•Compute Boolean operations between concave polygons with holes and several contours.•Compute the holes of every contour.•The algorithm is efficient and simple.
In this paper a simple and efficient ...algorithm for computing Boolean operations on polygons is presented. The algorithm works with almost any kind of input polygons: concave polygons, polygons with holes, several contours and self-intersecting edges. Important topological information, as the holes of the result polygon, is computed.
This work aims at developing a topology optimization method for the design of self-supporting structures that can be directly realized by additive manufacturing (AM). In the proposed method, ...polygon-featured holes are introduced as basic design primitives whose movements, deformations and intersections allow to control the structural topology. In addition, overhang constraints are introduced directly into the geometry description, so that the inclined angles along the overhang boundaries are not less than the critical overhang angle (COA). Meanwhile, unprintable V-shaped areas caused by polygon intersections are gradually eliminated through polygon modifications and re-optimization. Numerical examples are tested to demonstrate the effectiveness of the proposed method.
•This method includes overhang constraints directly in the geometry description.•Polygon holes act as basic design primitives to realize the structural topology.•Various critical overhang angles and build directions are studied.
We describe the topology of the space L(n) of polygons with collinear vertices, and prove that it is contained in the boundary of the set of simple polygons. We show that the subspace Q(n)⊂L(n) ...corresponding to elements in the boundary of the set of convex polygons is a submanifold with boundary, and we describe its topology and its complement L(n)∖Q(n). We also describe the quotient manifolds of L(n), Q(n) and L(n)∖Q(n) by the diagonal action of the Complex Affine Group.
Recently friction stir processing (FSP) has shown keen interest to achieve superplasticity in different aluminum alloys. The pin profile of FSP tool is one of the important process parameter which ...controls the mechanical and metallurgical properties of stir zone (SZ), like other variables of tool rotational speed, travel speed, and tool tilt. The high strength 7075 aluminum (Al-Zn-Mg-Cu) alloy was subjected to FSP to investigate effects of pin profiles on the superplastic behavior. Three different polygonal pin profiles of square, pentagon and hexagon were used. Microstructure, microhardness and grain size measurements were performed for all FSP samples. Fine grain uniform microstructure without cavitation in the SZ was observed in sample produced by square pin only. All polygonal pin profiles indicated sticking of workpiece material around tool pin that resulted in non-uniform grain microstructure in the SZ. Hot tensile testing was carried out for square pin under the superplastic condition of 3×10−4s−1 and 400°C to study the superplastic behavior. Uniform superplastic elongation of 227% was obtained in the gage region of the square pin sample.
The uniformly asymptotical normality of frequency polygons for ψ-mixing samples is investigated under the given conditions. Moreover, the corresponding rate of convergence is also derived, which is ...nearly O(n- 1/6) for the given assumptions.
A regular polygon is a polygon that is both equilateral and equiangular. One of the properties of the regular polygon is has a circumcircle and an incircle. The analogy of a regular polygon on plane ...is a regular polyhedron in space, while the analogy of a circle on plane is a spehe in space. Relationship between regular polygon and circle on plane can be studied in space. The purpose of this paper is to show the existance and the characteristics of a circumsphere and an insphere of the regular polyhedron. The result show that each regular polyhedron has a circumsphere and an insphere. The center of the circumsphere of a regular polyhedron is intersection of the perpendicular bisector planes of the regular polyhedron and the center of the insphere of a regular polyhedron is intersection of the angle bisector planes of the regular polyhedron. The characteristics of a circumsphere and an insphere of a regular polyhedron are: 1) The center of a circumsphere and an insphere of a regular polyhedron is coincide; 2) The length of radius of a circumsphere and an insphere of a regular polyhedron meet the equation R=s2sin(180°2ev)sin(180°2en)cos(θ2);r=s2cot(180°2en)tan(θ2).
•The interface tracking method VOSET was extended to arbitrary 2D grid types.•A set of geometric methods were proposed for interface reconstruction and the calculation of level-set function in ...arbitrary convex polygons.•The VOSET method was used to simulate two-phase flows with large density ratio in complex domains.
This article presents an extension of coupled volume-of-fluid and level-set method (VOSET) for simulating free surfaces flows in arbitrary 2D polygon meshes. A series of techniques are introduced for geometric calculations in convex polygons. Newton iteration is adopted for the interface reconstruction in polygons, and incremental remapping approach is employed for the propagation of the volume fractions. The interface tracking test results suggested a second order accuracy in mixed and hexagonal grids. For the validation purpose, a Rayleigh-Taylor instability problem, a liquid column collapse problem and a single bubble rising problem were numerically studied, and the obtained results show excellent agreements with experimental data and benchmark solutions in literatures. Finally the proposed VOSET method was applied in simulating the working process of a flow-focusing microfluidic droplet generator.
Polygons: 8Circulenes were easily prepared by Pd‐catalyzed annulations of tetraiodotetraphenylenes with alkynes. Their saddle‐shaped structure with an 8radialene character was identified by X‐ray ...crystallography. Similar to 1,3,5,7‐cyclooctatetraene, they have a tub‐shaped eight‐membered ring, but all of the bond lengths and bond angles are almost equal. Variable‐temperature NMR investigations showed interesting dynamic behavior.
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