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Evaluating the element mass matrix

The element mass matrix is evaluated from

For a 3-noded bar element, the shape function is defined as


The mass matrix is given by

After expanding, becomes

Multiplying out the bracketed terms gives

This shows that integration of order four is required for the mass matrix of such a bar element. Because a Gauss integration rule of order n integrates exactly a polynomial of order (2n-1), the mass matrix will be integrated accurately using a 3-point rule ([2*3]-1 = 5)

This is in comparison with the stiffness matrix, given by

Which is of order two and only requires a Gauss rule of order 2 ([2*2]-1 = 3).

Hence, for a consistent mass matrix the full integration rule as used for the stiffness is appropriate. This is the case because the consistent mass matrix is obtained from the shape functions [N] directly rather than their derivatives (i.e. the [B] matrix), as in the case of the stiffness and force matrices.

For lumped mass matrices, only the volume strictly needs to be accurately obtained. In LUSAS, however, the lumped mass matrix is typically evaluated from a summation of the consistent mass terms - hence the same order of integration rule is used regardless of the type of mass matrix specified. The few exceptions to this rule are typically indicated when the default and fine integration rules are the same – notably in the case of the QSI4/SHI4 shell elements. For these elements, the lumped mass matrix is evaluated directly; hence higher order rules do not need to be invoked for mass evaluations.

It is for this reason that the fine integration rule is invoked automatically for dynamic analyses to ensure that the mass matrix is evaluated correctly. Note that this is not necessary when using Constant Body Force to apply self weight loading in a static analysis, since this effectively applies a lumped mass - the exact volume being evaluated readily with the default integration scheme used.

See Theory Manual 1 for more information on mass matrices, specifically the difference between consistent and lumped matrices.

Finite Element Theory Contents

Shape Functions

Isoparametric Finite Element Formulation

Numerical Integration

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