A semi-analytical scheme for highly oscillatory integrals over tetrahedra

This is the peer reviewed version of the following article: [Hospital-Bravo, R., Sarrate, J., and Díez, P. (2017) A semi-analytical scheme for highly oscillatory integrals over tetrahedra. Int. J. Numer. Meth. Engng, 111: 703–723. doi: 10.1002/nme.5474], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5474/full. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.This paper details a semi-analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the Partition of Unity Method (PUM) enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non-oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared to standard Gauss-Legendre quadratures.Peer ReviewedPostprint (author's final draft

Some Applications of the Generalized Multiscale Finite Element Method

Many materials in nature are highly heterogeneous and their properties can vary at different scales. Direct numerical simulations in such multiscale media are prohibitively expensive and some types of model reduction are needed. Typical model reduction techniques include upscaling and multiscale methods. In upscaling methods, one upscales the multiscale media properties so that the problem can be solved on a coarse grid. In multiscale method, one constructs multiscale basis functions that capture media information and solves the problem on the coarse grid. Generalized Multiscale Finite Element Method (GMsFEM) is a recently proposed model reduction technique and has been used for various practical applications. This method has no assumption about the media properties, which can have any type of complicated structure. In GMsFEM, we first create a snapshot space, and then solve a carefully chosen eigenvalue problem to form the offline space. One can also construct online space for the parameter dependent problems. It is shown theoretically and numerically that the GMsFEM is very efficient for the heterogeneous problems involving high-contrast, no-scale separation. In this dissertation, we apply the GMsFEM to perform model reduction for the steady state elasticity equations in highly heterogeneous media though some of our applications are motivated by elastic wave propagation in subsurface. We will consider three kinds of coupling mechanism for different situations. For more practical purposes, we will also study the applications of the GMsFEM for the frequency domain acoustic wave equation and the Reverse Time Migration (RTM) based on the time domain acoustic wave equation

Identifying the wavenumber for the inverse Helmholtz problem using an enriched finite element formulation

We investigate the inverse problem of identifying the wavenumber for the Helmholtz equation. The problem solution is based on measurements taken at few points from inside the computational domain or on its boundary. A novel iterative approach is proposed based on coupling the secant and the descent methods with the partition of unity method. Starting from an initial guess for the unknown wavenumber the forward problem is solved using the partition of unity method. Then the secant/descent methods are used to improve the initial guess by minimizing a predefined objective function based on the difference between the solution and a set of data points. In the next round of iterations the improved wavenumber estimate is used for the forward problem solution and the partition of unity approximation is improved by adding more enrichment functions. The iterative process is terminated when the objective function has converged and a set of two predefined tolerances are met. To evaluate the estimate accuracy we propose to utilize extra data points. To validate the approach and test its efficiency two wave applications with known analytical solutions are studied. The results show that the proposed approach can achieve high accuracy for the studied applications even when the considered data is contaminated with noise. Despite the clear advantages that were previously shown in the literature for solving the forward Helmholtz problem, this work presents a first attempt to solve the inverse Helmholtz problem with an enriched finite element approach

Generalized finite element method for Helmholtz equation

This dissertation presents the Generalized Finite Element Method (GFEM) for the scalar Helmholtz equation, which describes the time harmonic acoustic wave propagation problem. We introduce several handbook functions for the Helmholtz equation, namely the planewave, wave-band, and Vekua functions, and we use these handbook functions to enrich the Finite Element space via the Partition of Unity Method to create the GFEM space. The enrichment of the approximation space by these handbook functions reduces the pollution effect due to wave number and we are able to obtain a highly accurate solution with a much smaller number of degrees-of-freedom compared with the classical Finite Element Method. The q-convergence of the handbook functions is investigated, where q is the order of the handbook function, and it is shown that asymptotically the handbook functions exhibit the same rate of exponential convergence. Hence we can conclude that the selection of the handbook functions from an admissible set should be dictated only by the ease of implementation and computational costs. Another issue addressed in this dissertation is the error coming from the artificial truncation boundary condition, which is necessary to model the Helmholtz problem set in the unbounded domain. We observe that for high q, the most significant component of the error is the one due to the artificial truncation boundary condition. Here we propose a method to assess this error by performing an additional computation on the extended domain using GFEM with high q

Efficient implementation of high-order finite elements for Helmholtz problems

Computational modeling remains key to the acoustic design of various applications, but it is constrained by the cost of solving large Helmholtz problems at high frequencies. This paper presents an efficient implementation of the high-order Finite Element Method for tackling large-scale engineering problems arising in acoustics. A key feature of the proposed method is the ability to select automatically the order of interpolation in each element so as to obtain a target accuracy while minimising the cost. This is achieved using a simple local a priori error indicator. For simulations involving several frequencies, the use of hierarchic shape functions leads to an efficient strategy to accelerate the assembly of the finite element model. The intrinsic performance of the high-order FEM for 3D Helmholtz problem is assessed and an error indicator is devised to select the polynomial order in each element. A realistic 3D application is presented in detail to demonstrate the reduction in computational costs and the robustness of the a priori error indicator. For this test case the proposed method accelerates the simulation by an order of magnitude and requires less than a quarter of the memory needed by the standard FEM

Efficient finite element methods for aircraft engine noise prediction

Aircraft noise has a negative environmental impact. One of the ways in which it can be mitigated is by placing acoustic liners inside the aircraft's engines. These liners can be optimised for noise reduction. A cost effective way to optimise acoustic liners is to make use of numerical modelling. However, there is room for improvement of the efficiency of current modelling methods. This thesis is concerned with the efficient numerical prediction of noise emitted from modern aircraft engines. Four high order finite element methods are used to solve the convected wave equation, and their performances are compared. The benefit of using the hierarchic Lobatto finite element method to solve this type of problem is demonstrated. A scheme which optimises the efficiency of the high order method is developed. The scheme automatically chooses the most efficient order for a given element, depending on the element size, and the problem parameters on that element. The computational cost of using the standard quadratic finite element method to solve a typical engine intake noise problem, is compared to the cost of the proposed adaptive-order method. A significant improvement in terms of efficiency is demonstrated when using the proposed method over the standard method. Furthermore, a new formulation based on potential flow theory for the solution of vortex sheet problems (typically encountered when dealing with exhaust noise problems) is presented.

Analysis of high-order finite elements for convected wave propagation

In this paper, we examine the performance of high-order finite element methods (FEM) for aeroacoustic propagation, based on the convected Helmholtz equation. A methodology is presented to measure the dispersion and amplitude errors of the p-FEM, including non-interpolating shape functions, such as ‘bubble’ shape functions. A series of simple test cases are also presented to support the results of the dispersion analysis. The main conclusion is that the properties of p-FEM that make its strength for standard acoustics (e.g., exponential p-convergence, low dispersion error) remain present for flow acoustics as well. However, the flow has a noticeable effect on the accuracy of the numerical solution, even when the change in wavelength due to the mean flow is accounted for, and an approximation of the dispersion error is proposed to describe the influence of the mean flow. Also discussed is the so-called aliasing effect, which can reduce the accuracy of the solution in the case of downstream propagation. This can be avoided by an appropriate choice of mesh resolution

A Modal-Based Partition of Unity Finite Element Method for Elastic Wave Propagation Problems in Layered Media

Financiado para publicación en acceso aberto: Universidade da Coruña/CISUG[Abstract] The time-harmonic propagation of elastic waves in layered media is simulated numerically by means of a modal-based Partition of Unity Finite Element Method (PUFEM). Instead of using the standard plane waves or the Bessel solutions of the Helmholtz equation to design the discretization basis, the proposed modal-based PUFEM explicitly uses the tensor-product expressions of the eigenmodes (the so-called Love and interior modes) of a spectral elastic transverse problem, which fulfil the coupling conditions among layers. This modal-based PUFEM approach does not introduce quadrature errors since the coefficients of the discrete matrices are computed in closed-form. A preliminary analysis of the high condition number suffered by the proposed method is also analyzed in terms of the mesh size and the number of eigenmodes involved in the discretization. The numerical methodology is validated through a number of test scenarios, where the reliability of the proposed PUFEM method is discussed by considering different modal basis and source terms. Finally, some indicators are introduced to select a convenient discrete PUFEM basis taking into account the observability of cracks located on a coupling boundary between two adjacent layers.This work has been supported by Xunta de Galicia project “Numerical simulation of high-frequency hydro-acoustic problems in coastal environments - SIMNUMAR” (EM2013/052), co-funded with European Regional Development Funds (ERDF). Moreover, the second and fifth authors have been supported by MICINN projects MTM2014-52876-R, MTM2017-82724-R, PID2019-108584RB-I00, and also by ED431C 2018/33 - M2NICA (Xunta de Galicia & ERDF) and ED431G 2019/01 - CITIC (Xunta de Galicia & ERDF). Additionally, the third author has been supported by Junta de Castilla y León under projects VA024P17 and VA105G18, co-financed by ERDF funds. This work has been funded for open access charge by Universidade da Coruña/CISUGXunta de Galicia; EM2013/052Xunta de Galicia; ED431C 2018/33Xunta de Galicia; ED431G 2019/01Junta de Castilla y León; VA024P17Junta de Castilla y León; VA105G1