Dual Boundary Element Method Research Articles

An advanced fast multipole dual boundary element method for analyzing multiple cracks propagation

A new fast multipole dual boundary element method (FMDBEM) is developed to analyze multiple crack propagations. To evaluate accurately the stress fields around the crack tip, a variable-order asymptotic element (VAE) is first proposed to express the singular behavior. This VAE is also suitable for the V-notches with different opening angles, requiring only minor adjustments of stress exponents. Then the VAE is introduced into the FMDBEM framework, where several singularity problems of integrals are solved. Finally, the FMDBEM with VAEs, combined with an adaptive scheme, is used to determine the crack propagation paths. Numerical examples show that the present method is accurate and easy to implement making it an appealing tool for analyzing large complex structures that include randomly distributed cracks and notches.

Hydrodynamic Performance of an Inverted Trapezoidal Breakwater with Permeable Retrofit

Abstract The scattering of incident waves by a surface-piercing inverted trapezoidal breakwater (SPTB) encircled by retrofit is numerically examined based on the assumptions of potential flow theory. The dual boundary element method is adopted to evaluate the hydrodynamic performance of the retrofitted SPTB breakwater. The scattering coefficients (i.e., wave transmission, wave reflection, energy loss), and force coefficients acting on the inner SPTB and outer retrofit are reported against relative water depth for various input values of breakwater and incident waves. The SPTB with 10% porosity, spacing S/h=0.25, width varied within 1.5≤B/W≤2, and depth d/h=0.2 is suggested against the incident waves to secure the coastal infrastructure.

Bayesian-informed fatigue life prediction in shallow shell structures with the dual boundary element method

This study introduces an innovative approach for statistically inferring the Equivalent Initial Flaw Size Distribution (EIFSD) in shallow shell structures with the aim of predicting fatigue life. The proposed approach integrates Bayesian inference and surrogate modelling techniques, and utilizes a long crack growth model. The EIFSD is crucial for estimating fatigue life and ensuring structural durability. To showcase the proposed methodology, a numerical example featuring a fuselage window structure under multi-axial loading, employing the Dual Boundary Element Method (DBEM) for crack growth modelling, is presented. Efficiency gains in the inference of the EIFSD are achieved through the development of surrogate models, notably a Co-Kriging model that significantly reduces computational time and outperforms a Kriging model in error reduction. The study explores two assumptions for inspection data sourcing, leading to enhanced methodological effectiveness across varying uncertainty levels. It considers the separate effects of Paris Law constants and loading conditions under increased uncertainty, presenting cases that demonstrate the Co-Kriging model’s high accuracy and computational efficiency. The proposed methodology’s robustness is further validated by assessing the impact of prior distribution quality on Bayesian inference, showing the method’s adaptability to high uncertainty while maintaining a good level of accuracy. This study also demonstrates an approach for fatigue life estimation of structures using the inferred EIFSD. By evaluating the structure up to a critical crack length, the methodology allows for the accurate estimation of fatigue life, which, in turn, facilitates the determination of optimal inspection intervals. Notably, under conditions of high uncertainty, the method showcased a maximum difference of 1.51% in the estimated fatigue life when compared to the true EIFSD, underscoring the technique’s high accuracy and reliability.

Optimal design of an inclined barrier for performance improvement of an oscillating water column device with an artificial neural network model

An oscillating water column (OWC) device with a surface-piercing inclined barrier has been studied in irregular waves. This study is to determine the optimal shape of an inclined barrier utilizing an Artificial Neural Network (ANN) model for the performance enhancement of the conventional OWC system. We employed the dual boundary element method (DBEM) as our numerical approach, which resolves two separate boundary integral equations (singular and hyper-singular) formulated at the coincident source points situated on either side of the degenerate barrier boundary. The design parameters affecting the performance of the OWC system have been optimized to maximize the power extraction using the trained ANN model. The training dataset was prepared using the DBEM, which is then used to train and validate the ANN model. Subsequently, the trained model was employed to make predictions using a broader dataset to identify the optimal combination of design parameters that maximize the conversion efficiency. The findings reveal that the well-trained ANN model can learn complex nonlinear relationships, make good predictions with high accuracy using a larger dataset of input conditions, and offer the optimal values of an inclined barrier appropriate for the wave conditions at the sea installation site.

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

In this study, numerical solutions of the unsteady convection–diffusion type equations of variable convective coefficients are investigated by two effective techniques alternative to direct boundary element method (BEM). That is, the domain boundary element and the dual reciprocity BEMs with the fundamental solution of convection–diffusion equation are used in space to transform the governing differential equations into equivalent integral equations while a backward finite difference scheme is utilized in time discretization. The fruitfulness of these combined techniques are shown by the implementation of the techniques for some widely investigated fluid dynamics problems. In fact, the lid-driven cavity flow is solved, and precise results are attained as benchmarks for assessing the accuracy of the aforementioned methods. Further, the application of the methods is extended for the natural convection flow which is governed mainly by the convection–diffusion type equations accompanied by the velocity components as variable convective coefficients. The obtained numerical results reveal that the domain BEM which uses the fundamental solution of convection–diffusion equation captures the characteristics and physical advancement of the fluid flow quite well at various combined values of the problem physical parameters.

Reliability analysis of fatigue crack growth in shallow shell structures using the Dual Boundary Element Method

This work presents a novel methodology for fatigue life reliability analysis using a newly developed Dual Boundary Element Method-based Implicit Differentiation Method (DBEM-IDM) in shallow shell structures. The proposed DBEM formulations evaluate stress intensity factors and fatigue life sensitivities in shallow shells considering geometrical variables and curvature. The IDM formulations, obtained through direct differentiation of the shallow shell DBEM formulations, uses the First-Order Reliability Method (FORM) to assess the reliability index and probability of failure of shallow shell structures with multiple cracks. Two numerical examples were simulated to illustrate the effectiveness of the proposed methodology. The first example examines a shallow shell with a centre hole and cracks subjected to various loads. The critical crack length required to ensure structural reliability and inspectability is determined. Fatigue reliability assessment is performed for the same example, employing limit state functions expressed in terms of fatigue life. FORM results are compared with the results of Monte Carlo Simulations (MCS), and provided a maximum difference of 3.67%. Parametric analyses are conducted to identify significant variables that affect reliability, and it was found that precise determination of design parameters is necessary to reduce the probability of failure, particularly for the Paris law constants. The second example, featuring more complex geometry, involved assessing fatigue reliability in a cylindrical shallow shell structure with multiple cracks. The results of this example indicate a 3.49% maximum difference between FORM and MCS. The results exhibit a low error compared to the MCS, and the inherent efficiency of the IDM-based FORM underscores its suitability for addressing uncertainty in reliability analysis.

An extended isogeometric boundary element formulation for three-dimensional linear elastic fracture mechanics

This paper presents a novel extended isogeometric boundary element formulation (XIGABEM) for three-dimensional linear elastic fracture mechanics. The formulation utilises the Dual BEM to accommodate coincident geometries for opposing crack surfaces, and inherits the well-known advantages of the NURBS basis as other isogeometric implementations. The originality herein involves the extension of the above-mentioned scheme to 3D using enrichment functions derived from asymptotic solutions for near-field crack tip displacements, in which Williams’ expansions are used on the crack surfaces and on the boundaries crossed by the crack front. Besides, Heaviside functions enrich external boundaries and allow the displacement discontinuity modelling. As with most enriched formulations, additional degrees of freedom are introduced; novel strategies are presented for the generation of auxiliary equations to recover a square system.Another key element of the proposed scheme is that the stress intensity factors are recovered directly from the solution vector and no post processing is required. Four applications demonstrate the formulation robustness, with results of models having comparatively few degrees of freedom comparing well against classical and other published results.

A general DBEM for mixed-mode cohesive crack problems

This work proposes a general dual boundary element method (DBEM) for cohesive mixed-mode crack problems. Local cohesive stiffnesses are defined to yield the boundary element nonlinear equations for any mixed-mode cohesive model in a direct way. The main novelty resides on the fact that the formulation can take into account all possible cohesive surface conditions: (i) contact, (ii) softening, (iii) unloading/reloading and (iv) complete failure, which may occur independently of the chosen cohesive model. Besides, the well-established Park–Paulino–Roesler (PPR) model was incorporated into DBEM methods. The tangent matrix of the discrete model is explicitly derived and the solution is sought by a proper degree-of-freedom (DOF) control method. The proposed formulation is validated reproducing mode I and mode II patch-tests. Some other relevant problems, including a mixed-mode fracture test, are also presented to demonstrate that the above conditions are essential to reproduce complex mixed-mode cohesive fractures. The results also shown that the control of a monotonically increasing DOF yields reliable solutions in problems with snap-back instabilities. In order to facilitate the validation of the method, predefined crack paths were assumed in the examples. Further developments may include incorporating arbitrary crack growth and damage-based cohesive models for fatigue crack growth analysis.

Reliability-based Fracture Analysis for Shallow Shell Structure with the Dual Boundary Element Method

Reliability analysis has gained prominence as a stochastic approach to incorporate uncertainties in structural analysis. This study presents a novel methodology for evaluating the sensitivity of the crack tip stress intensity factor in a shallow shell structure. The work focuses on the reliability analysis of a shallow shell structure containing a crack originating from the corner of a fuselage window. The analysis incorporates uncertainties in both geometrical and loading parameters. The sensitivity of the crack tip stress intensity factors with respect to the considered uncertainties is determined. The Implicit Differentiation Method (IDM)-based First-Order Reliability Method (FORM) is utilized, and the results are compared to the results obtained from Monte Carlo Simulation (MCS), with a maximum difference of 2.99%. The reliability analysis aids in determining an appropriate inspection crack size that satisfies safety requirements, consequently facilitating the selection of the corresponding inspection technique.

Dual boundary element method for comparative studies on fatigue crack growth models

Fatigue crack growth studies require models that accurately predict component life with low uncertainty. Despite the large number of proposed models, there is no clarity on their applicability, which justifies a comparative analysis between some of them. The dual boundary element method (DBEM) was applied for cracked bodies, whereby the stress intensity factors (SIF), the growth rate, and the number of cycles were computed. Three crack increment models were studied under constant amplitude fatigue loads: the Paris, the Klesnil-Lucas, and the Forman models. Results were validated with experimental literature and through the finite element method, indicating that each model represents a specific zone of the crack growth curve. Klesnil-Lucas model reproduces the region near the fracture threshold, Paris fits the controlled crack growth zone, whereas Forman’s model recreates the unstable fracture zone, i.e., when the stress intensity factor approaches the material’s fracture toughness. The J-integral with stress field decomposition gave errors below 0.8% for mode I. Results were similar for the propagation path and the number of cycles to those obtained with the finite element method, with errors of about 3% considering different K-effective approaches. Klesnil-Lucas accurately predicts the number of cycles with an error margin below 3%, considering the curved region in the growth rate at the propagation onset, while the Paris model becomes very conservative, predicting values up to 50% lower than experimental data. The Klesnil-Lukas model is advised for simulating the entire crack propagation.

Analysis of three-dimensional cracks horizontally placed in transversely isotropic FGMs under interior compressive stresses

This paper develops the dual boundary element method (DBEM) using a new fundamental singular solution. This new solution is the elastic fields of transversely isotropic multilayered materials of infinite extent under point concentrated loads. The discretization technique is utilized to approximate arbitrary variations of functionally graded materials (FGMs) with transverse isotropy in a graded direction. The proposed DBEM is used to analyze the three-dimensional crack problems in FGM halfspaces. The penny-shaped and elliptical cracks are horizontally placed on the interface between the FGM layer and homogeneous halfspace and are subjected to uniform compressive stresses on crack surfaces. The stress intensity factors are provided to understand the fracture behaviors of transversely isotropic FGMs. An influential factor is defined to describe the influence of the anisotropy of the FGMs on stress intensity factors. Results show that the anisotropy and heterogeneity of materials exert an obvious influence on the fracture properties of crack problems.

Optimization of parameters of the OWC wave energy converter device using MLP and XGBoost models

In the present work, we have studied the performance of a breakwater-integrated quarter-circle-shaped front wall OWC device under the influence of irregular incident waves. Firstly, the boundary value problem associated with the hydrodynamics of OWC device is handled for solution using the dual boundary element method (BEM). To examine the complex relationships between all input features and the target variable in a time-efficient manner, supervised machine learning models are developed. Here, two different models: (i) multilayer perceptron (MLP) model based on an artificial neural network, and (ii) a tree ensemble model, namely the XGBoost model are developed. The submergence depth of the front wall of the OWC device, chamber length, rotational speed, and diameter of the turbine blade are considered as input attributes, and the average annual power generated by the OWC device is considered as the output attribute. The MLP model is employed to optimize these input parameters, leveraging the insights provided by the XGBoost model to maximize the annual average power generation. From the dual BEM based numerical results, and using the Latin hypercube sampling technique, 3750 samples were generated to train, validate, and test the machine learning models. Using the XGBoost model with the support of accumulated local effect plots, we find four specific regions of the input space in which the annual average power extraction will be maximum. Hereafter, an extended input database is generated with twenty equally spaced levels for each parameter and the dataset is passed through the developed MLP model to find the optimized values of the parameters of the OWC device which maximizes the power generation. It is obtained that the maximum power generation is attained for y0/h=−0.65, r/h=3, 2.8≤D≤3 and 70≤N≤80∪105≤N≤116.

Methodology for simulating the elastoplastic behavior in crack propagation problems using dual formulations of the boundary element method

Due to the application of cyclic loads, the existing crack length in a structure increases with time, resulting in greater stress concentration around the crack tip and therefore causing an increase in propagation velocity and a decrease in residual strength of the structure which, at a given time, becomes so low that the structure can no longer support the service loads. This paper presents a new approach to evaluate two-dimensional elastoplastic models in a crack propagation scenario, based on the boundary element method and its dual formulations. The methodology adopted consists of two stages: firstly, the simulation of the elastoplastic behavior, which considers the process of initial stresses and allows the treatment of several flow criteria, without taking into account the incompressibility of inelastic deformations. In this step, the domain integral due to non-homogeneous terms in the plastification region is transformed into a boundary integral by the Dual Reciprocity Method (DRM) using the polyharmonic splines functions due to its local behavior and, from the plastic calculus, the J-Integral is used to compute the SIF's. The second stage aims to simulate, in an incremental way, the crack propagation path using homemade software for crack modeling and analysis based on ElastoPlastic Fracture Mechanics (EPFM) and the Dual Boundary Element Method (DBEM). To validate the adopted methodology, as well as correctly simulate the mechanical behavior of cracks in the plastic regime of the material, three 2D models with straight and inclined cracks are used. Initially, the von Mises criterion for perfectly plastic materials is used and the numerical results compared with classic models from the literature, showing the effectiveness of the program in crack treatment and in the prediction of the propagation path, as well as the non-linear algorithm in the region plastification using the DRM.

Numerical instability and its treatment for steady state heat conduction problems in exchanger tubes using the dual boundary element method

In this paper, we employed the boundary element method (BEM) to determine the heat conduction shape factor (CSF) for the steady-state heat conduction problem in the heat exchanger tube. Various doubly-connected geometric shapes were considered. Numerical results of the conduction shape factor matched well with the theoretical data. Numerical instability was found for a special size. Degenerate scale occurred for all cases of regular N-gons in determining the conduction shape factor and matched the data as analytically predicted. Numerical results showed that a degenerate scale is only dependent on the outer boundary instead of the inner boundary. Three regularization techniques, the dual BEM, the CLEEF(UT), and the CLEEF(LM), were provided to avoid numerical instability. It was found that the larger number of sides for the N-gon shape is given, the lower the CSF is obtained. As N approaches infinity, it converges to the minimum value of the solution for an annulus.

Application of quadratic pressure drop condition in hydroelastic analysis of floating structures protected by perforated barrier

This paper presents the application of the quadratic pressure drop condition in hydroelastic analysis of large floating structures protected by perforated barrier. An iterative procedure is proposed to incorporate the quadratic pressure drop condition into the recently developed finite element – dual boundary element method. Numerical examples are carried out to compare the proposed quadratic model with the existing model involving the conventional linear pressure drop condition. It is found that the proposed quadratic model agrees rather well with the linear model for small wave amplitudes. However, for strong waves, there are significant differences between the quadratic model and linear model in terms of the resulting excitation force on the barrier and hydroelastic responses of floating structures. The excitation force from the quadratic model may be up to 100% higher than that from the linear model.

Surrogate modeling and global sensitivity analysis for the simultaneous growth of multiple hydraulic fractures

Hydraulic fracturing has been widely used in enhancing the conductivity of the rock formations and increasing the rate of extraction of unconventional resources. During hydraulic fracturing treatments, there are considerable uncertainties in the completion operations and measurement of the geomechanical parameters such as the perforation geometry and elastic modulus of the rock matrix, which may result in a large error in fracture pattern prediction and parameter optimization when deterministic models are implemented. Therefore, a good understanding of the impact of the input uncertainties on the output predictions, also referred to as uncertainty quantification, has become a major issue. In this paper, a non-intrusive stochastic model is developed by combining a fully coupled hydraulic fracturing model based on the dual boundary element method with a surrogate model using polynomial chaos expansion. The perforation geometry, mechanical parameters of the rock formation, orientation and magnitude of the in-situ stress are treated as random variables. The surrogate model is constructed using data from the hydraulic fracturing model and is capable of providing fast approximations of the objectives, thereby making Monte Carlo simulations and sensitivity studies feasible. The stochastic model is validated against the random responses predicted by semi-analytical solutions for the single fracture case and shows excellent agreements with a relative error less than 0.1% for the injection pressure and 1% for the maximum aperture. The impact of the geomechanical and geometric parameter randomness on the growth of multiple fractures is investigated. The results show that the randomness in the fracture toughness and Young’s modulus plays an important role in fracture growth among geomechanical parameters, and the uncertainty of the fracture pattern is mostly attributed to the randomness in the initial length and aperture of the inner fracture among geometric uncertainties.

Mathematical modelling of nonlinear pressure drops in arbitrarily shaped port utilizing dual boundary element method

A mathematical model is developed to analyze the wave-trapping characteristics of different types of porous and non-porous barriers in arbitrarily shaped domains utilizing the iterative Dual Boundary Element Method (DBEM) under the resonance conditions. In this analysis, Vertical, Wavy, Inverted V-shaped, and Inverted L-shaped barriers are utilized to investigate the effect of incident waves over the boundary of the harbor. The harbor is divided into an interconnected domain with variable bathymetry, and the Helmholtz equation is solved in each subdomain utilizing the prescribed boundary conditions. Further, the semi-empirical discharge equation with quadratic pressure drop is considered on the porous barrier to incorporate the effect of incident wave amplitude on the energy loss. The present numerical scheme is validated with existing studies and verified with convergence analysis. Further, the mathematical model is implemented on the Visakhapatnam port at four key locations with improved performance of different types of barriers. Moreover, the influence of the partial reflection coefficient is also investigated by considering the various combination of partially reflecting boundaries of the port. The wave trapping or absorbing characteristics of porous barriers proves an efficient tool to reduce the wave resonance inside the port.

Statistical Inference of Equivalent Initial Flaw Size Distribution for Fatigue Analysis of an Anisotropic Material

A novel methodology for the fatigue life uncertainty quantification of anisotropic structures is presented in this work. The concept of the equivalent initial flaw size distribution (EIFSD) is employed to overcome the difficulties in small cracks detection and fatigue prediction. This EIFSD concept is combined with the dual boundary element method (DBEM) to provide an efficient methodology for modelling the fatigue crack growth. Bayesian inference is used to infer the EIFSD based on inspection data from the routine maintenance of the structure, simulated with the DBEM. A large amount of DBEM simulations were required for the Bayesian inference. Therefore, surrogate models are used as part of the inference to further improve computational efficiency. A numerical example featuring an anisotropic plate is investigated for demonstrating the proposed methodology. When considering a low level of uncertainty in the crack propagation parameters, an error of 0.12% was found between the estimated fatigue life obtained using the proposed method compared to actual fatigue life, and only 0.35% error when considering high level of uncertainty. The application of the estimated fatigue life can be used to determine an appropriate inspection interval for aircraft maintenance.

Wave attenuation by inclined porous plates using dual boundary element method

In this study, the wave attenuation by the inclined porous plates has been evaluated through a numerical approach. As a numerical method, we used the dual boundary element method (DBEM) that solves two distinct boundary integral equations (singular and hyper-singular) derived at the coincident source points on each side of the degenerate plate boundary. A quadratic velocity model with the porosity-dependent drag coefficient is adopted to account for energy dissipation through the porous plate. The developed DBEM tool is validated through the comparison with self-conducted experiments at a two-dimensional wave tank. Then it is used to assess the reflection coefficients and wave forces with various design parameters such as porosities, inclined angles, plate arrangements, and wave characteristics. The dual inclined porous plates arranged in either a serial or parallel configuration did not enhance the wave-absorbing efficiency significantly, compared to a single inclined porous plate with optimal values of porosity (P = 0.1) and inclined angle (θ = 10°).

Finite element - Dual boundary element method for hydroelastic analysis of very large floating structures protected by perforated barrier

This paper presents a Finite Element – Dual Boundary Element Method (FE-DBEM) for hydroelastic analysis of very large floating structures protected by vertical perforated barrier. The structure and barrier are modelled based on the shell theory, while the linear potential flow theory is employed for modelling wave structure interactions. The algebraic equations of the fluid-structure interaction are obtained by using the Finite Element Method (FEM) and Dual Boundary Element Method (DBEM). In DBEM, an approximation method is proposed for calculating the hypersingular integrals associated with the perforated barrier. The proposed method is validated by comparing with published experimental and mathematical results. As compared to the previously proposed Finite Element – Multi-Domain Boundary Element Method, the method presented in this paper has an advantage that the fluid domain does not need to be decomposed into sub-domains in the formulation. In addition, the same mesh of elements can be adopted in the DBEM and FEM; and thus, these methods can be coupled easily for fluid-structure-interaction analysis.