# A Size-Dependent Cost Function to Solve the Inverse Elasticity Problem
## 4 Discussion  
This paper presents a novel inverse method to characterize the shear modulus distribution of solids. This method modifies the standard cost function that minimizes the absolute error between the measured and computed displacements, and it is capable of improving the shear modulus contrast of inclusions with varying sizes. We have performed a number of 2-D numerical simulations to test the feasibility of this approach, and compared the performance of the standard and proposed approaches. The reconstruction results revealed that the proposed approach performs much better than the standard method, and improves the shear modulus contrast in the inclusions significantly. Additionally, a 1-D theoretical analysis was conducted to validate what we have observed in 2-D simulations.  
The simulations presented in Section 3.1 revealed that the solution to the inverse problem utilizing the standard approach is size-dependent. More specifically, the smaller inclusion was underestimated, and the underestimation became more significant with the reduction of the size of the inclusion. To address this size-sensitivity issue, a modified cost function depending on the size of the inclusion was proposed in this paper. The new approach improved the shear modulus contrast in inclusions remarkably. In fact, the limitation of the standard L2 norm-based optimization inverse approaches was also discussed in [16]. The authors in [16] observed that the solution to the inverse problem by minimizing the standard L2 norm cost function is boundary-sensitive (see Figure 1). Therefore, minimizing L2 norm cost function might not be an optimal way to solve the inverse problem in practical engineering to characterize mechanical property distribution of solids.  
We also employed an 1-D theoretical analysis to compare the new and standard approaches with a wide range of size ratios of the two inclusions (see Figure 5). The main merit of this 1-D analysis is that a large number of cases can be tested within seconds. This analysis showed that the new approach reduced the relative error in the shear modulus contrast in the inclusions with different sizes. We should note that we utilized the exact simulated displacement data to evaluate Equations (6) and (8) without introducing noise, since the underestimation of the shear modulus contrast in inclusions  
Figure 5: (a)The variation of the relative error defined above with respect to the size of the right inclusion, \(a^{2}\), with underestimation \(\varepsilon=40\%\). (b) The variation of the relative error defined above with respect to \(a^{2}\) with \(\varepsilon=20\%\).  
using the standard inverse approach is induced by the regularization. Though the new approach is capable of improving the shear modulus contrast in inclusions, the recovered shear modulus of the smaller inclusion is still underestimated, particularly in cases with a larger size ratio of the inclusions. In addition, we merely restricts our attention in the size sensitivity issue in this paper, but do not consider the boundary sensitivity issue. Thus, an improved and more general approach should be proposed in future work to completely resolve the issue. Besides, the displacement data can be obtained in high resolution by digital imaging correlation (DIC) [15; 17; 18; 19]. In the future, we will test the proposed approach using experimental data from the DIC system.