Yashchuk Y. Computer modeling of elasticity problems based on new h-adaptive scheme

The thesis is concerned with the development and efficiency investigations of error estimation methodology in finite element method for elasticity problems, based on comparison of finite and boundary element methods results. We proved that under certain conditions the difference of stresses obtained with finite and boundary element methods estimates the real error of finite element stresses. We constructed an h-adaptive scheme using this estimation. The scheme does not sustain the conformity of finite element mesh. For solving problems with nonconforming meshes we use the mortar element method. Therefore the continuity of the displacements is not preserved and weak continuity is imposed using the Lagrange multipliers. As a part of the h-adaptive scheme we present the methodology of construction of boundary element mesh as a trace of finite element one. The possibility of application of these methodologies is proved with corresponding lemmas and theorems. A couple of numerical experiments approve the correctness of the estimator. Considering the Lame problem (thick-walled pipe under pressure), for which the analytical solution is known, we found that the efficiency index of the estimator is close to 1, i. e. the efficiency is high. Also the L-form deformation problem was investigated. The numerical results show that the adaptive scheme effectively reveals singularity near the inner corner and other stress concentrators in the object. The implementation of this approach in elastic body contact problems is presented. We developed a combined iterative adaptive algorithm for solving this problem. The problem is transformed into an iterative process using domain decomposition algorithm. The nonpenetration condition is applied with penalty method. We solve the problem using finite element method. Each iteration is accompanied with one step of adaptation of finite element mesh according to the h-adaptive scheme we presented. The numerical experiments show that this modification greatly decreases the complexity of the discretized problem. The adapted mesh explicitly reveals the singularity near the contact area. The new h-adaptive scheme, presented in this thesis, allows saving computing resources while solving elasticity problems, reveals singularities in solutions, is suitable for parallel computations and can be adapted to other types of problems in mathematical physics.