# A Size-Dependent Cost Function to Solve the Inverse Elasticity Problem
## 3 Results  
### Modulus Reconstruction Acquired by Simulated Data  
We test the standard and newly proposed inverse algorithms by the simulated data. The displacement fields are acquired by solving forward problems with a known shear modulus distribution (see Figure 2a), assuming that the solid is in the state of the incompressible plane strain. The forward problem is solved by the stabilized finite element method [13]. In the target shear modulus distribution, there are two inclusions embedded in the soft background. The stiffness ratio between the two inclusions and the background is 5:1. The size ratio between the right and left inclusions varies from 1 to 2. We applied 1% uniform compression on the top edge and restricted the vertical motion of the bottom edge. To prevent rigid body motion, we fix the center point of the bottom edge. The problem domain is discretized uniformly by 14,400 bilinear elements (120 elements along the \(x\) and \(y\) directions, respectively). To mimic the experimental data, we add 3% noise into the simulated displacement data. In addition, we utilize the vertical displacement component as the measured data in solving the inverse problem, since the displacements perpendicular to the direction of the compressive loading are usually measured with low precision in practice [14].  
When these two inclusions have the same size (see Figure 2b), we observe that these two inclusions could be detected clearly from the background utilizing the standard inverse scheme. However, the shear modulus values of these two inclusions are roughly half of the target. This is probably due to the fact that the sizes of the inclusions are too small compared to the overall dimension of the problem domain. Thus, the shear modulus reconstruction is highly sensitive to the noise. We also find that the recovered shear modulus of the left is larger than that of the right inclusion, due to the noise effect.  
When we increase the radius of the right inclusion (see Figure 3), the reconstructed results utilizing the standard approach demonstrate that the left inclusion is remarkably underestimated (see Figure 3b,e). Moreover, the underestimation became more significant when the size ratio between the two inclusions increases. More specifically, the recovered shear modulus contrast between the right and left inclusions increases from 1.3 to 1.4. The reason for this will be explained in the next section. When we utilize the proposed inverse approach (see Figure 3c,f), it is observed that the stiffness values of the left inclusion improve significantly. That is, the recovered shear modulus contrasts between the right and left inclusions for Figure 3c,f are 0.99 and 0.95, respectively. However, we also observe that the shape of left inclusion is slightly distorted compared to the shear modulus reconstruction by the standard approach. This is because the weight of the nodes in the inclusion is inversely proportional to its area. Thus, the smaller inclusion gain more weight in the new cost function (Equation (4)), leading to the full minimization of displacements of the left inclusion in the new method.