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Complex multi-scale Finite Element (FE) analyses always involve high number of elements and therefore require very long time of computations. This is caused by the fact, that considered effects on smaller scales have greater influences on the whole model and larger scales. Thus, mesh density should be as high as required by the smallest scale factor. New submodelling routine has been developed to sufficiently decrease the time of computation without loss of accuracy for the whole solution. The presented approach allows manipulation of different mesh sizes on different scales and, therefore total optimization of mesh density on each scale and transfer results automatically between the meshes corresponding to respective scales of the whole model. Unlike classical submodelling routine, the new technique operates with not only transfer of boundary conditions but also with volume results and transfer of forces (current density load in case of electromagnetism), which allows the solution of full Maxwell's equations in FE space. The approach was successfully implemented for electromagnetic solution in the forward problem of Magnetic Field Tomography (MFT) based on Magnetoencephalography (MEG), where the scale of one neuron was considered as the smallest and the scale of whole-brain model as the largest. The time of computation was reduced about 100 times, with the initial requirements of direct computations without submodelling routine of 10 million elements.

A new submodelling technique for multi-scale finite element computation of electromagnetic fields: Application in bioelectromagnetism