Solution of the time‐harmonic viscoelastic inverse problem with interior data in two dimensions
Solution of the time‐harmonic viscoelastic inverse problem with interior data in two dimensions
复制标题
二维内部数据时调和粘弹性反问题的求解
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
I. Harari
中科院分区：
文献类型：
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作者：
Yixiao Zhang;A. Oberai;P. Barbone;I. Harari
在已知不可压缩粘弹性材料内部位移场的情况下，我们考虑了不可压缩线性粘弹性材料在无限小的时间谐变作用下，确定其复值剪切模数分布的问题。特别是，我们重点讨论了反平面剪切和平面应力的二维问题。这些问题是由生物力学成像中的应用引起的，在生物力学成像中，材料的模数分布被用来检测和/或诊断癌症肿瘤。我们分析了这些问题的强形式的适定性，并得出结论：要存在解，所测得的位移场必须满足相当严格的相容条件。我们提出了一个弱的或变分的公式，并证明了在较温和的条件下，测量数据解的存在唯一性。这个公式是通过作用于权函数的复共轭的伴随算子对剪切模量值的原始偏微分方程进行加权而得到的。为此，我们将其称为复伴随加权方程。我们考虑用全变分正则化对这些方程进行直接的有限元离散，并用合成的和实验测量的数据测试它的性能。我们发现，一般而言，CAWE方法比相应的最小二乘解的扩散性要小，并且总变分正则化显著改善了其在噪声存在时的性能。版权所有©2012 John Wiley&Sons，Ltd.
We consider the problem of determining the distribution of the complex‐valued shear modulus for an incompressible linear viscoelastic material undergoing infinitesimal time‐harmonic deformation, given the knowledge of the displacement field in its interior. In particular, we focus on the two‐dimensional problems of anti‐plane shear and plane stress. These problems are motivated by applications in biomechanical imaging, where the material modulus distributions are used to detect and/or diagnose cancerous tumors. We analyze the well‐posedness of the strong form of these problems and conclude that for the solution to exist, the measured displacement field is required to satisfy rather restrictive compatibility conditions. We propose a weak, or a variational formulation, and prove the existence and uniqueness of solutions under milder conditions on measured data. This formulation is derived by weighting the original PDE for the shear modulus by the adjoint operator acting on the complex‐conjugate of the weighting functions. For this reason, we refer to it as the complex adjoint weighted equation (CAWE). We consider a straightforward finite element discretization of these equations with total variation regularization, and test its performance with synthetically generated and experimentally measured data. We find that the CAWE method is, in general, less diffusive than a corresponding least squares solution, and that the total variation regularization significantly improves its performance in the presence of noise. Copyright © 2012 John Wiley & Sons, Ltd.
影响因子：
56.9
作者：
MUTHUPILLAI, R;LOMAS, DJ;EHMAN, RL
通讯作者：
EHMAN, RL