Maxwell's Equations in Accelerated Reference Frames and their Application in Computational Electromagnetism

Abstract In many engineering applications the interaction between the electromagnetic field and moving bodies is of great interest. E.g., motional induced eddy currents have to be taken into account correctly for the modelling and simulation of high-speed solenoid actuators. In connection with computational electromagnetism, it seems natural to use a Lagrangian (also called material) description. The unknowns are defined on the mesh, which moves and deforms together with the considered objects. What is the correct form of Maxwell's and the constitutive equations under such circumstances? Since the bodies might undergo accelerated motion, this question cannot in general be answered by the application of Lorentz transforms. Consequently, Maxwell's equations do not necessarily have their usual form in accelerated frames of reference. This was demonstrated in a classical paper by Schiff [1], where it is shown that a significant difference occurs even at "low" velocities, which are small compared to the velocity of light. In contrast, it is convenient to perform the analysis of rotating induction machines from the rotor's point of view. Despite the acceleration, starting from the usual form of Maxwell's equations yields the correct results. How could that be possible? There are only few publications that address the subject from a general point of view and not only for a restricted class of examples, e.g. Moreover, DFs allow separating the topological from the metric part of the theory. Using a noninertial frame induces a metric that is different from the standard Lorentz metric. This metric enters the formulation only through the coordinate expression for the four-dimensional Hodge operator. A localization transform can be introduced, to revert to a (3+1)-dimensional description. This is connected to the concept of a co-moving observer. The result is a relativistically correct Lagrangian form of Maxwell's and the constitutive equations. For "small" accelerations, i.e. if the extension of the system is neglectable compared to the radii of curvature, a concise set of transforms for all the relevant field quantities can be derived. These transforms are well suited for the implementation into numerical field computation codes

Accuracy of fully coupled and sequential approaches for modeling hydro- and geomechanical processes

Subsurface flow and geomechanics are often modeled with sequential approaches. This can be computationally beneficial compared with fully coupled schemes, while it requires usually compromises in numerical accuracy, at least when the sequential scheme is non-iterative. We discuss the influence of the choice of scheme on the numerical accuracy and the expected computational effort based on a comparison of a fully coupled scheme, a scheme employing a one-way coupling, and an iterative scheme using a fixed-stress split for two subsurface injection scenarios. All these schemes were implemented in the numerical simulator DuMux. This study identifies conditions of problem settings where differences due to the choice of the model approach are as important as differences in geologic features. It is shown that in particular transient and multiphase flow, effects can be causing significant deviations between non-iterative and iterative sequential schemes, which might be in the same order of magnitude as geologic uncertainty. An iterated fixed-stress split has the same numerical accuracy as a fully coupled scheme but only for a certain number of iterations which might use up the computational advantage of solving two smaller systems of equations rather than a big monolithical one.Deutsche ForschungsgemeinschaftH2020 European Research Counci

Numerical Methods for Flow in Fractured Porous Media

In this work we present the mathematical models for single-phase flow in fractured porous media. An overview of the most common approaches is considered, which includes continuous fracture models and discrete fracture models. For the latter, we discuss strategies that are developed in literature for its numerical solution mainly related to the geometrical relation between the fractures and porous media grids

A hybrid-dimensional coupled pore-network/free-flow model including pore-scale slip and its application to a micromodel experiment

Modeling coupled systems of free flow adjacent to a porous medium by means of fully resolved Navier-Stokes equations is limited by the immense computational cost and is thus only feasible for relatively small domains. Coupled, hybrid-dimensional models can be much more efficient by simplifying the porous domain, e.g., in terms of a pore-network model. In this work, we present a coupled pore-network/free-flow model taking into account pore-scale slip at the local interfaces between free flow and the pores. We consider two-dimensional and three-dimensional setups and show that our proposed slip condition can significantly increase the coupled model’s accuracy: compared to fully resolved equidimensional numerical reference solutions, the normalized errors for velocity are reduced by a factor of more than five, depending on the flow configuration. A pore-scale slip parameter βpore required by the slip condition was determined numerically in a preprocessing step. We found a linear scaling behavior of βpore with the size of the interface pore body for three-dimensional and two-dimensional domains. The slip condition can thus be applied without incurring any run-time cost. In the last section of this work, we used the coupled model to recalculate a microfluidic experiment where we additionally exploited the flat structure of the micromodel which permits the use of a quasi-3D free-flow model. The extended coupled model is accurate and efficient.Projekt DEALDeutsche Forschungsgemeinschaf

A monotone multigrid solver for two body contact problems in biomechanics

The purpose of the paper is to apply monotone multigrid methods to static and dynamic biomechanical contact problems. In space, a finite element method involving a mortar discretization of the contact conditions is used. In time, a new contact-stabilized Newmark scheme is presented. Numerical experiments for a two body Hertzian contact problem and a biomechanical application are reported

Modeling and discretization of flow in porous media with thin, full-tensor permeability inclusions

When modeling fluid flow in fractured reservoirs, it is common to represent the fractures as lower-dimensional inclusions embedded in the host medium. Existing discretizations of flow in porous media with thin inclusions assume that the principal directions of the inclusion permeability tensor are aligned with the inclusion orientation. While this modeling assumption works well with tensile fractures, it may fail in the context of faults, where the damage zone surrounding the main slip surface may introduce anisotropy that is not aligned with the main fault orientation. In this article, we introduce a generalized dimensional reduced model which preserves full-tensor permeability effects also in the out-of-plane direction of the inclusion. The governing equations of flow for the lower-dimensional objects are obtained through vertical averaging. We present a framework for discretization of the resulting mixed-dimensional problem, aimed at easy adaptation of existing simulation tools. We give numerical examples that show the failure of existing formulations when applied to anisotropic faulted porous media, and go on to show the convergence of our method in both two-dimensional and three-dimensional.publishedVersio