Male accessory breast cancer is an extremely rare tumor. Several risk factors have been identified, including genetic and hormonal abnormalities. Accessory breast carcinoma usually occurs under the axilla or in the inguinal region. Clinical diagnosis is frequently delayed due to the general lack of awareness among physicians and patients. In the present study, the case of a 63-year-old male patient who was diagnosed with accessory breast cancer at a local advanced stage was reported. However, the patient was successfully treated with endocrine therapy.

We develop a new spectral-element method by incorporating a domain-decomposition technique, referred to as the dual-primal spectral-element tearing and interconnecting method (SETI-DP), for high-precision forward simulation of the 3D direct current (DC) resistivity method. To handle the terrain and complex underground structures, we use unstructured tetrahedral meshes to divide the computational domain. These meshes are then partitioned into nonoverlapping subdomains. By applying a continuity condition of the current density to be solved, all subdomains become coupled to each other. Using a Lagrange multiplier, the global equation system for our forward modeling can be simplified into that for the subdomain interfaces, and the continuity conditions on the subdomain interfaces are obtained by solving these subdomain interface equations. Finally, the forward problem for each subdomain is solved independently using these boundary conditions. Numerical experiments indicate that our algorithm can greatly improve computational efficiency and reduce memory consumption without losing accuracy when choosing the appropriate number of subdomains. We further apply our method to the forward simulation of a tunnel detection problem. The numerical results demonstrate that our new SETI-DP method can effectively simulate the tunnel’s complex structures, and the DC resistivity method can detect the adverse geologic bodies in front of the tunnel face.

The traditional three-dimensional (3D) magnetotelluric (MT) forward modeling using Krylov subspace algorithms has the problem of low modeling efficiency. To improve the computational efficiency of 3D MT forward modeling, we present a novel geometric multigrid algorithm for the finite element method. We use the vector finite element to discretize Maxwell’s equations in the frequency domain and apply the Dirichlet boundary conditions to obtain large sparse complex linear equations for the solution of EM responses. To improve the convergence of the solution at low frequencies we use the divergence correction to correct the electric field. Then, we develop a V-cycle geometric multigrid algorithm to solve the linear equations system. To demonstrate the efficiency and effectiveness of our geometric multigrid method, we take three synthetic models (COMMEMI 3D-2 model, Dublin test model 1, modified SEG/EAEG salt dome model) and compare our results with the published ones. Numerical results show that the geometric multigrid algorithm proposed in this paper is much better than the commonly used Krylov subspace algorithms (such as SOR-GMRES, ILU-BICGSTAB, SOR-BICGSTAB) in terms of the iteration number, the solution time, and the stability, and thus is more suitable for large-scale 3D MT forward modeling.

As the structure of the underground space becomes increasingly complex, traditional two-dimensional seismoelectric methods are no longer adequate for the comprehensive exploration. To achieve precise imaging of the underground space, it is in urgent need to develop three-dimensional full-waveform modeling techniques. In this paper, we propose a three-dimensional time-domain finite-element method to solve the seismoelectric wave field in saturated porous media. Since the electroosmotic feedback is very small, we can ignore the mechanical disturbance caused by the electromagnetic fields induced by seismic waves, and thereby can decouple the electrokinetic coupling equations and separately solve the seismic and electromagnetic waves. For the simulation of seismic wavefield, we employ the explicit finite-element method, and utilize a lumped mass matrix instead of a consistent mass matrix to facilitate explicit recursion. Additionally, we apply the complex frequency-shifted unsplit perfectly matched layer technique to effectively handle seismic boundary conditions. Then, the velocity fields obtained by solving the poroelastic equations serve as the source term of the electromagnetic equations, and the finite-element method is used to solve the electromagnetic wavefield. Considering that the huge velocity difference exists between the electromagnetic and seismic waves, we adopt an unconditionally stable implicit method for the solution of the electromagnetic wavefield. By combining explicit and implicit recursion, the computational efficiency can be improved significantly. The accuracy of our time-domain finite-element algorithm is validated by checking our results against the analytical solutions for a half-space model. Furthermore, we conduct numerical simulations and analyses on a typical block model and a modified SEG/EAEG salt dome model.