Scientific Objectives

The main scientific objective in our research part of the Project was to develop 3D modelling of the Strain-stress State (SSS) in pneumatic tires. First, numerical 3D tire modelling puts interesting theoretical problems, especially for the Computational Mechanics. It is a hard test for an accuracy of Finite Element approximation because of rubber small compressibility and for algebraic system solution methods. One of the project objectives was to investigate practical efficiency of some modern iterative methods, including multigrid techniques and domain decomposition methods as well as methods for small compressible materials. Secondly, 3D modelling is interesting in the practical sense. By now, computational techniques based on simple tire models are spread widely in Russian tire industry. Hence, the project research initiated because of INTAS financial support contributes in setting up modern computational techniques in tire design. Thus, next objective was to develop programs package for evaluation of SSS in tires. In 3D model tire is considered as rubber-cord composite and the model is based on the equations of non-linear Elasticity. Such approach gets the increasing sense due to the recent development in microprocessors. This is especially true for PC's that dominate in Russia and are used for scientific computations.

The significance of 3D modelling seems to be growing up as a sequence of the multiprocessor systems developing. Special software like PVM or MPI is used worldwide to compute on network in parallel. One more objective was to set up parallel computations on PCs network available in Faculty of Mechanics and Mathematics, Lomonosov Moscow State University.

 Research Activities

During the work under the project, research was conducted in both theoretical and practical aspects. Theoretical research aimed to develop mechanical model of rubber-cord composite. The model developed is based on equations of non-linear Elasticity and asymptotic averaging method that was used to calculate effective elastic modulae of rubber-cord composite. Finite Element Method (FEM) was used for boundary value problem discretisation. The boundary-value problems that were considered are listed below:
 

• Static loading by pressure (2D problem).
• SSS under the loading by pressure and vertical force (3D problem).
• Loading by centrifugal forces and forces of friction while braking (3D problem).

Computer code was written in FORTRAN to implement 3D ire model on MS Windows workstations. Many efforts were made to debug the programs. It is important to emphasise that only INTAS grant support allowed maintaining that work. The debugging process is completed now. In part, the results of computations were compared with ANSYS program package. Satisfactory coincidence of the results was observed. Negotiations on the usage of the developed program package in NIISHP (Research Institute of Tire Industry, Moscow) were started.

Theoretical research and computer experiments were used to create effective Solver for bad conditioned systems of linear equations. The problem is that rubber-cord composite is complex system because of its internal heterogeneous structure. In addition, the complexity is caused by small thickness of a tire, large distortions of geometry and small compressibility of rubber and high elastic modulae of rubber-cord carcass. All that means that the FEM matrix condition number is high. It does not make significant difficulty in the case of two-dimensional problem (axis symmetrical stress State) but causes hard problem in the case of 3D formulation. In 2D case, direct solver is quite good.

Contrary in 3D case, direct solver requires too much computer memory so the fundamental problem is to develop fast iterative Solver for 3D FEM equations. Modern iterative techniques like PCG (Preconditioned Conjugate Gradients) method, Domain Decomposition method, MG (Multi Grid) method were compared in practice. Combined Chebyshev-Gradient method was suggested as alternate to PCG. Research on efficiency of iterative methods was conducted at the department of Mechanics of Composites about ten years ago. Now financial INTAS support has allowed renewing that research.

Special iterative methods were developed for materials with small compressibility. They were studied theoretically and practically. Unfortunately, it was revealed that they have significant advantage not for all boundary condition types.

Deformation of thin plate was computed using Kirchhoff - Love and Timoshenko - Reissner shell models. These solutions were compared with the solution based on 3D model. Thus the program of shell theories verification was started. That verification process has not completely fulfilled by now. Hence, there is deviation from initial scientific program that assumed to verify shell theories for layered materials. That deviation was caused the delay in computer code debugging. Now verification process is continuing.

Investigation of the efficiency of parallel computations on PCs network was set up. The parallel version of Domain decomposition technique was developed and implemented on the Department local network.

The results received during work under the project were reported at 8^th, 9^th and 10^th All-Russian symposiums ⌠The Problems of Tires and Rubber-Cord Composites■, that held in NIISHP Institute of Tire Industry, Moscow, in 1997, 98, 99. The reports were published in proceedings of the symposiums. Two students of the Mechanics of Composites Department (Faculty of Mechanics and Mathematics, Moscow State University) defended Diploma thesis's fulfilled in accordance with project scientific program in 1998, 1999. Graduate student S. A. Margarjan prepared Candidate thesis. Special seminar on tire Mechanics held at the Department in 1998, 1999.

Scientific Results

The step-by-step method for solution of non-linear boundary problems of Elasticity for heterogeneous medium was developed and implemented as software for Win32 platform. The developed method uses non-linear Elasticity boundary problem formulation in initial configuration, i.e. the configuration before loading. That means that both partial derivatives equations of equilibrium and boundary conditions are set up on the same domain and the same boundary correspondingly at each step.

Asymptotic multi-scale technique was applied for solution of linear boundary problem of Elasticity at each step. The technique gives the capability to obtain the effective elastic module of rubber-cord composite as well as stresses and strains fluctuations in layers of rubber-cord composite.

Computer programs were fully debugged for 2D and 3D cases. Test computations were made to investigate accuracy of the developed technique.

The mesh generation algorithm is adopted and the computer program is developed to generate regular grids in domains, typical for tires meridian section. Three-dimensional grid in the domain occupied by the tire before loading can be easily obtained from the grid in meridian section because of axial symmetry of the initial domain.

The method for solution of linear boundary problem at each step in couple with asymptotic technique was realized as computer program. Computations were made to evaluate the accuracy gained for some tire structure. The direct Holesky method is used for linear algebraic system solution in 2D case. It is rather effective since number of knots in the direction across the rubber-cord layers is small enough. The direct methods in 3D case implemented on low and middle power computers are not effective because the matrix band is too wide. The Combined Gradient-Chebyshev Iterative method is proposed for more effective solution of linear systems.

The research on efficiency of some modern iterative methods was conducted. The 3D boundary problem for rubber-cord material is difficult test for iterative method. The project research team has a background in this field gained earlier with Elasticoplastic problems. Therefore, INTAS financial support allowed renewing that interesting investigation. Practical comparison of modern iterative processes was fulfilled. They are Combined Gradient-Chebyshev, Multi-grid, Conjugate Gradient and Domain Decomposition methods. Special iterative method was developed for improving iterations rate in the case of small compressible material. It works for boundary conditions of Dirichlet type. Efficiency of that method was justified theoretically and verified practically.

Verification of Kirchhoff √ Love and Timoshenko √ Reissner shell theories was fulfilled. It was shown that Timoshenko √ Reissner theory gives the results that are close to the results obtained by means of 3D theory.

The research on efficiency of parallel implementation of Domain Decomposition method on PC▓s network was done. The result of that investigation is that efficient usage of iterative Domain Decomposition method can be only when the number of the sub domains is equal to the number of processors working in parallel.

The results of research fulfilled were published in:

B. E. Pobedria, S. V. Sheshenin, Three-dimensional Modelling of Stress-Strain State of Pneumatic Tires.  In The Problems of Tires and Rubber-Cord Composites, vol. 2, 1997, p.p. 320-326.

B. E. Pobedria, S. V. Sheshenin, S. A. Margarjan, M. Yu. Melnikov, Three-dimensional Modelling of Stress-Strain State of Pneumatic Tires, Part 2. In  The Problems of Tires and Rubber-Cord Composites, vol. 2, 1998, p.p. 290-295.

S. V. Sheshenin, S. A. Margarjan, Iterative solution method for 3D Model of SSS of pneumatic Tires. In The Problems of Tires and Rubber-Cord Composites, 1999, p.p. 272-277.

B. E. Pobedria, S. V. Sheshenin, S. A. Margarjan, Numerical 3D modelling of the rubber √ cord composites. (Submitted in Izv. RAS, Mechanics of Solids).

S. V.  Sheshenin, On Iterative Methods for solution some boundary value problems of Solids, Izv. RAS, MTT, 1997, #2, p.p. 21-26