Boundary Element Applications Research Articles

Meshfree reduced order model for turbomachinery blade flutter analysis

The paper introduces a development and application of boundary element based meshfree reduced order model for subsonic flutter analysis of turbomachinery blade cascade. The prime objective of the present work is to demonstrate the efficient use of meshfree hybrid boundary element method for subsonic flutter analysis of low pressure stage steam turbine blades . The meshfree incompressible fluid solver is developed employing 3D inviscid panel method . To model the flow separation inviscid-PM is hybridized with integral boundary layer approach using inviscid-viscous coupling mechanism along with discrete vortex particle method. A 3D annular test cascade is modeled using proposed meshfree solver for the flutter analysis. The comparison of estimated pressure coefficient and aerodynamic damping highlights a good agreement with experimental data and mesh based CFD-model’s results. Moreover, it have observed that the computational time for flutter analysis using proposed method is significantly lower compared to traditionally used CFD based numerical models for subsonic flutter in 3D blade cascade.

NUMERICAL SIMULATION OF WAVE PROCESSES IN PARTIALLY SATURATED POROELASTIC MEDIA (REVIEW)

The review of a number of scientific works in the field of modeling wave processes in partially saturated porous media is proposed. Review consists of three parts. In the first part of overview, classification of porous media according to the degree of saturation with a wetting filler is stated. The fundamental works of the modern theory of porous continuums, which resulted in the Biot's theory and the mixture theory, are considered. The advantages, disadvantages, as well as the conditions for the correspondence between the two approaches are indicated. The second part is devoted to scientific publications of an applied and academic nature, which are mostly devoted to the study of the features of wave processes in poroelastic media by analytical and semi-analytical methods. Special attention is paid to the works devoted to the modeling of slow longitudinal waves as a distinctive feature of wave processes in partially saturated poroelastic media. The third part gives an overview of the application of boundary element and finite element methods to study the stress-strain state of saturated and partially saturated porous media, the behavior of waves during their propagation in the media under consideration, and the analysis of local dynamic effects. The two methods are compared. Experimental works confirming theoretical results obtained within the Biot's theory are indicated. A brief analysis of modern achievements in the problems of numerical simulation of dynamic effects in semi-infinite porous media with a layered structure such as the seabed is carried out. Works on the derivation of fundamental solutions for systems of poroelasticity equations and the construction of the corresponding integral equations in time and space of the Fourier and Laplace integral transformations are considered. Works are given that consider various aspects of boundary element modeling of unsaturated porous media and the application of boundary and finite element methods on specific examples.

Construction and application of adaptive semi-Infinite boundary element with dynamic problems on half-plane

For thedynamic problem ofhalf-plane, if the radiation condition at the semi-infinite boundary is not taken into account in the numerical calculation, the accuracy of the result will be affected.In this paper, the basic theory of time-domain boundary element method (TD-BEM) and the propagation characteristics of stress waves in elastic media are used to transform a semi-infinite boundary into a semi-infinite boundary element which can adjust the size of the element automatically with time-space parameters.Enablingthe element to simulate radiative damping effects in the far field.At last, the efficiency of theelement is verified with a half-plane example under dynamic load by comparing its results with the results of the finite element method (FEM). Theverificationshows thatthe adaptive semi-infinite element can effectively simulate the radiation conditions in the far area. And it is convenient to use TD-BEM to solve the half-plane dynamics problem.

An interaction integral and a modified crack closure integral for evaluating piezoelectric crack-tip fracture parameters in BEM

To evaluate the crack-tip field intensity factors of a piezoelectric crack with any inclined angle, the current widely-used interaction integral method (I-integral) is here extended to the boundary element applications under some coordinate transformations. As well, a new modified crack closure integral method (MCCI) is proposed by considering the discontinuous quarter-point singular elements for the crack-face discretization arising from the dual boundary element method (BEM). This dual BEM involves the strongly singular displacement boundary integral equations (BIEs) for the external boundary and the hypersingular traction BIEs for the crack faces. The crack-tip fracture parameters evaluated by the I-integral and MCCI are verified by the existing analytical solutions and meanwhile, compared with those results achieved by the classical displacement extrapolation method and the J-integral. Three examples are presented to show the high accuracy of the interaction integral method and the improvement of MCCI for the piezoelectric crack problems.

BOUNDARY ELEMENT METHOD MODYFICATIONS FOR USE IN SOME IMPEDANCE AND OPTICAL TOMOGRAPHY APPLICATIONS

The article presents two elements associated with the practice of application of the boundary element method. The first is associated with BEM ability to analyze an open boundary objects and application of infinite boundary elements in the area of mammography. The second element is associated with the damped wall investigations. Wall humidity and moisture represents heterogeneous environment (Functionally Graded Materials) which has to be treated in a special way.

Modification of the boundary integral equation for the Navier–Lamé equations in 2D elastoplastic problems and its numerical solving

SummaryThe paper presents the generalization of the modification of classical boundary integral equation and obtaining parametric integral equation system for 2D elastoplastic problems. The modification was made to obtain such equations for which numerical solving does not require application of finite or boundary elements. This was achieved through the use of curves and surfaces for modeling introduced at the stage of analytical modification of the classic boundary integral equation. For approximation of plastic strains the Lagrange polynomials with various number and arrangement of interpolation nodes were used. Reliability of the modification was verified on examples with analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.

THE FMM-IBEM SOLUTION OF THREE-DIMENSIONAL SCATTERING OF ELASTIC WAVES

Boundary element method has unique advantages to solve elastic wave scattering problems, while the asymmetric and dense matrix equation greatly restrict the application of traditional boundary element in practical engineering. Therefore, based on the single-layer potential theory, combined with the fast multipole expansion technique, a new rapid indirect boundary element method is developed by using the Taylor series expansion of compressible and shear wave potential function for solving large scale elastic wave scattering problems efficiently and accurately. It is verified that this method greatly improves the computing speed and reduces the calculation storage. Then the elastic wave scattering problem for millions of DOFs can be solved effectively and accurately on an ordinary personal workstation. Finally, the scattering problem of P wave by ellipsoidal cavities group in elastic full-space is solved, and some important characteristics for the displacement and stress fields in steady state are discussed based on the numerical results. This method is especially suitable for solving the multiple scattering problems at low dimensionless frequency (ka<5.0).

A combined application of boundary-element and Runge-Kutta methods in three-dimensional elasticity and poroelasticity

The report presents the development of the time-boundary element methodology and a description of the related software based on a stepped method of numerical inversion of the integral Laplace transform in combination with a family of Runge-Kutta methods for analyzing 3-D mixed initial boundary-value problems of the dynamics of inhomogeneous elastic and poro-elastic bodies. The results of the numerical investigation are presented. The investigation methodology is based on direct-approach boundary integral equations of 3-D isotropic linear theories of elasticity and poroelasticity in Laplace transforms. Poroelastic media are described using Biot models with four and five base functions. With the help of the boundary-element method, solutions in time are obtained, using the stepped method of numerically inverting Laplace transform on the nodes of Runge-Kutta methods. The boundary-element method is used in combination with the collocation method, local element-by-element approximation based on the matched interpolation model. The results of analyzing wave problems of the effect of a non-stationary force on elastic and poroelastic finite bodies, a poroelastic half-space (also with a fictitious boundary) and a layered half-space weakened by a cavity, and a half-space with a trench are presented. Excitation of a slow wave in a poroelastic medium is studied, using the stepped BEM-scheme on the nodes of Runge-Kutta methods.

A Boundary Element Application of Plastic Hinges Development in a Circular Ring

Circular ring is the most popular specimen in industry. The location of a crack in a circular ring can strongly affect the sequence of plastic hinge development which in turn affects stability of a circular ring which has been obtained theoretically in Young [1] but not any numerically. Boundary element method is employed to investigate these effects. In fact, the ring will collapse if the number of plastic hinges is up to four. The results show the potential application of the boundary element method in this project.

The Application of Boundary Element on Pressure Transient of Carbonat Reservoir

ABSTACT: The effect of the shape of reservoir boundaries on the well-bore pressure response is evident. The conventional analytical methods can only be used to calculate well-bore pressure response in regularly shaped reservoirs. A mathematical model for well test interpretation of arbitrary shaped triple porosity reservoir was established and transformed to modified Helmholtz equation in this paper. By using the boundary element method and Duhamel principle, well-bore pressure with effects of skins and well-bore storage is obtained. The type curves are plotted and analyzed considering effects of the dimension fracture storage parameter, the inter-porosity flow parameter, arbitrary shape and impermeable barries. At last, the boundary element method is verified against the analytical solution and is applicable to transient pressure analysis of arbitary shaped reservoir. KETWORDS: carbonate reservoir, triple porosity, boundary element method, transient pressure, type curve

On NGF Applications to LBIE Potential and Displacement Discontinuity Analyses

This work aims at extending the concept of the Numerical Green's Function (NGF), known from boundary element applications to potential and fracture mechanics problems, to the Local Boundary Integral Equation (LBIE) context. As a "companion" solution, the NGF is used to remove the integrals of the main discontinuities over the crack boundary and is to be introduced only for source points whose support touches or contains the actual crack surfaces.

An hpr-mesh refinement algorithm for BEM

This paper describes a mesh refinement scheme for boundary element method in which the number of elements, the size of elements, element end locations and the element polynomial order are determined to meet the user specified accuracy. The use of grading function in conjunction with the L 1 norm makes the mesh refinement scheme applicable to a variety of boundary element formulation and applications. The algorithm is stable for smooth, discontinuous, as well as singular density functions. Numerical results for mathematical test functions as well as for elastostatic problems demonstrate the viability and versatility of the algorithm for BEM applications.

Chebyshev acceleration of criticality calculations in boundary element applications to neutron diffusion

Abstract In this study, Chebyshev polynomial method has been applied in the acceleration of the outer iterations for the numerical solution of the two dimensional neutron diffusion equation by the boundary element method. Boundary element discretization of the diffusion equation results in a full and nonsymmetric coefficient matrix which is quite different than the sparse and symmetrical matrices of the finite element and finite difference methods to which Chebyshev acceleration has been classically applied. To assess the merit of the Chebyshev method in the case of boundary element discretization of the multigroup diffusion equation constitutes the main objective of the study. Numerical experimentation has established that the Chebyshev acceleration is effective also in case of boundary element discretization despite the dissimilarity of the coefficient matrices compared to the more conventional methods.

The dual reciprocity method applied to free vibrations of 2D structures using compact supported radial basis functions

This paper presents a new boundary element application for free vibration analysis of 2D elastic structures. The dual reciprocity method is applied using four compact supported radial basis functions for approximating the domain inertia terms. The eigen-problem of displacement is then solved considering the traction contribution by means of static condensation. The formulation is also extended to consider additional internal nodes to improve accuracy. Three numerical problems are studied to demonstrate the validity and accuracy of the developed formulation. The results are compared to those obtained from analytical and other numerical solutions. A parametric study is set up to demonstrate the effect of the compact support radius on the final results and on the sparsity of system matrices.

Lamb mode conversion at edges. A hybrid boundary element–finite-element solution

Two general and flexible numerical techniques based on the finite-element and boundary element methods developed by the authors in a previous paper are applied to study Lamb wave propagation in multilayered plates and Lamb mode conversion at free edges for frequencies beyond the first cutoff frequency. Both techniques are supported by a meshing criterion which guarantees the accuracy of the results when a condition is fulfilled. A finite-element formulation is directly applicable to study Lamb wave propagation and reflection by simple obstacles such as a flat edge. In order to tackle Lamb wave diffraction problems by defects with more complex geometries, a hybrid boundary element-finite-element formulation is used. This technique provides a major improvement with respect to the only previous boundary element application on Lamb waves: the connecting boundary might be placed as close to the reflector as desired, reducing greatly the requirement on mesh size. Two main application problems on practical metallic plates are studied and compared with reported numerical, theoretical, and experimental results: (1) Lamb wave propagation in degraded titanium diffusion bonds, and (2) Lamb mode conversion at inclined or perpendicular free edges of steel plates for frequencies beyond the first cutoff frequency.

Hierarchical LU Decomposition-based Preconditioners for BEM

The adaptive cross approximation method can be used to efficiently approximate stiffness matrices arising from boundary element applications by hierarchical matrices. In this article an approximative LU decomposition in the same format is presented which can be used for preconditioning the resulting coefficient matrices efficiently. If the LU decomposition is computed with high precision, it may even be used as a direct yet efficient solver.

ℋ2-matrices – Multilevel methods for the approximation of integral operators

Multigrid methods are typically used to solve partial differential equations, i.e., they approximate the inverse of the corresponding partial differential operators. At least for elliptic PDEs, this inverse can be expressed in the form of an integral operator by Green's theorem. This implies that multigrid methods approximate certain integral operators, so it is straightforward to look for variants of multigrid methods that can be used to approximate more general integral operators. ?2-matrices combine a multigrid-like structure with ideas from panel clustering algorithms in order to provide a very efficient method for discretizing and evaluating the integral operators found, e.g., in boundary element applications.

Adaptive Geometrically Balanced Clustering of H-Matrices

In [8], a class of (data-sparse) hierarchical (*-) matrices is introduced that can be used to efficiently assemble and store stiffness matrices arising in boundary element applications. In this paper, we develop and analyse modifications in the construction of an *-matrix that will allow an efficient application to problems involving adaptive mesh refinement. In particular, we present a new clustering algorithm such that, when an *-matrix has to be updated due to some adaptive grid refinement, the majority of the previously assembled matrix entries can be kept whereas only a few new entries resulting from the refinement have to be computed. We provide an efficient implementation of the necessary updates and prove for the resulting *-matrix that the storage requirements as well as the complexity of the matrix-vector multiplication are almost linear, i.e., *.

A time-dependent Green's function-based model for stream-unconfined aquifer flows

A numerical formulation that is based on the Green element method (GEM), which incorporates a time-dependent Green's function, is used to solve transient two-dimensional flows of stream-unconfined aquifer interaction. The Green's function comes from the fundamental solution to the linear diffusion differential operator in two spatial dimensions. In classical boundary element applications, this Green's function has found use primarily in linear heat transfer and flow problems; its use here for the nonlinear stream-unconfined aquifer flow problem represents the computational flexibility that is achieved with a Green element sense of implementing the singular integral theory. The nonlinear discretised element equations obtained from numerical calculations are linearised by the Picard and Newton-Raphson methods, while the global coefficient matrix, which is banded and sparse, is readily amenable to matrix solution routines. Using four numerical examples, the accuracy of the current formulation is assessed as against an earlier one that incorporates the Logarithmic fundamental solution. It is observed that comparable accuracy is achieved between both formulations, indicating that the current formulation is a viable numerical solution strategy for the stream-aquifer flow problem. Water SA Vol.29(3) 2003: 273-280

Integration over simplexes for accurate domain and boundary integral evaluation in boundary element methods

The accuracy of boundary element methods is heavily dependent on the accurate determination of singular domain and boundary integrals on element domains. Many methods have been developed for accurate integral determination but none that is universally applicable across the range of boundary element applications. This paper is concerned with a semi-analytical method that applies to simplexes of arbitrary dimension. The method involves the careful employment of multiple integration where the inner integral is performed along a radial direction. Evaluation of the radial-inner integral on a simplex of dimension n provides n+1 integrals on simplex domains of dimension n−1. It is shown in the paper that provided closed form solutions exist for the inner integrals the method can be repeated on these n+1 simplexes providing a recursive method of integration. Performing radial inner integration on a simplex is shown to be ideal for the correct treatment of the radial singularities present. If a closed form solution is not available then a radial function is added and subtracted to facilitate the use of numerical integration. In addition to singularity annihilation the form of this radial function is selected to facilitate the continued recursive application of radial integration. The method is tested on domain and boundary integrals present in thermoelastostatic, elastodynamic and on simple test problems for which known analytical solutions exist. The results obtained using the semi-analytical simplex approach are shown to be considerably more accurate than those obtained using standard quadrature methods.