Solver 3D

Abstract

The presented article contains a 3D mesh generation routine optimized with the Metropolis algorithm. The procedure enables to produce meshes of a prescribed volume V0 of elements. The finite volume meshes are used with the Finite Element approach. The FEM analysis enables to deal with a set of coupled nonlinear differential equations that describes the electrodiffusional problem. Mesh quality and accuracy of FEM solutions are also examined. High quality of FEM type space – dependent approximation and correctness of discrete approximation in time are ensured by finding solutions to the 3D Laplace problem and to the 3D diffusion equation, respectively. Their comparison with analytical solutions confirms accuracy of obtained approximations.

Introduction

One from the most important physical processes is electrodiffusion. It describes both diffusional motion of mass and charge flow due to applied electric field. The electric potential distribution is govern by the Poisson equation and total transport of particles is given in terms of the continuity equation[1]. The significance of this equation is broadly described in existing physical, chemical and biological literature[2] and lots of scientific articles, particularly, those which concern properties of nano and micro transport.
Mathematically, equations of electrodiffusion constitute a set of coupled nonlinear equations where the Laplace operator[3, 4, 5] appears together with the first order partial time derivative. The Laplace operator is the basic operator met in many physical situations. Thus the first step to deal with the electrodiffusional problem is to approximate the solution of the Laplace equation with help of the Finite Element Method. Practically, it means that an appropriate mesh should be designed for a prescribed 3D domain. The mesh must fit well to the physical conditions like e. g. symmetry of the problem.
Therefore, different mesh shapes could be desired (spherical, cylindrical, conical or cubic) up to problem. After having accurate basic spatial solutions on appropriate meshes, the problem should be extended to the time – dependent case of the diffusion equation by finding discrete approximation in time. It could be done by means of truncated Taylor series or other single – step procedures like the Crank – Nicolson scheme[6, 7] or the Gurtin's approach to finite element approximation in terms of variational principle[8]. From now, further extension of above – presented computations involving non – linear terms could be easily implemented and numerically solved using the Newton's method[9].

finite element method, fem, numerical integration, differential equations, lde, 3D mesh generator, Metfem3D, taketechease

References

  1. ^ M. Toda and R. Kubo, Statistical Physics II. Nonequilibrium Statistical Mechanics, Springer – Verlag 1985
  2. ^ N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North – Holland, Amsterdam 1981; E. Nelson, Dynamical Theories of Brownian Motion, University Press, Princeton, NY, 1967;
  3. ^ R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Addison Wesley Publishing Company, 2005
  4. ^ L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998
  5. ^ R. Courant and D. Hilbert, Methods of mathematical physics vol. 1, Interscience Publishers Ltd., London, 1953
  6. ^ O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, The Finite Element Method. Its Basis & Fundamentals 6-th eds., Elsevier Ltd., 2005
  7. ^ J. Crank and P. Nicolson, A practical method for numerical integration of solutions of partial differential equations of heat conduction type., Proc. Camb. Phil. Soc., 43, p. 50 1947
  8. ^ M. E. Gurtin, Q. Appl. Math., 22, pp. 252-256 1964; M. E. Gurtin, Arch. Rat. Mech. Anal., 16, 34-50 1969; E. L. Wilson and R. E. Nickell, Nucl. Eng. Design, 4, pp. 1-11 1966
  9. ^ C. T. Kelley, Solving Nonlinear Equations with Newton's Method in Fundamentals of Algorithms SIAM, 2003; J. Brzozka, L. Dorobczyński, MATLAB. Srodowisko obliczeń naukowo-technicznych. PWN, Warszawa, 2008