scholarly journals Towards a combined perfectly matching layer and infinite element formulation for unbounded elastic wave problems

2021 ◽  
pp. 108128652110408
Author(s):  
Joseph S. Pettigrew ◽  
Anthony J. Mulholland ◽  
Katherine M. M. Tant
Keyword(s):  
Finite Element ◽  
Elastic Wave ◽  
Three Dimensions ◽  
Element Formulation ◽  
Infinite Element ◽  
Matching Layer ◽  
Plane Wave Incident ◽  
The Time Domain ◽  
Wave Problems ◽  
Stress Free Boundary

This paper presents a framework for implementing a novel perfectly matching layer and infinite element (PML+IE) combination boundary condition for unbounded elastic wave problems in the time domain. To achieve this, traditional hexahedral finite elements are used to model wave propagation in the inner domain and IE test functions are implemented in the exterior domain. Two alternative implementations of the PML formulation are studied: the case with constant stretching in all three dimensions and the case with spatially dependent stretching along a single direction. The absorbing ability of the PML+IE formulation is demonstrated by the favourable comparison with the reflection coefficient for a plane wave incident on the boundary achieved using a finite-element-only approach where stress free boundary conditions are implemented at the domain edge. Values for the PML stretching function parameters are selected based on the minimisation of the reflected wave amplitude and it is shown that the same reduction in reflection amplitude can be achieved using the PML+IE approach with approximately half of the number of elements required in the finite-element-only approach.

A Combined Perfectly Matching Layer and Infinite Element Formulation for Unbounded Wave Problems in the Frequency Domain

2014 ◽  
Author(s):  
Joseph S. Pettigrew ◽  
Anthony J. Mulholland ◽  
Jeffrey L. Cipolla ◽  
John Mould ◽  
Robert Banks
Keyword(s):  
Frequency Domain ◽  
Marked Improvement ◽  
Test Problem ◽  
Near Field ◽  
Element Formulation ◽  
Infinite Element ◽  
Matching Layer ◽  
Modal Response ◽  
New Type ◽  
Wave Problems

In this paper, Berenger’s Perfectly Matching Layer (PML) and Bettess’ Infinite Element (IE) scheme are combined to create a new type of element for unbounded acoustic wave problems. An assessment of this new element formulation is made through its use in the calculation of the acoustic modal response of a spherical radiator in the frequency domain. The performance of the PML+IE approach is contrasted with the IE only methodology by comparing them to the exact solution of this test problem in terms of the surface inertia and resistance in the near field. The results are encouraging and the PML+IE approach shows a marked improvement in performance, particularly at lower frequencies.

Finite Element Formulation of Exact Dirichlet-to-Neumann Radiation Conditions on Elliptic and Spheroidal Boundaries

1999 ◽  
Author(s):  
Lonny L. Thompson ◽  
Runnong Huan ◽  
Cristian Ianculescu
Keyword(s):  
Finite Element ◽  
Iterative Solution ◽  
Three Dimensions ◽  
Symmetric System ◽  
Element Formulation ◽  
Local Boundary ◽  
Radiation Boundary Conditions ◽  
Matrix Partition ◽  
Non Local ◽  
Spheroidal Coordinates

Abstract Exact Dirichlet-to-Neumann (DtN) radiation boundary conditions are derived in elliptic and spheroidal coordinates and formulated in a finite element method for the Helmholtz equation in unbounded domains. The DtN map matches the first N wave harmonics exactly at the artificial boundary. The use of elliptic and spheroidal boundaries enables the efficient solution of scattering from elongated objects in two- and three-dimensions respectively. Modified DtN conditions based on first and second order local boundary operators are also derived in elliptic and spheroidal coordinates, in a form suitable for finite element implementation. The modified DtN conditions are more accurate than the DtN boundary condition, yet require no extra memory and little extra cost. Direct implementation involves non-local spatial integrals leading to a dense, fully populated submatrix. A matrix-free interpretation of the non-local DtN map for elliptic and spheroidal boundaries, suitable for iterative solution of the resulting complex-symmetric system is described. For both the DtN and modified DtN conditions, we describe efficient and effective SSOR preconditioners with Eisenstat’s trick based on the matrix partition associated with the interior mesh and local boundary operator. Numerical examples of scattering from elliptic and spheroidal boundaries are computed to demonstrate the efficiency and accuracy of the boundary treatments for elongated structures.

Application of Finite Element Method in Time Domain to a Rotor System With Nonlinear Supports

2001 ◽  
Author(s):  
Liguo Wang ◽  
Wenhu Huang ◽  
Chao Hu
Keyword(s):  
Finite Element ◽  
Time Domain ◽  
Nonlinear Differential Equations ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Periodic Response ◽  
Jeffcott Rotor ◽  
Rotor Bearing ◽  
The Time Domain ◽  
Bearing System

Abstract A new method for analyzing periodic response of rotor dynamic system with nonlinear supports is presented in this paper. Based on a finite element formulation in the time domain, this method transforms nonlinear differential equations governing the dynamic behavior of rotor-bearing system into a set of nonlinear algebraic equations that can be reduced and calculated by the characteristic set of Wu elimination method. The analytic solution of the nodal displacement has been obtained finally. According to this result the behavior of periodic response is analyzed. The feasibility and advantage of the proposed method are illustrated with an example of flexible Jeffcott rotor-bearing system with nonlinear supports.

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