Systems and methods of generating an orthodontic model are disclosed. The method may include: generating an initial model of a patient dentition; generating a target model of the patient dentition; defining a plurality of caps and a plurality of links, wherein each link connects two of the plurality of caps; generating a relaxed model of a dental appliance from the plurality of caps and the plurality of links; generating a deformed model of a dental appliance from the plurality of caps and plurality of links; and determining a plurality of movements, wherein the plurality of moments transform the relaxed model to the deformed model and wherein the moments are configured to direct the patient dentition from the initial model to the target model.