Goenezen Sevan, Barbone Paul, Oberai Assad A
Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute,110, 8th St., Troy, NY 12180, USA.
Comput Methods Appl Mech Eng. 2011 Mar 1;200(13-16):1406-1420. doi: 10.1016/j.cma.2010.12.018.
We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya-Babuska-Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress-strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.
我们最近开发并测试了一种高效算法，用于求解可压缩超弹性材料的非线性逆弹性问题。该问题的数据是在两个不同的整体应变水平下测量得到的固体内部的准静态变形场。我们算法的主要组成部分包括基于梯度的拟牛顿最小化策略、伴随方程的使用以及材料参数延续的新策略。在本文中，我们展示了对该算法的几种扩展。首先，我们将其扩展到不可压缩介质，从而将其适用性扩展到在缓慢变形下几乎不可压缩的组织。我们通过使用基于残差的、稳定的、混合有限元公式来求解正向问题来实现这一点，该公式规避了拉迪任斯卡娅 - 巴布斯卡 - 布雷zzi 条件。其次，我们展示了如何通过对方程组进行预处理或通过将问题拆分为两个不同步骤来改善非线性参数空间分布的恢复。最后，我们提出了一种具有指数应力 - 应变行为的新应变能密度函数，该函数产生偏应力张量，与其他指数函数相比，简化了压力的解释。我们通过从有噪声的合成变形场中求解材料参数的空间分布来测试整体方法。
