Efficient Construction of B-Spline Curves with Minimal Internal Energy

In this paper,we propose an efficient method to construct energy-minimizing B-spline curves by using discrete mask method.The linear relations between control points are firstly derived for different energy-minimization problems,then the construction of B-spline curve with minimal internal energy can be addressed by solving a sparse linear system.The existence and uniqueness of the solution for the linear system are also proved.Experimental results show the efficiency of the proposed approach,and its application in 1 G blending curve construction is also presented.

Electronic structure and geometric construction modulation of carbon-based single/dual atom catalysts for electrocatalysis

Both carbon-based single atom catalysts(SACs)and dual atom catalysts(DACs)have garnered significant attention in the field of electrochemical reactions because of the impressive attributes,including exceptional catalytic activity,selectivity,and cost-effectiveness.The ability to modulate the electronic structure and geometric construction of active sites within SACs/DACs is paramount for unleashing their complete potential,which in turn can ultimately dictate catalytic behavior with unprecedented precision.In this review,the recent major developments of the regulation strategies for modulating electronic structure and geometric construction of carbon-based SACs/DACs are summarized.For the SACs,the recently reported modulation methods are categorized into four strategies,including adjusting the density of single atoms,defect engineering,confinement effect and strain engineering.And for the DACs,the five methods contain bonded dual-atom adjustment,non-bonded and bridged dual-atom adjustment,metal and nonmetal dual-atom adjustment,bilayer dual-atom adjustment and homogeneous dual-atom adjustment.The recently developed synthetic strategies are comprehensively summarized,especially their electronic structure and geometric configuration are discussed in detail,the different catalytic applications of electrochemical reactions,and their unique catalytic mechanism are highlighted.Finally,the challenges and prospects of SACs/DACs for tailoring their electronic structures and geometric arrangements are further discussed.

Construction of Regular Heptagon by Rhombic Bicompasses and Ruler

We discuss a new possible construction of the regular heptagon by rhombic bicompasses explained in the text as a new geometric mean of constructions in the spirit of classical constructions in connection with an unmarked ruler (straightedge). It avoids the disadvantages of the neusis construction which requires the trisection of an angle and which is not possible in classical way by compasses and ruler. The rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points (arms) on one circle determines the position of the points on the other circle. This means that the positions of all points (arms) on both circles are determined in unique way.

Geometric basis:a geometric solving cell for geometric computing

Geometric computing is an important tool in design and manufacturing and in arts. Conventionally, geometric computing is taken by algebraic computing. The vivid intuition of objects in visualization is lost in numeric functions, which is however very useful to human cognition as well as emotion. In this paper, we proposed a concept and theory of geometric basis （GB） as the solving cell for geometric computing. Each GB represents a basic geometric operation. GB works as both expressing and solving cell just like the concept of basis in linear algebra by which every element of the vector space can be expressed. For 3D problems, with a procedure of a projections reduction, the problem can be reduced to plane and the reduction function can be designed as a GB. A sequence of GB can construct a higher layer GB. Then, by the traversal of tree, a sequence of GB is got and this sequence is just the construction process and also the solution of this geometric problem.

Geometry model construction in infrared image theory simulation of buildings

Geometric model construction is the basis of infrared image theory simulation. Taking the construction of the geometric model of one building in Harbin as an example, this paper analyzes the theoretical groundings of simplification and principles of geometric model construction of buildings. It then discusses some particular treatment methods in calculating the radiation transfer coefficient in geometric model construction using the Monte Carlo Method.

Discrete element simulation of crushable rockfill materials

A discrete element method was used to study the evolution of particle crushing in a rockfill sample subjected to triaxial shear. A simple procedure was developed to generate clusters with arbitrary shapes, which resembled real rockfill particles. A theoretical method was developed to define the failure criterion for an individual particle subjected to an arbitrary set of contact forces. Then, a series of numerical tests of large-scale drained triaxial tests were conducted to simulate the behaviors of the rockfill sample. Finally, we examined the development of micro-characteristics such as particle crushing, contact characteristics, porosity, deformation, movement, and energy dissipation. The simulation results were partially compared with the laboratory experiments, and good agreement was achieved, demonstrating that the particle crushing model proposed can be used to simulate the drained triaxial test ofrockfill materials. Based on a comparison of macro behaviors of the roekfill sample and micro structures of the particles, the microscopic mechanism of the rockfill materials subjected to triaxial shear was determined qualitatively. It is shown that the crushing rate, rather than the number of crushed particles, can be used to reflect the relationship between macro- and micro-mechanical characteristics of rockfill materials. These research results further develop our understanding of the deformation mechanism of rockfill materials.