3D BEM Research Articles

3D isogeometric indirect BEM solution based on virtual surface sources on the boundaries of Helmholtz acoustic problems

A solution for 3D Helmholtz acoustic problems is introduced based on an indirect boundary element method (indirect BEM) coupled with isogeometric analysis (IGA). The novelty of this work arises from using virtual surface sources placed directly on the scatterer boundaries, producing robust results. These virtual surface sources are discretized by the same Non-Uniform Rational B-Splines (NURBS) approximating the scatterer CAD model. This allows modeling of general irregular geometries. The proposed solution has the same features of BEM approaches, which do not need any domain discretization or truncation boundaries at the far-field. It shows an additional merit by arranging the linear system of equations directly depending on a single coefficient matrix, consuming less computational time compared to other BEM methods. A Greville abscissae collocation scheme is proposed with offsets at C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^0$$\\end{document}-continuities. This collocation scheme allows for easy evaluation for both free-terms and normals at the collocation points. The performance of the proposed solution is discussed on 3D numerical exterior problems and compared against other BEM methods. Then, the practical interior muffler problem with internal extended thin tubes is studied and the obtained results are compared against other numerical methods in addition to the available experimental data, showing the capability of the proposed solution in handling thin-walled geometries.

A 3D BEM-Coupled Mode Model for the Performance Analysis of Wave Energy Converter Parks in Nearshore-Coastal Regions

Estimation on the production capacity of wave energy converter arrays (WECs) of the type of simple floaters deployed in nearshore locations highly depends on the evaluation of their performance. The latter also depends on various factors, including the dimensions and inertial characteristics of the devices, their relevant positioning, and the power take-off (PTO) system characteristics. Studying the system operation, based on the prevailing sea conditions in the region considered for deployment, can ensure that such WEC farms are sized and designed in an effective way. Furthermore, the wavelength and propagation direction of incoming wave fields can be significantly impacted by wave-seabed interactions in coastal areas, which can alter the WECs’ response pattern and ultimately the array’s power output. In this work, a 3D BEM hydrodynamic model is proposed aiming to assess the energy-capturing capacity of WEC arrays, accounting for the hydrodynamic interactions between various identical floating devices, as well as the local seabed topography. The model is supplemented by a Coupled Mode System (CMS) to calculate the incident wave field propagating over variable bathymetry, in order to simulate realistic nearshore environments. Finally, a case study is performed for an indicative geographical area, north of the coast of the island of Ikaria, located in the Eastern Aegean Sea region, where the wave potential is high, using long-term data. The latter study highlights the applicability of the proposed method and suggests its usage as a tool to support optimal WEC park design.

An adaptive binary-tree subdivision method for evaluation of nearly singular integrals in 3D BEM

PurposeThe purpose of this paper is to present an adaptive binary-tree element subdivision method (BTSM) for the evaluation of nearly singular integrals in three-dimensional boundary element method, which can facilitate automatic and high-quality patch generation.Design/methodology/approachIn this method, the nearly singular element is split into two sub-elements. Each sub-element is then examined to determine if it is to be subdivided based on a specific subdivision criterion. The specific subdivision ensures that those sub-elements far from the source point are sparse. And then those sub-elements in close proximity to the source point are replaced by regular triangular elements.FindingsWith the proposed method, the sub-elements obtained are automatically refined as they approach the projection point, and they are “good” in shape and size for standard Gaussian quadrature. Thus, the proposed method can be used to evaluate nearly singular integrals accurately for cases of different element shapes and various locations of the source point.Originality/valueNumerical examples for surface elements with various relative locations of the source point are presented. The results demonstrate that the proposed method has much better accuracy and robustness than some other methods.

A Modified Mild-Slope Model for the Hydrodynamic Analysis of Arrays of Heaving WECs in Variable Bathymetry Regions

A simplified model based on the Modified Mild-Slope Equation with inclusions is developed for modelling the scattering of waves from multiple heaving point absorbers arranged in an array in general bottom topography. The model is used, in conjunction with a 3D BEM, in order to estimate the parameters modelling the energy extraction of the devices using data obtained from the hydrodynamic responses and performance of the single floating WEC. Subsequently, the present model is used for specific examples to calculate the wave field and the hydrodynamic performance of arrays of heaving WECs in constant depth and variable bathymetry regions and illustrate the effect of bottom slope and variation on the calculated wave field in the domain and in the vicinity of the devices. The present simplified model provides a low-cost first estimation of the wave conditions in the domain, which could be exploited as a supporting tool for best arrangement and park design purposes.

WAVE FIELD GENERATED BY FINITE-SPAN HYDROFOILS OPERATING BENEATH A FREE SURFACE

The present paper focuses on the numerical investigation of the flow around the fully submerged 2D and 3D hydrofoils operating close to a free surface. Iterative boundary element method is implemented to predict the flow field. This study aims to investigate the aspect ratio effect on the free surface interactions and hydrodynamic performance of the hydrofoils under a free surface by using potential flow theory. Three different submergence depths and aspect ratios are studied in the wide range of Froude Numbers. In 3D cases, spanwise width of the numerical wave tank model is selected both equal and wider to the foil span, to observe the sidewall effects. Wave field seems to be two dimensional at low Froude numbers. On the other hand, signs of three dimensionalities are observed on the free surface structure for higher Fn, even the predicted wave elevations are very close to 2D calculations in the midsection. Increment in the Fn give a rise to the amplitude of the generated waves first, however a further increase in Fn has a lowering effect with the beginning of waves spill in the spanwise direction in the form of Kelvin waves. Free surface proximity and resultant wave field are also seeming to be linked with the lift force on the hydrofoil. As aspect ratio of the foil increase, 3D lift values are getting closer to those of 2D calculations. However, it is seen that, 3D BEM predictions of a hydrofoil under free surface effect cannot be considered two-dimensional even the aspect ratio is equal to 8.

The BEM based on conformal Duffy-distance transformation for three-dimensional elasticity problems

Here, we describe the robust and efficient application of the conventional 3D BEM in solving elasticity problems. We have focused on the precise computation of weakly singular integrals. The conformal Duffy-distance transformation was employed to alleviate near singularities caused from two aspects: (1) the large aspect ratio of elements, i.e., element shape distortions; and (2) the closeness of element boundaries to field points, i.e., ill-shaped patches. Then, the rigid body motion method was employed to evaluate strongly singular integrals. Numerical solutions of 3D elastostatic problems demonstrated the high accuracy of the proposed method with coarse meshes and high convergence rates with mesh refinement. Compared with the Duffy transformation and original polar coordinate transformations, the proposed method is insensitive to element shapes.

Multi-DOF WEC Performance in Variable Bathymetry Regions Using a Hybrid 3D BEM and Optimization

In the present work a hybrid boundary element method is used, in conjunction with a coupled mode model and perfectly matched layer model, for obtaining the solution of the propagation/diffraction/radiation problems of floating bodies in variable bathymetry regions. The implemented methodology is free of mild-slope assumptions and restrictions. The present work extends previous results concerning heaving floaters over a region of general bottom topography in the case of generally shaped wave energy converters (WECs) operating in multiple degrees of freedom. Numerical results concerning the details of the wave field and the power output are presented, and the effects of WEC shape on the optimization of power extraction are discussed. It is demonstrated that consideration of heave in combination with pitch oscillation modes leads to a possible increase of the WEC performance.

Efficient evaluation of weakly singular integrals with Duffy-distance transformation in 3D BEM

Weakly singular integrals are commonly encountered in the application of BIEs. In general, the 3D integral boundary element is divided into several triangle patches using the image of singular point in the local coordinate system, which transforms the integral into vertex singularity problem over each triangle patch. In this paper, firstly the classic Duffy transformation used for vertex singularity problem are modified into an optimized form termed as ‘Duffy-distance transformation’, taking into account the near singularity caused by the integral patch shape. Besides, the previously proposed conformal mapping is coupled with the Duffy-distance transformation to eliminate the near singularity caused by element shape distortion. Planar quadrilateral elements with different inclined angles and aspect ratios are given to verify the accuracy and efficiency in detail, and another two curved elements extracted from cylinder and sphere surfaces are presented to demonstrate the applicability for higher-ordered elements.

Implementation of Different Formulas for Special Green’s Functions in a 3D Half-Space BEM

The numerical stability of different formulas for the correction term of the half-space Green’s function is investigated. The formula with complex monopoles is taken as a reference. The expressions of Koh and Yook, tested in a previous publication [R. Piscoya and M. Ochmann, Acoustical Green’s function and boundary element techniques for 3D half-space problems, J. Comput. Acoust. (2017), https://doi.org/10.1142/S0218396X17300018 ], are rewritten to improve their range of application. The formulas of Sommerfeld and Thomasson are analyzed and its suitability for a BEM implementation is evaluated by comparing their accuracy against our reference. For the sake of completeness, the first and second derivatives of the formulas are explicitly written.

A method of treating the nearly singular integral in calculation of sound radiation with BEM

For the nearly singular integral of three-dimensional acoustic boundary element method (BEM), based on the 6-noded triangular isoparametric element, a new semi-analytical algorithm of three-dimensional high order element is proposed in this paper. Using Taylor expansion of trigonometric functions in the three dimensional acoustic fundamental solutions, the singular part of the fundamental solutions is separated. Based on the geometric characteristics of the 6-noded triangular element, an approximate singular kernel function is constructed which has the same singularity as singular integral kernel function. Subtracting the approximate kernel function from the kernel function of the singular integral, the latter is decomposed into a regular kernel function and an approximate singular kernel function. The integral of the regular kernel function can be calculated accurately by using the conventional Gauss numerical quadrature. The integral of the new singular part is calculated by the semi-analytic formula derived in this paper. In the surface of the integral element, the local coordinate system ρθ is established and the approximate singular integral is transformed into the integrals of variables ρ and θ which are already separated in ρθ system. The integral with respect to polar variable ρ is expressed by the analytic formulations first. Then the new singular integral which is a surface integral is transformed into the line integral with respect to variable θ , which can be evaluated by the Gaussian quadrature. Consequently, the new semi-analytic algorithm is established to calculate the nearly singular surface integrals in 3D acoustic BEM. Some examples are given in the last part of this paper to show the accuracy and the effectiveness of the present algorithm. The computed results demonstrate that the semi-analytic algorithm with high order element presented in this paper is more effective than linear regularization BEM to solve nearly singular integrals for 3D acoustic BEM.

Hydroelastic response of 19,000 TEU class ultra large container ship with novel mobile deckhouse for maximizing cargo capacity

This paper is related to structural design evaluation of 19,000 TEU ultra large container ship, dealing with hydroelastic response, i.e. springing and whipping. It illustrates application of direct calculation tools and methodologies to both fatigue and ultimate strength assessment, simultaneously taking into account ship motions and her elastic deformations. Methodology for springing and whipping assessment within so called WhiSp notation is elaborated in details, and in order to evaluate innovative container ship design with increased loading capacity, a series of independent hydroelastic computations for container ship with mobile deckhouse and conventional one are performed with the same calculation setup. Fully coupled 3D FEM – 3D BEM model is applied, while the ultimate bending capacity of hull girder is determined by means of MARS software. Beside comparative analysis of representative quantities for considered ships, relative influence of hydroelasticity on ship response is addressed.

A new semi-analytic algorithm of nearly singular integrals on higher order element in 3D potential BEM

By analyzing the geometric characteristics of 8-noded quadrilateral surface elements in three dimensional boundary element method (3D BEM), the relative distance from a source point to the integral element is defined. For the nearly singular integrals on higher order elements in 3D potential BEM, the equivalent integral kernels are constructed by the geometric analysis between the source point and the element in ρθ system. Subtracting the equivalent kernels from and adding them back to the nearly singular kernels, the nearly singular surface integrals are transformed into the sum of both the non-singular integrals and the singular integrals. So the leading singular parts are separated. The former are computed efficiently by the Gaussian quadrature and the latter are performed with respect to the integral variables ρ and θ, respectively, in which the integrations with respect to ρ are expressed by analytical formulations. Consequently, a new semi-analytic algorithm is established to calculate the nearly strongly singular and hyper-singular surface integrals on higher order element in 3D BEM.Several examples about 3D heat conduction are given to demonstrate the efficiency and accuracy of the present semi-analytic algorithm in BE analysis. Moreover, the present algorithm is used to analyze very thin structures in 3D BEM.

A combined conformal and sinh–sigmoidal transformations method for nearly singular boundary element integrals

Accurate and efficient evaluation of nearly singular integrals is a major concern in 3D BEM. Most existing widely-used non-linear transformations are only performed in radial direction. Actually, the near singularity may derive from three aspects: element shape, radial direction and angular direction. In this paper, a combined conformal and sinh–sigmoidal transformations method is proposed to evaluate nearly singular integrals arising in 3D BEM. The method can be decomposed into three steps: firstly, a conformal transformation is introduced to eliminate the shape effect caused by large aspect ratios and peak/big obtuse angles; secondly, the classical sinh transformation is applied in radial direction to cluster more Gaussian points towards the nearly singular point; finally, an improved sigmoidal transformation is utilized to rearrange Gaussian points in angular direction more reasonably. Extensive numerical examples including unit triangular element, elements with different aspect ratios, elements with different angles and curved triangular element are given to verify the robustness and competitiveness of presented method.

A 3D FEM/BEM code for ground–structure interaction: Implementation strategy including the multi-traction problem

The purpose of this paper is to describe the development of a 3D BEM–FEM code for ground–structure interaction. The technical choices and difficulties are reported. In particular, the multitraction problem has been implemented in 3D following a technique recently published in 2D for elastodynamics. It is showed that the separation of tractions is mandatory at corners but not at edges. The free surface and infinite interlayers are meshed by means of finite planes of varying dimensions. The paper also focuses on the validity of 2.5 approaches suggesting that in many situations the 2.5D model is well adapted. Reference situations are used for validation. The case of a pile joining a free surface and an interlayer between two different soils is described in detail. Finally computations are validated against measurements.

Use of the sinh transformation for evaluating 2D nearly singular integrals in 3D BEM

This paper presents a new strategy for the numerical evaluation of 2D nearly singular integrals that arise in the solution of 3D BEM using eight-node second-order quadrilateral surface elements. The strategy is an extension of the previous sinh transformation techniques, which is used to evaluate the 1D or 2D nearly singular integrals on simple geometry elements, such as linear or planar elements. The novel feature of the proposed method is that a new distance formula is introduced here, and based on this an ingenious combination of the distance formula and the sinh transformation is developed, and hence, the rapid variation of the distance formula on the integration interval can been smoothed out. Several numerical examples involving boundary layer effect and thin-body problems in 3D potential theory are given to verify the accuracy and efficiency of the presented method.

Calculation of 2D nearly singular integrals over high-order geometry elements using the sinh transformation

The accurate evaluation of nearly singular integrals plays an important role in the implementation of BEM. In general, these include evaluating the solution near the boundary or treating problems with thin domains, which are respectively named the boundary layer effect and the thin-body effect in BEM. Although many methods of evaluating two-dimensional (2D) nearly singular integrals have been developed in recent years with varying degrees of success, questions still remain. In this paper, we present an efficient strategy for numerical evaluation of 2D nearly singular integrals that arise in the solution of 3D BEM using eight-node second-order quadrilateral surface elements. The strategy is an extension of the sinh transformation, which is used to evaluate the 1D or 2D nearly singular integrals on simple geometry elements, such as usual linear or planar elements. Several numerical examples involving boundary layer effect and thin body problems in 3D potential problems are investigated to verify the proposed scheme, yielding very promising results.

An adaptive element subdivision method for evaluation of weakly singular integrals in 3D BEM

A general adaptive element subdivision method is presented for the numerical evaluation of weakly singular integrals in three-dimensional boundary element analyses. In our method, the element is subdivided into a number of patches through a sequence of spheres with decreasing radius. The patches obtained by our method are automatically refined as they approach the source point. Consequently, each patch is “good” in shape and size for standard Gaussian quadrature, and hence high accuracy can be achieved by a small number of Gaussian sample points. Our method is applicable to any shape of element with arbitrary location of the source point inside, at vertices or on edges of the element. Numerical examples are presented for planar and curved surface elements. The results demonstrate that our method can provide much better accuracy and efficiency than the conventional subdivision method.

A new method for numerical evaluation of nearly singular integrals over high-order geometry elements in 3D BEM

This work presents a new method for numerical computation of the two-dimensional nearly singular integrals by using the eight-node second-order quadrilateral surface elements in 3D BEM. A new indirect regularized boundary element formulation excluding the CPV (Cauchy Principal Value) and HFP (Hadamard-Finite-Part) integrals is proposed. Based on this, a new approximation formula of the distance from the fixed calculation point to a generic point of the aforementioned surface geometry elements is developed firstly, and then the exponential transformation, which has been widely employed in 2D BEM, is extended to 3D BEM to remove the near singularities of integrands for considered integrals. Several numerical examples are given to verify the high efficiency and the stability of the proposed scheme.

Computation of nearly singular integrals in 3D BEM

This paper presents a general methodology for numerical evaluation of the nearly singular 2D integrals over the eight-node second-order quadrilateral geometry elements arising in 3D BEM. An accurate formula of distance between the source and the field point is proposed firstly. And then an extended form of the exponential transformation, which was firstly proposed by present author to regularize nearly singular integrals arising in 2D BEM, is developed to smooth out the rapid variation of the aforementioned formula of distance. Finally, several numerical examples involving boundary layer effect and thin body problems in 3D elastostatics are investigated to verify the proposed scheme, yielding very promising results. Moreover, it should be stressed that the proposed scheme is suitable for any high-order surface elements.

The distance sinh transformation for the numerical evaluation of nearly singular integrals over curved surface elements

This paper presents a new transformation termed as the distance sinh transformation for the numerical evaluation of nearly singular integrals arising in 3D BEM. The new transformation is an improvement of the previous sinh transformation. The original sinh transformation is only limited to planar elements. Moreover, when the nearly singular point is located outside the element, results obtained by the original sinh transformation combined with conventional subdivision method are not quite accurate. In the presented work, the sinh transformation combined with the distance function is proposed to overcome the drawbacks of the original sinh transformation. With the improved transformation, nearly singular integrals on the curved surface elements can be accurately calculated. Furthermore, an alternative subdivision method is proposed using an approximate nearly singular point, which is quite simple for programming and accurate results can be obtained. Numerical examples for both curved triangular and quadrangular elements are given to verify the accuracy and efficiency of the presented method.