Topology optimization applied to the acoustic medium inverse problem in the time domain using integer linear programming - Moreira, João B. D.; Gonçalves, Juliano F.; Sivapuram, Raghavendra; Carmo, Bruno S.; Silva, Emílio C. N. (Structural and multidisciplinary optimization, 2023)
In this paper, the acoustic inverse problem modeled in the time domain featuring wave velocity reconstruction in the presence of sharp interfaces is addressed using an integer design variable approach. The medium being reconstructed is assumed piecewise constant, with single-material obstacles embedded in a homogeneous background. The wave equation is modeled using the Finite Element Method (FEM). The inversion procedure aims at finding the parameter field that minimizes a least-squares misfit function with respect to data generated from a synthetic model. The proposed optimization methodology is based on a sequential Integer Linear Programming (ILP) formulation used in the field of Topology Optimization (TO). Since this is a gradient-based technique, the sensitivity with respect to the integer design variable is evaluated by the adjoint method. Sensitivities are modified using both damping filters and Helmholtz-type Partial Differential Equation (PDE) filters to deal with the ill-posedness that is inherent to this class of inverse problem. The integer design variable is binary, associating each point of the domain either to the homogeneous background or to the embedded obstacles. This description naturally incorporates the sharp interface hypothesis, whereas a continuous design variable may generate transition regions with intermediate values and no clearly defined boundary. The damping filter is successful in controlling instabilities by incorporating the whole optimization history to the design update. Furthermore, the generality and effectiveness of the proposed framework are evaluated by addressing 2D problems from the literature and a proposed 3D case, all featuring sharp interfaces.