Séance Séminaire

NEMESIS Seminar

An Arbitrary-Order Augmented Lagrangian Discrete De Rham Scheme for Contact Mechanics Tresca Friction

We will discuss the arbitrary-order augmented Lagrangian Discrete de Rham\r\nscheme for contact mechanics with Tresca friction at the matrix--fracture\r\ninterfaces of a co-dimension-one fracture network embedded in a polyhedral domain,\r\nwithin a static, isotropic, linear elasticity model. Both the normal Signorini-type/traction constraint and the Tresca friction law are recast as projection identities, onto the half-line and onto the friction ball of $\\gamma g$~($\\gamma>0$ is a parameter and $g\\geq 0$ is the Tresca threshold), respectively, and combined into a single augmented Lagrangian bilinear form. The displacement is discretised by a DDR space of arbitrary degree $k\\ge 0,$ enriched on the fracture faces by bubble degrees of freedom that ensure the inf--sup compatibility between the Lagrange multiplier space and the discrete jump of the displacement along the fracture. We establish a discrete Korn inequality adapted to the bubble\r\nenrichment and the fracture network, and prove the well-posedness of the scheme --\r\nexistence via a topological degree argument and uniqueness via monotonicity -- and derive an abstract error estimate yielding optimal order $h^{k+1}$ convergence for the displacement and the Lagrange multiplier variables