Automatic generation of finite-element code by simultaneous optimization of expressions. The paper presents a MATHEMATICA package SMS (Symbolic Mechanics System) for the automatic derivation of formulas needed in nonlinear finite element analysis. Symbolic generation of the characteristic arrays of nonlinear finite elements (e.g. nodal force vectors, stiffness matrices, sensitivity vectors) leads to exponential behavior, both in time and space. A new approach, implemented in SMS, avoids this problem by combining several techniques: symbolic capabilities of Mathematica, automatic differentiation technique, simultaneous optimization of expressions and a stochastic evaluation of the formulas instead of a conventional pattern matching technique. SMS translates the derived symbolic formulas into an efficient compiled language (FORTRAN or C). The generated code is then incorporated into an existing finite element analysis environment. SMS was already used to developed several new, geometrically and materially nonlinear finite elements with up to 72 degrees of freedom. The design and implementation of SMS are presented. Efficiency of the new approach is compared with the efficiency of the manually written code on an example.

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  1. Neto, Alfredo Gay: Simulation of mechanisms modeled by geometrically-exact beams using Rodrigues rotation parameters (2017)
  2. Viebahn, Nils; Pimenta, Paulo M.; Schröder, Jörg: A simple triangular finite element for nonlinear thin shells: statics, dynamics and anisotropy (2017)
  3. Kiran, Ravi; Khandelwal, Kapil: Automatic implementation of finite strain anisotropic hyperelastic models using hyper-dual numbers (2015)
  4. Šolinc, Urša; Korelc, Jože: A simple way to improved formulation of $\textFE^2$ analysis (2015)
  5. Alnæs, Martin S.; Logg, Anders; Ølgaard, Kristian B.; Rognes, Marie E.; Wells, Garth N.: Unified form language: a domain-specific language for weak formulations of partial differential equations (2014)
  6. Zupan, Eva; Saje, Miran; Zupan, Dejan: Dynamics of spatial beams in quaternion description based on the Newmark integration scheme (2013)
  7. Seabra, Mariana R.R.; Cesar de Sa, Jose M.A.; Šuštarič, Primož; Rodič, Tomaž: Some numerical issues on the use of XFEM for ductile fracture (2012)
  8. Stupkiewicz, Stanisław; Lengiewicz, Jakub; Korelc, Jože: Sensitivity analysis for frictional contact problems in the augmented Lagrangian formulation (2010)
  9. Korelc, Jože: Automation of primal and sensitivity analysis of transient coupled problems (2009)
  10. Brank, Boštjan; Ibrahimbegovic, Adnan; Bohinc, Uroš: On prediction of 3d stress state in elastic shell by higher-order shell formulations (2008)
  11. Kiousis, D.E.; Gasser, T.C.; Holzapfel, G.A.: Smooth contact strategies with emphasis on the modeling of balloon angioplasty with stenting (2008)
  12. Gams, M.; Planinc, I.; Saje, M.: A heuristic viscosity-type dissipation for high-frequency oscillation damping in time integration algorithms (2007)
  13. Gams, M.; Planinc, I.; Saje, M.: Energy conserving time integration scheme for geometrically exact beam (2007)
  14. Kunc, Robert; Žerovnik, Andrej; Prebil, Ivan: Verification of numerical determination of carrying capacity of large rolling bearings with hardened raceway (2007)
  15. Brank, Boštjan: Nonlinear shell models with seven kinematic parameters (2005)
  16. Stadler, M.; Holzapfel, G.A.: Subdivision schemes for smooth contact surfaces of arbitrary mesh topology in 3D (2004)
  17. Wriggers, Peter; Krstulović-Opara, Lovre: The moving friction cone approach for three-dimensional contact simulations (2004)
  18. Stupkiewicz, Stanisław; Korelc, Jože; Dutko, Martin; Rodič, Tomaž: Shape sensitivity analysis of large deformation frictional contact problems (2002)
  19. Wriggers, P.: Finite element methods for rolling contact (2000)
  20. Wriggers, P.; Krstulović-Opara, L.: On smooth finite element discretizations for frictional contact problems (2000)

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