Showing papers in "Engineering Analysis With Boundary Elements in 2012"

Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation

Abstract: Hardy’s multiquadric and its related interpolators have been found to be highly efficient for interpolating continuous, multivariate functions, as well as for the solution of partial differential equations. Particularly, the interpolation error can be dramatically reduced by varying the shape parameter to make the interpolator optimally flat. This improvement of accuracy is accomplished without reducing the fill distance of collocation points, that is, without the increase of computational cost. There exist a number of mathematical theories investigating the multiquadric family of radial basis functions. These theories are often not fully tested due to the computation difficulty associated with the ill-conditioning of the interpolation matrix. This paper overcomes this difficulty by utilizing arbitrary precision arithmetic in the computation. The issues investigated include conditional positive definiteness, error estimate, optimal shape parameter, traditional and effective condition numbers, round-off error, derivatives of interpolator, and the edge effect of interpolation.

An application of fast multipole method to isogeometric boundary element method for Laplace equation in two dimensions

Abstract: According to the concept of isogeometric analysis, we have developed a boundary element method (BEM) using B-spline basis functions for the two-dimensional Laplace equation, focusing on external Neumann problems. Further, we have applied the fast multipole method (FMM) to the present isogeometric BEM to reduce the computational complexity from O ( n 2 ) to O ( n ), where n is the number of control points to define the closed boundary of the computational domain. In a benchmark test, we confirmed that the FMM can accelerate the isogeometric BEM successfully. In addition, the proposed fast BEM can be an alternative of the standard fast BEM using the piecewise-constant elements. Finally, the feasibility of the proposed method for solving large-scale problems was demonstrated through numerical examples.

Meshless simulations of the two-dimensional fractional-time convection–diffusion–reaction equations

Abstract: The aim of this work is to propose a numerical approach based on the local weak formulations and finite difference scheme to solve the two-dimensional fractional-time convection–diffusion–reaction equations. The numerical studies on sensitivity analysis to parameter and convergence analysis show that our approach is stable. Moreover, numerical demonstrations are given to show that the weak-form approach is applicable to a wide range of problems; in particular, a forced-subdiffusion–convection equation previously solved by a strong-form approach with weak convection is considered. It is shown that our approach can obtain comparable simulations not only in weak convection but also in convection dominant cases. The simulations to a subdiffusion–convection–reaction equation are also presented.

An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation

Abstract: For the method of fundamental solutions (MFS), a trial solution is expressed as a linear combination of fundamental solutions. However, the accuracy of MFS is heavily dependent on the distribution of source points. Two distributions of source points are frequently adopted: one on a circle with a radius R , and another along an offset D to the boundary, where R and D are problem dependent constants. In the present paper, we propose a new method to choose the best source points, by using the MFS with multiple lengths R k for the distribution of source points, which are solved from an uncoupled system of nonlinear algebraic equations. Based on the concept of equilibrated matrix, the multiple-length R k is fully determined by the collocated points and a parameter R or D , such that the condition number of the multiple-length MFS (MLMFS) can be reduced smaller than that of the original MFS. This new technique significantly improves the accuracy of the numerical solution in several orders than the MFS with the distribution of source points using R or D . Some numerical tests for the Laplace equation confirm that the MLMFS has a good efficiency and accuracy, and the computational cost is rather cheap.

Bone tissue remodelling analysis considering a radial point interpolator meshless method

Abstract: In this work the natural neighbour radial point interpolation method (NNRPIM), an improved meshless method, is extended to the bone remodelling analysis. A biomechanical model for predicting the bone density distribution was developed. The proposed gradient remodelling algorithm considers an anisotropic material law for the mechanical behaviour of the bone tissue, based on experimental data available in the literature, allowing to gradually correlate the bone density with the obtained level of stress. The viability and efficiency of the model were successfully tested on the classical femur bone example and a novel calcaneus bone example under multiple loading conditions.

Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions

Abstract: A comparison of the performance of the global and the local radial basis function collocation meshless methods for three dimensional parabolic partial differential equations is performed in the present paper. The methods are structured with multiquadrics radial basis functions. The time-stepping is performed in a fully explicit, fully implicit and Crank–Nicolson ways. Uniform and non-uniform node arrangements have been used. A three-dimensional diffusion–reaction equation is used for testing with the Dirichlet and mixed Dirichlet–Neumann boundary conditions. The global methods result in discretization matrices with the number of unknowns equal to the number of the nodes. The local methods are in the present paper based on seven-noded influence domains, and reduce to discretization matrices with seven unknowns for each node in case of the explicit methods or a sparse matrix with the dimension of the number of the nodes and seven non-zero row entries in case of the implicit method. The performance of the methods is assessed in terms of accuracy and efficiency. The outcome of the comparison is as follows. The local methods show superior efficiency and accuracy, especially for the problems with Dirichlet boundary conditions. Global methods are efficient and accurate only in cases with small amount of nodes. For large amount of nodes, they become inefficient and run into ill-conditioning problems. Local explicit method is very accurate, however, sensitive to the node position distribution, and becomes sensitive to the shape parameter of the radial basis functions when the mixed boundary conditions are used. Performance of the local implicit method is comparatively better than the others when a larger number of nodes and mixed boundary conditions are used. The paper represents an extension of our recently made similar study in two dimensions.

A scaled boundary finite element method applied to electrostatic problems

Abstract: The scaled boundary finite element method (SBFEM) is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. In this paper, the SBFEM is firstly extended to solve electrostatic problems. Two new SBFE coordination systems are introduced. Based on Laplace equation of electrostatic field, the derivations (based on a new variational principle formulation) and solutions of SBFEM equations for both bounded domain and unbounded domain problems are expressed in details, the solution for the inclusion of prescribed potential along the side-faces of bounded domain is also presented in details, then the total charges on the side-faces can be semi-analytically solved, and a particular solution for the potential field in unbounded domain satisfying the constant external field is solved. The accuracy and efficiency of the method are illustrated by numerical examples with complicated field domains, potential singularities, inhomogeneous media and open boundaries. In comparison with analytic solution method and other numerical methods, the results show that the present method has strong ability to resolve singularity problems analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom. The method in electromagnetic field calculation can have broad application prospects.

Radial basis functions methods for solving Fokker–Planck equation

Abstract: In this paper two numerical meshless methods for solving the Fokker–Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker–Planck equation by using collocation method are applied. The first is based on the Kansa's approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and Ne errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa's approach.

A fully nonlinear implicit model for wave interactions with submerged structures in forced or free motion

Abstract: The purpose of this work is to develop advanced numerical tools for modeling two-way fully nonlinear interactions of ocean surface waves (irregular waves in the general situation) with submerged structures undergoing large amplitude motion, that could represent Wave Energy Converters (WECs). In our modeling approach, an existing two-dimensional Numerical Wave Tank (NWT), based on potential flow theory, is extended to include a submerged horizontal cylinder of arbitrary cross-section. The mathematical problem and related numerical solution are first introduced. Then, conservation of volume and conservation of energy are checked, respectively, in the case of a circular cylinder in a prescribed large amplitude motion and in the case of a circular cylinder in a free motion. Interactions between waves and a submerged circular cylinder computed by the model are then compared to mathematical solutions for two situations: a cylinder in prescribed motion and a freely moving cylinder.

Investigation on near-boundary solutions by singular boundary method

Abstract: This study is to solve the dramatic drop of numerical solution accuracy of the singular boundary method (SBM) in near-boundary regions, also known as the boundary layer effect in the literature of boundary element method (BEM), where we encounter ‘nearly singular’ interpolation functions when the field point is close to the boundary source points. It is noted that the SBM uses the singular fundamental solution as its interpolation basis function. Different from singularity at origin, the fundamental solution interpolation at near-boundary regions remains finite. However, instead of being a flat function, the interpolation function develops a sharp peak as the field point approaches the boundary, namely, nearly singular behaviors. Consequently, the evaluation of the physical quantities at this point is much less accurate than at central region points. To remove or damp out the rapid variations of this nearly singular interpolation, this paper introduces a nonlinear transformation, based on the sinh function, to remedy this boundary layer effect. Our numerical experiments verify that the proposed approach can improve the SBM near-boundary solution accuracy by several orders of magnitude in terms of relative errors. The SBM solution appears accurate at a point as close to the boundary as 1.0E–10 scale.

Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function

Abstract: In this paper, an analysis is performed to find the numerical solution of a mathematical model for capillary formation in tumor angiogenesis. Firstly, a time stepping approach is employed for the time derivative, then a meshfree process based on a global collocation method using the radial basis functions (RBFs) is applied for solving the problem. Stability analysis of the method is investigated. Because of non-availability of the exact solutions, efficiency and accuracy of the method is demonstrated, by comparison with existing methods. Also the method is successfully applied for solving the problem with high values of the cell diffusion constant, which many of the available methods are not applicable for solving these cases.

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

Abstract: This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton–Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.

A time-stepping DRBEM for the transient magneto-thermo-visco-elastic stresses in a rotating non-homogeneous anisotropic solid

Abstract: The main objective of this paper is to study the transient magneto-thermo-visco-elastic stresses in a non-homogeneous anisotropic solid placed in a constant primary magnetic field acting in the direction of the z -axis and rotating about it with a constant angular velocity. The system of fundamental equations is solved by means of a dual-reciprocity boundary element method (DRBEM). In the case of plane deformation, a numerical scheme for the implementation of the method is presented and the numerical computations are carried out for the temperature, displacement components and thermal stress components. The validity of DRBEM is examined by considering a magneto-thermo-visco-elastic solid occupies a rectangular region and good agreement is obtained with the results obtained by other methods. The results obtained are presented graphically to show the effect of inhomogeneity on the displacement components and thermal stress components.

Two-dimensional contaminant transport modeling using meshfree point collocation method (PCM)

Abstract: Groundwater contamination is a severe problem in many parts of the world including India. The complex problem of groundwater flow and contaminant transport is studied generally by solving the governing equations of flow and transport using numerical models such as finite difference method (FDM) or finite element method (FEM). Meshfree (MFree) method is an alternative numerical approach to solve these governing equations in simple and accurate manner. MFree method does not require any grid and only makes the use of a set of scattered collocation points, regardless of the connectivity information between them. Kansa (1990) [9] developed a multi-quadratic (MQ) based MFree method for the solution of partial differential equations. Based on the Kansa’s method, the present study proposes a MFree point collocation method (PCM) with multi-quadric radial basis function (MQ-RBF) for the two-dimensional coupled groundwater flow and transport simulation in unconfined conditions. The accuracy of the developed model is verified with available analytical solutions in literature. The coupled model developed is further applied to a field problem to compute the groundwater head and concentration distribution and the results are compared with available finite element based simulation. The outcomes of the model results showed the applicability of the present approach.

Meshless local Petrov–Galerkin analysis of free convection of nanofluid in a cavity with wavy side walls

Abstract: The problem of laminar natural convection of Al2O3–water nanofluid in a cavity with wavy side walls has been investigated using the meshless local Petrov–Galerkin method. The considered cavity is a square enclosure having left and right wavy side walls. The left and right vertical wavy walls of the enclosure are maintained at constant temperatures Th and Tc, respectively, with Th>Tc. The horizontal top and bottom walls of the cavity are kept insulated. To carry out the numerical simulations, the developed governing equations are determined in terms of the stream function–vorticity formulation. The weighting function in the weak formulation of the governing equations is taken as unity, and the field variables are approximated using the MLS interpolation. Capability and adaptability of the proposed meshless technique is verified by comparisons of the obtained results through the present meshless method with those existing in the literature. Two different models proposed in the literature are considered for the effective dynamic viscosity of the nanofluid. Using the developed code, a parametric study is performed incorporating the two viscosity formulas, and the effects of the Rayleigh number and the volume fraction of the nanoparticles on the fluid flow and heat transfer inside the wavy enclosure are investigated in each case. The results show that significant differences exist between the rates of heat transfer in the cavity for the two viscosity models employed. At Ra=103 the average Nusselt number of the hot wall increases with increase in the volume fraction of the nanoparticles for both considered viscosity models. At other Rayleigh numbers (Ra=104, 105, and 106) the average Nusselt number estimated for Brinkman formula increases with increase in volume fraction of the nanoparticles while it decreases for Maiga's correlation.

An efficient Wave Based Method for solving Helmholtz problems in three-dimensional bounded domains

Abstract: This paper discusses the use of the Wave Based Method for the analysis of time-harmonic three-dimensional (3D) interior acoustic problems. Conventional element-based prediction methods, such as the Finite Element Method, are most commonly used for these types of problems, but they are restricted to low-frequency applications. The Wave Based Method is an alternative deterministic technique which is based on the indirect Trefftz approach. Up to now, this method's very high computational efficiency has been illustrated mainly for two-dimensional (2D) problem settings, allowing the analysis of problems at higher frequencies. The numerical validation examples presented in this work shows that the enhanced computational efficiency of the Wave Based Method in comparison with conventional element-based methods is kept when the method is extended to 3D case with and without the presence of material damping.

Isogeometric analysis in BIE for 3-D potential problem

Abstract: The isogeometric analysis is introduced in the Boundary Integral Equation (BIE) for solution of 3-D potential problems. In the solution, B-spline basis functions are employed not only to construct the exact geometric model but also to approximate the boundary variables. And a new kind of B-spline function, i.e., local bivariate B-spline function, is deducted, which is further applied to reduce the computation cost for analysis of some special geometric models, such as a sphere, where large number of nearly singular and singular integrals will appear. Numerical tests show that the new method has good performance in both exactness and convergence.

Numerical solutions of boundary detection problems using modified collocation Trefftz method and exponentially convergent scalar homotopy algorithm

Abstract: In this paper, the boundary detection problem, which is governed by the Laplace equation, is analyzed by the modified collocation Trefftz method (MCTM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the boundary detection problem, the Cauchy data is given on part of the boundary and the Dirichlet boundary condition on the other part of the boundary, whose spatial position is unknown a priori. By adopting the MCTM, which is meshless and integral-free, the numerical solution is expressed by a linear combination of the T-complete functions of the Laplace equation. The use of a characteristic length in MCTM can stabilize the numerical procedure and ensure highly accurate solutions. Since the coefficients of MCTM and the position of part of the boundary are unknown, to collocate the boundary conditions will yield a system of nonlinear algebraic equations; the ECSHA, which is exponentially convergent, is adopted to solve the system of nonlinear algebraic equations. Several numerical examples are provided to demonstrate the ability and accuracy of the proposed meshless scheme. In addition, the consistency of the proposed scheme is validated by adding noise into the boundary conditions.

On error control in the element-free Galerkin method

Abstract: The paper investigates discretisation error control in the element-free Galerkin method (EFGM) highlighting the differences from the finite element method (FEM). We demonstrate that the (now) conventional procedures for error analysis used in the finite element method require careful application in the EFGM, otherwise competing sources of error work against each other. Examples are provided of previous works in which adopting an FEM-based approach leads to dubious refinements. The discretisation error is here split into contributions arising from an inadequate number of degrees of freedom eh, and from an inadequate basis ep. Numerical studies given in this paper show that for the EFGM the error cannot be easily split into these component parts. Furthermore, we note that arbitrarily setting the size of nodal supports (as is commonly proposed in many papers) causes severe difficulties in terms of error control and solution accuracy. While no solutions to this problem are presented in this paper it is important to highlight these difficulties in applying an approach to errors from the FEM in the EFGM. While numerical tests are performed only for the EFGM, the conclusions are applicable to other meshless methods based on the concept of nodal support.

A radial basis function approach to compute the first-passage probability density function in two-dimensional jump-diffusion models for financial and other applications

Abstract: We consider the problem of computing the survival (first-passage) probability density function of jump-diffusion models with two stochastic factors. In particular the Fokker–Planck partial integro-differential equation associated to these models is solved using a meshless collocation approach based on radial basis functions (RBF). To enhance the computational efficiency of the method, the calculation of the jump integrals is performed using a suitable Chebyshev interpolation procedure. In addition, the RBF discretization is carried out in conjunction with an ad hoc change of variables, which allows to use radial basis functions with equally spaced centers and at the same time yields an accurate resolution of the gradients of the survival probability density function near the barrier. Numerical experiments are presented showing that the RBF approach is extremely accurate and fast, and performs significantly better than the conventional finite difference method.

Gaussian RBF-FD weights and its corresponding local truncation errors

Abstract: In this work we derive analytical expressions for the weights of Gaussian RBF-FD and Gaussian RBF-HFD formulas for some differential operators. These weights are used to derive analytical expressions for the leading order approximations to the local truncation error in powers of the inter-node distance h and the shape parameter ϵ . We show that for each differential operator, there is a range of values of the shape parameter for which RBF-FD formulas and RBF-HFD formulas are significantly more accurate than the corresponding standard FD formulas. In fact, very often there is an optimal value of the shape parameter ϵ + for which the local error is zero to leading order. This value can be easily computed from the analytical expressions for the leading order approximations to the local error. Contrary to what is generally believed, this value is, to leading order, independent of the internodal distance and only dependent on the value of the function and its derivatives at the node.

Nonlinear flexural analysis of functionally graded plates under different loadings using RBF based meshless method

Abstract: The nonlinear flexural response of functionally graded plates under different transverse loading is investigated using multiquadric radial basis function (MQRBF) method. The mathematical formulation of the actual physical problem of the plate subjected to mechanical loading is presented utilizing Levinson shear deformation theory and von Karman nonlinear kinematics. These non-linear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. The effect of the volume fraction exponent on the nonlinear flexural response of simply supported and clamped plates under uniformly distributed, sinusoidal, line and point loads is studied.

Probabilistic crack growth analyses using a boundary element model: Applications in linear elastic fracture and fatigue problems

Abstract: This paper addresses the numerical solution of random crack propagation problems using the coupling boundary element method (BEM) and reliability algorithms. Crack propagation phenomenon is efficiently modelled using BEM, due to its mesh reduction features. The BEM model is based on the dual BEM formulation, in which singular and hyper-singular integral equations are adopted to construct the system of algebraic equations. Two reliability algorithms are coupled with BEM model. The first is the well known response surface method, in which local, adaptive polynomial approximations of the mechanical response are constructed in search of the design point. Different experiment designs and adaptive schemes are considered. The alternative approach direct coupling, in which the limit state function remains implicit and its gradients are calculated directly from the numerical mechanical response, is also considered. The performance of both coupling methods is compared in application to some crack propagation problems. The investigation shows that direct coupling scheme converged for all problems studied, irrespective of the problem nonlinearity. The computational cost of direct coupling has shown to be a fraction of the cost of response surface solutions, regardless of experiment design or adaptive scheme considered.

Analyzing modified equal width (MEW) wave equation using the improved element-free Galerkin method

Abstract: The element-free Galerkin (EFG) method is a promising method for solving partial differential equations in which trial and test functions employed in the discretization process result from moving least-squares (MLS) approximation. In this paper, by employing the improved moving least-squares (IMLS) approximation, we derive formulae for an improved element-free Galerkin (IEFG) method for the modified equal width (MEW) wave equation. A variation of the method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method. Therefore, the IEFG method may result a better computing speed. In this paper, the effectiveness of the IEFG method for modified equal width (MEW) wave equation is investigated by numerical examples.

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

Abstract: This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.

A new radial basis function for Helmholtz problems

Abstract: In this paper, a new radial basis function (RBF) is proposed to solve Helmholtz problems in the traditional collocation method. Since the matrix equation arising from the RBF interpolation is ill-conditioned, a regularized singular value decomposition method is used to obtain a more accurate solution. Numerical examples of both direct and inverse problems are presented to demonstrate the effectiveness and applicability of the proposed RBF versus the traditional multiquadric RBF.

Comparison of local weak and strong form meshless methods for 2-D diffusion equation

Abstract: A comparison between weak form meshless local Petrov–Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.

Simultaneous determination for a space-dependent heat source and the initial data by the MFS

Abstract: In this paper we propose a numerical algorithm based on the method of fundamental solutions for recovering a space-dependent heat source and the initial data simultaneously in an inverse heat conduction problem. The problem is transformed into a homogeneous backward-type inverse heat conduction problem and a Dirichlet boundary value problem for Poisson's equation. We use an improved method of fundamental solutions to solve the backward-type inverse heat conduction problem and apply the finite element method for solving the well-posed direct problem. The Tikhonov regularization method combined with the generalized cross validation rule for selecting a suitable regularization parameter is applied to obtain a stable regularized solution for the backward-type inverse heat conduction problem. Numerical experiments for four examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed algorithm.

Numerical solution of stokes flow in a circular cavity using mesh-free local RBF-DQ

Abstract: This work reports the results of a numerical investigation of Stokes flow problem in a circular cavity as an irregular geometry using mesh-free local radial basis function-based differential quadrature (RBF-DQ) method. This method is the combination of differential quadrature approximation of derivatives and function approximation of Radial Basis Function. As a result, the method can be used to directly approximate the derivatives of dependent variables on a scattered set of knots. In this study knots were distributed irregularly in the solution domain using the Halton sequences. The method is applied on a two-dimensional geometry. The obtained results from the numerical simulations are compared with those gained by previous works. Outcomes prove that the current technique is in very good agreement with previous investigations and this fact that RBF-DQ method is an accurate and flexible method in solution of partial differential equations (PDEs).

A meshless analysis of three-dimensional transient heat conduction problems

Abstract: In this paper, we consider a numerical modeling of a three-dimensional transient heat conduction problem. The modeling is carried out using a meshless reproducing kernel particle (RKPM) method. In the mathematical formulation, a variational method is employed to derive the discrete equations. The essential boundary conditions of the formulated problems are enforced by the penalty method. Compared with numerical methods based on meshes, the RKPM needs only scattered nodes, rather than having to mesh the domain of the problem. An error analysis of the RKPM for three-dimensional transient heat conduction problem is also presented in this paper. In order to demonstrate the applicability of the proposed solution procedures, numerical experiments are carried out for a few selected three-dimensional transient heat conduction problems.