HYBRID NUMERICAL METHOD FOR NON-STATIONARY CONTINUUM MECHANICS N.G. Burago 1,3, I.S. Nikitin 2,3 1 IPMech RAS, 2 ICAD RAS, 3 Bauman MSTU Volodarka

Under consideration: Problems of Continuum Mechanics in Moving Regions of Complex Geometry Advantage of three method components: 1. Moving Adaptive Grids 2. Equilibrated Viscosity Scheme 3. Overlapping Grids

Types of Grid Adaption 1) Region boundaries of complex shape 2) Minimization of approximation errors ( min |hdy/dx| )

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

Grid is treated as isotropic thermo-elastic medium :

Governing variational equation Cell shape control parameter Cell compressibility parameter Thermal expantion (monitor function, “anti- temperature”)

Method for grid equation - stabilization method Algorithm: two-layered explicit scheme of stabilization which provides equilibrium (by norm) of members with L and f Details: ipmnet.ru/~burago

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

Уравнения для задач механики жидкости и газа Уравнения для твердых деформируемых сред аналогичны Variational Equations for Fluid Dynamics Equations for Solid Mechanics are analogical

Simplified explicit scheme SUPG FEM 1 (variant of stabilized Petrov-Galerkin method) иначе то если 1 Brooks 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 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

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

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 function

Adaptive grids for supersonic flows in channels with obstacles

maintenance of uniform distribution of nodes 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 holeMain 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:

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