This paper presents the formulation and implementation of an Error in Constitutive Equations (ECE) method suitable for large-scale inverse identification of linear elastic material properties in the context of steady-state elastodynamics. In ECE-based methods, the inverse problem is postulated as an optimization problem in which the cost functional measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses. Furthermore, in a more recent modality of this methodology introduced by Feissel and Allix (2007), referred to as the Modified ECE (MECE), the measured data is incorporated into the formulation as a quadratic penalty term. We show that a simple and efficient continuation scheme for the penalty term, suggested by the theory of quadratic penalty methods, can significantly accelerate the convergence of the MECE algorithm. Furthermore, a (block) successive over-relaxation (SOR) technique is introduced, enabling the use of existing parallel finite element codes with minimal modification to solve the coupled system of equations that arises from the optimality conditions in MECE methods. Our numerical results demonstrate that the proposed methodology can successfully reconstruct the spatial distribution of elastic material parameters from partial and noisy measurements in as few as ten iterations in a 2D example and fifty in a 3D example. We show (through numerical experiments) that the proposed continuation scheme can improve the rate of convergence of MECE methods by at least an order of magnitude versus the alternative of using a fixed penalty parameter. Furthermore, the proposed block SOR strategy coupled with existing parallel solvers produces a computationally efficient MECE method that can be used for large scale materials identification problems, as demonstrated on a 3D example involving about 400,000 unknown moduli. Finally, our numerical results suggest that the proposed MECE approach can be significantly faster than the conventional approach of L(2) minimization using quasi-Newton methods.

中文翻译：

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使用本构方程泛函误差的频域弹性动力学中的大规模参数估计问题

本文介绍了本构方程误差 (ECE) 方法的公式化和实现，该方法适用于稳态弹性动力学背景下线性弹性材料特性的大规模逆识别。在基于 ECE 的方法中，逆问题被假定为优化问题，其中成本函数测量连接运动学容许应变和动态容许应力的本构方程中的差异。此外，在 Feissel 和 Allix (2007) 引入的这种方法的更新模式中，称为修改后的 ECE (MECE)，将测量数据作为二次惩罚项纳入公式。我们展示了一个简单有效的惩罚项延续方案，由二次惩罚方法的理论提出，可以显着加快 MECE 算法的收敛速度。此外，还引入了（块）连续过松弛 (SOR) 技术，允许使用现有的并行有限元代码进行最小修改来求解由 MECE 方法中的最优条件产生的耦合方程组。我们的数值结果表明，所提出的方法可以成功地从部分和噪声测量中重建弹性材料参数的空间分布，在 2D 示例中只需 10 次迭代，在 3D 示例中则需要 50 次迭代。我们（通过数值实验）表明，与使用固定惩罚参数的替代方案相比，所提出的延续方案可以将 MECE 方法的收敛速度提高至少一个数量级。此外，所提出的块 SOR 策略与现有的并行求解器相结合，产生了一种计算效率高的 MECE 方法，可用于大规模材料识别问题，如涉及约 400,000 个未知模量的 3D 示例所示。最后，我们的数值结果表明，所提出的 MECE 方法可以比使用拟牛顿方法的 L(2) 最小化的传统方法快得多。