An efficient and high‐order frequency‐domain approach for transient acoustic–structure interactions in three dimensions

This paper is concerned with numerical solution of the transient acoustic–structure interaction problems in three dimensions. An efficient and higher‐order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency‐domain. The exponential window applied to the acoustic–structure system yields an artificial damping to the system, which eliminates the wrap‐around errors brought by the discrete Fourier transform. The frequency‐domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast Fourier transform. Numerical studies are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

A remark on the numerical solution of singular integral equations and the determination of stress-intensity factors

Summary As is well-known, an efficient numerical technique for the solution of Cauchy-type singular integral equations along an open interval consists in approximating the integrals by using appropriate numerical integration rules and appropriately selected collocation points. Without any alterations in this technique, it is proposed that the estimation of the unknown function of the integral equation is further achieved by using the Hermite interpolation formula instead of the Lagrange interpolation formula. Alternatively, the unknown function can be estimated from the error term of the numerical integration rule used for Cauchy-type integrals. Both these techniques permit a significant increase in the accuracy of the numerical results obtained with an insignificant increase in the additional computations required and no change in the system of linear equations solved. Finally, the Gauss-Chebyshev method is considered in its original and modified form and applied to two crack problems in plane isotropic elasticity. The numerical results obtained illustrate the powerfulness of the method.     

An Analytical Study on a Model Describing Heat Conduction in Rectangular Radial Fin with Temperature-Dependent Thermal Conductivity

The coupling of the homotopy perturbation method (HPM) and the variational iteration method (VIM) is a strong technique for solving higher dimensional initial boundary value problems. In this article, after a brief explanation of the mentioned method, the coupled techniques are applied to one-dimensional heat transfer in a rectangular radial fin with a temperature-dependent thermal conductivity to show the effectiveness and accuracy of the method in comparison with other methods. The graphical results show the best agreement of the three methods; however, the amount of calculations of each iteration for the combination of HPM and VIM was reduced markedly for multiple iterations. It was found that the variation of the dimensionless temperature strongly depends on the dimensionless small parameter ${\varepsilon_1}$ . Moreover, as the dimensionless length increases, the thermal conductivity of the fin decreases along the fin.     

具有不对称间隙的二元机翼颤振研究

利用数值方法分析了有预荷载情况下具有不对称间隙的气动弹性非线性与结构非线性耦合的二元机翼颤振问题的动力学响应,研究了当速度比U*/UL*从零逐渐增大到1时系统的全局演化规律,分析了频率比ω、机翼质量密度比μ、预荷载M0、初偏值αf、间隙δ及初始条件等参数对颤振临界点的影响,发现在有预荷载的不对称分段线性系统中不存在混沌运动形式,而且系统幅值的变化规律不受初值条件的影响,并根据引起俯仰幅值变化的敏感参数,提出几种控制和减小颤振的方法,可用于指导飞机的飞行结构设计及飞机结构的日常维护。     

A meshless solution to two-dimensional convection–diffusion problems

An integral equation domain decomposition method has been implemented in a meshless fashion. The method exploits the advantage of placing the source point always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green's identities and the remaining equations are the derivatives of the first equation in respect to space coordinates. Radial basis function interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). Dual reciprocity method (DRM) has been applied to convert the domain integrals into boundary integrals, though the approach is general and can be applied without the DRM. The accuracy and robustness of the method has been tested on a convection–diffusion problem. The results obtained using the current approach have been compared with previously reported results obtained using the finite element method (FEM), and the DRM multi-domain approach (DRM-MD) showing similar level of accuracy.     

Analytical treatment of boundary integrals in direct boundary element analysis of plate bending problems

A direct-type Boundary Element Method (BEM) for the analysis of simply supported and built-in plates is employed. The integral equations due to a combined biharmonic and harmonic governing equations are first established. The boundary integrals developed are then evaluated analytically. The domain integrals due to external body forces are also transformed over the boundary and subsequently evaluated analytically. Thus, it needs only the boundary to be discretized. Without loss of generality, the exact expression for the integrals would enhance the solution accuracy of the BEM. This is due to the fact that at locations where the fundamental solutions approach their singular points the value determined by numerical quadrature may be inconsistent and inaccurate. Also, another major advantage of the exact expressions for integrations is the insensitivity to the geometrical location of the source point on the boundary. The distribution of boundary quantities is approximated either over linear or quadratic boundary elements. General type of plate bending problems, with plates of different geometrical shapes supported simply or fixed can be handled. Loading may be applied point concentrated, uniformly distributed within the domain or over the boundary. Also, hydrostatic pressure can be applied. The results obtained by BEM in comparison with those obtained by analytical or other approximate solutions are found to be very accurate and the solution method is efficient.     

Implementation of thermoelastic forces in boundary element analysis of elastic contact and fracture mechanics problems

This paper presents a pseudo-body-force approach multi-domain boundary integral equation method for the analysis of thermoelastic and body-force type elastic contact and fracture mechanics problems. Using this approach only the boundaries of the bodies involved have to be discretized. The transformation of the domain integrals due to body-force and pseudo-force to their equivalent boundary integrals are shown. Also, it is shown that by employing the initial strain approach the same set of equivalent boundary integrals would be obtained. Isoparametric quadratic elements are employed to represent the geometries and the functions. This two-dimensional BEM thermoelastic implementation can be found very simple and can be applied to both harmonic and nonharmonic temperature distributions. The accuracy is asserted by applying it to several thermoelastic fracture mechanics and contact problems.     

Application of meshfree collocation method to a class of nonlinear partial differential equations

In this work, an algorithm for the numerical solution of the generalized Hirota–Satsuma equations and Jaulent–Miodek equations based on meshless radial basis functions (RBFs) method using collocation points, called Kansa's method, is presented. Four model problems with six different initial conditions are considered for the computation. A fairly explicit scheme is used to approximate the solution. The comparison is made with the exact solutions of each problem of the generalized Hirota–Satsuma coupled Korteweg–de Vries equations. A system consisting highly nonlinear partial differential equations known as Jaulent–Miodek equations and generalized Hirota–Satsuma coupled modified-Korteweg–de Vries equations are considered for comparison with the work already published. The multiquadric RBF results are compared with homotopy perturbation method (HPM) and variational iteration method (VIM) to highlight the excellent performance of the method.     

A numerical procedure for multiple circular holes and elastic inclusions in a finite domain with a circular boundary

This paper describes a numerical procedure for solving two-dimensional elastostatics problems with multiple circular holes and elastic inclusions in a finite domain with a circular boundary. The inclusions may have arbitrary elastic properties, different from those of the matrix, and the holes may be traction free or loaded with uniform normal pressure. The loading can be applied on all or part of the finite external boundary. Complex potentials are expressed in the form of integrals of the tractions and displacements on the boundaries. The unknown boundary tractions and displacements are approximated by truncated complex Fourier series. A linear algebraic system is obtained by using Taylor series expansion without boundary discretization. The matrix of the linear system has diagonal submatrices on its diagonal, which allows the system to be effectively solved by using a block Gauss-Seidel iterative algorithm.     

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials,with an application to acoustic scattering problems

This paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions. The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.     

Impact response of a cracked orthotropic medium-revisited

Elastodynamics response of an infinite orthotropic medium containing a central crack under impact loading has been investigated. Laplace and Fourier transforms have been employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform domain which has finally been solved by the method of iteration in the low frequency case. Analytic expressions for the stress intensity factors and crack opening displacement are also obtained for low frequency.     

Method for integrating the absorption cross sections of spheres over wavelength or diameter

The absorption cross sections of spherical particles and droplets must be integrated over frequency or droplet size or both for various applications. Morphology-dependent resonances (MDRs) of the spheres can make evaluation of such integrals difficult because the MDRs can contribute significantly to the integrals even when their linewidths are extremely narrow, especially when the absorption is weak. A method of evaluating these integrals by use of Lorentzian approximations near MDRs is described. Calculated frequency-integrated absorption cross sections illustrate how the method obtains accurate cross sections with far fewer integration points than a method that uses equally spaced points. The method reported here suggests a way to integrate over frequency in more-complicated scattering and emission problems and should also be useful for integrating scattering and absorption by other shapes, e.g., spheroids and cylinders, for which the MDR positions and linewidths can be calculated.     

A methodology for solving inverse coefficient problems of determining nonlinear thermophysical characteristics of anisotropic bodies

A methodology is proposed for solving inverse coefficient thermal-conductivity problems of defining the thermal-conductivity tensor components that depend on the temperature by introducing a quadratic residual functional, its linearization, a minimization iteration algorithm, and a method of parametric identification considering errors in determining the experimental temperature values. The existence and uniqueness of the solution to inverse coefficient problems of nonlinear thermal conductivity in anisotropic bodies at moderate constraints on the descent parameters and the sensitivity matrix norms are proven. The results obtained for carbon-carbon composites support the entire methodology for numerical solution to inverse coefficient problems with an allowable error of the experimental temperature values. The proposed methodology can be applied to define both linear and nonlinear characteristics of anisotropic heat-protection materials used in aircraft and space engineering.     

Using the iterated sinh transformation to evaluate two dimensional nearly singular boundary element integrals

Recently, sinh transformations have been proposed to evaluate nearly weakly singular integrals which arise in the boundary element method. These transformations have been applied to the evaluation of nearly weakly singular integrals arising in the solution of Laplace's equation in both two and three dimensions and have been shown to evaluate the integrals more accurately than existing techniques.More recently, the sinh transformation was extended in an iterative fashion and shown to evaluate one dimensional nearly strongly singular integrals with a high degree of accuracy. Here the iterated sinh technique is extended to evaluate the two dimensional nearly singular integrals which arise as derivatives of the three dimensional boundary element kernel. The test integrals are evaluated for various basis functions and over flat elements as well as over curved elements forming part of a sphere.It is found that two iterations of the sinh transformation can give relative errors which are one or two orders of magnitude smaller than existing methods when evaluating two dimensional nearly strongly singular integrals, especially with the source point very close to the element of integration. For two dimensional nearly weakly singular integrals it is found that one iteration of the sinh transformation is sufficient.     

An iterative solution of the two-dimensional electromagnetic inverse scattering problem

A new method, based on an iterative procedure, for solving the two-dimensional inverse scattering problem is presented. This method employs an equivalent Neumann series solution in each iteration step. The purpose of the algorithm is to provide a general method to solve the two-dimensional imaging problem when the Born and the Rytov approximations break down. Numerical simulations were calculated for several cases where the conditions for the first order Born approximation were not satisfied. The results show that in both high and low frequency cases, good reconstructed profiles and smoothed versions of the original profiles can be obtained for smoothly varying permittivity profiles (lossless) and discontinuous profiles (lossless), respectively. A limited number of measurements around the object at a single frequency with four to eight plane incident waves from different directions are used. The method proposed in this article could easily be applied to the three-dimensional inverse scattering problem, if computational resources are available.     

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.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations

This paper presents the combination of new mesh-free radial basis function network (RBFN) methods and domain decomposition (DD) technique for approximating functions and solving Poisson's equations. The RBFN method allows numerical approximation of functions and solution of partial differential equations (PDEs) without the need for a traditional ‘finite element’-type (FE) mesh while the combined RBFN–DD approach facilitates coarse-grained parallelisation of large problems. Effect of RBFN parameters on the quality of approximation of function and its derivatives is investigated and compared with the case of single domain. In solving Poisson's equations, an iterative procedure is employed to update unknown boundary conditions at interfaces. At each iteration, the interface boundary conditions are first estimated by using boundary integral equations (BIEs) and subdomain problems are then solved by using the RBFN method. Volume integrals in standard integral equation representation (IE), which usually require volume discretisation, are completely eliminated in order to preserve the mesh-free nature of RBFN methods. The numerical examples show that RBFN methods in conjunction with DD technique achieve not only a reduction of memory requirement but also a high accuracy of the solution.     

Acoustic calculations with the hybrid boundary element method in time domain

The boundary element method (BEM) is an efficient tool for the calculation of acoustic wave propagation in fluids. Transient waves can be solved by either using a formulation in frequency domain along with an inverse Fourier transformation or a time domain formulation. To increase the efficiency for the solver and allow for an efficient coupling with finite element domains the symmetry of the system matrices is advantageous. If Hamilton's principle is used, a symmetric variational formulation can be established with the velocity potential as field variable. The single field principle is generalized as multifield principle as basis of a hybrid BEM for the calculation of acoustic fields in compressible fluids in time domain. The state variables are separated into boundary variables, which are approximated by piecewise polynomials and domain variables, which are approximated by a superposition of weighted fundamental solutions. In both approximations the time and space dependency is separated. This is why static fundamental solution can be used for the field approximation. The domain integrals are eliminated, respectively, transformed into boundary integrals and an equation of motion with symmetric mass and stiffness matrix is obtained, which can be solved by a direct time integration scheme or by mode superposition. The time derivative of the equation of motion leads to a formulation with pressure and acoustic flux on the boundary for an easier interpretation of the variables.     

Frequency domain analysis by the exponential window method and SGBEM for elastodynamics

Dynamic analysis of a system can be carried out either in the time or frequency domain. Time responses/ histories of this system may be directly obtained using time-domain analysis. In case of frequency domain analysis in the Fourier space, the inverse fast Fourier transform (inverse FFT) would naturally be an appropriate choice for converting frequency solutions to the desired time responses. However, the standard FFT can not be applied to undamped systems as the free-vibration terms of these systems never decay which violates the periodic nature of the standard FFT algorithm. In addition, the FFT may be computationally expensive for lightly damped systems. An alternative to overcome the above limitations is the so-called exponential window method (EWM) commonly used in digital signal processing. This paper presents a combination of the EWM and the symmetric-Galerkin boundary element method for 2-D elastodynamic analysis in the frequency domain of undamped and lightly damped systems. Several numerical examples, including fracture problems, are given to illustrate the efficiency and accuracy of the proposed frequency domain analysis.