General fractal topography: an open mathematical framework to characterize and model mono-scale-invariances

Abstract

In this work, we reported there are two kinds of independent complexities in mono-scale-invariance, namely to be behavioral complexity determined by fractal behavior and original one wrapped in scaling object. Quantitative characterization of the complexities is fundamentally important for mechanism exploration and essential understanding of nonlinear systems. Recently, a new concept of fractal topography ($\mathrm {\Omega }$) was emerged to define the fractal behavior in self-similarities by scaling lacunarity (P) and scaling coverage (F) as $\mathrm {\Omega }(P, F)$. It is, however, a “special fractal topography” because fractals are rarely deterministic and always appear stochastic, heterogeneous, and even anisotropic. For that, we reviewed a novel concept of “general fractal topography”, ${\varvec{\Omega }}(\mathrm {{\mathbf {P}}}, \mathrm {{\mathbf {F}}})$. In ${\varvec{\Omega }}$, $\mathrm {{\mathbf {P}}}$ is generalized to a set accounting for direction-dependent scaling behaviors of the scaling object G, while $\mathrm {{\mathbf {F}}}$ is extended to consider stochastic and heterogeneous effects. And then, we decomposed G into $\mathrm {{\mathbf {G}}}\left( G_+, G_-\right)$ to ease type controlling and measurement quantification, with $G_+$ wrapping the original complexity while $G_-$ enclosing behavioral complexity. Together with ${\varvec{\Omega }}$ and $\mathrm {{\mathbf {G}}}$, a mathematical model $F_\mathrm {3S}\left( {\varvec{\Omega }}, \mathrm {{\mathbf {G}}}\right)$ was then established to unify the definition of deterministic or statistical, self-similar or self-affine, single- or multi-phase properties. For demonstration, algorithms are developed to model natural scale-invariances. Our investigations indicated that the general fractal topography is an open mathematic framework which can admit most complex scaling objects and fractal behaviors.

Appendices

Appendix I: Modeling algorithm for pore–solid fractal

1.1 Preliminary work

For simplicity and without loss of generality, the x-direction size of G is assumed to be ranged in $\left[ l_{x_\mathrm {min}}, l_{x_\mathrm {max}}\right]$, and the sample number of G is denoted by ${\mathcal {N}}_{G}$.

At first, the fractal topography is determined accounting for the stochastic nature and anisotropic scale-invariant properties (inset $\textcircled {\small {1}}$ in Fig. 9) as:

(1) Determine $\mathrm {{\mathbf {F}}}$ of $\{F_1, \ldots , F_{N_\mathrm {e}}\}$ and their probability $\{\mathrm {{Pr}}_1, \ldots , \mathrm {{Pr}}_{N_\mathrm {e}}\}$ with $\sum _{i=1}^{N_\mathrm {e}}\mathrm {{Pr}}_{i}=1$. The requirements for these parameters have been described in Eq. (8).

(2) Determine the set of scaling lacunarities $\mathrm {{\mathbf {P}}}$ and denote by $P_\mathrm {min}$ the minimum scaling lacunarity.

(3) Therefore, the general scaling coverage $C_\mathrm {F}$ is obtained as $\left( \prod _{i=1}^dP_i/P_{\mathrm {min}}^d\right) \times \langle \mathrm {{\mathbf {F}}}\rangle$.

And then, a space of linear size $L_{0_x}\times {L_{0_y}}$ is discretized into a grid system of ${\mathbb {G}}_0$, in terms of a rectangle cell with spacings of $l_{x_\mathrm {max}}$ and $l_{y_\mathrm {max}}$ in x- and y-direction which are the linear sizes of the maximum scaling object G in different direction (inset $\textcircled {\small {2}}$ in Fig. 9). The sample number ${\mathcal {N}}_{G}$ could be therefore initialized as $\langle \mathrm {{\mathbf {F}}}\rangle \times \left( L_{0_x}\times {L_{0_y}}\right) /\left( l_{x_\mathrm {max}}\times {l_{y_\mathrm {max}}}\right)$.

1.2 Generation process

The fractal modeling process basically demonstrated in Fig. 9 follows

• (i) Each cell in ${\mathbb {G}}_0$ ($C_\mathrm {dad}$ for short) will be assigned a stochastic number by a uniform distribution function within $\left[ 0, 1\right)$. For $C_\mathrm {dad}$ with stochastic number ranged in $\left( \sum _{j=1}^{i-1}\mathrm {{Pr}}_{i-1}, \sum _{j=1}^{i}\mathrm {{Pr}}_i\right]$ in a population with sample number of ${\mathcal {N}}_{G}$, it is then replaced by $\mathrm {Evt}_i\left( l_{x_\mathrm {max}}, l_{y_\mathrm {max}}\right)$ and new subgrains ($C_\mathrm {son}$) with pore, solid, and fractal phase yield (inset $\textcircled {\small {3}}$ in Fig. 9).

• (ii) If $l_{x_\mathrm {max}}>l_{x_\mathrm {min}}$, set $l_{x_\mathrm {max}}=l_{x_\mathrm {max}}/P_x$ and $l_{y_\mathrm {max}}=l_{y_\mathrm {max}}/P_y$ and subdivide each $C_\mathrm {son}$ with fractal phase (cells with gray color in inset $\textcircled {\small {3}}$) into a $P_x\times P_y$ sub grid system ${\mathbb {G}}_1$ (as those bounded by blue lines in insets $\textcircled {\small {4}}$ and $\textcircled {\small {5}}$ in two successive iterations); else jump to Step (iv).

• (iii) Take the new cells in all ${\mathbb {G}}_1$s as a whole population and denote them as the role of $C_\mathrm {dad}$. Therefore, the sample size of $C_\mathrm {dad}$ population changes into ${\mathcal {N}}_{G}={\mathcal {N}}_{G}\times {C_\mathrm {F}}$. Repeat Steps (i) and (ii) for all $C_\mathrm {dad}$s.

• (iv) Finally, assign solid (inset $\textcircled {\small {7}}$) or pore phase (inset $\textcircled {\small {8}}$) to all $C_\mathrm {son}$s with fractal phase as we like to.

Appendix II: Modeling algorithm for fractal tight porous media

1.1 Preliminary work

Definitions of original spatial and scaling parameters need by a $F_\mathrm {3S}$, respectively, to be:

• (1) A space of size $l_{x_\mathrm {max}}\times {l_{y_{\mathrm {max}}}}$, denoted by ${\mathbb {R}}_0$, is considered as the representative unit of a fractal tight porous media.

• (2) Denote the minimum scale of G in the direction of x by $l_{x_\mathrm {min}}$ and initialize to $l_{x_\mathrm {max}}$.

• (3) ${\mathcal {N}}_\mathrm {sub}$ (the number of subregions in the G for a fractal network) is provided to portion a target space.

• (4) Define the scaling lacunarity set $\mathrm {{\mathbf {P}}}_\mathrm {net}$ of $\left( P_{x}, P_{y}\right)$.

1.2 Modeling process

In modeling of self-affine fractal tight porous media, an important step is to stretch the space in a determined direction before the space portion, as Fig. 15 demonstrates. Therefore, the modeling process is then modified into:

Step I: mathematical definition of$F_\mathrm {3S}\left\{ {\varvec{\Omega }}, \mathrm {{\mathbf {G}}}, \mathrm {{\mathbf {L}}}\right\}$

• (i) A set of Poisson points with number of ${\mathcal {N}}_\mathrm {sub}$ were generated in the full domain of ${\mathbb {R}}_0$ to portion it into a set of polygons following the Voronoi scheme similar to the way proposed by Wu et al. [70] (see inset $\textcircled {\small {1}}$ in Fig. 15).

• (ii) Stretch ${\mathbb {R}}_0$ into ${\mathbb {R}}_1$ of size $l_{x_\mathrm {max}}\times {l_{y_\mathrm {max}}}$ with $l_{y_\mathrm {max}}$ equal to $P_{yx}l_{y_\mathrm {max}}$.

• (iii) Determine $\lambda _\mathrm {max}$: According to Eq. (27), $x_p=1-{{\mathcal {N}}_\mathrm {pt}}/{P_{y}^d}$. And, $x_p=L_\mathrm {t}\lambda _\mathrm {max}/S_0$ in ${\mathbb {R}}_1$ with $L_\mathrm {t}$ the total length of all new edges and $S_0=l_{x_\mathrm {max}}\times {l_{y_\mathrm {max}}}$. Therefore, $\lambda _\mathrm {max}$ is obtained.

• (iv) Construct scaling object G: All new edges were buffered into a network of channels with a two-side buffering distance of $\lambda _\mathrm {max}/2$ except for the space boundaries. The generated isolated polygons and the channels represent the grains and the intersected capillaries, respectively, as shown by inset $\textcircled {\small {2}}$ in Fig. 15.

• (v) Determine the expectation of scaling coverage: randomly choose ${\mathcal {N}}_f$ grains to be fractal phase (gray regimes in inset $\textcircled {\small {2}}$), others to solid one. Therefore, $x_p$, $x_s$, and $x_f$ are determined. Consequently, the scaling coverage $C_\mathrm {F}$ is quantified as per Eq. (30).

Thus, $F_\mathrm {3S}\left\{ \mathbf {\Omega }(\mathrm {{\mathbf {P}}}_\mathrm {net}, \mathrm {{\mathbf {F}}}_\mathrm {net}), \mathrm {{\mathbf {G}}}(G_+, G_-), \mathrm {{\mathbf {L}}}[\mathrm {{\mathbf {l}}}_\mathrm {min}, \mathrm {{\mathbf {l}}}_\mathrm {max}]\right\}$ is quantitatively defined if $\mathrm {{\mathbf {l}}}_\mathrm {min}$ had been preset, with $\langle \mathrm {{\mathbf {F}}}_\mathrm {net}\rangle =C_\mathrm {F}$.

Step II: fractal iteration

• (i) Each grains with fractal phase is portioned into a set of polygons with a fixed number of points (${\mathcal {N}}_\mathrm {sub}^{xy}$ by Eq. (32)) randomly distributed in. Set $l_{y_\mathrm {max}}=P_{yx}l_{y_\mathrm {max}}$ and stretch ${\mathbb {R}}_i$ and all elements into ${\mathbb {R}}_{i+1}$ of size $l_{x_\mathrm {max}}\times {l_{y_\mathrm {max}}}$. After setting $\lambda _\mathrm {max}$ to $\lambda _\mathrm {max}/P_{x}$, all new edges are then buffered into channels with a two-side buffering distance of $\lambda _\mathrm {max}/2$, and the new subgrains were generated, as the inset $\textcircled {\small {3}}$ demonstrates.

• (ii) Set ${\mathcal {N}}_f=C_\mathrm {F}\times {\mathcal {N}}_f$ and randomly choose ${\mathcal {N}}_f$ new subgrains to be fractal phase, others to solid phase. Assign the regime of new channels to be pore phase and set $l_{x_\mathrm {min}}=l_{x_\mathrm {min}}/P_{x}$;

• (iii) Repeat step (ii) if the given hydraulic diameter $\lambda _\mathrm {min}>\lambda _\mathrm {max}$, otherwise go to Step III.

Step III: final disposition

• (i) Set all new subgrains to be solid phase, see the insets $\textcircled {\small {4}}$ and $\textcircled {\small {5}}$.

• (ii) Stretch the final ${\mathbb {R}}$ with all elements back to its original size and the self-affine fractal networks is obtained, as shown in inset $\textcircled {\small {6}}$ of Fig. 15.