Magnetic Vector Potential Formulation Research Articles

Matrix-Free Edge-Domain Decomposition Method for Massively Parallel 3-D Finite Element Simulation With Field-Circuit Coupling

In this article, a novel edge-domain decomposition (EDD) method is proposed to solve 3-D nonlinear finite element (FE) problems of electromagnetic devices and transient field circuit co-simulation. The method applies reduced magnetic vector potential formulation to discretize the physical problem based on 3-D edge elements, and the solution region is divided into many sub-domains that only contain one edge unknown. The solution of lightweight nonlinear sub-domain systems can be massively parallelized, and the neighbor-to-neighbor communication scheme eliminates the need to assemble the global FE matrix. This article also introduces an indirect coupling scheme to handle large eddy currents to interface the EDD FE system with external circuits. The abovementioned algorithms are then implemented on a many-core GPU for transient field circuit co-simulation. The result shows an auto-gauging property, and the comparison with a commercial FE software indicates a speedup of over 43 times with relative error less than 2%.

Hybrid Formulation Domain-Decomposition Finite-Element Method for Simulating Eddy-Current Testing

Simulation of eddy-current testing (ECT) with the scanning of a ferrite-core coil using the conventional finite-element method (FEM) requires re-meshing, as it uses one solution domain to cover all the conducting materials and magnetic materials, which is cumbersome and results in low computation efficiency. In this article, we present the hybrid formulation domain-decomposition FEM to resolve the problem. The solution domain is decomposed into two subdomains, and the test sample and the ferrite core are included in different subdomains. The subdomains are discretized separately, and the discretization is performed only once. The reduced magnetic vector potential formulation is used in the subdomain of the test sample, whereas the magnetic scalar potential formulation is used in the subdomain of the ferrite core. The proposed method is applied in simulating the ECT of the aluminum plates having through holes with or without defects, and the efficiency of the method is demonstrated. Finally, experimental validation is performed.

3-D Numerical Modeling of Claw-Pole Alternators With its Electrical Environment

This article describes a methodology for modeling a six-phase claw-pole alternator with its electrical environment. Magnetic nonlinearities, eddy currents, and rectifiers are taken into account. To solve magnetodynamic problems, we use the modified magnetic vector potential formulation. The complex structure of the machine requires a 3-D finite-element analysis. To limit the mesh size, we introduced a refinement strategy based on the calculation of the time derivative of the magnetic vector potential, a solution to the magnetostatic case. In addition, we propose to reduce the transient state by improving the initial solution from the solution of a magnetostatic problem. These different numerical techniques drastically reduce the computational time and memory resources. To validate the proposed approach, some results are compared with experimental ones.

A Symmetric Field-Circuit Coupled Formulation for 3-D Transient Full-Wave Maxwell Problems

In this paper, a symmetric field-circuit coupled finite element method for low-frequency full-wave Maxwell problems using a magnetic vector potential formulation is proposed. The resultant fully discrete coefficient matrix is made symmetric for the first time by introducing the so-called source electric scalar potential for solid conductors, where the terminal currents are converted from the surface integrations of the current density vectors to volumetric integrations. The numerical examples, including a benchmark capacitor charging problem with external circuit connections, are solved, and the numerical results match well with the reference solutions. The proposed formulation is useful when analyzing electromagnetic fields with coupled inductive–capacitive effects and external circuit connections.

2-D Numerical Modeling of a Bulk HTS Magnetization Based on <bold>H</bold> Formulation Coupled With Electrical Circuit

Bulk high temperature superconductors (HTS) can be magnetized and act as permanent magnet much stronger than conventional ones as NdFeB. The design of the inductor is a key point to perform the desired magnetization of the HTS bulk. In this paper, we focus on modeling a pulsed field magnetization (PFM) process of an HTS bulk using a coil powered with a magnetizer. The built model is a 2-D axisymmetric problem, based on the H formulation and coupled with electrical equations though the magnetic flux seen by the magnetizing coil. The calculation of this magnetic flux in the H formulation is not trivial and was validated using magnetic vector potential formulation on a coil in the air. Assuming different operating conditions, the bulk HTS is then modeled using four different properties corresponding to air, perfect diamagnetic, copper, and HTS. It was shown that simulating a PFM process could lead to different value of peak current and applied magnetic field to the bulk HTS, depending on the critical current density of the bulk, for example. These variations are in the range of the air and diamagnetic cases. Therefore, the proposed method should be used in order to predict a realistic trapped magnetic field in the HTS bulk by taking into account its reaction seen by the coil during the PFM process.

Simulation of keyhole plasma arc welding with electro-magneto-thermo-hydrodynamic interactions

Keyhole-mode welding is an advantage of plasma arc welding (PAW), and much research has been conducted on keyhole formation and weld pool geometry. However, researchers usually constructed equivalent heat source models to represent the actual thermal arc energy. A unified keyhole PAW model was constructed to make a comprehensive analysis of the keyhole-mode heat transfer originating from the electro-magneto-thermo conversion within the plasma arc. The magnetic vector potential formulation was applied to obtain a more accurate electromagnetic field. Joule heating, electron energy transfer, and electromagnetic force were all taken into account. Temperature field and velocity field in arc region and workpiece were simultaneously predicted. It is found that the radial heat conduction flux is about two times of that in the axis direction along the keyhole wall. All heat fluxes jump dramatically near the joint of keyhole wall and top metal surface. Current density and electromagnetic force distributions revealed that there are highest Joule heating and electromagnetic effects near the electrode tip, and they decrease sharply along the radial direction. The experimental test was carried out on the 304 stainless steel, and the experimental images agree well with the calculated results. The orthogonal test shows that distance from torch to the workpiece, shrinkage of tungsten electrode, and orifice diameter are of the primary effect on the weld bead geometry, and the shielding gas flow rate has the least impact.

Modelling and experimental demonstration of a litz coil‐based high‐temperature induction heating system for melting application

This study presents analytical and numerical modelling and experimental demonstration of a litz wire-based induction heating system suitable for high-temperature application such as melting. The use of litz wire in induction heating coil minimises the skin effect and proximity effect and in turn improves the system efficiency. This makes it superior compared with a solid wire and copper tube induction coil. However, the limitation of litz wire is temperature withstanding capacity of the strand insulation material as a result of which it is rarely used in high-temperature applications (>700°C). Numerical modelling of induction heating process was done by using magnetic vector potential formulation and Fourier equations. Temperature-dependent material properties were taken account to precisely model the induction heating process. Analytically estimated and measured equivalent impedances of litz coil were compared at different frequency (1-20 kHz) for initial validation of simulation. The number of turns and frequency were selected as per required litz coil efficiency and induced power in crucible. Finally during the experimental demonstration, crucible and litz coil temperatures were measured by `K' type thermocouples and were compared with simulation results. The maximum temperature of ~750°C could be attained in the design while limiting the litz coil temperature within 85°C.

A 2-D Finite-Element Model for Electrothermal Transients in Accelerator Magnets

Superconducting accelerator magnets require sophisticated monitoring and means of protection due to the large energy stored in the magnetic field. Numerical simulations play a crucial role in understanding transient phenomena occurring within the magnet, and can, therefore, help to prevent disruptive consequences. We present a 2-D FEM model for the simulation of electro-thermal transients occurring in superconducting accelerator magnets. The magnetoquasistatic problem is solved with a modified magnetic vector potential formulation, where the cable eddy currents are resolved in terms of their equivalent magnetization. The heat balance equation is then investigated, and the relevant heat sources are discussed. The model implements a two-port component interface and is resolved, as part of an electrical circuit, in a cooperative simulation scheme with a lumped-parameter network.

Explicit time integration of eddy current problems using a selective matrix update strategy

PurposeDiscretizing the magnetic vector potential formulation of eddy current problems in space results in an infinitely stiff differential algebraic equation system that is integrated in time using implicit time integration methods. Applying a generalized Schur complement to the differential algebraic equation system yields an ordinary differential equation (ODE) system. This ODE system can be integrated in time using explicit time integration schemes by which the solution of high-dimensional nonlinear algebraic systems of equations is avoided. The purpose of this paper is to further investigate the explicit time integration of eddy current problems.Design/methodology/approachThe resulting magnetoquasistatic Schur complement ODE system is integrated in time using the explicit Euler method taking into account the Courant–Friedrich–Levy (CFL) stability criterion. The maximum stable CFL time step can be rather small for magnetoquasistatic field problems owing to its proportionality to the smallest edge length in the mesh. Ferromagnetic materials require updating the reluctivity matrix in nonlinear material in every time step. Because of the small time-step size, it is proposed to only selectively update the reluctivity matrix, keeping it constant for as many time steps as possible.FindingsNumerical simulations of the TEAM 10 benchmark problem show that the proposed selective update strategy decreases computation time while maintaining good accuracy for different dynamics of the source current excitation.Originality/valueThe explicit time integration of the Schur complement vector potential formulation of the eddy current problem is accelerated by updating the reluctivity matrix selectively. A strategy for this is proposed and investigated.

Robust full‐wave Maxwell solver in time‐domain using magnetic vector potential with edge elements

Magnetic vector potential (MVP) formulations are widely used in low-frequency eddy-current computation. For full-wave Maxwell problems with coupled resistive, inductive and capacitive effects, existing MVP formulations are either ungauged or with too complicated stabilisation procedure or with non-symmetric system matrix. Since sophisticated preconditioners are necessary to ensure good convergence rates of iterative solvers when used to solve the resultant linear system, it is also important to investigate symmetric gauged formulations which can be solved by more robust state-of-the-art sparse direct solvers. Traditional Coulomb gauged formulation using penalty technique is not feasible for edge elements since the divergence of the edge element basis function is zero within each element. In this study, a novel MVP formulation with Coulomb Gauge for full-wave Maxwell problems using edge elements is proposed. The proposed formulation is symmetric and stable for all frequencies, easy to implement and uniquely solvable. Numerical results obtained using the proposed MVP formulation are demonstrated to showcase its stability and accuracy. The method is very promising in further engineering applications.

Influence of the rotor eccentricity on the torque of a cage induction machine

AbstractThe non-uniform air gap in an electrical machine caused by rotor eccentricity creates an asymmetrical flux-density distribution in the air gap. This can affect the nominal torque produced by the machine. Eccentricity also produces forces that act on the rotor which may also have an effect on the torque. Thus, it is important to know how the torque of the machine behaves. In this paper, the torque of a cage induction machine is studied when the machine has dynamic eccentricity. The study is performed using the finite element method and a magnetic vector potential formulation. The torque is calculated by the method of energy balance. The harmonic components of the torque are also analyzed. The results show that the machine under eccentricity does not exhibit the same torque as a normal healthy machine. The harmonic components around the first principal slot harmonic is most affected.

A Novel Coulomb-Gauged Magnetic Vector Potential Formulation for 3-D Eddy-Current Field Analysis Using Edge Elements

Magnetic vector potential (MVP) formulation combined with finite edge elements has gained substantial application in 3-D eddy-current field computation. Compared with its magnetic scalar potential counterpart, the MVP formulation is much easier to treat multiply-connected conductors. With the availability of advanced direct linear solvers with high performance on parallel or multi-core platforms, it becomes more and more important to develop gauged MVP formulations. To this end, the gauging technique is essential to ensure that the MVP formulation is uniquely solvable by direct solvers. In this paper, a novel method for imposing Coulomb gauge to transient eddy-current problems using edge elements for approximating the MVP is proposed. The resultant linear matrix equation is symmetric and uniquely solvable. Current excitations and voltage excitations for stranded conductors and solid conductors are formulated, respectively. Field-circuit coupled scheme for solid conductors is also presented. Extensive numerical examples are computed using the proposed formulation, which has proved to be promising in further engineering applications.

A New Stable Full-Wave Maxwell Solver for All Frequencies

Magnetic vector potential (MVP) formulations are widely used in the electromagnetic (EM) field computation. For problems with both inductive and capacitive effects, existing full-wave MVP formulations are mostly nonuniquely solvable, and sophisticated preconditioners are indispensable when solving these resultant linear matrix equations. To find a stable formulation with a unique solution, it is important to incorporate the necessary physics equations and the gauge condition properly. Traditional Coulomb-gauged formulation using the penalty technique does not work for edge elements, since the divergence of the edge element basis function is zero within each element. In this paper, a novel MVP formulation with Coulomb gauge for full-wave Maxwell equations using edge elements is proposed in the frequency domain. Extensive numerical examples are computed to show that the proposed formulation is stable even for tiny frequencies, which is very useful in full-wave engineering EM field computation.

A Novel Gauged Vector Potential Formulation for 3-D Motional Eddy-Current Problems Using Edge Elements

The accurate computation of motional eddy currents is essential for the design of electromagnetic devices such as eddy-current brakes. When the conductors are uniform in the moving direction, it is convenient to formulate and solve the problem within a fixed coordinate system without needing to use two or more coordinate systems and connecting the fields in each region. Because ungauged magnetic vector potential (MVP) formulation can be solved using only iterative linear solvers, which are not so robust as direct solvers, it is desirable to impose proper gauge condition to ensure that the formulation has a unique solution. In this paper, a novel gauged formulation for motional eddy-current problems using finite edge element approximation of the MVP is proposed. Both transient and steady-state motional eddy-current problems are studied. A dummy scalar variable is introduced in the non-conducting region to enforce the Coulomb gauge, together with the gauge condition in the conducting region, to ensure that the resultant linear system can be solved using direct linear solvers. Three-dimensional numerical examples are presented to showcase the accuracy and usefulness of the proposed method for practical engineering computation.

A Novel Formulation With Coulomb Gauge for 3-D Magnetostatic Problems Using Edge Elements

A novel magnetic vector potential (MVP) formulation with Coulomb gauge is proposed for solving 3-D magnetostatic problems. A dummy scalar variable is introduced in the whole computation domain to impose the Coulomb gauge. Unlike traditional Coulomb-gauged formulation using the penalty technique, which works only for nodal elements approximation of the MVP, the new formulation proposed in this paper allows the use of edge element approximation of the MVP. It can avoid possible large errors at material interfaces or re-entrant corners of the problem domain when using nodal discretizations for the MVP. This formulation is easy to implement, and the resultant matrix system is symmetric with a unique solution. Both linear and nonlinear numerical examples are presented to showcase the accuracy and usefulness of the proposed method for engineering computation.

Multiple Right-Hand Side Techniques in Semi-Explicit Time Integration Methods for Transient Eddy-Current Problems

The spatially discretized magnetic vector potential formulation of magnetoquasistatic field problems is transformed from an infinitely stiff differential algebraic equation system into a finitely stiff ordinary differential equation (ODE) system by application of a generalized Schur complement for nonconducting parts. The ODE can be integrated in time using explicit time integration schemes, e.g. the explicit Euler method. This requires the repeated evaluation of a pseudo-inverse of the discrete curl-curl matrix in nonconducting material by the preconditioned conjugate gradient (PCG) method which forms a multiple right-hand side problem. The subspace projection extrapolation method and proper orthogonal decomposition are compared for the computation of suitable start vectors in each time step for the PCG method which reduce the number of iterations and the overall computational costs.

Efficient finite element method to estimate eddy current loss due to random interlaminar contacts in electrical sheets

AbstractElectrical sheets of electrical machines are laminated to reduce eddy current loss. However, a series of punching and pressing processes form random galvanic contacts at the edges of the sheets. These galvanic contacts are random in nature and cause an additional eddy current loss in the laminated cores. In this paper, a stochastic Galerkin finite element method is implemented to consider random interlaminar contacts in the magnetic vector potential formulation. The random interlaminar conductivities at the edges of the electrical sheets are approximated using a conductivity field and propagated through the finite element formulation. The spatial random variation of the conductivity causes the solution to be random, and hence, it is approximated by using a polynomial chaos expansion method. Finally, the additional eddy current losses due to the interlaminar contacts are estimated from a stochastic Galerkin method and compared with a Monte Carlo method. Accuracy and computation time of both models are discussed in the paper.

Design and simulation of the electromagnetic heating of a biological tissue

PurposeThe purpose of this paper is to present the design and the numerical calculation of the electromagnetic heating system for the ablation therapy. Hence, the heating of the tumor cells must be processed very carefully to achieve a localized coagulative necrosis and to avoid too high temperatures inside the tissue.Design/methodology/approachThe non-invasive method of the ablation therapy is implemented due to the inductive power transmission between the generator and implant. The ferromagnetic implant has a small size and can be placed intravenously into tumor cells. High-frequency driving currents are necessary to obtain high induced eddy currents within the ferromagnetic implant.FindingsFinite element analysis has been used for the design and numerical calculation of the electromagnetic heating system. The electromagnetic analysis is done in the time domain due to the nonlinearity of the ferromagnetic implant. Magnetic fields are computed based on a magnetic vector potential formulation. The thermal analysis is done in the time domain as well. The temperature computation in biological tissue is based on a heat balance equation.Research limitations/implicationsThis paper is focused on the design and simulation of the inductive system for the ablation therapy.Practical implicationsThe designed system can be practically implemented. It can be used for the clinical study of the immune response by the thermal ablation therapy.Originality/valueThe common method of thermal ablation is combined with an inductive power transmission. It enables a repetitive application of this method to study the immune response.

Explicit time integration of transient eddy current problems

SummaryFor time integration of transient eddy current problems, commonly implicit time integration methods are used, where in every time, step 1 or several nonlinear systems of equations have to be linearized with the Newton‐Raphson method because of ferromagnetic materials involved. In this paper, a generalized Schur complement is applied to the magnetic vector potential formulation, which converts a differential‐algebraic equation system of index 1 into a system of ordinary differential equations with reduced stiffness. For the time integration of this ordinary differential equations system of equations, the explicit Euler method is applied. The Courant‐Friedrich‐Levy stability criterion of explicit time integration methods may result in small time steps. Applying a pseudoinverse of the discrete curl‐curl operator in nonconducting regions of the problem is required in every time step. For the computation of the pseudoinverse, the preconditioned conjugate gradient method is used. The cascaded subspace extrapolation method is presented to produce suitable start vectors for these preconditioned conjugate gradient iterations. The resulting scheme is validated using the nonlinear TEAM 10 benchmark problem.

Using Reduced Support to Enhance Parallel Strong Scalability in 3D Finite-Element Magnetic Vector Potential Formulations with Circuit Equations

ABSTRACTQuasi-static electromagnetic problems involving, e.g., inductors, are often solved numerically using the finite-element method with magnetic vector and electric scalar potentials. Coupling the inductors to external circuits may however, lead, to large matrix equations that could become bottlenecks in parallel computation systems, particularly for strong scaling when more processors are introduced to scale down the total computation time. It is argued and shown by numerical simulations that the use of reduced support in the finite-element model improves the strong scalability of multiprocessor simulations due to the reduced communication between the global constraint owner processes and the finite-element equation owner processes.