Abstract: Classical compatible discretizations on staggered Lagrangian grids do not explicitly provide the pressure on cell faces, which makes the work done on internal interfaces difficult to evaluate by a face-pressure formula. This paper introduces an external corner-force method. Under the assumption that the nodal momentum update can be decomposed into corner pressure vectors, the method constructs a compatible internal-energy update without changing the original momentum scheme. The external corner forces cancel around each node and therefore preserve total energy. A discrete formula for the work done on closed sub-domain boundaries is then derived and proved to be exactly consistent with the discrete energy balance. Numerical examples demonstrate the effectiveness of the method for global and sub-domain energy-conservation diagnostics.