Showing papers in "Engineering Analysis With Boundary Elements in 2015"
TL;DR: In this paper, the authors proposed a numerical method for the solution of time fractional nonlinear sine-Gordon equation that appears extensively in classical lattice dynamics in the continuum media limit and Klein-Gordon equations which arises in physics.
Abstract: In this paper, we propose a numerical method for the solution of time fractional nonlinear sine-Gordon equation that appears extensively in classical lattice dynamics in the continuum media limit and Klein–Gordon equation which arises in physics. In this method we first approximate the time fractional derivative of the mentioned equations by a scheme of order O ( τ 3 − α ) , 1 α 2 then we will use the Kansa approach to approximate the spatial derivatives. We solve the two-dimensional version of these equations using the method presented in this paper on different domains such as rectangular and non-rectangular domains. Also, we prove the unconditional stability and convergence of the time discrete scheme. We show that convergence order of the time discrete scheme is O ( τ ) . We solve these fractional PDEs on different non-rectangular domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear time fractional PDEs. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.
110 citations
TL;DR: The method of approximate particular solutions (MAPS) as mentioned in this paper is an alternative radial basis function (RBF) meshless method, which is defined in terms of a linear combination of the particular solutions of the inhomogeneous governing equations with traditional RBFs as the source term.
Abstract: The method of approximate particular solutions (MAPS) is an alternative radial basis function (RBF) meshless method, which is defined in terms of a linear combination of the particular solutions of the inhomogeneous governing equations with traditional RBFs as the source term. In this paper, we apply the MAPS to both constant- and variable-order time fractional diffusion models. In the discretization formulation, a finite difference scheme and the MAPS are used respectively to discretize time fractional derivative and spatial derivative terms. Numerical investigation examples show the present meshless scheme has highly accuracy and computationally efficiency for various fractional diffusion models.
105 citations
TL;DR: In this article, an element-free based improved moving least squares-Ritz (IMLSRitz) method is proposed to study the buckling behavior of functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs).
Abstract: An element-free based improved moving least squares-Ritz (IMLS-Ritz) method is proposed to study the buckling behavior of functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) resting on Winkler foundations. The first-order shear deformation theory (FSDT) is employed to account for the effect of shear deformation of plates. The IMLS is used for construction of the two-dimensional displacement field. We derive the energy functional for moderately thick plates. By minimizing the energy functional via the Ritz method, solutions for the critical buckling load of the functionally graded carbon nanotube (FG–CNT) reinforced composite plates on elastic matrix are obtained. Numerical experiments are carried out to examine the effect of the Winkler modulus parameter on the critical buckling loads. The influences of boundary condition, plate thickness-to-width ratio, plate aspect ratio on the critical buckling loads are also investigated. It is found that FG–CNT reinforced composite plates with top and bottom surfaces of CNT-rich have the highest critical buckling loads.
103 citations
TL;DR: In this paper, a meshless local Petrov-Galerkin weak form and moving least squares (MLS) approximation is proposed to simulate 3D nonlinear wave equation with Dirichlet boundary conditions.
Abstract: This paper proposes an approach based on the Galerkin weak form and moving least squares (MLS) approximation to simulate three space dimensional nonlinear wave equation of the form u tt + α u t + β u = u xx + u yy + u zz + δ g ( u ) u t + f ( x , y , z , t ) , 0 x , y , z 1 , t > 0 subject to given appropriate initial and Dirichlet boundary conditions. The main difficulty of methods in fully three-dimensional problems is the large computational costs. In the proposed method, which is a kind of Meshless local Petrov–Galerkin (MLPG) method, meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The moving least squares approximation is proposed to construct shape functions. A two-step time discretization method is employed to approximate the time derivatives. To treat the nonlinearity, a kind of predictor–corrector scheme combined with one-step time discretization and Crank–Nicolson technique is adopted. Several numerical examples are presented and satisfactory agreements are achieved.
94 citations
TL;DR: In this paper, a spectral meshless radial point interpolation (SMRPI) technique is proposed and, as a test problem, applied to the two-dimensional diffusion equation with an integral (non-classical) condition.
Abstract: In this article, a new spectral meshless radial point interpolation (SMRPI) technique is proposed and, as a test problem, applied to the two-dimensional diffusion equation with an integral (non-classical) condition. This innovative method is based on erudite combination of meshless methods and spectral collocation techniques but it is not traditional meshless collocation method. As the meshless method, the point interpolation method with the help of radial basis functions is used to construct shape functions as basis functions in the frame of spectral collocation methods. These basis functions (shape functions) have Kronecker delta function property. In the proposed method, high order derivatives (in high order partial differential equations) are evaluated by constructing and using operational matrices. In SMRPI method, it does not require any kind of integration locally over small quadrature domains as it is essential in Galerkin weak form meshless methods such as element free Galerkin (EFG) and meshless local Petrov–Galerkin (MLPG) methods. Also, it is not needed to determine shape parameter which is often played important role in these methods, for instance recall test (weight functions) for moving least squares (MLS) approximation in the frame of MLPG. Therefore, computational costs of SMRPI method is less expensive. A comparison study of the efficiency and accuracy of the present method and meshless local Petrov–Galerkin (MLPG) method is given by applying on mentioned diffusion equation. Convergence studies in the numerical examples show that SMRPI method possesses reasonable rate of convergence.
81 citations
TL;DR: In this paper, an element-free computational framework based on the improved moving least squares Ritz (IMLS-Ritz) method is first explored for solving two-dimensional elastodynamic problems.
Abstract: An element-free computational framework based on the improved moving least-squares Ritz (IMLS-Ritz) method is first explored for solving two-dimensional elastodynamic problems. Employing the IMLS approximation for the field variables, discretized governing equations of the problem are derived via the Ritz procedure. Using the IMLS approximation, an orthogonal function system with a weight function is employed to construct the two-dimensional displacement fields. The resulting algebraic equation system from the IMLS-Ritz algorithm is solved without a matrix inversion. Numerical time integration for the dynamic problems is performed using the Newmark-β method. The involved essential boundary conditions are imposed through the penalty method. To examine the numerical stability of the IMLS-Ritz method, convergence studies are carried out by considering the influences of support sizes, number of nodes and time steps involved. The applicability of the IMLS-Ritz method is demonstrated through solving a few selected examples and its accuracy is validated by comparing the present results with the available solutions.
73 citations
TL;DR: In this paper, a contour integral method is used to convert the nonlinear eigenproblems caused by the boundary element method into ordinary eigen-problems, and all fictitious eigenfrequencies corresponding to the related interior problem are observed.
Abstract: This paper is concerned with the fictitious eigenfrequency problem of the boundary integral equation methods when solving exterior acoustic problems. A contour integral method is used to convert the nonlinear eigenproblems caused by the boundary element method into ordinary eigenproblems. Since both real and complex eigenvalues can be extracted by using the contour integral method, it enables us to investigate the fictitious eigenfrequency problem in a new way rather than comparing the accuracy of numerical solutions or the condition numbers of boundary element coefficient matrices. The interior and exterior acoustic fields of a sphere with both Dirichlet and Neumann boundary conditions are taken as numerical examples. The pulsating sphere example is studied and all fictitious eigenfrequencies corresponding to the related interior problem are observed. The reasons are given for the usual absence of many fictitious eigenfrequencies in the literature. Fictitious eigenfrequency phenomena of the Kirchhoff–Helmholtz boundary integral equation, its normal derivative formulation and the Burton–Miller formulation are investigated through the eigenvalue analysis. The actual effect of the Burton–Miller formulation on fictitious eigenfrequencies is revealed and the optimal choice of the coupling parameter is confirmed.
69 citations
TL;DR: In this paper, the authors apply two different meshless methods based on radial basis functions (RBFs) for numerical solution of the Cahn-Hilliard (CH) equation in one, two and three dimensions.
Abstract: The present paper is devoted to the numerical solution of the Cahn–Hilliard (CH) equation in one, two and three-dimensions. We will apply two different meshless methods based on radial basis functions (RBFs). The first method is globally radial basis functions (GRBFs) and the second method is based on radial basis functions differential quadrature (RBFs-DQ) idea. In RBFs-DQ, the derivative value of function with respect to a point is directly approximated by a linear combination of all functional values in the global domain. The main aim of this method is the determination of weight coefficients. GRBFs replace the function approximation into the partial differential equation directly. Also, the coefficients matrix which arises from GRBFs is very ill-conditioned. The use of RBFs-DQ leads to the improvement of the ill-conditioning of interpolation matrix RBFs. The boundary conditions of the mentioned problem are Neumann. Thus, we use DQ method directly on the boundary conditions, which easily implements RBFs-DQ on the irregular points and regions. Here, we concentrate on Multiquadrics ( MQ ) as a radial function for approximating the solution of the mentioned equation. As we know this radial function depends on a constant parameter called shape parameter. The RBFs-DQ can be implemented in a parallel environment to reduce the computational time. Moreover, to obtain the error of two techniques with respect to the spatial domain, a predictor–corrector scheme will be applied. Finally, the numerical results show that the proposed methods are appropriate to solve the one, two and three-dimensional Cahn-Hilliard (CH) equations.
62 citations
TL;DR: In this paper, a meshless Local Radial Basis Function Collocation Method (LRBFCM) was proposed for the solution of coupled heat transfer and fluid flow problems with a free surface.
Abstract: This paper explores the application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) for the solution of coupled heat transfer and fluid flow problems with a free surface. The method employs the representation of temperature, velocity and pressure fields on overlapping five-noded sub-domains through collocation by using Radial Basis Functions (RBFs). This simple representation is then used to compute the first and second derivatives of the fields from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time integration scheme. For numerical efficiency, the Artificial Compressibility Method (ACM) with Characteristic Based Split (CBS) technique is firstly adopted to solve the pressure–velocity coupled equations. The performance of the method is assessed based on solving the classical two-dimensional De Vahl Davis steady natural convection benchmark problem with an upper free surface for Rayleigh number ranged from 103 to 105 and Prandtl number equals to 0.71.
57 citations
TL;DR: In this paper, a meshless local Petrov-Galerkin weak form and moving least squares (MLS) approximation is proposed to construct shape functions for a two-dimensional time-fractional telegraph equation defined by Caputo sense for (1 α 2 ).
Abstract: In this paper, a classical type of two-dimensional time-fractional telegraph equation defined by Caputo sense for ( 1 α 2 ) is analyzed by an approach based on the Galerkin weak form and moving least squares (MLS) approximation subject to given appropriate initial and Dirichlet boundary conditions. In the proposed method, which is a kind of the Meshless local Petrov–Galerkin (MLPG) method, meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares. The moving least squares approximation is proposed to construct shape functions. Two numerical examples are presented and satisfactory agreements are achieved.
52 citations
TL;DR: In this article, the authors solved the one and two-dimensional time-dependent coupled sine-Gordon equations using RBFs collocation and RBF-QR methods and showed how one can overcome the ill-conditioning of coefficient matrix for the small shape parameters using RBF -QR method.
Abstract: Radial basis function (RBF) approximation is an extremely powerful tool for solving various types of partial differential equations, since the method is meshless and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. In this paper, the authors solve the one and two-dimensional time-dependent coupled sine-Gordon equations using RBFs collocation and RBF-QR methods and show how one can overcome the ill-conditioning of coefficient matrix for the small shape parameters using RBF-QR method. The main aim of the current paper is to show that the meshless techniques based on the collocation methods are also suitable for solving the system of coupled nonlinear equations especially sine-Gordon equation. Several test problems are employed and results of numerical experiments are presented and also are compared with analytical solutions. The obtained results confirm the acceptable accuracy of the new methods.
TL;DR: In this paper, an implementation of the kp-Ritz method with the nonlocal continuum model as an element-free computational framework has been performed to investigate the free vibration behavior of a single-layered graphene sheet (SLGS).
Abstract: In this paper, an implementation of the kp-Ritz method with the nonlocal continuum model as an element-free computational framework has been performed to investigate the free vibration behavior of a single-layered graphene sheet (SLGS). The nonlocal continuum model, which combines the Eringen nonlocal constitutive equation with the classical plate theory, has the ability to take the small scale effect into account. The study has shown that the element-free kp-Ritz method is an efficient approach for solving nonlocal continuum model for the SLGS vibration. The accuracy of the kp-Ritz method has been validated through comparison with the MD computation. The values of nonlocal parameter used in the study have been derived by matching the frequency of nonlocal continuum model with that of the MD model for different sizes and boundary conditions. The simulated results have illustrated that the value of nonlocal parameter depends on the sizes and boundary conditions of SLGS.
TL;DR: Three-dimensional selective smoothed finite element method with edge-based and node-based strain smoothing techniques for nonlinear anisotropic large deformation analyses of nearly incompressible cardiovascular tissues and outperforms the standard FEM and other S-FEMs.
Abstract: This paper presents a three-dimensional selective smoothed finite element method with edge-based and node-based strain smoothing techniques (3D-ES/NS-FEM) for nonlinear anisotropic large deformation analyses of nearly incompressible cardiovascular tissues. 3D-ES/NS-FEM owns several superior advantages, such as the robustness against the element distortions and superior computational efficiency, etc. To simulate the large deformation experienced by cardiovascular tissues, the static and explicit dynamic 3D-ES/NS-FEMs are derived correspondingly. Performance contest results show that 3D-ES/NS-FEM-T4 outperforms the standard FEM and other S-FEMs. Furthermore, this 3D-ES/NS-FEM-T4 is applied to analyze intact common carotid artery undergo mean blood pressure and passive inflation of anatomical rabbit bi-ventricles. The results are validated with the reference solutions, and also demonstrate that present 3D-ES/NS-FEM-T4 is a powerful and efficient numerical tool to simulate the large deformation of anisotropic tissues in cardiovascular systems.
TL;DR: In this article, the problem of heat transfer in the 3D domain of heating tissue is described by dual-phase lag equation supplemented by adequate boundary and initial conditions, and the general boundary element method is proposed.
Abstract: Heat transfer processes proceeding in the 3D domain of heating tissue are discussed. The problem is described by dual-phase lag equation supplemented by adequate boundary and initial conditions. To solve the problem the general boundary element method is proposed. The examples of computations are presented in the final part of the paper. The efficiency and exactness of the algorithm proposed are discussed and the conclusions are also formulated.
TL;DR: In this paper, a local radial basis functions (LRBF) collocation method is proposed for solving the (Patlak-) Keller-Segel model, where the Crank-Nicolson difference scheme is used to obtain a finite difference scheme with respect to the time variable.
Abstract: In this paper local radial basis functions (LRBFs) collocation method is proposed for solving the (Patlak-) Keller–Segel model. We use the Crank–Nicolson difference scheme for the time derivative to obtain a finite difference scheme with respect to the time variable for the Keller–Segel model. Then we use the local radial basis functions (LRBFs) collocation method to approximate the spatial derivative. We obtain the numerical results for the mentioned model. As we know, recently some approaches presented for preventing the blow up of the cell density. In the current paper we use the multiquadric (MQ) radial basis function. The aim of this paper is to show that the meshless methods based on the local RBFs collocation approach are also suitable for solving models that have the blow up of the cell density. Also, six test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.
TL;DR: In this article, a simple accurate formula is presented to evaluate the origin intensity factor of the singular boundary method (SBM) for two-dimensional Dirichlet potential problems, which is considered as an improved version of the method of fundamental solutions and remedies the controversial auxiliary boundary outside the computational domain in the latter.
Abstract: In this work, a simple accurate formula is presented to evaluate the origin intensity factor of the singular boundary method (SBM) for two-dimensional Dirichlet potential problems The SBM is considered as an improved version of the method of fundamental solutions and remedies the controversial auxiliary boundary outside the computational domain in the latter The origin intensity factor is a central concept in the SBM to overcome the source singularity of the fundamental solution while placing source points on the physical boundary In literature, the origin intensity factor for the Dirichlet boundary condition is numerically obtained which may cause numerical instability in large-scale simulations This work proposes a simple formula to calculate the origin intensity factor for two-dimensional Dirichlet potential problems Numerical experiments show that it is feasible and perform robustly for problems under various irregular domains
TL;DR: In this paper, a numerical technique is proposed for solving the stochastic advection-diffusion equations, which directly simulates the noise terms at the collocation points in each time step.
Abstract: In this paper, a numerical technique is proposed for solving the stochastic advection–diffusion equations. Firstly, using the finite difference scheme, we transform the stochastic advection–diffusion equations into elliptic stochastic partial differential equations (SPDEs). Then the method of radial basis functions (RBFs) based on pseudospectral (PS) approach has been used to approximate the resulting elliptic SPDEs. In this study, we have used generalized inverse multiquadrics (GIMQ) RBFs, to approximate functions in the presented method. The main advantage of the proposed method over traditional numerical approaches is directly simulating the noise terms at the collocation points in each time step. To confirm the accuracy of the new approach and to show the performance of the selected RBFs, four examples are presented in one, two and three dimensions in regular and irregular domains. For test problems the statistical moments such as mean, variance and standard deviation are computed.
TL;DR: In this article, the effect of porosity in attenuating surface gravity wave scattering and trapping by bottom-standing and surface-piercing porous structures of finite width in two-layer fluid is analyzed based on the linearized water wave theory in water of uniform depth.
Abstract: The present study deals with oblique surface gravity wave scattering and trapping by bottom-standing and surface-piercing porous structures of finite width in two-layer fluid. The problems are analyzed based on the linearized water wave theory in water of uniform depth. Both the cases of interface piercing and non-piercing structures are considered to analyze the effect of porosity in attenuating waves in surface and internal modes. Eigenfunction expansion method is used to deal with wave past porous structures in two-layer fluid assuming that the associated eigenvalues are distinct. Further, the problems are analyzed using boundary element method and results are compared with the analytic solution derived based on the eigenfunction expansion method. Efficiency of the structures of various configuration and geometry on scattering and trapping of surface waves are studied by analyzing the reflection and transmission coefficients for waves in surface and internal modes, free surface and interface elevations, wave loads on the structure and rigid wall. The present study will be of significant importance in the design of various types of coastal structures used in the marine environment for reflection and dissipation of wave energy at continental shelves dominated by stratified fluid which is modeled here as a two-layer fluid.
TL;DR: In this paper, a coupled smoothed finite element method (S-FEM) is developed to deal with the structural-acoustic problems consisting of a shell configuration interacting with the fluid medium.
Abstract: In this paper, a coupled smoothed finite element method (S-FEM) is developed to deal with the structural-acoustic problems consisting of a shell configuration interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements that can be generated automatically for any complicated geometries are adopted to discretize the problem domain. A gradient smoothing technique (GST) is introduced to perform the strain smoothing operation. The discretized system equations are obtained using the smoothed Galerkin weakform, and the numerical integration is applied over the further formed edge-based and face-based smoothing domains, respectively. To extend the edge-based smoothing operation from plate structure to shell structure, an edge coordinate system is defined local on the edges of the triangular element. Numerical examples of a cylinder cavity attached to a flexible shell and an automobile passenger compartment have been conducted to illustrate the effectiveness and accuracy of the coupled S-FEM for structural-acoustic problems.
TL;DR: In this paper, a high-order implicit technique is associated with a meshless method to model material mixing observed in friction stir welding (FSW) process, which allows obtaining very large time steps and reducing computation time by minimizing the number of tangent matrix decompositions.
Abstract: In the present work, a high order implicit technique is associated with a meshless method to model material mixing observed in friction stir welding (FSW) process. This new algorithm combines the following mathematical procedures: a time discretization, a space discretization, a homotopy transformation, a perturbation technique and a continuation method. The perturbation technique, after a time discretization, a space discretization, a homotopy transformation, transforms the nonlinear problem into a sequence of linear ones at each time. By comparison to the classical iterative algorithms, the proposed one allows obtaining very large time steps and reducing computation time by minimizing the number of tangent matrix decompositions. The strong formulation is considered to avoid the drawback of numerical integration. The resulting algorithm is well adapted to large deformations in the mixing zone nearly the welding tool. We limit ourselves to bidimensional visco-plastic problems to show the performance of the proposed algorithm by comparison to the classical incremental iterative methods.
TL;DR: In this paper, the authors proposed an optimum design method for two-dimensional heat conduction problem with heat transfer boundary condition based on the boundary element method (BEM) and the topology optimization method.
Abstract: This paper proposes an optimum design method for two-dimensional heat conduction problem with heat transfer boundary condition based on the boundary element method (BEM) and the topology optimization method. The level set method is used to represent the structural boundaries and the boundary mesh is generated based on iso-surface of the level set function. A major novel aspect of this paper is that the governing equation is solved without ersatz material approach and approximated heat convection boundary condition by using the mesh generation. Additionally, the objective functional is defined also on the design boundaries. First, the topology optimization method and the level set method are briefly discussed. Using the level set based boundary expression, the topology optimization problem for the heat transfer problem with heat transfer boundary condition is formulated. Next, the topological derivative of the objective functional is derived. Finally, several numerical examples are provided to confirm the validity of the derived topological derivative and the proposed optimum design method.
TL;DR: The interpolating Galerkin boundary node method (IGBNM) as discussed by the authors combines an improved interpolating moving least square (IIMLS) scheme and a variational formulation of boundary integral equations.
Abstract: Combining an improved interpolating moving least-square (IIMLS) scheme and a variational formulation of boundary integral equations, a symmetric and boundary-only meshless method, which is called the interpolating Galerkin boundary node method (IGBNM), is developed in this paper for 2D and 3D Stokes flow problems. The IIMLS is used to form shape functions with delta function property. So unlike the Galerkin boundary node method (GBNM), the IGBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables. Besides, to obtain uniqueness of unknown boundary functions and to retain symmetry of system matrices, a Lagrange multiplier is introduced and then a variational formulation with side conditions is gained. Consequently, in the IGBNM, boundary conditions can be applied directly and easily, and the resulting system matrices are symmetric. Thus, the IGBNM gives greater computational precision than the GBNM. The numerical formulae are valid for 2D and 3D Stokes flows and also valid for both interior and exterior problems simultaneously. The capability of the IGBNM is illustrated and assessed by some numerical examples.
TL;DR: In this paper, numerical solution of a two-dimensional fractional evolution equation has been investigated by using two different aspects of strong form meshless methods, namely, a time discretization approach and a numerical technique based on the convolution sum.
Abstract: In the current work, numerical solution of a two-dimensional fractional evolution equation has been investigated by using two different aspects of strong form meshless methods. In the first method a time discretization approach and a numerical technique based on the convolution sum are employed to approximate the appearing time derivative and fractional integral operator, respectively. It has been proven analytically that the time discretization scheme is unconditionally stable. Then a meshfree collocation method based on the radial basis functions is used for solving resulting time-independent discretization problem. As the second approach, a fully Kansa׳s meshfree method based on the Gaussian radial basis function is formulated and well-used directly for solving the governing problem. In this technique an explicit formula to approximate the fractional integral operator is computed. The given techniques are used to solve two examples of problem. The computed approximate solutions are reported through the tables and figures, also these results are compared together and with the other available results. The presented results demonstrate the validity, efficiency and accuracy of the formulated techniques.
TL;DR: In this paper, the authors proposed a numerical method based on the dual reciprocity boundary elements method (DRBEM) to solve the stochastic partial differential equations (SPDEs).
Abstract: This paper proposes a numerical method based on the dual reciprocity boundary elements method (DRBEM) to solve the stochastic partial differential equations (SPDEs). The concept of dual reciprocity method is used to convert the domain integral to the boundary. The conventional DRBEM starts with approximation of the source term of the original PDEs with radial basis functions (RBFs). Due to the fact that the nonhomogeneous term of SPDEs considered in this paper involves Wiener process, the traditional DRBEM cannot be applied. So a modification of it is suggested that has some advantages in comparison with the traditional DRBEM and can be developed for solving the SPDEs. The time evolution is discretized by using the finite difference method, while the modified DRBEM is proposed for spatial variations of field variables. The noise term is approximated at the collocation points at each time step. We employ the generalized inverse multiquadrics (GIMQ) RBFs to approximate functions in the presented technique. To confirm the accuracy of the new approach, several examples are employed and simulation results are reported. Also the convergence of the new technique is studied numerically.
TL;DR: In this paper, an efficient method for numerical evaluation of all kinds of singular curved boundary integrals from 2D/3D BEM analysis is proposed based on an operation technique on a projection line/plane.
Abstract: In this paper, an efficient method for numerical evaluation of all kinds of singular curved boundary integrals from 2D/3D BEM analysis is proposed based on an operation technique on a projection line/plane. Firstly, geometry variables on a curved line/surface element are expressed in terms of parameters on the projection line/plane, and then all singularities are analytically removed by expressing the non-singular part of the integration kernel as a power series in a local distance defined on the projection line/plane. Also, a set of crucial relationships computing derivatives of intrinsic coordinates with respect to local orthogonal coordinates is derived. A few examples are provided to demonstrate the correctness and the stability of the proposed method.
TL;DR: In this paper, a finite element and indirect boundary element coupling method is presented for the time-harmonic response of a laterally loaded floating pile embedded in a transversely isotropic multilayered half-space.
Abstract: A finite element and indirect boundary element coupling method is presented for the time-harmonic response of a laterally loaded floating pile embedded in a transversely isotropic multilayered half-space. The floating pile is modeled as a Bernoulli-Euler beam using the finite element method (FEM), while the soil is modeled by using an indirect boundary element method (BEM) based on the fundamental solution for a transversely isotropic multilayered half-space. Then the governing equation of the interaction between the pile and transversely isotropic multilayered half-space is deduced by coupling FEM and BEM. Numerical examples are performed to validate the presented theory and to investigate the impact degree of anisotropy and layering arrangement on the dynamic response of a pile.
TL;DR: In this article, a fully nonlinear numerical model for a floating body in the open sea is developed based on velocity potential together with a higher-order boundary element method (BEM), where the total wave elevation and the total velocity potential are separated into two parts, based on the incoming wave from infinity and the disturbed potential by the body.
Abstract: A fully nonlinear numerical model for a floating body in the open sea has been developed based on velocity potential together with a higher-order boundary element method (BEM). The total wave elevation and the total velocity potential are separated into two parts, based on the incoming wave from infinity and the disturbed potential by the body. The mesh is generated only once at the initial time and the element nodes are rearranged subsequently without changing their connectivity by using a spring analogy method. Through some auxiliary functions, the mutual dependence of fluid/structure motions are decoupled, which allows the body acceleration to be obtained without the knowledge of the pressure distribution. Numerical results are provided for forces and run-ups of a fixed cylinder with flare and the comparison is made with the second order theory in the frequency domain. Simulations are also made for a freely floating body responding to wave excitation. Resonance related to ringing excited by the high order force at the triple wave frequency is discussed. Further results are provided for motions, forces and run-ups of a floating cylinder with flare. Comparison with the results for the fixed body and body in single degree of freedom is made.
TL;DR: In this paper, a new numerical method for solving nonlinear fractional integro-differential equations is presented based upon hybrid functions approximation, and the properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented.
Abstract: In this paper, a new numerical method for solving nonlinear fractional integro-differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann–Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the nonlinear fractional integro-differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
TL;DR: In this paper, a new technique is presented for transforming the domain integral related to the source term that characterizes the Poisson Equation, within the scope of the boundary element method, for two-dimensional problems.
Abstract: In this paper a new technique is presented for transforming the domain integral related to the source term that characterizes the Poisson Equation, within the scope of the boundary element method, for two-dimensional problems. Similarly to the Dual Reciprocity Technique, the proposed scheme avoids domain discretization using primitive radial basis functions; however, it transforms the domain integral into a single boundary integral directly. The proposed procedure is simpler, more versatile and some useful and modern techniques related to radial basis function theory can be applied. Numerical tests show the accuracy of the proposed technique for a simple class of complete radial interpolation functions, pointing out the importance of internal poles and the potential of applying fitting interpolation schemes to minimize the computational storage, particularly considering more complex future approaches, in which a mass matrix may be generated. For the analysis of the accuracy and convergence of the proposed method, results are compared with those obtained using Dual Reciprocity, using known analytical solutions for reference.
TL;DR: In this paper, a direct interpolation technique that uses radial basis functions is applied to the boundary element method integral term, which refers to inertia, in the Helmholtz equation; consequently, free vibration frequencies and corresponding amplitudes can be determined from an eigenvalue problem solution.
Abstract: In the present study, a direct interpolation technique that uses radial basis functions is applied to the boundary element method integral term, which refers to inertia, in the Helmholtz equation; consequently, free vibration frequencies and corresponding amplitudes can be determined from an eigenvalue problem solution. The proposed method, which has already been successfully applied to scalar problems governed by the Poisson equation, does not require standard domain integration procedures, which employ cell discretisation, and is more robust than the dual-reciprocity technique. Although similar to the latter in some aspects, because it uses radial basis functions and their primitives for interpolation, the proposed methodology is more general. It allows the immediate use of interpolation functions of any type, and there are no convergence or monotonicity problems as the number of basis points is increased.