The meshless analog equation method: I. Solution of elliptic partial differential equations

Katsikadelis, John T.

doi:10.1007/s00419-008-0294-6

Cited by 14 publications

(6 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…According to developed AEM (MAEM), the nonlinear governing equation is replaced with equivalent nonhomogeneous linear one (analog equation) with known fundamental solution and under the same boundary conditions. The MAEM has been successfully used for a wide variety of PDEs such as: soap bubble problem [31], heat flows in bodies with nonlinear material properties, determination of a surface with constant mean curvature or with constant Gaussian curvature and the problem of minimal surface [30], nonlinear static and dynamic problems in general bodies [32], nonlinear dynamic analysis of heterogeneous orthotropic members [33], 2D elastostatic problem in inhomogeneous anisotropic bodies [34], for some type of elliptic problems [35] and etc.…”

Section: Analog Equation Methodsmentioning

confidence: 99%

A boundary-only meshless method for numerical solution of the Eikonal equation

Dehghan

Salehi

2010

Comput Mech

View full text Add to dashboard Cite

The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method.

show abstract

Section: Analog Equation Methodsmentioning

confidence: 99%

A boundary-only meshless method for numerical solution of the Eikonal equation

Dehghan

Salehi

2010

Comput Mech

View full text Add to dashboard Cite

show abstract

“…A numerical solution based on the concept of the AEM has been applied for linear multi‐term fractional differential equations with variable coefficients. Also, this method has been employed to solve the soap bubble problem , the heat flows in bodies with nonlinear material properties, the determination of a surface with constant mean curvature or with constant Gaussian curvature and the problem of minimal surface for some type of elliptic problems and so on.…”

Section: Analog Equation Methodsmentioning

confidence: 99%

A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation

Dehghan

Salehi

2012

Math Methods in App Sciences

View full text Add to dashboard Cite

Communicated by S. GeorgievIn the current article, we investigate the RBF solution of second-order two-space dimensional linear hyperbolic telegraph equation. For this purpose, we use a combination of boundary knot method (BKM) and analog equation method (AEM). The BKM is a meshfree, boundary-only and integration-free technique. The BKM is an alternative to the method of fundamental solution to avoid the fictitious boundary and to deal with low accuracy, singular integration and mesh generation. Also, on the basis of the AEM, the governing operator is substituted by an equivalent nonhomogeneous linear one with known fundamental solution under the same boundary conditions. Finally, several numerical results and discussions are demonstrated to show the accuracy and efficiency of the proposed method.where @ is the boundary of , which is a closed subset of R 2 and T is a positive number.When˛andˇare constants and˛>ˇ> 0, A D B D 1, Eqn (1) represents the two-dimensional telegraph equation which will be studied in the current investigation.Equation (1), referred as the second-order telegraph equation with constant coefficients, models the mixture between diffusion and wave propagation by introducing a term that accounts for the effects of finite velocity to standard heat or mass transport equation [1]. Problem (1) is commonly used in signal analysis for transmission and propagation of the electrical signals [2].Authors of [33,34] introduced the BKM, which eliminates the disadvantage of applying fictitious boundary and preserves all benefits of the MFS. The BKM employs the non-singular general solutions in place of the singular fundamental solutions. This method is truly meshfree, integration-free, mathematically simple, boundary-type and easy to implement. The BKM has been employed for a wide range of physical and engineering problems. In [35], the BKM is extended to solve the 2D Helmholtz and convection-diffusion problems under rather complex-shaped interior and exterior contours, which shows that unlike the MFS, the BKM is insensitive to geometric irregularity. This method is used in [36] for solving the Poisson equation . In that article, authors used the high-order general solutions of the Helmholtz and modified Helmholtz equations to evaluate the particular solution. In [37], the BKM is employed for solving the Cauchy problem associated with the inhomogeneous Helmholtz equation. For problems with noisy boundary data, the standard methods for solving matrix equations yield unstable results; thus, these authors [37] used truncated singular value decomposition to solve the resulting matrix equation and obtained a stable and accurate numerical solution. Authors of [38] investigated the regularization techniques for BKM, and their experiments show that Thikhonov regularization with generalized cross-validation approach is more efficient than other regularization techniques for the BKM solution of Helmholtz and modified Helmholtz equations. Authors of [39] used the geodesic distance instead of the standard isotropic Euc...

show abstract

“…The regular distribution schemes employed in the examples are selected from a sequence of increasingly refined grids until the results converge satisfactorily. Using these finer grids of collocation points, stable solutions are expected since small perturbations in the location of the points have little influence on the obtained results [27,28].…”

Section: Numerical Examplesmentioning

confidence: 98%

Transient elastodynamic analysis of nonhomogeneous anisotropic plane bodies

Kandilas

2012

Acta Mech

View full text Add to dashboard Cite

A boundary-only BEM procedure is employed to solve the transient dynamic analysis of nonhomogeneous anisotropic plane elastic bodies. The response of such bodies is governed by two coupled linear, second-order hyperbolic PDEs with spatially dependent coefficients. The lack of a reliable 2D time-domain elastodynamic fundamental solution is overcome using the principle of the Analog Equation, a method by which the equations of motion of the problem are substituted by two coupled quasi-static Poisson-type equations having as nonhomogeneous terms the components of a fictitious time-dependent load distribution in the specified domain. The standard BEM is employed for the solution of the substitute equations. To avoid the appearance of the domain integral in the integral representation of the solution, the fictitious load distribution is approximated by multiquadrics with unknown time-dependent expansion coefficients, which are calculated at discrete timepoints by collocating the equations of motion at a predefined set of domain interpolation nodes. The obtained numerical results by the proposed method demonstrate its stability and accuracy over other numerical methods.

show abstract

The meshless analog equation method: I. Solution of elliptic partial differential equations

Cited by 14 publications

References 4 publications

A boundary-only meshless method for numerical solution of the Eikonal equation

A boundary-only meshless method for numerical solution of the Eikonal equation

A method based on meshless approach for the numerical solution of the two‐space dimensional hyperbolic telegraph equation

Transient elastodynamic analysis of nonhomogeneous anisotropic plane bodies

Contact Info

Product

Resources

About