^{†}Corresponding author. Email: sunj@sdu.edu.cn
^{*}Project supported by the National Natural Science Foundation of China (Grant Nos. 61273088, 61473173, and 61473174).
We present a directional region control (DRC) model of thermal diffusion fractal growth with active heat diffusion in threedimensional space. This model can be applied to predict the space body heat fractal growth and study its directional region control. When the nonlinear interference term and the inner heat source term are generalized functions, the relationship between the particle aggregation probability and the interference terms can be obtained using the norm theory. We can then predict the aggregation form of particles in different regions. When the nonlinear interference terms in the model are expressed as a trigonometric function and its composite function, our simulations show that the DRC method of thermal fractal diffusion is effective and has reference value for the directional control of actual fractal growth systems.
Fractal geometry theory plays an important role in the nonlinear field.^{[1– 9]} The diffusion limited aggregation (DLA) model proposed by Witten and Sander, which explains the farfrom equilibrium natural phenomenon, is an especially hot research topic.^{[10– 13]} Various improved DLA models have been proposed to better explain specific growth processes such as dendritic growth, ^{[14]} electrodeposition, ^{[15]} and fluid displacement.^{[16]} The DLA model allows researchers to obtain the features of fractal growth and derive the morphology control theory under some twodimensional growth conditions.^{[17– 19]} However, natural fractal growth is conducted in the threedimensional space and the actual system is usually expected to control the fractal growth form in a fixed region, such as crystal growth occurring along specified axes, nonuniform bacterial colony growth, and particles aggregating along a certain orientation. At present, the orientation restriction issue for fractal growth is usually addressed only in experimental research, ^{[20– 22]} and it is seldom tackled theoretically. Meanwhile, many external environmental factors may interfere with the process of fractal growth, which makes it complicated and difficult to predict the morphology of fractal growth. Therefore, the research on the directed prediction and control of threedimensional fractal growth is of great significance.
We consider a space body in a real environment and provide a directional region control model for the thermal fractal diffusion system. The quantitative relationship between the particle aggregation probability and the interference term is obtained using the norm theory, in which the nonlinear interference term and the internal heat source term are generalized functions. Under certain conditions, the source term can suppress the nonlinear interference term, meaning that the source term controls the growth particles to aggregate in a different direction or region. This situation is called directional region control (DRC) in this paper. We predict the aggregation morphology of particles in differently directed regions, and achieve the DRC of space body thermal diffusion growth.
Considering the thermal diffusion of a space body W (as shown in Fig. 1) in an actual complex environment, we take the thermal diffusion process as a distributed parameter system meeting the boundary conditions, ^{[23]}
where V(x) refers to the space body W ’ s temperature (the probability of thermal diffusion) at certain point x, and u(x) refers to the density of the inner heat resource. The f is a nonlinear function, referred to as the forced term in physics; u(x) is a quantitative initial condition, referred to as the source term. By assuming G = u(x) + f (V(x), ∂ V(x)/∂ x), both the nonlinear interference term f (V(x), ∂ V(x)/∂ x) and the source term u(x) can be regarded as a part of the environmental interference term G, which is assumed to be an arbitrary function. Figure 2 shows the thermal fractal diffusion aggregation cluster without environmental interference.
Generalized functions can be regarded as continuous linear functionals of function space and are applied extensively in mathematics, physics, mechanics, and engineering.^{[24]} Let us assume that both f (V(x), ∂ V(x)/∂ x) and u(x) are generalized functions, and that these generalized functions are all elements in a basic function space K. In general, it is very difficult to find an analytical solution to the nonlinear partial differential equation (1). By studying the weak solution to system (1), we discuss the quantitative relationship between the particle aggregation probability and the environmental interference term G.
Definition 1 (L^{2} space) Assume that Ω is a bounded domain of R^{3}. We use L^{2}(Ω ) to express the function class consisting of all the measurable functions V(x) which are defined in Ω and meet ∫ _{Ω }  V(x) ^{2} dx < ∞ . The L^{2}(Ω ) is a vector space with norms defined as
Definition 2 (Sobolev space H^{k}) Assume that Ω is a bounded domain of R^{3}. We use H^{k}(Ω ) to express the function class consisting of all the measurable functions which are defined in Ω and meet
Definition 3 Assume that a(V, v) is a bilinear functional in Hilbert space H, i.e., V and v are both linear.
(i) a(V, v) is bounded, if there is a positive constant M ≥ 0, such that
(ii) a(V, v) is compulsory, if there is a positive constant δ > 0, such that
Now we can consider the existence and uniqueness of the weak solution of the following system:
Theorem 1 Assume that u(x) ∈ L^{2}(Ω ), f (V, DV) ∈ L^{2}(Ω ). There exists a unique weak solution
where λ is a positive constant.
Proof Denote
Obviously, a(V, ψ ) is bilinear. First, we prove the boundedness of a(V, ψ ),
Next, due to the ellipticity condition, we have
Therefore, according to the Lax– Milgram Theorem, ^{[25]} there exists a unique
To improve the regularity of the weak solution for system (2), we introduce the difference operator. If V(x) is a function defined on R^{3}, then
Proposition 1 We have the following basic properties of the difference operator.
(i) The adjoint operator of
(ii) The differential operator and
(iii) The difference of products can be expressed as
where T_{i} is the shift operator in the direction of
Proposition 2 Assume that Ω ⊂ R^{3} is a region.
(i) Assume that V ⊂ H^{1}(Ω ), Ω _{1} ⊂ ⊂ Ω . For h > 0, when sufficiently small,
(ii) Assume that V ⊂ L^{2}(Ω ) and there exists a constant K such that Ω _{1} ⊂ ⊂ Ω . When h > 0 is small enough,
We can obtain Theorem 2 using the basic properties of the difference operator.
Theorem 2 If
where C is a positive constant that only depends on the distance dist {Ω _{1}, ∂ Ω } from Ω _{1} to ∂ Ω .
Proof Supposing Ω _{1} ⊂ Ω , we note that d = dist {Ω _{1}, ∂ Ω }/4. The cutoff function
Based on the definition of the weak solution and the density of
If h meets the condition
By applying the exchangeability of the difference operator and the differential operator, and the relationship between the difference operator and its adjoint operator, the above integral relationship can be rewritten as
The following formula is obtained from formula (4) by calculation:
For further calculations, we need to estimate the third term on the right side of formula (5). From Proposition 1, we conclude that
Substituting formula (6) into formula (5), we obtain
and then
Next, if we estimate the first term in formula (7) by using the cutoff function and Proposition 1, then we have
Substituting formula (8) into formula (7), we conclude that
Based on the above formula and using the cutoff function, we have
Based on Proposition 2, we can conclude that formula (3) is correct. The proof is completed.
Based on Theorem 1, we can conclude that when f (V(x, y, z), DV(x, y, z) ∈ L^{2}(Ω ), u(x, y, z) ∈ L^{2}(Ω ), and
By assuming that u(x) is a constant k, f(V(x), ∂ V(x)/∂ x) = a sin^{2}(V(x)), and f(V(x), ∂ V(x)/∂ x) = bV(x)^{2} + c sin(V(x)), the control region H of the source item u(x) can be taken as a oneeighth space, one half space, cuboid region, spherical domain, or sectional region.
The thermal diffusion of a space body is a random process induced by a temperature gradient. In our recent work, ^{[23]} the thermal diffusion fractal growth was simulated. The growthprobability function of every possible aggregation point has a nonlinear relationship with the temperature gradient. With the environmental disturbance, the boundary condition V(x) is set to be zero at any point on the aggregation surface, and for the boundary at infinity, V(x) is set to be one, which provides a stable particle flow. By modifying the movementboundary conditions of the mathematical model, the temperature gradient can be calculated from the Poisson equation. Furthermore, by using the roulette wheel selection algorithm, the particle aggregation location can be determined.
With the environmental interference considered, the thermal diffusions in different regions are simulated by using the simulation method in the literature.^{[23]} As shown in Figs. 2– 5, the interference source item controls the aggregation of the thermal particles in different directions. This result shows that our control method is valid.
A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. It has also been considered as a measure of the spacefilling capacity of a pattern, which tells how a fractal scales differently from the space it is embedded in.^{[26]} As the thermal diffusion system of a space point source has an obvious central point, the Sandbox method is used to calculate the fractal dimension (D_{f}) of the thermal aggregation cluster with different interference actions.^{[27]} Figure 6 shows the comparison between the thermal diffusion fractal dimensions of different sectional control regions. Figure 5 shows the results predicted by our DRC model for spherical domain, where the spherical interference control region is
and the sectional control region is
According to Fig. 6, the aggregations of thermal particles based on the spherical control region are more complex than that controlled in different directions in addition to the spherical domain. Although they share the same control spherical domain, the directed region control beyond spherical domain determines the growth direction and also reduces the growth complexity. Figure 6 also shows that the fractal dimensions of clusters aggregated in the multidirectional region are larger than those of clusters aggregated in the unidirectional region for the same spherical domain. This indicates that the complexity of thermal aggregation clusters increases along with the increase in multidirectional control.
We have presented a DRC model of thermal fractal diffusion with actual environmental interference. Based on the norm theory, the control analysis of this model is given. When the nonlinear interference terms and the inner heat source terms are generalized functions, the quantitative relationship between the particle aggregation probability and the disturbance term can be obtained using the norm theory. Our simulations show that this DRC model for the thermal diffusion fractal growth of a space body is valid. This paper addresses only the generalized form of the disturbance term in L^{2} space. If the generalized function belongs to the space L^{p}(p ≥ 1), then the DRC of space fractal growth will be of universal significance. The more complex theoretical structure will be subject to greater change and we will address the topic in a separate paper.
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