commit a846cd78265af269dd0705b0782b65e83b494e05
parent 06311ada020897c923caac7aea289a9dd545e58f
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Mon, 31 Oct 2016 22:53:54 +0300
Add outline for the second part!
Diffstat:
| phd-diss-ru.org | | | 37 | +++++++++++++++++++++++-------------- |
| phd-diss.org | | | 247 | +++++++++++++++++++++++++++++++++++++++++-------------------------------------- |
2 files changed, 150 insertions(+), 134 deletions(-)
diff --git a/phd-diss-ru.org b/phd-diss-ru.org
@@ -281,23 +281,14 @@ $\zeta_z=k\zeta$, где $k$ --- волновое число. Формула я
Формула дифференцируется для получения производных потенциала, а полученные
значения подставляются в динамическое граничное условие для вычисления давлений.
-* Применение модели АРСС в задаче имитационного моделирования морского волнения
+* Модель АРСС в задаче имитационного моделирования морского волнения
** Предпосылки к поиску новой модели ветрового волнения
-** Форма АКФ для разных волновых профилей
-*** Два метода для определения формы АКФ
-**** Аналитический метод.
-**** Эмпирический метод.
-*** Примеры АКФ для различных волновых профилей
-**** АКФ стоячей волны.
-**** АКФ прогрессивной волны.
-*** Сравнение изученных методов
** Основные формулы трехмерного процесса AРСС
*** Три возможных процесса
**** Процесс авторегрессии (АР).
**** Процесс скользящего среднего (СС).
**** Смешанный процесс авторегрессии скользящего среднего (АРСС).
*** Критерии выбора процесса для моделирования разных профилей волн
-** Верификация интегральных характеристик взволнованной поверхности
** Моделирование нелинейности морских волн
Модель АРСС позволяет учесть асимметричность распределения волновых аппликат,
т.е. сгенерировать морские волны, закон распределения аппликат которых имеет
@@ -341,7 +332,7 @@ $y_k|_{k=0}^N$ сетки сгенерированной поверхности
где
\begin{equation*}
C_m = \frac{1}{\sqrt{2\pi}}
- \int\limits_{0}^\infty
+ \int\limits_{0}^\infty
f(y) H_m(y) \exp\left[ -\frac{y^2}{2} \right],
\end{equation*}
$H_m$ --- полином Эрмита, а $f(y)$ --- решение
@@ -376,9 +367,27 @@ $\epsilon$:
бисекции. Использование полиномиальной аппроксимацией в формулах для
коэффициентов ряда Грама---Шарлье не приводит к аналогичным ошибкам.
-** Нефизическая природа модели
-* Постановка численного эксперимента
-* Определение поля давлений под дискретно заданной взволнованной поверхностью
+** Определение поля давлений под дискретно заданной взволнованной поверхностью
+* Численные методы и результаты экспериментов
+** Форма АКФ для разных волновых профилей
+*** Два метода для определения формы АКФ
+**** Аналитический метод.
+**** Эмпирический метод.
+*** Примеры АКФ для различных волновых профилей
+**** АКФ стоячей волны.
+**** АКФ прогрессивной волны.
+*** Сравнение изученных методов
+** Дополнительные формулы, методы и алгоритмы для модели АРСС
+*** Аппроксимация распределения аппликат.
+*** Генерация белого шума.
+*** Генерация взволнованной поверхности.
+** Верификация модели АРСС
+*** Методика постановки численных экспериментов
+*** Верификация интегральных характеристик взволнованной поверхности
+*** Верификация полей потенциалов скоростей
+**** Отличие от формул линейной теории.
+**** Отличие от формул теории волн малой амплитуды.
+*** TODO Нефизическая природа модели
* Высокопроизводительный программный комплекс для моделирования морского волнения
* Заключение
* Благодарности
diff --git a/phd-diss.org b/phd-diss.org
@@ -240,7 +240,126 @@ MA process for propagating waves. With new formulae for 3 dimensions a single
mixed ARMA process might be a better choice, but this is the objective of the
future research.
-** Verification of wavy surface integral characteristics
+** Modeling non-linearity of ocean waves
+** Determining wave pressures for discretely given wavy surface
+* Numerical methods and experimental results
+** The shape of ACF for different types of waves
+*** Two methods to find ocean wave's ACF
+**** Analytic method of finding the ACF.
+The simplest way to find auto-covariate function for a particular ocean wave
+profile is to apply Wiener---Khinchin theorem. According to this theorem the
+autocorrelation $K$ of a function $\zeta$ is given by the Fourier transform of
+the absolute square of the function:
+\begin{equation}
+ K(t) = \Fourier{\left| \zeta(t) \right|^2}.
+ \label{eq:wiener-khinchin}
+\end{equation}
+When $\zeta$ is replaced with actual wave profile, this formula gives you
+analytic expression for the corresponding ACF.
+
+For three-dimensional wave profile (2D in space and 1D in time) analytic
+expression is a polynomial of high order and is best obtained via computer
+algebra software. Then for practical usage it can be approximated by
+superposition of exponentially decaying cosines (which is how ACF of a
+stationary ARMA process looks like cite:box1976time).
+
+**** Empirical method of finding the ACF.
+However, for three-dimensional case there exists simpler empirical method which
+does not require sophisticated software to determine shape of the ACF. It is
+known that ACF represented by exponentially decaying cosines of a wave profile
+satisfies first order Stokes' equations for gravity waves cite:boccotti1983wind.
+So, if the shape of the wave profile is the only concern, then one can simply
+multiply it by a decaying exponent to get appropriate ACF. This ACF will not
+reflect other wave profile parameters such as wave height and period, but opens
+possibility to simulate waves of a particular non-analytic shape by "drawing"
+their profile, then multiplying it by an exponent and using the resulting
+function as ACF. So, this empirical method is imprecise but offers simpler
+alternative to Wiener---Khinchin theorem; it is mainly useful to test ARMA
+model.
+
+*** Examples of ACFs for various types of wave profiles
+**** Standing wave.
+For three-dimensional standing wave the profile is approximated by
+\begin{equation}
+ \zeta(t, x, y) = A \sin (k_x x + k_y y) \sin (\sigma t).
+ \label{eq:standing-wave}
+\end{equation}
+In order to get ACF via analytic method one needs to multiply this expression by
+a decaying exponent, because Fourier transform is defined for a function $f$ that
+$f \underset{x \rightarrow \pm \infty}{\longrightarrow} 0$. The formula of the
+profile then transforms to
+\begin{equation}
+ \zeta(t, x, y) =
+ A
+ \exp\left[-\alpha (|t|+|x|+|y|) \right]
+ \sin (k_x x + k_y y) \sin (\sigma t).
+ \label{eq:decaying-standing-wave}
+\end{equation}
+Then, if one takes 3D Fourier transform of this expression via any capable
+computer algebra software, the resulting polynomial may be fitted to the
+following ACF approximation.
+\begin{equation}
+ K(t,x,y) =
+ \gamma
+ \exp\left[-\alpha (|t|+|x|+|y|) \right]
+ \cos \beta t
+ \cos \left[ \beta x + \beta y \right].
+ \label{eq:standing-wave-acf}
+\end{equation}
+So, after applying Wiener---Khinchin theorem we get the same formula but with
+sines replaced with cosines. This replacement is important because the value of
+ACF at $(0,0,0)$ equals to the variance of wave elevation, and if one used sines
+the value would be wrong.
+
+If one tries to replicate the same formula via empirical method, the usual way
+is to adapt eqref:eq:decaying-standing-wave to match eqref:eq:standing-wave-acf.
+This can be done by changing the phase of the sine, or by replacing sine with
+cosine to move the maximum of the function to $(0,0,0)$.
+
+**** Propagating wave.
+Three-dimensional profile of this type of wave is approximated by
+\begin{equation}
+ \zeta(t, x, y) = A \cos (\sigma t + k_x x + k_y y).
+ \label{eq:propagating-wave}
+\end{equation}
+For the analytic method one may repeat steps from the previous two paragraphs
+with ACF approximated by
+\begin{equation}
+ K(t,x,y) =
+ \gamma
+ \exp\left[-\alpha (|t|+|x|+|y|) \right]
+ \cos\left[\beta (t+x+y) \right].
+ \label{eq:propagating-wave-acf}
+\end{equation}
+For the empirical method propagating wave profile is simply multiplied by
+a decaying exponent without need to adapt the maximum value of ACF.
+
+*** Comparison of studied methods
+To summarise, the analytic method of finding ocean wave's ACF reduces to the
+following steps:
+- Make wave profile decay when approach $\pm \infty$ by multiplying it by
+ a decaying exponent.
+- Take Fourier transform of absolute square of the decaying wave profile using
+ computer algebra software.
+- Fit the resulting polynomial to the appropriate ACF approximation.
+
+Two examples in this section showed that in case of standing and propagating
+waves their decaying profiles resemble the corresponding ACFs with the exception
+that the origin should be moved to the function's maximal value for the ACF to
+be useful in ARMA model simulations. So, using the empirical method the ACF is
+found in the following steps:
+- Make wave profile decay when approach $\pm \infty$ by multiplying it by
+ a decaying exponent.
+- Move maximum value to the origin by adjusting phases or using trigonometric
+ identities to shift the phase of the resulting function.
+
+** Additional formulae, methods and algorithms for ARMA model
+*** Wave elevation distribution approximation
+*** White noise generation
+*** Wavy surface generation
+** ARMA model verification
+*** Numerical experiments implementation methodology
+*** Verification of wavy surface integral characteristics
Research shows cite:рожков1990вероятностные that several ocean wave
characteristics (e.g. wave height, wave period, wave length etc.) have Weibull
distribution differing only in shape parameter (tab. [[tab:weibull-shape]]), and
@@ -324,11 +443,11 @@ exit
#+caption: Time slices of ACF function for standing (left column) and propagating waves (right column).
#+name: fig:acf-plots
-| \includegraphics{standing-acf-0} | \includegraphics{propagating-acf-00} |
-| \includegraphics{standing-acf-1} | \includegraphics{propagating-acf-01} |
-| \includegraphics{standing-acf-2} | \includegraphics{propagating-acf-02} |
-| \includegraphics{standing-acf-3} | \includegraphics{propagating-acf-03} |
-| \includegraphics{standing-acf-4} | \includegraphics{propagating-acf-04} |
+| \includegraphics{standing-acf-0} | \includegraphics{propagating-acf-00} |
+| \includegraphics{standing-acf-1} | \includegraphics{propagating-acf-01} |
+| \includegraphics{standing-acf-2} | \includegraphics{propagating-acf-02} |
+| \includegraphics{standing-acf-3} | \includegraphics{propagating-acf-03} |
+| \includegraphics{standing-acf-4} | \includegraphics{propagating-acf-04} |
#+caption: Quantile-quantile plots for standing waves.
#+name: fig:standing-wave-distributions
@@ -341,119 +460,8 @@ exit
| \includegraphics{propagating-wave-length-x} | \includegraphics{propagating-wave-period} |
*** TODO Discuss graphs
-
-** The shape of ACF for different types of waves
-*** Two methods to find ocean wave's ACF
-**** Analytic method of finding the ACF.
-The simplest way to find auto-covariate function for a particular ocean wave
-profile is to apply Wiener---Khinchin theorem. According to this theorem the
-autocorrelation $K$ of a function $\zeta$ is given by the Fourier transform of
-the absolute square of the function:
-\begin{equation}
- K(t) = \Fourier{\left| \zeta(t) \right|^2}.
- \label{eq:wiener-khinchin}
-\end{equation}
-When $\zeta$ is replaced with actual wave profile, this formula gives you
-analytic expression for the corresponding ACF.
-
-For three-dimensional wave profile (2D in space and 1D in time) analytic
-expression is a polynomial of high order and is best obtained via computer
-algebra software. Then for practical usage it can be approximated by
-superposition of exponentially decaying cosines (which is how ACF of a
-stationary ARMA process looks like cite:box1976time).
-
-**** Empirical method of finding the ACF.
-However, for three-dimensional case there exists simpler empirical method which
-does not require sophisticated software to determine shape of the ACF. It is
-known that ACF represented by exponentially decaying cosines of a wave profile
-satisfies first order Stokes' equations for gravity waves cite:boccotti1983wind.
-So, if the shape of the wave profile is the only concern, then one can simply
-multiply it by a decaying exponent to get appropriate ACF. This ACF will not
-reflect other wave profile parameters such as wave height and period, but opens
-possibility to simulate waves of a particular non-analytic shape by "drawing"
-their profile, then multiplying it by an exponent and using the resulting
-function as ACF. So, this empirical method is imprecise but offers simpler
-alternative to Wiener---Khinchin theorem; it is mainly useful to test ARMA
-model.
-
-*** Examples of ACFs for various types of wave profiles
-**** Standing wave.
-For three-dimensional standing wave the profile is approximated by
-\begin{equation}
- \zeta(t, x, y) = A \sin (k_x x + k_y y) \sin (\sigma t).
- \label{eq:standing-wave}
-\end{equation}
-In order to get ACF via analytic method one needs to multiply this expression by
-a decaying exponent, because Fourier transform is defined for a function $f$ that
-$f \underset{x \rightarrow \pm \infty}{\longrightarrow} 0$. The formula of the
-profile then transforms to
-\begin{equation}
- \zeta(t, x, y) =
- A
- \exp\left[-\alpha (|t|+|x|+|y|) \right]
- \sin (k_x x + k_y y) \sin (\sigma t).
- \label{eq:decaying-standing-wave}
-\end{equation}
-Then, if one takes 3D Fourier transform of this expression via any capable
-computer algebra software, the resulting polynomial may be fitted to the
-following ACF approximation.
-\begin{equation}
- K(t,x,y) =
- \gamma
- \exp\left[-\alpha (|t|+|x|+|y|) \right]
- \cos \beta t
- \cos \left[ \beta x + \beta y \right].
- \label{eq:standing-wave-acf}
-\end{equation}
-So, after applying Wiener---Khinchin theorem we get the same formula but with
-sines replaced with cosines. This replacement is important because the value of
-ACF at $(0,0,0)$ equals to the variance of wave elevation, and if one used sines
-the value would be wrong.
-
-If one tries to replicate the same formula via empirical method, the usual way
-is to adapt eqref:eq:decaying-standing-wave to match eqref:eq:standing-wave-acf.
-This can be done by changing the phase of the sine, or by replacing sine with
-cosine to move the maximum of the function to $(0,0,0)$.
-
-**** Propagating wave.
-Three-dimensional profile of this type of wave is approximated by
-\begin{equation}
- \zeta(t, x, y) = A \cos (\sigma t + k_x x + k_y y).
- \label{eq:propagating-wave}
-\end{equation}
-For the analytic method one may repeat steps from the previous two paragraphs
-with ACF approximated by
-\begin{equation}
- K(t,x,y) =
- \gamma
- \exp\left[-\alpha (|t|+|x|+|y|) \right]
- \cos\left[\beta (t+x+y) \right].
- \label{eq:propagating-wave-acf}
-\end{equation}
-For the empirical method propagating wave profile is simply multiplied by
-a decaying exponent without need to adapt the maximum value of ACF.
-
-*** Comparison of studied methods
-To summarise, the analytic method of finding ocean wave's ACF reduces to the
-following steps:
-- Make wave profile decay when approach $\pm \infty$ by multiplying it by
- a decaying exponent.
-- Take Fourier transform of absolute square of the decaying wave profile using
- computer algebra software.
-- Fit the resulting polynomial to the appropriate ACF approximation.
-
-Two examples in this section showed that in case of standing and propagating
-waves their decaying profiles resemble the corresponding ACFs with the exception
-that the origin should be moved to the function's maximal value for the ACF to
-be useful in ARMA model simulations. So, using the empirical method the ACF is
-found in the following steps:
-- Make wave profile decay when approach $\pm \infty$ by multiplying it by
- a decaying exponent.
-- Move maximum value to the origin by adjusting phases or using trigonometric
- identities to shift the phase of the resulting function.
-
-** Modeling non-linearity of ocean waves
-** Non-physical nature of ARMA model
+*** Verification of velocity potential fields
+*** Non-physical nature of ARMA model
ARMA model, owing to its non-physical nature, does not have the notion of ocean
wave; it simulates wavy surface as a whole instead. Motions of individual waves
and their shape are often rough, and the total number of waves can not be
@@ -464,7 +472,6 @@ In theory, ocean waves themselves can be chosen as ACFs, the only pre-processing
step is to make them decay exponentially. This is required to make AR model
stationary and MA model parameters finding algorithm to converge.
-* Determining wave pressures for discretely given wavy surface
* High-performance software implementation of ocean wave simulation
* Conclusion
* Acknowledgments
