A finite element approach to a coupled structural-acoustic radiation system with application to loudspeaker characteristic calculation

The finite element method is applied to an elastic shell which is vibrating to generate acoustic radiation. A vibrating shell of revolution backed by an enclosure in its rear side is to radiate sound waves into a semi-infinite space in front. As a numerical example, some characteristics of a direct radiator type loudspeaker model are calculated and discussed. The driving point impedance and the far field sound pressure frequency characteristics are shown, together with the effects of the radiation and the enclosure.     

A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation

A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lamb's problem and plane wave nonlinear propagation.     

Domain compression via anisotropic metamaterials designed by coordinate transformations

We introduce a spatial coordinate transformation technique to compress the excessive white space (i.e. free-space) in the computational domain of finite methods. This approach is based on the form-invariance property of Maxwell’s equations under coordinate transformations. Clearly, Maxwell’s equations are still satisfied inside the transformed space, but the medium turns into an anisotropic medium whose constitutive parameters are determined by the coordinate transformation. The proposed technique can be employed to reduce the number of unknowns especially in high-frequency applications wherein a finite method requires an electrically-large computational domain. After developing the analytical background of this technique, we report some numerical results for finite element simulations of electromagnetic scattering problems.     

Least-squares Legendre spectral element solutions to sound propagation problems

This paper presents a novel algorithm and numerical results of sound wave propagation. The method is based on a least-squares Legendre spectral element approach for spatial discretization and the Crank-Nicolson [Proc. Cambridge Philos. Soc. 43, 50-67 (1947)] and Adams-Bashforth [D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (CBMS-NSF Monograph, Siam 1977)] schemes for temporal discretization to solve the linearized acoustic field equations for sound propagation. Two types of NASA Computational Aeroacoustics (CAA) Workshop benchmark problems [ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics, edited by J. C. Hardin, J. R. Ristorcelli, and C. K. W. Tam, NASA Conference Publication 3300, 1995a] are considered: a narrow Gaussian sound wave propagating in a one-dimensional space without flows, and the reflection of a two-dimensional acoustic pulse off a rigid wall in the presence of a uniform flow of Mach 0.5 in a semi-infinite space. The first problem was used to examine the numerical dispersion and dissipation characteristics of the proposed algorithm. The second problem was to demonstrate the capability of the algorithm in treating sound propagation in a flow. Comparisons were made of the computed results with analytical results and results obtained by other methods. It is shown that all results computed by the present method are in good agreement with the analytical solutions and results of the first problem agree very well with those predicted by other schemes.     

Transient torsional vibration responses of finite, semi-infinite and infinite hollow cylinders

Torsional guided waves are often used to detect the defects in a hollow cylinder. To realize the excitation of the torsional guided waves with high efficiency, the transient vibration responses of finite, semi-infinite and infinite hollow cylinders to external torsional forces must be clarified theoretically. In this study, the method of eigenfunction expansion is employed to solve the above problems. The exact analytical solutions derived by this method are not only explicit but also concise. Furthermore, the analytical solution of the transient torsional vibration of the finite hollow cylinder is numerically evaluated. The results obtained agree well with those simulated by the finite element method.     

Application of the FEM and the BEM to compute the field of a transducer mounted in a rigid baffle (3D case)

A three-dimensional finite element model has been developed which allows the harmonic analysis of a piezoelectric structure mounted on a rigid baffle and radiating into water. The solution of this problem consists of coupling a finite element method to a boundary element method. The first one enables the modelling of the vibrating structure and the second one the modelling of propagating waves in the semi-infinite fluid medium surrounding the structure. In this way, the near-field and the far-field pressures are calculated as well as the displacement field of the piezoelectric structure taking into account the acoustical interaction. Numerical and experimental results are provided which validate the numerical procedure. The good agreement obtained indicates that this three-dimensional model is a very useful tool to optimise the design of transducer arrays used in medical imaging.     

An investigation of transmission coefficients for finite and semi-infinite coupled plate structures

This paper introduces a method for determining the transmission coefficient for finite coupled plates using an analytical waveguide model combined with a scattering matrix. In the scattering matrix method, the amplitudes of the structural waves impinging on a junction are separated into incident, reflected, and transmitted components. The energy flow due to each of these waves is obtained using a wave impedance method, which is subsequently used to determine the transmission coefficient. Transmission coefficients for semi-infinite and finite L-shaped plates are investigated for single and multiple point force excitations, and for controlled incident wave sources. It is shown that the transmission coefficients can also be calculated from details of the modal transmission coefficients and the modal composition of the energy incident on the junction. Results show that the modal transmission coefficients are largely independent of whether the plates have finite or semi-infinite boundary conditions, and are only dependent on the details of the coupling. Finally, frequency averaged transmission coefficients are compared for semi-infinite and finite structures. In the cases considered, it is found that the semi-infinite system is a good approximation for finite systems after frequency averaging, especially if the system is excited with multiple point force excitation.     

Mode propagation in curved waveguides and scattering by inhomogeneities: application to the elastodynamics of helical structures

This paper reports on an investigation into the propagation of guided modes in curved waveguides and their scattering by inhomogeneities. In a general framework, the existence of propagation modes traveling in curved waveguides is discussed. The concept of translational invariance, intuitively used for the analysis of straight waveguides, is highlighted for curvilinear coordinate systems. Provided that the cross-section shape and medium properties do not vary along the waveguide axis, it is shown that a sufficient condition for invariance is the independence on the axial coordinate of the metric tensor. Such a condition is indeed checked by helical coordinate systems. This study then focuses on the elastodynamics of helical waveguides. Given the difficulty in achieving analytical solutions, a purely numerical approach is chosen based on the so-called semi-analytical finite element method. This method allows the computation of eigenmodes propagating in infinite waveguides. For the investigation of modal scattering by inhomogeneities, a hybrid finite element method is developed for curved waveguides. The technique consists in applying modal expansions at cross-section boundaries of the finite element model, yielding transparent boundary conditions. The final part of this paper deals with scattering results obtained in free-end helical waveguides. Two validation tests are also performed.     

Wave mode diffusion and propagation in structural wave guide under Varying Temperature

A numerical approach is presented to study the guided wave propagation through periodic specimen with thermal dependence of material properties. There is a great interest in extending the skills of the wave finite element (WFE) method to figure out the variations in the wave propagation properties due to temperature fluctuations. Thermal effects on the dispersion curves thereby on group velocity are discussed. Comparisons between numerical results and analytical developments for various temperatures are given to prove the effectiveness of the proposed approach to predict the sensitivity of guided wave propagation characteristics in presence of temperature variations.     

An element-free Galerkin(EFG) method for numerical solution of the coupled Schrdinger-KdV equations

The present paper deals with the numerical solution of the coupled Schrdinger-KdV equations using the elementfree Galerkin(EFG) method which is based on the moving least-square approximation.Instead of traditional mesh oriented methods such as the finite difference method(FDM) and the finite element method(FEM),this method needs only scattered nodes in the domain.For this scheme,a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method.In numerical experiments,the results are presented and compared with the findings of the finite element method,the radial basis functions method,and an analytical solution to confirm the good accuracy of the presented scheme.     

Flows with suspended and floating particles

The evolution of the configuration of a set of particles dispersed in a flowing liquid is crucial in many applications such as sedimentation, slurry transport, rheology and structured arrays of micro- and nano-particles. Direct simulation based on what is called fictitious domain method coupled with finite element method has been used to study particulate flows and sedimentation process. Here we extend the previously proposed formulations to naturally include buoyancy force and the capillary driven attraction or repulsion of particles located at fluid interfaces. The set of differential equations is discretized using a fully implicit-fully coupled fictitious domain/finite element approach, avoiding numerical instabilities that may arise from explicit integration. The proposed formulation and implementation are validated by comparing the predictions of simple 2D flows to available numerical or analytical solutions. The method is then used to analyze the flotation of 2D particles and capillary driven aggregation at fluid interfaces.     

Crack Tip Opening Angle as a Fracture Resistance Parameter to Describe Ductile Crack Extension and Arrest in Steel Pipes under Service Pressure

The angle between two element sides representing the crack tip is defined as the crack tip opening angle (CTOA). Its critical value is used as a criterion of fracture resistance for characterizing stable tearing in thin metallic materials. Various methods are used for determination of the CTOA. Optical microscopy is one of the most common methods as well as fitting of experimental load-displacement diagrams by the finite element method (DIC). Additionally, analytical analysis using the experimental load-displacement curve method (SSM) derived from the plastic hinge model of deflection in three-point bending of a ductile specimen is applied. This approach assumes a constant rotation centre distance. Values of CTOA for API 5L X65 pipe steel found by three methods—DIC, CNM, and SSM—are given. Values of CTOA given by these three methods are similar and close to 20°. A discussion on the different parameters used to characterize the fracture resistance of running cracks in a pipe under service pressure is presented. The energy of fracture at impact determined by Charpy or drop-weight tear test (DWTT) tests and the critical J energy parameter are considered as well as the yield locus after damage, cohesive zone energy, and CTOA is another approach. One notes that CTOA is assumed to be constant during stable crack extension and decreases linearly with crack length during the instable and primary phase. A numerical technique to describe a ductile running crack using the node release technique and using CTOA as the fracture resistance criterion is presented. This method is compared with three different two-curve methods (TCMs): the Battelle, high strength line pipe (HLP), and HLP-Sumitomo methods. The Batelle TCM, as the oldest method, based on Charpy energy, gives a strongly conservative prediction. Predictions by the CTOA method are close to those obtained by the HLP-Sumitomo one.     

Numerical investigation of spin-torque using the Heisenberg model

We present numerical calculations of the spin transfer torque resulting in current-induced domain wall motion. Rather than the conventional micromagnetic finite difference or finite element method, we use an atomistic/classical Heisenberg spin model approach, which is well suited to study geometrically confined domain walls. We compute the behaviour of domain walls in a one dimensional chain when currents are injected using adiabatic and non-adiabatic spin torque terms. Our results are compared to analytical calculations and are found to agree very well for small current densities. At larger current densities deviations are observed, which can be attributed to the approximations used in the analytical calculations.     

Space–time discontinuous Galerkin finite element method for two-fluid flows

A novel numerical method for two-fluid flow computations is presented, which combines the space–time discontinuous Galerkin finite element discretization with the level set method and cut-cell based interface tracking. The space–time discontinuous Galerkin (STDG) finite element method offers high accuracy, an inherent ability to handle discontinuities and a very local stencil, making it relatively easy to combine with local hp-refinement. The front tracking is incorporated via cut-cell mesh refinement to ensure a sharp interface between the fluids. To compute the interface dynamics the level set method (LSM) is used because of its ability to deal with merging and breakup. Also, the LSM is easy to extend to higher dimensions. Small cells arising from the cut-cell refinement are merged to improve the stability and performance. The interface conditions are incorporated in the numerical flux at the interface and the STDG discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative. The numerical method is applied to one and two dimensional two-fluid test problems using the Euler equations.     

Semi-analytical determination of natural frequencies and mode shapes of multi-span bridge decks

The determination of the natural frequencies and mode shapes of structures requires an analytical, semi-analytical or numerical method. This paper presents a new semi-analytical approach to determine natural frequencies and mode shapes of a multi-span, continuous, orthotropic bridge deck. The suggested approach is based on the modal method, which differs from other approaches in its decomposition of the admissible functions defining the mode shapes. Implementation of this technique is simple and enables avoidance of cumbersome mathematical calculations. In this paper, application of the semi-analytic approach to a three-span, orthotropic roadway bridge deck is compared in the first 16 modes of previously published fully analytical results and to a finite element method analysis. The simplified implementation matches within 2 percent in all cases, with the additional benefit of including intermodal coupling. The approach can be extended to similar bridges with more than three spans.     

Finite element prediction of wave motion in structural waveguides

A method is presented by which the wavenumbers for a one-dimensional waveguide can be predicted from a finite element (FE) model. The method involves postprocessing a conventional, but low order, FE model, the mass and stiffness matrices of which are typically found using a conventional FE package. This is in contrast to the most popular previous waveguide/FE approach, sometimes termed the spectral finite element approach, which requires new spectral element matrices to be developed. In the approach described here, a section of the waveguide is modeled using conventional FE software and the dynamic stiffness matrix formed. A periodicity condition is applied, the wavenumbers following from the eigensolution of the resulting transfer matrix. The method is described, estimation of wavenumbers, energy, and group velocity discussed, and numerical examples presented. These concern wave propagation in a beam and a simply supported plate strip, for which analytical solutions exist, and the more complex case of a viscoelastic laminate, which involves postprocessing an ANSYS FE model. The method is seen to yield accurate results for the wavenumbers and group velocities of both propagating and evanescent waves.     

Statistical characteristics of photon distributions in a semi-infinite turbid medium: Analytical expressions and their application to optical tomography

The paper focuses on the determination of statistical characteristics of photon distributions in a semi-infinite turbid medium, specifically the photon average trajectory and the root-mean-square deviation of photons from the average trajectory, with an approach based on the diffusion approximation to the radiative transfer equation. We show that the Dirichlet and Robin boundary conditions used for this purpose give close results. We derive exact analytical expressions for the case of the Dirichlet boundary condition. To demonstrate the practical value of our results we consider approximate solution of the inverse problem of time-domain diffuse optical tomography with the flat layer transmission geometry. The problem is solved with the method of photon average trajectories which are constructed with analytical expressions derived for a semi-infinite medium.     

On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach

In finite element methods that are based on position and slope coordinates, a representation of axial and bending deformation by means of an elastic line approach has become popular. Such beam and plate formulations based on the so-called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories. Examining the existing planar absolute nodal coordinate element, which uses a curvature proportional bending strain expression, it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and, even more serious, the normal force is incorrect. A correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived. The strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the St. Venant-Kirchhoff material: therefore, the strain energy is derived for the latter case and compared to classical nonlinear rod theories. The error in the original absolute nodal coordinate formulation is documented by numerical examples. The numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach, while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications. The numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre-curved static and dynamic problems, including buckling and eigenvalue analysis. The resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy-momentum conservation.     

弧长法在斜直井内管柱非线性屈曲分析中的应用

对于斜直井内底部-段管柱的后屈曲问题,基于受径向约束管柱的微分求积(DQ,Differential Quadrature)单元,构建了弧长迭代法.给出详细的迭代步骤和迭代初值的确定方法,对不同端部侧向约束条件下的管柱非线性屈曲进行迭代计算.并与现有文献中的近似解析解、实验结果和纯载荷增量迭代法的数值计算结果进行比较.结果显示,本文方法克服了有限单元法在处理管柱自重时的困难,同时能自动调节增量步长,跟踪管柱非线性后屈曲平衡路径的全过程.计算效率高、收敛性好、易于实施,可以用来分析斜直井内管柱的非线性屈曲问题.     

Two-dimensional radiative equilibrium: A semi-infinite medium subjected to a finite strip of radiation

Exact numerical results are presented for the emissive power and radiative flux at the boundary of a two-dimensional, absorbing-emitting, semi-infinite medium bounded by (1) a strip of collimated radiation and (2) a constant temperature black strip. The method of super-position is used to obtain the finite strip solutions in terms of cosine varying solutions. The infinite integrals arising in the solutions are converted to an alternating series of finite integrals. The Euler transformation is then applied to speed up convergence. Error bounds are determined whereby the two-dimensional finite strip analysis can be approximated by the simpler one-dimensional solution.