no mesh reconstruction) Euler (grid nodes are returned to original positions at the
n-th time step) Local (only eight neighbor nodes are used for new node
coordinates determination) Algebraic (new nodes coordinates are calculated by
bilinear interpolation of boundaries nodes coordinates in mathematic coordinate system)
Poison equation solution is used for new coordinates determination
System of equations in arbitrary moving coordinates I:
System of equations in arbitrary moving coordinates I:
Equation of continuity:
Euler equations for velocity components:
x component:
y component:
z component:
System of equations in arbitrary moving coordinates II:System of equations in arbitrary moving coordinates II:
Equation for A:
Equation for z component of magnetic field:
Equation for total energy:
Equation for the square root of the metric tensor:
Here uk is a projection of mass velocity vector to k-th vector of a local basis, uk=uk, where k is a covariant local basis vector, vk,Bk are determined in the same way, v - coordinate system velocity, g is a determinant of metric tensor with covariant component gij,
k - mathematical coordinates.
System of equations in arbitrary moving coordinates III:System of equations in arbitrary moving coordinates III:
= const. - equation of continuity,g = const. - equation for determinant of metric tensor,u = const. - Euler equation for velocity.
Equation for A:
Equation for Bz:
Equation for energy:
Difference scheme for single diffusive equation:Difference scheme for single diffusive equation:
Where: kk
g
D
Diffusive equations system:Diffusive equations system:
Difference sheme for diffusive equations system:Difference sheme for diffusive equations system:
Z-pinch simulationZ-pinch simulation
Anod
Cathod
Experimental Setup
If plasma liner implosion is used as plasma radiation source one needs to receive the uniform plasma column. The ideal configuration for such type of radiation source could be an annular pinch (see Fig.1, left). The plasma, imploded from large radius, reaches the axis and the efficient radiator is formed. But this scheme is unsuitable because of MHD instabilities which present the great danger to uniformity of the liner implosion. So the compression achieved in corresponding experiments is substantially lower than one predicted by one-dimensional (1D) MHD calculations.
Simulation
Hollow gas puff simulation:Hollow gas puff simulation:
Density [g/cc]
Magnetic field, -component
[10 MGs]
Pinhole image reconstruction throw steady equation of radiation transfer resolving
Pinhole image reconstruction throw steady equation of radiation transfer resolving
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The equation of radiation transfer along a ray has the following form:
Pinhole Image Reconstruction on Simulation ofGas Puff Implosion (I)
Pinhole Image Reconstruction on Simulation ofGas Puff Implosion (I)
Pinhole Im age
Density[g/cc]
Temperature[keV]
193.5 ns
180 ns
Pinhole Image Reconstruction on Simulation ofGas Puff Implosion (II)
Pinhole Image Reconstruction on Simulation ofGas Puff Implosion (II)
196.5 ns
204.5 ns
One Shell Gas PuffMass 100 g/cmR adius 2.2 cm 10% random initial density disturbance
Pinhole Im age
Density[g/cc]
Temperature[keV]
Simulation parameters:
Current:1.7 MA
100 ns rise time
Laser produced plasma jet expansion into vacuum
Laser produced plasma jet expansion into vacuum
Temperature evolution, time = 0.1, 0.3, 0.5, 0.8
ns. Energy of laser pulse is4 kJ/cm2. Triangle pulse
with duration time of 1 ns. Focal spot 50 m.
Laser produced plasma jet expansion into background plasma
Laser produced plasma jet expansion into background plasma
Temperature evolution. time = 0.1, 0.3, 0.5, 0.8
ns. Energy of laser pulse
4 kJ/cm2. Triangle pulse shape with duration time
of 1 ns. Focal spot 50 m.Background plasma
density was 10-6 g/cm3
Experimental SetupExperimental Setup
Laser B eam
G as JetVa lve
P la sm a is c rea ted du e to in te rac tion o f la se r p u lse w ith a g a s p u ff ta rge t.
In itia l D en s ity P ro file
L a se r P aram ete rs :N d :g la ss 1 n s
1 0 J5 0 m
la se r sy s tem p ro d uc ing p u lse s w ith e n erg y u p to a n d fo ca l rad iu s m
Plasma Channel Formation. Density Evolution.Plasma Channel Formation. Density Evolution.
Tim e: ns0.4 Tim e: ns0.6
Tim e: ns0.8 Tim e: ns1.0
Plasma Channel Formation. Temperature Evolution.Plasma Channel Formation. Temperature Evolution.
Tim e: ns0.4 Tim e: ns0.6
Tim e: ns0.8 Tim e: ns1.0
The plasma is heated up to high temperature (temperature reaches
450 eV).
A motion of the SW should be spherical but the SW dynamics
propagating along the laser pulse direction differs from the perpendicular SW one.
As the velocity of SWs is approximately equal to 107 cm/sec,
the perpendicular SW leaves the region of the laser absorption (the
focal radius) after 0.3-0.5 ns.
2D Irregular Grid for Mathematical Modeling
Gas Dynamics Poisson Equation Solver Maxwell Equation Solver Heat Transfer
Oleg V. Diyankov
Sergei S. Kotegov
Vladislav Yu. Pravilnikov
Yuri Yu. Kuznetsov
Aleksey A. Nadolskiy
RFNC – ARITPh
supported by LLNL grant B329117
IGM CodeIGM Code
Overview
The IGM code was created to perform 2D flows modeling. The main feature of the code is the possibility of large deformations accounting.
3 physical processes are taken into account now: gas dynamics, heat conduction and Poisson equation.
The main advantage before well-known finite element codes is the possibility of arbitrary deformations description.
Features & Benefits
The flow region is initially covered by a set of Voronoi cells, and then at each time step the grid is reconstructed.
This allows neighbor points to move free in any direction, so they may move very far from each other.
The GUI interface for the IGM code (it is called CELLS) allows to put in initial data (geometry, matters, initial distribution of the values), and to look through the received results.
Applications
Plasma physics (instability study, laser produced plasma, and so on).
High velocity impact. Heat transfer. Electrostatic fields.
Voronoi diagram
Benefits: Local orthogonality Local uniformity
Gas-dynamic equations with heat conductivity
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The development of Raleigh-Taylor instability
The grid at the last moment for RT instability test.
High velocity impact problem (the angle equals to 90 degrees)
High velocity impact problem (the angle equals to 45 degrees)
Poisson problems f(r)= 10x8(x-2)8[36(x-1)2+2x(x-2)]y10(2-y)10+
10y8(y-2)8[36(y-1)2+2y(y-2)]x10(2-x)10
(x,y)=x10(2-x)10y10(2-y)10
Discharger(distribution of potential, 105 V)
Discharger (distribution of electric field strength, 105 V/cm)
Specifications
The source code, written in C++, has approximately 100000 lines.
Typical calculation takes from 10 minutes up to 1 hour on the SGI R10000.
RAM needed: 800 bytes per cell.
Nonstationary Preconditioned Iterative Methods for Large Linear Systems
Solving
Oleg V. Diyankov
Vladislav Y. Pravilnikov
Natalia N. Kuznetsova
Sergei V. Koshelev
Igor V. Krasnogorov
Dmitriy V. Gorshkov
Aleksey A. Nadolsky
Introduction Iterative Methods Preconditioners The PLS Code
Introduction
The work has been performed according to the contract with ExxonMobil Upstream Technology Corporation.
The main goal was to select methods which are the most appropriate ones to the ExxonMobil specific problems.
The results of many scientists and especially prof. Yousef Saad, Dr. Edmond Chow, prof. Kolotilina, Dr. Eremin have been used.
The goal of the work is to solve the linear system of equations
Ax=b, (1)
where A is a large sparse matrix, b is a right hand side vector, x is a vector of unknowns. Preconditioned Iterative Methods could be used for solving iteratively such systems.
Iterative MethodsIterative methods can be written in the general form as follows:
bxAxx
F j
j
jj
j
1
jF
j
(2)
where - the sequence of nonsingular matrices, - the sequence of real parameters.
jjj FH
bAxHxx jj
jj 1
Or, if we denote , formula (2) canbe expresses in the following form:
(3)
jHIterative methods are called stationary, if don’t depend upon the iteration count j. Otherwise,iterative methods are called nonstationary.
The most known stationary iterative methods are the Jacobi method, the Gauss-Seidel method and Successive Overrelaxation (SOR) method.The most known nonstationary iterative methods are the Conjugate Gradient (CG) method and Minimal Residual (MinRes) method and their modifications.As a rule, for nonstationary iterative methods the solution xj on J-th iteration is searched through minimization of a quadraticfunctional from xj.
bAxxx jj
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The Minimal Residual (MinRes) methodThe MinRes iterations can be written in the following form:
where
Modifications: The General Minimal Residual (GMRES) method and Quasi-Minimal Residual (QMR) method.In general the minimal residual method can be expressed in the following form:
bAxHxx jj
jj 1
where Hj is a matrix, which depends from some parameters and which is chosen to minimize |b-Axj|. This method can be applied to systems with nonsymmetrical matrix A.
The Conjugate Gradient (CG) method
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The BiConjugate Gradient (BiCG) methodThe BiCG method generate two CG-type bi-orthogonal sequences of vectors, one based on A, and one on AT.
This method can be used for systems with nonsymmetrical and nonsingular matrix A.
PreconditionersbMAxM 11 (4)
The preconditioner is called incomplete, if it is constructed on a subset of indices:
0|, ijM mjiP
There is three basic type of preconditioners:• The Incomplete Cholesky (IC) factorization;• The Incomplete LU (ILU) factorization;• The Approximate Inverse (AI) factorization.
The Incomplete Cholesky (IC) factorization
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The algorithm of the ILU(0) (ILU without extension of stencil) construction can be described as follows.
For each row of A we should fulfill the followingsteps:1. For the I-th row we select all nonzeros
having indices less than I;2. For each J we calculate aij = aij / ajj;3. For only nonzeros in I-th row with indices K
greater than I we fulfill the elimination aik = aik – aij * ajk.
There is two main strategies for construction of a preconditioners with extended stencil:• structural;• fill-in with drop tolerance.The algorithm of the FILU (fill-in ILU with drop tolerance eps ) can be described as follows.For each row of A we should fulfill the followingsteps:1. For the I-th row we select all nonzeros
having indices less than I;2. For each such J we calculate aij = aij / ajj;3. For all nonzeros in I-th and J-th rows with
indices K greater than I we fulfill the elimination aik = aik – aij * ajk if aik exists and for all other nonzero ajk we check the drop condition | aij * ajk| > eps * |Ai|.
The Approximate Inverse (AI) factorization1AM
The algorithm of the AI construction can be described as follows.
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For each row of matrix A we should fulfill following steps: • The determination of the fill stencil;• The calculation of the Mi coefficients from auxiliary
linear system P * mi = ai, where mi and ai consists from nonzeros of rows Mi and Ai respectively.
The PLS code
The PLS (Parallel Linear Solver) is the C++ object-oriented code for solving of a large sparse linear systems. Linear systems are considered as multi-block systems. For solving of these systems we use preconditioned iterative methods.General steps of solution process are:• Transformation of a multi-block system to a single block system;• Generation of a Preconditioner;• Execution of an Iterative Method;• Inverse transformation of obtained solution to a multi-block structure.The general object-oriented program languages concepts such as the polymorphism and the derivation has been used in the code.
Brief description of the PLS code
We’ve realized four version of the code:• Serial Unix version;• Parallel Unix version based on MPI;• Threads Unix version;• Serial Windows-NT version.The PLS code can be used both as command-line utility and as library with program interface. The user can use these interfaces in his application to choose the iterative method, the preconditioner and call solver.We’ve implemented the following iterative methods:CG, BiCG, CGS, BiCGStab, MMR(l) (the Modified Minimal Residual method),BiCGStab(L).We’ve implemented the following preconditioners: ILU, MILU, FILU, MFILU, AIP, AIP(L).Moreover, the special solver of the block Gauss-Siedel type has been realized for systems with a 3x3 block matrix.
The results, presented here, have been obtained on Pentium PC cluster system (PII/400).
Equations marked as ExxonMobil ones are samples of real equations, appearing in ExxonMobil applications. Their structure and specific characteristics is ExxonMobil proprietary information.
Numerical results
Task 1.
The solution of Poisson equation with discretization on irregular grid constructed on Voronoi cells.
Matrix size: N=50249, NNZ=349945.
Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.02, 0.2) 0.65 BiCGStab 16.81 17.80
1 AIP(1) 7.74 BiCGStab 39.87 47.953 ParAIP(1) 3.14 ParBiCGStab 21.35 24.497 ParAIP(1) 1.35 ParBiCGStab 10.18 11.53
Task 2ExxonMobil’s Mixed Cube problem (mc30).Matrix size: N=110700, NNZ=593100
Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.02, 0.2) 9.79 BiCGStab 26.59 37.19
1 AIP(1) 17.81 BiCGStab 69.85 88.48
3 ParAIP(1) 7.57 ParBiCGStab 43.89 51.46
7 ParAIP(1) 3.13 ParBiCGStab 21.65 24.78
Task 3ExxonMobil’s Mixed Cube problem. (mc40)Matrix size: N=260800, NNZ=1406400
Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.02, 0.2) 26.56 BiCGStab 102.95 131.41
1 AIP(1) 42.50 BiCGStab 237.43 281.88
3 ParAIP(1) 18.27 parBiCGStab 152.93 171.20
7 ParAIP(1) 7.79 parBiCGStab 100.67 108.46
15 ParAIP(1) 3.61 parBiCGStab 55.62 59.23
Task 4
ExxonMobil’s Mixed Cube problem. (mc50)Matrix size: N=507500 , NNZ=2747500 Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.02, 0.2) 58.86 BiCGStab 242.32 304.97
1 AIP(1) 83.78 BiCGStab 722.97 810.52
3 ParAIP(1) 36.46 parBiCGStab 411.47 447.93
7 ParAIP(1) 15.71 parBiCGStab 233.10 248.81
15 ParAIP(1) 7.11 parBiCGStab 124.04 131.15
Task 5
ExxonMobil’s Mfem problem. (mfem_32_64_16)Matrix size: N=134656, NNZ=704000
Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.02, 0.2) 2.47 BiCGStab 31.99 35.61
1 AIP(0) 2.8 BiCGStab 271.87 275.82
1 AIP(1) 20.80 BiCGStab 114.61 136.53
1 AIP(2) 261.08 BiCGStab 144.24 406.48
3 ParAIP(1) 8.91 parBiCGStab 71.10 80.01
7 ParAIP(0) 0.62 parBiCGStab 116.81 117.42
7 ParAIP(1) 3.62 parBiCGStab 34.23 37.85
7 ParAIP(2) 50.53 parBiCGStab 49.43 99.96
15 ParAIP(1) 1.75 parBiCGStab 21.69 23.44
Task 6
ExxonMobil’s Mfem problem. (mfem_48_96_24)Matrix size: N=450432, NNZ=2395008
Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.02, 0.2) 6.41 BiCGStab 139.16 149.33
1 AIP(1) 72.21 BiCGStab 810.89 887.03
3 ParAIP(1) 30.89 parBiCGStab 412.42 443.31
7 ParAIP(1) 13.23 parBiCGStab 196.56 209.79
15 ParAIP(1) 6.17 parBiCGStab 153.08 159.25
Task 7
ExxonMobil’s Geo problem. (geo20)Matrix size: N=60983, NNZ=1430843
Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.02, 0.2) 18.19 BiCGStab 105.48 124.89
1 FILU(0.02, 0.2) 6.21 Geo(BiCGStab) 87.71 94.35
3 ParAIP(1) 2.84 parGeo parBiCGStab
36.97 39.81
7 ParAIP(1) 1.40 parGeo parBiCGStab
20.23 21.63
15 ParAIP(1) 0.74 parGeo parBiCGStab
11.87 12.61
Task 8
ExxonMobil’s Geo problem. (geo30)Matrix size: N=200073, NNZ=5203713
Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.01, 0.1) 100.23 BiCGStab 130.88 234.18
1 FILU(0.01, 0.1) 31.76 Geo(BiCGStab) 341.21 374.44
3 ParAIP(1) 10.47 parGeo(parBiCGStab) 224.51 234.98
7 ParAIP(1) 4.80 parGeo(parBiCGStab) 105.95 110.75
15 ParAIP(1) 2.71 parGeo(parBiCGStab) 63.03 65.74
Task 9
ExxonMobil’s Geo problem. (geo_32_64_16)Matrix size: N=244081 , NNZ=6218735
Nproc Preconditioner Tprec Iterative Method Tim Ttotal
1 FILU(0.02, 0.2) 208.22 BiCGStab 125.75 337.89
1 FILU(0.02,0.2
13.74 Geo(BiCGStab) 338.18 353.52
3 ParAIP(1) 12.65 parGeo parBiCGStab
249.75 262.40
7 ParAIP(1) 5.91 parGeo parBiCGStab
120.34 126.25
15 ParAIP(1) 3.40 parGeo parBiCGStab
65.41 68.81
Difference Schemes for Hyperbolic Systems Treatment
Central difference schemes for Euler gas dynamics (Kurganov-Tadmor difference schemes)
Application of KT difference scheme to gas pipe simulation problem
KT scheme for 1D Lagrangian gas dynamics KT scheme for multi-dimensional gas
dynamics irregular grid simulations
Euler equationsThe non-steady-state flow of heat-conductive viscous gas in a constant section pipe is described by system of differential equations in partial derivatives [1]:
0
z
u
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p
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Equation of state: ,TR
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1
1
8314gR (J/kg К) – universal gas constant,
- molecular weight, - Poisson index.
(1)
zt, - time and space coordinates along pipeline axis.
Tpu ,,,, - density, mass velocity, pressure, specific internal energy and gas temperature
tzTT ee , - temperature of environment (ground, water, air)
RD 2 - pipe radius
- given function, describing relief along a pipeline.
- pipeline length (simulation region).
- resistance coefficient.
- pipe diameter, R
L
zhh
)(Re, 0d
uD
Re - Reynold's number.
)(T - dynamic gas viscousity
)(00 zdd - effective rough inner pipe surface
),( tz - heat transfer coefficient
Difference schemeThe uniform grid is made along a pipeline section. Density, speed, specific internal energy and temperature of gas are defined in middle of intervals. In addition one accounting interval is attached to external borders of the pipeline, so-called fictitious cells. The known spatial distribution of appropriate values is assign in middle of intervals in the initial time moment.
Second order semi-discrete central difference scheme [2] is used for solution of equations (1):
z
tHtHtu
dt
d jjj
)()(
)( 2121 , where
22u
uu j
Difference boundary conditionsIt was supposed, that the condition of inflow gas is put on a left border, and the pressure is set on a right border. Linear dependencies of basic values chosen as initial approximation.
1. Inflow condition.),( 00 p ),( 00 TpCompositions or are set on inflow. Let's set for
definiteness, then temperature we shall express from the equation of state:),( 00 p
0
00
p
RT
g
Values with index "0" refer to border (node of accounting cell), "R" (right) - to center of the first accounting cell, "L" (left) - to center of the fictitious cell. Let's find values at the center of the fictitious cell. Linear extrapolation of density, impulse and pressure is used for this purpose.
RL 02
1)()(2)( RRL uuu
0
20
2
)(
1
2
LR
L
uppE
Accordingly, flows on boundary edge will be written down as:
001 uH
2000
2 upH
21
200
003 u
puH
The value of mass velocity is necessary for finding flows.
0 1 2 3 4 5 6 7 8 9 10
x 104
487.1506
487.1507
487.1508
487.1509
487.151
487.1511
487.1512
487.1513Ro * U
1 variant: linear extrapolation of mass velocity
1
10
)(5.0
)(5.1
R
R
R
R uuu
Small oscillation arise at such assignment order %, as it is visible from the dependence of stationary flow via coordinate, given in figure.
0u410
)( u
2 variant: linear extrapolation of impulse
0
10 2
)()(3
RR uuu
In this case flow is monotone at the left border.
0 1 2 3 4 5 6 7 8 9 10
x 104
480
482
484
486
488
490
492Ro * U, Time: 12979.166667
2. Outflow condition.Only pressure is known on outflow.0p
Values with index "0" refer to border (node of accounting cell), "L" (left) - to center of the last accounting cell, "R" (right) - to center of the fictitious cell. Let's carry out linear extrapolation of density and impulse. If we set boundary conditions and flows to
10 5.05.1 LL
10 )(5.0)(5.1)( LL uuu
01 )( uH
0
20
02 )(
u
pH
0
20
00
03
2
)(
1
)(
u
pu
H
then the large entropy trace arises on the right border (up to 1-2%), which scale does not depend on number of grid intervals.
1. Therefore we have tried to find additional difference boundary conditions being a consequence of the basic boundary condition and equations (1). For this purpose the first equation of system (1) multiply on “u” and subtract from second equation.
uuDdz
dhg
z
p
z
uu
t
u 2
Received equation multiply on , combine with the first equation (1), multiplied on and subtract the third equation. In result we have,
u
22u
2
2uu
Ddz
dhgu
z
up
z
u
t
Subtract last equation from the first equation (1) multiplied on :
2
2uu
Ddz
dhgu
z
up
zu
t
Multiply this equation by and combine with the first equation (1) multiplied by :
p
p
22
2uu
Ddz
dhgu
p
z
uc
z
pu
t
p
, where c is a sound speed
The equation below is received using the equation of state of ideal gas
2
2)1())(1( uu
Ddz
dhgu
z
up
z
pu
t
p
Last equation write down in difference form and take into account, that at ,
follow , then
Lz 0
t
p
2
2)1())(1( uu
Ddz
dhgu
z
up
z
pu
(2)
The law of entropy conservation is used as third difference boundary condition:
02
t
p
tt
sT
The law of entropy conservation in finite differences has the following form:
uuDdz
dhg
z
p
z 22
3. Entropy equation
(3)
Using (2) and (3), one can obtain a system of equations with respect to and 0 0u
08
1
022
1
1
1
822
200
200000
000
2000
302
uuuuD
uuppuupp
ppuuuu
D
pp
LLLLLLL
LLLL
L
L
L
L
Using the found values and , it is possible to find the other unknowns0 0u
21
2000
0
upE
LR uuu 02
LR 02
RRR uu )(
LR EEE 02
001 uH
2000
2 upH
21
200
003 u
puH
Error in temperature at the right boundary decreases with the increasing of grid points number.
Numerical resultsThe test task given in the report [1], we solve for evaluation of the numerical technique. Gas compressibility, heat exchange with walls, friction on internal surface of a pipe, viscosity essentially influence gas dynamics in these tasks.
Stationary gas flow in heat-insulated pipeline100N - number of difference intervals. Calculations were carried out in interval Lz ,0
F ix e d v a lu e s : 1D , 510L , 273eT , 210 .
B o u n d a r y c o n d i t io n s : 6106 Leftp , 273LeftT , 6105.1 Rightp .
In i t ia l c o n d i t io n s : l in e a r in te rp o la t io n b e tw e e n th e f o l lo w in g v a lu e s o n b o u n d a r y
6106 Leftp a n d 6105.1 Rightp ; 8.42Left a n d 4/LeftRight ;
5.11Leftu a n d LeftRight uu 4 . T e m p e ra tu r e i s c o n s ta n t : 273oT .
The stationary task 1 was solved by a method of establishment flow under given boundary conditions.
Heat exchange with an environment is completely absent in this task: . Calculation results in comparison by exact solutions are given in Fig. 1. Good consent is observed with exact solutions.
0
0 1 2 3 4 5 6 7 8 9 10
x 104
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6x 10
6
p
z0 1 2 3 4 5 6 7 8 9 10
x 104
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
T-2
73
z
0 1 2 3 4 5 6 7 8 9 10
x 104
10
15
20
25
30
35
40
45
50
u
z0 1 2 3 4 5 6 7 8 9 10
x 104
487.12
487.122
487.124
487.126
487.128
487.13
487.132
487.134
487.136
487.138
487.14
Ro
* u
z
Pressure Temperature
Velocity Mass flow
Fig. 1. Stationary gas flow distribution of pressure, temperature, velocity andmass flow in in heat-insulated pipeline. Solid line – numerical results,stars – exact solutions.
In this task the convergence of numerical solution to analytical was analyzed.
101
102
103
10-3
10-2
10-1
100
Con
verg
ence
Amount of spatial cells is plotted on the blue diagram on X-axis in logarithmic scale, logarithm of norm of a difference numerical solutions and analytical – on Y-axis.
The green colour plots the cubic law: .3
30
0 x
xyy
It is possible to make a conclusion that the solution converges to analytical with the third order depending on amount of spatial cells.
Literature
1. Anuchin M.G., Dremov V.V. Model for calculate flow of natural gas on gas pipeline. RFNC-VNIITF report, № ПС.99.7432, Snezhinsk, 1999.2. Kurganov A., Tadmor E. High-resolution central schemes. UCLA Report №16, 1999.
3D Elastic-Plastic Modeling of Processes of Ceramics
Formation Problem Statement Material Model (by Bychenkov) Lagrange Irregular Voronoi Grids Central Difference Scheme for Irregular
Grids in 2D and 3D Cases
Problem Statement
Model of Elastic-Plastic Deformations I
SSlS
x
gSugpt
gE
gSgpt
gu
t
g
s
iTi
ii
ii
ii
2
u
)(u
)()(
0
Model of Elastic-Plastic Deformations II
x
u
y
u
SSS
SS
SS
yx
xxyyxy
xyyy
xyxx
)(2
2
2
Model of Elastic-Plastic Deformations III
PK
J
c
P
PlsE
P
tcP
psp
p
pijij
2
0
1
2
2
The additional modes of the model
Porosity accounting Material crash accounting Welding of the material accounting
The KT Difference Scheme for Lagrangian Gas Dynamics
ux
u
t
g
upug
p
t
ugt
g
0
0))2
((
0
0
2
Difference scheme for 1D Lagrangian Gas Dynamics I
0)(1
0)(1
0)(1
0
12/1
2/12/1
2/12/1
gk
gk
k
Ek
Ek
k
Ik
Ik
k
k
FFht
g
FFht
E
FFht
It
m
Difference scheme for 1D Lagrangian Gas Dynamics II
)(5.0))()((5.0
)(5.0))()((5.0
)(5.0)(5.0
2/12/1111
2/12/12/12/12/1
2/12/12/12/12/1
kkkkg
k
kkkkI
k
kkkkI
k
ggcuuF
EEcupupF
IIcppF
Difference scheme for 1D Lagrangian Gas Dynamics III
2/12/1
2/12/1
2/1
112/1
2/1
112/1
2
2
2
2
2
2
kkk
kkk
kkk
kkk
kkk
kkk
Dgh
gg
Dgh
gg
DEh
EE
DEh
EE
DIh
II
DIh
II
Difference scheme for 1D Lagrangian Gas Dynamics IV
),mod(min
),mod(min
),mod(min
2/12/12/12/32/1
11
11
h
gg
h
ggDg
h
EE
h
EEDE
h
II
h
IIDI
kkkkk
kkkkk
kkkkk
Difference scheme for multidimensional irregular grid
The multidimensional Voronoi Grid is used To obtain scheme for multi-dimensional
case one should reformulate original 1D KT scheme to make generalization more obvious (see the next 2 slides)
Difference scheme for 1D Lagrangian Gas Dynamics (Variation for generalization for
multidimensional irregular grid)
))(2
1(
2
)(2
))(2
1(
2
)(2
))()((5.0
))(2
1(
2
)(2
)(5.0
2/12/32/12/3
2/12/311
11
112/1
11
112/1
kkkk
kkkg
k
kkkk
kkkkE
k
kkkk
kkkkI
k
DgDgggc
DuDuh
uF
DEDEEEc
DpuDpuh
upupF
DIDIIIc
DpDph
ppF
Difference scheme for 1D Lagrangian Gas Dynamics (Variation for generalization for
multidimensional irregular grid)
)))(,)mod((min))(,)mod(((min2
1)(
)2()(
))()((2
)(2
112
11
22/12/1
kkkkk
kkkk
kkk
kkkk
xxxxxD
xxxx
xDxg
cDuDu
hu
t
x
A test problem for the lagrangian difference scheme
In the next five slides the results of a test problem numerical modeling are presented
The initial conditions are: the first region – density 4, specific internal energy – 0.5, velocity – 1; the second region – density – 1, specific internal energy – 1e-6, velocity – 0.
Boundary conditions: the left boundary – piston with velocity 1, the right one – rigid wall.
The plots present density and velocity for 10 successive times.
density
velocity
density
velocity
density
velocity
density
velocity
density
velocity
Numerical results discussion
The numerical results show, that the scheme has inherited its good quality from the Euler parent KT scheme
The smoothness is minimal (even after more, than 10 shock reflections the number of point at the shock is not more then five)
Some entropy traces are presented (at the boundaries and at the place, where initially the shock has been located)
Explanation of some details of the multidimensional scheme, presented on the
next slide. is a set of numbers of cells, which are
neighbor cells to the k-th one DI, DE are the results of minmod procedure
applied to the gradients (in common case six ones), calculated in the Delauneux triangles, including k-th point
Dr is a result of minmod operation applied to the corresponding laplacians, calculated in the neighbor points
k
Lagrangian KT Scheme for Irregular Grid
k k
k k k j jkj
j k j k jkj
k k k j jkj
j k j k jkj
k k
j k
g
t
tI g p p n
cI I DI DI n
tE g pu pu n
cE E DE DE n
tr u
gr Dr
rc
r r
k k
k k
0
1
2 2
1
2
1
2 2
1
2
1
2
( ) ( ) ( ( ))
( ) (( ) ( ) ) ( ( ))
( )
( )
n jkj k
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