The physics of deformation for biological soft-tissue is best described by nonlinear continuum mechanics-based models, which then can be discretized by the FEM for a numerical solution. However, computational complexity of such models have limited their use in applications requiring real-time or fast response. In this work, we propose a graphic processing unit-based implementation of the FEM using implicit time integration for dynamic nonlinear deformation analysis. This is the most general formulation of the deformation analysis. It is valid for large deformations and strains and can account for material nonlinearities. The data-parallel nature and the intense arithmetic computations of nonlinear FEM equations make it particularly suitable for implementation on a parallel computing platform such as graphic processing unit. In this work, we present and compare two different designs based on the matrix-free and conventional preconditioned conjugate gradients algorithms for solving the FEM equations arising in deformation analysis. The speedup achieved with the proposed parallel implementations of the algorithms will be instrumental in the development of advanced surgical simulators and medical image registration methods involving soft-tissue deformation.

Real-time simulation of haptic interaction with deformable objects is computationally demanding. In particular in finite-element (FE) based analysis of such interactions, a large system of equations must be solved at an update rate of 100-1,000 Hz for simulation fidelity and stability. A new hardware-based parallel implementation of a Preconditioned Conjugate Gradient (PCG) algorithm is proposed for solving the linear systems of equations arising from FE-based deformation models. Concurrent utilization of a large number of fixed-point computing units on a Field-Programmable Gate Array (FPGA) device yields a very fast solution to these equations. Quantization and overflow errors in the fixed-point implementation of the iterative solver are minimized through dynamic scaling and preconditioning. Numerical accuracy of the solution, the architecture design, and issues pertaining to the degree of parallelism and scalability of the architecture are discussed in detail. The implementation of the solver on an Altera EP3SE110 FPGA device has enabled real-time simulation of three-dimensional linear elastic deformation models with 1,500 nodes at an update rate of up to 2,500 Hz.

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