Numerical solution of stochastic Ito^(^)-Volterra integral equations based on Bernstein multi-scaling polynomials

In this paper,first,Bernstein multi-scaling polynomials(BMSPs)and their properties are introduced.These polynomials are obtained by compressing Bernstein polynomials(BPs)under sub-intervals.Then,by using these polynomials,stochastic operational matrices of integration are generated.Moreover,by transforming the stochastic integral equation to a system of algebraic equations and solving this system using Newton’s method,the approximate solution of the stochastic Ito^(^)-Volterra integral equation is obtained.To illustrate the efficiency and accuracy of the proposed method,some examples are presented and the results are compared with other methods.

A review on the multi-scaled structures and mechanical/thermal properties of tool steels fabricated by laser powder bed fusion additive manufacturing

The laser powder bed fusion(LPBF) process can integrally form geometrically complex and high-performance metallic parts that have attracted much interest,especially in the molds industry.The appearance of the LPBF makes it possible to design and produce complex conformal cooling channel systems in molds.Thus,LPBF-processed tool steels have attracted more and more attention.The complex thermal history in the LPBF process makes the microstructural characteristics and properties different from those of conventional manufactured tool steels.This paper provides an overview of LPBF-processed tool steels by describing the physical phenomena,the microstructural characteristics,and the mechanical/thermal properties,including tensile properties,wear resistance,and thermal properties.The microstructural characteristics are presented through a multiscale perspective,ranging from densification,meso-structure,microstructure,substructure in grains,to nanoprecipitates.Finally,a summary of tool steels and their challenges and outlooks are introduced.

Enhanced Thermoelectric Performance of Non-equilibrium Synthesized Fe0.4Co3.6Sb12-xGex Skutterudites via Randomly Distributed Multi-scaled Impurity Dots

The p-type Ge doped Fe0.4Co3.6Sb12-xGex skutterudites with multi-scaled impurity dots（500 nm-2 mm） were successfully prepared by using melt-quenching（MQ） and subsequent spark plasma sintering（SPS） technique. Compared with traditional method, the new technology significantly shortened the processing time from several days to less than 24 hours. The phase of impurity dots was demonstrated to be CoSb through analysis of X-ray diffraction（XRD） and energy-dispersive spectrum（EDS）. Impurity dots were induced by Ge substitution of Sb in the non-equilibrium synthesized process. Due to the abandonment of the long reaction of annealing crystallization, a few of Ge atoms would fail to substitute Sb site of skutterudite in this non-equilibrium synthesized process, leading to that the multi-scaled impurity dots randomly distributed in the matrix of skutterudite Fe0.4Co3.6Sb12-xGex. The combination of multi-scaled impurity dots scattering long wavelength heat-carrying phonons and the point defect scattering short and middle wavelength heat-carrying phonons dramatically made the 22.2% reduction of lattice thermal conductivity. As a result, compared with unsubstituted sample of Fe0.4Co3.6Sb12, the maximum ZT value was increased by 30.5%. Thus, the two marked features of this new synthesis process, the shortened preparation time and the enhanced thermoelectric performance, would make a promising commercial application in the future.

Multi-range Fractals in Materials

A new model of multi-range fractals is proposed to explain the experimental results observed on the fractal dimensions of the fracture surfaces in materials.The relationship of multi-range fractals with multi-scaling fractals has been also discussed.

A two scaled numerical method for global analysis of high dimensional nonlinear systems

This paper first analyzes the features of two classes of numerical methods for global analysis of nonlinear dynamical systems, which regard state space respectively as continuous and discrete ones. On basis of this understanding it then points out that the previously proposed method of point mapping under cell reference (PMUCR), has laid a frame work for the development of a two scaled numerical method suitable for the global analysis of high dimensional nonlinear systems, which may take the advantages of both classes of single scaled methods but will release the difficulties induced by the disadvantages of them. The basic ideas and main steps of implementation of the two scaled method, namely extended PMUCR, are elaborated. Finally, two examples are presented to demonstrate the capabilities of the proposed method.

Pore scale numerical investigation of counter-current spontaneous imbibition in multi-scaled pore networks

The multi-scaled pore networks of shale or tight reservoirs are considerably different from the conventional sandstone reservoirs.After hydraulic fracturing treatment,the spontaneous imbibition process plays an important role in the productivity of the horizontal wells.Applying the color-gradient model of Lattice Boltzmann Method(LBM)accelerated with parallel computing,we studied the countercurrent spontaneous imbibition process in two kinds of pore structures with different interlacing distributions of large and small pores.The effect of geometry configuration of pore arrays with different pore-scale and the capillary number Ca on the mechanism of counter-current spontaneous imbibition as well as the corresponding oil recovery factor are studied.We found that the wetting phase tends to invade the small pore array under small Ca in both types of geometry configurations of different pore arrays of four pore arrays zones.The wetting phase also tends to invade the pore array near the inlet for injecting the wetting phase no matter if it is a large pore array or small pore array except for the situation when the Ca is large to a certain value.In this situation,the small pore arrays show resistance to the wetting phase,so the wetting phase doesn't invade the small pore near the inlet,but invades the large pore preferentially.Both the geometry configurations of different pore arrays and Ca have a significant effect on the oil recovery factor.This work will help to solve the doubt about the selectivity of the multi-scaled pores of the wetting phase and the role of pores with different sizes in imbibition and oil draining in countercurrent spontaneous imbibition processes.