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Center for Nonlinear Studies |
Multi-material Eulerian formulations were initially designed to solve hydrodynamic problems. Their ability to automatically generate new interfaces and solve non-linear large deformation problems make these formulations attractive for solving solid mechanics problems. However, the development of multi-material elements and contact algorithms for Eulerian formulations has traditionally been challenging. Mixture theories have worked well for ``bonded'' contacts, but they fail to correctly represent different contact types (e.g. frictionless slip).Enforcing prescribed contact constraints upon the nodal velocities and accelerations has proven successful for ``Lagrange + remapping'' Eulerian methods employing one set of degree of freedom per material (e.g. Vitali and Benson 2006). However, this formulation is not readily applied to Godunov methods that do not employ staggered mesh and where nodal values are replaced by face fluxes. This talk focuses on the development of a multi-material formulation for Eulerian frameworks employing Godunov methods. In order to obtain material independency, the formulation assigns one set of degrees of freedom to each material present in the problem. The basis for the contact scheme is the non-penetration condition, which is enforced upon the stresses and velocities of the materials present in multi-material elements and results in a frictionless slip contact. Frictional contacts can also be included by adding constraints to the scheme. Materials stresses are equilibrated in the direction normal to the interface by employing a relaxation algorithm, while the velocities are balanced in the same direction satisfying the conservation of linear momentum. A challenge inherent to Godunov methods is the calculation of the solution variables at the faces of the elements. The proposed method is applied locally and interpolates the ``material'' Riemann solution with the ``interface'' Riemann solution. The material solution corresponds to the classic Riemann solution, which uses the values of the material being analyzed present in the elements adjacent to the face. The interface solution is obtained by solving the Riemann problem with the values of the material being analyzed and an average of the remaining values as left and right states, respectively. The effectiveness of the new method is presented with example calculations.
Host: Mikhail Shashkov. shashkov@lanl.gov, 667-4400