  One of the major issues in the computational mechanics is to take into
account the geometrical complexity. To overcome this difficulty and to avoid
the expensive mesh generation, geometrically unfitted methods, i.e. the
numerical methods using the simple computational meshes that do not fit the
boundary of the domain, and/or the internal interfaces, have been widely
developed. In the present work, we investigate the performances of an unfitted
method called $\phi$-FEM that converges optimally and uses classical finite
element spaces so that it can be easily implemented using general FEM
libraries. The main idea is to take into account the geometry thanks to a level
set function describing the boundary or the interface. Up to now, the
$\phi$-FEM approach has been proposed, tested and substantiated mathematically
only in some simplest settings: Poisson equation with Dirichlet/Neumann/Robin
boundary conditions. Our goal here is to demonstrate its applicability to some
more sophisticated governing equations arising in the computational mechanics.
We consider the linear elasticity equations accompanied by either pure
Dirichlet boundary conditions or by the mixed ones (Dirichlet and Neumann
boundary conditions co-existing on parts of the boundary), an interface problem
(linear elasticity with material coefficients abruptly changing over an
internal interface), a model of elastic structures with cracks, and finally the
heat equation. In all these settings, we derive an appropriate variant of
$\phi$-FEM and then illustrate it by numerical tests on manufactured solutions.
We also compare the accuracy and efficiency of $\phi$-FEM with those of the
standard fitted FEM on the meshes of similar size, revealing the substantial
gains that can be achieved by $\phi$-FEM in both the accuracy and the
computational time.
