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首页> 外文期刊>Computer Methods in Applied Mechanics and Engineering >Solution of the nonlinear elasticity imaging inverse problem:The incompressible case
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Solution of the nonlinear elasticity imaging inverse problem:The incompressible case

机译：非线性弹性成像反问题的解决：不可压缩的情况

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摘要

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.
机译：我们最近开发并测试了一种有效的算法，用于解决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总应变水平下测得的固体内的准静态变形场。我们算法的主要成分是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连续的新颖策略。在本文中，我们提出了对该算法的一些扩展。首先，我们将其扩展到不可压缩的介质，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，混合的有限元公式解决了Ladydyskayaskaya-Babuska-Brezzi条件的正向问题来实现这一目标。其次，我们演示了如何通过预处理材料参数方程组或将问题分成两个不同的步骤来改善非线性参数空间分布的恢复。最后，我们提出了一种新的具有指数应力-应变行为的应变能密度函数，该函数产生了偏应力张量，从而与其他指数函数相比简化了压力的解释。我们通过从嘈杂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。

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