high-performance-Multiscale Finite Element Methods hp-MSFEMs

The iron core of electrical devices is laminated to reduce the eddy current (EC) losses. The geometric dimensions are extremely different. The thickness of iron laminates is about 0.3mm and separated by quite small air gaps. On the other hand, the overall dimensions of the core are in the meter range consisting of up to several thousands of laminates. Finite element (FE) simulations are indispensable for the optimal design of the devices. However, modeling of each laminate by FEs leads to an extremely large nonlinear system of equations, well above hundreds of millions, which cannot reasonably be solved with present computer capacities. The development of multiscale finite element methods (MSFEMs) for ECs in laminated iron has brought the simulation capabilities a major step forward. Nevertheless, MSFEMs suffer from various shortcomings and the computational costs are still too high. There is no error estimator for MSFEMs which is a big problem to trust in MSFEM solutions. MSFEMs in 3D are severely restricted to simple problems. These difficulties are major challenges in computational electromagnetics. Completely new methods will be developed to achieve the breakthrough of MSFEMs. Adaptive MSFEMs facilitating MSFEMs with different potential formulations, higher order MSFEMs, harmonic balance MSFEMs etc. are absolutely unique and novel. Efficient and reliable equilibrated local error estimators with a computable constant and based on the theorem of Prager and Synge to allow both h- and p-refinement will be developed. The simulation of one laminate often suffices for electrical machines assuming common simplifications. To avoid expensive 3D FE simulations, completely new space splitting 2-D/1- D methods will be developed considering in particular the edge effect and Biot-Savart-fields, leading to a major reduction of computational costs. The missing models for interfaces with large stray fields are a very serious shortcoming of MSFEMs in 3D. Feasible solutions in 3D are of exceptional importance, because stray fields are present in almost all electrical devices. Current MSFEM approaches vanish in air which represents a serious problem. Thus, radically new approaches have to be found. Novel nonlinear model order reduction (MOR) schemes have to be developed for different MSFEMs to achieve tremendous savings in computational costs. MOR schemes exploiting the specific nature of MSFEMs of ECs will be provided to cope with large nonlinear systems of equations. The aims of the project are a strong reduction of the high computational costs of MSFEMs to run such simulations on personal computers without any difficulty. Unique adaptive MSFEMs will guarantee highly accurate and efficient MSFEM solutions. Novel MSFEMs will solve the still existing shortcomings in 3D.

The iron core of electrical devices is laminated to reduce the eddy current (EC) losses. The geometric dimensions are extremely different. The thickness of iron laminates is about 0.3mm and separated by quite small air gaps. On the other hand, the overall dimensions of the core are in the meter range consisting of up to several thousands of laminates. Finite element (FE) simulations are indispensable for the optimal design of the devices. However, modeling of each laminate by FEs leads to an extremely large nonlinear system of equations, well above hundreds of millions, which cannot reasonably be solved with present computer capacities. The development of multiscale finite element methods (MSFEMs) for ECs in laminated iron has brought the simulation capabilities a major step forward. Nevertheless, MSFEMs suffer from various shortcomings and the computational costs are still too high. There is no error estimator for MSFEMs which is a big problem to trust in MSFEM solutions. MSFEMs in 3D are severely restricted to simple problems. These difficulties are major challenges in computational electromagnetics. Completely new methods will be developed to achieve the breakthrough of MSFEMs. Adaptive MSFEMs facilitating MSFEMs with different potential formulations, higher order MSFEMs, harmonic balance MSFEMs etc. are absolutely unique and novel. Efficient and reliable equilibrated local error estimators with a computable constant and based on the theorem of Prager and Synge to allow both h- and p-refinement will be developed. The simulation of one laminate often suffices for electrical machines assuming common simplifications. To avoid expensive 3D FE simulations, completely new space splitting 2-D/1- D methods will be developed considering in particular the edge effect and Biot-Savart-fields, leading to a major reduction of computational costs. The missing models for interfaces with large stray fields are a very serious shortcoming of MSFEMs in 3D. Feasible solutions in 3D are of exceptional importance, because stray fields are present in almost all electrical devices. Current MSFEM approaches vanish in air which represents a serious problem. Thus, radically new approaches have to be found. Novel nonlinear model order reduction (MOR) schemes have to be developed for different MSFEMs to achieve tremendous savings in computational costs. MOR schemes exploiting the specific nature of MSFEMs of ECs will be provided to cope with large nonlinear systems of equations. The aims of the project are a strong reduction of the high computational costs of MSFEMs to run such simulations on personal computers without any difficulty. Unique adaptive MSFEMs will guarantee highly accurate and efficient MSFEM solutions. Novel MSFEMs will solve the still existing shortcomings in 3D.