Contents
How to use a lumped mass matrix in dynamics?
Explicit dynamics simulations require the usage of lumped mass matrices i.e. diagonal mass matrices for which inversion can be done explicitly. We show how to do this for P1 and P2 Lagrange elements.
How are diagonal terms used in lumped mass matrix?
To avoid this, a lumped mass matrix is adopted. This lumped mass matrix is a diagonal matrix of which the diagonal terms are given by where Eq. ( 37) was used in the second equality and the partition of unity (PU) property of FE shape functions, ∑ JϕJ (x) = 1, ∀x, was used in the third equality.
How to find the lumped mass matrix of a truss element?
Thus, the lumped mass matrix of a space truss element is given by Derive the consistent and lumped mass matrices of a planar truss element shown in Fig. 12.3. Figure 12.3. A planar truss element.
How is the lumped mass matrix used in Fe solution?
For more details of mathematical methods used to lumping the mass matrix and its consequences for the FE solution accuracy, the reader is referred to Hughes (1987), among many others. By using the lumped mass matrix, the DE solution strategy proceeds as follows: Lump the mass matrix and set initial conditions;
How to find the lumped mass of an element?
E.g. for element T with number of vertices n, the diagonal lumped mass i is obtained as: for linear shape functions. We can obtain the same result by integrating the mass form with a “vertex” scheme (i.e. quadrature points are located at the vertices of the simplex with equal weights | T | n: Lumped mass matrix (‘vertex’ scheme): [ [0.167 0.
What can you do with a mass matrix?
For explicit dynamics simulation, the mass matrix can then be manipulated using the diagonal vector, to compute for instance M − 1 w where w is some Function. The above trick consists in summing all rows and affecting the corresponding mass to the corresponding diagonal degree of freedom.