Calculation of impedance in time ============================== As mentioned earlier, the Laplace-Temps method is a numerical approach that makes it possible to discretize a convolutional integral in time. Implementation in code_aster ---------------------------- Equation () shows that the numerical evaluation of weights:math: `{\ Psi} _ {\ Psi} _ {k} ^ {j}` first goes through that of:math: `{\ widehat {Z}}} _ {m}} _ {m} ^ {k}`. In turn, this can be obtained from a Cauchy integral applied to a contour:math: `|z|=\ mathrm {\ rho} `: .. csv-table:: ":math:`{Z}_{m}(k\mathrm{\Delta }t)=\frac{1}{2\mathrm{\pi }i}{\int }_{", "z", "=\mathrm{\rho }}{\widehat{Z}}_{m}\left(\frac{\mathrm{\delta }(z)}{\mathrm{\Delta }t}\right){z}^{-k-1}dz` ", "(22)" If we express this integral in polar coordinates :math:`z=\mathrm{\rho }{e}^{i\mathrm{\theta }}`, then we apply the trapezoidal rule to the phase to discretize it into :math:`L` not identical with a value of :math:`\mathrm{\Delta }\mathrm{\theta }=\frac{2\mathrm{\pi }}{L}`, we obtain: .. math:: :label: eq-23 {Z} _ {m} (k∆t)\ approx {Z} _ {Z} _ {m} _ {m} ^ {k} =\ frac {{\ rho}} {L}\ sum _ {l=0} ^ {0} ^ {L-1} ^ {L-1} ^ {L-1} ^ {L-1} ^ {L-1} ^ {L-1} ^ {L-1} ^ {L-1} ^ {L-1} {L-1} {L-1} {\ L-1} {\ L-1} {\ widehat {Z}}} _ {m} ({s} _ {l}) {e} ^ {-i\ frac {2\ frac {2\ pi l} {L} k} where the operator defined in the Laplace domain is sampled on :math:`{s}_{l}=\frac{\mathrm{\delta }(\mathrm{\rho }{e}^{2\mathrm{\pi }il∕L})}{\mathrm{\Delta }t}` with :math:`\mathrm{\delta }(z)=\frac{3}{2}-2z+\frac{1}{2}{z}^{2}` the characteristic polynomial of a second-order multi-step linear method. Note that :math:`N` and :math:`L` are not necessarily equal, :math:`N` being the number of time steps in the time window of interest (for example, as a function of the duration of the seismic signal) and :math:`L` being the number of time steps for calculating the impedance in time. If we consider that :math:`{\widehat{Z}}_{m}({s}_{l})` is evaluated with variability [2] _ :math:`{ϵ}_{\mathit{CQM}}`, the :math:`{\widehat{Z}}_{m}^{k}` coefficients will be calculated with precision [3] _ :math:`O(\sqrt{{ϵ}_{\mathit{CQM}}})` if :math:`L=N` and :math:`{\rho }^{N}=\sqrt{{ϵ}_{\mathit{CQM}}}`. With the values of :math:`L` and :math:`\rho` fixed, the weights of the series can be obtained from a classical FFT, which reduces the complexity of the calculation algorithm to :math:`O(L\mathrm{log}L)` instead of :math:`O({L}^{2})`. However, the value of :math:`{ϵ}_{\mathit{CQM}}` is not an easy step to calibrate and depends in part on the nature of the problem to be solved. In fact, values that are too low of :math:`{ϵ}_{\mathit{CQM}}`, such as :math:`{10}^{-20}` or :math:`{10}^{-30}`, may result in integrative radii :math:`\mid z\mid =\rho` that are too small. Since the Cauchy integrand contains a zero singularity, too small values of the radius may result in a poor evaluation of the integral. As such, parametric studies were carried out in to conclude that a good compromise is reached when :math:`{ϵ}_{\mathit{CQM}}={10}^{-10}` [4] _ . Likewise, it will be noted that the dependence in :math:`{\rho }^{-k}` leads, for large :math:`k` values, to inaccuracies in the calculated :math:`{\widehat{Z}}_{m}^{k}` coefficients. Knowing that the use of more precise integration methods, such as Simpson for example, does not correct this problem, the idea is to extend the time range (i.e. :math:`L=\mathit{mN},m\mathrm{\in }\mathrm{ℝ}`) over which the impedance is calculated in order to reduce disturbances on the solution of interest. As such, it has been observed that, in general, the inaccuracies of :math:`{\widehat{Z}}_{m}^{k}` only spread to the response beyond :math:`t\approx \mathrm{0,7}{T}_{\mathit{FIN}}`, where :math:`{T}_{\mathit{FIN}}` is the final moment for calculating the impedance [5] _ . Key numerical considerations ------------------------------------- We saw earlier that, in general, the singular part of the ground impedance can be approximated by a polynomial of order two. If the coefficients of this polynomial are real, it is easy to demonstrate that the ground impedance meets :math:`\widehat{Z}(\sigma -i\omega )=\mathit{conj}[\widehat{Z}(\sigma +i\omega )]` in the analytical plane of the Laplace domain [6] _ . It will be noted that when hysteretic damping is used in the ground (case of MISS3D), the imaginary part of its static stiffness is non-zero and the hypothesis of real coefficients is no longer verified. A second interesting property coming directly from Laplace's definition of unilateral transform is the following: .. math:: :label: eq-24 \ widehat {Z} (s=0) = {\ int} _ {0} _ {0} ^ {0} ^ {0} ^ {N} ^ {N} Z (k\ Delta t)\ approx\ sum _ {k} ^ {N} Z (k\ Delta t) for :math:`N` chosen big enough (~ a hundred steps of time). Note that the ground impedance evaluated at :math:`s=0` corresponds to the static ground stiffness (real part), which is very different from the instantaneous ground stiffness. This must be taken into account when a dynamic calculation (:math:`Z(t=0)]` is assembled at the first limb) is performed following a static calculation (:math:`\mathrm{\Re }[\widehat{Z}(s=0)]` is assembled at the first limb) [7] _ . .. [1] Referring to the concept of reproducibility (precision in English). .. [2] Referring to the concept of accuracy (accuracy in English). .. [3] By default, this value initializes the PRECISION keyword for the CALC_MISS operator (cf. [U7.03.12]) when the expected result type is' FICHIER_TEMPS '. .. [4] To do this, CALC_MISS (TYPE_RESU = 'FICHIER_TEMPS') provides for the keyword COEF_SURECH (cf. [U7.03.12]) initialized to 1.35, which is the equivalent of doing :math:`L=\mathrm{1.35N}` in equation 23. .. [5] This property is used in code_aster: the impedance is sampled on the upper half-plane, then completed numerically with the conjugate. Therefore, care must be taken with soils with very strong hysteretic damping (greater than 25%). .. [6] Code_aster has a macro-command (PRE_SEISME_NONL, cf. [U4.63.02])) that makes it possible to perform the static-dynamic transition of a non-linear soil-structure interaction calculation.