LFA-tuned matrix-free multigrid method for the elastic Helmholtz equation

We present an efficient matrix-free geometric multigrid method for the elastic Helmholtz equation, and a suitable discretization. Many discretization methods had been considered in the literature for the Helmholtz equations, as well as many solvers and preconditioners, some of which are adapted for the elastic version of the equation. However, there is very little work considering the reciprocity of discretization and a solver. In this work, we aim to bridge this gap. By choosing an appropriate stencil for re-discretization of the equation on the coarse grid, we develop a multigrid method that can be easily implemented as matrix-free, relying on stencils rather than sparse matrices. This is crucial for efficient implementation on modern hardware. Using two-grid local Fourier analysis, we validate the compatibility of our discretization with our solver, and tune a choice of weights for the stencil for which the convergence rate of the multigrid cycle is optimal. It results in a scalable multigrid preconditioner that can tackle large real-world 3D scenarios.Comment: 20 page

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

International audienceThis article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green's tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green's tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process

3D simulation of magneto-mechanical coupling in MRI scanners using high order FEM and POD

Magnetic Resonance Imaging (MRI) scanners have become an essential tool in the medi-cal industry due to their ability to produce high resolution images of the human body. To generate an image of the body, MRI scanners combine strong static magnetic fields with transient gradient magnetic fields. The interaction of these magnetic fields with the con-ducting components present in superconducting MRI scanners gives rise to an important problem in the design of new MRI scanners. The transient magnetic fields give rise to the appearance of eddy currents in conducting components. These eddy currents, in turn, result in electromagnetic stresses, which cause the conducting components to deform and vibrate. The vibrations are undesirable as they lead to a deterioration in image quality (with image artefacts) and to the generation of noise, which can cause patient discomfort. The eddy currents, in addition, lead to heat being dissipated and deposited into the cryo-stat, which is filled with helium in order to maintain the coils in a superconducting state. This deposition of heat can cause helium boil off and potentially result in a costly magnet quench. Understanding the mechanisms involved in the generation of these vibrations and the heat being deposited into the cryostat are, therefore, key for a successful MRI scanner design. This involves the solution of a coupled magneto-mechanical problem, which is the focus of this work.In this thesis, a new computational methodology for the solution of three-dimensional (3D) magneto-mechanical coupled problems with application to MRI scanner design is presented. To achieve this, first an accurate mathematical description of the magneto-mechanical coupling is presented, which is based on a Lagrangian formulation and the assumption of small displacements. Then, the problem is linearised using an AC-DC splitting of the fields, and a variational formulation for the solution of the linearised prob-lem in a time-harmonic setting is presented. The problem is then discretised using high order finite elements, where a combination of hierarchical H1 and H(curl) basis func-tions is used. An efficient staggered algorithm for the solution of the coupled system is proposed, which combines the DC and AC stages and makes use of preconditioned iter-ative solvers when appropriate. This finite element methodology is then applied to a set of challenging academic and industrially relevant problems in order to demonstrate its accuracy and efficiency.This finite element methodology results in the accurate and efficient solution of the magneto-mechanical problem of interest. However, in the design stage of a new MRI scanner, this coupled problem must be solved repeatedly for varying model parameters such as frequency or material properties. Thus, even if an efficient finite element solver is available for the solution of the coupled problem, the need for these repeated simulations result in a bottleneck in terms of computational cost, which leads to an increase in design time and its associated financial implications. Therefore, in order to optimise this process, the application of Reduced Order Modelling (ROM) techniques is considered. A ROM based on the Proper Orthogonal Decomposition (POD) method is presented and applied to a series of challenging MRI configurations. The accuracy and efficiency of this ROM is demonstrated by performing comparisons against the full order or high fidelity finite element software, showing great performance in terms of computational speed-up, which has major benefits in the optimisation of the design process of new MRI scanners

The development of a fast multipole boundary element method for coupled acoustic and elastic problems

This thesis presents a dual fast multipole boundary element method (FMBEM) for modelling 3D acoustic coupled fluid-structure interaction problems in the frequency domain. Boundary integral representations are used to represent both the exterior fluid and interior elastic solid domains and the fast multipole method is employed to accelerate the calculations in both domains. The dual FMBEM yields a similar solution accuracy to the conventional models, while its solution times and memory requirements are substantially reduced

Modeling, Discretization, Optimization, and Simulation of Multiphysics Problems (IIT Indore)

The goal of this winter school is to give an introduction to numerical modeling of multiphysics problems. These are nonstationary, nonlinear, coupled partial differential equations. The philosophy of this school is to provide a mixture of very basic techniques that are immediately applied to `complicated' practical and/or current research problems

Computational Multiscale Methods

Almost all processes in engineering and the sciences are characterised by the complicated relation of features on a large range of nonseparable spatial and time scales. The workshop concerned the computer-aided simulation of such processes, the underlying numerical algorithms and the mathematics behind them to foresee their performance in practical applications

Aeroelastic Analysis of a Wind Turbine Blade Using the Harmonic Balance Method

Most current wind turbine aeroelastic codes rely on the blade element momentum method with empirical corrections to compute aerodynamic forces on the wind turbine blades. While efficient, this method relies on experimental data and does not allow designers much flexibility for alternative blade designs. Unsteady solutions to the Navier-Stokes equations offer a significant improvement in aerodynamic modeling, but these are currently too computationally expensive to be useful in a design situation. However, steady-state solutions to the Navier-Stokes equations are possible with reasonable computation times. The harmonic balance method provides a way to represent unsteady, periodic flows through coupled a set of steady-state solutions. This method offers the possibility of unsteady flow solutions at a computational cost on the order of a few steady-state solutions. By coupling a harmonic balance driven aerodynamic model with a mode shape-based structural dynamics model, an efficient aeroelastic model for a wind turbine blade driven by the Navier-Stokes equations is developed in this dissertation. For wind turbine flows, turbulence modeling is essential, especially in the transition of the boundary layer from laminar to turbulent. As part of this dissertation, the Spalart-Allmaras turbulence model and the gamma-Re\_theta-t transition model are included in the aerodynamic model. This marks the first time that this transition model, turbulence model, and the harmonic balance method have been coupled to study unsteady wind turbine aerodynamics. Results show that the transition model matches experimental data more closely than a fully turbulent model for the onset of both static and dynamic stall. Flutter is of particular interest as turbines continue to increase in size, and longer and softer blades continue to enter the field. In this dissertation, flutter is investigated for the 1.5 MW WindPACT rotor blade. The aeroelastic model created, which incorporates the harmonic balance method and a fully turbulent aerodynamic model, is the first of its kind for wind turbine flutter analysis. Predictions match those of other aeroelastic models for the 1.5 MW WindPACT blade, and the first flapwise and edgewise modes are shown to dominate flutter for the rotor speeds considered