A hybrid boundary element method (BEM)/finite element method (FEM) approach is proposed in order to properly consider the anisotropic properties of the cardiac muscle in the magneto- and electrocardiographic forward problem. Within the anisotropic myocardium a bidomain model based FEM formulation is applied. In the surrounding isotropic volume conductor the BEM is adopted. Coupling is enabled by requesting continuity of the electric potential and the normal of the current density across the boundary of the heart. Here, the BEM part is coupled as an equivalent finite element to the finite element stiffness matrix, thus preserving in part its sparse property. First, continuous convergence of the coupling scheme is shown for a spherical model comparing the computed results to an analytic reference solution. Then, the method is extended to the depolarization phase in a fibrous model of a dog ventricle. A precomputed activation sequence obtained using a fine mesh of the heart was downsampled and used to calculate body surface potentials and extracorporal magnetic fields considering the anisotropic bidomain conductivities. Results are compared to those obtained by neglecting in part or totally (oblique or uniform dipole layer model) anisotropic properties. The relatively large errors computed indicate that the cardiac muscle is one of the major torso inhomogeneities.

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Since the storage requirements of the BEM are proportional to 2 , only relative small problems can be solved on a PC or a workstation. In this paper we present an adaptive multilevel fast multipole method for the solution of electrostatic problems with the BEM. We will show, that in practice the storage requirements and the computational costs are approximately proportional to and therefore even large three dimensional problems can be solved on a relative small computer.Index Terms-Boundary element methods, fast multipole method, iterative methods, Laplace equations.

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PurposeThe purpose of this paper is to present an application of augmented reality (AR) in the context of teaching of electrodynamics. The AR visualization technique is applied to electromagnetic fields. Carrying out of numerical simulations as well as preparation of the AR display is shown. Presented examples demonstrate an application of this technique in teaching of electrodynamics.Design/methodology/approachThe 3D electromagnetic fields are computed with the finite element method (FEM) and visualized with an AR display.FindingsAR is a vivid method for visualization of electromagnetic fields. Students as well as experts can easily connect the characteristics of the fields with the physical object.Research limitations/implicationsThe focus of the presented work has been on an application of AR in a lecture room. Then, easy handling of a presentation among with low‐hardware requirements is important.Practical implicationsThe presented approach is based on low‐hardware requirements. Hence, a presentation of electromagnetic fields with AR in a lecture room can be easily done. AR helps students to understand electromagnetic field theory.Originality/valueWell‐known methods like FEM and AR have been combined to develop a visualization technique for electromagnetic fields, which can be easily applied in a lecture room.