Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

Explicit local time-stepping methods for time-dependent wave propagation

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leap-frog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations validate the theory and illustrate the usefulness of these local time-stepping methods.Comment: overview paper, typos added, references updated. arXiv admin note: substantial text overlap with arXiv:1109.448

Nonstandard Finite Element Methods

[no abstract available

Development of a Sensible Reduced-Order Modeling Framework for Geomechanics Simulation: With Application to Coupled Flow and Geomechanics Simulation

With the recent development of unconventional reservoirs, attention has been geared towards the integration of the geomechanical models with traditional flow simulation. A case in point is quantifying rock-fluid interactions in hydraulic fracturing operations. Although much effort has gone into the creation and advancement of commercial simulation software for coupled flow and geomechanics, it is still in its infancy. The models are considerably oversimplified and poorly representative of the problem’s complex nature. Throughout history, several contributions have been made into the development of efficient model-order reduction (MOR) techniques for “flow only” simulations. Yet – to date – contributions to the mechanical models in coupled simulations have been minimal. This study tackles this challenging aspect, by proposing a novel model reduction adaptive workflow, especially for the mechanics simulators, that (1) can be coupled with any simulator that can export mass, stiffness, and load matrices; (2) can achieve 2 orders of magnitude in computational time reduction; and (3) do not add more complexity to the solution. In the first part of this research, several – widely used – reduction techniques for structural mechanics were implemented based on the construction of the dynamic condensation matrix. Single-step reduction methods were first executed; in particular, Guyan DOFs based reduction techniques. Following that, two-step methods were implemented; where corrections were made to the results obtained from the former. Finally, iterative (three-step) reduction methods were applied; handling the problem of master DOFs selection through consistent updates of the dynamic condensation matrix until convergence is achieved. To that end, two schemes are presented; based on the convergence of the dynamic condensation matrix, as well as, the eigenvalues of the reduced-order model. In the second part of this research, we provide a rigorous framework for testing the completeness, efficiency, and convergence for all the presented reduction techniques. Regarding the completeness of the reduced models, two main criteria were investigated; namely, modal assurance criterion (MAC) and singular value decomposition (SVD). For efficiency testing, percent error (PE) of natural frequencies and the correlation coefficient for modal vector (CCFMV) values were considered. Finally, the efficiency of the convergent criterion was demonstrated through the errors associated with the column vectors of the condensation matrix. Several numerical examples are presented to show the efficiency of the presented framework, particularly for coupled simulations. Based on the adopted framework, we managed to reduce the scale of the finite element models to less than 9% of the full model with error as low as 1%. In terms of computational speed and runtime, we achieved substantial speedups; up to 20X. Given the proposed workflow, large-scale complex simulations – similar to those associated with hydraulic fracturing – could be more feasible and less costly. This, ultimately, would give allowance for incorporating the complex physics pertinent to unconventional reservoirs and motivate the advent of their development at no additional cost

Numerical Methods for the Estimation of the Impact of Geometric Uncertainties on the Performance of Electromagnetic Devices

This work addresses the application of Isogeometric Analysis to the simulation of particle accelerator cavities and other electromagnetic devices whose performance is mainly determined by their geometry. By exploiting the properties of B-Spline and Non-Uniform B-Spline basis functions, the Isogeometric approximation allows for the correct discretisation of the spaces arising from Maxwell's equations and for the exact representation of the computational domain. This choice leads to substantial improvements in both the overall accuracy and computational effort. The suggested framework is applied to the evaluation of the sensitivity of these devices with respect to geometrical changes using Uncertainty Quantification methods and to shape optimisation processes. The particular choice of basis functions simplifies the construction of the geometry deformations significantly. Finally, substructuring methods are proposed to further reduce the computational cost due to matrix assembly and to allow for hybrid coupling of Isogeometric Analysis and more classical Finite Element Methods. Considerations regarding the stability of such methods are addressed. The methods are illustrated by simple numerical tests and real world device simulations with particular emphasis on particle accelerator cavities

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells