A posteriori error analysis for contact problems with application to dam modeling
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Speaker: Ilaria Fontana, Dept of Mathematics, University of Arizona
Title: A posteriori error analysis for contact problems with application to dam modeling
Abstract: The finite element method is a widely adopted computational tool for solving problems formulated as partial differential equations in computational mechanics. For instance, engineering teams often perform finite element numerical simulations to analyze the behavior of large hydraulic structures. The nonlinear behavior of concrete dams, particularly within their weakest zones, such as the concrete-rock contact in the foundation or the interfaces between blocks, significantly influences their stability. In practice, these nonlinearities are predominantly due to contact, which generates significant numerical challenges for achieving convergence in finite element computations. Therefore, developing efficient numerical methods for handling contact problems is crucial for robust industrial numerical simulations.
We focus on frictional unilateral contact problems within a framework including both Tresca and Coulomb friction conditions. Numerically, we adopt a finite element discretization with weak enforcement of nonlinear contact conditions called the Nitsche method. This choice enables an easy implementation of the contact conditions in a weak sense, without the introduction of additional unknowns such as Lagrange multipliers. For this problem, we present an a posteriori error analysis based on equilibrated stress reconstruction, providing a guaranteed fully-computable upper bound of the error that distinguishes the different components of the error by defining some local and global error estimators. The key idea behind these estimators is to provide a numerical correction of the stress tensor with continuous normal trace across the edges of the mesh, and locally in equilibrium with the volumetric and surface forces. We also propose a practical way to achieve this reconstruction by assembling the solutions of local problems defined on patches around mesh vertices using the Arnold-Falk-Winther mixed finite element space. We then use the local estimators to introduce a fully-automated adaptive algorithm for the refinement of the mesh with a stopping criterion for the linearization algorithm. Numerical results compare this adaptive algorithm with a uniform approach, showing the efficiency of the algorithm.