Fast Sweeping Method Research Articles

Calculating traveltimes in 2D general tilted transversely isotropic media using fast sweeping method

Traveltime calculations play an important role in the field of exploration seismology, such as traveltime tomography and seismic imaging and so on. Seismic anisotropy poses a challenge for traveltime calculation, because anisotropic eikonal solvers are more complex than the isotropic counter part. To solve the eikonal equations in 2D tilted transversely isotropic (TTI) media, we have developed a fast algorithm combine with fast sweeping method to compute the first arrival traveltimes of quasi-P (qP)-, quasi-SV (qSV)-, and quasi-SH(qSH)-waves. For the qP- and qSV-waves, we analyzed the quartic coupled slowness surface equation derived from the Christoffel equation. Then, we constructed a local solver to relate traveltime and slowness. We found that in the local solver, one component of the slowness vector is known and the corresponding slowness equation is monotonic. This provides a strong basis for the fast iterative algorithm we proposed, where we use the Newton method to solve the qP- and qSV-wave slowness equation to determine the related traveltimes. For the qSH wave, the slowness equation is quadratic and simple to solve. Numerical experiments demonstrate that the proposed method can obtain accurate traveltimes for simple and complicated 2D TTI models.

Multi-Objective Reverse Design and Pattern Analysis of Solid Propellant Grains

Solid propellant grain reverse design aims to discover optimal grain geometries by shape optimization methods to match the desired solid motor performance curves. To maximize the performance matching degree and the propellant loading fraction simultaneously, this study develops a multi-objective evolutionary neural network for the grain reverse design, where the burning surface regression calculation is efficiently employed using the fast-sweeping method. Then, grain shape feature extraction and pattern analysis are achieved through image singular value decomposition and self-organizing mapping, respectively. Finally, the design case of a dual-thrust motor and a Mars ascent vehicle show that the method can well balance the performance-matching degree and propellant loading fraction. Moreover, without any training data set, it can generate dozens of grain shape patterns, highlighting their diversity and providing new ideas for solid rocket motor designers. Our method can offer a new pathway for the research field of solid rocket motor design.

An absolutely convergent fixed-point fast sweeping WENO method on triangular meshes for steady state of hyperbolic conservation laws

High order accuracy fast sweeping methods have been developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs) on rectangular meshes, while they were not available yet on unstructured meshes. In this paper, we extend high order accuracy fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. Similar as other fast sweeping methods, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. This also provides certain convenience in designing high order fast sweeping methods on unstructured meshes. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils, specifically here a fifth-order finite volume unstructured WENO scheme for spatial discretization. Extensive numerical experiments, including problems with complex domain geometries, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler time marching method and the popular third order total variation diminishing Rung-Kutta (TVD-RK3) time-marching method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 in order to converge to steady state solutions.

Evaluating a distance function

Computing the distance function to some surface or line is a problem that occurs very frequently. There are several ways of computing a relevant approximation of this function, using for example technique originating from the approximation of Hamilton Jacobi problems, or the fast sweeping method. Here we make a link with some elliptic problem and propose a very fast way to approximate the distance function.

Modelling crowd pressure and turbulence through a mixed-type continuum approach

Empirical studies of large gatherings and natural disasters have revealed two important features of dense crowds: extremely high crowd pressure and crowd turbulence. In this study, a mixed-type continuum model for multidirectional pedestrian flow was developed that explicitly considered the phase transition of different anticipation characteristics under different densities. Non-hyperbolicity was used to model the strong instabilities during crowd turbulence. In addition, by estimating the aggregated crowd pressure, the proposed model could clarify the effects of both force chains and panic sentiment, phenomena commonly observed during crowd disasters. The non-hyperbolic partial differential equations were solved using the mixed-type finite difference method, and Eikonal equations were solved using the fast sweeping method. Subsequently, the continuum model was applied to simulations of two real-world scenarios – the 2015 Hajj crowd disaster and the 2010 Love Parade crowd disaster – and validated through comparison with empirical observations. Overall, the proposed model is an efficient tool for evaluating crowd management strategies to predict and assess the crowd state.

Direct Simple Computation of Middle Surface Between 3D Point Clouds and/or Discrete Surfaces by Tracking Sources in Distance Function Calculation Algorithms

Abstract In this paper, we introduce novel methods for computing middle surfaces between various 3D data sets such as point clouds and/or discrete surfaces. Traditionally the middle surface is obtained by detecting singularities in computed distance function such as ridges, triple junctions, etc. It requires to compute second order differential characteristics, and also some kinds of heuristics must be applied. Opposite to that, we determine the middle surface just from computing the distance function itself which is a fast and simple approach. We present and compare the results of the fast sweeping method, the vector distance transform algorithm, the fast marching method, and the Dijkstra-Pythagoras method in finding the middle surface between 3D data sets.

Unconditionally stable higher order semi-implicit level set method for advection equations

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of advection by an external velocity and by a speed in the normal direction that are applicable in level set methods. The recommended numerical scheme is third order accurate for the linear advection in the 2D case with a space dependent velocity. Using a combination of analytical and numerical tools in the von Neumann stability analysis, the third order scheme is claimed to be unconditionally stable. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function in the 1D case. In the case of nonlinear advection, a semi-implicit discretization is proposed to linearize the problem. The compact implicit part of the stencil of numerical schemes contains unknowns only in the upwind direction. Consequently, algebraic solvers like the fast sweeping method can be applied efficiently to solve the resulting algebraic systems. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of the velocity confirm the good accuracy of the third order scheme even in the case of very large Courant numbers. The advantage of the high-resolution scheme is documented for examples where the advected level set functions contain large jumps in the gradient.

Early Identification of Abnormal Regions in Rock-Mass Using Traveltime Tomography

Early identification of abnormal regions is crucial in preventing the occurrence of underground geotechnical disasters. To meet the high-accuracy detection requirements of underground engineering, this paper proposes a tomography method for abnormal region identification in complex rock-mass structures that utilizes traveltime tomography combined with the damped least-squares method and Gaussian filtering. The proposed method overcomes the limitation of velocity difference in empty region detection and mitigates the impact from isolated velocity mutation in the iteration. Numerical and laboratory experiments were conducted to evaluate the identification accuracy and computational efficiency of forward modeling, including the shortest-path method (SPM), dynamic SPM (DSPM), and fast sweeping method (FSM). The results show that DSPM and FSM can clearly detect abnormal regions in numerical and laboratory experiments. Field experiments were conducted in the Shaanxi Zhènào mine and achieve the reconstruction of the underground roadway distribution. This paper not only realizes the application of abnormal region identification using traveltime tomography but also provides new insight into potential hazards detection in underground geotechnical engineering.

A topography-dependent eikonal solver for accurate and efficient computation of traveltimes and their derivatives in 3D heterogeneous media

Accurate and efficient computation of traveltimes and their spatial derivatives for 3D heterogeneous media is essential in many areas of seismology, such as traveltime/slope tomography and quantitative migration. A popular method of obtaining this information is to solve the eikonal equation using a finite-difference (FD) scheme. However, the complex surface of the earth causes numerical complications when applying FD methods based on rectangular grids. To overcome the challenge of taking into account topographic effects, we make use of the 3D topography-dependent eikonal equation (TDEE) which is a 3D surface-flattening scheme that uses curvilinear grids to transform irregular surfaces in the physical domain into regular surfaces in the computational domain. Unfortunately, solving the TDEE in three dimensions starting from the initial point-source condition needs to get around the point-source singularity. Therefore, we formulate the 3D factored TDEE (FTDEE) according to a factorization principle. Furthermore, to numerically solve the 3D original TDEE, as well as the FTDEE, we apply a lock sweeping method, which avoids the problem of repetitive computation when using the fast sweeping method. Several numerical examples demonstrate the efficiency of the new lock scheme in terms of computation time and accuracy of the 3D FTDEE solver for the calculation of traveltimes and their derivatives. This scheme provides a good choice for the calculation of traveltimes and their spatial derivatives using models with irregular surfaces, which means greater potential for seismic data processing under conditions of complex topography.

Adjoint-state traveltime tomography for azimuthally anisotropic media in spherical coordinates

SUMMARYTong has proposed an adjoint-state traveltime tomography method to determine velocity heterogeneity and azimuthal anisotropy. This method, however, ignores the Earth’s curvature when deriving the eikonal equation for azimuthally anisotropic media. Thus, further coordinate transformation or approximation is required to ensure the accuracy of traveltime prediction in large-scale tomography. To address this problem, we derive the eikonal equation for azimuthally anisotropic media in spherical coordinates, which naturally considers the Earth’s curvature. Another key ingredient is the forward modelling algorithm, whose accuracy and efficiency dominate the numerical error and computational cost of the inversion. In this study, we apply a modified fast sweeping method to solve the eikonal equation in spherical coordinates. Two approaches, including the third-order weighted essentially non-oscillatory approximation and multiplicative factorization technique, are applied to improve the accuracy. According to the numerical experiments, this new eikonal solver achieves a second-order accuracy and is about two orders of magnitude more accurate than the commonly used first-order fast sweeping method with similar runtime. Taking advantage of the two improvements, we develop a novel eikonal equation-based adjoint-state traveltime tomography method for azimuthally anisotropic media in spherical coordinates. This method is applicable for large-scale tomography, and its performance is verified by a synthetic checkerboard test and a practical seismic tomographic inversion in central California near Parkfield.

Eikonal equation-based earthquake location with irregular surfaces

SUMMARYEarthquake location is a basic seismological problem and has a key role in many quantitative seismic analyses. The significant anomaly of traveltimes observed in mountainous areas with irregular surfaces has made earthquake location a challenge. To cope with this problem, we develop an effective eikonal equation-based earthquake location method based on unstructured mesh for 2-D/3-D isotropic and anisotropic media with irregular surfaces. First, the location misfit function is established by the reciprocity principle. Then, we use a global search algorithm to find the optimal origin times and hypocentres. Afterward, we apply eikonal equation-based master-event relocation method to relocate earthquakes on lateral boundaries of the region where the location results may be biased. To accurately compute the traveltime in 3-D vertically transversely isotropic and tilted transversely isotropic models with irregular surfaces or interfaces, we also propose a 3-D iterative fast sweeping method for eikonal equation on the unstructured tetrahedral mesh. Finally, we verify the proposed method by performing numerical experiments in 2-D/3-D irregular isotropic and anisotropic models. The numerical tests indicate that the proposed eikonal equation-based earthquake location method provides an effective way to find accurate hypocentre and origin time in 2-D/3-D isotropic and anisotropic media with irregular surfaces, even for inhomogeneous complex structures. In addition, the eikonal equation-based master-event location method has also yielded promising relocation results.

An efficient hybrid mesh method for seismic traveltime computation with irregular surfaces

Accurate and efficient traveltime computation with irregular surfaces is a prerequisite for traveltime tomography, seismic location, and seismic exploration in mountainous areas. The eikonal equation method is computationally favourable and has gained increasing popularity for obtaining traveltime in recent years. Nowadays most algorithms that can solve eikonal equation with irregular surfaces are based on fully unstructured triangular mesh or surface-fitting mesh. However, these methods have poor performance in terms of computational cost and accuracy. To cope with this problem, we develop a hybrid mesh method based on fast-sweeping-based eikonal solver with rectangular and unstructured triangular mesh for obtaining traveltime with irregular surfaces. In other words, we propose to divide the model into several different regions, which are discretized by rectangular mesh and unstructured triangular mesh respectively. Near the irregular surfaces of model, we use fast sweeping method based on unstructured triangular mesh which releases the full potential of unstructured triangular mesh’s flexibility in accurately describing the irregular geometry of computation domain. For the rest part of the model, we adopt rectangular mesh to make full use of high efficiency of the corresponding high-order fast sweeping algorithm. At the interface of two types of mesh, we need interpolate the known nodes’s traveltimes to get the boundary condition of another mesh, so as to start the fast sweeping method therein. It is found that the accuracy of hybrid mesh method can be increased by 83%, the computation time is reduced by 72%, and the storage requirement is reduced by 72%, compared with the fast sweeping method with full triangular mesh, when the rectangular mesh proportion is 80% in our model. Therefore, this proposed hybrid mesh method can greatly enhance the computational efficacy and serve as an potential forward engine in seismic inversions.

A second-order distributed memory parallel fast sweeping method for the Eikonal equation

The Eikonal equation is used to calculate wave propagation and distance fields, and due to its complexity requires numerical treatment for its solution. We present a second-order distributed memory parallel fast sweeping method. The second-order solution switches on a two-point stencil when two upwind points are available, and reverts to first-order otherwise. In all examples, the second-order method improves the solution over the first-order, allowing for significant savings in memory while achieving the same accuracy. Parallelization over distributed memory saw good weak scaling with optimal convergence. The computational time for second-order was approximately 2.5 times slower than first-order, where the largest amount of mesh points ran on 144 cores (512 GB) was ≈20 billion. The savings in memory from the second-order method combined with the distributed memory algorithm result in the ability to solve problems much larger than are possible with the serial first-order method.

Modeling pedestrian flow through a bottleneck based on a second-order continuum model

Large numbers of pedestrians moving in a two-dimensional continuous walking facility can be regarded as a continuum fluid that is governed by the conservation laws of fluid dynamics. Pedestrian flow differs from common flows driven by a pressure difference in that it has special path-choice behaviors associated with the purpose of the trip. This paper presents a new second-order continuum model for pedestrian flow to study the collective behaviors of a large-scale crowd in front of a bottleneck that are important for the planning and design of walking facilities. A pressure term is introduced to describe how pedestrians react to the state of traffic in the desired direction of motion of pedestrian flow. The desired direction of motion is determined by an Eikonal equation on the instantaneous walking cost caused by pedestrians walking to the final destination in a reactive manner. The solution algorithm is composed of a third-order fast sweeping method for the Eikonal equation and a third-order WENO method for the continuum model in a conservation-law form. Numerical experiments of pedestrian flow passing through a bottleneck of varying widths are carried out to verify the proposed model and algorithm.

Effects of Air Quality on Housing Location: A Predictive Dynamic Continuum User-Optimal Approach

Recent decades have seen increasing concerns regarding air quality in housing locations. This study proposes a predictive continuum dynamic user-optimal model with combined choice of housing location, destination, route, and departure time. A traveler’s choice of housing location is modeled by a logit-type demand distribution function based on air quality, housing rent, and perceived travel costs. Air quality, or air pollutants, within the modeling region are governed by the vehicle-emission model and the advection-diffusion equation for dispersion. In this study, the housing-location problem is formulated as a fixed-point problem and the predictive continuum dynamic user-optimal model with departure-time consideration is formulated as a variational inequality problem. The Lax-Friedrichs scheme, the fast-sweeping method, the Goldstein-Levitin-Polyak projection algorithm, and self-adaptive successive averages are adopted to discretize and solve these problems. A numerical example is given to demonstrate the characteristics of the proposed housing-location choice problem with consideration of air quality and to demonstrate the effectiveness of the solution algorithms.

Estimation of traffic emissions in a polycentric urban city based on a macroscopic approach

This paper presents a macroscopic dynamic modeling framework for road traffic emissions in an urban city that has numerous central business districts (CBDs). The framework combines a two-dimensional (2D) continuum dynamic traffic assignment (DTA) model that captures macroscopic features of traffic flow and a microscopic instantaneous emission model associated with the instantaneous speed and acceleration of traffic flow. A heterogeneous traveling population continuously distributed in space is classified according to the trip destinations chosen by travelers. Each copy of flow tends to select an optimal route with the minimum cost to destination by means of instantaneous traffic information. For each copy of flow, an unstructured finite volume method as well as a fast sweeping method is designed to solve the proposed model. In order to validate the feasibility of the model and algorithm, a case study is conducted as an example to study the macroscopic characteristics of traffic flow and predict traffic-related pollutants like NOx, VOC, CO2 and PM in urban areas.

An efficient path planning algorithm for the biped robot in a static environment using fast sweeping method

Obtaining the collision-free path for the biped walking machine/robot in an obstacle clustered environment is a challenging task. To overcome this difficulty, researchers around the world are implementing several collision-free path planning algorithms. It is significant to note that an effective path-planning methodology helps the biped robot to achieve a collision-free path with a minimum travel path. In the current investigation, the authors have proposed a new path planning procedure, that is, Fast Sweeping Method (FSM), to obtain the optimal path for the biped robot in a static environment. To check the effectiveness of the developed procedure, in the present investigation, the authors have tested it in both simulations as well as on an actual biped robot in a static domain. Further, the results of the current algorithm are compared with two established path planning algorithms, namely the Fast Marching Method (FMM) and A-star algorithms. In the end, it has been detected that the proposed algorithm obtains the best collision-free optimal path when compared with the Fast Marching Method (FMM) and A-star algorithms.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show high-order accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

Non-Destructive Testing for Cavity Damages in Automated Machines Based on Acoustic Emission Tomography

Damage detection is important for the maintenance of automated machines. General non-destructive testing techniques require static equipment and complex analysis processes, which restricts the maintenance of automated machines. Therefore, this paper proposes an acoustic emission (AE) tomography method for detecting cavity damage in automated machines, combining the fast sweeping method (FSM) and the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method. This approach overcomes the limitations of real-time AE detection for cavity damage in continuous and homogeneous materials. The proposed method has been applied in numerical and laboratory experiments to validate its feasibility. The results show that the inversed low-velocity regions correspond to the actual cavity regions, and the sources of cavity damage can be effectively detected. This paper provides a new perspective for AE testing technologies, and also lays the foundation for other non-destructive testing techniques, in terms of cavity damage detection.

High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations

In this paper, we combine the nonlinear HWENO reconstruction in \cite{newhwenozq} and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in \cite{mehweno1}. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function $\phi$. They are now computed from the current value of $\phi$ and the previous spatial derivatives of $\phi$. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO scheme. In addition, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method is used to update the numerical approximation. It is an explicit method and does not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some known issues, including that the iterative number is very sensitive to the parameter $\varepsilon$ used in the nonlinear weights, as observed in previous studies. Finally, in order to further reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the good performance of the proposed fixed-point fast sweeping HWENO methods.