  The FE$^2$ method is a very flexible but computationally expensive tool for
multiscale simulations. In conventional implementations, the microscopic
displacements are iteratively solved for within each macroscopic iteration
loop, although the macroscopic strains imposed as boundary conditions at the
micro-scale only represent estimates. In order to reduce the number of
expensive micro-scale iterations, the present contribution presents a
monolithic FE$^2$ scheme, for which the displacements at the micro-scale and at
the macro-scale are solved for in a common Newton-Raphson loop. In this case,
the linear system of equations within each iteration is solved by static
condensation, so that only very limited modifications to the conventional,
staggered scheme are necessary. The proposed monolithic FE$^2$ algorithm is
implemented into the commercial FE code Abaqus. Benchmark examples demonstrate
that the monolithic scheme saves up to ~60% of computational costs.
