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 ( $var element = document.getElementById("moose-equation-9b393191-849d-4d42-b7f5-55c69734b7a4");katex.render("\\Delta t_{crit}", element, {displayMode:false,throwOnError:false});$ ) is mathematically defined by:

$(1)var element = document.getElementById("moose-equation-a8301e4d-1648-4efc-8cbf-c16112716022");katex.render(" \\Delta t_{crit} = \\frac{2}{\\omega^{G}_{max}}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-9dccfa82-2df1-4b20-96e1-68caf8d0e905");katex.render("\\omega^{G}_{max}", element, {displayMode:false,throwOnError:false});$ is the maximum Eigen frequency of the entire mesh. Since it is numerically expensive to solve for $var element = document.getElementById("moose-equation-db62440b-1cb5-4437-bd86-6e2e3511a87d");katex.render("\\omega^{G}_{max}", element, {displayMode:false,throwOnError:false});$ , 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):

$(2)var element = document.getElementById("moose-equation-fa3b268f-27ea-4bde-ac85-24358e6ce3ea");katex.render(" \\Delta t^e_{crit} = min\\Bigg(\\frac{\\Delta x}{c_d},~\\frac{\\Delta x}{c_s}\\Bigg)", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-16f307e4-9971-475f-828f-77c62d33fc65");katex.render("\\Delta x", element, {displayMode:false,throwOnError:false});$ is the minimum element size, $var element = document.getElementById("moose-equation-33255a8e-f70c-4253-b178-7c3138ffaadc");katex.render("c_d", element, {displayMode:false,throwOnError:false});$ is the dilatational wave speed, and $var element = document.getElementById("moose-equation-4c9fa635-f89a-4a64-a621-b827f8fb52ea");katex.render("c_s", element, {displayMode:false,throwOnError:false});$ is the shear wave speed. Further, these wave speeds are given by:

$(3)var element = document.getElementById("moose-equation-b35f0780-2157-4e7c-b341-283e069f893d");katex.render(" \\begin{aligned} c_d &= \\sqrt{\\frac{E~(1-\\nu)}{(1+\\nu)~(1-2~\\nu)~\\rho}}\\\\ c_s &= \\sqrt{\\frac{G}{\\rho}}\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-6be2e0b4-d417-4ff6-9d7c-4f19310e9df0");katex.render("E", element, {displayMode:false,throwOnError:false});$ is the material elasticity modulus, $var element = document.getElementById("moose-equation-e1adbb9d-c430-428a-8209-97a20508c205");katex.render("G", element, {displayMode:false,throwOnError:false});$ is the shear modulus, $var element = document.getElementById("moose-equation-026be384-26b2-4835-8eaa-8aad7f331de9");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ is the Poisson's ratio and $var element = document.getElementById("moose-equation-d32d56a6-8098-4aee-8197-9432e7585ea1");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ is the density. Since for most applications, $var element = document.getElementById("moose-equation-d7fe0e20-59a6-4663-b24d-d0e81dbdde03");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ is positive, the $var element = document.getElementById("moose-equation-bd75e683-c73b-4564-bb93-044a6c60436b");katex.render("\\frac{\\Delta x}{c_d}", element, {displayMode:false,throwOnError:false});$ 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:

$(4)var element = document.getElementById("moose-equation-beb92ab1-8abd-4944-be0f-262c4e94420a");katex.render(" \\Delta t^e_{crit} = min\\Bigg(\\frac{\\Delta x}{c_1},~\\frac{\\Delta x}{c_2}\\Bigg)", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-01024fb4-3657-472f-a2ef-a2a64f4f9828");katex.render("\\Delta x", element, {displayMode:false,throwOnError:false});$ is the minimum element size and $var element = document.getElementById("moose-equation-512dfde8-14fb-43ef-9962-5c9200b52af7");katex.render("c_1", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-3116e0e5-739d-401e-8964-4a4df76f47bf");katex.render("c_2", element, {displayMode:false,throwOnError:false});$ are given by:

$(5)var element = document.getElementById("moose-equation-e6b5c391-7db9-4659-9bf9-a08b4c154470");katex.render(" \\begin{aligned} c_1 &= \\sqrt{\\frac{E}{\\rho}}\\\\ c_2 &= \\sqrt{\\kappa~\\frac{G}{\\rho}}\\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-3e152f3d-8cea-4003-8887-88cf84f87279");katex.render("E", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-4a27a8ad-c43f-4bc8-87a6-0e7575833435");katex.render("G", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-613b83fa-555e-492f-a6cd-fdf54b78bb28");katex.render("\\kappa", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-b355ba6e-02e8-4edf-859b-20e63e125895");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ are the beam's young's modulus, shear modulus, shear factor, and density, respectively.

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

$(6)var element = document.getElementById("moose-equation-07b4017d-49d6-40f1-935c-1e6ae28585e3");katex.render(" \\Delta t^e_{crit} = \\frac{2}{\\sqrt{\\frac{\\kappa~G~A}{\\rho~I}}}\\\\", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-e4b24c99-e927-429d-a84f-162575d3cfa8");katex.render("A", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-fbeb4780-a636-498c-9c36-5f07e89f37cd");katex.render("I", element, {displayMode:false,throwOnError:false});$ are the cross-sectional area and moment of interia, respectively. For implementation, equations (4) and (6) are computed, and the minimum of these is treated as the critical time step.

References

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.

@article{Askes2015,
    author = "Askes, Harm and Rodriguez-Ferran, Antonio and Hetherington, Jack",
    title = "The effects of element shape on the critical time step in explicit time integrators for elasto-dynamics",
    journal = "International Journal for Numerical Methods in Engineering",
    volume = "101",
    number = "11",
    pages = "809--824",
    year = "2015",
    publisher = "Wiley"
}

T. J. R Hughes. The Finite Element Method, Linear Static and Dynamic Finite Element Analysis. New Jersey:Prentice-Hall, 1987.

@book{hughes1987finite,
    author = "Hughes, T. J. R",
    title = "The Finite Element Method, Linear Static and Dynamic Finite Element Analysis",
    publisher = "New Jersey:Prentice-Hall",
    year = "1987"
}

DDSA Junior and JEM Rivera. Stability criterion to explicit finite difference applied to the bresse system. Afrika Matematika, 26(5):761–778, 2015.

@article{Junior2015,
    author = "Junior, DDSA and Rivera, JEM",
    title = "Stability criterion to explicit finite difference applied to the Bresse system",
    journal = "Afrika Matematika",
    volume = "26",
    number = "5",
    pages = "761--778",
    year = "2015",
    publisher = "Springer"
}

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.

@article{Krieg1973,
    author = "Krieg, RD",
    title = "On the behavior of a numerical approximation to the rotatory inertia and transverse shear plate",
    journal = "Journal of Applied Mechanics",
    volume = "40",
    number = "4",
    pages = "977--992",
    year = "1973",
    publisher = "ASME"
}

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.

@article{Wright1998,
    author = "Wright, JP",
    title = "Numerical stability of a variable time step explicit method for Timoshenko and Mindlin type structures",
    journal = "Communications in Numerical Methods in Engineering",
    volume = "14",
    number = "2",
    pages = "81--86",
    year = "1998",
    publisher = "Wiley"
}