Boundary Integral Formula Research Articles

A generalized isogeometric boundary element method for the uncertain analysis of infinite domain two-dimensional acoustic problems

The key aim of this paper is to provide a new nth generalized order perturbed isogeometric fast multistage technique of boundary elements to compute the propagation of time harmonics in an infinite region. Structural geometry and boundary integral equations are constructed by using non-uniform rational B-splines. The source of system uncertainty is believed to be the incident plane wave number’s unpredictability. The actual field, depending on the input random variables, is simulated using the extended nth-order perturbation method. The field and kernel values for boundary integral formulas are generated via the nth-order generalized series of Taylor expansions using perturbation parameters. The fast multipole method (FMM) is utilized to speed up the process. The effectiveness and correctness of the proposed algorithm are verified by Monte Carlo simulations (MCs) with numerical examples.

Efficient and Rigorous Modeling of Light Scattering From Complex Plasmonic Nanostructures With Multiple Incident Excitations

Optical responses to incident lights with different incident directional angles or polarization angles are often of concern in the analysis of light scattering from plasmonic nanostructures. This leads to a multiple-excitation problem for the modeling and prediction of light scattering. An efficient and rigorous numerical method is presented to deal with this problem. In this method, the hybrid finite element and boundary integral (FEBI) formula is adopted to model the interior complex plasmonic materials and the exterior scattering space rigorously, and a low-rank compressed matrix factorization algorithm based on the FEBI framework is proposed to deal with the multiple-excitation problem efficiently. Numerical experiments are performed to illustrate the effectiveness of the proposed method.

A fast multipole boundary element method for half-space acoustic problems in a subsonic uniform flow

This paper presents a fast multipole boundary element method (FMBEM) for half-space acoustic problems in a subsonic uniform flow. First, the fundamental solution, i.e., Green's function for half-space acoustic wave problems in a subsonic uniform flow is derived. By employing the Burton-Miller method, a convected hybrid boundary integral formula is achieved to overcome the non-uniqueness difficulty. The developed formula does not need to consider the integral on the infinite plane and it is convenient for numerical implementation. The numerical evaluation of singular integrals in the convected boundary integral formulae is also introduced. Then, the fast multipole method (FMM) for convected BEM is derived and adopted to improve the calculation efficiency and increase the computational scale. Different from the previous FMM for full-space problems, the multipole expansion and translation of the mirror-image model are the key to the FMM for half-space problems. The impact of the infinite plane on the results can be computed by building the mirror-image model. And only the structural boundary in the real domain needs to be discretized and constructed for the tree structure. Numerical examples including a pulsating half-sphere problem, a sphere model and an aircraft model in moving flows are given to validate the accuracy and efficiency of the developed method.

Regularization in X-ray Tomography by a Cauchy Type Boundary Integral Formula

We propose a choice of stabilization parameters in x-ray computerized tomography by a Cauchy type boundary integral formula. Since most of the inverse problems are ill-posedness in the sense of Hadamard, accurate numerical proceduresalsoapproximateitsinstability. Thereforecomputationalandobservationerrorsinfluencesnumericalresults seriously. In order to relax it, regularization methods are introduced with some artificial stabilization parameters which control accuracy and stability. The proposed criteria gives a posteriori choice of the stabilization parameter, and is computable from only measurement data, and no a priori information are required. We will also show a feasibility of the proposed method by numerical reconstruction from measurement data.

A high-order BEM for acoustic problems in a subsonic uniform flow

The present work is to develop a boundary element method (BEM) with curved surface elements for 3D acoustic problems in a subsonic uniform flow. For exterior sound propagation problems in moving flows, the convected conventional boundary integral equation (CBIE) suffers from the irregular frequency issue which becomes more severe in the presence of moving flows. In this paper, a hybrid convected boundary integral formula based on the Burton-Miller method is implemented to avoid the fictitious eigenfrequency problem. However, the hypersingular integral involved in the convected normal derivative boundary integral equation (NDBIE) is one of the crucial troublesomeness in the implementation of BEM with curved continuous elements for 3D acoustic problems. Moreover, the presence of the Mach number further complicates the evaluation of the singular integrals. To resolve this dilemma, an effective method based on polar coordinate transformation (PCT) is developed to compute the singular integrals. By mapping physical coordinates to the intrinsic local coordinates, this method can effectually remove various kinds of singularities appearing in the CBIE and NDBIE. Additionally, a complete scheme on the high-order convected BEM including the scheme of collocation points is also constructed for acoustic problems in a subsonic uniform flow. Several numerical examples are elaborately designed to verify the accuracy of the developed method.

Correction to: Boundary Integral Formula for Harmonic Functions on Riemann Surfaces

We present a corrected proof of Proposition 2.3 in the article “Boundary integral formula for harmonic functions on Riemann surfaces” [2].

SGBEM Analysis of Diffraction of P- and SV-Waves by a Plane Crack in an Infinite Domain

Diffraction of elastic waves by a plane crack in an unbounded domain was investigated in this work using the symmetric Galerkin boundary element method (SGBEM) for elastodynamics in the frequency domain. The types of wave considered herein are the incident plane harmonic compressional wave (P-wave) and the incident plane harmonic shear wave (SV-wave). By using the scalar and vector potential functions of these incident waves, the wave displacement field and thereby, the wave traction field can be identified. To solve for the scattered field problem of these incoming waves on a crack in an infinite plate using the SGBEM, the crack faces were loaded with tractions equal and opposite to those produced at the same position by the incident waves. The scattering displacements and tractions found were then employed in a new boundary integral formula developed for computing the dynamic T-stress (DTS) for cracks in an infinite domain, while only the scattering displacements were needed for determining the dynamic stress intensity factors (DSIFs) by the displacement correlation technique. Numerous numerical examples were presented in this work. The SGBEM results for the DSIFs agreed very well with those obtained from using the integral transform method, and to the best knowledge of the authors, the SGBEM results for the DTS were presented for the first time in this work.

Boundary Integral Formula for Harmonic Functions on Riemann Surfaces

We construct a boundary integral formula for harmonic functions on smoothly-bordered subdomains of Riemann surfaces embeddable into $${\mathbb {C}}{\mathbb {P}}^2$$ . The formula may be considered as an analogue of the Green’s formula for domains in $${\mathbb {C}}$$ .

A time-domain finite element boundary integral approach for elastic wave scattering

The response of complex scatterers, such as rough or branched cracks, to incident elastic waves is required in many areas of industrial importance such as those in non-destructive evaluation and related fields; we develop an approach to generate accurate and rapid simulations. To achieve this we develop, in the time domain, an implementation to efficiently couple the finite element (FE) method within a small local region, and the boundary integral (BI) globally. The FE explicit scheme is run in a local box to compute the surface displacement of the scatterer, by giving forcing signals to excitation nodes, which can lie on the scatterer itself. The required input forces on the excitation nodes are obtained with a reformulated FE equation, according to the incident displacement field. The surface displacements computed by the local FE are then projected, through time-domain BI formulae, to calculate the scattering signals with different modes. This new method yields huge improvements in the efficiency of FE simulations for scattering from complex scatterers. We present results using different shapes and boundary conditions, all simulated using this approach in both 2D and 3D, and then compare with full FE models and theoretical solutions to demonstrate the efficiency and accuracy of this numerical approach.

An improved axisymmetric convected boundary element method formulation with uniform flow

This paper presents an improved form of the convected Boundary Element Method (BEM) for axisymmetric problems in a subsonic uniform flow. The proposed formulation based on the axisymmetric Convected Helmholtz Equation (CHE) and its fundamental solution that describes the sound radiation from a monopole source. The variables in the new axisymmetric boundary integral formulation can be expressed explicitly in terms of the acoustic pressure and its particular normal derivative. Also, the constant coefficients are expressed only in terms of the axisymmetric convected Green's function and its convected normal derivative. The particular and convected derivatives reduce the flow effects of the normal and the flow direction derivatives incorporated in the conventional convected boundary integral formulas. The advanced form of the axisymmetric boundary integral representation with flow is a similar form of the axisymmetric boundary element method without flow. Precisely, the two new operators significantly reduce the computational burden of the classical BEM and then becomes the CPU time of BEM without flow. The formula is verified comparing to both analytical and Finite Element Methods (FEM) of an axisymmetric infinite rigid duct in a subsonic uniform flow.

Derivation and implementation of the boundary integral formula for the convective acoustic wave equation in time domain.

Kirchhoff's formula for the convective wave equation is derived using the generalized function theory. The generalized convective wave equation for a stationary surface is obtained, and the integral formulation, the convective Kirchhoff's formula, is derived. The formula has a similar form to the classical Kirchhoff's formula, but an additional term appears due to a moving medium effect. For convenience, the additional term is manipulated to a final form as the classical Kirchhoff's formula. The frequency domain boundary integral can be obtained from the current time domain boundary integral form. The derived formula is verified by comparison with the analytic solution of source in the uniform flow. The formula is also utilized as a boundary integral equation. Time domain boundary element method (BEM) analysis using the boundary integral equation is conducted, and the results show good agreement with the analytical solution. The formula derived here can be useful for sound radiation and scattering by arbitrary bodies in a moving medium in the time domain.

An efficient hybrid method for scattering from arbitrary dielectric objects buried under a rough surface: TM case

A hybrid method combining the finite element method (FEM) with the boundary integral equation (BIE) is presented in this paper to investigate two-dimensional (2D) electromagnetic scattering properties of multiple dielectric objects buried beneath a dielectric rough ground for TM case. In traditional FEM simulation, the artificial boundaries, such as perfectly matched layer (PML) and the absorbing boundary conditions (ABC), are usually adopted as truncated boundaries to enclose the whole model. However, the enclosed computational domain increases quickly in size for a rough surface with a large scale, especially for the scattering model of objects away from the rough surface. In the hybrid FEM-BIE method, one boundary integral equation is adopt to depict the scattering above the rough surface based on Green's function. Based on the domain decomposition technique, the computational region below the rough ground is divided into multiple isolated interior regions containing each object and the exterior region. Finite element formulations are only applied inside interior regions to derive a set of linear systems, and another boundary integral formula is developed below the rough surface which also acts as the boundary constraint of the FEM region. Compared with traditional FEM, the hybrid technique presented here is highly efficient in terms of computational memory, time, and versatility. Numerical simulations are carried out based on hybrid FEM-BIE to study the scattering from multiple dielectric objects buried beneath a rough ground.

A non-singular boundary integral formula for frequency domain analysis of the dynamic T-stress

A non-singular 2-D boundary integral equation (BIE) in the Fourier-space frequency domain for determining the dynamic T-stress (DTS) is presented in this paper. This formulation, based upon the Fourier transform of the asymptotic expansion for the stress field in the vicinity of a crack tip, can be conveniently implemented as a post-processing step in a frequency-domain boundary element analysis of cracks. The proposed BIE is accurate as it can be directly collocated at the crack tip in question. The technique is also computationally effective as it simply requires a similar computing effort as that used in determining the dynamic stress components at an interior point of a domain. Five numerical examples involving both straight and curved cracks are studied to validate the proposed technique. For the frequency domain analysis of the DTS in these examples, the exponential window method is employed to obtain its time history.

Higher Order Boundary Integral Formula and Integro-Differential Equation on Stein Mainfolds

Fundamental Research Funds for the Central Universities [2010121002]; National Science Foundation of China [10771174]; University of Macau [RG079/04-05S/QT/FST]; Macao Science and Technology Development Fund [051/2005/A]

A non-singular boundary integral formula for determining the T-stress for cracks of arbitrary geometry

A simple, yet accurate 2-D boundary integral equation (BIE) for determining the T-stress for cracks of arbitrarily geometry is introduced in this paper. The formulation is based upon the asymptotic expansion for the stress field in the vicinity of a crack tip. It can be conveniently implemented in the post-processing stage of a boundary element fracture analysis. As demonstrated in this work, the proposed BIE is non-singular, and thus it can directly be collocated at the crack tip under consideration. The technique requires a similar computational effort as that used in calculating the stress components at an interior point of a domain. Consequently, this new approach is very computationally effective and accurate for evaluating the elastic T-stress. Five test examples, involving straight, kink and curved cracks, are studied to validate the proposed technique and to assess its accuracy.

Thermal Bending of Circular Plates for Non-Axisymmetrical Problems

Using the Green function, the boundary integral formula and natural boundary integral equation for thermal elastic problems are obtained. Then based on bending solutions to circular plates subjected to the non-axi- symmetrical load, by utilizing the Fourier series and convolution formulae, the bending solutions under non-axisymmetrical thermal conditions have been obtained. The calculating process is simple. Examples show the discussed methods are effective.

A Test on Two Codes for Extrapolating Solar Linear Force-free Magnetic Fields

We evaluate the performances of two newly-implemented codes for extrapolating the solar linear force-free magnetic fields, by measuring their quantified responses to the lower boundary vector field data on a finite region using analytical models. The codes are based on two boundary integral formulas with different mechanisms in utilizing the transverse boundary field: the first one only utilizes the transverse boundary field to derive the value of the force-free parameter, while the other one explicitly utilizes the whole transverse boundary field, in addition to the vertical field component. Studies on the test cases show that both of the codes could reproduce the analytical model fields with reasonable accuracy within the valid domain, provided a sufficient amount of data is available at the lower boundary. The code utilizing explicitly all three components of the boundary field shows generally better performances and requires a smaller boundary-data area in order to achieve the same degree of accuracy in the metric quantities such as the normalized vector error, vector correlation, vector angular difference, and magnetic energy; however, the accuracy in the divergence-free metric and especially the force-free metric are less ideal.

Geothermal refraction problem for a 2-D body of polygonal cross-section buried in the two-layered earth

We present the exact boundary integral formulae  for calculation of the geothermal  anomaly due  to  a two  dimensional body  whose  thermal conductivity  is λT   and  its cross-section is bounded by the closed general  polygonal  contour.  This  body  is buried  in the  superficial layer  of conductivity λ1 . The  half-space  z > h with  the  thermal conductivity  λ2  is considered as  a substratum of models.   The  boundary integral technique for the solution  of this  problem requires  the application of logarithmic potential  terms  in infinite  series.   Numerical calculations  based  on derived  formulae  revealed  that the  surface anomaly heat flow reflects  the “topography”  mainly  of the upper  boundary of the perturbing body.   The derived  algorithm and the developed  numerical program enable calculations for a number of interesting models:  intrusions, protrusions of the substratum etc.

Domain dependent Dirac's delta and derivatives with an application to electromagnetic boundary integral representations

The domain dependent versions of derivatives and Dirac's delta are defined in distributional sense. These operations enable to obtain domain dependent fundamental solutions and global boundary integral representation formulae. A global representation formula is defined everywhere, also on the boundary, and includes the jump relations of the boundary. The use of the domain dependent objects can be interpreted as taking the boundary limit in prior to integrating by parts when deriving the familiar boundary integral equations. As an application, the representation formulae are obtained for the solutions of the Helmholtz equation and the Maxwell equations.

Second-Order Optimality Conditions for Constrained Domain Optimization

This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ? Ω g(x)?dx=1, a domain is sought which maximizes either $\mathcal{F}_{x_{0}}(\Omega )=F(x_{0},u(x_{0}))$ , fixed x 0?Ω, or ?(Ω)=? Ω F(x,u(x))?dx, where u solves the Dirichlet problem Δu(x)=?f(x), x?Ω, u(x)=0, x??Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals $\mathcal{F}_{x_{0}}$ and ?. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω.