A Transient Infinite Element for Multi-Dimensional Acoustic Radiation

1995 ◽  
Author(s):  
R. Jeremy Astley
Keyword(s):  
Exact Solution ◽  
Full Range ◽  
Three Dimensional ◽  
Test Problems ◽  
Transient Pressure ◽  
Infinite Domain ◽  
Transient Wave ◽  
Exterior Region ◽  
Infinite Elements ◽  
Transient Solutions

Abstract A novel family of infinite “wave envelope” elements is proposed for the solution of transient wave problems in unbounded regions. The elements are formed by applying an inverse Fourier transformation to a discrete wave envelope model in the frequency domain. This gives a coupled system of second-order equations which are readily integrated in time to yield transient pressure histories at nodal points on the surface of the radiating body and — in retarded form — at discrete points within the infinite domain. The infinite elements formed in this way can be applied quite generally to two-dimensional and three-dimensional problems and are fully compatible with conventional finite acoustical elements. They can be used to model radiating bodies of arbitrary shape but are demonstrated in the current instance in application to test problems which involve sound fields generated by spherical surfaces excited from rest, the exterior region being modeled by finite and infinite elements with explicit transverse interpolation. The computed transient solutions obtained from this formulation are compared to analytic solutions and shown to yield accurate results over a full range of exciting frequencies. The utility of the method for problems which involve broadband excitation is confirmed by comparisons of computed and analytic surface impedances for the steady harmonic case. These indicate that the accuracy of the scheme is limited only by a requirement to match element order to the highest order multi-pole component present in the radiated field. That is to say, elements of radial order 1 give an exact solution for monopole fields at all frequencies; elements of order 2 give an exact solution for dipole fields; elements of order 3 give an exact solution for quadrupole fields and so on. Similar results are presented for the fully three dimensional case. These support the extension of this hypothesis to three dimensional transient solutions subject only to the normal limitations imposed by spatial resolution in the transverse direction.

Integral Equation Method via Domain Decomposition and Collocation for Scattering Problems

1995 ◽  
Vol 62 (1) ◽  
pp. 186-192 ◽  
Author(s):  
Xiaogang Zeng ◽  
Fang Zhao
Keyword(s):  
Domain Decomposition ◽  
Large Scale ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Computational Effort ◽  
Boundary Integral ◽  
Infinite Domain ◽  
Scattering Problems ◽  
Exterior Region ◽  
Plane Incident Wave

In this paper an exterior domain decomposition (DD) method based on the boundary element (BE) formulation for the solutions of two or three-dimensional time-harmonic scattering problems in acoustic media is described. It is known that the requirement of large memory and intensive computation has been one of the major obstacles for solving large scale high-frequency acoustic systems using the traditional nonlocal BE formulations due to the fully populated resultant matrix generated from the BE discretization. The essence of this study is to decouple, through DD of the problem-defined exterior region, the original problem into arbitrary subproblems with data sharing only at the interfaces. By decomposing the exterior infinite domain into appropriate number of infinite subdomains, this method not only ensures the validity of the formulation for all frequencies but also leads to a diagonalized, blockwise-banded system of discretized equations, for which the solution requires only O(N) multiplications, where N is the number of unknowns on the scatterer surface. The size of an individual submatrix that is associated with a subdomain may be determined by the user, and may be selected such that the restriction due to the memory limitation of a given computer may be accommodated. In addition, the method may suit for parallel processing since the data associated with each subdomain (impedance matrices, load vectors, etc.) may be generated in parallel, and the communication needed will be only for the interface values. Most significantly, unlike the existing boundary integral-based formulations valid for all frequencies, our method avoids the use of both the hypersingular operators and the double integrals, therefore reducing the computational effort. Numerical experiments have been conducted for rigid cylindrical scatterers subjected to a plane incident wave. The results have demonstrated the accuracy of the method for wave numbers ranging from 0 to 30, both directly on the scatterer and in the far-field, and have confirmed that the procedure is valid for critical frequencies.

Tire Modeling by Finite Elements

1992 ◽  
Vol 20 (1) ◽  
pp. 33-56 ◽  
Author(s):  
L. O. Faria ◽  
J. T. Oden ◽  
B. Yavari ◽  
W. W. Tworzydlo ◽  
J. M. Bass ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Rolling Contact ◽  
Test Problems ◽  
State Behavior ◽  
Normal Contact ◽  
New Developments ◽  
Applied Torque ◽  
Dimensional Finite Element ◽  
Tire Modeling ◽  
Three Dimensional Finite Element

Abstract Recent advances in the development of a general three-dimensional finite element methodology for modeling large deformation steady state behavior of tire structures is presented. The new developments outlined here include the extension of the material modeling capabilities to include viscoelastic materials and a generalization of the formulation of the rolling contact problem to include special nonlinear constraints. These constraints include normal contact load, applied torque, and constant pressure-volume. Several new test problems and examples of tire analysis are presented.

Effective Convergence Checks for Verifying Finite Element Stresses at Three-Dimensional Stress Concentrations

2018 ◽  
Vol 3 (3) ◽  
Author(s):  
J. R. Beisheim ◽  
G. B. Sinclair ◽  
P. J. Roache
Keyword(s):  
Finite Element ◽  
Error Estimation ◽  
A Posteriori Error Estimates ◽  
Three Dimensional ◽  
Posteriori Error ◽  
Test Problems ◽  
A Posteriori ◽  
Stress Concentrations ◽  
Element Analysis ◽  
A Posteriori Error

Current computational capabilities facilitate the application of finite element analysis (FEA) to three-dimensional geometries to determine peak stresses. The three-dimensional stress concentrations so quantified are useful in practice provided the discretization error attending their determination with finite elements has been sufficiently controlled. Here, we provide some convergence checks and companion a posteriori error estimates that can be used to verify such three-dimensional FEA, and thus enable engineers to control discretization errors. These checks are designed to promote conservative error estimation. They are applied to twelve three-dimensional test problems that have exact solutions for their peak stresses. Error levels in the FEA of these peak stresses are classified in accordance with: 1–5%, satisfactory; 1/5–1%, good; and <1/5%, excellent. The present convergence checks result in 111 error assessments for the test problems. For these 111, errors are assessed as being at the same level as true exact errors on 99 occasions, one level worse for the other 12. Hence, stress error estimation that is largely reasonably accurate (89%), and otherwise modestly conservative (11%).

Adaptivity and Three Dimensional Hierarchical Boundary Elements for Rigid Acoustic Scattering Problems

1995 ◽  
Author(s):  
Steven J. Newhouse ◽  
Ian C. Mathews
Keyword(s):  
Boundary Element ◽  
Acoustic Analysis ◽  
Three Dimensional ◽  
Acoustic Scattering ◽  
Error Estimator ◽  
Infinite Domain ◽  
Scattering Problems ◽  
Effective Form ◽  
Mesh Density ◽  
Acoustic Scattering Problems

Abstract The boundary element method is an established numerical tool for the analysis of acoustic pressure fields in an infinite domain. There is currently no well established method of estimating the surface pressure error distribution for an arbitrary three dimensional body. Hierarchical shape functions have been used as a highly effective form of p refinement in many finite and boundary element applications. Their ability to be used as an error estimator in acoustic analysis has never been fully exploited. This paper studies the influence of mesh density and interpolation order on several acoustic scattering problems. A hierarchical error estimator is implemented and its effectiveness verified against the spherical problem. A coarse cylindrical mesh is then refined using the new error estimator until the solution has converged. The effectiveness of this analysis is shown by comparing the error indicators derived during the analysis to the solution generated from a very fine cylindrical mesh.

The generalized Burgers and Zabolotskaya-Khokhlov equations

1994 ◽  
Vol 447 (1930) ◽  
pp. 253-270 ◽  
Keyword(s):  
Exact Solution ◽  
Plane Waves ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Compressive Wave ◽  
Similarity Reduction ◽  
Weakly Nonlinear ◽  
Cylindrically Symmetric ◽  
Generalized Burgers Equation ◽  
Long Time

A point transformation between forms of the generalized Burgers equation (g b e) first given by Cates (1989) is investigated. Applications include generalizations of Scott’s (1981) classification of long-time behaviour for compressive wave solutions of the GBE and the equivalence of the exponential and cylindrical forms of the GBE, yielding an exact solution for the exponential GBE. Applications to nonlinear diffractive acoustics are considered by using a similarity reduction of the dissipative Zabolotskaya-Khokhlov (dzk) equation (describing the evolution of nearly plane waves in a weakly nonlinear medium with allowance for transverse variation effects) onto the GBE. The result is that waves from parabolic sources may be described by the cylindrical GBE in the case of two dimensions, and by the spherical GBE in the three-dimensional, cylindrically symmetric case. Furthermore, results on the formation of shocks and caustics in the context of the ZK equation are presented, along with an exact solution to the DZK equation. Exact solutions with caustic singularities are studied, along with a possible mechanism for their control. Finally, results on the evolution of a shock approaching a caustic are given through the identification of a series of parameter regimes dependent on the diffusivity.

Three-Dimensional CFD Analysis of Meshing Losses in a Spur Gear Pair

2018 ◽  
Author(s):  
T. Fondelli ◽  
D. Massini ◽  
A. Andreini ◽  
B. Facchini ◽  
F. Leonardi
Keyword(s):  
High Speed ◽  
Numerical Study ◽  
Fluid Dynamic ◽  
Three Dimensional ◽  
Spur Gear ◽  
Computational Domain ◽  
Gear Pair ◽  
Transient Pressure ◽  
Numerical Effort ◽  
Free Environment

The reduction of fluid-dynamic losses in high speed gearing systems is nowadays increasing importance in the design of innovative aircraft propulsion systems, which are particularly focused on improving the propulsive efficiency. Main sources of fluid-dynamic losses in high speed gearing systems are windage losses, inertial losses resulting by impinging oil jets used for jet lubrication and the losses related to the compression and the subsequent expansion of the fluid trapped between gears teeth. The numerical study of the latter is particularly challenging since it faces high speed multiphase flows interacting with moving surfaces, but it paramount for improving knowledge of the fluid behavior in such regions. The current work aims to analyze trapping losses in a gear pair by means of three-dimensional CFD simulations. In order to reduce the numerical effort, an approach for restricting computational domain was defined, thus only a portion of the gear pair geometry was discretized. Transient calculations of a gear pair rotating in an oil-free environment were performed, in the context of conventional eddy viscosity models. Results were compared with experimental data from the open literature in terms of transient pressure within a tooth space, achieving a good agreement. Finally, a strategy for meshing losses calculation was developed and results as a function of rotational speed were discussed.

Efficient Structural Acoustic Analysis Using Finite and Infinite Elements and Parallel Processing Architectures

1997 ◽  
Author(s):  
Stewart W. Moore ◽  
Henno Allik
Keyword(s):  
Acoustic Analysis ◽  
Three Dimensional ◽  
Infinite Element ◽  
Analytical Technique ◽  
Infinite Elements ◽  
Infinite Element Method ◽  
Modeling Practices ◽  
Processing Architectures ◽  
Memory Architectures ◽  
Element Theory

Abstract The analysis of three-dimensional shell structures submerged in an infinite fluid and subjected to arbitrary loadings is a computationally demanding problem regardless of the analytical technique used. Over the past several years, we have developed a combined finite/infinite element method of solving this class of problems that is more efficient than other available techniques, and have implemented it in a comprehensive set of computer programs called SARA. This paper presents an overview of our work in parallizing this software. In the first part of the paper, we describe our method for solving the fluid-structure interaction equations including infinite element theory, and modeling practices that have evolved for solving cylindrical geometries. The second part of the paper addresses parallalization of SARA-3D on both shared and distributed memory architectures. The SARA implementation of the method is described along with sample problems, and a comparison to a SARA-3D solution is provided.

Modelling Thousands of Fractures Using Analytic Elements

2021 ◽  
Author(s):  
Erik Toller ◽  
Otto Strack
Keyword(s):  
Exact Solution ◽  
Plane Strain ◽  
Numerical Models ◽  
Element Model ◽  
Global Scale ◽  
Hydraulic Fractures ◽  
Infinite Domain ◽  
Novel Approach ◽  
Analytic Elements ◽  
Kolosov Functions

&lt;p&gt;Understanding and modelling hydraulic fractures and fracture networks have a fundamental role in mapping the mechanical behaviour of rocks. A problem arises in the discontinuous behaviour of the fractures and how to accurately and efficiently model this. We present a novel approach for modelling many cracks randomly using analytic elements placed under plane strain conditions in an elastic medium. The analytic elements allow us to model the assembly computationally efficiently and up to machine precision. The crack element is the first step in the development of a model suitable for investigating the effect of fissures on tunnels in rock. The model can be used to validate numerical models and more.The solution for a single hydraulic pressurized crack in an infinite domain in plane strain was initially developed by Griffith (1921). We demonstrate that it is possible, by using series expansions in terms of complex variables, based on the Muskhelisvili-Kolosov functions, to generalize this solution to the case of an assembly of non-intersecting pressurized cracks. The solution consists of infinite series for each element Strack &amp; Toller (2020). The expressions for the displacements and stress tensor components approach the exact solution, as the number of terms in the series approaches infinity.We present the case where two cracks approach each other orthogonally to less than 1/2000th of the cracks length. We show the effect of increasing the number of terms in the expansion and how this influences the precision, demonstrating that the result approaches the exact solution. We also present a case with 10,000 cracks; the coefficients are determined using an iterative solver. By using analytic elements, we can both present the corresponding stress and deformations field for the global scale and for small scales in the close proximity of individual cracks.ReferencesGriffith, A. A. (1921). The phenomena of rupture and &amp;#64258;ow in solids. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 221(582-593):163&amp;#8211;198.Strack, O. D. L. and Toller, E. A. L. (2020). An analytic element model for highly fractured elastic media, manuscript submitted for publication in International Journal for Numerical and Analytical Methods in Geomechanics.&lt;/p&gt;

Geometrical characterisation of fault arrays in three dimensions

2021 ◽  
Author(s):  
Vincent Roche ◽  
Giovanni Camanni ◽  
Conrad Childs ◽  
Tom Manzocchi ◽  
John Walsh ◽  
...  
Keyword(s):  
Full Range ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Normal Faults ◽  
Fault Geometry ◽  
Surface Geometry ◽  
Fault Surface ◽  
Mechanical Stratigraphy ◽  
Quantitative Basis ◽  
Geometry Analysis

&lt;p&gt;Normal faults are often complex three-dimensional structures comprising multiple sub-parallel segments separated by intact or breached relay zones. In this study we outline geometrical characterisations capturing this 3D complexity and providing a semi-quantitative basis for the comparison of faults and for defining the factors controlling their geometrical evolution. Relay zones are classified according to whether they step in the strike or dip direction and whether the relay zone-bounding fault segments are unconnected in 3D or bifurcate from a single surface. Complex fault surface geometry is then described in terms of the relative numbers of different types of relay zones to allow comparison of fault geometry between different faults and different geological settings. A large database of 87 fault arrays compiled primarily from mapping 3D seismic reflection surveys and classified according to this scheme, reveals the diversity of 3D fault geometry. Analysis demonstrates that mapped fault geometries depend on geological controls, primarily the heterogeneity of the faulted sequence and the presence of a pre-existing structure. For example, relay zones with an upward bifurcating geometry are prevalent in faults that reactivate deeper structures, whereas the formation of laterally bifurcating relays is promoted by heterogeneous mechanical stratigraphy. In addition, mapped segmentation depends on resolution limits and biases in fault mapping from seismic data. In particular, the results suggest that the proportion of bifurcating relay zones increases as data resolution increases. Overall, where a significant number of relay zones are mapped on a single fault, a wide variety of relay zone geometries occurs, demonstrating that individual faults can comprise segments that are both bifurcating and unconnected in three dimensions. Models for the geometrical evolution of fault arrays must therefore account for the full range of relay zone geometries that appears to be a characteristic of all faults.&lt;/p&gt;

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