Institute of Computational Mathematics and Mathematical Geophysics SB RAS

Abstracts

High-Performance Computing, Parallel Methods and Algorithms in Geophysical Problems

Processes of the propagation of weak shock waves are analyzed within the framework of nonlinear model of granular material, possessing elastic and plastic properties, with different resistances to tension and compression. Mathematical model uses the Mises-Schleiher strength criterion to describe the effect of loosening of granular material and the Mises yield criterion to determine the transition from elastic state to plastic one.

Mathematical model of the Cosserat continuum, taking into account the microstructure of material, is applied to the analysis of wave motions, too. Within the framework of this model the independent rotations of particles of microstructure are considered together with translational motion of medium, asymmetrical stress tensor and moment stress tensor are applied for the description of stressed state. The symmetry conditions are formulated, which allow to reduce the volume of computations many times. In view of linearity the problem in general formulation is reduced to four subproblems, the solutions of which form fundamental system of solutions. There are two traditional problems, which settings are known from linear (momentless) elasticity theory, about the propagation of stress waves, caused by the action of normal and tangential loads. And two non-standard problems, which are characteristic for moment medium, about the propagation of torsional waves and waves of rotational motion of particles due to the action of concentrated torsional moment and moment of the rotation force at the boundary.

Efficient algorithm of numerical realization of models is worked out on the basis of a method of splitting onto physical processes and onto spatial variables. In this algorithm one-dimensional hyperbolic systems of equations are solved by means of an explicit monotone ENO-scheme, and special correction of stresses is used to take into account the plasticity. Numerical solution of three-dimensional problems with the help of this algorithm at uniprocessor computer is restricted on the volume of main memory with a moderate dimension of grid. The accuracy of received results are not always satisfactory, especially if computational domain has complex structure - contains a lot of inside interfaces of materials with essentially different mechanical properties, rigid inclusions of small size, etc. The parallel version of proposed algorithm is worked out. This algorithm is realized as a parallel program system for numerical solution of plane and spatial problems of the dynamics on multiprocessor computers. Program system allows simulating the propagation of stress waves and strain waves produced by mechanical impacts in a mass of medium, aggregated of arbitrary number of heterogeneous blocks with curvilinear boundaries. It may be applied for the solution of direct seismic problems taking into account complicated mechanical properties of geomaterials. Vectors of displacement, velocity and stress are supposed to be continuous at internal interfaces of blocks in agreement to the conditions of rigid connection. The boundary conditions in terms of velocities as far as in stresses, and also the unreflecting boundary conditions are allowed at external boundaries of blocks.

The programming is fulfilled by means of the SPMD technology in Fortran-95 with the use of MPI library. Suggested program system consists of the pre-processor program, the main program of velocities and stresses computation, the subroutines for the realization of boundary conditions and pasting together conditions in interior boundaries, and the post-processor program. The universality of programs is achieved by means of special packing of variables, used on each computational node of cluster, into one-dimensional arrays of high dimensionality. Parallelization of computations is carried out at the stage of a problem splitting with respect to spatial variables. The pre-processor is intended for the initial data preparation in the form of binary files and for their uniform distribution between processors on principle of uniform load. It also constructs curvilinear differential grids in blocks. The uniformity of load of processors is provided with subroutines, realizing the algorithms of automatic 1D, 2D or 3D decomposition of domain. The main program makes computations consisting of mutually coordinated step-by-step realization of the space-variable splitting method with the subsequent solution correction on each time step. Data interchange between processes is realized by means of the function send-receive under the solution of one-dimensional systems of equations, standard technology of the contour meshes is used. Under numerical realization of the pasting together conditions the solution at refinement grid, obtained by the intersection of grids on sides of interface, is defined with the help of equations on characteristics, and then this solution is carried onto initial grids by an averaging method. The post-processor performs special compression of files, containing the results of computations in breakpoints for their following transportation along the global network. Graphical representation of results is realized with the help of special programs, meant for personal computer. The program for representation of the results of computations of wave problems, received on clusters, in SEG-Y format is worked out with the purpose of further processing of data in the system SeisView.

The verification of main program was fulfilled on one-dimensional tests - exact solutions of problems with plane elastic and plastic waves. Numerical computations of the spatial interaction of signotons (shock waves of small amplitude, on fronts of which the sign of strain is changed) in heterogeneously loosened granular material with a cumulative splash formation were carried out by means of described program system at the cluster MVS-1000M of Institute of Computational Modeling of SB RAS (Krasnoyarsk). Computations of the waves propagation into layered granular medium under action of the periodic perturbation source on one of its boundaries were fulfilled. At the cluster MVS-15000BM of Joint Supercomputer Center of RAS (Moscow) the Lamb problem about the action of normal concentrated impulsive load onto the boundary of half-space, consisting of two heterogeneous elastic layers (compact ground and strong rock), separated by interface in the form of hyperbolic paraboloid, was solved. Three-dimensional problems about the action of concentrated impulsive load on the normal to the surface of elastic medium with rigid inclusion and about the action of load under the angle to the surface of homogeneous elastic half-space for moment medium were solved numerically, too. The level surfaces of velocities and stresses, seismograms of the particles displacement, on which one can see incident longitudinal and transverse waves, conical, reflected and refracted longitudinal waves, surface Rayleigh waves, were constructed on results of computations. On seismograms of the particles rotation angles in moment medium one can see four types of waves, which are typical for spatial model, and also the oscillations, which size depends on the size of particles of microstructure.

Results of computations have shown that multiprocessor computer systems possess of essential advantages in comparison with personal computers, if number of used processors is selected in reasonable accordance with the dimensionality of a spatial grid. Besides, parallel computational technologies are unique facility of the analysis of solutions with acceptable accuracy for most of three-dimensional problems.

This work was supported by the Russian Foundation for Basic Research (project no. 08-01-00148) and the Complex Program of Basic Research of the Presidium of RAS no. 14 “Fundamental Problems of Informatics and Informational Technologies”.

Note. Abstracts are published in author's edition

Mail to Webmaster

[Home] [Conferences]

© 1996-2008, Siberian Branch of Russian Academy of Sciences, Novosibirsk
    Last update: 06-Jul-2012 (11:52:47)