Hello everyone, I read fundamental books of FEM about the penalty method to impose constraints. I found that is posible to employ such a method in the galerkin formulation. However, I would like to employ it for transient problems. Do I have to penalize the mass matrix?? Any comments and references will be welcome.
Hello Julio. Actually, the penalty procedure must be adressed in terms of an additional constraint to your weak formulation. Regardless of whether the problem is stationary or transient, the aforementioned constraint introduces an additional vector and matrix in the FEM equations system.
i. e.
MÜ+KU+KpU=F+Fp
or in the case of Heat Transfer problems
C\dot{T}+KT+KpT=Q+Qp
Accordingly, the penalty procedure is not intended to penalize the mass, capacitance, stiffness or conduction matrix involved in the FEM formulations. On the other hand, such a procedure gives rise to additional terms related to the Dirichlet condition constraint (Kp, Qp, Fp).
Please check the following references:
Zienkiewicz O. C., Taylor R. L. and Zhu J.Z. The Finite Element Method-Its Basis and Fundamentals, 753, Butterworth-Heinemann, Oxford (2014)
G.R. Liu. The Finite Element Method, a practical course, 348, Butterworth-Heinemann, Oxford (2003)
Álvarez Hostos J.C., Bencomo A.D., Puchi Cabrera E.S., Guérin J.-D. and Dubar L.: Modelling the viscoplastic flow behavior of a 20MnCr5 steel grade deformed under hot-working conditions, employing a meshless technique, International Journal of Plasticity, (2018), DOI:10.1016/j.ijplas.2018.01.005
Chen Y., Lee J. D. and Eskandarian A. Meshless Methods in Solid Mechanics, 210, Springer Science, New York (2014)
G.R. Liu. MESHFREE METHODS, Moving Beyond The Finite Element Method, 693, CRC Press LLC, Boca Ratón-Fl (2003)
Li X. and Dong H.: The element-free Galerkin method for the non linear p-Laplacian equation, Computers and Mathematics with Applications, 75, (2018), 2459-2560
Thank you so much Prof Alvarez for you very worthwhile comment.
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