Dynamic contact problems in bone neoplasm analyses and the primal-dual active set (PDAS) method. (English) Zbl 1374.92071

Nedoma, Jiří

Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, November 18–21, 2015. In honor of the birthday anniversaries of Ivo Babuška (90), Milan Práger (85), and Emil Vitásek (85). Prague: Czech Academy of Sciences, Institute of Mathematics (ISBN 978-80-85823-65-3). 158-183 (2015).

Summary: The growth of neoplasms (benign and malignant tumors and cysts), located in a system of loaded bones, is simulated. Because the geometry of the system of loaded and possible fractured bones with enlarged neoplasms changes in time, the corresponding mathematical models of tumor’s and cyst’s evolutions lead to coupled free boundary problems and dynamic contact problems with or without friction. The discussed parts of these models are based on the theory of dynamic contact problems without or with Tresca or Coulomb frictions in the visco-elastic rheology. The numerical solution of the problem with Coulomb friction is based on the semi-implicit scheme in time and the finite element method in space, where the Coulomb law of friction at every time level is approximated by its value from the previous time level. The algorithm for the corresponding model of friction is based on the discrete mortar formulation of the saddle point problem and the primal-dual active set algorithm. The algorithm for the Coulomb friction model is based on the fixpoint algorithm, that is an extension of the PDAS algorithm for the Tresca friction. In this algorithm the friction bound is iteratively modified using the normal component of the Lagrange multiplier. Thus, the friction bound and the active and inactive sets are updated in every step of the iterative algorithm and at every time step corresponding to the semi-implicit scheme.
For the entire collection see [Zbl 1329.00187].

MSC:

92C50	Medical applications (general)
65K10	Numerical optimization and variational techniques
65N15	Error bounds for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74M15	Contact in solid mechanics

Keywords:

dynamic contact problems; tumors and cysts; Coulomb and Tresca frictions; variational formulation; semi-implicit scheme; FEM; mortar approximation; PDAS algorithm