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Inverse design has greatly expanded nanophotonic devices and brought optimized performance. However, the use of inverse design for plasmonic structures has been challenging due to local field concentrations that can lead to errors in gradient calculation when the continuum adjoint method is used. On the other hand, with the discrete adjoint method one can achieve the exact gradient. Historically the discrete version is exclusively used with a Finite Element model, and applying the Finite-Difference Time-Domain (FDTD) method in inverse design of plasmonic structures is rarely attempted. Due to the popularity of using FDTD in simulating plasmonic structures, we integrate the discrete adjoint method with FDTD and present a framework to carry out inverse design of plasmonic structures using density-based topology optimization. We demonstrate the exactness of the gradient calculation for a plasmonic block structure with varying permittivity. Another challenge that is unique with plasmonic structures is that non-physical amplification caused by poorly chosen material interpolation can destroy a stable convergence of the optimization. To avoid this, we adopt a non-linear material interpolation scheme in the FDTD solver. In addition, filtering-and-projection regularization is incorporated to ensure manufacturability of the designed structures. As an example of this framework, successful reconstruction of electric fields of a plasmonic bowtie aperture is presented.