Institute of Mechanics, Ruhr-University Bochum, Bochum, Germany.
Phys Med Biol. 2011 Jul 21;56(14):4239-65. doi: 10.1088/0031-9155/56/14/004. Epub 2011 Jun 23.
A novel finite element formulation suitable for computing efficiently the stiffness distribution in soft biological tissue is presented in this paper. For that purpose, the inverse problem of finite strain hyperelasticity is considered and solved iteratively. In line with Arnold et al (2010 Phys. Med. Biol. 55 2035), the computing time is effectively reduced by using adaptive finite element methods. In sharp contrast to previous approaches, the novel mesh adaption relies on an r-adaption (re-allocation of the nodes within the finite element triangulation). This method allows the detection of material interfaces between healthy and diseased tissue in a very effective manner. The evolution of the nodal positions is canonically driven by the same minimization principle characterizing the inverse problem of hyperelasticity. Consequently, the proposed mesh adaption is variationally consistent. Furthermore, it guarantees that the quality of the numerical solution is improved. Since the proposed r-adaption requires only a relatively coarse triangulation for detecting material interfaces, the underlying finite element spaces are usually not rich enough for predicting the deformation field sufficiently accurately (the forward problem). For this reason, the novel variational r-refinement is combined with the variational h-adaption (Arnold et al 2010) to obtain a variational hr-refinement algorithm. The resulting approach captures material interfaces well (by using r-adaption) and predicts a deformation field in good agreement with that observed experimentally (by using h-adaption).
本文提出了一种适用于计算软生物组织刚度分布的新型有限元公式。为此，考虑并迭代求解有限应变超弹性的反问题。与 Arnold 等人（2010 年，《物理医学与生物学》55,2035）一致，通过使用自适应有限元方法，可以有效地减少计算时间。与以前的方法形成鲜明对比的是，新颖的网格自适应依赖于 r 自适应（在有限元三角剖分中重新分配节点）。这种方法可以非常有效地检测健康组织和患病组织之间的材料界面。节点位置的演化由特征化超弹性反问题的相同最小化原理正则驱动。因此，提出的网格自适应是变分一致的。此外，它保证了数值解的质量得到提高。由于所提出的 r 自适应仅需要相对粗糙的三角剖分来检测材料界面，因此基础有限元空间通常不够丰富，无法足够准确地预测变形场（正向问题）。出于这个原因，将新颖的变分 r 细化与变分 h 自适应（Arnold 等人，2010 年）相结合，以获得变分 h-r 细化算法。所得到的方法可以很好地捕捉材料界面（使用 r 自适应），并预测与实验观察到的变形场非常吻合（使用 h 自适应）。
