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Add undamped oscillator test function
The six-dimensional undamped oscillator test function has been added to the codebase. The function is often used as a test funtion in reliability analysis; it is also used a metamodeling test function. The documentation has been accordingly updated. This commit should resolve Issue #325.
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| Original file line number | Diff line number | Diff line change |
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| --- | ||
| 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 | ||
| --- | ||
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| (test-functions:undamped-oscillator)= | ||
| # Undamped Oscillator | ||
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| ```{code-cell} ipython3 | ||
| import numpy as np | ||
| import matplotlib.pyplot as plt | ||
| import uqtestfuns as uqtf | ||
| ``` | ||
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| The undamped oscillator function (`UndampedOscillator`) is a six-dimensional, | ||
| scalar-valued test function that models a non-linear, undamped, | ||
| single-degree-of-freedom, forced oscillating mechanical system. | ||
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| This function is frequently used as a test function for reliability analysis | ||
| methods (see {cite}`Bucher1990, Rajashekhar1993, Gayton2003, Schueremans2005, Echard2011, Echard2013`). | ||
| Additionally, in {cite}`Luethen2021`, the function is employed | ||
| as a test function for metamodeling exercises. | ||
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| ## Test function instance | ||
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| To create a default instance of the test function: | ||
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| ```{code-cell} ipython3 | ||
| my_testfun = uqtf.UndampedOscillator() | ||
| ``` | ||
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| Check if it has been correctly instantiated: | ||
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| ```{code-cell} ipython3 | ||
| print(my_testfun) | ||
| ``` | ||
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| ## Description | ||
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| The system under consideration is a single-degree-of-freedom mechanical system | ||
| that undergoes undamped forced oscillation. | ||
| The performance function is analytically defined as follows[^location]: | ||
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| $$ | ||
| g(\boldsymbol{x}) = 3 r - \lvert z_{\text{max}} \rvert, | ||
| $$ | ||
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| where $z_{\text{max}}$ is the maximum displacement response of the system | ||
| given by | ||
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| $$ | ||
| z_{\text{max}} (\boldsymbol{x}) = \frac{2 F_1}{m \omega_0^2} \sin{\left( \frac{\omega_0 t_1}{2} \right)} | ||
| $$ | ||
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| and | ||
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| $$ | ||
| \omega_0 = \sqrt{\frac{c_1 + c2}{m}}. | ||
| $$ | ||
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| $\boldsymbol{x} = \{ m, c_1, c_2, r, F_1, t_1 \}$ is the six-dimensional vector | ||
| of input variables probabilistically defined further below. | ||
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| The failure state and the failure probability are defined as | ||
| $g(\boldsymbol{x}; \boldsymbol{p}) \leq 0$ | ||
| and $\mathbb{P}[g(\boldsymbol{X}; \boldsymbol{p}) \leq 0]$, respectively. | ||
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| ## Probabilistic input | ||
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| The available probabilistic input models are shown in the table below. | ||
| The different specifications alter the failure probability of the system | ||
| (as expected). | ||
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| ```{table} Available probabilistic input of the undamped oscillator function | ||
| :name: undamped-oscillator-inputs | ||
| | No. | Remark | Keyword | Source | | ||
| |:---:|:------------------:|:----------------------:|:----------------------------:| | ||
| | 1. | $\mu_{F_1} = 1.00$ | `Gayton2003` (default) | {cite}`Gayton2003` (Table 9) | | ||
| | 2. | $\mu_{F_1} = 0.60$ | `Echard2013-1` | {cite}`Echard2013` (Table 4) | | ||
| | 3. | $\mu_{F_1} = 0.45} | `Echard2013-2` | {cite}`Echard2013` (Table 4) | | ||
| ``` | ||
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| The default input is shown below. | ||
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| ```{code-cell} ipython3 | ||
| :tags: [hide-input] | ||
| print(my_testfun.prob_input) | ||
| ``` | ||
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| ## Reference results | ||
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| This section provides several reference results of typical UQ analyses involving | ||
| the test function. | ||
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| ### Sample histogram | ||
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| Shown below is the histogram of the output based on $10^6$ random points: | ||
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| ```{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("$g(\mathbf{X})$") | ||
| plt.gcf().set_dpi(150); | ||
| ``` | ||
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| ### Failure probability ($P_f$) | ||
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| Some reference values for the failure probability $P_f$ from the literature | ||
| are summarized in the tables below according to the chosen input specification. | ||
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| ::::{tab-set} | ||
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| :::{tab-item} Gayton2003 | ||
| | Method | $N$ | $\hat{P}_f$ | $\mathrm{CoV}[\hat{P}_f]$ | Source | | ||
| |:--------------------------------------:|:-----------------:|:----------------------:|:-------------------------:|:---------------------------------:| | ||
| | {term}`DS` | $1281$ | $3.5 \times 10^{-2}$ | — | {cite}`Schueremans2005` (Table 5) | | ||
| | {term}`DS` + Polynomial | $62$ | $3.4 \times 10^{-2}$ | — | {cite}`Schueremans2005` (Table 5) | | ||
| | {term}`DS` + Splines | $76$ | $3.4 \times 10^{-2}$ | — | {cite}`Schueremans2005` (Table 5) | | ||
| | {term}`DS` + Neural network (NN) | $86$ | $2.8 \times 10^{-2}$ | — | {cite}`Schueremans2005` (Table 5) | | ||
| | {term}`MCS` + {term}`IS` | $6144$ | $2.7 \times 10^{-2}$ | — | {cite}`Schueremans2005` (Table 5) | | ||
| | {term}`MCS` + {term}`IS` + Polynomials | $109$ | $2.5 \times 10^{-2}$ | — | {cite}`Schueremans2005` (Table 5) | | ||
| | {term}`MCS` + {term}`IS` + Splines | $67$ | $2.7 \times 10^{-2}$ | — | {cite}`Schueremans2005` (Table 5) | | ||
| | {term}`MCS` + {term}`IS` + NN | $68$ | $3.1 \times 10^{-2}$ | — | {cite}`Schueremans2005` (Table 5) | | ||
| | {term}`MCS` | $7 \times 10^{4}$ | $2.834 \times 10^{-2}$ | $2.2\%$ | {cite}`Echard2011` (Table 6) | | ||
| | Adaptive Kriging + {term}`MCS` + EFF | $58$ | $2.834 \times 10^{-2}$ | — | {cite}`Echard2011` (Table 6) | | ||
| | Adaptive Kriging + {term}`MCS` + U | $45$ | $2.851 \times 10^{-2}$ | — | {cite}`Echard2011` (Table 6) | | ||
| ::: | ||
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| :::{tab-item} Sobol1999-2 | ||
| | Method | $N$ | $\hat{P}_f$ | $\mathrm{CoV}[\hat{P}_f]$ | Source | | ||
| |:-----------------------------:|:--------------------:|:---------------------:|:-------------------------:|:----------------------------:| | ||
| | {term}`MCS` | $1.8 \times 10^{8}$ | $9.09 \times 10^{-6}$ | $2.47\%$ | {cite}`Echard2013` (Table 5) | | ||
| | {term}`FORM` | $29$ | $9.76 \times 10^{-6}$ | — | {cite}`Echard2013` (Table 5) | | ||
| | {term}`IS` | $20 + \times 10^{4}$ | $9.13 \times 10^{-6}$ | $2.29\%$ | {cite}`Echard2013` (Table 5) | | ||
| | Adaptive Kriging + {term}`IS` | $29 + 38$ | $9.13 \times 10^{-6}$ | $2.29\%$ | {cite}`Echard2013` (Table 5) | | ||
| ::: | ||
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| :::{tab-item} Moon2012-1 | ||
| | Method | $N$ | $\hat{P}_f$ | $\mathrm{CoV}[\hat{P}_f]$ | Source | | ||
| |:-----------------------------:|:--------------------:|:---------------------:|:-------------------------:|:----------------------------:| | ||
| | {term}`MCS` | $9 \times 10^{8}$ | $1.55 \times 10^{-8}$ | $2.68\%$ | {cite}`Echard2013` (Table 5) | | ||
| | {term}`FORM` | $29$ | $1.56 \times 10^{-8}$ | — | {cite}`Echard2013` (Table 5) | | ||
| | {term}`IS` | $29 + \times 10^{4}$ | $1.53 \times 10^{-8}$ | $2.70\%$ | {cite}`Echard2013` (Table 5) | | ||
| | Adaptive Kriging + {term}`IS` | $29 + 38$ | $1.54 \times 10^{-8}$ | $2.70\%$ | {cite}`Echard2013` (Table 5) | | ||
| ::: | ||
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| :::: | ||
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| ## References | ||
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| ```{bibliography} | ||
| :style: unsrtalpha | ||
| :filter: docname in docnames | ||
| ``` | ||
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| [^location]: see Section 6.4.1 in {cite}`Gayton2003`. |
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