We present the solution of the inverse problem for partially known elastic modulus values, e.g., the elastic modulus is known in some small region on the boundary of the domain from measurements. The inverse problem is posed as a constrained minimization problem and regularized with two different regularization types. In particular, the total variation diminishing (TVD) and the total contrast diminishing (TCD) regularizations are employed. We test both regularization strategies with theoretical diseased tissues, such as a stiff tumor surrounded by healthy background tissue and an atherosclerotic plaque having a soft inclusion surrounded by a stiff cap. In the present study, it is assumed that no traction data is available and the absolute elastic modulus distribution is calibrated from partially known elastic modulus values. We observe that this calibration fails with TVD regularization, while TCD regularization yields well-recovered absolute elastic modulus reconstructions in the presence of high noise levels in the displacement data. Finally, we investigate this problem analytically and provide an explanation for these observations. This work will advance efforts in parameter identification of heterogeneous materials as it provides a methodology to incorporate partially known parameters into the inverse problem formulation that will ultimately drive the inverse solution to an absolute and unique parameter distribution. This has great importance in classifying breast tumors based on their elastic modulus values and in planning surgical interventions of atherosclerotic plaques.
An inverse problem formulation of the immersed‐boundary method