Critical Time Step

Explicit time integration schemes are simple to implement, and they require minimal CPU time and memory per time step. However, explicit schemes are only conditionally stable. That is, the time step used for integration should be small than the so called critical time step so the solution remains valid.

Critical time step is defined as the smallest time step which ensures that the speed of propagating waves within an element is less than what it can theoretically transmit. As per Hughes (1987), critical time step () is mathematically defined by:

(1)

where is the maximum Eigen frequency of the entire mesh. Since it is numerically expensive to solve for , maximum Eigen frequency of an element with the smallest size is usually computed as a conservative estimate. This computation, as can be expected, depends on the element type and is further discussed for different element types below.

3D Isotropic Elastic Element

For a 3D isotropic elastic element, the critical time step is given by Askes et al. (2015):

(2)

where is the minimum element size, is the dilatational wave speed, and is the shear wave speed. Further, these wave speeds are given by:

(3)

where is the material elasticity modulus, is the shear modulus, is the Poisson's ratio and is the density. Since for most applications, is positive, the term in equation (2) is significant.

Elastic Beam Element

Critical time step for an elastic beam element is based on the works: Krieg (1973), Wright (1998), and Junior and Rivera (2015). When either pure axial or pure shear deformations govern, the critical time step is given by:

(4)

where is the minimum element size and and are given by:

(5)

where , , , and are the beam's young's modulus, shear modulus, shear factor, and density, respectively.

When flexure governs, the critical time step is given by:

(6)

where and are the cross-sectional area and moment of inertia, respectively. For implementation, equations (4) and (6) are computed, and the minimum of these is treated as the critical time step.

References

  1. Harm Askes, Antonio Rodriguez-Ferran, and Jack Hetherington. The effects of element shape on the critical time step in explicit time integrators for elasto-dynamics. International Journal for Numerical Methods in Engineering, 101(11):809–824, 2015.[BibTeX]
  2. T. J. R Hughes. The Finite Element Method, Linear Static and Dynamic Finite Element Analysis. New Jersey:Prentice-Hall, 1987.[BibTeX]
  3. DDSA Junior and JEM Rivera. Stability criterion to explicit finite difference applied to the bresse system. Afrika Matematika, 26(5):761–778, 2015.[BibTeX]
  4. RD Krieg. On the behavior of a numerical approximation to the rotatory inertia and transverse shear plate. Journal of Applied Mechanics, 40(4):977–992, 1973.[BibTeX]
  5. JP Wright. Numerical stability of a variable time step explicit method for timoshenko and mindlin type structures. Communications in Numerical Methods in Engineering, 14(2):81–86, 1998.[BibTeX]