Wave Envelope Elements Research Articles

A weak-form meshfree coupled with infinite element method for predicting acoustic radiation

To improve the computational accuracy and eliminate the element matching in the traditional method based on the finite and infinite elements, the coupling of the meshfree weak-form method with the variable order infinite acoustic wave envelope element (WEE) is studied in this paper and achieved by forming continuous acoustic shape function on an artificial interface, by which the infinite domain is divided into two subdomains where the meshfree and WEE methods are applied respectively. A coupled meshfree weak-form method is developed for use in predicting acoustic radiation in infinite domain. A key feature of the present method is the ability of properly modelling the amplitude decay of the outgoing travelling wave and significantly improving the prediction accuracy. The impacts of the meshfree shape function approximation/interpolation scheme and the space scale of the region used for the coupling on the proposed method are investigated. Simulation and experiment are performed to test the applicability and performance of the method, and show that the method is a valuable approach for predicting exterior acoustic radiation in infinite domain.

An Element-Free Galerkin Coupled with Improved Infinite Element Method for Exterior Acoustic Problem

It is known that the variable-order infinite acoustic wave envelope element (WEE) must be coupled with finite element method (FEM) by element matching for computing acoustic field radiated from radiators with complex geometric shapes. Therefore, the WEEs have to be reconstructed when the finite elements are refined or changed. To overcome the shortcoming, the element-free Galerkin (EFG) coupled with improved WEE (IWEE) method is presented to compute acoustic problems in the infinite domain. The continuity and compatibility of the acoustic pressure are maintained by IWEE that is composed of a standard WEE and a fictitious finite mesh. A key feature of the method is the introduction of the EFG method which is employed to eliminate the element matching and improve the accuracy of predicted acoustic pressure. The factors that affect the performance of the method are investigated by numerical examples, which include shape function construction, the weight function and the size of the influence domain. The numerical results show that the present method provides more accurate results compared to the coupled FEM-WEE method. The experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain.

A Meshfree Radial Point Interpolation Coupled with Infinite Acoustic Wave Envelope Element Method for Computing Acoustic Fields

A Meshfree Radial Point Interpolation Coupled with Infinite Acoustic Wave Envelope Element Method for Computing Acoustic Fields

A coupled interpolating meshfree method for computing sound radiation in infinite domain

SummaryIn this paper, the coupling of the improved interpolating element‐free Galerkin (IIEFG) method and the variable‐order infinite acoustic wave envelope element (WEE) method is studied. A coupled IIEFG‐WEE method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages. The coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the IIEFG and WEE domains. In the IIEFG domain, the improved interpolating moving least‐squares (IIMLS) method is used to form the shape functions satisfying the Kronecker delta condition while nonsingular weight functions can be used. The impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated. The numerical results show that the coupled IIEFG‐WEE method can take full advantage of both the IIEFG and WEE methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the WEE. The experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain.

Calculation of sound radiation in infinite domain using a meshless method

A meshless method coupling with a variable order infinite acoustic wave envelope element for sound radiation calculation in infinite domain is presented with the aim of accurately calculating the acoustic radiation and improving computational efficiency. It is based on using the element-free Galerkin method in the inner region enclosing the radiator and a variable order infinite acoustic wave envelope element in the outer region for the proper modeling of the pressure amplitude decay. The details are provided for the derivation and implementation of this method. The factors of influencing the performance of the method, which include the shape function constructing, the number of integration points, the weight functions, and the support domain, are discussed. A hybrid adaptive Gauss-Legendre quadrature is devised to obtain good integration accuracy. The suitable radius of the support domain for the acoustic field calculation in free space is also determined by use of numerical experiments. A complex structure is designed for simulation to validate the method. The results illustrate the accuracy, applicability, and effectiveness of this method.

Finite element analysis of piezoelectric underwater transducers for acoustic characteristics

This paper presents a simulation technique for analyzing acoustic characteristics of piezoelectric underwater transducers. A finite element method is adopted for modeling piezoelectric coupled problems including material damping and fluid-structure interaction problems by taking system matrices in complex form. For the finite element modeling of unbounded acoustic fluid, infinite wave envelope element (IWEE) is adopted to take into account the infinite domain. An in-house finite element program is developed and technical issues for implementing the program are explained. Using the simulation program, acoustic characteristics of tonpilz transducer are analyzed in terms of modal analysis, radiated pressure distribution, pressure spectrum, transmitting-voltage response and impedance analysis along with experimental comparison. The developed simulation technique can be used for designing ultrasonic transducers in the areas of nondestructive evaluation, underwater acoustics and bioengineering.

A new infinite element for unbounded water wave problems

A new mapped infinite wave envelope element is developed to model two-dimensional unbounded water wave problems in conjunction with wave finite elements. The infinite elements may be used on the exterior of an ellipse or other shapes. Examples are implemented to test the model for the two special cases of circular and elliptical cylinder diffraction problems. Comparisons are made between the analytical and finite/infinite element solutions. The errors are small, around 2% for a very coarse mesh and decrease to around 0.5% when the wave envelope infinite elements are located reasonably far from the diffracting object, i.e. at a distance equal to the semi-major axis of the ellipse.

Improved conditioning of infinite elements for exterior acoustics

AbstractAn optimized version of the so‐called mapped wave envelope elements, also known as Astley–Leis elements, is introduced. These elements extend to infinity in one dimension and therefore provide an approach to the simulation of exterior acoustical problems in both frequency and time domains. Their formulation is improved significantly through the proper choice of polynomial bases in the direction of radiation. In particular, certain Jacobi polynomials are identified which behave well with respect to conditioning of the system matrices. As a consequence, the size of the finite element discretization may reduce considerably without any loss of accuracy. In addition, the new polynomial bases lead to superior performance of the infinite elements in conjunction with iterative solvers. Copyright © 2003 John Wiley & Sons, Ltd.

A Comparison of Numerical Methods for Active Sonar Array Performance

Low frequency active sonar (LFAS) arrays are complicated devices requiring careful design. Prototype LFAS arrays are expensive to construct and test. Accurate prediction of acoustic and electrical performance is therefore of great interest to LFAS designers. This generally involves solving a fully coupled problem relating the electrical drive to the resulting acoustic field. To derive results a numerical solution method is clearly the only recourse. This paper compares various numerical techniques in terms of accuracy, efficiency and overall applicability for the solution of LFAS problems. These are based around finite element (FE) and boundary element (BE) descriptions of the surrounding acoustic medium. Here we consider a pure FE approach based on wave envelope elements and a combined FE/BE scheme using an approximate BE formulation. These are contrasted with a pure BE approach that has been demonstrated to provide accurate predictions of LFAS array performance over a number of years. A piston stack transducer and a line array of free-flooding ring projectors are considered as example LFAS problems. The acoustic, structural and electrical responses are considered.

A COMPARISON OF NUMERICAL METHODS FOR ACTIVE SONAR ARRAY PERFORMANCE

Low frequency active sonar (LFAS) arrays are complicated devices requiring careful design. Prototype LFAS arrays are expensive to construct and test. Accurate prediction of acoustic and electrical performance is therefore of great interest to LFAS designers. This generally involves solving a fully coupled problem relating the electrical drive to the resulting acoustic field. To derive results a numerical solution method is clearly the only recourse. This paper compares various numerical techniques in terms of accuracy, efficiency and overall applicability for the solution of LFAS problems. These are based around finite element (FE) and boundary element (BE) descriptions of the surrounding acoustic medium. Here we consider a pure FE approach based on wave envelope elements and a combined FE/BE scheme using an approximate BE formulation. These are contrasted with a pure BE approach that has been demonstrated to provide accurate predictions of LFAS array performance over a number of years. A piston stack transducer and a line array of free-flooding ring projectors are considered as example LFAS problems. The acoustic, structural and electrical responses are considered.

Far-Field Calculations for Turbofan Noise

The development of variable-order mapped infinite wave envelope elements for finite element modeling of acoustic radiation from turbofan nacelles is reviewed. These elements can be used outside of the near field as a nonreflecting boundary condition and as a finite-dimensional discretization of the infinite exterior domain for calculation of far-field acoustic radiation. The Kirchhoff method previously has been investigated in the context of the turbofan inlet radiation problem to extend a near-field solution to the far field. When the results of the mapped infinite wave envelope element analysis are compared with results of the Kirchhoff method, it is found that the mapped element formulation efficiently provides the acoustic far field, obviating the need to carry out an additional Kirchhoff integral formulation. Relative efficiencies are considered, and it is found that finite-dimensional discretization of the far field using mapped infinite elements is superior to the Kirchhoff formulation when radiation directivity in the entire far field is required.

P-adaptive finite element analysis of unbounded electromagnetic wave problems

The boundary that artificially truncates an unbounded domain in the finite element analysis of an electromagnetic wave problem necessarily introduces a numerical error. The adaptive technique proposed reduces this boundary error, as well as the usual finite element discretisation error. The scattering or radiating object and its immediate surroundings are meshed with hierarchal finite elements. Outside, a thick layer of free space is meshed with relatively few hierarchal wave-envelope elements. During p-adaption, increasing the order of the wave-envelope elements reduces the boundary error. Moreover, this reduction in boundary error is selective: in directions of strong radiation, the error reduction is greater.

A Reflection Free Boundary Condition for Propagation in Uniform Flow Using Mapped Infinite Wave Envelope Elements

Variable order mapped infinite wave envelope elements are developed for finite element modeling of acoustic radiation in a uniformly moving medium. These elements are used as a nonreflecting boundary condition for computations on an infinite domain in which a radiating body is immersed in a moving medium which is essentially undisturbed outside of the near field. The mapped elements provide a boundary condition equivalent to element stiffness, mass, and damping matrices appended to an inner standard FEM mesh. A demonstration of the performance of mapped elements as influenced by element order is given in the context of acoustic radiation from a turbofan inlet and exhaust.

A REFLECTION FREE BOUNDARY CONDITION FOR PROPAGATION IN UNIFORM FLOW USING MAPPED INFINITE WAVE ENVELOPE ELEMENTS

Variable order mapped infinite wave envelope elements are developed for finite element modeling of acoustic radiation in a uniformly moving medium. These elements are used as a nonreflecting boundary condition for computations on an infinite domain in which a radiating body is immersed in a moving medium which is essentially undisturbed outside of the near field. The mapped elements provide a boundary condition equivalent to element stiffness, mass, and damping matrices appended to an inner standard FEM mesh. A demonstration of the performance of mapped elements as influenced by element order is given in the context of acoustic radiation from a turbofan inlet and exhaust.

MAPPED INFINITE WAVE ENVELOPE ELEMENTS FOR ACOUSTIC RADIATION IN A UNIFORMLY MOVING MEDIUM

Variable order mapped infinite wave envelope elements are developed for finite-element modelling (FEM) of acoustic radiation in a uniformly moving medium. These elements can be used as a non-reflecting boundary condition for computations on an infinite domain in which a radiating body is immersed in a moving medium which is essentially undisturbed outside of the near field. An additional result of this study shows that the mapped wave envelope elements provide a boundary condition equivalent to stiffness, mass, and damping matrices appended to the inner mesh. By choosing the transition between the standard FEM mesh and the mapped infinite wave envelope as a surface of constant phase the mass matrix is caused to vanish identically. This has implications for transient FEM modelling of acoustic radiation. A demonstration of the characteristics of mapped infinite wave envelope elements is given in the context of acoustic radiation from a turbofan inlet for which benchmark results are known.

A reflection free boundary condition for propagation in uniform flow using mapped infinite wave envelope elements

A new reflection free boundary condition based on mapped infinite wave envelope elements is developed for finite-element modeling of acoustic radiation in a uniformly moving medium. These elements are useful for obtaining closure for computations on an infinite domain in which a radiating body is immersed in a moving medium which is essentially undisturbed outside of the near field. The application described is for harmonic excitation which leads to a model independent of time. An additional result of this study shows that the mapped wave envelope elements provide a boundary condition equivalent to stiffness, mass, and damping matrices appended to the inner mesh. By choosing the transition between the standard FEM mesh and the mapped WE as a surface of constant phase, the mass matrix is caused to vanish identically. This has implications for transient FEM modeling of acoustic radiation. A demonstration of the characteristics of the mapped WE elements is given in the context of acoustic radiation from a turbofan inlet for which benchmark results are known. It is found that for similar computational cost mapped infinite wave envelope elements are superior to conventional wave envelope elements for time-independent formulations.

Transient spheroidal elements for unbounded wave problems

A wave-envelope element numerical scheme is applied to the solution of unbounded wave problems. The scheme is based on a Fourier transformation of a discrete model formulated in the frequency domain. This yields a discrete system of ordinary differential equations in time which are local in space. Oblate and prolate spheroidal element geometries are used. The accuracy of the scheme is demonstrated by a comparison of computed and analytic solutions for axisymmetric test cases. Time-harmonic and transient solution are presented. An indirect solution procedure is also presented which permits the sparsity of the transient equations to be utilised more effectively.

Mapped spheroidal wave-envelope elements for unbounded wave problems

This paper describes a family of axisymmetric, spheroidal ‘wave envelope’ elements for modelling exterior wave problems. They are of variable radial order and can be used to represent steady and transient wave fields. The formulation is presented for the axisymmetric case using elements which are based on oblate and prolate spheroidal geometries. These offer the prospect of reduced dimensionality—in comparison to conventional, spherically formulated elements—when used to represent wave fields in the vicinity of slender or flat objects. Conjugated weighting functions are used to give frequency-independent acoustic ‘mass’, ‘stiffness’ and ‘damping’ matrices. This facilitates a simple extension of the method to transient problems. The effectiveness and accuracy of the method is demonstrated by a comparison of computed and analytic solutions for sound fields generated by a rigid sphere in steady harmonic oscillation, by a rigid sphere excited from rest, and by a circular plate vibrating in a plane baffle.

Three-dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part II. Formulation in the time domain

A variable-order, infinite “wave-envelope” element scheme is formulated for transient, unbounded acoustical problems. The transient formulation which is local in space and time is obtained by applying an inverse Fourier transformation to a time-harmonic wave-envelope model whose formulation is described in a companion article. This procedure yields a coupled system of second-order differential equations which can be integrated in time to yield transient pressure histories at discrete nodal points. Far-field transient pressures can also be obtained at adjusted times. The method can be applied quite generally to two-dimensional and three-dimensional problems and is compatible with a conventional finite element model in the near field. The utility of the method is confirmed by the presentation of transient solutions for axisymmetric and fully three-dimensional test problems. An implicit time integration scheme is used and computed results are compared to analytic solutions and to solutions obtained from alternative numerical schemes. Close correspondence is demonstrated and the scheme is shown to be stable for the problems which are presented. CPU times for a large three-dimensional problem are shown to compare favorably with those required for an equivalent transient boundary element computation.

Three-dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part I. Formulation in the frequency domain

Mapped wave-envelope elements of variable radial order are presented for the computation of time-harmonic, unbounded, three-dimensional acoustical fields. Their application to transient problems is described in a companion article (Part II). Accuracy is assessed by a comparison of computed and analytic solutions for multi-pole fields generated by a vibrating sphere. Solutions are also presented for plane wave scattering. Elements of radial order m+l are shown to be capable of modeling multi-pole components of order m, although the provision of adequate transverse resolution is shown to be a stringent requirement, particularly at high frequencies. Ill-conditioning of the coefficient matrix limits the practical implementation of the method to elements of radial order eleven or less. The utility of the method for more general geometries is demonstrated by the presentation of computed solutions for the sound field generated by the vibration of a cylindrical piston in a plane baffle and of an idealised engine casing under anechoic conditions. The computed results are shown to be in close agreement with the analytic solution in the case of the cylindrical piston, and with a boundary element solution in the case of the engine casing.