Coupling and computation of electromagnetism and mechanics

Accurate coupling of electromagnetism and mechanics is of interest in computations of transducers such as piezoelectric, pyroelectric, electromagnetic sensors and actuators. Balance equations in mechanics as well as the MAXWELL equations for electromagnetism have been established in science. However, if the coupling between these governing equations are necessary, several difficulties arise. Herein we identify the challenges and propose possible solutions for computational analysis.TU Berlin, Open-Access-Mittel - 201

Theory and computation of electromagnetic fields and thermomechanical structure interaction for systems undergoing large deformations

For an accurate description of electromagneto-thermomechanical systems, electromagnetic fields need to be described in a Eulerian frame, whereby the thermomechanics is solved in a Lagrangean frame. It is possible to map the Eulerian frame to the current placement of the matter and the Lagrangean frame to a reference placement. We present a rigorous and thermodynamically consistent derivation of governing equations for fully coupled electromagneto-thermomechanical systems properly handling finite deformations. A clear separation of the different frames is necessary. There are various attempts to formulate electromagnetism in the Lagrangean frame, or even to compute all fields in the current placement. Both formulations are challenging and heavily discussed in the literature. In this work, we propose another solution scheme that exploits the capabilities of advanced computational tools. Instead of amending the formulation, we can solve thermomechanics in the Lagrangean frame and electromagnetism in the Eulerian frame and manage the interaction between the fields. The approach is similar to its analog in fluid structure interaction, but more challenging because the field equations in electromagnetism must also be solved within the solid body while following their own different set of transformation rules. We additionally present a mesh-morphing algorithm necessary to accommodate finite deformations to solve the electromagnetic fields outside of the material body. We illustrate the use of the new formulation by developing an open-source implementation using the FEniCS package and applying this implementation to several engineering problems in electromagnetic structure interaction undergoing large deformations

Theory and computation of higher gradient elasticity theories based on action principles

In continuum mechanics, there exists a unique theory for elasticity, which includes the first gradient of displacement. The corresponding generalization of elasticity is referred to as strain gradient elasticity or higher gradient theories, where the second and higher gradients of displacement are involved. Unfortunately, there is a lack of consensus among scientists how to achieve the generalization. Various suggestions were made, in order to compare or even verify these, we need a generic computational tool. In this paper, we follow an unusual but quite convenient way of formulation based on action principles. First, in order to present its benefits, we start with the action principle leading to the well-known form of elasticity theory and present a variational formulation in order to obtain a weak form. Second, we generalize elasticity and point out, in which term the suggested formalism differs. By using the same approach, we obtain a weak form for strain gradient elasticity. The weak forms for elasticity and for strain gradient elasticity are solved numerically by using open-source packages—by using the finite element method in space and finite difference method in time. We present some applications from elasticity as well as strain gradient elasticity and simulate the so-called size effect

An Accurate Finite Element Method for the Numerical Solution of Isothermal and Incompressible Flow of Viscous Fluid

Despite its numerical challenges, finite element method is used to compute viscous fluid flow. A consensus on the cause of numerical problems has been reached; however, general algorithms—allowing a robust and accurate simulation for any process—are still missing. Either a very high computational cost is necessary for a direct numerical solution (DNS) or some limiting procedure is used by adding artificial dissipation to the system. These stabilization methods are useful; however, they are often applied relative to the element size such that a local monotonous convergence is challenging to acquire. We need a computational strategy for solving viscous fluid flow using solely the balance equations. In this work, we present a general procedure solving fluid mechanics problems without use of any stabilization or splitting schemes. Hence, its generalization to multiphysics applications is straightforward. We discuss emerging numerical problems and present the methodology rigorously. Implementation is achieved by using open-source packages and the accuracy as well as the robustness is demonstrated by comparing results to the closed-form solutions and also by solving well-known benchmarking problems

Thin‐layer inertial effects in plasticity and dynamics in the Prandtl problem

Especially in metal forming, large plastic deformation occurs in thin plates. The problem of compressing dies is analyzed to evaluate the spreading of a thin layer in between. The velocity of dies is a given function in time so that the kinematics of the process is known. This problem can be considered as a generalization of the classical Prandtl problem by taking inertial effects into account and introducing dimensionless parameters as internal variables depending on time. The first parameter is purely geometric corresponding to the thin‐layer approximation; the second and the third parameters are dimensionless velocity and acceleration during the dies getting pressed. We use singular asymptotic expansions of unknown functions and study how these parameters vary preceding the dies of moment. Depending on this relation, the dynamic corrections to the quasistatic solution is a part of various terms of the asymptotic series. The corresponding analytical investigation both for general case and for particular typical regimes of plates motion is carried out.TU Berlin, Open-Access-Mittel - 201

Three-dimensional elastic deformation of functionally graded isotropic plates under point loading

Acknowledgement Financial support of this research by The Royal Society (UK) under grant number JP090633 is gratefully acknowledged.Peer reviewedPostprin