Dual Boundary Element Method Research Articles

Numerical analysis of the influence of crack surface roughness on the crack path

Abstract The influence of the 3D frictional crack surface interaction on the fracture mechanical parameters as well as on the crack path is numerically investigated. For the solution of the boundary value problem the 3D dual boundary element method in terms of the discontinuous formulation is utilized. This method is especially suited for contact problems because it directly deals with the discontinuities at the crack surfaces. The contact problem is solved by the application of the penalty method. Coulomb’s frictional law is utilized for the consideration of the dissipative nature of friction. For discrete steps within one load cycle the stress intensity factors are determined by an extrapolation procedure from the stress field. Based on the analysis of a load cycle, the cyclic stress intensity factors are obtained. For the simulation of crack propagation an implicit time integration scheme of a crack propagation law implemented in terms of a predictor–corrector scheme is applied. The influence of the crack surface roughness on the crack path is shown by numerical examples.

Comparison of DBEM and FEM crack path predictions in a notched shaft under torsion

Abstract The rather complex 3D fatigue crack growth behaviour of two anti-symmetric “bird wing” cracks, initiated from the two crack front corner points of a notched shaft undergoing torsion, is investigated by the Dual Boundary Element Method (DBEM) and by the Finite Element Method (FEM). Different criteria for the crack path assessment (Minimum Strain Energy Density, Maximum Principal Stress and Approximate Energy Release Rate) and for the Stress Intensity Factor (SIF) evaluation (COD and J -integral) are adopted. The SIF’s and the crack path, calculated by such different approaches, turn out to be well consistent with each other. Moreover the simulated crack path qualitatively agrees with experimental findings available from literature.

Axisymmetric and two-dimensional crack identification using boundary elements and coupled quasi-random downhill simplex algorithms

In this paper, a crack identification method is developed for axisymmetric and two-dimensional structures. The dual boundary element method is coupled with a quasi-random sequence and the downhill simplex algorithm to minimize the error function, which is expressed as the difference between strains at the boundary sensor points in the guessed crack shape and ones measured in the actual crack identity. The measures are randomly perturbed and the residual norm is regularized to provide an efficient and numerically stable algorithm that leads to stabilized solutions. The accuracy of this method is investigated in the identification of single two-dimensional and axisymmetric cracks on several cases. Through the treated examples, we showed that the quasi-random sequence presents a solid tool, allowing the choice of the optimal initial solution necessary to the convergence of the downhill simplex algorithm.

Application of Two Dimensional Extended Boundary Node Method to Potential Problem

The boundary node method has been reformulated without using any integration cells and its performance has been investigated by comparing with the dual reciprocal boundary element method (DRM). The results of computations show that the accuracy of the proposed method is superior to that of the DRM regardless of the number of a boundary node, a boundary condition and a boundary shape. In addition, when the number of boundary nodes exceeds a certain limit, the calculation speed of the proposed method becomes almost equal to that of the DRM.

Evaluation of fatigue crack growing in cortical bone using the BEM

An effective and efficient methodology for the fatigue cracks growth evaluation in human cortical bone is presented. The solution is based on the elastic linear fracture mechanics. For the evaluation of cracked boundary the dual boundary element method is used. The solution is based on the development and application of a computational algorithm to obtain stresses and displacements on the boundary and the crack-tip stress intensity factor for elements under fatigue load. For the fatigue regimen, the principles of Paris's law with cracks increments are employed. Finally, the results in cortical bone are compared with available experimental results.

CV dual reciprocity BEM for transient flow in blocky systems with singular points and lines of discontinuities

The advantages of the complex variable (CV) and dual reciprocity method (DRM) are combined to effectively solve plane problems for heat (fluid, charge) flow in blocky systems with potential, flux, and/or conductivity discontinuity at contacts and with singular points such as crack tips, common apexes of structural elements and thin inclusions. We establish that in the problems considered it is reasonable to (i) distinguish the holomorphic “quasi-steady” part of the solution, (ii) use Gauss-type radial basis functions (RBF) to simplify further the employment of a found solution of a potential problem for solving thermo- (poro-) elastic problems, (iii) locate the centers of RBF outside of the contour of integration, (iv) account for the sensitivity of the solution to the distance between the centers of RBF by taking the normalized distance as d = t * / 1 . 2 8 to have the accuracy of one percent at a normalized time t *, and account for the sensitivity to the scaling parameter a sc by taking it within the range 1.6 d< a sc <8 d to avoid growing inaccuracy of the solution for small values of a sc ( a sc <1.6 d) and growing instability for large values of a sc ( a sc >8 d), and (v) use the solution of the ordinary differential equations (ODE) with respect to approximation parameters rather than with respect to the temperature at the centers of RBF. Numerical tests and examples illustrate the accuracy and abilities of the developed technique.

Modeling of fatigue crack propagation using dual boundary element method and Gaussian Monte Carlo method

This paper studies the modeling of fatigue crack propagation on a multiple crack site of a finite plate using deterministic and probabilistic methods. Stress intensity factor has been calculated by the combined deterministic approach of the dual boundary element method (DBEM) and the probabilistic approach of the Gaussian Monte Carlo method. The Gaussian Monte Carlo method has been incorporated to simulate the random process of the fatigue crack propagation. A finite plate of aluminum alloy 2024-T3 with a thickness of 1.6 mm and 14 holes is analyzed and the fatigue life of the plate is predicted by following a linear elastic law of fracture mechanics. The results of fatigue life predicted by DBEM-Monte Carlo method are in good agreement with experimental ones. The same approach is also applied to two other engineering applications of a gear tooth and a bracket.

A fast BEM for the analysis of damaged structures with bonded piezoelectric sensors

A fast boundary element method for the analysis of three-dimensional solids with cracks and adhesively bonded piezoelectric patches, used as strain sensors, is presented. The piezoelectric sensors, as well as the adhesive layer, are modeled using a 3D state-space finite element approach. The piezoelectric patch model is formulated taking into account the full electro-mechanical coupling and embodying the suitable boundary conditions and it is eventually expressed in terms of the interface variables, to allow a straightforward coupling with the underlying host structure, which is modeled through a 3D dual boundary element method, for accurate analysis of cracks. The technique is computationally enhanced, in terms of memory storage and solution time, using the hierarchical format in conjunction with a GMRES solver. An original strategy retaining the advantages of the fast hierarchical solution without increasing the implementation complexity to take into account the piezoelectric patches is proposed for the solution of the final system. The presented work is a step towards modeling of structural health monitoring systems.

Fictitious frequency revisited

The nonexistence and nonuniqueness problems associated with integral equation methods for exterior acoustics are revisited. The Fredholm alternative theorem in conjunction with the singular value decomposition updating technique is used to simultaneously determine the fictitious frequencies and corresponding modes in exterior acoustics. After selecting the combined Helmholtz interior integral equation formulation (CHIEF) points, the influence row vectors are obtained. A criterion in selecting the minimum number of CHIEF points and their positions is proposed by testing the orthogonal condition between the influence row vector and the right unitary vector. It is proved in the discrete system for arbitrary-shape problems that the source of numerical instability of irregular frequency originates from the zero divided by zero using the generalized coordinates of unitary vectors. The mathematical structures of the four influence matrices in the dual boundary element method (BEM) are examined by using the left and right unitary matrices. Extracting the true eigenvalue and filtering out the fictitious frequency can be unified by using the updating term and updating document, respectively. Radiation problems of a cylinder and a square rod are demonstrated to see the validity of the present formulation.

A fast dual boundary element method for 3D anisotropic crack problems

AbstractIn the present paper a fast solver for dual boundary element analysis of 3D anisotropic crack problems is formulated, implemented and tested. The fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix. The admissible low rank blocks are computed by adaptive cross approximation (ACA). The performance of ACA against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated, focusing on the balance between the kernel representation accuracy and the accuracy required for ACA. The system solution is computed by a preconditioned GMRES and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. The effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated, highlighting the accuracy as well as the significant reduction in memory storage and analysis time. In particular, it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom, obtaining up to solution speedups of order 10 for systems of order 104. Moreover, the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Acceleration of 3D crack propagation simulation by the utilization of fast BEM-techniques

For the acceleration of the 3D simulation of fatigue crack propagation fast BEM techniques are combined in the present paper. The so-called dual BEM is utilized to separate the coincident crack surfaces. In the present context, the relevant boundary integral equations are evaluated in the framework of a collocation procedure. For the reduction of the numerical complexity of the stress analysis fast BEM techniques are applied. The domain is decomposed in order to obtain a band-structured system matrix. The crack is modeled in terms of the dual discontinuity method. Therewith, the relevant boundary of the domain is reduced by one crack surface. Finally, the adaptive cross approximation, an algebraic matrix compression technique, is utilized to compress the system matrix. The efficiency of this procedure is shown on complex industrial applications.

Dual Boundary Element Method Applied to Antiplane Crack Problems

This paper is concerned with an efficient dual boundary element method for 2d crack problems under antiplane shear loading. The dual equations are the displacement and the traction boundary integral equations. When the displacement equation is applied on the outer boundary and the traction equation on one of the crack surfaces, general crack problems with anti‐plane shear loading can be solved with a single region formulation. The outer boundary is discretised with continuous quadratic elements; however, only one of the crack surfaces needs to be discretised with discontinuous quadratic elements. Highly accurate results are obtained, when the stress intensity factor is evaluated with the discontinuous quarter point element method. Numerical examples are provided to demonstrate the accuracy and efficiency of the present formulation.

Three-dimensional stress intensity factors of a central square crack in a transversely isotropic cuboid with arbitrary material orientations

In this paper, we present the dual boundary element method (dual-BEM) or single-domain BEM to analyze the mixed three-dimensional (3D) stress intensity factors (SIFs) in a finite and transversely isotropic solid containing an internal square crack. The planes of both the transverse isotropy and square crack can be oriented arbitrarily with respect to a fixed global coordinate system. A set of four special nine-node quadrilateral elements are utilized to approximate the crack front as well as the outer boundary, and the mixed 3D SIFs are evaluated using the asymptotic relation between the SIFs and the relative crack opening displacements (COD) via the Barnett–Lothe tensor. Numerical examples are presented for a cracked cuboid which is transversely isotropic with any given orientation and is under a uniform vertical traction on its top and bottom surfaces. The square crack is located in the center of the cuboid but is oriented arbitrarily. Our results show that among the selected material and crack orientations, the mode-I SIF reaches the largest possible value when the material inclined angle ψ 1=45° and dig angle β 1=45°, and the crack inclined angle ψ 2=0° and dig angle β 2=0°. It is further observed that when the crack is oriented vertically or nearly vertically, the mode-I SIF becomes negative, indicating that the crack closes due to an overall compressive loading normal to the crack surface. Variation of the SIFs for modes II and III along the crack fronts also shows some interesting features for different combinations of the material and crack orientations.

A two-parameter model for crack growth simulation by combined FEM–DBEM approach

This paper describes the application of a two-parameters crack growth model, based on the usage of two threshold material parameters (Δ K th and K max,th) and on the allowance for residual stresses, introduced at the crack tip by a fatigue load spectrum or by material plastic deformations. The coupled usage of finite element method (FEM) and dual boundary element method (DBEM) is proposed in order to take advantage of the main capabilities of the two methods. The procedure is validated by comparison with available experimental results, in order to assess its capability to predict the retardation phenomena, introduced by a variable load spectrum or by a plastic deformation introduced with a tool on the panel (indentation). In particular two different tests are made: the first test involve a CT specimen undergoing a load spectrum and the second one involve a dented panel undergoing a constant amplitude fatigue load. In both cases a satisfactory numerical–experimental correlation will be proved. The main advantages of the aforementioned procedure are: the simplicity of the crack growth law calibration (few constant amplitude tests are sufficient without the need for any non-physical calibration parameters), and the possibility to simulate residual stress effects on crack propagation with a simplified approach, based on linear elastic fracture mechanics.

Non-linear MSD crack growth by DBEM for a riveted aeronautic reinforcement

A special specimen is created cutting a rectangular notched area from the surrounding of the upper left corner of a wide body aircraft door. Then a constant amplitude fatigue traction load is applied by a special servo-hydraulic machine, in order to induce a Multi Site Damage (MSD) scenario. The Dual Boundary Element method (DBEM), as implemented in a commercial code, is adopted for a three-dimensional MSD crack growth simulation of such multi-layer and multi-material component. To this aim, the cracked part of a pre-existing global two-dimensional model is extracted and “extruded” in order to generate a three-dimensional submodel, whose boundary conditions are imposed displacements, calculated by the two-dimensional model, along a virtual line corresponding to the submodel boundary. Non-linear contact conditions are applied between the mating plate surfaces in the area surrounding the cracks, in order to precisely model the plate interactions in the area of interest. The three-dimensional approach is aimed to improve, with respect to the two-dimensional approach, the correlation between numerical and experimental results (e.g. by an accurate assessment of the secondary bending effects). The obtained improvements on crack growth rates, in the initial part of the crack propagation, justify the increased computational effort that a three-dimensional non-linear approach involves. The proposed numerical procedure, based on DBEM, is successfully validated for the virtual testing of a complex aeronautic reinforcement.

Dual Boundary Element Analysis for Time-Dependent Fracture Problems in Creeping Materials

This paper presents the development of a new boundary element formulation for analysis of fracture problems in creeping materials. For the creep crack analysis the Dual Boundary Element Method (DBEM), which contains two independent integral equations, was formulated. The implementation of creep strain in the formulation is achieved through domain integrals in both boundary integral equations. The domain, where the creep phenomena takes place, is discretized into quadratic quadrilateral continuous and discontinuous cells. The creep analysis is applied to metals with secondary creep behaviour. This is con ned to standard power law creep equations. Constant applied loads are used to demonstrate time e¤ects. Numerical results are compared with solutions obtained from the Finite Element Method (FEM) and others reported in the literature.

Dynamic analysis of 3-D structures with adaptivity in RBF of dual reciprocity BEM

A new adaptive dual reciprocity boundary element method for dynamic analysis of 3-D structures is presented in this paper. It is based on finding the best approximation function of a radial basis function (RBF) group <TEX>$f=r^n+c$</TEX> which minimize the error of displacement field expansion. Also, the effects of some parameters such as the existence of internal points, number of RBF functions and position of collocation nodes in discontinuous elements are investigated in this adaptive procedure. Three numerical examples show improvement in dynamic response of structures with adaptive RBF in dual reciprocity with respect to ordinary BEM.

Influence of Temperature Gradients on the Sound radiated from Flames

The far field pressure of a turbulent flame can be determined using the standard boundary element method (BEM) if the sound pressure or its derivative is known at a closed surface (control surface) surrounding the flame, as long as the medium outside the control surface is homogeneous. If temperature gradients are present, the homogeneous Helmholtz equation is no more valid. In that case, the wave equation can be rewritten in form of an inhomogeneous Helmholtz equation with a source term that depends also on the unknown pressure. Using the "Dual Reciprocity BEM" the integral form of this wave equation can be solved involving only surface integrals, so that the sound field can still be computed from field values at the control surface. The cases under study consider a volume of hot gas with a temperature distribution that is prescribed or obtained from a CFD simulation. The influence of the temperature gradients on the sound field can be evaluated by comparison of characteristic quantities like sound power and radiation patterns, with and without temperature gradient.

A coupled numerical scheme of dual reciprocity BEM with DQM for the transient elastodynamic problems

AbstractThe two‐dimensional transient elastodynamic problems are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in spatial domain with the differential quadrature method (DQM) in time domain. The DRBEM with the fundamental solution of the Laplace equation transforms the domain integrals into the boundary integrals that contain the first‐ and the second‐order time derivative terms. Thus, the application of DRBEM to elastodynamic problems results in a system of second‐order ordinary differential equations in time. This system is then discretized by the polynomial‐based DQM with respect to time, which gives a system of linear algebraic equations after the imposition of both the boundary and the initial conditions. Therefore, the solution is obtained at any required time level at one stroke without the use of an iterative scheme and without the need of very small step size in time direction. The numerical results are visualized in terms of graphics. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Development of “event-to-event” nonlinear technique to lightly reinforced concrete beams by simplified constitutive modeling

The two-dimensional dual boundary element method is applied in this paper for the static analysis of lightly reinforced concrete beams, along with a nonlinear noniterative approach, called “event-to-event” technique. This numerical technique put some restrictions on the model development, such as continuity of the process of discrete crack growth and necessity of converting the constitutive law curves in a multi-linear form. The steel–concrete interaction is taken into account by employing a modification of a simplified bond slip model, known from the literature. A number of numerical examples are solved, analyzed and compared with results from experimental studies. A parametric study is carried out in order to reveal the specific features of the theory and to assess the potential of the numerical method, nonlinear technique and constitutive modeling employed.