Analytical solution for real-time simulation of membrane deformation

Abstract

Despite considerable efforts in simulating the behavior of soft tissues in the last decades, real-time modeling of soft tissue deformation remains a challenge. Current deformable models face several challenges that limit their use in interactive applications. In particular, the primary limitations with regard to effciency and stability should be addressed to achieve graphic and haptic interactivity. Herein, a new analytical approach for real-time simulation of object deformation is proposed. This approach utilizes physically motivated analytical equations to achieve deformations. Moreover, we developed an analytical solution for elastic two-dimensional (2D) membranes and three-dimensional (3D) objects enclosed by a fluid-filled membrane. We solved Poisson’s equation with a Dirichlet boundary condition using Green’s functions to obtain the analytical solutions. We obtained the equilibrium configurations of a deformed 2D membrane in contact with a probe and compared the results to the experimental data through statistical validation. Subsequently, the proposed approach was extended to simulate the surface deformation of a 3D object with volume conservation. Furthermore, the methodology was successfully implemented in a virtual-reality environment with haptic force feedback to enhance the user’s immersion. Our results demonstrated that the proposed analytical approach enables real-time modeling of membrane deformations, and the statistical analysis provides a deep understanding of the model’s strengths and limitations.