Stelmashchuk V. Analysis of generalized thermopiezoelectricity problems and projection-mesh schemes for their solution

The thesis is devoted to investigation and development of numerical FEM-based schemes for problems of classical and generalized thermopiezoelectricity. In the introduction the relevance of the thesis topic is substantiated, the scientific novelty and practical value of the work are defined, the subject and the goal of the research are formulated. Besides, the list of the applicant`s publications on the thesis topic is given here. In Chapter 1 the problem of the interaction of mechanical, electrical and heat fields in pyroelectric materials is considered. Firstly, the classical mathematical model of thermopiezoelectricity is discussed, and its drawbacks are pointed out. Then the generalized models of thermopiezoelectricity, namely Lord-Shulman and Green-Lindsay theories, are described. The corresponding initial boundary value problems of these models of thermopiezoelectricity are formulated. Various methods of solving such kind of problems are discussed, in particular, the finite element method (FEM). Chapter 2 is dedicated to the numerical solution of the problem of forced harmonic vibrations of pyroelectric materials in case of the classical thermopiezoelectricity models. The existing numerical schemes have been verified by a set of numerical experiments and their analysis. Besides, a special h-adaptive FEM scheme has been proposed for solving such kind of problems. Chapter 3 is devoted to the numerical solution of non-stationary problems of classical thermopiezoelectricity. The existing numerical schemes have been verified by a set of numerical experiments. The obtained numerical results have been compared to the solutions of piezoelectricity problem and are in agreement with the solutions of the classical thermopiezoelectricity problem provided by other researchers. In Chapter 4 the Lord-Shulman thermopiezoelectricity problems are considered. For forced harmonic vibrations of pyroelectrics under the Lord-Shulman model the well-posedness of the corresponding variational problem has been proved. Based on FEM, the numerical scheme for its solution has been constructed. The analysis of this scheme robustness and convergence has been performed. For non-stationary Lord-Shulman thermopiezoelectricity problem the well-posedness of the corresponding variational problems has been proved too. Based on FEM and one step recurrent scheme, the numerical scheme for its solution has been constructed. A set of numerical experiments has been carried out and the obtained results are in agreement with the ones obtained by other researchers using different solving methodologies. In Chapter 5 the Green-Lindsay thermopiezoelectricity problems are considered. Like in case of Lord-Shulman model, the well-posedness of the corresponding variational problems of forced harmonic vibrations and the non-stationary one has been proved. Numerical schemes, based on FEM and one step recurrent scheme, for solving such kind of problems have been constructed. The robustness and convergence of the numerical scheme for Green-Lindsay forced harmonic vibrations problem have been proved. A set of numerical experiments has been performed. The basic provisions and the results of the theoretical research have been confirmed by the results of the numerical experiments, which were carried out using self-developed software.