  We use numerical relativity simulations to describe the spacetime evolution
during nonlinear structure formation in $\Lambda$CDM cosmology. Fully nonlinear
initial conditions are set at an initial redshift $z\approx 300$, based
directly on the gauge invariant comoving curvature perturbation $\mathcal{R}_c$
commonly used to model early-universe fluctuations. Assigning a simple 3-D
sinusoidal structure to $\mathcal{R}_c$, we then have a lattice of
quasi-spherical over-densities representing idealised dark matter halos
connected through filaments and surrounded by voids. This structure is
implemented in the synchronous-comoving gauge, using a pressureless perfect
fluid (dust) description of CDM, and then it is fully evolved with the Einstein
Toolkit code. With this, we look into whether the Top-Hat spherical and
homogeneous collapse model provides a good description of the collapse of
over-densities. We find that the Top-Hat is an excellent approximation for the
evolution of peaks, where we observe that the shear is negligible and collapse
takes place when the linear density contrast reaches the predicted critical
value $\delta^{(1)}_C =1.69$. Additionally, we characterise the outward
expansion of the turn-around boundary and show how it depends on the initial
distribution of matter, finding that it is faster in denser directions,
incorporating more and more matter in the infalling region. Using the EBWeyl
code [1] we look at the distribution of the electric and magnetic parts of the
Weyl tensor, finding that they are stronger along and around the filaments,
respectively. We introduce a method to dynamically classify different regions
in Petrov types. With this, we find that the spacetime is of Petrov type I
everywhere, as expected, but we can identify the leading order type, finding a
transition between different types as non-linearity grows, with production of
gravitational waves.
