N.G. Bourago (Burago) : code ASTRA

Code ASTRA (Automated System for Theoretical Research and Analysis)

1. What is ASTRA?

The code ASTRA provides numerical solution of 1,2,3D unsteady nonlinear problems in continuum mechanics using moving adaptive Eulerian-Lagrangian unstructured grids and the finite element method. It is not universal, but rather general (see Demo ) and flexible.

ASTRA simulates nonlinear processes

• in solid mechanics
• in fluid and gas dynamics
• in shallow water theory
• in filtration and diffusion theory.

Following phenomenas are taken into account:

• compressibility as well as incompressibility,
• convection forced, natural, solutal, Marangoni et cetera,
• momentum, heat, mass and dopant transfer,
• shock waves, boundary layers, moving and free boundaries,
• contact interactions,
• large deformations,
• complex rheology of materials (nonlinear elastic, plastic and viscous properties),
• strain/temperature history influence,
• fatique, damage and vice versa, sintering and consolidation,
• filtration of a contaminant(s) in deformable porous media,
• elastic plastic damagable rock media,
• localization of strains,
• gravity and microgravity effects and,
• many other effects.

By using ASTRA it is possible to solve

• dynamical (unsteady, fast),
• quasi-statical (steady, slow)
• strain-stress (solids) and,
• convection - diffusion (fluids, gases) processes.

2. What is the mathematical basis of ASTRA?

Code ASTRA is based upon the special version [1] of the finite element method. The major features are

• no matrix operations,
• FEM style remake of good finite difference techniques by using Bubnov-Galerkin finite element approximations,
• a preference is given to more simple and robust methods,
• identical numerical algorithms

3. Physical basement of ASTRA

All equations are formulated in actual configuration settings, which are common for solid and fluid mechanics, and the arbitrary moving adaptive coordinates and adaptive unstructured grids are used. The Eulerian, Lagrangian or mixed descriptions of motion for various parts of the solution region are used simultaneously in accordance with physical statement of particular problem. For proper description of continuum motion in case of multiple moving and free boundaries the mathematical techniques of continuous and discrete markers are implemented.

While the code ASTRA was in progress (since 1976) the appropriate constitutive equations were developed on the basis of thermodynamical laws for irreversible nonequilibrated processes. So the set of new constitutive equations is used to get an adequate description of a variety of processes such as sintering, damage or/and softening of materials. Due to convenient arrangement and flexibility of the code ASTRA it is easy to quickly make multiple checkups of many strange and even "mad" physical ideas, arised in attempts to simulate various mechanical phenomenas.

4. Numerical methods.

Quasi-second order explicit and implicit finite difference schemes in time are implemented. It is important to remark that the major line of ASTRA which states that the preference always should be given to most simple, identical and robust algorithms has the very pleasant outcome: algorithms for very different areas in continuum mechanics such as solid mechanics, incompressible fluid and compressible gas dynamics, theory of shallow water, filtration and diffusion problems appeared finally to be almost the same.

All implicit schemes use the conjugate gradient non-matrix method with preconditioning using the diagonal approximation of stiffness matrix.

It:

• provides extremely fast solution of the auxiliary boundary value problems for each time step,
• significantly saves computer memory because there is no need to condense and conserve stiffness matrices,
• makes the programming process to be easy and,
• gives to the creator the freedom to take care only about the physical issues of particular problem.

Each conjugate gradient iteration is realized like a routine time step in explicit methods. Finite and not big number of such iterations is required to reach high accuracy of the solution. The conservative Lagrange multiplier and penalty techniques are used to calculate contact interactions in solid mechanics and to satisfy the incompressibility condition for incompressible solid and liquid medii.

Nonlinear artificial viscosity terms are added into equations to provide the stability and the accuracy of the algorithms. The analysis of Hirt-Shokin approximations of finite element schemes and exponential correction technique lay in the basement of the choice of such terms. In addition moving adaptive grids also serve to decrease approximation errors.

All algorithms contain mainly arithmetical operations with vectors of high dimension, which relates to the number of unknowns. Therefore the code ASTRA was easily vectorized more then decade ago.

All algorithms have on-line accuracy control.

5. Why ASTRA is a good code?

In contrast to many application codes the code ASTRA:

• WORKS;
• works FAST;
• works WITHOUT FATAL ERRORS such as: "...the error is occured... you must restart your computer...";
• GIVES A RESULT anyway,
• COVERS A WIDE (outstandingly) PROBLEM AREA IN a frame of CONTINUUM MECHANICS.

ALSO code ASTRA has following important features:

• On-line preprocessors for generation of unstructured finite element grids in two and three dimensional regions. Sure these algorithms are not universal, but nevertheless they allow to feel free in respect of geometrical conditions of many practical applications.
• On-line editor for preparation of the geometrical and physical starting data. It uses free format rules (NAMELIST style). It has on-line help for keywords and on-line diagnostics of errors.
• On-line symbolic debugger is permanently available to provide the on-line detailed analysis and management of code data, breakpoints, dumping, graphical processing and restart.
• An in-built post-processor provides on-line visualization and monitoring of the results in form of tables, graphs, contour lines, color zones. With code ASTRA it is easy to produce movies (it has an in-built animator) and to get hard copies of pictures in PCL, HPGL or PostScript formats.
• The possibility of modification and replacement of the code components is supplied. The code is very flexible.

6. Computer platforms and basic software.

Code ASTRA is known since 1977 (by name ASTRA since 1983). It works (worked) on the following computers:

• BESM-6 (Computing Center AS USSR, 1976-1983)
• IBM 360/370 (IPMech AS USSR), 1984-1989)
• VAX 750 (IPMech AS USSR, 1990)
• IBM PC: 486 or higher: under MS DOS/MS WINDOWS/LINUX environment. (IPMech RAS, 1991-now)
• Work stations HP, SUN, IBM, Silicon Graphics: under X-WINDOWS (1993-now in: Chalmers Univ. of Technol.; Moscow State Univ.; SRL of Ford Motor Co.; Free Brussel Univ. (ULB))

Main ASTRA language is FORTRAN-77. Some other languages (C and Assembler) are used only for low level subroutines to support hardware interface. Any 32-bit compiler can be used to make the code ASTRA alive.

The code ASTRA is not commercial. All rights reserved by the creator N.G.Bourago.

7. References

Bourago N.G., Kukudzhanov V.N. "Solution of elasto-plastic problems by finite element method. HydroCode ASTRA", Preprint of the Institute for Problems in Mechanics of AS USSR, n.326, 1988,pp. 1-63

Bourago N.G., Kukudzhanov V.N. "Solution of elasto-plastic problems by finite element method. HydroCode ASTRA", in book: Computer Mechanics of Solids, vol. 2, Nauka, 1991, pp.78-122

Bourago N., Computer code "ASTRA" for nonlinear problems in continuum mechanics, Proc. of NORDIC-7 seminar, Trondheim, 1994 (Short description in English)

Complete list of references can be found on my WebPage: http://www.ipmnet.ru/~burago

Contact information:
Email: burago@ipmnet.ru Web: http://www.ipmnet.ru/~burago Tel: +7 (495) 434-41-35 Fax: +7 (495) 739-95-31
Mail address: Institute for Problems in Mechanics of RAS, prospect Vernadskogo 101, b1, Moscow, 119526, Russia