--- jupytext: formats: ipynb,md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.14.1 kernelspec: display_name: Python 3 (ipykernel) language: python name: python3 --- (test-functions:damped-oscillator-reliability)= # Damped Oscillator Reliability ```{code-cell} ipython3 import numpy as np import matplotlib.pyplot as plt import uqtestfuns as uqtf ``` The damped oscillator reliability problem is an eight-dimensional reliability analysis test function {cite}`DerKiureghian1990, DerKiureghian1991, Bourinet2011, Dubourg2011`. ```{note} The reliability analysis variant differs from the {ref}`base model `. The base model computes the mean-square relative displacement of the secondary spring without reference to the performance of the system. ``` ## Test function instance To create a default instance of the test function: ```{code-cell} ipython3 my_testfun = uqtf.DampedOscillatorReliability() ``` Check if it has been correctly instantiated: ```{code-cell} ipython3 print(my_testfun) ``` ## Description The problem poses the reliability of a damped oscillator system as defined in {cite}`Igusa1985` (see {ref}`Damped Oscillator `). The reliability of the system depends on the secondary spring as described by the following performance function {cite}`DerKiureghian1990, Dubourg2011` [^location]: $$ g(\boldsymbol{x}; p) = F_s - p k_s \mathcal{M}(M_p, M_s, K_p, K_s, \zeta_p, \zeta_s, S_0), $$ $$ \mathcal{M}(M_p, M_s, K_p, K_s, \zeta_p, \zeta_s, S_0) = \left( \pi \frac{S_0}{4 \zeta_s \omega_s^3} \frac{\zeta_a \zeta_s}{\zeta_p \zeta_s (4 \zeta_a^2 + \theta^2) + \gamma \zeta_a^2} \frac{(\zeta_p \omega_p^3 + \zeta_s \omega_s^3) \omega_p}{4 \zeta_a \omega_a^4} \right)^{0.5} $$ $$ \begin{aligned} \omega_p & = \left( \frac{k_p}{m_p}\right)^{0.5} & \omega_s & = \left(\frac{k_s}{m_s}\right)^{0.5} & \omega_a & = \frac{\omega_p + \omega_s}{2}\\ \gamma & = \frac{m_s}{m_p} & \zeta_a & = \frac{\zeta_p + \zeta_s}{2} & \theta & = \frac{\omega_p - \omega_s}{\omega_a} \\ \end{aligned} $$ where $\boldsymbol{x} = \{ M_p, M_s, K_p, K_s, \zeta_p, \zeta_s, S_0, F_s \}$ is the eight-dimensional vector of input variables probabilistically defined further below and $p$ is a deterministic parameter of the function (i.e., the peak factor). The failure state and the failure probability are defined as $g(\boldsymbol{x}; p) \leq 0$ and $\mathbb{P}[g(\boldsymbol{X}; p) \leq 0]$, respectively. ## Probabilistic input Based on {cite}`DerKiureghian1991`, the probabilistic input model for the damped oscillator reliability consists of eight independent random variables with marginal distributions shown in the table below. ```{code-cell} ipython3 :tags: [hide-input] print(my_testfun.prob_input) ``` In the literature, the force capacity of the secondary spring (i.e., $F_s$) have different probabilistic specifications as summarized in the table below. | $F_S$ | Keyword | Source | |:----------------------------------------:|:------------------------------:|:-------------------------:| | $\mathcal{N}_{\mathrm{log}}(15.0, 1.50)$ | `DerKiureghian1990a` (default) | {cite}`DerKiureghian1990` | | $\mathcal{N}_{\mathrm{log}}(21.5, 2.15)$ | `DerKiureghian1990b` | {cite}`DerKiureghian1990` | | $\mathcal{N}_{\mathrm{log}}(27.5, 2.75)$ | `DerKiureghian1990c` | {cite}`DerKiureghian1990` | Note that the parameters of the log-normal distribution given above correspond to the mean and standard deviation of the log-normal distribution (and not the mean and standard deviation of the underlying normal distribution). With the higher mean of $F_s$, the failure probability becomes smaller. ## Parameters The performance function contains a single parameter whose default value is shown below {cite}`DerKiureghian1991`. ```{code-cell} ipython3 :tags: [hide-input] print(my_testfun.parameters) ``` ## Reference results This section provides several reference results of typical UQ analyses involving the test function. ### Sample histogram Shown below is the histogram of the output based on $10^6$ random points: ```{code-cell} ipython3 :tags: [hide-input] xx_test = my_testfun.prob_input.get_sample(1000000) yy_test = my_testfun(xx_test) idx_pos = yy_test > 0 idx_neg = yy_test <= 0 hist_pos = plt.hist(yy_test, bins="auto", color="#0571b0") plt.hist(yy_test[idx_neg], bins=hist_pos[1], color="#ca0020") plt.axvline(0, linewidth=1.0, color="#ca0020") plt.grid() plt.ylabel("Counts [-]") plt.xlabel("$\mathcal{M}(\mathbf{X})$") plt.gcf().set_dpi(150); ``` ### Failure probability Some reference values for the failure probability $P_f$ and from the literature are summarized in the table below ($\mu_{F_s}$ is the log-normal distribution mean of $F_s$). | $\mu_{F_s}$ | Method | $N$ | $\hat{P}_f$ | $\mathrm{CoV}[\hat{P}_f]$ | Source | |:-----------:|:------------------------:|:---------------:|:---------------------:|:-------------------------:|:---------------------------------------:| | $15$ | {term}`FORM` | $1'179$ | $2.19 \times 10^{-2}$ | — | {cite}`DerKiureghian1990, Bourinet2011` | | | {term}`SS` | $3 \times 10^5$ | $4.63 \times 10^{-3}$ | $< 3\%$ | {cite}`Dubourg2011` | | | Meta-IS[^meta-is] | $464 + 600$ | $4.80 \times 10^{-3}$ | $< 5\%$ | {cite}`Dubourg2011` | | | {term}`SVM` + {term}`SS` | $1'719$ | $4.78 \times 10^{-3}$ | $< 4\%$ | {cite}`Bourinet2011` | | $21.5$ | {term}`FORM` | $2'520$ | $3.50 \times 10^{-4}$ | — | {cite}`DerKiureghian1990, Bourinet2011` | | | {term}`SS` | $5 \times 10^5$ | $4.75 \times 10^{-5}$ | $< 4\%$ | {cite}`Dubourg2011` | | | Meta-{term}`IS` | $336 + 400$ | $4.46 \times 10^{-5}$ | $< 5\%$ | {cite}`Dubourg2011` | | | {term}`SVM` + {term}`SS` | $2'865$ | $4.42 \times 10^{-5}$ | $< 7\%$ | {cite}`Bourinet2011` | | $27.5$ | {term}`FORM` | $2'727$ | $3.91 \times 10^{-6}$ | — | {cite}`DerKiureghian1990, Bourinet2011` | | | {term}`SS` | $7 \times 10^5$ | $3.47 \times 10^{-7}$ | $< 5\%$ | {cite}`Dubourg2011` | | | Meta-IS | $480 + 200$ | $3.76 \times 10^{-7}$ | $< 5\%$ | {cite}`Dubourg2011` | | | {term}`SVM` + {term}`SS` | $4'011$ | $3.66 \times 10^{-7}$ | $< 10\%$ | {cite}`Bourinet2011` | Note that in the table above the total number of model evaluations for metamodel-based Importance Sampling (Meta-IS) is the sum of training runs and the correction runs {cite}`Dubourg2011`. ## References ```{bibliography} :style: unsrtalpha :filter: docname in docnames ``` [^location]: see, for instance, Eqs. (5.5) and (5.7), pp. 184-185 in {cite}`Dubourg2011`. [^meta-is]: Metamodel-based Importance Sampling