A higher order polynomial reproducing radial basis function neural network (HOPR-RBFN) for real-time interactive simulations of nonlinear deformable bodies with haptic feedback

Interactive simulation of nonlinear deformable bodies with haptic feedback is particularly challenging as the corresponding coupled nonlinear equations must be solved at a high rate of 1 kHz. With increasing demand on the complexity of the models and scenarios that must be simulated to develop interactive applications such as surgical simulators, and given the current state of hardware, a nai¿ve solution strategy based on iterative finite element algorithms is not feasible. However, much advantage may be gained if the deformation response of the computational models may be characterized a priori and a radial basis function neural network (RBFN) is trained based upon this data. Once trained, the RBFN may be used during real time computations to calculate deformation fields and reaction forces with minimal computational cost. However, traditional RBFNs have zeroth order polynomial accuracy, implying that they cannot recreate polynomial fields. Since the error in the RBFN approximation is governed by the highest order of the polynomial that the approximant can reproduce, we have developed and successfully tested a higher order polynomial reproducing RBFN (HOPR-RBFN) which, compared to traditional RBFN, reduces the approximation error significantly and allows much fewer neurons to be used for comparable accuracy. Results are provided for realistic surgical scenarios with hyperelastic (neo-Hookean) material models within a fully nonlinear large deformation simulation framework.