User:Lothar.brendel - Revision history

Lothar.brendel: /* Scratch Pad */ +Florian's 'sjkr

2026-06-24T08:57:52Z

Scratch Pad: +Florian's 'sjkr

← Older revision		Revision as of 10:57, 24 June 2026
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	<nowiki>*</nowiki>: even though in general \(n_{i-1}\ne n_i-1\)		<nowiki>*</nowiki>: even though in general \(n_{i-1}\ne n_i-1\)

			== Florian's ''sjkr'' ==

			... in src/interactions/sjkr.rs corresponds to [https://www.cfdem.com/media/DEM/docu/gran_cohesion_sjkr2.html LIGGGHTS' ''sjkr2''], i.e.

			\[
			\begin{align}
			F_n &= -4\gamma A\\[2ex]
			\text{with}\quad
			A &= \pi\xi R_\text{eff}
			\end{align}
			\]

			Hence, \(\gamma\) is an energy/''volume'' instead of energy/area like the \(\gamma\) in the actual JKR model.

	== Hamiltons principle ==		== Hamiltons principle ==

Lothar.brendel: /* Transformation of probability density */ strictly decreasing

2026-04-24T15:12:59Z

Transformation of probability density: strictly decreasing

← Older revision		Revision as of 17:12, 24 April 2026
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	and		and
	\[		\[
	P_y(f(a+\text dx))-P_y(f(a)) = P_y(f(a)+f'(a)\text dx)-P_y(f(a)) = ~~\underbrace{~~p_x(f(a))~~}_{p_y(a)}~~\underbrace{f'(a)\text dx}_{\text dy}		P_y(f(a+\text dx))-P_y(f(a)) = P_y(f(a)+f'(a)\text dx)-P_y(f(a)) = p_x(f(a))\underbrace{f'(a)\text dx}_{\text dy}
	\]		\]

			Comparison \(\Rightarrow p_y(f(a))=p_x(a)/f'(a)\)

			For strictly decreasing \(f\) the order is swapped, i.e. \(P_x(b)-P_x(a) =P_y(f(a))-P_y(f(b))\), yielding the positive \(-f'(a)\).

	== 2D flow in polar coordinates \((\rho,\phi)\) ==		== 2D flow in polar coordinates \((\rho,\phi)\) ==

Lothar.brendel: transf. of prob-density

2026-04-24T09:28:21Z

transf. of prob-density

← Older revision		Revision as of 11:28, 24 April 2026
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	= Scratch Pad =		= Scratch Pad =

			== Transformation of probability density ==

			Let \(x\) be stochastic and \(y=f(x)\) with \(f\) being strictly increasing.

			\[
			\text{Prob}(x\in[a,b)) = P_x(b)-P_x(a) = \text{Prob}(y\in[f(a),f(b)))=P_y(f(b))-P_y(f(a))
			\]

			For \(b=a+\text dx\):
			\[
			P_x(a+\text dx)-P_x(a) = p_x(a)\text dx
			\]
			and
			\[
			P_y(f(a+\text dx))-P_y(f(a)) = P_y(f(a)+f'(a)\text dx)-P_y(f(a)) = \underbrace{p_x(f(a))}_{p_y(a)}\underbrace{f'(a)\text dx}_{\text dy}
			\]


	== 2D flow in polar coordinates \((\rho,\phi)\) ==		== 2D flow in polar coordinates \((\rho,\phi)\) ==

Lothar.brendel: /* Uncertainty of a crossing point */ big()

2026-04-24T09:00:42Z

Uncertainty of a crossing point: big()

← Older revision		Revision as of 11:00, 24 April 2026
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	To leading order, the fluctuations of the measured location are thus given by those of \(y(x_0)\), while those of \(y'(x_0)\) contribute only as a higher-order correction:		To leading order, the fluctuations of the measured location are thus given by those of \(y(x_0)\), while those of \(y'(x_0)\) contribute only as a higher-order correction:
	\[		\[
	\langle(x-x_0)^2\rangle\approx\frac{\langle y^2\rangle (x_0)-\langle y\rangle^2(x_0)} {\langle y'\rangle^2(x_0)}=\frac{\Xi_{p'q}x_0^{p'+q}}{(p\langle a_p\rangle x_0^{p-1})^2}		\langle(x-x_0)^2\rangle\approx\frac{\langle y^2\rangle (x_0)-\langle y\rangle^2(x_0)} {\langle y'\rangle^2(x_0)}=\frac{\Xi_{p'q}x_0^{p'+q}}{\big(\langle a_p\rangle p x_0^{p-1}\big)^2}
	\]		\]

Lothar.brendel: /* Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\) */ engl.

2026-04-24T08:58:33Z

Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\): engl.

← Older revision		Revision as of 10:58, 24 April 2026
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	== Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\) ==		== Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\) ==

	~~verschwindende partielle Ableitung nach~~ \(a_q\):		vanishing partial derivative with respect to \(a_q\):
	\[		\[
	\begin{align}		\begin{align}
	0&=\sum_i \frac{y_i-\sum_p a_p x_i^p}{\sigma_i^2}x_i^q\\		0&=\sum_i \frac{y_i-\sum_p a_p x_i^p}{\sigma_i^2}x_i^q\\
	\Leftrightarrow\quad		\Leftrightarrow\quad
	\underbrace{\sum_i\frac{y_i x_i^q}{\sigma_i^2}}_{\displaystyle\psi_q}&=\sum_i\sum_p \frac{a_p x_i^{p+q}}{\sigma_i^2}=\sum_p a_p\underbrace{\sum_i\frac{x_i^{p+q}}{\sigma_i^2}}_{\displaystyle X_{pq}}		\underbrace{\sum_i\frac{y_i x_i^q}{\sigma_i^2}}_{\displaystyle\psi_q}& =\sum_i\sum_p \frac{a_p x_i^{p+q}}{\sigma_i^2}=\sum_p a_p\underbrace{\sum_i\frac{x_i^{p+q}}{\sigma_i^2}}_{\displaystyle X_{pq}}
	\end{align}		\end{align}
	\]		\]
	~~Unter der Voraussetzung, dass die~~ (~~nicht~~-~~stochastische~~) ~~Matrix~~ \(X\) ~~eine Inverse~~ \(\Xi\) ~~besitzt~~, ~~gilt also~~ (~~ab jetzt Summenkonvention~~)		Assuming that the (non-stochastic) matrix \(X\) has an inverse \(\Xi\), the following holds true (from now on, summation convention)
	\[		\[
	a_p = \Xi_{pq}\psi_q		a_p = \Xi_{pq}\psi_q
	\]		\]
	~~und wegen der Linearität und der Tatsache~~, ~~dass~~ \(\Xi\) ~~nicht von den stochastischen~~ \(y_i\) ~~abhängt~~, ~~auch~~		and, due to linearity and the fact that \(\Xi\) does not depend on the stochastic \(y_i\), also
	\[		\[
	\langle a_p\rangle = \Xi_{pq}\langle\psi_q\rangle		\langle a_p\rangle = \Xi_{pq}\langle\psi_q\rangle
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	=== Uncertainty of the fitting parameters ===		=== Uncertainty of the fitting parameters ===

	~~Für die Korrelationen benötigen wir die Produkte~~		For the correlations, we need the products
	\[		\[
	\langle\psi_q\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i y_j\rangle		\langle\psi_q\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i y_j\rangle
	\quad\text{~~und~~}\quad		\quad\text{and}\quad
	\langle\psi_q\rangle\langle\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i\rangle\langle y_j\rangle		\langle\psi_q\rangle\langle\psi_{q'}\rangle = \frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\langle y_i\rangle\langle y_j\rangle
	\]		\]
	so ~~dass~~		so that
	\[		\[
	\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle=\frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\big(\underbrace{\langle y_i y_j\rangle-\langle y_i\rangle\langle y_j\rangle}_{\displaystyle\sigma^2_i\delta_{ij}}\big)=\sum_i\frac{x_i^{q+q'}}{\sigma^2_i}=X_{qq'}		\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle=\frac{x_i^q x_j^{q'}}{\sigma^2_i\sigma^2_j}\big(\underbrace{\langle y_i y_j\rangle-\langle y_i\rangle\langle y_j\ rangle}_{\displaystyle\sigma^2_i\delta_{ij}}\big)=\sum_i\frac{x_i^{q+q'}}{\sigma^2_i}=X_{qq'}
	~.		~.
	\]		\]
	~~Damit und mit~~		Thus, and using
	\[		\[
	a_p a_{p'}=\Xi_{pq}\Xi_{p'q'}\psi_q\psi_{q'}		a_p a_{p'}=\Xi_{pq}\Xi_{p'q'}\psi_q\psi_{q'}
	\]		\]
	~~können wir die Varianz der Fitparameter~~ (\(p'=p\)) ~~berechnen zu~~		we can calculate the variance of the fit parameters (\(p'=p\)) as
	\[		\[
	\langle a_p^2\rangle-\langle a_p\rangle^2=\Xi_{pq}\Xi_{pq'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)=x^{p+p'}\Xi_{pq}\Xi_{pq'}X_{qq'}=\Xi_{pq}\delta_{pq}=\Xi_{pp}		\langle a_p^2\rangle-\langle a_p\rangle^2=\Xi_{pq}\Xi_{pq'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)=x^{p+p'}\ \Xi_{pq}\Xi_{pq'}X_{qq'}=\Xi_{pq}\delta_{pq}=\Xi_{pp}
	~.		~.
	\]		\]
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	=== Uncertainty of the fitting function's values ===		=== Uncertainty of the fitting function's values ===

	~~Für ein vorgegebenes~~ \(x\) ~~gilt für das prognostizierte~~ \(y=a_p x^p\):		For a given \(x\), the following holds for the predicted \(y=a_p x^p\):
	\[		\[
	\begin{align}		\begin{align}
	\langle y\rangle &=\langle a_p\rangle x^p=\langle\psi_q\rangle\Xi_{qp}x^p\\		\langle y\rangle &=\langle a_p\rangle x^p=\langle\psi_q\rangle\Xi_{qp}x^p\\
	\langle y^2\rangle-\langle y\rangle^2 &= x^{p+p'}\big(\langle a_p a_{p'}\rangle-\langle a_p\rangle\langle a_{p'}\rangle\big)=x^{p+p'}\Xi_{pq}\Xi_{p'q'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)\\		\langle y^2\rangle-\langle y\rangle^2 &= x^{p+p'}\big(\langle a_p a_{p'}\rangle-\langle a_p\rangle\langle a_{p'}\rangle\big)=x^{p+p'}\ \Xi_{pq}\Xi_{p'q'}\big(\langle\psi_q\psi_{q'}\rangle-\langle\psi_q\rangle\langle\psi_{q'}\rangle\big)\\
	&=x^{p+p'}\Xi_{pq}\Xi_{p'q'}X_{qq'}=x^{p+p'}\Xi_{pq}\delta_{p'q}=\Xi_{pp'}x^{p+p'}		&=x^{p+p'}\Xi_{pq}\Xi_{p'q'}X_{qq'}=x^{p+p'}\ \Xi_{pq}\delta_{p'q}=\Xi_{pp'}x^{p+p'}
	\end{align}		\end{align}
	\]		\]

	~~Die Varianz von~~ \(y(x)\) ~~ist also ein Polynom mit dem doppelten Grad des Fit-Polynoms~~.		The variance of \(y(x)\) is thus a polynomial of twice the degree of the fitting polynomial.

	=== Uncertainty of a crossing point ===		=== Uncertainty of a crossing point ===

	~~Es sei~~ \(x_0\) ~~durch~~ \(\langle y\rangle(x_0)=c\) ~~definiert~~. ~~Wir linearisieren um~~ \(x_0\), d.h.		Let \(x_0\) be defined by \(\langle y\rangle(x_0)=c\). We linearize about \(x_0\), i.e.,
	\[		\[
	\langle y\rangle(x) = \langle y\rangle(x_0)+(x-x_0)\langle y'\rangle(x_0)+\mathcal O((x-x_0)^2)		\langle y\rangle(x) = \langle y\rangle(x_0)+(x-x_0)\langle y'\rangle(x_0)+\mathcal O((x-x_0)^2)
	\]		\]
	~~und fragen nach der Auswirkung der Fluktuationen auf das gemessene \(x_0\)~~:		and ask about the effect of the fluctuations on the measured location:
	\[		\[
	\begin{align}		\begin{align}
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	\end{align}		\end{align}
	\]		\]
	~~In führender Ordnung sind die Fluktuationen des gemessenen \(x_0\) also durch die von~~ \(y(x_0)\) ~~gegeben~~, ~~die von~~ \(y'(x_0)\) ~~tragen als Korrektur höherer Ordnung bei~~:		To leading order, the fluctuations of the measured location are thus given by those of \(y(x_0)\), while those of \(y'(x_0)\) contribute only as a higher-order correction:
	\[		\[
	\langle(x-x_0)^2\rangle\approx\frac{\langle y^2\rangle(x_0)-\langle y\rangle^2(x_0)}{\langle y'\rangle^2(x_0)}=\frac{\Xi_{p'q}x_0^{p'+q}}{(p\langle a_p\rangle x_0^{p-1})^2}~,		\langle(x-x_0)^2\rangle\approx\frac{\langle y^2\rangle (x_0)-\langle y\rangle^2(x_0)} {\langle y'\rangle^2(x_0)}=\frac{\Xi_{p'q}x_0^{p'+q}}{(p\langle a_p\rangle x_0^{p-1})^2}
	\]		\]
	~~wobei die Summe im Nenner bei \(p=1\) startet.~~

	= Scratch Pad =		= Scratch Pad =

Lothar.brendel: /* Background */ engl.

2026-04-24T08:46:54Z

Background: engl.

← Older revision		Revision as of 10:46, 24 April 2026
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	== Background ==		== Background ==

	~~Falls die Messwerte~~ \(y_i\) ~~unabhängig aber mit selber Normalverteilung um einen~~ \(x_i\)~~-abhängigen Mittelwert streuen~~, ~~so maximieren~~		If the measured values \(y_i\) are statistically independent but follow the same normal distribution centered around a mean \(\langle y_i\rangle\) that depends on \(x_i\), then the parameters \(\{a_k\}\) which minimize
	~~die Parameter~~ \(\{a_k\}\)~~, welche~~
	\[		\[
	\sum_{i=1}^n(y_i-\hat y(x_i;a_1,\dots,a_m))^2		\sum_{i=1}^n(y_i-\hat y(x_i;a_1,\dots,a_m))^2
	\]		\]
	~~minimieren, die Wahrscheinlichkeitsdichte für das Auftreten des Datensatzes~~ \(\{y_i\}\); d.h., ~~benachbarte Datensätze sind weniger wahrscheinlich~~. ~~Falls auch die Breite der jeweiligen Normalverteilung~~ \(~~x_i~~\)~~-abhängig ist~~, \(~~\sigma_i~~\), ~~so ist stattdessen~~		maximize the probability density for the occurrence of the data set \(\{y_i\}\); i.e., adjacent data sets are less likely. If the variance of the respective normal distribution, \(\sigma_i\), is also \(x_i\)-dependent, then instead
	\[		\[
	\frac{\chi^2}{2}=\sum_{i=1}^n\frac{(y_i-\hat y(x_i;a_1,\dots,a_m))^2}{2\sigma_i^2}		\frac{\chi^2}{2}=\sum_{i=1}^n\frac{(y_i-\hat y(x_i;a_1,\dots,a_m))^2}{2\sigma_i^2}
	\]		\]
	~~zu minimieren~~.		must be minimized.

	== Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\) ==		== Polynomial fit \(\hat y=\sum_{p=0}^{m-1}a_p x^p\) ==

Lothar.brendel: +Polyfit

2026-04-21T16:02:29Z

+Polyfit

@@ Line 1: / Line 1: @@
-Lothar Brendel
+= Polynomial fitting and estimation of uncertainty =
-"Admin" of this Wiki
+== Background ==
 = Scratch Pad =
@@ Line 14: / Line 104: @@
 ** \(\sigma_{r\phi}=\sigma_{\phi r}=\mu\displaystyle\frac{\partial_\phi u_r-u_\phi}{r}+\mu\,\partial_r u_\phi\)
 ** \(\sigma_{\phi\phi}=-p+(\zeta-2\mu/3)\displaystyle\frac{\partial_r(ru_r)+\partial_\phi u_\phi}{r}+2\mu\displaystyle\frac{\partial_\phi u_\phi+u_r}{r}\)
 == Energy equation ==

Lothar.brendel: /* Discretizations */ f_0 in 2 und 4

2026-02-16T12:05:49Z

Discretizations: f_0 in 2 und 4

← Older revision		Revision as of 14:05, 16 February 2026
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	\begin{align}		\begin{align}
	f'_0 &= \frac{f_{1/2}-f_{-1/2}}{h}+\mathcal O(h^2)\\		f'_0 &= \frac{-f_{-1/2}+f_{1/2}}{h}+\mathcal O(h^2)\\
	&= \frac{f_{-3/2}-27 f_{-1/2}+27 f_{1/2}-f_{3/2}}{24h}+\mathcal O(h^4)\\		&= \frac{f_{-3/2}-27 f_{-1/2}+27 f_{1/2}-f_{3/2}}{24h}+\mathcal O(h^4)\\
	&= \frac{-9 f_{-5/2}+125 f_{-3/2}-2250 f_{-1/2}+2250 f_{1/2}-125 f_{3/2}+9 f_{5/2}}{1920h}+\mathcal O(h^6)\\		&= \frac{-9 f_{-5/2}+125 f_{-3/2}-2250 f_{-1/2}+2250 f_{1/2}-125 f_{3/2}+9 f_{5/2}}{1920h}+\mathcal O(h^6)\\[3ex]
	f_0 &= \frac{3 f_{-5/2}-25 f_{-3/2}+150 f_{-1/2}+150 f_{1/2}-25 f_{3/2}+3 f_{5/2}}{256}+\mathcal O(h^6)		f_0 &= \frac{f_{-1/2}+f_{1/2}}{2}+\mathcal O(h^2)\\
			&= \frac{-f_{-3/2}+9f_{-1/2}+9f_{1/2}-f_{3/2}}{16}+\mathcal O(h^4)\\
			&= \frac{3 f_{-5/2}-25 f_{-3/2}+150 f_{-1/2}+150 f_{1/2}-25 f_{3/2}+3 f_{5/2}}{256}+\mathcal O(h^6)
	\end{align}		\end{align}

Lothar.brendel: /* Values of ghost cells / Robin -1

2026-02-09T13:35:38Z

Values of ghost cells: Robin *-1

← Older revision		Revision as of 15:35, 9 February 2026
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	Robin \(af(0)+b f'(0)=c\) :		Robin \(af(0)+b f'(0)=c\) :
	\[		\[
	f(-x):=\frac{2x}{ax-b}c-\frac{~~ax+b~~}{ax-b}~~f(x)~~		f(-x):=\frac{b+ax}{b-ax}f(x)-c\frac{2x}{b-ax}
	\]		\]

Lothar.brendel: /* Values of ghost cells */ en

2026-02-08T15:51:01Z

Values of ghost cells: en

← Older revision		Revision as of 17:51, 8 February 2026
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	\]		\]

	~~Spezialfälle sind~~:		Special cases:

	Dirichlet \(f(0)\):		Dirichlet \(f(0)\):