Implementation of a generalized exponential basis functions method for linear and non‐linear problems

In this paper, we address shortcomings of the method of exponential basis functions by extending it to general linear and non‐linear problems. In linear problems, the solution is approximated using a linear combination of exponential functions. The coefficients are calculated such that the homogenous form of equation is satisfied on some grid. To solve non‐linear problems, they are converted to into a succession of linear ones using a Newton–Kantorovich approach. The generalized exponential basis functions (GEBF) method developed can be implemented with greater ease compared with exponential basis functions, as all calculations can be performed using real numbers and no characteristic equation is needed. The details of an optimized implementation are described. We compare GEBF on some benchmark problems with methods in the literature, such as variants of the boundary element method, where GEBF shows a good performance. Also, in a 3D problem, we report the run time of the proposed method compared with that of Kratos, a parallel, highly optimized finite element code. The results show that in this example, to obtain the same level of error, much less computational effort is needed in the proposed method. Practical limitations might be encountered, however, for large problems because of dense matrix operations involved. Copyright © 2015 John Wiley & Sons, Ltd.     

A mixed finite element method for non‐linear and nearly incompressible elasticity based on biorthogonal systems

We present a finite element method for non‐linear and nearly incompressible elasticity. The formulation is based on Petrov–Galerkin discretization for the pressure and is closely related to the average nodal pressure formulation presented earlier in the context of incompressible and nearly incompressible dynamic explicit applications (Commun. Numer. Meth. Engng 1998; 14 :437–449). Some numerical examples are presented to show the efficiency of the approach. Copyright © 2009 John Wiley & Sons, Ltd.     

A two‐dimensional stochastic algorithm for the solution of the non‐linear Poisson–Boltzmann equation: validation with finite‐difference benchmarks

This paper presents a two‐dimensional floating random walk (FRW) algorithm for the solution of the non‐linear Poisson–Boltzmann (NPB) equation. In the past, the FRW method has not been applied to the solution of the NPB equation which can be attributed to the absence of analytical expressions for volumetric Green's functions. Previous studies using the FRW method have examined only the linearized Poisson–Boltzmann equation. No such linearization is needed for the present approach. Approximate volumetric Green's functions have been derived with the help of perturbation theory, and these expressions have been incorporated within the FRW framework. A unique advantage of this algorithm is that it requires no discretization of either the volume or the surface of the problem domains. Furthermore, each random walk is independent, so that the computational procedure is highly parallelizable. In our previous work, we have presented preliminary calculations for one‐dimensional and quasi‐one‐dimensional benchmark problems. In this paper, we present the detailed formulation of a two‐dimensional algorithm, along with extensive finite‐difference validation on fully two‐dimensional benchmark problems. The solution of the NPB equation has many interesting applications, including the modelling of plasma discharges, semiconductor device modelling and the modelling of biomolecular structures and dynamics. Copyright © 2005 John Wiley & Sons, Ltd.