New Algorithm For Numerical Solution Research Articles

A calculation method for finite depth free-surface green function

ABSTRACT: An improved boundary element method is presented for numerical analysis of hydrodynamic behavior of marine structures. A new algorithm for numerical solution of the finite depth free-surface Green function in three dimensions is developed based on multiple series representations. The whole range of the key parameter R / h is divided into four regions, within which different representation is used to achieve fast convergence. The well-known epsilon algorithm is also adopted to accelerate the convergence. The critical convergence criteria for each representation are investigated and provided. The proposed method is validated by several well-documented benchmark problems.

Parallel solutions of static Hamilton-Jacobi equations for simulations of geological folds

Two new algorithms for numerical solution of static Hamilton-Jacobi equations are presented. These algorithms are designed to work efficiently on different parallel computing architectures, and numerical results for multicore CPU and GPU implementations are reported and discussed. The numerical experiments show that the proposed solution strategies scale well with the computational power of the hardware. The performance of the new methods are investigate for tow types of static Hamilton-Jacobi formulations are investigated, the isotropic eikonal equation and an anisotropic formulation used to simulate different types of geological folding. Simulations of geological folding is a key component in an industrial software used in oil and gas exploration. In particular, our experiments indicate that the new algorithms would be capable of accelerating an existing industrial simulator substantially. Direct comparison with the current industry code shows that computing times can be reduced from several minutes to a few seconds. The new methods are now being migrated to the industrial software.

A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.

New numerical algorithm for solving multidimensional heterogeneous model of the fixed bed reactor

Although there are many approaches to modeling of fixed-bed reactors none of the heterogeneous models is thoroughly solved in the space of all the four dimensions: the axial and radial directions of the reactor, catalyst particle radius and time. The full heterogeneous model of the fixed-bed reactor has been proposed and a new algorithm for numerical solution of the mathematical model developed in this paper. As a practical case study the proposed method is applied to simulate methanol synthesis fixed-bed reactor working under industrial conditions. The proposed algorithm of solving the set of differential equation for the momentum, mass and energy balances for the heterogeneous model proved to be very effective thanks to a universal coupling of macro- and micro-geometries represented by two finite element meshes system. The solution of the multidimensional model of methanol synthesis in a fixed bed reactor allows to follow concentration and temperature profiles not only along the catalyst height but also in radial direction as well as inside porous catalyst particle. The application of the numerical solution of the multidimensional heterogeneous model of methanol synthesis in the fixed bed revealed interesting phenomena inside the catalyst and in the reactor. An occurrence of the temperature maximum in a middle axial position due to change of both reversible reactions of methanol synthesis and water gas shift reaction from forward to backward direction was observed. The evolution in time of hot spots (temperature propagation) due to local decrease of catalyst diameter and bed porosity was demonstrated by using a transient simulation of the process.

A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law

3D elastoplastic frictional contact problems with orthotropic friction law belong to the unspecified boundary problems with nonlinearities in both material and geometric forms. One of the difficulties in solving the problem lies in the determination of the tangential slip states at the contact points. A great amount of computational efforts is needed so as to obtain high accuracy numerical results. Based on a combination of the well known mathematical programming method and iterative method, a finite element model is put forward in this paper. The problems are finally reduced to linear complementarity problems. A specially designed smoothing algorithm based on NCP-function is then applied for solving the problems. Numerical results are given to demonstrate the validity of the model and the algorithm proposed.

A new algorithm for numerical solution of dynamic elastic–plastic hardening and softening problems

The objective of this paper is to develop a new algorithm for numerical solution of dynamic elastic–plastic strain hardening/softening problems, particularly for the implementation of the gradient dependent model used in solving strain softening problems. The new algorithm for the solution of dynamic elastic–plastic problems is derived based on the parametric variational principle. The gradient dependent model is employed in the numerical model to overcome the mesh-sensitivity difficulty in dynamic strain softening or strain localization analysis. The precise integration method, which has been used for the solution of linear problems, is adopted and improved for the solution of dynamic non-linear equations. The new algorithm is proposed by taking the advantages of the parametric quadratic programming method and the precise integration method. Results of numerical examples demonstrate the validity and the advantages of the proposed algorithm.

A combined parametric quadratic programming and precise integration method based dynamic analysis of elastic-plastic hardening/softening problems

The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems.

Conjugate progector preconditioning for the solution of contact problems

AbstractWe present a new algorithm for numerical solution of contactproblems without friction. The method is based on enhancing conjugate projector preconditioning into the standard conjugate gradient method for the solution of quadratic programming problems with inequality constraints. We show that the algorithm is correct and discuss the efficiency of the algorithm. The relation of the algorithm to the domain decomposition methods is also explained. The paper is supplied with numerical experiment.

A new algorithm for numerical solution of ordinary differential equations

A numerical integration scheme which is particularly well suited to initial value problems having oscillatory or exponential solutions is proposed. The derivation of the algorithm is based on a representation of problems (that is problems having oscillatory or exponential solutions), the complex parameters have the real plane. The interpolating function has two complex parameters whose numerical estimates are obtained by using Newton-like scheme to solve three simultaneous nonlinear equations. For the above class of paoblems (that is problems having oscillatory or exponential solutions), the complex parameters have constant values throughout the interval of integration. Hence, the parameters are obtainable at the first integration step. As the approach is applicable to systems of equations, then for an initial value problem of order m, m sets of simultaneous equations have to be solved for the complex parameters.