IEEE TRANSACTIONS ON MAGNETICS, VOL. 52, NO. 3, MARCH 2016 7002004 Space-Time Field Projection: Finite-Element Analysis Coupled Between Different Meshes and Different Time-Step Settings Zifu Wang 1 , Thomas Henneron 2 , and Heath Hofmann 3 1 Jing-Jin Electric North America, Rochester Hills, MI 48309 USA 2 Lille Laboratory of Electrical Engineering and Power Electronics, Université Lille 1, Villeneuve d’ASCQ 59655, France 3 Department of Electrical Engineering and Computer Sciences, University of Michigan, Ann Arbor, MI 48109 USA In this paper, field projection methods are used to couple finite-element analysis carried out on different meshes and different temporal discretization bases. The prior-art mesh-to-mesh spatial field projection is extended to the time domain in this paper. We first define a space-time error norm between a given field distribution and the target field to be determined, and then minimize it using the Galerkin method. Biorthogonal test functions are also introduced into the projection process to replace inner product matrices with diagonal matrices and reduce the computation cost in terms of memory as well as the calculation time required. Index Terms—Biorthogonal functions, field projection, finite-element methods, Galerkin method, modeling. I. I NTRODUCTION I N ORDER to analyze a multi-physics problem using weak-coupling technique, the original problem is decom- posed into several subproblems of different physic natures (e.g., electromagnetic, thermal, or mechanical) and is then solved separately using finite-element methods. Such subprob- lems usually have different areas of interest and different discretization bases. For higher accuracy and lower compu- tation cost, each subproblem is preferred to be solved on a mesh optimized for its own. In this case, the communi- cation between the subproblems can be carried out through mesh-to-mesh field projection [1]–[3]. Similar spatial field transfer examples can be found in domain decomposition methods, remeshing, source field discretization, and so on [2], [4]–[7]. However, in multi-physics problems, we also want the flexibility to employ different time-step settings for the subproblems. For instance, the time constants for electro- magnetic and thermal analysis are often very different from each other. Sometimes variable time step can also be used to accelerate solution. Under such condition, the forced use of a common time-step size for all subproblems can be very expensive in terms of computation cost. To overcome such inconvenience, in this paper, we develop space-time field projection. It allows the unidirectional coupling of finite-element analysis (FEA) carried out on different meshes and different temporal discretization bases. Biorthogonal functions are then applied as test functions of the Petrov–Galerkin method, in order to reduce computation cost. II. MESH- TO-MESH FIELD PROJECTION First, allow us to briefly recall the mesh-to-mesh spatial projection of electromagnetic fields. In the following, we note Manuscript received July 5, 2015; revised September 4, 2015 and October 6, 2015; accepted October 12, 2015. Date of publication October 16, 2015; date of current version February 17, 2016. Corresponding author: Z. Wang (e-mail: zifu.wang@outlook.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2015.2490600 H for the magnetic field, B for the magnetic flux density, and μ is the magnetic permeability. In this paper, only linear materials are considered. We denote by H s the magnetic field obtained on the source mesh and by H t the magnetic field to be calculated on the target mesh. Here, H s is supposed to be calculated using H-conforming formulation (e.g. the formulation based on scalar potential ). The mesh-to-mesh spatial energetic norm of the interpolation error over domain D is defined as [8] ε H =  D μ 2 ‖H t - H s ‖ 2 d τ. (1) To project fields onto a target mesh with respect to the magnetic energy, weak formulation can be developed based on the minimization of the energetic norm (1). III. SPACE-TIME FIELD PROJECTION For illustration, for space-time field projection, we also consider the magnetic field H as example. The proposed projection methods can, however, be generalized to all electromagnetic fields. In the space-time projection, the given field on the source mesh H s and the field to be calculated on the target mesh H t are also discretized using different temporal dis- cretization bases. As source field, the spatial and temporal variation H s (x , t ) is known (using interpolation) on domain D during time T . We define the space-time error norm between H t and H s ε H =  T  D μ 2 ‖H t - H s ‖ 2 d τ dt . (2) Since H t ∈ H(curl, D), edge elements offer the most suitable spatial discretization base. In terms of temporal discretization, we use a linear interpolation function. Thus, H t writes H t (x , t ) =  i =1,..., E , j =1,..., N w e i (x )w t j (t ) X ij (3) 0018-9464 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.