3. Numerical method

The numerical method implemented in*code*_aster was proposed by Nigam and Jennings [bib2], which is more effective than the Newmark method (see [Bib3]) in the case of subcritical damping that corresponds to our initial seismic problem [R4.05.03].

We are therefore led to solve the differential equation:

(3.1)\[ \ ddot {q}\ left (t\ right) +2\ mathrm {\ right) +2\ mathrm {\ xi}\ mathrm {\ omega}\ left (t\ right) + {\ mathrm {\ omega}}} ^ {2} q\ left (t\ right) =\ mathrm {\ alpha}\ left (t\ right)\]

with zero initial conditions and subcritical damping, whose solution is written as:

(3.2)\[\begin{split} q\ left (t\ right) =-\ frac {1} {{\ mathrm {\ omega}} _ {d}} {\ int} _ {0} ^ {t} {e} ^ {-\ mathrm {\ xi}\ mathrm {\ xi}\ xi}\ {\ xi}\\\\\\\\\\\}\ left\ mathrm {\ omega}\ left (t-\ mathrm {\ tau}\ right)}\ text {sin}\ e} ^ {-\ mathrm {\ xi}}\ left [{\ mathrm {\ mathrm {\ xi}}\ left m {\ omega}} _ {d}\ left (t-\ mathrm {\ tau}\ right)\ right]\ mathrm {\ alpha}\ left (\ mathrm {\ tau}\ right) d\ mathrm {\ tau}\ right) d\ mathrm {\ tau}\end{split}\]

With \({\mathrm{\omega }}_{d}=\mathrm{\omega }\sqrt{1-{\mathrm{\xi }}^{2}}\). Assuming that \(\alpha (t)\) varies linearly within each \(\Delta (t)\) interval, we can write:

(3.3)\[ \ mathrm {\ alpha}\ left (\ mathrm {\ tau}\ right) =\ mathrm {\ alpha}\ left (t-\ mathrm {\ Delta} t\ right) +\ frac {\ mathrm {\ tau}} {\ mathrm {\ tau}}} {\ mathrm {\ tau}\ right}} {\ mathrm {\ delta} t\ right) +\ frac {\ mathrm {\ delta} t\ right) +\ frac {\ mathrm {\ tau}}} {\ mathrm {\ tau}}} {\ mathrm {\ tau}}} {\ mathrm {\ tau}}} {\ mathrm {\ delta} t\ right) -\ mathrm m {\ alpha}\ left (t-\ mathrm {\ Delta} t\ right)\ right]\ right]\ text {for}\ mathrm {\ tau}\ in\ left [\ mathrm {0,}\ mathrm {\ Delta} t\ right]\]

Note: the \(\mathrm{\Delta }t\) time step is not necessarily constant.

Hence the equation to be solved (expressed in the new variable \(\tau\)):

(3.4)\[ \ ddot {q}\ left (\ mathrm {\ tau}\ right) +2\ mathrm {\ tau}\ right) +2\ mathrm {\ xi}\ left (\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) + {\ mathrm {\ tau}\ right) tau}\ text {for}\ mathrm {\ tau}\ in\ left [\ mathrm {0,}\ mathrm {\ Delta} t\ right]\]

Where \(a=\alpha (t-\Delta t )\) and \(b=\left[\alpha (t)-\alpha (t-\Delta t )\right]/\Delta t\). With the initial conditions:

(3.5)\[\begin{split} \ {\ begin {array} {c} q\ left (0\ right) =q\ left (t-\ mathrm {\ Delta} t\ right)\\\ dot {q}\ left (0\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right) =\ dot {q}\ right)\end{split}\]

The solution to this equation is the superposition of a particular solution:

(3.6)\[ {q} _ {p} (t) =-\ frac {a} {{\ omega} {{\ omega} ^ {2}} +\ frac {2\ xib} {{\ omega} ^ {3}}} -\ frac {3}}} -\ frac {b} {b} {b} {b} {b} {b} {b} {b} {b} {b} {b} {b} {{b}} {\]

and solutions to the homogeneous problem:

(3.7)\[ {q} _ {h} (t) = {e} ^ {-\ xi\ omega\ tau}\ left [{C} _ {1}\ text {.} \ text {cos} ({\ omega} _ {d}\ tau) + {C} _ {2}\ text {.} \ text {sin} ({\ omega} _ {d}\ tau)\ right]\]

As a result:

(3.8)\[ q (\ tau) = {e} ^ {-\ xi\ omega\ tau}\ left [{C} _ {1}\ text {.} \ text {cos} ({\ omega} _ {d}\ tau) + {C} _ {2}\ text {.} \ text {sin} ({\ omega} _ {d}\ tau)\ tau)\ right] -\ frac {a} {{\ omega} ^ {2}} +2\ frac {\ xib} {{\ omega} {\ omega} ^ {3}}} -\ frac {b\ text {.} \ tau} {{w} ^ {2}}\]

By deriving \(q\) (with respect to \(\mathrm{\tau }\)) we have:

(3.9)\[\begin{split} \ begin {array} {c}\ dot {q}\ left (\ mathrm {\ tau}\ right) =\ left (-\ mathrm {\ xi}\ mathrm {\ omega}\ right) {e} ^ {\ right) {e} ^ {-\ mathrm {\ xi}\ left ({C} _ {1}\ text {cos} {\ mathrm {\ omega}} _ {d}\ mathrm {\ tau} + {C} _ {2}\ text {sin} {\ mathrm {\ omega}} _ {d}\ mathrm {\ tau}\ right) +\\ {tau}\ right) +\\ {e}\ right) +\\ {e} ^ {-\ mathrm {\ xi}\ mathrm {\ omega}} _ {d}\ mathrm {\ omega}} _ {d}\ mathrm {\ omega}}\ mathrm {\ tau}\ right) +\\ {e}\ right) +\\ {e} ^ {-\ mathrm {\ xi}\ mathrm {\ omega}} _ {d}\ mathrm {\ omega}}}\ left (- {C} _ {1} {\ mathrm {\ omega}}} _ {d}\ text {sin} {\ mathrm {\ omega}} _ {d}\ mathrm {\ tau} + {\ mathrm {\ tau}} + {\ C} _ {2} {2} {\ mathrm {\ omega}} {2} {\ mathrm {\ omega}} {2} {\ mathrm {\ omega}} {2} {\ mathrm {\ omega}} {2} {\ mathrm {\ omega}} {2} {\ mathrm {\ omega}} {2} {\ mathrm {\ omega}} {2} {\ mathrm {\ omega}} {2} {\ mathrm {\ omega}} mathrm {\ tau}\ right) -\ frac {b} {{\ mathrm {\ omega}} ^ {2}}\ end {array}}\end{split}\]

The coefficients \({C}_{1}\) and \({C}_{2}\) are then determined by the initial conditions at the beginning of the interval (i.e. for \(\tau =0\)):

(3.10)\[ {C} _ {1} =q\ left (t-\ mathrm {\ Delta} t\ right) +\ frac {a} {{\ mathrm {\ omega}} ^ {2}} -\ frac {2}} -\ frac {2}} -\ frac {2}} -\ frac {2}} -\ frac {2}} -\ frac {2}} -\ frac {2}} -\ frac {2}} -\ frac {2}} -\ frac {2}}\]

And:

(3.11)\[ {C} _ {2} =\ frac {1} {{\ mathrm {\ omega}}} _ {d}}\ left [\ dot {q}\ left (t-\ mathrm {\ Delta} t\ right) +\ mathrm {\ delta} t\ right) +\ mathrm {\ xi}\ right) +\ frac {\ mathrm {\ xi} a} {\ mathrm {\ omega}}} -\ frac {2 {\ mathrm {\ xi}} ^ {2} -1} {{\ mathrm {\ omega}}} ^ {2}} b\ right]\]

By bringing \({C}_{1}\) and \({C}_{2}\) into the expression for \(q\) and \(\dot{q}\) we get matrix equality for \(\tau =\Delta t\):

\[\]

: label: eq-20

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