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Periodic Report Summary 2 - CONTACT (Nonlinear inclusions, hemivariational inequalities with applications to contact mechanics)

The project has developed a joint training and cutting-edge research program based on state-of-the-art technologies that has strengthen the research partnership among Jagiellonian University in Krakow, Poland, University of Perpignan, France, Guangxi University of Nationalities in Nanning, P.R. China, Zhejiang University in Hangzhou, P.R. China, University of Iowa, USA and Oakland University in Rochester, USA in the area of common research interest, nonlinear inclusions, hemivariational inequalities and modeling of contact mechanics. The duration of the project was 48 months. This aim has been achieved through short- and long-term periods of staff exchange between partners in Europe, United States and China, and networking activities between the six institutions. The ultimate goal of this project was to achieve more rapid progress in advancing current knowledge and concepts through combined endeavour leading to a book monograph and joint-author high citation index publications. In this way, we have established a long-term research cooperation between six institutions based on active technology and scientific knowledge application and transfer. The scientific aim of the research exchange was to develop new and non-standard mathematical and numerical tools directly motivated by the needs of the analysis of various classes of contact problems which are of fundamental importance in technology, industry and real engineering applications. A throughout research of contact processes allowed to gain an improved fundamental understanding of contact mechanics and advance current knowledge that can ultimately be used for the improvement of industrial applications of economic benefits. Trainings, meetings, seminars and two workshops have been implemented in this research project in order to share all the knowledge and information gained throughout the work and to form the basis of long lasting collaborations.

All goals of the joint exchange collaboration has been successfully completed. Project has delivered results and provided impacts beyond the planned goals. We have created a network of research centers in Europe, Unites States and P.R. China, and applied the state-of-art knowledge and methods of nonlinear and functional analysis, differential equations and numerical analysis. The joint collaboration realized the following tasks:
a) the study of several classes of nonlinear inclusions, and systems of variational and hemivariational inequalities, including existence, uniqueness, regularity and asymptotic behaviour of the solution;
b) the construction of mathematical models which describe the contact, motivated by engineering applications;
c) the numerical analysis of the models, including the analysis of approximation schemes, convergence results of the discrete solutions, optimal error estimates;
d) the construction of reliable and efficient algorithms for the numerical approximations, their implementation in a numerical code and their validation in the study of academic test problems and real engineering applications.

We have also obtained new results for static, quasistatic and evolution inclusions. We have employed the theories of monotone, pseudomonotone and quasimonotone multivalued operators in Banach spaces and delivered surjectivity results for these classes of operators, and applied them to problems of mechanics described by hemivariational inequalities. In this way, the project has expanded Mathematical Theory of Contact Mechanics to include new effects. As industry is moving toward math-based, computer-aided product design, the project provided the construction of general, reliable and effective numerical algorithms for problems under consideration. The theoretical setting we have developed would be useful for the optimization and optimal control of the industrial processes. The methods elaborated can be useful for investigating other scientific problems, such as plate tectonics, aimed at earthquakes predictions, smart materials or the emerging field of biomechanics of the human body. A considerable effort in the modeling, analysis and numerics for contact problems have been made and the results published can be used in academy and in industry. The project was a big success, the progress we made in Mathematical Theory of Contact Mechanics is impressive.

The project was very productive, it resulted in three research monographs and high quality publications. New collaborations were successfully built on the basis of the undertaken research. The project has enhanced the production of knowledge and scientific excellence by enabling European universities to establish and maintain contacts with their partners outside Europe. We have promoted mobility of early-stage researchers, participation in research and innovative activities and multicultural dialogue. Exchange visits have allowed to establish institutional and methodological frames for cooperation and gave an opportunity to personal contacts.

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