We present GRAMSES, a new pipeline for nonlinear cosmological $N$-body simulations in General Relativity (GR). This code adopts the Arnowitt-Deser-Misner (ADM) formalism of GR, with constant mean curvature and minimum distortion gauge fixings, which provides a fully nonlinear and background independent framework for relativistic cosmology. Employing a fully constrained formulation, the Einstein equations are reduced to a set of ten elliptical equations which are solved using multigrid relaxation with adaptive mesh refinements (AMR), and three hyperbolic equations for the evolution of tensor degrees of freedom. The current version of GRAMSES neglects the latter by using the conformal flatness approximation, which allows it to compute the two scalar and two vector degrees of freedom of the metric. In this paper we describe the methodology, implementation, code tests and first results for cosmological simulations in a $Lambda$CDM universe, while the generation of initial conditions and physical results will be discussed elsewhere. Inheriting the efficient AMR and massive parallelisation infrastructure from the publicly-available $N$-body and hydrodynamic simulation code RAMSES, GRAMSES is ideal for studying the detailed behaviour of spacetime inside virialised cosmic structures and hence accurately quantifying the impact of backreaction effects on the cosmic expansion, as well as for investigating GR effects on cosmological observables using cosmic-volume simulations.
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