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This thesis introduces a novel deformation method for computational meshes. It is based on the numerical path following for the equations of nonlinear elasticity. By employing a logarithmic variation of the neo-Hookean hyperelastic material law, the method guarantees that the mesh elements do not become inverted and remain well-shaped. In order to demonstrate the performance of the method, this thesis addresses two areas of active research in isogeometric analysis: volumetric domain parametrization and fluid-structure interaction. The former concerns itself with the construction of a parametrization for a given computational domain provided only a parametrization of the domain’s boundary. The proposed mesh deformation method gives rise to a novel solution approach to this problem. Within it, the domain parametrization is constructed as a deformed configuration of a simplified domain. In order to obtain the simplified domain, the boundary of the target domain is projected in the \(L^2\)-sense onto a coarse NURBS basis. Then, the Coons patch is applied to parametrize the simplified domain. As a range of 2D and 3D examples demonstrates, the mesh deformation approach is able to produce high-quality parametrizations for complex domains where many state-of-the-art methods either fail or become unstable and inefficient. In the context of fluid-structure interaction, the proposed mesh deformation method is applied to robustly update the computational mesh in situations when the fluid domain undergoes large deformations. In comparison to the state-of-the-art mesh update methods, it is able to handle larger deformations and does not result in an eventual reduction of mesh quality. The performance of the method is demonstrated on a classic 2D fluid-structure interaction benchmark reproduced by using an isogeometric partitioned solver with strong coupling.