A direct method to identify Young’s moduli and boundary conditions of the heterogeneous material

Identifying unknown Young’s moduli and boundary conditions of the heterogeneous material using locally observed boundary data is the inverse problem which is generally solved by iterative methods. In this paper, a two-steps direct method is proposed for the first time to solve this inverse problem without iterations. The proposed method innovatively decomposes the heterogeneous elasticity inverse problem to two homogeneous elasticity sub-inverse problems. The single-data and multiple-data based direct methods are applied to identify background Young’s modulus and displacement boundary conditions, while the Maxwell–Betti principle based direct method and the equivalent boundary force based direct method are proposed to identify Young’s moduli of inclusions. In addition, an optimal experimental design method with a goal-oriented criterion is proposed to improve the accuracy of the two-steps direct method by optimizing the force application positions in observation data acquisition. Both numerical and physical experiments were conducted. The results demonstrate the feasibility of the proposed two-steps direct method and its optimal experimental design method.