The inerter is a mechanical component that generates a force proportional to the relative acceleration between its terminals, significantly amplifying inertial forces while contributing minimal self-weight. When combined with a one-way clutch and dampers, the clutched-inerter damper (CID) transitions into energy harvesting mode, wherein energy is captured and stored through the rotation of its flywheel. However, the inherent nonlinearity and discontinuity associated with clutch mechanisms present challenges for their numerical modelling and design. Most existing studies to date have concentrated on small-scale systems with a limited number of inerters, typically relying on simplified models due to a lack of robust tools to simulate their response. In this work, we propose a numerical modelling method based on Mixed Lagrangian Formalism (MLF) and adjoint-sensitivity analysis for CIDs. Compared to conventional methods, our MLF approach demonstrates superior accuracy, efficiency, and stability. The modelling and optimization strategy described in this paper, enable researchers to simulate and design systems with a larger number of degrees of freedom and a substantial number of inerter-based devices, while significantly reducing computational effort and improving performance.