Boundary Slope Control in Topology Optimization for Additive Manufacturing: For Self-Support and Surface Roughness

This paper studies how to control boundary slope of optimized parts in density-based topology optimization for additive manufacturing (AM). Boundary slope of a part affects the amount of support structure required during its fabrication by additive processes. Boundary slope also has a direct relation with the resulting surface roughness from the AM processes, which in turn affects the heat transfer efficiency. By constraining the minimal boundary slope, support structures can be eliminated or reduced for AM, and thus, material and postprocessing costs are reduced; by constraining the maximal boundary slope, high-surface roughness can be attained, and thus, the heat transfer efficiency is increased. In this paper, the boundary slope is controlled through a constraint between the density gradient and the given build direction. This allows us to explicitly control the boundary slope through density gradient in the density-based topology optimization approach. We control the boundary slope through two single global constraints. An adaptive scheme is also proposed to select the thresholds of these two boundary slope constraints. Numerical examples of linear elastic problem, heat conduction problem, and thermoelastic problems demonstrate the effectiveness and efficiency of the proposed formulation in controlling boundary slopes for additive manufacturing. Experimental results from metal 3D printed parts confirm that our boundary slope-based formulation is effective for controlling part self-support during printing and for affecting surface roughness of the printed parts.

A result on the Slope conjectures for 3-string Montesinos knots

The Slope Conjecture and the Strong Slope Conjecture predict that the degree of the colored Jones polynomial of a knot is matched by the boundary slope and the Euler characteristic of some essential surfaces in the knot complement. By solving a problem of quadratic integer programming to find the maximal degree and using the Hatcher–Oertel edgepath system to find the corresponding essential surface, we verify the Slope Conjectures for a family of 3-string Montesinos knots satisfying certain conditions.

Boundary Slope Control in Topology Optimization for Additive Manufacturing

The paper studies how to control boundary slope of optimized parts in density-based topology optimization for additive manufacturing (AM). Boundary slope of a part affects the amount of support structure required during its fabrication by additive processes. Boundary slope also has direct relation with the resulting surface roughness from the AM processes, which in turn affects the heat transfer efficiency. By constraining the minimal boundary slope, support structures can be eliminated or reduced for AM, and thus material and post-processing costs are reduced; by constraining the maximal boundary slope, high surface roughness can be attained, and thus the heat transfer efficiency is increased. In this paper, the boundary slope is controlled through a constraint between the density gradient and the given build direction. This allows us to explicitly control the boundary slope through density gradient in the density-based topology optimization approach. We control the boundary slope through a single global constraint. Numerical examples on heat conduction problem, and coupled 2D and 3D thermoelastic problems demonstrate the effectiveness and efficiency of the proposed formulation in controlling boundary slopes for additive manufacturing.

Development of gravity currents on rapidly changing slopes

Gravity currents often occur on complex topographies and are therefore subject to spatial development. We present experimental results on continuously supplied gravity currents moving from a horizontal to a sloping boundary, which is either concave or straight. The change in boundary slope and the consequent acceleration give rise to a transition from a stable subcritical current with a large Richardson number to a Kelvin–Helmholtz (KH) unstable current. It is shown here that depending on the overall acceleration parameter$\overline{T_{a}}$, expressing the rate of velocity increase, the currents can adjust gradually to the slope conditions (small$\overline{T_{a}}$) or go through acceleration–deceleration cycles (large$\overline{T_{a}}$). In the latter case, the KH billows at the interface have a strong effect on the flow dynamics, and are observed to cause boundary layer separation. Comparison of currents on concave and straight slopes reveals that the downhill deceleration on concave slopes has no qualitative influence, i.e. the dynamics is entirely dominated by the initial acceleration and ensuing KH billows. Following the similarity theory of Turner 1973 (Buoyancy Effects in Fluids. Cambridge University Press), we derive a general equation for the depth-integrated velocity that exhibits all driving and retarding forces. Comparison of this equation with the experimental velocity data shows that when$\overline{T_{a}}$is large, bottom friction and entrainment are large in the region of appearance of KH billows. The large bottom friction is confirmed by the measured high Reynolds stresses in these regions. The head velocity does not exhibit the same behaviour as the layer velocity. It gradually approaches an equilibrium state even when the acceleration parameter of the layer is large.

Road Boundary Detection Based on Random Forests and Particle Filter

Road detection is a primary problem for autonomous vehicle. There have been many approaches attempting to solve this problem. However, most of the approaches tend to be affected by shadow, occlusion and not always robust. In this paper, we propose a new approach to detect the road boundary, our approach has two parts: firstly, we detect the road boundary by random forests on the pixel level, then we track the road boundary by particle filter with the parameters of vanish point position and road boundary slope. The experiment shows our approach is an effective way to detect road boundary fast and robustly.

THE JONES POLYNOMIAL AND BOUNDARY SLOPES OF ALTERNATING KNOTS

We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is given by the signature plus twice the maximum degree of the Jones polynomial. For alternating Montesinos knots, these are the minimal and maximal boundary slopes.

ROOTS OF UNITY ASSOCIATED TO STRONGLY DETECTED BOUNDARY SLOPES

We found a family of infinitely many hyperbolic knot manifolds each member of which has a strongly detected boundary slope with associated root of unity of order 4.