Finite Elements with Nonreflecting Boundary Conditions Formulated by the Helmholtz Integral Equation

1999 ◽  
Vol 121 (2) ◽  
pp. 214-220
Author(s):  
Shu-Wei Wu
Keyword(s):  
Integral Equation ◽  
Finite Elements ◽  
Surrounding Medium ◽  
Constraint Equation ◽  
Thin Body ◽  
Source Surface ◽  
Element Formulation ◽  
Artificial Boundary ◽  
Acoustic Domain ◽  
Helmholtz Integral

In the proposed approach, an acoustic domain is split into two parts by an arbitrary artificial boundary. The surrounding medium around the vibrating surface is discretized with finite elements up to the artificial boundary. The constraint equation specified on the artificial boundary is formulated with the Helmholtz integral equation straightforwardly, in which the source surface coincides with the vibrating surface discretized with boundary elements. To ensure the uniqueness of the numerical solution, the composite Helmholtz integral equation proposed by Burton and Miller was adopted. Due to the avoidance of singularity problems inherent in the boundary element formulation, this method is very efficient and easy to implement in an isoparametric element environment. It should be noted that the present method also can be applied to thin-body problems by using quarter-point elements.

scholarly journals Polygonal finite elements for three-dimensional Voronoi-cell-based discretisations

2012 ◽  
Author(s):  
Kaliappan Jayabal ◽  
Andreas Menzel
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Three Dimensional ◽  
Specific Property ◽  
Voronoi Cell ◽  
Polycrystalline Materials ◽  
Element Formulation ◽  
Hybrid Finite Element ◽  
Grain Structures ◽  
Polygonal Elements

Hybrid finite element formulations in combination with Voronoi-cell-based discretisation methods can efficiently be used to model the behaviour of polycrystalline materials. Randomly generated three-dimensional Voronoi polygonal elements with varying numbers of surfaces and corners in general better approximate the geometry of polycrystalline microor rather grain-structures than the standard tetrahedral and hexahedral finite elements. In this work, the application of a polygonal finite element formulation to three-dimensional elastomechanical problems is elaborated with special emphasis on the numerical implementation of the method and the construction of the element stiffness matrix. A specific property of Voronoi-based discretisations in combination with a hybrid finite element approach is investigated. The applicability of the framework established is demonstrated by means of representative numerical examples.

scholarly journals International Workshop on Finite Elements for Microwave Engineering

2016 ◽  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Dominant Role ◽  
Piecewise Linear ◽  
International Workshop ◽  
American Mathematical Society ◽  
Element Formulation ◽  
Microwave Engineering ◽  
History Of ◽  
Element Activity

When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018.

Application of the Complex Envelope Vectorization to a Boundary Element Formulation

2008 ◽  
Author(s):  
Oliviero Giannini ◽  
Aldo Sestieri
Keyword(s):  
Finite Elements ◽  
External Field ◽  
Boundary Element ◽  
High Frequency ◽  
Computational Cost ◽  
Field Variable ◽  
Element Formulation ◽  
Complex Envelope ◽  
Reduced Dimensions ◽  
Core Solution

The complex envelope vectorization (CEV) is a recent method that has been successfully applied to structural and internal acoustic problems. Unlike other methods proposed in the last two decades to solve high frequency problems, CEV is not an energy method, although it shares with all the other techniques a variable transformation of the field variable. By such transformation involving a Hilbert transform, CEV allows the representation of a fast oscillating signal through a set of low oscillating signals. Thanks to such transformation it is possible to solve a high frequency dynamic problem at a computational cost that is lower than that required by finite elements. In fact, by using finite elements, a high frequency problem usually implies large matrices. On the contrary the CEV formulation is obtained by solving a set of linear problems of highly reduced dimensions. Although it was proved that CEV is in general a successful procedure, it was shown that it is particularly appropriate when the modes of the system have a negligible role on the solution. Moreover, the numerical advantage of the CEV formulation is much more pronounced when full matrices are used. Thus, for the first time it is applied to a boundary element formulation (BEM). Both external and internal acoustic fields of increasing complexity are considered: the internal and external field generated by a pulsating sphere; the external field of a forced box, where the velocity field is determined by finite elements; a set of 4 plates that form an open cavity. The results are compared with those obtained by a BEM procedure (SYSNOISE), highlighting the good quality of the proposed approach. An estimate of the computational advantage is also provided. Finally it is worthwhile to point out that the reduction of the BE matrices allows for an in-core solution even for large problems.

scholarly journals The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter

2016 ◽  
Vol 24 (01) ◽  
pp. 1550016 ◽  
Author(s):  
Steffen Marburg
Keyword(s):  
Integral Equation ◽  
Literature Review ◽  
Coupling Parameter ◽  
Solution Process ◽  
Normal Derivative ◽  
The Neumann Problem ◽  
Spurious Modes ◽  
Specific Formulation ◽  
Wrong Sign ◽  
Helmholtz Integral

The phenomenon of irregular frequencies or spurious modes when solving the Kirchhoff–Helmholtz integral equation has been extensively studied over the last six or seven decades. A class of common methods to overcome this phenomenon uses the linear combination of the Kirchhoff–Helmholtz integral equation and its normal derivative. When solving the Neumann problem, this method is usually referred to as the Burton and Miller method. This method uses a coupling parameter which, theoretically, should be complex with nonvanishing imaginary part. In practice, it is usually chosen proportional or even equal to [Formula: see text]. A literature review of papers about the Burton and Miller method and its implementations revealed that, in some cases, it is better to use [Formula: see text] as coupling parameter. The better choice depends on the specific formulation, in particular, on the harmonic time dependence and on the fundamental solution or Green’s function, respectively. Surprisingly, an unexpectedly large number of studies is based on the wrong choice of the sign in the coupling parameter. Herein, it is described which sign of the coupling parameter should be used for different configurations. Furthermore, it will be shown that the wrong sign does not just make the solution process inefficient but can lead to completely wrong results in some cases.

Modelling of rotors with three-dimensional solid finite elements

2001 ◽  
Vol 36 (4) ◽  
pp. 359-371 ◽  
Author(s):  
A Nandi ◽  
S Neogy
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cross Sections ◽  
Three Dimensional ◽  
Element Model ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Two Dimensional ◽  
Inertia Effects ◽  
Solid Finite Elements

A shaft is modelled using three-dimensional solid finite elements. The shear-deformation and rotary inertia effects are automatically included through the three-dimensional elasticity formulation. The formulation allows warping of plane cross-sections and takes care of gyroscopic effect. Unlike a beam element model, the present model allows the actual rotor geometry to be modelled. Shafts with complicated geometry can be modelled provided that the shaft cross-section has two axes of symmetry with equal or unequal second moment of areas. The acceleration of a point on the shaft is determined in inertial and rotating frames. It is found that the finite element formulation becomes much simpler in a rotating frame of reference that rotates about the centre-line of the bearings with an angular velocity equal to the shafts spin speed. The finite element formulation in the above frame is ideally suited to non-circular shafts with solid or hollow, prismatic or tapered sections and continuous or abrupt change in cross-sections. The shaft and the disc can be modelled using the same types of element and this makes it possible to take into account the flexibility of the disc. The formulation also allows edge cracks to be modelled. A two-dimensional model of shaft disc systems executing synchronous whirl on isotropic bearings is presented. The application of the two-dimensional formulation is limited but it reduces the number of degrees of freedom. The three-dimensional solid and two-dimensional plane stress finite element models are extensively validated using standard available results.

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