Sabine Zaglmayr

High order Nedelec elements with local complete sequence properties

Purpose – The goal of the presented work is the efficient computation of Maxwell boundary and eigenvalue problems using high order H(curl) finite elements.Design/methodology/approach – Discusses a systematic strategy for the realization of arbitrary order hierarchic H(curl)‐conforming finite elements for triangular and tetrahedral element geometries. The shape functions are classified as lowest order Nedelec, higher‐order edge‐based, face‐based (only in 3D) and element‐based ones.Findings – Our new shape functions provide not only the global complete sequence property but also local complete sequence properties for each edge‐, face‐, and element‐block. This local property allows an arbitrary variable choice of the polynomial degree for each edge, face, and element. A second advantage of this construction is that simple block‐diagonal preconditioning gets efficient. Our high order shape functions contain gradient shape functions explicitly. In the case of a magnetostatic boundary value problem, the gradien...

Finite-element simulation of wave propagation in periodic piezoelectric SAW structures

Many surface acoustic wave (SAW) devices consist of quasiperiodic structures that are designed by successive repetition of a base cell. The precise numerical simulation of such devices, including all physical effects, is currently beyond the capacity of high-end computation. Therefore, we have to restrict the numerical analysis to the periodic substructure. By using the finite-element method (FEM), this can be done by introducing periodic boundary conditions (PBCs) at special artificial boundaries. To be able to describe the complete dispersion behavior of waves, including damping effects, the PBC has to be able to model each mode that can be excited within the periodic structure. Therefore, the condition used for the PBCs must hold for each phase and amplitude difference existing at periodic boundaries. Based on the Floquet theorem, our two newly developed PBC algorithms allow the calculation of both, the phase and the amplitude coefficients of the wave. In the first part of this paper we describe the basic theory of the PBCs. Based on the FEM, we develop two different methods that deliver the same results but have totally different numerical properties and, therefore, allow the use of problem-adapted solvers. Further on, we show how to compute the charge distribution of periodic SAW structures with the aid of the new PBCs. In the second part, we compare the measured and simulated dispersion behavior of waves propagating on periodic SAW structures for two different piezoelectric substrates. Then we compare measured and simulated input admittances of structures similar to SAW resonators.

8E-4 Perfectly Matched Layer Finite Element Simulation of Parasitic Acoustic Wave Radiation in Microacoustic Devices

Classical finite element methods are only capable of describing a limited computation area; the substrate of a micro-acoustic device must therefore be described by suitable boundary conditions. This is obtained by absorbing boundary conditions (ABC) or perfectly matched layers (PML) which both suppress reflections from the substrate. PML was implemented into a high-performance finite element code. The employed variant of the PML approach relies on a complex variable transformation of the basic piezoelectric equations in the PML layer. Harmonic admittances of a system with aluminum electrodes on a 42deg YX- LiTaO3 substrate are determined with PML and ABC methods and compared to FEM/BEM results, which can be considered as exact solution of the piezoelectric half space problem. The results of PML approach FEM/BEM, while the ABC results deviate. The correct adjustment of PML parameters to minimize reflection at the substrate/PML interface is illustrated by visualizations of the wave fields.

A Macro Finite-Element Formulation for Cardiac Electrophysiology Simulations Using Hybrid Unstructured Grids

Abstract-Electrical activity in cardiac tissue can be described by the bidomain equations whose solution for large-scale simulations still remains a computational challenge. Therefore, improvements in the discrete formulation of the problem, which decrease computational and/or memory demands are highly desirable. In this study, we propose a novel technique for computing shape functions of finite elements (FEs). The technique generates macro FEs (MFEs) based on the local decomposition of elements into tetrahedral subelements with linear shape functions. Such an approach necessitates the direct use of hybrid meshes (HMs) composed of different types of elements. MFEs are compared to classic standard FEs with respect to accuracy and RAM memory usage under different scenarios of cardiac modeling, including bidomain and monodomain simulations in 2-D and 3-D for simple and complex tissue geometries. In problems with analytical solutions, MFEs displayed the same numerical accuracy of standard linear triangular and tetrahedral elements. In propagation simulations, conduction velocity and activation times agreed very well with those computed with standard FEs. However, MFEs offer a significant decrease in memory requirements. We conclude that HMs composed of MFEs are well suited for solving problems in cardiac computational electrophysiology.

Simulation of Diffraction in Periodic Media with a Coupled Finite Element and Plane Wave Approach

The aim of this paper is to discuss simulation methods of diffraction of electromagnetic waves on biperiodic structures. The region with complicated structures is discretized by Nedelec finite elements. In the unbounded homogeneous regions above and below, a plane wave expansion containing the exact far-field pattern is applied. A consistent coupling is achieved by the method of Nitsche. By numerical experiments we investigate the speed of convergence depending on the mesh refinement, the element order, and the number of evanescent waves.

Finite element calculation of the dispersion relations of infinitely extended SAW structures including bulk wave radiation

In the design procedure of surface acoustic wave (SAW) devices simple models like equivalent circuit models or the Coupling of Modes (COM) model are used to achieve short calculation times. Therefore, these models can be used for iterative component optimization. However, they are subject to many simplifications and restrictions. In order to improve the parameters required for the simpler models and to achieve better insight ot the physics of SAW devices analysis tools solving the constitutional partial differential equations are needed. We have developed an efficient calculation scheme based on the finite element method. It makes use of newly established periodic boundary conditions (PBCs) allowing the simulation of an infinitely extended SAW device. This is a good approximation of many SAW devices which show a large number of periodically arranged electrodes. We have developed two different formulations for the PBCs: One leads to a small quadratic eigenvalue problem operating on a larger matrix. These formulations allow the calculation of the complete dispersion relation. Bulk acoustic waves (BAWs) which are generated due to mode conversion at electrode edges are allowed to leave the calculation area nearly without reflection. Therefore, the calculation scheme also considers damping coefficients caused by the conversion of surface waves into bulk waves. This behavior coincides well with real SAW devices in which the substrate thickness is large compared to the used wavelengths and, additionally, the bulk waves are scattered in all directions at the rough substrate bottom. In the paper, a short introduction to the basic theory of the numerical calculation scheme will be given first. The applicability of the calculation scheme will be demonstrated by comparing analytical, measured and simulated results.

Schwarz Preconditioning for High Order Simplicial Finite Elements

This paper analyzes two-level Schwarz methods for matrices arising from the p-version finite element method on triangular and tetrahedral meshes. The coarse level consists of the lowest order finite element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a model problem.

Eigenvalue Problems in Surface Acoustic Wave Filter Simulations

Surface acoustic wave filters are widely used for frequency filtering in telecommunications. These devices mainly consist of a piezoelectric substrate with periodically arranged electrodes on the surface. The periodic structure of the electrodes subdivides the frequency domain into stop-bands and pass-bands. This means only piezoelectric waves excited at frequencies belonging to the pass-band-region can pass the devices undamped.

A Symmetric Low-Frequency Stable Broadband Maxwell Formulation for Industrial Applications

A new monolithic, symmetric, low-frequency stable, broadband formulation of the time-harmonic Maxwells equations is presented. The approach is based on a variational formulation of Maxwells equations which accounts for the frequency dependencies of the underlying physical phenomena and does not require auxiliary variables. The formulation allows for stable evaluation at low frequencies, even if the frequency equals zero. It has a broadband validity and correctly comprises both high-frequency effects and (quasi-)static behaviors. This is demonstrated by using different test examples from scientific to large-scale real-world applications.

A reduced basis approach for broadband Maxwell simulations with validity in any frequency

Based on a low-frequency stable formulation of Maxwells equations, we propose a finite element and reduced basis scheme which allows for a robust and accurate evaluation of the system in any frequency. The approach considers in particular the limit cases, as the frequency tends to zero or to a maximal frequency for which a discretization is reasonable.