Resulting Boundary Integral Equation Research Articles

An efficient algorithm for time-domain acoustic scattering in three dimensions by layer potentials

In this paper, we develop a simple and fast algorithm for the time-domain acoustic scattering by a sound-soft or an impedance obstacle in three dimensions. We express the solution to the scattering problem by layer potentials, and then a time-domain boundary integral equation is derived. To numerically solve the resulting boundary integral equation, we propose a full discretization scheme by combining the convolution splines with a Galerkin method. In time, we approximate the density in a backward manner in terms of the convolution splines. In space, we project the density at each time onto the space of spherical harmonics, and then use the spatial discretization of a Nyström type on the surface of an obstacle which is homeomorphic to a sphere. A gallery of numerical examples are presented to show the efficiency of our algorithm. The stability, convergence and accuracy of the algorithm are discussed.

An alternative potential method for mixed steady‐state elastic oscillation problems

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.

MIXED BOUNDARY VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION

We consider a new approach to investigating the mixed boundary value problem (BVP) for the Helmholtz equation in the case of a three-dimensional unbounded domain Ω−⊂ℝ3 with compact boundary surface S=∂Ω−, which is divided into two disjoint parts, SD and SN, where the Dirichlet and Neumann type boundary conditions are prescribed respectively. Our approach is based on the classical potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP to a system of integral equations containing neither extensions of the Dirichlet or Neumann data nor a Steklov–Poincaré type operator involving the inverse of the single layer boundary integral operator, which is not available explicitly for arbitrary boundary surfaces. The right-hand sides of the resulting boundary integral equations system are functions coinciding with the given Dirichlet and Neumann data of the problem under consideration. We show that the corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L2-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the class of functions belonging to the Sobolev space W2, loc1(Ω−) and satisfying the Sommerfeld radiation conditions. We also show that the pseudodifferential matrix operator thus obtained is invertible in the Lp-based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses the Cα-Hölder continuity property in the closed domain Ω¯− with α=12−𝜀, where 𝜀>0 is an arbitrarily small number.

A numerical scheme for the time-fractional diffusion equation by layer potentials

We develop a numerical scheme for solving an initial-boundary value problem for the time-fractional diffusion equation, based on the boundary integral equation method. By expressing the solution as a single-layer potential, the initial-boundary value problem is transformed into a boundary integral equation for the unknown density function. To numerically solve the resulting boundary integral equation, we develop a stable discretization scheme for layer potentials. First, we rewrite the layer potential operators as generalized Abel integral operators in time, where the kernels are time-dependent boundary integrals. Then, the asymptotic expansions of those kernels at the initial time are derived by carefully analyzing the fundamental solution of the time-fractional diffusion equation. Consequently, we establish a stable time discretization scheme, using the composite trapezoidal rule with a correction at the endpoint to deal with the singularity of the integrand. The spatial discretization is performed by a standard quadrature rule for boundary integrals of smooth functions. Finally, we present several numerical examples to show the efficiency and accuracy of the proposed numerical scheme.

Numerical solution for the stress near a hole with corners in an infinite plate under biaxial loading

We consider the elastic stress near a hole with corners in an infinite plate under biaxial stress. The elasticity problem is formulated using complex Goursat functions, resulting in a set of singular integro-differential equations on the boundary. The resulting boundary integral equations are solved numerically using a Chebyshev collocation method which is augmented by a fractional power term, derived by asymptotic analysis of the corner region, to resolve stress singularities at corners of the hole. We apply our numerical method to the test case of the hole formed by two partially-overlapping circles, which can include either a corner pointing into the solid or a corner pointing out of the solid. Our numerical results recover the exact stress on the boundary to within relative error $10^{-3}$ for modest computational effort.

Analytical preconditioners for Neumann elastodynamic boundary element methods

Recent works in the boundary element method (BEM) community have been devoted to the derivation of fast techniques to perform the matrix-vector product needed in the iterative solver. Fast BEMs are now very mature. However, it has been shown that the number of iterations can significantly hinder the overall efficiency of fast BEMs. The derivation of robust preconditioners is now inevitable to increase the size of the problems that can be considered. Analytical preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead of discretization. The main contribution of this paper is to propose new analytical preconditioners to treat Neumann exterior scattering problems in 2D and 3D elasticity. These preconditioners are local approximations of the adjoint Neumann-to-Dirichlet map. We propose three approximations with different orders. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). An analytical spectral study confirms the expected behavior of the preconditioners, i.e., a better eigenvalue clustering especially in the elliptic part contrary to the standard CFIE of the first-kind. We provide various 2D numerical illustrations of the efficiency of the method for smooth and non smooth geometries. In particular, the number of iterations is shown to be independent of the density of discretization points per wavelength which is not the case for the standard CFIE. In addition, it is less sensitive to the frequency.

FEATURES OF BOUNDARY ELEMENT METHOD APPLICATION FOR SOLUTIONS OF BOUNDARY-VALUE PROBLEMS IN MOVING BOUNDARY DOMAINS

Summary. The motion of a fluid with a free surface is the subject of a separate section of theoretical hydromechanics. The development of this direction is stimulated by numerous applications to modern technologies, especially for the design and operation of hydraulic structures, shipbuilding and navigation, the study of ebbs and flows, tsunamis and other natural phenomena. The complexity of setting and solving this class of problems lies in the presence of a moving boundary of the solution area, called the free surface. From a mathematical point of view, the peculiarity of the considered class of problems lies in a specific non-linearity: a solution-dependent change in time of the shape of the solution area. Assuming the fluid to be incompressible and in viscid, under the conditions of the potentiality of the field of body forces and the potentiality of the flow at the initial moment of time, we obtain that the flow is described by the Laplace equation for the velocity potential with the traditional boundary conditions of non-flow at the solid boundaries of the flow region, as well as special boundary conditions at the free boundary, one of which, the Cauchy-Lagrange integral, contains a quadratic nonlinearity in the flow velocity. The assumption made about the potentiality of the alternating field of body forces greatly simplifies the formulation of the problem and allows us to consider the problem formulated above as a model one for a wide class of fluid flows with a free surface. For the numerical solution of the problem described above, a regular version of the boundary element method was used. The use of regular algorithms of the boundary element method makes it possible to bypass numerous computational limitations and improve in one way or another the approximations of the shape of the moving boundary, known and unknown functions on it. To do this, the Laplace equation was transformed into a boundary integral equation of potential theory, and the time interval in which the flow field is studied was divided into the corresponding number of sufficiently small intervals. At each time step, the resulting boundary integral equation was solved using the standard algorithm of the boundary element method. The calculation of the new position of the free boundary of the flow was carried out using the Euler scheme.

A multi-domain boundary element method to analyze the reflection and transmission of oblique waves from double porous thin walls

In the present paper, we examine the performance of an efficient type of wave-absorbing porous marine structure under the attack of regular oblique waves by using a Multi-Domain Boundary Element Method (MDBEM). The structure consists of two perforated vertical thin barriers creating what can be called a wave absorbing chamber system. The barriers are surface piercing, thereby eliminating wave overtopping. The problem of the interaction of obliquely incident linear waves upon a pair of perforated barriers is first formulated in the context of linear diffraction theory. The resulting boundary integral equation, which is matched with far-field solutions presented in terms of analytical series with unknown coefficients, as well as the appropriate boundary conditions at the free surface, seabed, and barriers, is then solved numerically using MDBEM. Dissipation of the wave energy due to the presence of the perforated barriers is represented by a simple yet effective relation in terms of the porosity parameter appropriate for thin perforated walls. The results are presented in terms of reflection and transmission coefficients. The effects of the incident wave angles, relative water depths, porosities, depths of the walls, and other major parameters of interest are explored.

PERFORMANCE OF POROUS MARINE STRUCTURES OF SINGLE AND DOUBLE PERFORATED SEAWALLS IN REGULAR OBLIQUE WAVES

In the present paper we examine the performance of two very common types of wave absorbing porous marine structures under regular oblique waves. The first structure consists of a single perforated vertical seawall, whereas the second consists of two perforated vertical seawalls creating what is called a chamber system (Jarlan type breakwater). The structures are surface piercing thus eliminating wave overtopping. Two methods are used for the present investigation. In the first method the problem of the interaction of obliquely incident linear waves upon a pair of porous vertical seawalls is first formulated in the context of the linear diffraction theory. The resulting boundary integral equation, which is matched with far-field solutions represented in terms of analytical series with unknown coefficients, and appropriate boundary conditions at the free surface, seabed and seawalls, is then solved numerically using the multi-domain boundary element method. In the second method a semi-analytical solution is developed by means of the eigenfunction expansions and a minimization approach using a least square method. In both methods the dissipation of the wave energy due to the presence of the perforated seawalls is represented by a simple yet effective relation in terms of the porosity parameter appropriate for thin perforated walls. The results are presented in terms of reflection and transmission coefficients, and the wave energy dissipation. Effects of the incident wave angles, porosities and depths of the walls and other major parameters of interest are explored.

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non-smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

Heat conduction in convectively cooled eccentric spherical annuli: A boundary integral moment method

In this paper heat conduction equation for an eccentric spherical annulus with the inner surface kept at a constant temperature and the outer surface subjected to convection is solved analytically. Eccentric problem domain is first transformed into a concentric domain via formulating the problem in bispherical co-ordinate system. Since an analytical Green?s function for the heat conduction equation in bispherical co-ordinate for an eccentric sphere subject to boundary condition of third type can not be found, an analytical Green's function obtained for Dirichlet boundary condition is employed in the solution. Utilizing this Green's function yields a boundary integral equation for the unknown normal derivative of the surface temperature distribution. The resulting boundary integral equation is solved analytically using method of moments. The method has been applied to heat generating eccentric spherical annuli and results are compared to the simulation results of FLUENT CFD code. A very good agreement was observed in temperature distribution computations for various geometrical configurations and a wide range of Biot number. Variation of heat dissipation with radii and eccentricity ratios are studied and a very good agreement with FLUENT has been observed

Computations of linear and nonlinear ship waves by higher-order boundary element method

A practical method for computing ship waves accurately and efficiently is favorable for hull form optimization in early stage of ship design. In the present study, Rankine source method incorporated with high-order boundary element (HOBEM) discretization is applied to solve linear ship waves at first. Numerical implementation is described in detail. An incomplete LU factorization preconditioned Generalized Minimal Residual method (GMRES) is employed to solve the resulting boundary integral equation in order to improve efficiency. A corresponding Fortran program is developed for the validation study. It is applied to solve ship waves of different hulls, including slender Wigley hull, KRISO Container Ship (KCS) with relatively full form and a fishery patrol boat. Ship wave drag, sinkage, trim and wave pattern over a wide range of Froude numbers are all well predicted.In order to further investigate nonlinear effects on ship waves, a fully nonlinear potential flow method based on stationary iteration is proposed. The same numerical approach of HOBEM is employed. The combined nonlinear free surface condition is solved in each iteration to evaluate the free surface. Numerical investigation for Wigley and KCS hull shows the present nonlinear method are accurate. Comparison study with linear and partial nonlinear solution are also carried out and nonlinear effects on ship waves are detailedly discussed.

A dual reciprocity multiwavelet Galerkin method for the numerical solution of Poisson׳s equation

A dual reciprocity multiwavelet Galerkin method is developed for the solution of Poisson׳s equation in this paper, which combines the dual reciprocity boundary element method (DRBEM) and the multiwavelet Galerkin method (MGM). The DRBEM is used to transform the domain integral in the boundary element formulation of Poisson׳s equation into the boundary of the domain, which is based on compactly supported positive definite radial basis function. Then, the MGM is employed for solving the resulting boundary integral equation, in which Alpert multiwavelets are employed to construct the trial and test functions of Galerkin variational formulation. Because of the use of multiwavelets, the resulting system matrix can be approximated by a sparse matrix. Compared to the DRBEM based on radial basis functions, the present method reduces the memory spaces and computational costs of the system matrix significantly. Numerical results show the efficiency of the present method.

Ship-hull shape optimization with a T-spline based BEM–isogeometric solver

In this work, we present a ship-hull optimization process combining a T-spline based parametric ship-hull model and an Isogeometric Analysis (IGA) hydrodynamic solver for the calculation of ship wave resistance. The surface representation of the ship-hull instances comprise one cubic T-spline with extraordinary points, ensuring C2 continuity everywhere except for the vicinity of extraordinary points where G1 continuity is achieved. The employed solver for ship wave resistance is based on the Neumann–Kelvin formulation of the problem, where the resulting Boundary Integral Equation is numerically solved using a higher order collocated Boundary Element Method which adopts the IGA concept and the T-spline representation for the ship-hull surface. The hydrodynamic solver along with the ship parametric model are subsequently integrated within an appropriate optimization environment for local and global ship-hull optimizations against the criterion of minimum resistance.

Comparison of different BEM+Born series modeling schemes for wave propagation in complex geologic structures

Complex geologic structures generally consist of irregular subregions with piecewise constant properties. Two different boundary-element method (BEM) plus Born-series schemes have been formulated for wave-propagation simulation in such piecewise homogeneous media by incorporating a Born series and boundary integral equations. Both schemes decompose the resulting boundary integral equation matrix into two parts: (1) the self-interaction operator, handled with a fully implicit BEM, and (2) the extrapolation operator, approximated by a Born series. The first scheme associates the self-interaction operator with each boundary itself and interprets the extrapolation operator as cross-interactions between different boundaries in a subregion. The second scheme relates the self-interaction operator toeach subregion itself and expresses the extrapolation operator as cross-interactions between different subregions in a whole model. In the second scheme, the matrix dimension of the resulting boundary integral equation is reduced by eliminating the traction field. Both numerical schemes have been validated by dimensionless frequency responses to a semicircular alluvial valley, compared with the full-waveform BEM numerical solution. We then extended these schemes to a complex fault model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments indicate that the [Formula: see text] series modeling schemes significantly improve computational efficiency, especially for high frequencies and with multisource seismic surveys. The tests also confirmed that the second modeling scheme has a faster convergence but may cause more noise in higher-order iterations than the first scheme.

BEM+Born series modeling schemes for wave propagation and their convergence analysis

This paper formulates two different boundary-element+Born series schemes for wave propagation simulation in multilayered media by incorporating a Born series and boundary integral equations. The first scheme directly decomposes the resulting boundary integral equation matrix into the self-interaction operators associated with each boundary itself and the extrapolation operators expressing cross-interactions between different boundaries in a subregion. For the second scheme, the matrix dimension is firstly reduced to a half by the elimination of the traction field in the equations. The resulting new matrix can also be split into the self-interaction matrices associated each subregion itself and the extrapolation matrices interpreting cross-interactions between different subregions in a whole model. Both the numerical schemes avoid the inversion of the relatively much larger boundary integral equation matrix of a full-waveform BE method and hence save computing time and memory greatly. The two schemes are validated by calculating synthetic seismograms for a homogeneous layered model, compared with the full-waveform BE numerical solution. Numerical experiments indicate that both the BEM+Born series modeling schemes are valid and effective. The tests also confirm that the second modeling scheme has a faster convergence in comparison with the first one.

Modified Combined Field Integral Equations for Electromagnetic Scattering

The boundary integral formulation of exterior boundary value problems for the Maxwell system may not be equivalent to the original uniquely solvable problem if the wave number corresponds to an eigenvalue of an associated interior eigenvalue problem. To avoid these spurious modes one may use a combined boundary integral approach. To analyze the resulting boundary integral equations in the energy function spaces suitable regularizations have to be introduced. Here we formulate and analyze a modified boundary integral equation which is based on the use of standard boundary integral operators only. A first numerical example shows the applicability of the proposed approach.

トランサム船尾流場の解析法について

An analytical method in so-called Neumann-Kelvin problem is proposed to represent 2-D flow about semi-displacement type ship having blunt stem and transom stern. At first, the usual Neumann-Kelvin solution, complex potential, is deduced from Green's integral formula by using the boundary conditions on the linearized free surface and on the exact body surface and is represented by the velocity distribution along the body surface and by wave sources located at the fore and aft cross points. This new representation shows explicitly the non-uniqueness of Neumann-Kelvin solutions. Next, the transom wall is removed and the representation is reduced to the sum of integration along the body surface, the unknown being the velocity differences, and the wave source located at the fore cross point. The resulting boundary integral equation has two unkowns which are determined from planning conditions, Kutta velocity condition and the other wave height condition at the transom point. The numerically obtained solutions show good flows around the body with blunt stem and transom stern. It is reported to be able to assess the Froude number when planing starts by combining wave breaker indices, the values of resistance, lift and trim moment when the rise of center of gravity and the trim are varied and the Froude number when heave and pitch are unstable. It is also shown that a concave camber over the bottom of planing hull makes a profit for attitude and resistance. At last, a possibility is suggested to obtain a good evalution of resistance for body with transom stern by using high-speed approximation of inverse mirror solution.

Symmetric Galerkin BEM for bodies with unconstrained contours

The solution via symmetric Galerkin boundary element method of problems in linear elasticity presents a major difficulty when a whole connected component of the boundary is under Neumann condition. In fact, apart from some special cases here discussed in detail, the resulting boundary integral equations are in general insensitive to the rigid body displacement of unconstrained connected boundary-components, even when the displacement is prescribed on some other boundary portions. A simple procedure is here proposed to solve the aforementioned indeterminacy while preserving the symmetry of the integral operator. Representative examples confirm the numerical effectiveness of the proposed method.

A boundary element method applied to sandwich plates of arbitrary plan form

A study is made of the bending analysis of isotropic, linear elastic sandwich plates with arbitrary shape and boundary conditions. The problem is examined by means of the indirect boundary element method (IBEM) in the context of the small-deflection, first-order shear deformation theory. The behavior of the plate is governed by a system of the differential equations expressed in terms of the potential and stream functions. The resulting boundary integral equations are formulated. The numerical solutions of these equations follow the routine procedure of the BEM. The numerical results obtained demonstrate the efficiency and accuracy of the proposed technique.