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<p>This dissertation addresses interface problems simulated with the finite element
method (FEM) with mesh adaptivity. More specifically, we concentrate on the strategies
that adaptively modify the mesh and the associated data transfer issues. </p><p>In
finite element simulations there often arises the need to change the mesh and continue
the simulation on a new mesh. Analysts encounter such an issue when they adaptively
refine the mesh to reduce the computational cost, smooth distorted elements to improve
system conditioning, or introduce new surfaces and change the domain in simulations
of fracture problems. In such circumstances, the transfer of data from the old mesh
to the new one is of crucial importance, especially for nonlinear problems. We are
concerned in this work with contact problems with adaptive re-meshing and fracture
problems modeled with the eXtended finite element method (X-FEM). For the former ones,
the transfer of surface data is built upon the technique of parallel transport, and
the error of such a transfer strategy is investigated through classic benchmark tests.
A transfer scheme based on a least squares problem is also proposed to transfer the
bulk data when nearly incompressible hyperelastic materials are employed. For the
latter type of problems, we facilitate the transfer of internal variables by making
partial elements utilize the same quadrature points from the uncut parent elements
and meanwhile adjusting the quadrature weights via the solution of moment fitting
equations. The proposed scheme helps avoid the complicated remapping procedure of
internal variables between two different sets of quadrature points. A number of numerical
examples are presented to demonstrate the robustness and accuracy of our proposed
approaches.</p><p>Another renowned technique to simulate fracture problems is based
upon the phase-field formulation, where a set of coupled mechanics and phase-field
equations are solved via FEM without modeling crack geometries. However, losing the
ability to model distinct surfaces in the phase-field formulation has drawbacks, such
as difficulties simulating contact on crack surfaces and poorly-conditioned stiffness
matrices. On the other hand, using the pure X-FEM in fracture simulations mandates
the calculation of the direction and increment of crack surfaces at each step, introducing
intricacies of tracing crack evolution. Thus, we propose combining phase-field and
X-FEM approaches to utilize their individual benefits based on a novel medial-axis
algorithm. Consequently, we can still capture complex crack geometries while having
crack surfaces explicitly modeled by modifying the mesh with the X-FEM.</p>
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