In the contribution growths of the neoplasms (benign and malignant tumors and cysts), located in a system of loaded bones, will be simulated. The main goal of the contribution is to present the useful methods and efficient algorithms for their solutions. 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 the coupled free boundary problems and the dynamic contact problems with or without friction. The discussed parts of these models will be 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 will be approximated by its value from the previous time level. The algorithm for the corresponding model of friction will be 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 will be based on the fixpoint algorithm, that will be 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.

EUDML-ID : urn:eudml:doc:287830
Mots clés:
Mots clés:

@article{702974,
     title = {Dynamic contact problems in bone neoplasm analyses and the primal-dual active set (PDAS) method},
     booktitle = {Application of Mathematics 2015},
     series = {GDML\_Books},
     publisher = {Institute of Mathematics CAS},
     address = {Prague},
     year = {2015},
     pages = {158-183},
     zbl = {06669928},
     url = {http://dml.mathdoc.fr/item/702974}
}

Nedoma, Jiří. Dynamic contact problems in bone neoplasm analyses and the primal-dual active set (PDAS) method, dans Application of Mathematics 2015, GDML_Books,  (2015), pp. 158-183. http://gdmltest.u-ga.fr/item/702974/