A Unified Definition of Electrostatic and Magnetic Fields

This article originates from the observation that field lines are drawn using distinctive rules in magnetic field and electrostatic fields. It aims at reconciliating the definitions of these fields and thus reaching a consensus on the interpretation of field lines. Our unified field definition combines three orthogonal vectors and a unique scalar value. Field lines are then defined as isovalue lines of the scalar value, rendering it simpler to interpret in both field types. Specific to our field definition is the use of square root of vector’s cross product so that all vectors have the same physical unit. This enhanced field definition also enables a more efficient calculation of Biot-Savart law. This article is the first of a series allowing the drawing of isovalue contour lines.

Approximate analytical solution for seepage field of drained tunnel in vertically stratified phreatic aquifer

To explore the water table and water inflow after tunnel excavation in a vertically stratified phreatic aquifer,approximate analytical solutions for the steady-state water table and water inflow of a drained tunnel in a vertically stratified phreatic aquifer were obtained based on the Dupuit assumptions and the integral method.By comparing the approximate analytical solutions with numerical solutions,it was found that the relative error of the approximate analytical solution for the water table elevation is less than 10%,and the relative error of the approximate analytical solution for the water inflow is approximately 25%.The sources of the above errors are as follows:(1)At the lateral boundary of water replenishment,the water surface should be tangent to the horizontal line,but the water surface for the approximate analytical solutions has a gradient.(2)At the vertical boundaries near the tunnel,the total head is variable,but the total head for the approximate analytical solutions is assumed to be constant.(3)The Dupuit assumptions are applied in the flow domain near the tunnel.Although the relative errors of the approximate analytical solutions for the water table elevation and water inflow are evident,the lowered water table is reflected in the approximate analytical solutions.

EPL-PRM:Equipotential line sampling strategy for probabilistic roadmap planners in narrow passages

Path planning is a crucial concern in the field of mobile robotics,particularly in complex scenarios featuring narrow passages.Sampling-based planners,such as the widely utilized probabilistic roadmap(PRM),have been extensively employed in various robot applications.However,PRM’s utilization of random node sampling often results in disconnected graphs,posing a significant challenge when dealing with narrow passages.In order to tackle this issue,we present equipotential line sampling strategy for probabilistic roadmap(EPL-PRM),a novel approach derived from PRM.This paper initially proposes a sampling potential field,followed by the construction of equipotential lines that are denser in the proximity of obstacles and narrow passages.Random sampling is subsequently conducted along these lines.Consequently,the sampling strategy enhances the likelihood of sampling nodes around obstacles and narrow passages,thereby addressing the issue of sparsity encountered in traditional sampling-based planners.Furthermore,we introduce a nodal optimization method based on an artificial repulsive field,which prompts sampled nodes to move in the direction of repulsion.As a result,nodes around obstacles are distributed more uniformly,while nodes within narrow passages gravitate toward the middle of the passages.Finally,extensive simulations are conducted to evaluate the proposed method.The results demonstrate that our approach achieves path planning with superior efficiency,lower cost,and higher reliability compared with traditional algorithms.