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<p>In this work, a stabilized finite element framework is developed to simulate small
and large deformation solid mechanics problems involving complex geometries and complicated
constitutive models. In particular, the focus is on solid dynamics problems involving
nearly and fully incompressible materials. The work is divided into three main themes,
the first is concerned with the development of stabilized finite element algorithms
for hyperelastic materials, the second handles the case of viscoelastic materials,
and the third focuses on algorithms for J2-plastic materials. For all three cases,
problems in the small and large deformation regime are considered, and for the J2-plasticity
case, both quasi-static and dynamic problems are examined.</p><p>Some of the key features
of the algorithms developed in this work is the simplicity of their implementation
into an existing finite element code, and their applicability to problems involving
complicated geometries. The former is achieved by using a mixed formulation of the
solid mechanics equations where the velocity and pressure unknowns are represented
by linear shape functions, whereas the latter is realized by using triangular elements
which offer numerous advantages compared to quadrilaterals, when meshing complicated
geometries. To achieve the stability of the algorithm, a new approach is proposed
in which the variational multiscale approach is applied to the mixed form of the solid
mechanics equations written down as a first order system, whereby the pressure equation
is cast in rate form.</p><p>Through a series of numerical simulations, it is shown
that the stability properties of the proposed algorithm is invariant to the constitutive
model and the time integrator used. By running convergence tests, the algorithm is
shown to be second order accurate, in the $L^2$-nrom, for the displacements, velocities,
and pressure. Finally, the robustness of the algorithm is showcased by considering
realistic test cases involving complicated geometries and very large deformation.</p>
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