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Boundary Conditions

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Book cover Finite Element Applications

Part of the book series: Springer Tracts in Mechanical Engineering ((STME))

Abstract

No FEM solution is possible without defining the behaviour/conditions of the nodes at the boundaries of the virtual domain, the so-called boundary conditions of the FEM model. Therefore, the solution of any FEM problem is highly dependent on the boundary conditions implemented. This chapter explores this extremely important feature of an FEM model. Although loads and boundary conditions are distinguished in certain FEM solvers, such as ABAQUS, the presentation here considers both of them under the broad description of boundary conditions. This chapter introduces the different types of boundary conditions commonly in use in many FEM solvers. Special attention is given to periodic boundary conditions since it has been shown in literature that they can quickly arrive at the convergent solution of an FEM problem. A methodology for imposing this type of boundary condition for a 2D representative volume element (RVE) is shown too. The ability to define correctly an appropriate boundary condition for a given problem is a vital skill for anyone who wants to use the FEM process to determine reliable solutions of any physical problem.

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Notes

  1. 1.

    Unstructured or free meshing refers to a random discretization of a virtual domain such that the resulting mesh is made in an irregular pattern. Its commonly used when the domain has a complex and irregular shape.

  2. 2.

    Note that node N 3 is excluded from the list of retained nodes as its deformation is influenced by the deformations of nodes N 2 and N 4. Node N 3 is commonly regarded as a slave/dummy node since its deformation degrees of freedom are constrained and kinematically linked to the N 2 and N 4 retained nodes.

  3. 3.

    Different FEM solvers have different implementations and keyword commands for enforcing the multi-freedom constraints equations. Interested readers should consult documentations of the said FEM solver to identify the format of the solver’s MFC commands.

  4. 4.

    If the user does not have a MATLAB™ license, visit the extra resources website to download a compiled executable of PBC2DGEN and use this instead.

  5. 5.

    The convert*.p, information.p and get_option_information.p scripts of Table 8.1 were developed in 2009 by Dr. Ben Elliot of the Impact Engineering Team, Department of Engineering Science, University of Oxford. His contribution here is gratefully acknowledged.

    Table 8.1 Description of the component *.P files of PBC2DGEN
    Full size table
  6. 6.

    Note: In ABAQUS, the XY Z reference frame is equivalent to the 123 −material reference frame. Hence, ε 11 ≡ ε XX

  7. 7.

    This term should not be confused with asymptotic homogenization – an analytical homogenization method – which requires mathematical stress-strain derivations of simplistic RVEs of the microstructure under investigation. For a UD composite, such representation can include simple ordered arrangement of the fibre within the matrix medium.

  8. 8.

    For this example, N = 1, since the chosen RVE window encloses only one single fibre as stated in the question.

  9. 9.

    The e-book version of these contour plots will show these regions as bright red shear bands.

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Authors and Affiliations

  1. Senior Lecturer in Engineering Mechanics, Department of Engineering Science, University of Greenwich, Chatham Maritime, UK

    M. Okereke

  2. Deputy Pro Vice Chancellor Faculty of Engineering & Science, University of Greenwich, Chatham Maritime, UK

    S. Keates

Authors
  1. M. Okereke
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  2. S. Keates
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Okereke, M., Keates, S. (2018). Boundary Conditions. In: Finite Element Applications. Springer Tracts in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-67125-3_8

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  • DOI: https://doi.org/10.1007/978-3-319-67125-3_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67124-6

  • Online ISBN: 978-3-319-67125-3

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