Curvature Constraints Research Articles

The Competition of Roughness and Curvature in Area-Constrained Polymer Models

The competition between local Brownian roughness and global parabolic curvature experienced in many random interface models reflects an important aspect of the KPZ universality class. It may be summarised by an exponent triple $(1/2,1/3,2/3)$ representing local interface fluctuation, local roughness (or inward deviation) and convex hull facet length. The three effects arise, for example, in droplets in planar Ising models (Alexander, '01, Hammond, '11,'12). In this article, we offer a new perspective on this phenomenon. We consider directed last passage percolation model in the plane, a paradigmatic example in the KPZ universality class, and constrain the maximizing path under the additional requirement of enclosing an atypically large area. The interface suffers a constraint of parabolic curvature as before, but now its local structure is the KPZ fixed point polymer's rather than Brownian. The local interface fluctuation exponent is thus two-thirds rather than one-half. We prove that the facet lengths of the constrained path's convex hull are governed by an exponent of $3/4$, and inward deviation by an exponent of $1/2$. That is, the exponent triple is now $(2/3,1/2,3/4)$ in place of $(1/2,1/3,2/3)$. This phenomenon appears to be shared among various isoperimetrically extremal circuits in local randomness. Indeed, we formulate a conjecture to this effect concerning such circuits in supercritical percolation, whose Wulff-like first-order behaviour was recently established (Biskup, Louidor, Procaccia and Rosenthal, '12).

Median Statistics Analysis of Deuterium Abundance Measurements and Spatial Curvature Constraints

Zavarygin et al. compiled a list of 15 deuterium abundance measurements; two were discarded because the remaining 13 measurements are then consistent with Gaussianity. Those authors found that the weighted-mean baryon density (Ωbh2) determined from the 13 measurements is mildly discrepant (1.6σ) with that determined from the Planck 2015 cosmic microwave background anisotropy data in a flat cosmogony. We find that a median statistic central estimate of Ωbh2 from all 15 deuterium abundance measurements is a more accurate estimate; is very consistent with Ωbh2 estimated from Planck 2015 data in a flat cosmogony, but is about 2σ lower than that found in a closed cosmogonical model from the Planck 2015 data.

Motion planning in Irreducible Path Spaces

The motion of a mechanical system can be defined as a path through its configuration space. Computing such a path has a computational complexity scaling exponentially with the dimensionality of the configuration space. We propose to reduce the dimensionality of the configuration space by introducing the irreducible path — a path having a minimal swept volume. The paper consists of three parts: In part I, we define the space of all irreducible paths and show that planning a path in the irreducible path space preserves completeness of any motion planning algorithm. In part II, we construct an approximation to the irreducible path space of a serial kinematic chain under certain assumptions. In part III, we conduct motion planning using the irreducible path space for a mechanical snake in a turbine environment, for a mechanical octopus with eight arms in a pipe system and for the sideways motion of a humanoid robot moving through a room with doors and through a hole in a wall. We demonstrate that the concept of an irreducible path can be applied to any motion planning algorithm taking curvature constraints into account.

Magnonic quantum spin Hall state in the zigzag and stripe phases of the antiferromagnetic honeycomb lattice

We investigated the topological property of magnon bands in the collinear magnetic orders of zigzag and stripy phases for the antiferromagnetic honeycomb lattice and identified Berry curvature and symmetry constraints on the magnon band structure. Different symmetries of both zigzag and stripy phases lead to different topological properties, in particular, the magnon bands of the stripy phase being disentangled with a finite Dzyaloshinskii-Moriya (DM) term with non-zero spin Chern number. This is corroborated by calculating the spin Nernst effect. Our study establishes the existence of the non-trivial magnon band topology for all observed collinear antiferromagnetic honeycomb lattice in the presence of the DM term.

An aerodynamic design optimization framework using a discrete adjoint approach with OpenFOAM

Advances in computing power have enabled computational fluid dynamics (CFD) to become a crucial tool in aerodynamic design. To facilitate CFD-based design, the combination of gradient-based optimization and the adjoint method for computing derivatives can be used to optimize designs with respect to a large number of design variables. Open field operation and manipulation (OpenFOAM) is an open source CFD package that is becoming increasingly popular, but it currently lacks an efficient infrastructure for constrained design optimization. To address this problem, we develop an optimization framework that consists of an efficient discrete adjoint implementation for computing derivatives and a Python interface to multiple numerical optimization packages. Our adjoint optimization framework has the following salient features: (1) The adjoint computation is efficient, with a computational cost that is similar to that of the primal flow solver and scales up to 10 million cells and 1024 CPU cores. (2) The adjoint derivatives are fully consistent with those generated by the flow solver with an average error of less than 0.1%. (3) The adjoint framework can handle optimization problems with more than 100 design variables and various geometric and physical constraints such as volume, thickness, curvature, and lift constraints. (4) The framework includes additional modules that are essential for successful design optimization: a geometry-parametrization module, a mesh-deformation algorithm, and an interface to numerical optimizations. To demonstrate our design-optimization framework, we optimize the ramp shape of a simple bluff geometry and analyze the flow in detail. We achieve 9.4% drag reduction, which is validated by wind tunnel experiments. Furthermore, we apply the framework to solve two more complex aerodynamic-shape-optimization applications: an unmanned aerial vehicle, and a car. For these two cases, the drag is reduced by 5.6% and 12.1%, respectively, which demonstrates that the proposed optimization framework functions as desired. Given these validated improvements, the developed techniques have the potential to be a useful tool in a wide range of engineering design applications, such as aircraft, cars, ships, and turbomachinery.

An improved non-uniformity correction algorithm and its GPU parallel implementation

The performance of SLP-THP based non-uniformity correction algorithm is seriously affected by the result of SLP filter, which always leads to image blurring and ghosting artifacts. To address this problem, an improved SLP-THP based non-uniformity correction method with curvature constraint was proposed. Here we put forward a new way to estimate spatial low frequency component. First, the details and contours of input image were obtained respectively by minimizing local Gaussian curvature and mean curvature of image surface. Then, the guided filter was utilized to combine these two parts together to get the estimate of spatial low frequency component. Finally, we brought this SLP component into SLP-THP method to achieve non-uniformity correction. The performance of proposed algorithm was verified by several real and simulated infrared image sequences. The experimental results indicated that the proposed algorithm can reduce the non-uniformity without detail losing. After that, a GPU based parallel implementation that runs 150 times faster than CPU was presented, which showed the proposed algorithm has great potential for real time application.

Minimal curvature-constrained networks

This paper introduces an exact algorithm for the construction of a shortest curvature-constrained network interconnecting a given set of directed points in the plane and a gradient descent method for doing so in 3D space. Such a network will be referred to as a minimum Dubins tree, since its edges are Dubins paths (or slight variants thereof). The problem of constructing a minimum Dubins tree appears in the context of underground mining optimisation, where the objective is to construct a least-cost network of tunnels navigable by trucks with a minimum turning radius. The Dubins tree problem is similar to the Steiner tree problem, except the terminals are directed and there is a curvature constraint. We propose the minimum curvature-constrained Steiner point algorithm for determining the optimal location of the Steiner point in a 3-terminal network. We show that when two terminals are fixed and the third varied in the planar version of the problem, the Steiner point traces out a limacon.

A Method on Dynamic Path Planning for Robotic Manipulator Autonomous Obstacle Avoidance Based on an Improved RRT Algorithm.

In a future intelligent factory, a robotic manipulator must work efficiently and safely in a Human–Robot collaborative and dynamic unstructured environment. Autonomous path planning is the most important issue which must be resolved first in the process of improving robotic manipulator intelligence. Among the path-planning methods, the Rapidly Exploring Random Tree (RRT) algorithm based on random sampling has been widely applied in dynamic path planning for a high-dimensional robotic manipulator, especially in a complex environment because of its probability completeness, perfect expansion, and fast exploring speed over other planning methods. However, the existing RRT algorithm has a limitation in path planning for a robotic manipulator in a dynamic unstructured environment. Therefore, an autonomous obstacle avoidance dynamic path-planning method for a robotic manipulator based on an improved RRT algorithm, called Smoothly RRT (S-RRT), is proposed. This method that targets a directional node extends and can increase the sampling speed and efficiency of RRT dramatically. A path optimization strategy based on the maximum curvature constraint is presented to generate a smooth and curved continuous executable path for a robotic manipulator. Finally, the correctness, effectiveness, and practicability of the proposed method are demonstrated and validated via a MATLAB static simulation and a Robot Operating System (ROS) dynamic simulation environment as well as a real autonomous obstacle avoidance experiment in a dynamic unstructured environment for a robotic manipulator. The proposed method not only provides great practical engineering significance for a robotic manipulator’s obstacle avoidance in an intelligent factory, but also theoretical reference value for other type of robots’ path planning.

NURBS Curve Interpolation Method with Flexibility and High Accuracy Based on Curvature Constraint and Displacement Compensation

NURBS Curve Interpolation Method with Flexibility and High Accuracy Based on Curvature Constraint and Displacement Compensation

A sharp scalar curvature estimate for CMC hypersurfaces satisfying an Okumura type inequality

We obtain a sharp estimate to the scalar curvature of stochastically complete hypersurfaces immersed with constant mean curvature in a locally symmetric Riemannian space obeying standard curvature constraints (which includes, in particular, a Riemannian space with constant sectional curvature). For this, we suppose that these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Melendez (Bull Braz Math Soc 45:385–404, 2014), which is a weaker hypothesis than to assume that they have two distinct principal curvatures. Our approach is based on the equivalence between stochastic completeness and the validity of the weak version of the Omori–Yau’s generalized maximum principle, which was established by Pigola et al. (Proc Am Math Soc 131:1283–1288, 2002; Mem Am Math Soc 174:822, 2005).

Buckling optimization of variable-stiffness composite panels based on flow field function

Due to the non-uniform in-plane stress distribution, variable-stiffness panel with curvilinear fiber paths is a promising structural concept for cutout reinforcement of composite structures under axial compression, due to the more diverse tailorability opportunities than simply choosing the best straight stacking sequence. However, traditional representation methods of curvilinear fiber path are usually not flexible for cutout reinforcement. In this study, the flow field function containing a uniform field and several vortex fields is utilized to represent the fiber path due to its inherent non-intersect and orthotropic features, and a bi-level optimization framework of variable-stiffness panels considering manufacturing constraints is then proposed. A typical rectangular composite panel with multiple cutouts is established to demonstrate the advantage of proposed framework by comparison with other fiber path functions. Results indicate that the flow fiber path only needs few variables to finely represent the fiber path, which can provide satisfying and manufacturable fiber paths by combination use of curvature constraint.

Semiparametric estimation under shape constraints

Substantial structure and restrictions, such as monotonicity and curvature constraints, necessary to give economic interpretation to empirical findings are often furnished by economic theories. Although such restrictions may be imposed in certain parametric empirical settings in a relatively straightforward fashion, incorporating such restrictions in semiparametric models is often problematic. A solution to this problem is provided via penalized splines, where monotonicity and curvature constraints are maintained through integral transformations of spline basis expansions. Large sample properties, implementation and inferential procedures are presented. Extension to multiple regressions under the framework of additive models is also discussed. A series of Monte Carlo simulations illustrate the finite sample properties of the estimator. The proposed method is employed to estimate a Lorenz curve of income and a production function with multiple inputs.

Feasibility study of robotic fibre placement on intersecting multi-axial revolution surfaces

Abstract In this paper, the first steps towards using a robotic workcell for the automated fibre placement (AFP) manufacturing of Y-shaped tubes are proposed. The proposed workcell is constituted of a standard serial manipulator holding the fibre placement toolhead combined to a rotary table on which the part where the fibres must be laid out is attached. The investigations carried out in this work explore the feasibility of this setup and more precisely the path planning aspect. To this aim two novel path planning algorithms are presented generalizing the techniques proposed in the literature for open-contoured and cylindrical surfaces. In the first, the maximal geodesic curvature typically allowed in AFP is disregarded to generate continuous paths with a constant placement angle on the branches without any gaps or overlaps. Subsequently, a second algorithm, taking into account this curvature constraint, is presented. These algorithms were implemented in a software using the MATLAB™ suite. Finally, an algorithm to optimize the motion of the robotic system is presented and simulations are discussed.

Nonlinear Transient Finite-Element Analysis of Delaminated Composite Shallow Shell Panels

The nonlinear transient behavior of the delaminated composite curved shell panel under different kinds of mechanical loading is investigated in this analysis. The delaminated shell panel model is developed mathematically using two higher-order midplane theories in conjunction with the Green–Langrage type of geometrical nonlinear strains including all the nonlinear higher-order terms. Further, the desired nonlinear responses are computed numerically with the help of a unique computer code developed in the MATLAB® environment. The nonlinear numerical responses are computed using Newmark’s time integration scheme together with the direct iterative method in conjunction with finite-element steps. Further, the convergence behavior of the present numerical results is checked. In addition, the validity of the present numerical responses is shown by comparing the results with those of the available published literature. Finally, the role of the size, location, and position of delamination as well as the effects of different design parameters (curvature ratio, aspect ratio, modular ratio, shell configuration, and constraint condition) on the nonlinear transient responses are computed through a wide variety of numerical examples and discussed in detail.

An integrated approach of reverse engineering aided remanufacturing process for worn components

Abstract The worn mechanical components/parts arrived in the remanufacturing system exhibit highly uncontrolled variabilities in failure conditions as well as structures and shape complexities. With the aid of reverse engineering (RE) technologies, a quick and accurate acquisition of the damaged areas of the worn part is attainable and thereby facilitates remanufacturing operations necessary to bring the parts back to like-new conditions. In this paper, a reverse engineering based approach is proposed to aid the remanufacturing processes of worn parts. The proposed approach integrates 3D surface data collection, nominal model reconstruction, fine registration, extraction of additive/subtractive repair, tool path generation and actual machining process, seeking to improve the reliability and efficiency of manual repair process. For nominal model reconstruction, a Prominent Cross-Section algorithm embedded with curvature constraint is proposed to automatically identify the boundary of the part's damaged area and thereby eliminate the defective point clouds from the reconstruction process. With the nominal reconstruction model and the 3D model of the worn part, a modified ICP algorithm integrating curvature and distance constraints is proposed to achieve a best-fit position of the two models by automatically identifying and eliminating the unreliable corresponding pairs through iterations. The proposed approach is demonstrated through remanufacturing of two different mechanical components and is approved to be efficient and effective.

Automatic method for estimation of in situ effective contact angle from X-ray micro tomography images of two-phase flow in porous media

Multiphase flow in porous media is strongly influenced by the wettability of the system, which affects the arrangement of the interfaces of different phases residing in the pores. We present a method for estimating the effective contact angle, which quantifies the wettability and controls the local capillary pressure within the complex pore space of natural rock samples, based on the physical constraint of constant curvature of the interface between two fluids. This algorithm is able to extract a large number of measurements from a single rock core, resulting in a characteristic distribution of effective in situ contact angle for the system, that is modelled as a truncated Gaussian probability density distribution. The method is first validated on synthetic images, where the exact angle is known analytically; then the results obtained from measurements within the pore space of rock samples imaged at a resolution of a few microns are compared to direct manual assessment. Finally the method is applied to X-ray micro computed tomography (micro-CT) scans of two Ketton cores after waterflooding, that display water-wet and mixed-wet behaviour. The resulting distribution of in situ contact angles is characterized in terms of a mixture of truncated Gaussian densities.

Driving on Point Clouds: Motion Planning, Trajectory Optimization, and Terrain Assessment in Generic Nonplanar Environments

We present a practical approach to global motion planning and terrain assessment for ground robots in generic three‐dimensional (3D) environments, including rough outdoor terrain, multilevel facilities, and more complex geometries. Our method computes optimized six‐dimensional trajectories compliant with curvature and continuity constraints directly on unordered point cloud maps, omitting any kind of explicit surface reconstruction, discretization, or topology extraction. We assess terrain geometry and traversability on demand during motion planning, by fitting robot‐sized planar patches to the map and analyzing the local distribution of map points. Our motion planning approach consists of sampling‐based initial trajectory generation, followed by precise local optimization according to a custom cost measure, using a novel, constraint‐aware trajectory optimization paradigm. We embed these methods in a complete autonomous navigation system based on localization and mapping by means of a 3D laser scanner and iterative closest point matching, suitable for both static and dynamic environments. The performance of the planning and terrain assessment algorithms is evaluated in offline experiments using recorded and simulated sensor data. Finally, we present the results of navigation experiments in three different environments—rough outdoor terrain, a two‐level parking garage, and a dynamic environment, demonstrating how the proposed methods enable autonomous navigation in complex 3D terrain.

Bounded distortion parametrization in the space of metrics

We present a framework for global parametrization that utilizes the edge lengths (squared) of the mesh as variables. Given a mesh with arbitrary topology and prescribed cone singularities, we flatten the original metric of the surface under strict bounds on the metric distortion (various types of conformal and isometric measures are supported). Our key observation is that the space of bounded distortion metrics (given any particular bounds) is convex, and a broad range of useful and well-known distortion energies are convex as well. With the addition of nonlinear Gaussian curvature constraints, the parametrization problem is formulated as a constrained optimization problem, and a solution gives a locally injective map. Our method is easy to implement. Sequential convex programming (SCP) is utilized to solve this problem effectively. We demonstrate the flexibility of the method and its uncompromised robustness and compare it to state-of-the-art methods.

On curvature approximation in 2D and 3D parameter–free shape optimization

Manufacturing constraints considered in shape optimization often need to be expressed in terms of curvature. Within the scope of a sensitivity---based parameter---free shape optimization approach, curvature constraints have to be formulated in terms of the FE node coordinates in order to derive the required first order gradients with respect to the design node coordinates. In this contribution we introduce approaches to approximate the curvature of a FE model using the coordinates of the FE nodes at the boundary of the geometry, as a smooth representation of the design boundary is not available. Therefore, in the 2D case we present two different smooth curves which represent the design boundary and for which the curvature can be computed analytically. In a third 2D, as well as in our 3D approach, we use geometric information of the discretization such as the distance to neighboring boundary nodes and edge normals to approximate the curvature at the respective boundary node under consideration.

Uniqueness of Spacelike Hypersurfaces in a GRW Spacetime via Higher Order Mean Curvatures

We study the problem of uniqueness concerning complete spacelike hypersurfaces immersed in a generalized Robertson–Walker (GRW) spacetime, whose fiber obeys suitable curvature constraints. In this setting, we apply some maximum principles in order to guarantee that such a spacelike hypersurface must be a slice of the ambient space, provided that some of their higher order mean curvatures satisfies appropriated controls. Furthermore, we also establish nonparametric results concerning entire spacelike graphs in GRW spacetimes.