This paper presents a measurement-based FEM (finite element method) modeling and haptic rendering framework for objects with hyper-elastic deformation property. A complete set of methods covering the whole process of the measurement-based modeling/rendering paradigm is newly designed and implemented, with a special emphasis on haptic feedback realism. To this end, we first build a data collection setup that accurately captures shape deformation and response forces during compressive deformation of cylindrical material samples. With this setup, training and testing sets of data are collected from four silicone objects having various material profiles. Then, an objective function incorporating both shape deformation and reactive forces is designed and used to identify material parameters based on training data and the genetic algorithm. For real-time haptic rendering, an optimization-based FEM solver is adopted, ensuring around 500 Hz update rate. The whole procedure is evaluated through numerical and psychophysical experiments. The numerical rendering error is calculated based on the difference between simulated and actually measured deformation forces. The errors are also compared to the human perceptual threshold and found to be perceptually negligible. Overall realism of the feedback from the system is also assessed through the psychophysical experiment. A total of twelve participants rated similarity between real and modeled objects, and the results reveal the rendering quality to be at a reasonable level of realism with positive user feedback. (C) 2020 Elsevier Ltd. All rights reserved.