V. Romero¹, S. Kanaun²

The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.4, pp. 233-242, 2007, DOI:10.3970/icces.2007.003.233

Abstract In this work Gaussian approximating functions proposed in the works of V. Maz'ya are used for the solution of the integral equations of elasto-plasticity for isotropic bodies. The use of this functions esentially simplify the calculation of the elements of the final matrix of the linear algebraic equations of the discretized problem. The elements of this matrix turn to be a combination of simple elementary functions. The method is applied to a 2D rectangular body that has a cut on a border and is subjected to axial tension. The convergence of the method is studied on this example. More >

Chih-Wen Chang¹, Chein-Shan Liu², Jiang-Ren Chang¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.2, pp. 69-80, 2007, DOI:10.3970/icces.2007.003.069

Abstract By using a quasi-boundary regularization we can formulate a two-point boundary value problem of the backward heat conduction equation. The ill-posed problem is analyzed by using the semi-discretization numerical schemes. Then, the resulting ordinary differential equations in the discretized space are numerically integrated towards the time direction by the Lie-group shooting method to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing initial conditions through a minimum… More >

Cosmin Anitescu¹, Elena Atroshchenko², Naif Alajlan³, Timon Rabczuk^3,*

CMC-Computers, Materials & Continua, Vol.59, No.1, pp. 345-359, 2019, DOI:10.32604/cmc.2019.06641

Abstract We present a method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy. In this procedure, a coarse grid of training points is used at the initial training stages, while more points are added at later stages based on the value of the residual at a larger set of evaluation points. This method increases the robustness of the neural network approximation and can result in significant computational savings, particularly when the solution is non-smooth. Numerical results are presented for benchmark problems for scalar-valued PDEs, namely Poisson and Helmholtz equations, as well as for an inverse… More >

Darin Koblick^1,2,3, Shujing Xu⁴, Joshua Fogel⁵, Praveen Shankar¹

CMES-Computer Modeling in Engineering & Sciences, Vol.111, No.1, pp. 1-27, 2016, DOI:10.3970/cmes.2016.111.001

Abstract The Modified Picard-Chebyshev Method (MPCM) is implemented as an orbit propagation solver for a numerical optimization method that determines minimum time orbit transfer trajectory of a satellite using a series of multiple impulses at intermediate waypoints. The waypoints correspond to instantaneous impulses that are determined using a nonlinear constrained optimization routine, SNOPT with numerical force models for both Two-Body and J2 perturbations. It is found that using the MPCM increases run-time performance of the discretized lowthrust optimization method when compared to other sequential numerical solvers, such as Adams-Bashforth-Moulton and Gauss-Jackson 8th order methods. More >

S. Abbasbandy^1,2, R.A. Van Gorder³, M. Hajiketabi¹

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.4, pp. 329-351, 2015, DOI:10.3970/cmes.2015.104.329

Abstract We transform the Yamabe equation on a ball of arbitrary dimension greater than two into a nonlinear singularly boundary value problem on the unit interval [0,1]. Then we apply Lie-group shooting method (LGSM) to search a missing initial condition of slope through a weighting factor r ∈ (0,1). The best r is determined by matching the right-end boundary condition. When the initial slope is available we can apply the group preserving scheme (GPS) to calculate the solution, which is highly accurate. By LGSM we obtain precise radial symmetric solutions of the Yamabe equation. These results are useful in demonstrating the… More >

W.M. Abd- Elhameed^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.3, pp. 159-185, 2014, DOI:10.3970/cmes.2014.101.159

Abstract This paper is concerned with developing two new algorithms for direct solutions of linear and nonlinear sixth-order two point boundary value problems. These algorithms are based on the application of the two spectral methods namely, collocation and Petrov-Galerkin methods. The suggested algorithms are completely new and they depend on introducing a novel operational matrix of derivatives which is expressed in terms of the well-known harmonic numbers. The basic idea for the suggested algorithms rely on reducing the linear or nonlinear sixth-order boundary value problem governed by its boundary conditions to a system of linear or nonlinear algebraic equations which can… More >

S.A. Khuri¹, A. Sayfy¹

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.2, pp. 81-96, 2014, DOI:10.3970/cmes.2014.101.081

Abstract In this article, we introduce a numerical scheme to solve a class of nonlinear two-point BVPs on a semi-infinite domain that arise in engineering applications and the physical sciences. The strategy is based on replacing the boundary condition at infinity by an asymptotic boundary condition (ABC) specified over a finite interval that approaches the given value at infinity. Then, the problem complimented with the resulting ABC is solved using a fourth order spline collocation approach constructed over uniform meshes on the truncated domain. A number of test examples are considered to confirm the accuracy, efficient treatment of the boundary condition… More >

E. Viola¹, F. Tornabene¹, E. Ferretti¹, N. Fantuzzi¹

CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.5, pp. 421-458, 2013, DOI:10.3970/cmes.2013.094.421

Abstract In this paper, an advanced version of the classic GDQ method, called the Generalized Differential Quadrature Finite Element Method (GDQFEM) is formulated to solve plate elastic problems with inclusions. The GDQFEM is compared with Cell Method (CM) and Finite Element Method (FEM). In particular, stress and strain results at fiber/matrix interface of dissimilar materials are provided. The GDQFEM is based on the classic Generalized Differential Quadrature (GDQ) technique that is applied upon each sub-domain, or element, into which the problem domain is divided. When the physical domain is not regular, the mapping technique is used to transform the fundamental system… More >

Xiaojing Liu¹, Jizeng Wang^1,2, Youhe Zhou^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.5, pp. 493-505, 2013, DOI:10.3970/cmes.2013.092.493

Abstract By combining techniques of boundary extension and Coiflet-type wavelet expansion, an approximation scheme for a function defined on a two-dimensional bounded space is proposed. In this wavelet approximation, each expansion coefficient can be directly obtained by a single-point sampling of the function. And the boundary values and derivatives of the bounded function can be embedded in the modified wavelet basis. Based on this approximation scheme, a modified wavelet Galerkin method is developed for solving two-dimensional nonlinear boundary value problems, in which the interpolating property makes the solution of such strong nonlinear problems very effective and accurate. As an example, we… More >

S.Yu. Reutskiy¹

CMES-Computer Modeling in Engineering & Sciences, Vol.87, No.4, pp. 355-386, 2012, DOI:10.3970/cmes.2012.087.355

Abstract The paper presents a new meshless numerical technique for solving nonlinear Poisson-type equation ∇²u = f (x) + F(u,x) for x ∈ R^d, d =1,2,3. We assume that the nonlinear term can be represented as a linear combination of basis functions F(u,x) = ∑_m^Mq_mφ_m. We use the basis functions φ_m of three types: the the monomials, the trigonometric functions and the multiquadric radial basis functions. For basis functions φ_m of each kind there exist particular solutions of the equation ∇²ϕ_m = φ_m in an analytic form. This permits to write the approximate solution in the form u_M = u_f… More >