Helmholtz Boundary Integral Equation Research Articles

Buckling and vibro-acoustic behavior analysis of laminated immersed cylindrical shells with hydrostatic pressure based on Chebyshev-Fourier spectral approach

The buckling and vibroacoustic behaviour of laminated cylindrical shell structures is vital for the safety and operational efficiency of underwater vehicles. This paper introduces a Chebyshev-Fourier spectral approach for the comprehensive analysis of buckling and vibro-acoustic behaviour in water-immersed laminated cylindrical shells under external hydrostatic pressure. The approach combines Chebyshev polynomials and Fourier series to approximate the displacement of shell and acoustic pressure of the fluid-structure-interaction (FSI) surface in cylindrical coordinates. An approximate solution is used in the axial direction, while an analytical solution is used in the circumferential direction, allowing for the analysis of different circumferential wave numbers n. The influence of external fluid pressure is characterized through the Helmholtz boundary integral equation, while the effect of hydrostatic pressure is deduced using linear elastic theory. From a theoretical derivation standpoint, hydrostatic pressure leads to a reduction in equivalent stiffness, while the external fluid contributes to an increase in the effective mass matrix. Validated against established literature analysis, the study finds that hydrostatic pressure decreases the natural frequencies, though the impact is less significant for lower circumferential wave numbers. For these lower wave numbers, the overall radiated sound power remains largely independent of external hydrostatic pressure, regardless of being in air or water. However, at higher circumferential wave numbers, the effect is more pronounced. This study not only advances the theoretical framework for analysing submerged cylindrical shells but also provides crucial insights into designing more robust structures for underwater applications.

Vibroacoustic analysis of submerged fluid-filled cylindrical shell

This study presents a comprehensive Chebyshev-Fourier spectral method for analyzing the free and forced vibroacoustic characteristics of submerged fluid-filled cylindrical shells in heavy fluids. The vibroacoustic interaction system of submerged cylindrical shells consists of three discrete regions: The Sanders shell theory is applied for the shell structure, whereas the wave equation is used to determine the fluid inside. The external fluid pressure on the shell is described using the Helmholtz boundary integral equation. The governing equations for the shell's motion, the formulation of the external fluid pressure on the fluid-structure-interaction (FSI) surface using first-class Chebyshev polynomial and Fourier series expansions, and the internal sound pressure employing two-dimensional Chebyshev polynomial and Fourier series expansions, are seamlessly integrated with Gauss-Chebyshev-Lobatto sampling techniques. Artificial springs are implemented to replicate the boundary conditions of the cylindrical shell. The precision of the proposed methodology is corroborated by comparisons with existing literature and finite element method (FEM) results. Furthermore, the research examines how the properties of the acoustical medium on the vibroacoustic responses of the cylindrical shell. Results indicate that the presence of air inside the shell slightly influence on its acoustic radiation. However, replacing the internal medium with water significantly alters the shell's original radiation characteristics.

A hybrid wave and finite element/boundary element method for predicting the vibroacoustic characteristics of finite-width complex structures

This paper presents a hybrid wave and finite element/boundary element method for predicting the vibroacoustic characteristics of complex panels characterised by an irregular cross-sectional shape with a non-flat upper surface. In this approach, the panels are treated as waveguides and only a small segment of the panel is modelled using finite elements. The mass and stiffness matrices of the segment are then post-processed to project the motion onto its cross-sectional boundaries interfacing with the surrounding fluids. The acoustic pressures in the fluids are modelled by applying Fourier transforms to the Helmholtz boundary integral equations in the wavenumber domain. For each wavenumber component, the pressures acting on the segment are transformed into external nodal forces by evaluating the virtual work performed by the surface pressure on the segment. The wave and finite element formulations are subsequently coupled to the discretised boundary integral equations to determine the response of the panel to acoustic wave excitation. The hybrid method for predicting dispersion curves and sound transmission is validated through comparisons with predictions obtained from both an analytical model and a wave-based method found in the literature. To demonstrate the practical utility of the approach, the hybrid method is employed to predict sound transmission through a corrugated sandwich panel, where developing an analytical model is difficult. The method provides accurate predictions at a low computational cost.

Isogeometric Fast Multipole Boundary Element Method Based on Burton-Miller Formulation for 3D Acoustic Problems

An isogeometric boundary element method is applied to simulate wave scattering problems governed by the Helmholtz equation. The NURBS (non-uniform rational B-splines) widely used in the CAD (computer aided design) field is applied to represent the geometric model and approximate physical field variables. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The singular integrals existing in Burton-Miller formulation are evaluated directly and accurately using Hadamard’s finite part integration. Fast multipole method is applied to accelerate the solution of the system of equations. It is demonstrated that the isogeometric boundary element method based on NURBS performs better than the conventional approach based on Lagrange basis functions in terms of accuracy, and the use of the fast multipole method both retains the accuracy for isogeometric boundary element method and reduces the computational cost.

Far field acoustic radiation and vibration analysis of combined shells submerged at finite depth from free surface

A completely submerged underwater vehicle is modeled as a conical-cylindrical-spherical combined shell, the vibro-acoustic response near free surface of which is rarely analyzed in existing studies. In this paper, a Ritz-Legendre spectral method is proposed to analyze the vibro-acoustic response of combined shells under different submerged depths from the free surface. A theoretical structural model is established based on the Love shell theory, virtual spring technology and coordination relationship between displacements and rotation angle of adjacent subshells. The half-space external acoustic field model with the free surface is formulated by the Legendre spectral method, image method and Kirchhoff–Helmholtz boundary integral equation. The unified displacement admissible function of each subshell is expanded as Legendre orthogonal polynomial along generatrix direction and Fourier series along circumferential direction. The free surface treated as an ideal infinite acoustic soft boundary is substituted into acoustic wave equation and the half-space Green's function is obtained. The key technology to construct final coupled governing equation is how to expand the half-space Green's function with two-dimensional Fourier transform. Computed results are compared with those of the references and finite element method. The convergence, applicability, and correctness of the present method are verified. Influence of the free surface on the vibration and far-field acoustic radiation of combined shells under different submerged depths is analyzed, providing engineering significance for predicting external acoustic radiation of complicated combined shells with free surface.

Spectral Galerkin method for solving Helmholtz boundary integral equations on smooth screens

Abstract We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin–Bubnov methods take as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the unit disk. By exactly depicting edge singular behaviors we show that these spectral or high-order bases yield super-algebraic error convergence in the corresponding energy norms whenever the screen is an analytic deformation of the unit disk. Moreover, we provide a fully discrete analysis of the method, including quadrature rules, based on analytic extensions of the spectral basis to complex neighborhoods. Finally, we include numerical experiments to support our claims as well as appendices with computational details for treating the associated singular integrals.

Fast Calderón preconditioning for Helmholtz boundary integral equations

Calderón multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds. However, when discretizing by local low-order basis functions as in standard Galerkin boundary element methods, their computational performance worsens as meshes are refined. This stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of L2-pairings. Based on coarser quadrature rules over dual cells and H-matrix compression, we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of Calderón preconditioning for the three-dimensional Helmholtz weakly and hyper-singular boundary integral operators. Several numerical experiments validate our claims and point towards further enhancements.

Noise Prediction of Traction Gear of High-Speed EMU Base on Acoustic-Structural Coupled Method

The dynamic characteristics of traction gear transmission system have great influence on the safety, comfort and reliability of EMU. Base on acoustic-structural coupled theory, the noise radiation characteristics of gear transmission system in high-speed train CRH380A are studied by finite element-boundary element method. The acoustic radiation analysis model is built by using acoustic BEM. The radiated noise is predicted by Helmholtz boundary integral equation. The research results can provide a theoretical basis for the optimization design of the low noise gear in EMU.

An isogeometric approach of two dimensional acoustic design sensitivity analysis and topology optimization analysis for absorbing material distribution

A study of structural shape optimization and absorbing material distribution optimization of noise barrier structures based on the recently proposed isogeometric analysis method with exact geometric definitions is presented. The acoustic scattering is approximated using the basis functions that represent the geometry. A fast multipole method is applied to accelerate the solution of the boundary element method (BEM). The Burton–Miller formulation is used to overcome the fictitious frequency problem when a single Helmholtz boundary integral equation is used for the exterior boundary–value problem. The strongly singular integrals in the Burton–Miller formulation using the isogeometric BEM are evaluated explicitly and directly, particularly for the sensitivity formulation with a hyper-singular integral. The optimality criteria method is used for two types of optimization analyses, shape optimization and material distribution topology optimization. For the shape optimization, the design variables can be set to the locations of the control points because the control points determine the shape of structure. For the second optimization, a new material interpolation scheme for acoustic problems based on the solid isotropic material with penalization (SIMP) method is given, where the interpolation variable is not the real structural density used in a conventional SIMP, but a fictitious material density that determines the normalized surface admittance. Several examples are presented to demonstrate the validity and efficiency of the proposed algorithm.

Minimization of sound radiation in fully coupled structural-acoustic systems using FEM-BEM based topology optimization

A topology optimization approach is proposed for the optimal design of bi-material distribution on underwater shell structures. The coupled finite element method (FEM) / boundary element method (BEM) scheme is used for the system response analysis, where the strong interaction between the structural and the acoustic domain is considered. The Burton-Miller formulation is used to overcome the fictitious eigen-frequency problem when using a single Helmholtz boundary integral equation for exterior acoustic problems. The design variables are the artificial densities of design material elements in a bi-material model constructed by the solid isotropic material with penalization (SIMP) method, and the minimization of sound power level (SWL) is chosen to be the design objective. In this study, the adjoint operator method is employed to calculate the sensitivity of the objective function with respect to the design variables. Based on the sensitivity information, the gradient-based optimization solver is finally applied for updating the design variables during the optimization process. Numerical tests are provided to illustrate the correctness of the sensitivity analysis approach and the validity of the proposed optimization procedure. Results show that the heavy fluid feedback has a big impact on the final design, and thus it is necessary to conduct a strong coupling scheme between the fluid and structures. In addition, the optimal design is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.

2D Structural Acoustic Analysis Using the FEM/FMBEM with Different Coupled Element Types

Abstract A FEM-BEM coupling approach is used for acoustic fluid-structure interaction analysis. The FEM is used to model the structure and the BEM is used to model the exterior acoustic domain. The aim of this work is to improve the computational efficiency and accuracy of the conventional FEM-BEM coupling approach. The fast multipole method (FMM) is applied to accelerating the matrix-vector products in BEM. The Burton-Miller formulation is used to overcome the fictitious eigen-frequency problem when using a single Helmholtz boundary integral equation for exterior acoustic problems. The continuous higher order boundary elements and discontinuous higher order boundary elements for 2D problem are developed in this work to achieve higher accuracy in the coupling analysis. The performance for coupled element types is compared via a simple example with analytical solution, and the optimal element type is obtained. Numerical examples are presented to show the relative errors of different coupled element types.

2D Acoustic Design Sensitivity Analysis Based on Adjoint Variable Method Using Different Types of Boundary Elements

Continuous linear and quadratic boundary elements are often applied to numerical solution. Discontinuous higher-order boundary elements are developed for 2D acoustic problems to achieve higher accuracy in this paper. The Burton–Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problem. The strong singular integrals in Burton–Miller formulation using different types of element discretization are evaluated explicitly and directly, respectively. An example of scattering by an infinite rigid cylinder is presented to compare the performance of different types of elements. The effect of the position of nodes on the performance of discontinuous elements is studied, and an empirical value for optimal nodal position is concluded in this paper. Adjoint variable method is applied to evaluate the sensitivity value of the objective function, and the method of moving asymptotes is used for structural optimization analysis of noise barrier.

Is the Burton–Miller formulation really free of fictitious eigenfrequencies?

This paper is concerned with the fictitious eigenfrequency problem of the boundary integral equation methods when solving exterior acoustic problems. A contour integral method is used to convert the nonlinear eigenproblems caused by the boundary element method into ordinary eigenproblems. Since both real and complex eigenvalues can be extracted by using the contour integral method, it enables us to investigate the fictitious eigenfrequency problem in a new way rather than comparing the accuracy of numerical solutions or the condition numbers of boundary element coefficient matrices. The interior and exterior acoustic fields of a sphere with both Dirichlet and Neumann boundary conditions are taken as numerical examples. The pulsating sphere example is studied and all fictitious eigenfrequencies corresponding to the related interior problem are observed. The reasons are given for the usual absence of many fictitious eigenfrequencies in the literature. Fictitious eigenfrequency phenomena of the Kirchhoff–Helmholtz boundary integral equation, its normal derivative formulation and the Burton–Miller formulation are investigated through the eigenvalue analysis. The actual effect of the Burton–Miller formulation on fictitious eigenfrequencies is revealed and the optimal choice of the coupling parameter is confirmed.

A wideband FMBEM for 2D acoustic design sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element method (FMBEM) for two dimensional acoustic design sensitivity analysis based on the direct differentiation method. The wideband fast multipole method (FMM) formed by combining the original FMM and the diagonal form FMM is used to accelerate the matrix-vector products in the boundary element analysis. The Burton---Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The strongly singular and hypersingular integrals in the sensitivity equations can be evaluated explicitly and directly by using the piecewise constant discretization. The iterative solver GMRES is applied to accelerate the solution of the linear system of equations. A set of optimal parameters for the wideband FMBEM design sensitivity analysis are obtained by observing the performances of the wideband FMM algorithm in terms of computing time and memory usage. Numerical examples are presented to demonstrate the efficiency and validity of the proposed algorithm.

Solving Acoustic Radiation and Scattering Based on Coiflet

In this paper, the acoustic Helmholtz boundary integral equation is solved using Coiflet scaling functions with interpolation approximation property. The scaling functions are utilized as base and test functions in Galerkin method and the expanded coefficients are the values of the function in sampling points, so the number of numerical integral is reduced. Two numerical examples are given and the calculation results agree well with the theoretical results, which show the high accuracy of the estimation and demonstrate validity and applicability of the method.

A new diagonal form fast multipole boundary element method for solving acoustic Helmholtz equation

The conventional exterior acoustic Helmholtz boundary integral equation is prohibitively expensive for solving large-scale engineering problems. In order to effectively overcome this problem, the fast multipole method is introduced to the BIE, and accelerates the iterative solution for the system matrix equation. Due to using the diagonal form multipole expansions of the fundamental solution in the BIE, the computational efficiency of the new fast multipole boundary element method (FMBEM) is improved significantly compared to the conventional BEM. Both the computational complexity and memory requirement of the FMBEM are drastically reduced to O(N), where N is the number of degrees of freedom (DOFs). Numerical examples including a large submarine model with more than 420000 DOFs demonstrate the accuracy and efficiency of the FMBEM, and clearly show the advantage of the new algorithm for solving the large-scale acoustic problems. The developed FMBEM would be potential for engineering applications.

A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures

We present a new nonlinear optimization procedure for the computation of generalized Gaussian quadratures for a broad class of square integrable functions on intervals. While some of the components of this algorithm have been previously published, we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works. The new algorithm successfully computes generalized Gaussian quadratures in a number of instances in which the previous algorithms fail. Four applications of our scheme to computational physics are presented: the construction of discrete plane wave expansions for the Helmholtz Green's function, the design of linear array antennae, the computation of a quadrature for the discretization of Laplace boundary integral equations on certain domains with corners, and the construction of quadratures for the discretization of Laplace and Helmholtz boundary integral equations on smooth surfaces.

A method for multi-frequency calculation of boundary integral equation in acoustics based on series expansion

The paper presents a method to solve the problem of multi-frequency calculation of Helmholtz boundary integral equation in acoustics. Based on series expansion, system matrices are independent of wavenumber and become the matrix power series of wavenumber. As a result, all matrices in the matrix power series are only dependent on the structure geometry. In addition, an element transform method to calculate the singular integral and Cauchy singular integral is also discussed because the singular integral need to be solved using the method. The convergence of the series expansion method is also proved in this paper. The effectiveness of the method is confirmed by two numerical examples.

Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations

Boundary integral equations are well suitable for the analysis of seismic waves propagation in unbounded domains. Formulations in elastodynamics are well developed. In contrast, for the dynamic analysis of viscoelastic media, there are very seldom formulations by boundary integral equations. In this Note, we propose a new and simple formulation of time harmonic viscoelasticity with the Zener model, which reduces to classical elastodynamics if a compatibility condition is satisfied by boundary conditions. Intermediate variables which satisfy the classical elastodynamic equations are introduced. It makes it possible to utilize existing numerical tools of time harmonic elastodynamics. To cite this article: S. Chaillat, H.D. Bui, C. R. Mecanique 335 (2007).

Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations

Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations