Application of the Coupled Boundary Element Method With Atomic Model in the Static Analysis

• Autorzy: Mrozek A. , Burczyński T.

Warianty tytułu

PL

Zastosowanie metody elementów brzegowych połączonej z dyskretnym modelem molekularnym w analizie statycznej układów mechanicznych

• Konferencja: 14th KomPlasTech Conference, Zakopane, January 14-17, 2007
• Języki publikacji: EN

Abstrakty

EN

1. Introduction The models of material behaviour in the atomistic scale allows us to understand the micro and macroscopic events. The algorithm based on the boundary element method (BEM) coupled with a discrete atomistic model is presented in this paper. In this approach, the material behaviour at the molecular level can be simulated and the total number of degrees of freedom is reduced, because in most cases only a small part of the mulitiscale model contains molecules and BEM doesn’t need discretization of the continuum’s domain. 2. The molecular model The discrete molecular model is applied to simulate deformations of the atomistic lattice under loads. This model is based on the equilibrium equations of atomic interaction forces. These pair-wise interaction forces between each two molecules in the lattice are computed as derivative of the interatomic potential respect to the distance between two atoms. In this work, the empirical Lennard-Jones and the Morse potentials were used to describe interatomic behaviour [4]. The force equilibrium equations are computed for all the nearest-neighbour atoms interactions and then, assembled into the global non-linear system of equations [2]. Constraints are applied using elimination method. The Newton-Raphson method is used to solve that system of equations. The main concept is to assume some initial positions of molecules (eg. undeformed lattice) and obtain final, stable equilibrium configuration of atoms. The equilibrium state of the lattice corresponds to the minimal value of the total potential energy of the atomic structure. The process of minimization of the potential energy can be also done by using the evolutionary algorithm [3]. The applications of these algorithms in prediction of atoms distribution give a great probability of finding the global optimal solutions. 3. The mulitiscale model The multiscale model is composed from three main blocks: the continuum model, the interface domain and the discrete atomic model. The boundary element method [1] is used to simulate material behaviour at the continuum level. The interface domain contains so-called embedded atoms which coordinates are equal to the corresponding nodes of boundary elements. Firstly, the macroscale boundary conditions are applied and the BEM model is solved. Displacements of the interface atoms are obtained and introduced as initial displacements of the outer boundary of the atomic lattice. In the next step, equilibrium positions of the atoms in the nanoscale model are computed, using the method described above. Finally, forces acting on interface atoms are computed and introduced as a nodal forces to the BEM model. These computations are repeated until achieved displacements are satisfactory small. 3. Final remarks Some simulations of a dislocation behaviour and deformations of the atomic lattice are performed. Both hexagonal and orthogonal configurations of the lattice are considered. The convergence of the Newton-Raphson method and the total number of iterations strongly depend on the initial positions of the atoms and their displacements taken form BEM. However, for small deformations of the atomic structure, the Newton-Raphson method is faster than the evolutionary algorithm. This kind of analysis gives possibility of simulation, e.g. slips, crack behaviour and fracture at the molecular level and also may be used in modelling some technological processes in material science. The application of loads and displacements to the BEM continuum model is easier then direct in the molecular level. 4. Acknowledgement: The research is financed by the Foundation for Polish Science (2005-2008). 5. References [1] T. Burczynski (1995), The Boundary Element Method in Mechanics, WNT, Warsaw (in Polish) [2] Y. W. Kwon (2003). Discrete atomic and smeared continuum modelling for static analysis Engineering Computations, Vol. 20 No. 8. 964–978. [3] A. Mrozek, W. Kus, P. Orantek, T. Burczynski (2005). Prediction of the aluminium atoms distribution using evolutionary algorithm, Recent Developments in Artificial Intelligence Methods, ed. T. Burczynski, W. Cholewa, M. Moczulski, AI-METH Series, Gliwice. 127-130. [4] R. Sunyk, P. Steinmann (2002). On higher gradients in continuum-atomistic modelling, International Journal of Solids and Structures, No. 40. 6877-6896.

Słowa kluczowe

EN

multiscale modelling; boundary element method; atomic model; static analysis

• Wydawca: Wydawnictwa AGH
• Czasopismo: Computer Methods in Materials Science
• Rocznik: 2007
• Tom: Vol. 7, No. 2
• Strony: 284--288
• Opis fizyczny: Bibliogr. 8 poz., rys.

Twórcy

autor

Mrozek A.

autor

Burczyński T.

• Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, ul. Konarskiego 18a, 44-100 Gliwice, Poland

Bibliografia

• Burczyński, T., 1995, The Boundary Element Method in Mechanics, WNT, Warsaw (in Polish).
• Habarta, M., Burczyński, T., 2006, Boundary element formulation for gradient elastostatics, Proc. IABEM 2006 Conference, Graz, 119-122.
• Kwon, Y. W., 2003, Discrete atomic and smeared continuum modelling for static analysis, Eng. Comput., 20, 8. 964-978.
• Liu, K., Karpov, E.G., Zhang, S., Park, H.S., 2004, An introduction to computational nanomechanics and materials, Comp. Meth. Appl. Mech. Eng., 193, 1529-1578.
• Mrozek, A., Burczyński, T., 2006, Analysis of the material behaviour at the nanoscale, Proc. 35th Solid Mechanics Conf., Kraków, 283-284.
• Mrozek, A., Kuś, W., Orantek, P., Burczyński, T., 2005, Prediction of the aluminium atoms distribution using evolutionary algorithm, in: Recent Developments in Artificial Intelligence Methods, eds, T. Burczyński, W. Cholewa, M. Moczulski, AI-METH Series, Gliwice, 127-130.
• Shenoy, V.B., Miler, R., Tadmor, E.B., Rodney, D., Philips, R., Ortiz, M., 1999, An adaptive finite element approach to atomic-scale mechanics - the quasicontinuum method. J. Mechanics and Physics of Solids, 47, 611-642.
• Sunyk, R., Steinmann, P., 2002, On higher gradients in continuum-atomistic modelling, Int. J. Solids and Structures, 40, 6877-6896.