Local convergence of the Levenberg-Marquardt method under H\"{o}lder metric subregularity

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under H\"{o}lder metric subregularity of the function defining the equation and H\"older continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the \L{}ojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the \L{}ojasiewicz gradient inequality assumption.Comment: 30 pages, 10 figure

Load flow calculation for droop-controlled islanded microgrids based on direct Newton-Raphson method with step size optimisation

Load flow calculation for droop-controlled islanded microgrids (IMGs) is different from that of transmission or distribution systems due to the absence of slack bus and the variation of frequency. Meanwhile considering the common three-phase imbalance condition in low-voltage systems, a load flow algorithm based on the direct Newton-Raphson (NR) method with step size optimisation for both three-phase balanced and unbalanced droop-controlled IMGs is proposed in this study. First, the steady-state models for balanced and unbalanced droop-controlled IMGs are established based on their operational mechanisms. Then taking frequency as one of the unknowns, the non-linear load flow equations are solved iteratively by the NR method. Generally, iterative load flow algorithms are faced with challenges of convergence performance, especially for unbalanced systems. To tackle this problem, a step-size-optimisation scheme is employed to improve the convergence performance for three-phase unbalanced IMGs. In each iteration, a multiplier is deduced from the sum of higher-order terms of Taylor expansion of the load flow equations. Then the step size is optimised by the multiplier, which can help smooth the iterative process and obtain the solutions. The proposed method is performed on several balanced and unbalanced IMGs. Numerical results demonstrate the correctness and effectiveness of the proposed algorithm

An indoor UWB 3D positioning method for coplanar base stations

As an indispensable type of information, location data are used in various industries. Ultrawideband (UWB) technology has been used for indoor location estimation due to its excellent ranging performance. However, the accuracy of the location estimation results is heavily affected by the deployment of base stations; in particular, the base station deployment space is limited in certain scenarios. In underground mines, base stations must be placed on the roof to ensure signal coverage, which is almost coplanar in nature. Existing indoor positioning solutions suffer from both difficulties in the correct convergence of results and poor positioning accuracy under coplanar base-station conditions. To correctly estimate position in coplanar base-station scenarios, this paper proposes a novel iterative method. Based on the Newton iteration method, a selection range for the initial value and iterative convergence control conditions were derived to improve the convergence performance of the algorithm. In this paper, we mathematically analyze the impact of the localization solution for coplanar base stations and derive the expression for the localization accuracy performance. The proposed method demonstrated a positioning accuracy of 5 cm in the experimental campaign for the comparative analysis, with the multi-epoch observation results being stable within 10 cm. Furthermore, it was found that, when base stations are coplanar, the test point accuracy can be improved by an average of 63.54% compared to the conventional positioning algorithm. In the base-station coplanar deployment scenario, the upper bound of the CDF convergence in the proposed method outperformed the conventional positioning algorithm by about 30%

Tailored composite microstructures via direct ink writing with acoustophoresis

Additive manufacturing techniques which enable control over the placement and orientation of particles within composite inks can produce structures with tailored gradients in structural and functional properties. One such technique is direct ink writing with acoustophoresis (DIWA), wherein a composite ink is extruded through a direct-write nozzle containing a standing bulk acoustic wave which aligns and positions particles. Driving force-based scaling relationships contextualize processing-structure relationships in DIWA. In a series of experiments which progress in geometric complexity from basic primitives to complete structures, a physical framework is constructed for controlling filament microstructures and external geometries in DIWA. In isolated filaments, there are trade-offs between focusing and form holding. Increasing the ink viscosity, increasing the print speed, and decreasing the acoustic wave amplitude widen the spatial distribution of particles in agreement with scaling relationships for acoustophoresis, but more viscous inks improve form holding. In the print bead between the nozzle and substrate, digital image analysis is used to measure filament stability, nozzle wetting, and rotational flows in the low-viscosity inks required for acoustophoresis. Viscocapillary lubrication theory accurately predicts the bounds of stability, and the contact line position and angle can be used to detect the beginnings of filament rupture, allowing for algorithms which prevent rupture in-situ. In polygonal prisms, the internal structure of filaments changes during deposition into layer-by-layer and bath support gels. Filament microstructures change during deposition, during relaxation, and when the nozzle returns to write neighboring lines. Experimental flow fields and particle distributions suggest that inertia and viscoplasticity influence the filament microstructure just after deposition and the microstructure of neighboring filaments, and interfacial energy and gravity cause filaments to spread after deposition. An analytical model is proposed to diagnose sources of direction dependent microstructures as a function of acoustics, inertia, viscous dissipation, and stage calibration. The support geometry can be used to accentuate or suppress aspects of this direction dependence. Finally, inertia swells written corners, and capillarity smooths written corners, leading to distortions in filament microstructures at corners. Bath support suppresses these corner defects