4. Seismic analysis

The seismic calculation of non-linear soil-structure interaction must be solved in the time domain. Some elements on the calculation of the seismic force and the discretization in time of the equations of the coupled system are provided below.

4.1. Calculation of the seismic force in time

The Laplace-Temps method could also be applied to the calculation of the seismic force, however the nature of the incident field developed in MISS3D invalidates the hypotheses on the solution fields sought in the Laplace domain. In this context, it is not possible to guarantee that the calculation of the seismic force at complex frequency with MISS3D is correct for any type of foundation (superficial and buried) and stratigraphy. The approach recommended in the studies thus involves the inverse Fourier transform of the seismic force evaluated by MISS3D in the frequency domain.

4.2. Time integration diagram

This part seeks to illustrate the numerical processing of the convolution integral in the context of a resolution with a Newmark family time integration diagram. However, the reasoning also applies to the integration scheme in time \(\mathrm{\alpha }\) - HHT, which is also available in code_aster.

We will thus seek to solve the following linear dynamic problem in time, which is supposed to be discretized in space with the classical finite element method:

(4.1)\[\begin{split} \ left [\ begin {array} {cc} {L} _ {L} _ {bb} (\ cdot) & {L} _ {b\ Gamma} (\ cdot)\\ (sym\ text {.}) & {L} _ {\ Gamma} {L} _ {\ Gamma}} {L} {L} _ {b} Gamma} (\ Gamma}} (\ Gamma}} {L}} {L} _ {b} Gamma} (\ cdot) (\ cdot)\ {u} _ {\ Gamma} (t)\ end {array}\ right] +\ left [\ begin {array} {c} 0\\ {R} _ {\ Gamma} (t)\ end {array}\ right] =\ left [\ array}\ right] =\ left [\ begin {array}\ right] =\ left [\ begin {array}\ right] =\ left [\ begin {array}\ right] +\ left [\ begin {array}\ right] +\ left [\ begin {array}\ right] +\ left [\ begin {array}\ right] +\ left [\ begin {array}\ right] +\ left [\ begin {array}\ right] +\ left [\ begin {array}\ right] +\ left\ left [\ begin {array}\ right] +\ left\]\end{split}\]

where a differentiation operator \({L}_{\mathit{\alpha \beta }}(\cdot )\) in time is introduced for \(\alpha ,\beta \in \text{{}b,\Gamma \text{}}\), \(\Gamma\) for the degrees of freedom of the interface in contrast to the rest of the degrees of freedom of the system noted \(b\) and where the soil-structure interaction effort vector noted \({R}_{\Gamma }(t)\), corresponds to the convolution integral that will be discretized with the Laplace-Temps method.

The resolution of the coupled soil-structure system involves discretizing all the quantities in time and isolating the unknowns at time \(t=n\mathrm{\Delta }t\), where the time step is noted \(\mathrm{\Delta }t\). In particular, for soil-structure interaction efforts, it is interesting to start from equation () and to group the non-unknown terms corresponding to the moments preceding \(t=n\Delta t\) in a single term in order to separate them from the unknowns to be solved in the \(n\) time step. The corresponding \({R}_{n}\approx {R}_{\Gamma }(n\Delta t)\) vector is written as follows:

(4.2)\[ {R} _ {n} = {\ Psi} _ {2} _ {2} ^ {1} ^ {1} {\ ddot {u}}} _ {\ Psi} _ {1} {\ dot {u}} {\ dot {u}}} _ {\ dot {u}}}} _ {\ Gamma, n}} _ {\ Gamma, n} + {\ dot {u}} {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {-1)}\]

With \({\Psi }_{0}^{1}\), \({\Psi }_{1}^{1}\), and \({\Psi }_{2}^{1}\) referring to instantaneous stiffness, damping, and inertia respectively [8] _ and \({R}_{\Sigma (n-1)}\) depending only on the quantities calculated at the previous moments [9] _ :

\[\]

: label: eq-27

{R} _ {Sigma (n-1)}} =sum _ {k=1} ^ {n-1} ({Psi} _ {2} ^ {n-k+1} {ddot {u}} _ {Gamma, k} _ {Gamma, k}} + {Gamma, k}} _ {Gamma, k}} _ {Gamma, k}} _ {Gamma, k}} _ {Gamma, k}} _ {Gamma, k}} _ {Gamma, k}} _ {Gamma, k}} + {Gamma, k}} _ {Gamma, k}} _ {Gamma, k}} + {Gamma, k}} _ {Gamma, k}} _ {Gamma {n-k+1} {u} _ {Gamma, k})

This grouping of terms makes it possible to introduce to the second member the part of the convolution coming from the preceding moments and in the first member the terms relating to unknowns. Indeed, this can easily be demonstrated by starting from an on-the-go formulation of an unconditionally stable Newmark integration diagram (\(\beta =0\text{.}25\), \(\gamma =0\text{.}5\)) applied to equation ():

\[\]

: label: eq-28

left [begin {array} {cc} {stackrel {} {K}} {K}}} _ {11} & {stackrel {} {K}}} _ {12}\ {stackrel {} {K} {K}}} {K}}} _ {22}end {array}right]left [begin {array} {K}}} _ {22}end {array}right]left [begin {array} {K}}} _ {22}end {array}right]left [begin {array} {K}} {c} {u} _ {b, n}\ {u} _ {Gamma, n}end {array}right] =left [begin {array} {c} {F} _ {b, n} _ {b, n}\ {F}}\ {F}} _ {F} _ _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {(n-1)}end {array}right]

where the matrix \(\stackrel{̃}{K}\) corresponds to the Newmark operator that will have to be inverted in each time step and the second member to the sum of the equivalent Newmark vector and the known part of the discrete convolution product. In particular, the term \(\stackrel{̃}{{K}_{22}}\), which applies to the degrees of freedom in the \(\Gamma\) interface, is written as:

(4.3)\[ {\ stackrel {} {K}}} _ {22} =\ frac {1} =\ frac {1}} {\ beta\ Delta {t} ^ {2}}} ({M} _ {22} + {\ Psi} _ {2}} _ {2} _ {2} + {2} _ {2} + {\ Psi} _ {2} _ {2} + {\ Psi} _ {2} _ {2} + {2} _ {2} _ {2} _ {2} + {1}} _ {1} ^ {1}) + ({K} _ {22} + {\ Psi} _ {0} ^ {1})\]

It should be noted that the terms \({\Psi }_{0}^{1}\), \({\Psi }_{1}^{1}\) and \({\Psi }_{2}^{1}\) play a role in the packaging of the Newmark operator \(\stackrel{̃}{K}\) and therefore, their value has an impact on the convergence of the solution. Note also that \(\stackrel{~}{{K}_{22}}\), remaining constant during the calculation, is only calculated once.

[1]

In code_aster, these instant terms are assembled to the first member via the MACR_ELEM_DYNA operator (cf. [U4.65.01]) and possibly a LIAISON_INTERF link in the AFFE_CHAR_MECA operator (cf. [U4.44.01]) if a reduced modal interface base was used.

[2]

The term \({R}_{\Sigma (n-1)}\) is modeled in code_aster during the nonlinear dynamic analysis as a load on the second member obtained with the « CHARGE_SOL » option of the AFFE_CHAR_MECA operator (cf. [U4.44.01]).