Narrow Results

This paper presents a robust and efficient boundary element method (BEM) for predicting underwater electromagnetic (EM) noise generated by permanent magnet synchronous motors (PMSMs) in rim-driven thrusters (RDTs). Compared to the Finite Element Method (FEM), the BEM has distinct advantages in handling exterior acoustic problems, such as easier implementation of Sommerfeld radiation condition, reduced mesh requirements, and lower discretization errors. However, when applied to the problem of EM noise in RDTs, the traditional BEM has two main drawbacks. First, the conventional BEM tends to produce fictitious modes at high frequencies. Second, due to the dense and frequency-dependent nature of the BEM matrices, the method suffers from low efficiency in large-scale frequency sweep calculations. To overcome the aforementioned drawbacks, the proposed method incorporates the Burton-Miller formulation to suppress fictitious modes and a Taylor expansion method for efficient frequency sweep calculations. Numerical experiments on a 96-slots, 16-poles PMSM validate the accuracy and efficiency of the proposed method. Additionally, this paper investigates fluid loading effects on the structural vibrations, providing insights for the design of vibration and noise reduction strategies in underwater motors.

The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.

The focusing action of refractive cylindrical microlens is investigated based on the rigorous electromagnetic theory with the use of the boundary element method. The focusing behaviors of these refractive microlenses with continuous and multilevel surface-envelope are characterized in terms of total electric-field patterns, the electric-field intensity distributions on the focal plane, and their diffractive efficiencies at the focal spots. The obtained results are also compared with the ones obtained by Kirchhoff's scalar diffraction theory. The present numerical and graphical results may provide useful information for the analysis and design of refractive elements in micro-optics.

A fast algorithm for the boundary element method is developed to handle problems in underwater acoustics. The algorithm employs the multipole and local expansions to approximate far-field potentials, and exploits the discrete convolution nature of mapping multipole to local expansions to accelerate the potential evaluation process. The speedup in the solution process is achieved by fast Fourier transform on the multipole and local expansion coefficients on a regular grid. The method is demonstrated by a three-dimensional underwater acoustics scattering problem, and it is found to achieve accurate results with relatively low order of expansion.

The complex interactions of two and three spark-generated bubbles are studied using high speed photography. The corresponding simulations are performed using a 3D Boundary Element Method (BEM) code. The bubbles generated are between 3 to 5 mm in radius, and they are either in-phase or out-of-phase with one another. The possible interaction phenomena between two identically sized bubbles are summarized. Depending on their relative distances and phase differences, they can coalesce, jet towards or away from one another, split into smaller bubbles, or 'catapult' away from one another. The 'catapult' effect can be utilized to generated high speed jet in the absence of a solid boundary or shockwave. Also three bubble interactions are highlighted. Complicated phenomena such as bubble forming an elliptical shape and bubble splitting are observed. The BEM simulations provide insight into the physics of the phenomena by providing details such as detailed bubble shape changes (experimental observations are limited by the temporal and spatial resolution), and jet velocity. It is noted that the well-tested BEM code [1,2] utilized here is computationally very efficient as compared to other full-domain methods since only the bubble surface is meshed.

Fully submerged sphere and cylinder point absorber (PA), wave energy converters (WECs) are analyzed numerically based on linearized potential flow theory. A boundary element method (BEM) (a radiation–diffraction panel program for wave-body interactions) is used for the basic wave-structure interaction analysis. In the present numerical model, the viscous damping is modeled by an equivalent linearized damping which extracts the same amount of wave energy over one cycle as the conventional quadratic damping term. The wave power capture width in each case is predicted. Comparisons are also made between the sphere and cylinder PAs which have identical geometrical scales and submerged depths. The results show that: (i) viscous damping has a greater influence on wave power performance of the cylinder PA than that of the sphere PA; (ii) the increasing wave height reduces wave power performance of PAs; (iii) the cylinder PA has a better wave power performance compared to the sphere PA in larger wave height scenarios, which indicates that fully submerged cylinder PA is a preferable prototype of WEC.

In this paper, we propose a two-dimensional (2D) boundary element method (BEM) to explore the coupling properties of two distorted graphene nanoribbons (GNRs) with wrinkles. From the calculation results, it is found that the resonant wavelengths are redshifted compared with a single GNR, and the scattering intensity is greatly increased, due to the coupling between the two GNRs. In the non-degenerate structure, the coupling resonance modes of the two GNRs are the superposition of the resonance modes of every single GNR, and the resonant wavelength and scattering intensity of different modes change differently with the gap between the two GNRs and their heights. When the coupling effect becomes stronger, some modes will split. We explain the results through the charge and current distribution of the GNRs. This work will help to understand the coupling mechanism of surface plasmon resonances of GNRs.

The major theme of this research is to develop the numerical scheme for the computation of nonlinear problems by the implementation of the boundary element method dependent on Taylor’s series. This paper deals with the problem of laminar flow in a semiporous channel in the presence of a transverse magnetic field and the homotopy analysis method (HAM) is employed along with the general boundary element method to compute an approximated solution of the system of nonlinear differential equation governing the problem concerned. A well-known and useful nonlinear differential equation of fluid mechanics $F^{(3)}(\eta )+\frac{1}{2}F(\eta )F^{(2)}(\eta )=0$ with the boundary conditions $F(0)=0=F^{(1)}(0)$, $F^1(\infty )=1$ is referred as an example to show a glimpse into the basic idea of the method and technique used in this paper. It shows the capabilities and wide range of applications of HAM using Taylor series expansion of the derived integral equation. A comparison of HAM with the results calculated previously has been discussed. The solution obtained with HAM in comparison with the numeric solution shows remarkable accuracy. Root mean square formula is computed for the convergence analysis at various values. An increment in the number of elements depicting the convergent results as error is decreased.

In this paper an optimal shape control problem dealing with heat transfer in enclosures is studied. We have considered an enclosure heated by a flame surface (taking into account radiation, conduction and convection effects), and we look for an optimal flame shape which minimizes a cost functional defined on the temperature field. This kind of problem arises in industrial furnaces optimization, as temperature uniformity is one of the most important aspects in industrial plant analysis and design. Analytical results (smoothness of the control-to-state mapping, existence of an optimal shape in a certain admissible class) as well as numerical optimization results by the boundary element method have been obtained; we have employed the gradient method to optimize the flame shape, exploiting the adjoint equation associated with our state equation and cost functional.

We propose a computational method to solve optimal swimming problems, based on the boundary integral formulation of the hydrodynamic interaction between swimmer and surrounding fluid and direct constrained minimization of the energy consumed by the swimmer.

We apply our method to axisymmetric model examples. We consider a classical model swimmer (the three-sphere swimmer of Golestanian et al.) as well as a novel axisymmetric swimmer inspired by the observation of biological micro-organisms.

In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the boundary partition and, moreover, steers the smoothness of the NURBS ansatz functions across elements. In particular and unlike prior work, the algorithm can increase and decrease the local smoothness properties and hence exploits the full potential of isogeometric analysis. We prove that the new adaptive strategy leads to linear convergence with optimal algebraic rates. Numerical experiments confirm the theoretical results. A short appendix comments on analogous results for the weakly-singular integral equation.

We study time-harmonic scattering in ${ℝ}^n$ ($n=2,3$) by a planar screen (a “crack” in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset $\mathrm{\Gamma }$ of the hyperplane ${\mathrm{\Gamma }}_{\infty }={ℝ}^{n-1}\times \{0\}$, on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, $\mathrm{\Gamma }$ can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such $\mathrm{\Gamma }$ we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regularity conditions, which hold automatically when $\partial \mathrm{\Gamma }$ is smooth. We show that the associated system of boundary integral operators is compactly perturbed coercive in an appropriate function space setting, strengthening previous results. This permits the use of Mosco convergence to prove convergence of boundary element approximations on smoother “prefractal” screens to the limiting solution on a fractal screen. We present accompanying numerical results, validating our theoretical convergence results, for three-dimensional scattering by a Koch snowflake and a square snowflake.

The vibrational analysis of ships involves the solution of a coupled fluid-structure interaction problem in order to simulate the inertia response of the water to the structural vibrations. The paper describes a coupled FE-BE procedure for the computational solution of this problem. Regarding the structural equations, special emphasis is put on the modeling of damping. Various models of damping, in particular local damping, are discussed and implemented into the FE-model of the ship structure. The computational results are compared to data from large-scale measurements.

Continuous interpolation of the sound pressure is favored in most applications of boundary element methods for acoustics. Few papers are known to the authors in which discontinuous elements are applied. Mostly they were used because they guarantee C¹ continuity of the geometry at element edges. The effect of superconvergence is known for boundary element collocation on discontinuous elements. This effect is observed if the collocation points are located at the zeroes of orthogonal functions, e.g. at the zeroes of the Legendre polynomials.

In this paper, we start with a review of related work. Then, formulation of discontinuous elements and position of nodes on the element are presented and discussed. One parameter controls the location of nodes on the element. The major part of the paper consists of the investigation of the computational example of a long duct. For that, the numeric solution is compared with the analytic solution of the corresponding one-dimensional problem. Error dependence in terms of frequency, element size and location of nodes on discontinuous elements is reported. It will be shown that the zeroes of the Legendre polynomials account for an optimal position of nodes. Similar results are observed for triangular elements. It can be seen that the error in the Euclidean norm changes by one or two orders of magnitude if the location of nodes is shifted over the element. It can be seen that the optimal location varies with the wave-number although remaining in the vicinity of the zeroes of orthogonal functions. The irregular mesh of a sedan cabin compartment accounts for the second example. Optimal choice of node position is confirmed for this example. One of the key results of this paper is that discontinuous boundary elements perform more efficiently than continuous ones, in particular for linear elements. This, however, further implies that nodes are located at the zeroes of orthogonal functions on the element.

Time dependent analysis of the dynamic damped behavior of continua are mathematically modelled by partial differential equations. One obtains uniqueness, existence and stability (well posed problems) by the implementation of the correct initial boundary conditions. However, by taking memory effects into consideration, any change in the past of the system changes the future dynamic behavior. Classical damping descriptions fail when describing the behavior of many materials, like teflon. This is because in classical theory the operators are local ones. The implementation of fractional time derivatives into the partial differential equations is an alternative technique to overcome these problems. Thereby the time derivative operator is a global one, memory effects in structure borne sound can be calculated. In this paper the theory of fractional time derivative operators and their application in continuum mechanics is presented. The main result when using this method for damping behavior is that a global operator is needed which takes the whole history into account. We call this theory the functional calculus method instead of the well-known fractional calculus with the use of initial conditions. In order to show the efficiency of this method the calculated impulse response of a viscoelastic rod is compared with measurements. It is shown that the damping behavior is described much better than by other models with comparably few parameters. Moreover, it is the only one that works in a wide frequency range and can describe the dispersion of the resonance frequencies. The implementation of this damping description in a Boundary Element Code is an application of dynamics of 3D continua in the frequency domain.

Theory and algorithm for the acoustic boundary element solver will be described. The solver represents the core of the package AField (acronym for acoustical field) and can be used for getting fast and stable solutions for huge as well as for small boundary elements models on PCs under Windows. The boundary element solver includes an early proposed GMRES iterative linear system solver (which can be used for big and huge models) and several traditional linear system solvers (for small and medium models). Implementation of the Burthon and Miller regularization method for constant boundary elements will be discussed. Approximations in proposed boundary element solver and their influence on solution errors are investigated. Results for several models are included. Several possibilities of Afield package are illustrated.

A boundary integral equation is described for the prediction of acoustic propagation from a monofrequency coherent line source in a cutting with impedance boundary conditions onto surrounding flat impedance ground. The problem is stated as a boundary value problem for the Helmholtz equation and is subsequently reformulated as a system of boundary integral equations via Green's theorem. The numerical solution of the coupled boundary integral equations by a simple boundary element method is then described. Predictions of A-weighted insertion losses for a traffic noise spectrum are made illustrating the effects of depth of the cutting and the profile of the associated noise barrier.

The fast multipole boundary element method (FMBEM) is an advanced BEM that leads to drastic reduction of processing time and memory requirements in a large-scale steady-state sound field analysis. In the FMBEM, hierarchical cell structure is employed to apply multipole expansion in multiple levels, and the setting of the hierarchical cell structure considerably affects the computational efficiency of the FMBEM. This paper deals with effective settings of hierarchical cell structure for taking full advantage of the FMBEM. A numerical study with objects of different shapes with the same DOF shows that both the computational complexity and the memory requirements with the FMBEM were greater for 1D-shaped objects than for 2D- or 3D-shaped ones, without a special setting of hierarchical cell structure for each problem. An effective setting for 1D-shaped objects is derived through theoretical and numerical studies, where special considerations are given to the arrangement of the cell structure and the treatment of translation coefficients between cells. This setting allows for efficient calculations not dependent on the shape of an analyzed object. A simple method to arrange hierarchical cell structure is proposed, which realizes the derived setting for arbitrarily-shaped problems.

The fast multipole boundary element method (FMBEM) is an advanced BEM, with which both the operation count and the memory requirements are O(N^alog^b N) for large-scale problems, where N is the degree of freedom (DOF), a ≥ 1 and b ≥ 0. In this paper, an efficient technique for analyses of plane-symmetric sound fields in the acoustic FMBEM is proposed. Half-space sound fields where an infinite rigid plane exists are typical cases of these fields. When one plane of symmetry is assumed, the number of elements and cells required for the FMBEM with this technique are half of those for the FMBEM used in a naive manner. In consequence, this technique reduces both the computational complexity and the memory requirements for the FMBEM almost by half. The technique is validated with respect to accuracy and efficiency through numerical study.

A broadband multilevel fast multipole algorithm (MLFMA) for the acoustic scattering from a sound-hard obstacle is presented. The formulation is based on the Burton–Miller boundary integral equation and Galerkin's method, avoiding any hypersingular integral operators. The resulting matrix equation has good iterative properties for all frequencies and avoids the interior resonance problem. The main novel feature is the use of a broadband MLFMA to accelerate the iterative generalized minimal residual (GMRES) solver. The algorithm is based on a combination of Rokhlin's translation formula for large division cubes and the spectral representation of the Green's function for cubes smaller than one half wavelength, thereby avoiding the sub-wavelength breakdown of the high-frequency MLFMA.