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Structural Engineering and Mechanics
  Volume 21, Number 5, November30 2005 , pages 539-551

A function space approach to study rank deficiency and spurious modes in finite elements
K. Sangeeta, Somenath Mukherjee and Gangan Prathap

    Finite elements based on isoparametric formulation are known to suffer spurious stiffness properties and corresponding stress oscillations, even when care is taken to ensure that completeness and continuity requirements are enforced. This occurs frequently when the physics of the problem requires multiple strain components to be defined. This kind of error, commonly known as locking, can be circumvented by using reduced integration techniques to evaluate the element stiffness matrices instead of the full integration that is mathematically prescribed. However, the reduced integration technique itself can have a further drawback - rank deficiency, which physically implies that spurious energy modes (e.g., hourglass modes) are introduced because of reduced integration. Such instability in an existing stiffness matrix is generally detected by means of an eigenvalue test. In this paper we show that a knowledge of the dimension of the solution space spanned by the column vectors of the strain-displacement matrix can be used to identify the instabilities arising in an element due to reduced/selective integration techniques a priori, without having to complete the element stiffness matrix formulation and then test for zero eigenvalues.
Key Words
    rank deficiency; zero energy mode; eigen value; function space; basis vectors; locking; reduced integration.
K. Sangeeta; CSIR Centre for Mathematical Modelling and Computer Simulation (C-MMACS), Bangalore 560 037, India
Somenath Mukherjee; Structures Division, National Aerospace Laboratories, Bangalore 560 017, India
Gangan Prathap; CSIR Centre for Mathematical Modelling and Computer Simulation (C-MMACS), Bangalore 560 037, India

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