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Analysis of novel finite element formulations for iterative solution of elastic inverse problems.

机译:弹性反问题迭代求解的新型有限元公式分析。

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摘要

Inverse problems are often formulated as optimization problems. One seeks the parameter distribution which, when used in a forward model of the problem, gives the best match possible to the measured data. We consider the problem of imaging the elastic modulus distributions of soft tissues in this context. These optimization formulations are usually ill-posed and hence some form of regularization is needed in order to guarantee uniqueness and stability for the inverse solution. In this dissertation, however, we present two stabilized optimization formulations for the plane stress inverse problem without any form of regularization. These formulations include novel Galerkin Least Squares (GLS) terms. The Flux-GLS and the Bspline-GLS are formulated and analyzed in this work. These GLS terms improve the stability of the optimization formulation without upsetting consistency. We prove that these two stabilized optimization formulations are well defined under reasonable conditions on the data. We prove further that these two stabilized methods have optimal rates of convergence with mesh refinement. Finally, computational examples are presented to demonstrate the analysis described in this dissertation. These numerical results show the accuracy and stability of these two methods in different conditions. In noisy conditions, we recover unbiased reconstructions of material property distributions without regularization in the presence of high levels noise in the data.
机译:逆问题通常被表述为优化问题。人们寻求一种参数分布,该参数分布在问题的正向模型中使用时,可以使所测得的数据达到最佳匹配。我们考虑在这种情况下成像软组织的弹性模量分布的问题。这些优化公式通常是不适定的,因此需要某种形式的正则化以确保逆解的唯一性和稳定性。然而,在本文中,我们针对平面应力反问题提出了两种稳定的优化公式,而没有任何形式的正则化。这些公式包括新颖的Galerkin最小二乘(GLS)术语。在这项工作中制定并分析了Flux-GLS和Bspline-GLS。这些GLS术语可提高优化公式的稳定性,而不会破坏一致性。我们证明这两个稳定的优化公式在合理的数据条件下定义良好。我们进一步证明,这两种稳定化方法在优化网格时具有最佳收敛速度。最后,通过算例说明了本文所进行的分析。这些数值结果表明了这两种方法在不同条件下的准确性和稳定性。在嘈杂的条件下,我们在数据中存在高水平噪声的情况下,无需进行正则化即可恢复材料属性分布的无偏重构。

著录项

  • 作者单位

    Boston University.;

  • 授予单位 Boston University.;
  • 学科 Engineering Mechanical.
  • 学位 Ph.D.
  • 年度 2010
  • 页码 151 p.
  • 总页数 151
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 机械、仪表工业;
  • 关键词

  • 入库时间 2022-08-17 11:37:20

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