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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
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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 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 O.V.Diyankov, I.V.Glazyrin, S.V.Koshelev, I.V.Krasnogorov, A.N.Slesareva, O.G.Kotova Russian Federal Nuclear Center - VNIITF P.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.
The MAG Code Model: The system of equations used in MAG code is determined by Braginskii model for one-temperature case: r
Details of Numeric: 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.
Boundary conditions: 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: 2. Conditions for heat conductivity: 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: 3. Conditions for magnetic field • 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 Bis used for the axial symmetry and Bz for the plane one; • given current: Bz=2p j /c – plane symmetry, B= 2p I /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: • 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: Equation of continuity: Euler equations for velocity components: x component: y component: z component:
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.
r = const. - equation of continuity, g = const. - equation for determinant of metric tensor, u = const. - Euler equation for velocity. System of equations in arbitrary moving coordinates III: Equation for A: Equation for Bz: Equation for energy:
Diffusive equations system: Difference sheme for diffusive equations system:
Z-pinch simulation Experimental Setup Simulation 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.
Hollow gas puff simulation: Magnetic field, -component [10 MGs] Density [g/cc]
Pinhole image reconstruction throw steady equation of radiation transfer resolving 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 (II)
Laser produced plasma jet expansion into vacuum Temperature evolution, time = 0.1, 0.3, 0.5, 0.8 ns. Energy of laser pulse is 4 kJ/cm2. Triangle pulse with duration time of 1 ns. Focal spot 50 mm.
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 mm. Background plasma density was 10-6 g/cm3
Plasma Channel Formation. Temperature Evolution. 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
IGM Code Oleg V. Diyankov Sergei S. Kotegov Vladislav Yu. Pravilnikov Yuri Yu. Kuznetsov Aleksey A. Nadolskiy RFNC – ARITPh supported by LLNL grant B329117
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
Equation of heat conductivity difference scheme: boundary condition:
Poisson equation Integrate over Voronoi cell where
Electric field strength where
High velocity impact problem (the angle equals to 90 degrees)
High velocity impact problem (the angle equals to 45 degrees)
Poisson problemsf(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 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.
