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**Total Lagrangian explicit dynamics finite element algorithm for computing soft tissue deformation.**
*(English)*
Zbl 1154.74045

Summary: We propose an efficient numerical algorithm for computing deformations of “very” soft tissues (such as the brain, liver, kidney etc.), with applications to real-time surgical simulation. The algorithm is based on finite element method using total Lagrangian formulation, where stresses and strains are measured with respect to the original configuration. This choice allows for pre-computing of most spatial derivatives before the commencement of time-stepping procedure. We use explicit time integration that eliminates the need for iterative equation solving during the time-stepping procedure. The algorithm is capable of handling both geometric and material nonlinearities. The total Lagrangian explicit dynamics (TLED) algorithm using eight-noded hexahedral under-integrated elements requires approximately 35% fewer floating-point operations per element and per time step than the updated Lagrangian explicit algorithm using the same elements. Stability analysis of the algorithm suggests that due to much lower stiffness of very soft tissues than that of typical engineering materials, integration time steps a few orders of magnitude larger than what is typically used in engineering simulations are possible. Numerical examples confirm the accuracy and efficiency of the proposed TLED algorithm.

### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74L15 | Biomechanical solid mechanics |

92C10 | Biomechanics |

### Keywords:

numerical stability; real-time surgical simulation; time-stepping procedure; eight-noded hexahedral under-integrated elements
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\textit{K. Miller} et al., Commun. Numer. Methods Eng. 23, No. 2, 121--134 (2007; Zbl 1154.74045)

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