Traditional Boundary Element Method Research Articles

A spherical source distribution method for calculating acoustic radiation of elastic underwater structures

In this paper, a spherical source distribution method is established, and two kinds of spherical sound sources of symmetric and antisymmetric, distributed on a line inside the structure are proposed, in order to realize the vibro-acoustic calculation of three-dimensional elastic underwater structure. The spherical source distribution method has strong applicability and is suitable for the case where the shape of the structure is not axisymmetric. This method is a new method, and its fundamental formula is similar to the traditional acoustic boundary integral method, but it also has obvious differences. In numerical calculation, the traditional boundary element method is to divide the surface elements on the wet surface of the object, and transform the three-dimensional acoustic problem into the two-dimensional discrete element problem to solve. However, the spherical source distribution method, whose source points are only distributed on a straight line inside the object, transforms the three-dimensional acoustic problem into the quasi-one-dimensional discrete element problem, which makes the complexity and computation amount of the whole programming significantly reduced. In this paper, the fundamental principle of the spherical source distribution method, the calculation formula and the verification results of several numerical examples are discussed.

Multi-domain wavelet boundary element method for calculating two-dimensional stress intensity factors

In order to improve the accuracy of stress intensity factors (SIFs) calculated by traditional boundary element methods (BEM), the multi-domain wavelet boundary element method (WBEM) is proposed. Firstly, by adjusting the nodes of the B-spline wavelet element on the interval, crack-tip elements are constructed. Since B-spline wavelet on the interval (BSWI) has excellent compact support characteristics and is particularly suitable for describing solution domains with large gradient changes, the constructed crack-tip can reduce the numerical oscillation effect near the crack tip. Secondly, the crack-tip elements are implemented into WBEM. And the combination of WBEM and multi-domain technology can effectively handle interface cracks. Thirdly, the crack problem solving strategy based on multi-domain WBEM can directly evaluate the SIFs of cracks. Finally, several numerical examples involving homogeneous media and bi-material models are given to verify that the proposed method is simple and highly accurate.

Discrete null field equation methods for solving Laplace's equation: Boundary layer computations

AbstractConsider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain , and use the null field equation (NFE) of Green's representation formulation, where the source nodes are located on a pseudo‐boundary outside but not close to its boundary . Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives of the solutions on the boundary can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise ‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing.

Isogeometric boundary element method for axisymmetric steady-state heat transfer

Isogeometric analysis (IGA) utilizes the Non-Uniform Rational B-Splines (NURBS) basis functions, which are commonly employed in Computer Aided Design (CAD), to both construct the problem’s geometry and approximate the field variables. In axisymmetric scenarios, the 3D problem can be simplified to 2D, requiring only the discretization of the 1D curve boundary for the boundary element method (BEM). This study proposes an isogeometric boundary element method (IGABEM) to simulate steady-state heat transfer in axisymmetric domains. Both the precise geometry of the boundary and the physical variables are approximated with the same NURBS basis functions, yielding higher-order continuity, superior accuracy per degree of freedom (DOF), and continuous nodal gradients with fewer DOFs than the traditional BEM. Additionally, a zoning method is introduced to handle heterogeneities. Five cases including a solid cylinder, a revolution hyperboloid with a coaxial cylindrical tube, an ellipsoid with a spherical cavity, a bimaterial ellipsoid and a complex geometry with a toroidal tube validate the accuracy, convergence, computational efficiency of IGABEM, and the applicability in addressing multiply-connected domains. The method outperforms Finite Element Method (FEM) and conventional BEM in accurately handling steady-state axisymmetric heat transfer issues under Dirichlet, Neumann, and Robin boundary conditions.

Data-driven analysis of parametrized acoustic systems in the frequency domain

A data-driven method combined with the formulations of boundary integral equations is developed for the frequency-domain analysis of parametrized acoustic systems, arising from the spatial discretization of the linear Helmholtz equation. The method derives surrogate models for the approximation of frequency response functions at selected field points via the construction of neural networks with radial basis function interpolation. This requires the offline collection of a training database consisting of discrete frequency-parameter samples and their associated response values, whereas the information about system matrices and interference with boundary element kernels are no longer necessary, leading to a matrix-free and non-intrusive nature. To ensure accurate fitting results, a tensor decomposition technique is employed to decouple the naturally formed tensor grid between the frequency and parameter inputs in the database. After establishing two sets of radial basis networks separately, the final reduced-order solutions as a continuous function of the frequency and parameters are represented by the linear combinations of tensor products. In the online phase, the original transfer function at any testing frequency-parameter location can be quickly predicted as direct output from the compact surrogate model. With the analytical solution, the traditional boundary element method and the fast multipole boundary element method as solvers for the generation of training data, two benchmark problems and one complex system are investigated. In all cases, the simplicity, versatility and efficiency of the proposed method are demonstrated through examples of multi-frequency solutions of acoustic systems with parametrized material and geometric properties.

On the efficient simulation of pass-by noise signals from railway wheels

The article presents an approach for calculating pass-by sound pressure radiated from railway wheels in the time domain using moving Green’s functions. The Green’s functions are obtained by using Finite Element (FE) and Boundary Element (BE) methods in the frequency domain, subsequent inverse Fourier transform, followed by convolution with a time series of rolling contact forces to obtain the pass-by time signals. However, traditional BE methods are computationally expensive due to the low structural damping of the wheel, necessitating a high frequency resolution.To overcome this issue, a modal approach is introduced in which the pass-by sound radiated by each wheel mode is calculated separately. This approach incorporates the dynamic response of the wheel in the time-domain processing and thus reduces the cost of the BE solution. A modal source signal is introduced to describe the excitation of each mode at each time step. The sound field radiated by unit modal amplitudes is calculated in BE and subsequently approximated by spherical harmonic (SH) equivalent sources, which allows for efficiently calculating acoustic transfer functions for varying relative positions of the wheel and a stationary receiver. Convolution of the source signal with the moving acoustic transfer function produces the pass-by pressure signal.The article investigates the directivity of the radiation from each mode and finds that most modes, including those with dominant radial deflection, radiate in mostly axial direction at high frequencies. Modes that dominate the pass-by pressure level are identified, both in frequency bands and with respect to the relative positioning of the wheel to the receiver. Finally, it is found that an SH expansion order of approximately 30 is required to satisfy the employed error measures, although lower orders may suffice for an auralisation of the signal.

Modified BEM for scattering analysis by a flaw at interface in an anisotropic multi-layered plate

In this paper, a modified boundary element method (BEM) for time-harmonic scattering computation analysis in two-dimensional (2-D), homogeneous, anisotropic and linear elastic plates is proposed. The aim of modification is to correct the spurious scattering introduced by inevitable model truncation at far-field in the traditional BEM model. The far-field wave displacement fields beyond the truncation points of BEM model are assumed to be the superposition of orthogonal propagating guided wave patterns, and are finally incorporated into BEM equation systems as the modified items to account for the contribution of infinite boundaries traditionally omitted. This method is simple and elegant, which has advantages on dimension reduction and needs no absorption medium or perfectly matched layer to suppress the reflected waves compared to FEM. Also, we can obtain the reflection and transmission coefficients of each mode directly without post-processing. This modified BEM is implemented to solve wave scattering problems due to a cavity-type defect at the interface in an infinite anisotropic multi-layered plate. The formulation is finally validated for several benchmark problems.

A Kernel-Free Boundary Integral Method for 2-D Magnetostatics Analysis

Performing magnetostatic analysis accurately and efficiently is very important for multi-objective optimization of electromagnetic device designs. In this research, a kernel-free boundary integral method (KFBIM) has been introduced for solving magnetic computations in a toroidal core geometry in 2-D. This study is very relevant in the design and optimization of toroidal inductors or transformers used in electrical systems, where lighter weight, higher inductance, higher efficiency, and lower leakage flux are required. The governing partial differential equations (PDEs) have been formulated as a system of the Fredholm integral equations of the second kind. Unlike traditional boundary integral methods or boundary element methods, KFBIM does not require an analytical form of Green’s function for evaluating integrals via numerical quadrature. Instead, KFBIM computes integrals by solving an equivalent interface problem on a Cartesian mesh. Compared with traditional finite difference methods for solving the governing PDEs directly, KFBIM produces a well-conditioned linear system. Therefore, the KFBIM requires only a fixed number of iterations when an iterative method [e.g., generalized minimal residual method (GMRES)] is applied for solving the linear system, and the numerical solution is not sensitive to computer round-off errors. The obtained results are then compared with a commercial finite element solver (ANSYS), which shows excellent agreement. It should be noted that, compared with FEM, the KFBIM does not require a body-fit mesh and can achieve high accuracy with a coarse mesh. In particular, the calculations of the magnetic potential and the tangential field intensity on the boundaries are more stable and exhibit almost no oscillations.

Isogeometric boundary element method for isotropic damage elastic mechanical problems

In this work, a complex isotropic damage mechanics structural model is developed using the isogeometric method to study the strain displacement and influence factors in isotropically damaged elastics. The proposed modeling method can be constructed dynamically by introducing control points and weight coefficients, which improves the accuracy and eliminates mesh errors, and the boundary element method is used for theoretical analysis and calculation of damage mechanics. For the description of irregular geometric models, the isogeometric method has unparalleled superiority. The isogeometric method can seamlessly connect the geometry generated by the advantages of CAD and CAE, and the combination of the two methods is very useful for the future extension of IGABEM to practical engineering applications. At the same time, we use the source point isolation method to establish the boundary-domain integral equations and use the radial integration method to solve the domain integrals. Compared with the traditional boundary element method, the isogeometric boundary element method makes the construction of complex models simple. The numerical results show that the numerical method has good stability and robustness, which provides strong support for further software integration.

Multi-domain physics-informed neural network for solving forward and inverse problems of steady-state heat conduction in multilayer media

In this paper, a novel deep learning technique, called multi-domain physics-informed neural network (M-PINN), is presented to solve forward and inverse problems of steady-state heat conduction in multilayer media. By adopting the domain decomposition technique, the multilayer media is first divided into several sub-domains. Then, the fully connected neural network is employed to approximate the temperature field on each sub-domain. Finally, a large total network framework is formed by combining subnetworks of all the mediums and using continuity conditions on interfaces. By training the total network, we can obtain the temperature distribution over the whole computational domain, including the interface between every two mediums. In the proposed method, the boundary conditions are introduced into the loss function, and the governing equation is used as a constrain item, which ensures the accuracy and stability of numerical approximation. As a meshless collocation technology, the M-PINN does not require tedious procedures such as meshing and numerical integration, and can freely address forward and inverse problems of thin body and coating structure. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results indicate that the Swish and the Sigmoid functions are two better activation functions for such problems. As the number of nodes increases, the number of hidden layers does not need to be increased. Even for the thin film at nanoscale, the M-PINN still obtains accurate results. Moreover, the proposed scheme shows better performance than the traditional boundary element method in solving nonlinear heat conduction problems.

A non-intrusive model order reduction boundary element method for frequency sweeps of 2D acoustics

The boundary element method (BEM) is one of the most commonly used prediction technique to deal with acoustic problems governed by the Helmholtz equation. Its use, however, suffers from the disadvantage that the system of equations have to be reformulated for each frequency of interest due to the frequency-dependent property. This undermines the inherent efficiency of frequency sweep analyses of BE models. Additionally, the resulting coefficient matrices are fully-populated, non-symmetric and complex-valued. In order to enable the traditional BEM to be applied for large-scale acoustic problems, a non-intrusive model order reduction boundary element method under the offline–online framework is proposed. In the offline phase, a proper reduced-order basis (ROB) which spans a projection subspace is constructed via the Arnoldi method. A newly developed cutoff technique is incorporated to reduce the involved computational complexity. Once a low-rank ROB is determined, the full-order frequency-decoupled system matrices based on the series expansions are projected onto the spanned subspace column-by-column to favor the memory requirement. In the online phase, fast frequency sweep analyses with a direct solver can be achieved because only the compact reduced-order model needs to be confronted. The high performance of the proposed solution approach in the context of single-frequency and multi-frequency analyses of two-dimensional (2D) BE models with the number of degrees of freedom ranging from a few hundred up to one hundred thousand is demonstrated by various problem settings, including both interior and exterior acoustic fields with different boundary conditions.

Performance assessment of internal porous structures on liquid sloshing in various 3D tanks by multi-domain IGABEM

An isogeometric boundary element method (IGABEM) is extended in this work to study liquid sloshing characteristics in various 3D tanks with arbitrary internal bottom-mounted porous structures. Unlike the classical boundary element method (BEM), the present method uses the non-uniform rational B-splines(NURBS) instead of the piecewise Lagrange polynomials as the shape function to approximate both the domain boundary and field variables, it completely inherits the advantages of the traditional BEM and the isogeometric analysis (IGA), such as the properties of only boundary discretization needing, higher order continuity, self-adaptability and so on. All the information required in the IGABEM for the mesh generation can be obtained from the CAD software, which may evidently reduce the time and memory consumptions of preprocessing, meanwhile, since the same basis functions (NURBS) are used for both the IGABEM and CAD models, the proposed method can exactly reconstruct the geometry of the analysis domain without any error, and this substantively contributes to the high calculation accuracy of IGABEM. In this paper, by using a zoning method the entire domains of fluid are divided into several sub-domains with the consideration of compatibility boundary conditions besides the porous structures. Owning to the constant cross section of the container, impermeable boundary condition at the tank bottom and the linear boundary condition at the free surface, the 3D Laplace equation (which governs the 3D sloshing problem) is transformed into a couple of Helmholtz equation and modified Helmholtz equations (corresponding to the propagating and evanescent modes, respectively). Green's theorem and the divergence theorem of Gauss are used in this paper to derive the IGABEM system equations for the two types of equations. Moreover, by introducing a series of eigen-functions associated with the vertical coordinate, the potential and velocity boundary conditions are expanded into those associated with the propagating and evanescent modes, which can be applied upon the Helmholtz equation and modified Helmholtz equations. Finally, the present problem can be solved by solving these equations and combining the results linearly. Accuracy and convergence of the present IGABEM technique are verified through numerical tests by comparing the obtained results with analytical, experimental solutions or those calculated with other numerical method, thereafter, liquid sloshing problems in the cubic or circular cylindrical vessels with the coaxial or eccentric elliptical and circular cylindrical porous structures are analyzed. The main influences of the geometric parameter, porous-effect parameter, number and arrangement of the porous structures are investigated in detail, some conclusions are outlined in the end of this paper.

A Simple, Accurate and Semi-Analytical Meshless Method for Solving Laplace and Helmholtz Equations in Complex Two-Dimensional Geometries

A localized virtual boundary element–meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex two-dimensional (2D) geometries. “Localized” refers to employing the moving least square method to locally approximate the physical quantities of the computational domain after introducing the traditional virtual boundary element method. The LVBE-MCM is a semi-analytical and domain-type meshless collocation method that is based on the fundamental solution of the governing equation, which is different from the traditional virtual boundary element method. When it comes to 2D problems, the LVBE-MCM only needs to calculate the numerical integration on the circular virtual boundary. It avoids the evaluation of singular/strong singular/hypersingular integrals seen in the boundary element method. Compared to the difficulty of selecting the virtual boundary and evaluating singular integrals, the LVBE-MCM is simple and straightforward. Numerical experiments, including irregular and doubly connected domains, demonstrate that the LVBE-MCM is accurate, stable, and convergent for solving both Laplace and Helmholtz equations.

Machine Learning Enhanced Boundary Element Method: Prediction of Gaussian Quadrature Points

This paper applies a machine learning technique to find a general and efficient numerical integration scheme for boundary element methods. A model based on the neural network multi-classification algorithm is constructed to find the minimum number of Gaussian quadrature points satisfying the given accuracy. The constructed model is trained by using a large amount of data calculated in the traditional boundary element method and the optimal network architecture is selected. The two-dimensional potential problem of a circular structure is tested and analyzed based on the determined model, and the accuracy of the model is about 90%. Finally, by incorporating the predicted Gaussian quadrature points into the boundary element analysis, we find that the numerical solution and the analytical solution are in good agreement, which verifies the robustness of the proposed method.

Energy-maximising experimental control synthesis via impedance-matching for a multi degree-of-freedom wave energy converter

We present, in this paper, an experimental framework for design and synthesis of impedance-matching-based (IM) controllers capable of maximising energy extraction in inherently multi degree-of-freedom wave energy converter (WEC) systems, and its subsequent application to the Intertial Sea Wave Energy Converter (ISWEC) device, by incorporating recent advances in IM-based theory. In particular, we consider a 1/20th scale prototype of the ISWEC system, tested as part of a larger experimental campaign conducted within the tank facilities available at Università degli Studi di Napoli Federico II, subject to a variety of wave conditions. We adopt two different control structures to realise an approximation of the IM principle, fully tuned based upon interpolation of a particular (experimentally obtained) non-parametric empirical transfer function estimate, which defines the optimal frequency-domain input-output response for energy-maximising behaviour. Furthermore, a performance comparison between controller tuning based upon traditional linear boundary element method models, and the presented experimental approach, is also offered, showing that the latter can consistently outperform the former in realistic scenarios, for the set of analysed sea-states.

Thermoelastic Mechanics Analysis Based on Isogeometric Boundary Element Method

Solid thermal stress caused by temperature has always been one of the important factors affecting the life of solid materials.The boundary element method, using boundary integrals instead of domain integrals, reduces the dimensionality of complex issue and the computational complexity. The non-uniform rational B-splines (NURBS) curve is used to construct the boundary of the calculation model, and the isogeometric idea is used to divide the unit, so as to overcome the geometric error caused by the unit discretization. Because of the existence of thermal stress, the constructed isogeometric boundary integral equation contains the domain integration term. The radial basis function method is used to convert the domain integration to the boundary integration. The advantage of the dimensionality reduction can be fully presented by using isogeometric boundary element method. The model calculation shows that the proposed calculation method is more reliable than the traditional boundary element method.

FMM-BEM algorithm based on improved free surface Green's function

In this paper, a Green's function satisfying free surface condition, which is more suitable for solving potential flow problems, is improved, proposed and applied to FMM-BEM to solve the problems of large amount of calculation and long calculation time in the traditional boundary element method. The results show that the FMM-BEM method is more efficient than the BEM method in hydrodynamic analysis of large-scale structures.

Generalized pressure-correction for viscous potential flow simulations of multiphase flows using boundary element method

A new formulation for correcting the pressure field in Viscous Potential Theory (VPT) is developed, generalizing previous Viscous Pressure Correction (VPC) methods to simulate multi-phase flows based on VPT. The latter theory is now applicable to a wider range of flow problems. Without limiting the general applicability of the formulation, the present work focuses on the dynamics of two-dimensional liquid blobs and liquid films in a void. The governing equations of the method are solved using the traditional Boundary Element Method together with fourth-order Runge-Kutta time-integration, thereby reducing the spatial dimension of the problem by one and increasing the computational efficiency dramatically as compared to general purpose solvers for interfacial flows. Verification and Validation of the new method and algorithm are carried out through comparison of the simulation results with those obtained from (semi-)analytical models, a general-purpose OpenFOAM solver with Volume-of-Fluid interface treatment, and a Smoothed-Particle Hydrodynamics solver. Several benchmark cases and a wide range of viscosities are considered. Significantly improved accuracy of the newly proposed method as compared to the original Viscous Pressure Correction of Viscous Potential Flow (VCVPF) approach is demonstrated for the case of a retracting free liquid film or ligament. In order to demonstrate the capacity of the present method and algorithm in handling complex multiphase flow problems and in providing physical insight into such phenomena, the disintegration mechanism of a strongly modulated discharging liquid sheet under the influence of gravity is analyzed.

RI-IGABEM based on PIM in transient heat conduction problems of FGMs

The isogeometric analysis boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation and good approximation properties. In this paper, a novel radial integration IGABEM (RI-IGABEM) based on precise integration method (PIM) is proposed to solve transient heat conduction problems of functionally gradient materials (FGMs) with heat source Similar to traditional boundary element method (BEM), in which the fundamental solution for the Laplace equation is used to derive the boundary-domain integral equations, the domain integrals caused by the heat source and the thermal conductivity varying with coordinates and the initial temperature appear in the transient heat conduction problems. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral. In addition, using a simple transformation method, the uniform potential method is successfully applied to solve the strongly singular integrals, and the Telles scheme and the element sub-division method are used to solve the weakly singular integrals in RI-IGABEM respectively. Furthermore, in order to improve the stability of numerical results, the PIM is adopted to solve the time domain problem. In order to validate the accuracy and convergence of the RI-IGABEM 2D and 3D numerical examples are used to discuss the influence of some factors, such as the number and the position of applied points, the order of basis functions, the number of model refinements and the length of time step.

Isogeometric boundary element method for steady-state heat transfer with concentrated/surface heat sources

An isogeometric boundary element method (IGABEM) is proposed for solving the steady-state heat transfer problems with concentrated/surface internal heat sources. The isogeometric boundary element method (IGABEM) possesses the advantages of both the isogeometric analysis (IGA) and the boundary element method (BEM), the non-uniform rational B-spline (NURBS) basis functions used in the (computer-aided design) CAD system are flexible and stable in dealing with irregular boundaries, due to which the NURBS basis functions are applied to exactly reconstruct the boundary geometry of the analysis domain, and the physical variables are also approximated by the NURBS as the standard IGA does. Dirac delta function is introduced in the present IGABEM for dealing with the concentrated point heat sources, and a local coordinate system is proposed for the domain integration over line heat source based on the NURBS basis functions. As to the domain integrals caused by the surface heat sources, the radial integration method (RIM) is used to transform the domain integrals to boundary ones without any internal cells for the original version of IGABEM. Several numerical examples involving Dirichlet, Neumann and Robin boundary conditions are analyzed, and circular region with point/line heat source(sources), global and local surface heat sources within multiply connected regions are also considered. Verification of the proposed method has been gained by comparing the present results with those obtained by several other researchers, and solutions achieved by analytical expression together with the fine-meshed ANSYS model are also utilized for the purpose of comparison. Meanwhile, the computational efficiency and convergence of the present method is evaluated by comparing the IGABEM with the finite element method (FEM) as well as the traditional BEM. Good performance of the IGABEM is observed. Finally, heat transfer within a simplified printed circuit board (PCB) with both concentrated (line and point) and surface heat generations is studied, the temperature distribution is achieved, which verifies the applicability of the proposed technoque in practical engineering.