The solution of Maxwell’s equations in multiphysics Klaus-Jürgen Bathe a,⇑ , Hou Zhang b , Yiguang Yan b a Massachusetts Institute of Technology, Cambridge, MA, USA b ADINA R&D, Inc., Watertown, MA, USA article info Article history: Received 20 August 2013 Accepted 29 September 2013 Available online 19 November 2013 Keywords: Electromagnetics Maxwell’s equations Structures Navier–Stokes equations Multiphysics Fully-coupled response abstract We consider the solution of the fully-coupled equations of electromagnetics with fluid flows and struc- tures. The electromagnetic effects are governed by the general Maxwell’s equations, the fluid flows by the Navier–Stokes equations, and the solids and structures by the general Cauchy equations of motion. We present an effective general finite element formulation for the solution of the Maxwell’s equations and demonstrate the coupling to the equations for fluids and structures. For the solution, we can use the elec- tric field and magnetic field intensities, or the electric and magnetic potentials, with advantages depend- ing on the problem solved. We give various example solutions that illustrate the use of the solution procedure. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction During the recent years, an increasing emphasis has been placed on the solution of multiphysics problems [1]. While the solution of problems considered separately in solids and struc- tures, in fluid flows, and in electromagnetics (EM) has been pur- sued for decades – and widely-used quite powerful computer programs are now available – the solution of problems in which general structures interact with fluid flows and electromagnetic waves has hardly been tackled and presents special difficulties. In- deed, only specific problems have been solved in which the solu- tion techniques have been developed specifically for the physical problem considered, see for example, refs. [2–11]. Considering the analysis of solids and structures coupled with fluid flows, many publications have recently appeared, and numer- ous applications are found, in particular, in biomechanics and the automotive and airplane industries. The next step for general mul- tiphysics solutions is clearly that electromagnetic effects should also be included. In today’s time, electrical devices are used daily by almost everybody in a multitude of applications, and to reach optimal designs the structural, fluid and electromagnetic fully-cou- pled effects would ideally be considered. These coupled effects can be particular important, for example, in problems of magneto-solid and fluid mechanics, in medical applications and biomedical engi- neering, metal processing, and plasma physics, see ref. [12] and the references therein. Numerous publications are also available on the numerical solution of electromagnetic field problems. In the most general cases, the general Maxwell’s equations are considered. However, while finite element solutions have been obtained for some dec- ades, the earliest attempts frequently showed spurious modes and in that sense were not reliable [13,14]. Thereafter, special fi- nite element schemes were designed, and in particular the edge- based elements [15]. These elements are more reliable but have the shortcomings that the edge degrees of freedom are difficult to couple with the usual nodal degrees of freedom used in the fi- nite element analyses of fluid flows and structures, divergence-free conditions are considered, the convergence is not optimal, and the elements do not directly fit into the usual post-processing schemes used. In more recent research, various discontinuous finite element schemes and meshless methods have been proposed, see for exam- ple, Nicomedes et al. [16] and Badia and Codina [17], but these pro- cedures are computationally quite costly or contain artificial numerical factors for general practical analyses. Our objective in this paper is to present a novel finite element scheme for the solution of the general Maxwell’s equations specif- ically developed to solve for electromagnetic effects coupled with fluid flows, solids and structures, while keeping our philosophy for the development of finite element procedures in mind [18]. Since we amply published on our solution procedures for fluid flows with structural interactions previously [19–22], we focus in this paper on the solution of Maxwell’s equations, to couple with the governing equations of fluids and structures. We consider the static and harmonic solutions of the Maxwell’s equations, includ- ing the solution of high-frequency problems, and present a general uniform procedure for solution in which either the primitive 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.09.006 ⇑ Corresponding author. Tel.: +1 6179265199. E-mail address: kjb@mit.edu (K.J. Bathe). Computers and Structures 132 (2014) 99–112 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc