Showing papers in "Engineering Analysis With Boundary Elements in 2006"
TL;DR: The basic concept and main procedures in the FMM for solving boundary integral equations are described in detail using the 2D potential problem as an example and the structure of a fast multipole BEM program is presented and the source code is made available that can help the development of fast multipoles BEM codes for solving other problems.
Abstract: The fast multipole method (FMM) has been regarded as one of the top 10 algorithms in scientific computing that were developed in the 20th century. Combined with the FMM, the boundary element method (BEM) can now solve large-scale problems with several million degrees of freedom on a desktop computer within hours. This opened up a wide range of applications for the BEM that has been hindered for many years by the lack of efficiencies in the solution process, although it has been regarded as superb in the modeling stage. However, understanding the fast multipole BEM is even more difficult as compared with the conventional BEM, because of the added complexities and different approaches in both FMM formulations and implementations. This paper is an introduction to the fast multipole BEM for potential problems, which is aimed to overcome this hurdle for people who are familiar with the conventional BEM and want to learn and adopt the fast multipole approach. The basic concept and main procedures in the FMM for solving boundary integral equations are described in detail using the 2D potential problem as an example. The structure of a fast multipole BEM program is presented and the source code is also made available that can help the development of fast multipole BEM codes for solving other problems. Numerical examples are presented to further demonstrate the efficiency, accuracy and potentials of the fast multipole BEM for solving large-scale problems.
199 citations
TL;DR: It is found that Gaussian elimination can be used reliably to solve the MFS equations and the use of the singular value decomposition shows no improvement overGaussian elimination provided that the boundary condition is non-noisy.
Abstract: In this paper, we consider the accuracy and stability of implementing the method of fundamental solutions. In contrast to the results shown in [5], we find that Gaussian elimination can be used reliably to solve the MFS equations and the use of the singular value decomposition shows no improvement over Gaussian elimination provided that the boundary condition is non-noisy. However, for noisy boundary conditions, there is evidence that the singular value decomposition with truncation is more accurate than Gaussian elimination.
153 citations
TL;DR: In this paper, the authors prove asymptotic feasibility for a generalized variant using separated trial and test spaces, and a greedy variation of this technique is provided, allowing a fully adaptive matrix-free and data-dependent meshless selection of the test and trial spaces.
Abstract: Though the technique introduced by Kansa is very successful in engineering applications, there were no proven results so far on the unsymmetric meshless collocation method for solving PDE boundary value problems in strong form. While the original method cannot be proven to be fail-safe in general, we prove asymptotic feasibility for a generalized variant using separated trial and test spaces. Furthermore, a greedy variation of this technique is provided, allowing a fully adaptive matrix-free and data-dependent meshless selection of the test and trial spaces.
146 citations
TL;DR: In this paper, a meshless method, based on coupling virtual boundary collocation method (VBCM) with the radial basis functions (RBF) and the analog equation method (AEM), was proposed to analyze generalized linear or nonlinear Poisson-type problems.
Abstract: This paper presents a meshless method, based on coupling virtual boundary collocation method (VBCM) with the radial basis functions (RBF) and the analog equation method (AEM), to analyze generalized linear or nonlinear Poisson-type problems. In this method, the AEM is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator, instead of other complicated ones which are needed in conventional BEM, can be employed. While global RBF is used to approximate fictitious body force which appears when the analog equation method is introduced, and VBCM are utilized to solve homogeneous solution based on the superposition principle. As a result, a new meshless method is developed for solving nonlinear Poisson-type problems. Finally, some numerical experiments are implemented to verify the efficiency of the proposed method and numerical results are in good agreement with the analytical ones. It appears that the proposed meshless method is very effective for nonlinear Poisson-type problems.
91 citations
TL;DR: In this article, the authors used a recently developed upgrade of the classical meshless Kansa method for solution of the transient heat transport in direct-chill casting of aluminium alloys.
Abstract: This paper uses a recently developed upgrade of the classical meshless Kansa method for solution of the transient heat transport in direct-chill casting of aluminium alloys. The problem is characterised by a moving mushy domain between the solid and the liquid phase and a moving starting bottom block that emerges from the mould during the process. The solution of the thermal field is based on the mixture continuum formulation. The growth of the domain and the movement of the bottom block are described by activation of additional nodes and by the movement of the boundary nodes through the computational domain, respectively. The domain and boundary of interest are divided into overlapping influence areas. On each of them, the fields are represented by the multiquadrics radial basis function collocation on a related sub-set of nodes. Time stepping is performed in an explicit way. The governing equation is solved in its strong form, i.e. no integrations are performed. The polygonisation is not present and the method is practically independent of the problem dimension. Realistic boundary conditions and temperature variation of material properties are included. An axisymmetric transient test case solution is shown at different times and its accuracy is verified by comparison with the reference finite volume method results.
81 citations
TL;DR: In this article, the meshless local Petrov-Galerkin (MLPG) method is used to solve stationary and transient heat conduction inverse problems in 2D and 3D axisymmetric bodies.
Abstract: The meshless local Petrov–Galerkin (MLPG) method is used to solve stationary and transient heat conduction inverse problems in 2-D and 3-D axisymmetric bodies. A 3-D axisymmetric body is generated by rotating a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduce the original 3-D boundary value problem to a 2-D problem. The analyzed domain is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIEs) on the boundaries of the chosen subdomains. The time integration schemes are formulated based on the Laplace transform technique and the time difference approach, respectively. The local integral equations are non-singular and take a very simple form. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. Singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.
77 citations
TL;DR: In this article, a meshless solution of the Reddy higher-order shell theory was proposed for static and free vibration analysis of laminated composite shells, based on the asymmetric global multiquadric radial basis function method proposed by Kansa.
Abstract: The higher-order shear deformation theory of laminated orthotropic elastic shells of Reddy accounts for parabolic distribution of the transverse shear strains through the thickness of the shell. The Reddy shell theory allows the fulfillment of homogeneous conditions (zero values) at the top and bottom surfaces of the shell. This paper deals with a meshless solution of the Reddy higher order shell theory in static and free vibration analysis. The meshless technique is based on the asymmetric global multiquadric radial basis function method proposed by Kansa. This paper demonstrates that this truly meshless method is very successful in the static and free vibration analysis of laminated composite shells.
71 citations
TL;DR: In this paper, a regularized meshless method (RMM) is developed to solve the two-dimensional Laplace problem with multiply-connected domain and the solution is represented by using the double-layer potential.
Abstract: In this paper, the regularized meshless method (RMM) is developed to solve two-dimensional Laplace problem with multiply-connected domain. The solution is represented by using the double-layer potential. The source points can be located on the physical boundary by using the proposed technique to regularize the singularity and hypersingularity of the kernel functions. The troublesome singularity in the traditional methods is avoided and the diagonal terms of influence matrices are easily determined. The accuracy and stability of the RMM are verified in numerical experiments of the Dirichlet, Neumann, and mixed-type problems under a domain having multiple holes. The method is found to perform pretty well in comparison with the boundary element method.
60 citations
TL;DR: In this article, the null-field integral equation for a medium containing circular cavities with arbitrary radii and positions under uniformly remote shear is derived, and the solution is formulated in a manner of a semi-analytical form since error purely attributes to the truncation of Fourier series.
Abstract: In this paper, we derive the null-field integral equation for a medium containing circular cavities with arbitrary radii and positions under uniformly remote shear. To fully capture the circular geometries, separate expressions of fundamental solutions in the polar coordinate and Fourier series for boundary densities are adopted. By moving the null-field point to the boundary, singular integrals are transformed to series sums after introducing the concept of degenerate kernels. The solution is formulated in a manner of a semi-analytical form since error purely attributes to the truncation of Fourier series. The two-hole problems are revisited to demonstrate the validity of our method. The bounded-domain approaches using either displacement or stress approaches are also employed. The proposed formulation has been generalized to multiple cavities in a straightforward way without any difficulty.
55 citations
TL;DR: A domain decomposition technique that allows to express a piecewise approximation of the solution using a method of fundamental solutions applied to each subdomain and an enriched approximation whereby singular functions are used.
Abstract: In this paper we consider the application of the method of fundamental solutions to solve crack problems. These problems present difficulties, which are not only related to the intrinsic singular nature of the problem, instead they are mainly related to the impossibility in choosing appropriate point sources to write the solution as a whole. In this paper we present: (1) a domain decomposition technique that allows to express a piecewise approximation of the solution using a method of fundamental solutions applied to each subdomain; (2) an enriched approximation whereby singular functions (fully representing the singular behaviour around the cracks or other sources of boundary singularities) are used. An application of the proposed techniques to the torsion of cracked components is carried out.
51 citations
TL;DR: In this paper, the authors analyzed electrostatic deformations of rectangular, annular circular, solid circular, and elliptic micro-electromechanical systems (MEMS) by modeling them as elastic membranes.
Abstract: We analyze electrostatic deformations of rectangular, annular circular, solid circular, and elliptic micro-electromechanical systems (MEMS) by modeling them as elastic membranes. The nonlinear Poisson equation governing their deformations is solved numerically by the meshless local Petrov–Galerkin (MLPG) method. A local symmetric augmented weak formulation of the problem is introduced, and essential boundary conditions are enforced by introducing a set of Lagrange multipliers. The trial functions are constructed by using the moving least-squares approximation, and the test functions are chosen from a space of functions different from the space of trial solutions. The resulting nonlinear system of equations is solved by using the pseudoarclength continuation method. Presently computed values of the pull-in voltage and the maximum pull-in deflection for the rectangular and the circular MEMS are found to agree very well with those available in the literature; results for the elliptic MEMS are new.
TL;DR: In this paper, a simple and less-costly MLPG method using the Heaviside step function as the test function in each sub-domain avoids the need for both a domain integral, except inertial force and body force integral in the symmetric weak form, and a singular integral for analysis of elasto-dynamic deformations near a crack tip.
Abstract: A simple and less-costly MLPG method using the Heaviside step function as the test function in each sub-domain avoids the need for both a domain integral, except inertial force and body force integral in the attendant symmetric weak form, and a singular integral for analysis of elasto-dynamic deformations near a crack tip. The Newmark family of the methods is applied into the time integration scheme. A numerical example, namely, a rectangular plate with a central crack with plate edges parallel to the crack axis loaded in tension is solved by this method. The results show that the stresses near the crack tip agree well with those obtained from another MLPG method using the weight function of the moving least square approximation as a test function of the weighted residual method. Time histories of dynamic stress intensity factors (DSIF) for mode-I are determined form the computed stress fields.
TL;DR: In this article, a modified meshless local Petrov-Galerkin (MLPG) method is presented for elasticity problems using the moving least squares (MLS) approximation, which does not need a mesh for the interpolation of the solution variables or for the integration of the energy.
Abstract: A modified meshless local Petrov–Galerkin (MLPG) method is presented for elasticity problems using the moving least squares (MLS) approximation. It is a truly meshless method because it does not need a mesh for the interpolation of the solution variables or for the integration of the energy. In this paper, a simple Heaviside test function is chosen to overcome the computationally expensive problems in the MLPG method. Essential boundary conditions are imposed by using a direct interpolation method based on the MLPG method establishes equations node by node. Numerical results in several examples show that the present method yielded very accurate solutions. And the sensitivity of the method to several parameters is also studied in this paper.
TL;DR: In this paper, a boundary element solution is implemented for magnetohydrodynamic (MHD) flow problem in ducts with several geometrical cross-section with insulating walls when a uniform magnetic field is imposed perpendicular to the flow direction.
Abstract: A boundary element solution is implemented for magnetohydrodynamic (MHD) flow problem in ducts with several geometrical cross-section with insulating walls when a uniform magnetic field is imposed perpendicular to the flow direction. The coupled velocity and induced magnetic field equations are first transformed into uncoupled inhomogeneous convection–diffusion type equations. After introducing particular solutions, only the homogeneous equations are solved by using boundary element method (BEM). The fundamental solutions of the uncoupled equations themselves (convection–diffusion type) involve the Hartmann number ( M ) through exponential and modified Bessel functions. Thus, it is possible to obtain results for large values of M ( M ≤300) using only the simplest constant boundary elements. It is found that as the Hartmann number increases, boundary layer formation starts near the walls and there is a flattening tendency for both the velocity and the induced magnetic field. Also, velocity becomes uniform at the center of the duct. These are the well-known behaviours of MHD flow. The velocity and the induced magnetic field contours are graphically visualized for several values of M and for different geometries of the duct cross-section.
TL;DR: In this article, the authors describe the applications of the method of fundamental solutions (MFS) for 2D and 3D unsteady Stokes equations and present a meshless numerical method with the concept of space-time unification.
Abstract: This paper describes the applications of the method of fundamental solutions (MFS) for 2D and 3D unsteady Stokes equations. The desired solutions are represented by a series of unsteady Stokeslets, which are the time-dependent fundamental solutions of the unsteady Stokes equations. To obtain the unknown intensities of the fundamental solutions, the source points are properly located in the time–space domain and then the initial and boundary conditions at the time–space field points are collocated. In the time-marching process, the prescribed collocation procedure is applied in a time–space box with suitable time increment, thus the solutions are advanced in time. Numerical experiments of unsteady Stokes problems in 2D and 3D peanut-shaped domains with unsteady analytical solutions are carried out and the effects of time increments and source points on the solution accuracy are studied. The time evolution of history of numerical results shows good agreement with the analytical solutions, so it demonstrates that the proposed meshless numerical method with the concept of space–time unification is a promising meshless numerical scheme to solve the unsteady Stokes equations. In the spirit of the method of fundamental solutions, the present meshless method is free from numerical integrations as well as singularities in the spatial variables.
TL;DR: It is expected that the domain decomposition approach coupled to parallel implementation should prove very competitive to alternatives proposed in the literature such as fast multipole acceleration methods that require a complete re-write of traditional BEM codes.
Abstract: This paper develops a parallel domain decomposition Laplace transform BEM algorithm for the solution of large-scale transient heat conduction problems. In order to tackle large problems the original domain is decomposed into a number of sub-domains, and a Laplace transform method is utilized to avoid time marching. A procedure is described which provides a good initial guess for the domain interface temperatures, and an iteration procedure is carried out to satisfy continuity of temperature and heat flux at the domain interfaces. The decomposition procedure significantly reduces the size of any single problem for the BEM, which significantly reduces the overall storage and computational burden and renders the application of the BEM to large transient conduction problems on modest computational platforms. The procedure is readily implemented in parallel and applicable to 3D problems. Moreover, as the approach described herein readily allows adaptation and integration of traditional BEM codes, it is expected that the domain decomposition approach coupled to parallel implementation should prove very competitive to alternatives proposed in the literature such as fast multipole acceleration methods that require a complete re-write of traditional BEM codes.
TL;DR: In this paper, the authors presented analytical Green's functions for the transient heat transfer phenomena by conduction, for an unbounded medium, half-space, slab and layered formation when subjected to a point heat source.
Abstract: This paper presents analytical Green's functions for the transient heat transfer phenomena by conduction, for an unbounded medium, half-space, slab and layered formation when subjected to a point heat source. The transient heat responses generated by a spherical heat source are computed as Bessel integrals, following the transformations proposed by Sommerfeld [Sommerfeld A. Mechanics of deformable bodies. New York: Academic Press; 1950; Ewing WM, Jardetzky WS, Press F. Elastic waves in layered media. New York: McGraw-Hill; 1957]. The integrals can be modelled as discrete summations, assuming a set of sources equally spaced along the vertical direction. The expressions presented here allow the heat field inside a layered formation to be computed without fully discretizing the interior domain or boundary interfaces. The final Green's functions describe the conduction phenomenon throughout the domain, for a half-space and a slab. They can be expressed as the sum of the heat source and the surface terms. The surface terms need to satisfy the boundary conditions at the surfaces, which can be of two types: null normal fluxes or null temperatures. The Green's functions for a layered formation are obtained by adding the heat source terms and a set of surface terms, generated within each solid layer and at each interface. These surface terms are defined so as to guarantee the required boundary conditions, which are: continuity of temperatures and normal heat fluxes between layers. This formulation is verified by comparing the frequency responses obtained from the proposed approach with those where a double-space Fourier transformation along the horizontal directions [Tadeu A, Antonio J, Simoes N. 2.5D Green's functions in the frequency domain for heat conduction problems in unbounded, half-space, slab and layered media. CMES: Computer Model Eng Sci 2004;6(1):43–58] is used. In addition, time domain solutions were compared with the analytical solutions that are known for the case of an unbounded medium, a half-space and a slab.
TL;DR: In this article, a boundary method for Helmholtz eigenproblems in simply and multiply connected domains is presented based on mathematically modelling the physical response of a system to external excitation over a range of frequencies.
Abstract: In this paper a new boundary method for Helmholtz eigenproblems in simply and multiply connected domains is presented. The method is based on mathematically modelling the physical response of a system to external excitation over a range of frequencies. The response amplitudes are then used to determine the resonant frequencies. So, contrary to the traditional scheme, the method described does not involve evaluation of determinants of linear systems. The solution of an eigenvalue problem is reduced to a sequence of inhomogeneous problems with the differential operator studied. The method shows a high precision in simply and multiply connected domains. The results of the numerical experiments justifying the method are presented.
TL;DR: In this article, a 3D BE model for the analysis of sediment effects on dynamic response of dam-reservoir systems and harbor structures is presented, and the results show the importance of a realistic representation of sediment and the influence of their consolidation degree, compressibility and permeability on the system dynamic response.
Abstract: Summary Sediment materials play an important role on the dynamic response of large structures where fluid–soil-structure interaction is relevant and materials of that kind are present. Dam-reservoir systems and harbor structures are examples of civil engineering constructions where those effects are significant. In those cases the dynamic response is determined by hydrodynamic water pressure, which depends on the absorption effects of bottom sediments. Sediments of very different mechanical properties may exist on the bottom. A three-dimensional BE model for the analysis of sediment effects on dynamic response of those structures is presented in this paper. One of the most extended models for sediment materials corresponds to Biot's fluid-filled poroelastic solid. The BE formulation for dynamics of poroelastic solids is reviewed including a weighted residual formulation more general and concise than those previously existing in literature. Systems consisting of water, other pressure wave propagating materials, viscoelastic solids and fluid-filled poroelastic zones, are studied. Coupling conditions at interfaces are taken into account in a rigorous way. A simple geometry coupled problem is first studied to asses the effects of sediments on its dynamic response and to determine the influence of parameters such as sediment depth, consolidation, compressibility and permeability. A fully 3D arch dam-reservoir-foundation system where sediments and radiation damping play an important role is also studied in this paper. Obtained results show the importance of a realistic representation of sediments and the influence of their consolidation degree, compressibility and permeability on the system dynamic response.
TL;DR: In this article, a new indirect radial-basis function (RBF) collocation method for numerically solving biharmonic boundary-value problems is reported. But this method is not suitable for the case of discretizing the governing differential equations on a set of scattered points.
Abstract: This paper reports a new indirect radial-basis-function (RBF) collocation method for numerically solving biharmonic boundary-value problems. The RBF approximations are constructed through an integration process. The new feature here is that integration constants are eliminated pointwise using the prescribed boundary conditions rather than employed as network weights. This treatment of integration constants is particularly well suited for the case of discretizing the governing differential equations on a set of scattered points. The proposed method is verified through the solution of benchmark test problems. Accurate results and high convergence rates are obtained.
TL;DR: In this article, a domain decomposition mesh-less approach for convecting fully viscous incompressible fluid interacting with conducting solids is presented. But it is not suitable for parallel computations.
Abstract: We develop an effective domain decomposition meshless methodology for conjugate heat transfer problems modeled by convecting fully viscous incompressible fluid interacting with conducting solids. The meshless formulation for fluid flow modeling is based on a radial basis function interpolation using Hardy inverse Multiquadrics and a time-progression decoupling of the equations using a Helmholtz potential. The domain decomposition approach effectively reduces the conditioning numbers of the resulting algebraic systems, arising from convective and conduction modeling, while increasing efficiency of the solution process and decreasing memory requirements. Moreover, the domain decomposition approach is ideally suited for parallel computation. Numerical examples are presented to validate the approach by comparing the meshless solutions to finite volume method (FVM) solutions provided by a commercial CFD solver.
TL;DR: In this article, the electrostatic properties of thin plate shaped structures relevant to the microelectro-mechanical systems (MEMS) have been computed using a nearly exact boundary element method (BEM) solver.
Abstract: The electrostatic properties of thin plate shaped structures relevant to the micro-electro-mechanical systems (MEMS) have been computed using a nearly exact boundary element method (BEM) solver. The solver uses closed form expressions for three-dimensional potential and force fields due to uniform sources/sinks distributed on finite flat surfaces. The expressions have been validated and, being analytical, have been found to be applicable throughout the physical domain. The solver has been applied to compute accurately and efficiently the charge densities on thin plate shaped conductors as used in MEMS components. We have presented results for the model problem of parallel plate capacitors and compared them with results obtained from several other BEM based solvers.
TL;DR: In this article, the basic concept and numerical implementation of a new local Petrov-Galerkin method for solving a dynamic problem are presented, which uses a radial basis function (RBF) coupled with a polynomial basis function as a trial function.
Abstract: The basic concept and numerical implementation of a new local Petrov-Galerkin method for solving a dynamic problem are presented in this paper. It uses a radial basis function (RBF) coupled with a polynomial basis function as a trial function, and uses the Heaviside function as a test function of the weighted residual method. The shape function has the Kronecker Delta properties for the trial-function-interpolation, and so no additional treatment to impose essential boundary conditions. The method does not involve any domain and singular integrals to generate the global effective stiffness matrix except for the mass and damping matrice; it only involves a regular boundary integral. It possesses a great flexibility in dealing with the numerical model of the elastic dynamic problem under various boundary conditions with arbitrary shapes. The Newmark family of methods is adopted in computation. The numerical results also show that using a multiquadrics (MQ) function with the polynomial basis function as the interpolation function can give quite accurate numerical results. The a Q and a S are investigated which are parameters of the radii of the sub-domain and influence domain, respectively.
TL;DR: In this article, a pure boundary element formulation is developed systematically for 2D initial-boundary value problems in the linear theory of transient heat conduction, where the time dependent fundamental solution of the diffusion operator is employed together with higher-order polyharmonic fundamental solutions.
Abstract: The paper deals with 2D initial-boundary value problems in the linear theory of transient heat conduction. A pure boundary element formulation is developed systematically. The time-dependent fundamental solution of the diffusion operator is employed together with higher-order polyharmonic fundamental solutions. The pseudo-initial temperature and/or heat sources density are approximated by using the triple-reciprocity formulation. All the time integrations are performed analytically in the time-marching scheme with integration within one time step and constant interpolation. The spatial discretization is reduced to boundary elements and free scattering of interior nodal points without any connectivity.
TL;DR: In this article, the influence of hydrodynamic behaviour of dielectric liquid media around a rapidly growing small spherical bubble in the process of electrical discharge machining between the tool and the workpiece on material removal from a workpiece is investigated.
Abstract: In this paper, the influence of hydrodynamic behaviour of dielectric liquid media around a rapidly growing small spherical bubble in the process of electrical discharge machining (EDM) between the tool and the workpiece on material removal from the workpiece is investigated. The tool and the workpiece are assumed as two parallel rigid boundaries with dielectric liquid between them. The boundary integral equation method is applied for numerical solution of the problem. Results illustrate the time dependent shapes of the bubble generated between the two parallel rigid boundaries due to the electrical discharge. Results also indicate that continuous growth of the gas bubble leads to a sharp drop of pressure within the bubble which results in the sharp pressure drop over the surfaces of the tool and the workpiece. This pressure drop over the surfaces of the tool and the workpiece causes expelling of the gas dissolved in the molten metal and helps the molten material to escape.
TL;DR: In this article, the analytical solution to the Rayleigh-plesset equation for a spherically symmetric oscillating bubble is extended to apply to the much more general (non-spherically-symmetric) bubble configuration.
Abstract: In this article the analytical solution to the Rayleigh–Plesset equation for a spherically symmetric oscillating bubble is extended to apply to the much more general (non-spherically symmetric) bubble configuration. An equivalent bubble radius and an equivalent bubble wall velocity are introduced in order to do so. The influence of gravity, surface tension, nearby solid walls, vapor bubbles, bubbles filled with adiabatic or isothermal gas have been considered in the model. An interesting outcome is that the equivalent bubble wall velocity is no longer the time derivative of the equivalent bubble radius. This observation can possibly explain why in various numerical and experimental observations the oscillation time of a bubble changes when compared to that of a standalone bubble; near a solid surface it increases while it decreases when the bubble is placed near a free surface. The current developed theory can be further employed to ascertain the accuracy of a numerical scheme simulating bubble dynamics in an incompressible surrounding flow approximation. An often used numerical technique to simulate such bubble dynamics is the boundary integral method (BIM).
TL;DR: In this paper, a new formulation is developed in a systematic way to solve generalized plane problems for anisotropic materials, with possible friction contact zones, as two-dimensional problems.
Abstract: It is in many cases very instructive and useful to have the possibility of treating three-dimensional problems by means of two-dimensional models. It always implies a reduction in computing cost which is particularly significant in presence of non-linearities, derived for instance from the presence of contact between the solids involved in the problem. The term generalized plane problem is adopted for a three-dimensional problem in a homogeneous linear elastic cylindrical body where strains and stresses are the same in all transversal sections. This concept covers many practical cases (for instance in the field of composites), a particular situation called generalized plane strain (strains, stresses and displacements are the same in all transversal sections) being the most frequently analyzed. In this paper, a new formulation is developed in a systematic way to solve generalized plane problems for anisotropic materials, with possible friction contact zones, as two-dimensional problems. The numerical solution of these problems is formulated by means of the boundary element method. An explicit expression of a new particular solution of the problem associated to constant body forces is introduced and applied to avoid domain integrations. Some numerical results are presented to show the performance and advantages of the formulation developed.
TL;DR: A wavelet transform based boundary element method (BEM) numerical scheme is proposed for the solution of the kinematics equation of the velocity-vorticity formulation of Navier-Stokes equations.
Abstract: A wavelet transform based boundary element method (BEM) numerical scheme is proposed for the solution of the kinematics equation of the velocity-vorticity formulation of Navier–Stokes equations. FEM is used to solve the kinetics equations. The proposed numerical approach is used to perform two-dimensional vorticity transfer based large eddy simulation on grids with 10 5 nodes. Turbulent natural convection in a differentially heated enclosure of aspect ratio 4 for Rayleigh number values Ra = 10 7 – 10 9 is simulated. Unstable boundary layer leads to the formation of eddies in the downstream parts of both vertical walls. At the lowest Rayleigh number value an oscillatory flow regime is observed, while the flow becomes increasingly irregular, non-repeating, unsymmetric and chaotic at higher Rayleigh number values. The transition to turbulence is studied with time series plots, temperature–vorticity phase diagrams and with power spectra. The enclosure is found to be only partially turbulent, what is qualitatively shown with second order statistics—Reynolds stresses, turbulent kinetic energy, turbulent heat fluxes and temperature variance. Heat transfer is studied via the average Nusselt number value, its time series and its relationship to the Rayleigh number value.
TL;DR: In this paper, a mixed time-harmonic boundary element procedure for the analysis of two-dimensional dynamic problems in cracked solids of general anisotropy is presented, where the fundamental solution is split into the static singular part plus dynamic regular terms.
Abstract: A mixed time-harmonic boundary element procedure for the analysis of two-dimensional dynamic problems in cracked solids of general anisotropy is presented. To the author's knowledge, no previous BE approach for time-harmonic two-dimensional crack problems in anisotropic solids exists. In the present work, the fundamental solution is split into the static singular part plus dynamic regular terms. Hypersingular integrals associated to the singular part in the traction boundary integral equation are transformed, by means of a simple change of variable, into regular ones plus very simple singular integrals with known analytical solution. Subsequently, only regular (frequency dependent) terms have to be added to the regularized static fundamental solution in order to solve the dynamic problem. The generality of this procedure permits the use of general straight or curved quadratic boundary elements. In particular, discontinuous quarter-point elements are used to represent the crack-tip behavior. Stress intensity factors are accurately computed from the nodal crack opening displacements at discontinuous quarter-point elements. The efficiency and robustness of the present time-harmonic BEM are verified numerically by several test examples. Results are also obtained for more complex configurations, not previously studied in the literature. They include curved crack geometry.
TL;DR: The results obtained by polynomial PICM show the presented schemes possess a considerable perfect stability and good numerical accuracy even for scattered models while matrix triangularization algorithm (MTA) adopted in the computed procedure.
Abstract: This paper presents a truly meshless method for solving partial differential equations based on point interpolation collocation method (PICM). This method is different from the previous Galerkin-based point interpolation method (PIM) investigated in the papers [G.R. Liu, (2002), mesh free methods, Moving beyond the Finite Element Method, CRC Press. G.R. Liu, Y.T. Gu, A point interpolation method for two-dimension solids, Int J Numer Methods Eng, 50, 937–951, 2001. G.R. Liu, Y.T. Gu, A matrix triangularization algorithm for point interpolation method, in Proceedings Asia-Pacific Vibration Conference, Bangchun Weng Ed., November, Hangzhou, People's Republic of China, 2001a, 1151–1154. 1–3.], because it is based on collocation scheme. In the paper, polynomial basis functions have been used. In addition, Hermite-type interpolations called as inconsistent PIM has been adopted to solve PDEs with Neumann boundary conditions so that the accuracy of the solution can be improved. Several examples were numerically analysed. These examples were applied to solve 1D and 2D partial differential equations including linear and non-linear in order to test the accuracy and efficiency of the presented method based on polynomial basis functions. The h-convergence rates were computed for the PICM based on different model of regular and irregular nodes. The results obtained by polynomial PICM show the presented schemes possess a considerable perfect stability and good numerical accuracy even for scattered models while matrix triangularization algorithm (MTA) adopted in the computed procedure.