# A Size-Dependent Cost Function to Solve the Inverse Elasticity Problem
## 1 Introduction  
The advancement of high-resolution imaging modalities like digital imaging correlation (DIC) makes it possible for us to measure full-field displacement fields with high accuracy [1; 2; 3]. The measured full-fields displacement fields can be utilized to identify the nonhomogeneous elastic distribution of solids. Mapping the nonhomogeneous elastic distribution usually requires solving an inverse problem in elasticity. There are many inverse approaches that have been developed to solve the inverse problem, including virtual fields methods [4; 5; 6], optimization-based approaches [7; 8], etc. Among them, optimization-based inverse approaches have been commonly used, as they are rigorous and highly robust. In the optimization-based inverse approach, the inverse problem in elasticity is posed to be a constrained minimization problem, where a cost function accounting for the discrepancy between measured and computed displacements is usually defined in the L2 norm and minimized iteratively. Since noise in the measured data will affect the quality of the final reconstructed results, regularization is introduced to prevent over-fitting. The regularization term is usually added as an additional termin the cost function (see Equation (3)). By setting an optimal weight for the regularization term, the inverse problem is solved without fully minimizing the displacement discrepancy term.  
However, recent studies have revealed that the cost function that minimizes the absolute error in the displacement discrepancy term does not perform well in some practical cases. In [9], the authors found that the solution to the inverse problem utilizing this standard cost function is boundary-sensitive. That is, the reconstructed elastic property distribution of solids is sensitive to the prescribed boundary conditions (see Figure 1).  
To address this issue, a modified cost function was proposed and significantly improved the reconstruction results--in particular, the stiffness contrast between different heterogeneous regions (reconstructed shear moduli in inclusions in Figure 1). The modified cost function was spatially weighted by the strain fields.  
In this paper, we will investigate another limitation of the standard optimization-based inverse approach. That is, the solution to the inverse problem depends on the size of the inclusions (see Figure 2). To address this issue, a novel cost function that is size-dependent will be proposed. The solid studied in this paper is assumed to be incompressible linear elasticity. We will adopt a number of simulations and a simple theoretical analysis to show why the proposed approach performs better than the standard approach.  
Figure 1: (**a**) Target shear modulus distribution. (**b**) Reconstructed shear modulus distribution when the loading is uniform. (**c**) Reconstructed shear modulus distribution when the loading is non-uniform.  
Figure 2: (**a**) Target shear modulus distribution where two inclusions are of the same size. (**b**) Reconstructed shear modulus distribution with a regularization factor of \(3.0\times 10^{-10}\).  
The paper is organized as follows. We will first discuss the standard and proposed inverse algorithms in **Methods** section. We will then solve the inverse problem of elasticity by both methods with the simulated displacement fields acquired by finite element simulations, in the **Results** section. To understand the performance of the standard and proposed inverse approaches, we will employ a one-dimensional (1-D) model composed of two elastic bars and perform a simple theoretical analysis. Afterwards, we will thoroughly elaborate the reconstruction results in the **Discussion** section.