Keywords: object-oriented finite elements, symbolic derivation, automated discretization of partial differential equations.

Mechanical engineering problems are becoming more and more complex. They may involve several types of physics at several time and space scale. In the 1990s the application of object-oriented design has proven to be very beneficial to the development of flexible programs. More recently, some authors present advanced object-oriented finite element codes e.g. Nikishkov [1], Mackie [2], VanderHeyden [3] and Eyheramendy [4] using modern programming languages such as Java or C#.

In the same time, the use of algebraic manipulation software has always been a point of interest for finite element developments (see Eyheramendy et al. [5] and references therein for a state of the art). Recently, Korelc [6] presented a MATHEMATICA package to symbolically derive finite models and to generate automatically finite element code. Similarly Logg et al. [7] presented the project FEniCS with the explicit goal of developing software for the automation of mathematical modeling.

In the approach we propose in this paper, we aim at developing a generic environment to automate discretization schemes in the context of PDEs. We aim at including all the ingredients needed to develop coupled mechanical problems, including Lagrangian concepts. Considering that the object-oriented programming can help to easily and naturally formulate a mathematical problem, we think it is easy to design a symbolic environment to automatically obtain the symbolic forms of the elements' contributions within a complex physical problem, and to automatically build the numerical model for the simulation tool. We present in this paper the basic object-oriented framework covering the elaboration of elemental contributions derived from a weak statement in the context of the finite element discretization. We propose a general mathematical approach to build discrete models based on a variational statement. We introduce an original object-oriented framework needed to obtain the discrete forms of complex coupled terms in variational statements. The approach is illustrated using the example of linear elasticity.

1

G.P. Nikishkov, "Object-oriented design of a finite element code in Java", Computer Modeling in Engineering and Sciences, 11, 81-90, 2006. doi:10.3970/cmes.2006.011.081

2

R.I. Mackie, "Object oriented implementation of distributed finite element analysis in .NET", Advances in Engineering Software, 38, 726-737, 2007. doi:10.1016/j.advengsoft.2006.08.028

3

W.B. VanderHeyden, E.D. Dendy, N.T. Padial-Collins, "CartaBlanca - A Pure-Java, Component-based Systems Simulation Tool for Coupled Non-linear Physics on Unstructured Grids", Joint ACM Java Grande - ISCOPE 2001 Conference Stanford, California June 2-4, 2001. doi:10.1145/376656.376836

4

D. Eyheramendy, "Advanced object models for mathematical consistency enforcement in scientific computing", WSEAS Transactions on Mathematics, 4(4), 457-463, 2005.

5

D. Eyheramendy, Th. Zimmermann, "The Object-oriented finite elements: II. A symbolic environment for automatic programming", Comput. Methods Appl. Mech. Engrg., 32, 259-276, 1996. doi:10.1016/0045-7825(96)01041-9

6

J. Korelc, "Multi-language and Multi-environment Generation of Nonlinear Finite Element Codes", Engineering with Computers, 18, 312-327, 2002. doi:10.1007/s003660200028

7

A. Logg, "Automating the Finite Element Method", Arch Comput Methods Eng., 14, 93-138, 2007. doi:10.1007/s11831-007-9003-9

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