The control variational method for beams in contact with deformable obstacles

• Published in: Zeitschrift für angewandte Mathematik und Mechanik 2012-01, Vol.92 (1), p.25-40
• Main Authors: Barboteu, M., Sofonea, M., Tiba, D.
• Format: Article
• Language: English
• Subjects: Calculus of variations and optimal control; control variational method; Euler-Bernoulli beam; Exact sciences and technology; finite difference method; Mathematical analysis; Mathematics; normal compliance; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Numerical methods in probability and statistics; numerical simulations; optimal state; Sciences and techniques of general use; Signorini's condition; weak solution
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201000161

We consider a mathematical model which describes the equilibrium of an elastic beam in contact with two obstacles. The contact is modeled with a normal compliance type condition in such a way that the penetration is allowed but is limited. We state the variational formulation of the problem and prove an existence and uniqueness result for the weak solution. Then, we provide an alternative approach to the model, based on the control variational method. Necessary and sufficient optimality conditions are derived, together with an approximation property. We also adapt our results to some versions of the model which describe the contact with a single obstacle. For this type of problems we present two methods of numerical approach, based on the iterative control variational method and the Newton method, respectively, and provide numerical simulations for the two algorithms. The authors consider a mathematical model which describes the equilibrium of an elastic beam in contact with two obstacles. The contact is modeled with a normal compliance type condition in such a way that the penetration is allowed but is limited. They state the variational formulation of the problem and prove an existence and uniqueness result for the weak solution. Then, they provide an alternative approach to the model, based on the control variational method. Necessary and sufficient optimality conditions are derived, together with an approximation property. They also adapt your results to some versions of the model which describe the contact with a single obstacle. For this type of problems they present two methods of numerical approach, based on the iterative control variational method and the Newton method, respectively, and provide numerical simulations for the two algorithms.