In this thesis we investigate eigenproblems arising in spectral geometry and in the theory of linear elasticity, and numerically study the mechanics of skeletal muscles in three dimensions. We first introduce a novel framework based on a combined finite element-Bayesian optimization strategy to tackle problems in spectral optimization. This method is used to investigate the Pólya-Szegö conjecture for the Dirichlet eigenvalues of the Laplacian on polygons and a similar conjecture for the Steklov eigenvalues for the Laplacian is proposed. We next explore four different eigenproblems for the Lamé operator in linear elasticity. For each we establish the existence of a countable (finite or infinite) set of eigenpairs. The first one relates to Steklov eigenpairs for elasticity where the spectral parameter appears on a Robin type boundary condition. The second eigenproblem concerns eigenmodes whose tangential traction and normal displacement on the boundary are constrained. The third eigenproblem refers to eigenmodes whose normal traction and tangential displacement are fixed on the boundary. We finally consider the Jones eigenvalue problem: elastic modes are assumed to be traction free on the boundary with the extra condition that they must be purely tangential to the boundary. For each eigenproblem, new versions of Korn's inequalities are proven to establish the existence of a point spectrum. Finally, we present a new numerical framework to computationally simulate the large deformations of skeletal muscles in three dimensions. A semi-implicit method in time and finite element method in space are coupled to approximate the solutions of the nonlinear dynamic system that governs the deformation of these tissues. This framework is later utilized to explore questions on the effects of density and inertia in muscle performance, to investigate the energy distribution in muscle during isometric contractions, and to study the energetics of muscles during compression tests on the surface of the tissues.
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