Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus G has a cut graph of length at most a given value. We prove a time lower bound for this problem of n Ω( g log g ) conditionally to the ETH. In other words, the first n O(g) -time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year-old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals , is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n Ω( gt + g 2 + t log ( g + t )) , conditionally to the ETH, for any choice of the genus g ≥ 0 of the graph and the number of terminals t ≥ 4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a gridlike structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value G of the genus.

Modeling of Critical Cutting Speed of White Layer Formation in Hard-Cutting Process

White layer exists on the machined surface of the hard-cutting and affects the surface quality and mechanical properties of a workpiece. Accurate predicting the critical cutting speed of white layer formation is of great significance for controlling the surface quality and selecting appropriate cutting parameters. In this work, an austenite transformation driving force calculation model of the white layer formation was established based on phase transformation thermodynamics theory, in which the influence of cutting temperature, stress and strain on the austenite transformation driving force in the hard-cutting process was taken into account. Second, a finite element (FE) model of the hard-cutting process was built by using hardened AISI52100 steel as cutting material. Then, a prediction model of critical cutting speed of the white layer formation was developed in combination with the austenite transformation driving force model and the hard-cutting FE model. Finally, the critical cutting speeds of the white layer formation at different chip thicknesses, tool rake angles and different levels of flank wear were simulated by using the critical cutting speed prediction model.

Tribological Performance of Micro-Groove Tools of Improving Tool Wear Resistance in Turning AISI 304 Process

AISI 304 has good physical and chemical properties and thus is widely used. However, due to the low thermal diffusivity, the cutting temperature of AISI 304 is high accelerating the wear of the tool. Therefore, tool wear is a major problem in machining hard cutting materials. In this study, we developed a new type of micro-groove tool whose rake surface was distributed with micro-groove by powder metallurgy based on the finite element temperature field morphology. We compared the wear of the proposed micro-groove tool with an untreated one by using a scanning electron microscope (SEM) and an X-ray energy spectrum. The abrasive, adhesive, and oxidation wear of the rake and the flank face of the micro-groove tool were lower than that of the untreated one. Due to the micro-groove on the rake face of the tool, the contact length between the tool and chip was reduced, leaving more extension space. Furthermore, chip extrusion deformation was avoided, and the energy caused by chip deformation was reduced. After 70 min of cutting, the counterpart reached the specified wear amount while the main cutting force, the feed resistance, and the cutting depth resistance of the proposed micro-groove tool were reduced by 16.1%, 33.9%, and 40.1%, respectively. With regard to steady state, the cutting temperature was reduced by 17.2% and the wear width of the flank face was reduced by 36.7%.