Rectangular Mesh Research Articles

Analytic formulas for complete and incomplete first-order TEM moments below ground in a conductive half-space

In the resistive limit, discrete conductors give rise to magnetic dipole fields. Magnetostatic modeling of time integrals, or moments, of transient electromagnetic (TEM) data therefore offers a means for fast approximate 3D modeling and inversion of TEM data sets. In our approximate inversion scheme, the net TEM moment response is represented as the combination of discrete conductor and uniform host responses. The inversion algorithm first estimates a homogeneous host conductivity, and then it subtracts the host response and fits the residual moment data by adjusting the conductivities of cells comprising a 3D rectangular mesh. To expedite calculation of the host response, we have derived analytic formulas for first-order TEM moments produced on and under the ground by a horizontal electric dipole on the surface of a homogeneous conductive half-space. We present analytic expressions for idealized all-time “complete” moments, or resistive limits, as well as for realizable finite-time “incomplete” moments. The moments produced by an arbitrary horizontal polygonal loop are determined by combining contributions from appropriately oriented electric dipoles. Downhole TEM moments computed with the new expressions reveal substantial differences between incomplete and complete moments when early time data are excluded and between step and impulse response incomplete moments. The role of the formulas in the first stage of our moment-based 3D inversion scheme is illustrated via analysis of downhole TEM data recorded at Santander, Peru. The host conductivity of best fit for early time B-field moments is 2.40 mS/m, consistent with apparent conductivities derived from ground TEM data recorded in the same area.

Dean flow of a Bingham fluid in a curved rectangular duct

In this paper we study the flow of a Bingham fluid on a curved (rectangular) pipe. Flows in this kind of geometries present secondary flows due to presence of centripetal forces in the radial direction. This so called Dean Flow has been extensively studied for Newtonian fluids. In considering a yield stress fluid in similar geometries the picture is less clear. Unfortunately, there is not an analytical solution, so the flow has been barely studied. Thus the purpose of this work. We consider only steady flow and develop a solution as a perturbation series in terms of the Dean number, D n . The leading order solution is an axisymmetric azimuthal flow, driven by the pressure gradient. We compute this flow numerically using an augmented Lagrangian method on a regular rectangular mesh. We have also studied the limiting case of zero flow and we give a general expression for the critical Bingham number, B c , for this to happen. The first order velocity perturbation produces 2 recirculating vortices, at top and bottom walls, analogous to those of the Newtonian flow. The size of the vortices appears to decrease slightly as B → B c − , and the magnitude of the secondary flow decays to zero significantly faster than that of the leading order flow. Finally we discuss the short comings of the perturbation series approach and propose how to overcome them. • Asymptotic expansion in Bingham Dean Flow. • Numerical solution of the primary flow for Bingham Dean Flow using Augmented Lagrangian method. • Zero flow Limit and calculation of critical Bingham number. • Numerical solution of secondary flows.

The Hamiltonian Properties of Rectangular Meshes with at Most Two Faulty Nodes

In this paper we consider the Hamilton cycle problem in the rectangular meshes with at most two faulty nodes. We prove that this problem is solvable in polynomial time with a corresponding algorithm. We provided an entirely new approach to this problem being different from the method early used on this problem.

Flow Field Study of Bio-Inspired Corrugated Airfoils at Low Reynolds Number with Different Peak Shapes

This study is intended to understand the fluid flow behaviour of a bio inspired corrugated wing obtained from the mid span of the dragonfly wing with different peak shapes of the corrugations. The aerodynamic effect due to variation of the shape of the first peak is studied with triangular and a curved peak shapes. The coordinates of the corrugated wing of the dragonfly were obtained from the existing literature and scaled up 1:50 to do the computational work on it. The corrugated wing was modeled by using a modeling software, the meshing was done by using ICEMCFD with a rectangular block meshing and simulated in Ansys Fluent software at 35000 Reynolds number and angles of attack ranging from 4° to 12°. The k-ε turbulence modeling was deployed to capture turbulence in the tested domain. The boundary conditions and size of the domain were selected as per available experimental wind tunnel setup. The flow characteristics like pressure and velocity of the triangular and curved peaks were obtained computationally and compared with each other having same geometrical parameters. The simulated results showed that the curved peak performed aerodynamically better than the triangular peak. The leading edge vortices were observed in both models trapped in the trough of the first valley with some different intensity. The validation of the computational flow results was done by existing experimental flow visualization in a wind tunnel and both results agree with each other.

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

Traveltime Calculations for qP, qSV, and qSH Waves in Two‐Dimensional Tilted Transversely Isotropic Media

AbstractThis paper presents a fast sweeping method (FSM) to calculate the first‐arrival traveltimes of the qP, qSV, and qSH waves in two‐dimensional (2D) transversely isotropic media, whose symmetry axis may have an arbitrary orientation (tilted transverse isotropy [TTI]). The method discretizes the anisotropic eikonal equation with finite difference approximations on a rectangular mesh and solves the discretized system iteratively with the Gauss‐Seidel iterations along alternating sweeping orderings. At each mesh point, a highly nonlinear equation is solved to update the numerical solution until its convergence. For solving the nonlinear equation, an interval that contains the solutions is first determined and partitioned into few subintervals such that each subinterval contains one solution; then, the false position method is applied on these subintervals to compute the solutions; after that, among all possible solutions for the discretized equation, a causality condition is imposed, and the minimum solution satisfying the causality condition is chosen to update the solution. For problems with a point‐source condition, the FSM is extended for solving the anisotropic eikonal equation after a factorization technique is applied to resolve the source singularities, which yields clean first‐order accuracy. When dealing with the triplication of the qSV wave, solutions corresponding to the minimal group velocity are chosen such that continuous solutions are computed. The accuracy, efficiency, and capability of the proposed method are demonstrated with numerical experiments.

A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem

In this article, we consider a class of singularly perturbed two-parameter parabolic partial differential equations with time delay on a rectangular domain. The solution bounds are derived by asymptotic analysis of the problem. We construct a numerical method using a hybrid monotone finite difference scheme on a rectangular mesh which is a product of uniform mesh in time and a layer-adapted Shishkin mesh in space. The error analysis is given for the proposed numerical method using truncation error and barrier function approach, and it is shown to be almost second- and first-order convergent in space and time variables, respectively, independent of both the perturbation parameters. At the end, we present some numerical results in support of the theory.

A Sequel of Inverse Lax–Wendroff High Order Wall Boundary Treatment for Conservation Laws

When solving CFD problems, the solver, or the numerical code, plays an important role. Depending on the phenomena and problem domain, designing such numerical codes can be hard work. One strategy is to start with simple problems and construct the code as building blocks. The purpose of this work is to provide a detailed review of the theory to compute analytical and exact solutions, and recent numerical methods to construct a code to solve compressible and inviscid fluid flows with high-resolution, arbitrary domains, non-linear phenomena, and on rectangular meshes. We also propose a modification to the inverse Lax–Wendroff procedure solid wall treatment and two-dimensional WENO-type extrapolation stencil selection and weights to handle more generic situations. To test our modifications, we use the finite difference method, Lax–Friedrichs splitting, WENO-Z+ scheme, and third-order strong stability preserving Runge-Kutta time discretization. Our first problem is a simple one-dimensional transient problem with periodic boundary conditions, which is useful for constructing the core solver. Then, we move to the one-dimensional Rayleigh flow, which can handle flows with heat exchange and requires more detailed boundary treatment. The next problem is the quasi-one-dimensional nozzle flow with and without shock, where the boundary treatment needs a few adjustments. The first two-dimensional problem is the Ringleb flow, and despite being smooth, it has a curved wall as the left boundary. Finally, the last problem is a two-dimensional conical flow, which presents an oblique shock and an inclined straight line wall being the cone surface. We show that the designed accuracy is being reached for smooth problems, that high-resolution is being attained for non-smooth problems, and that our modifications produce similar results while providing a more generic way to treat solid walls.

A GPU-Accelerated Filtered Density Function Simulator of Turbulent Reacting Flows

A new computational methodology is developed for large eddy simulation (LES) of turbulent reacting flows using graphic processing units (GPUs). The LES is based on the filtered density function (FDF) of the scalar-composition in conjunction with a discontinuous Galerkin (DG) discretisation scheme on a structured rectangular mesh. This hybrid solver is developed in a manner suitable for GPU computing. The simulator, as devised, is shown to be of the order of 200 times faster than the serial CPU-based calculations; facilitating the use of FDF for practical applications. The consistency and the accuracy of the methodology are demonstrated by simulations of a temporally developing mixing layer, under both non-reacting and reacting conditions.

Stability of Implicit Difference Schemes for a Linearized Hyperbolic Quasi-Gasdynamic System of Equations

We consider a multidimensional hyperbolic quasi-gasdynamic system of differential equations of the second order in time and space linearized on a constant solution (with an arbitrary velocity). For the linearized system with constant coefficients, we study the implicit three-level weighted and two-level vector difference schemes. The important domination property of the operator of viscous terms (with no allowance for the relaxation parameter) over the operator of convective terms is derived. We apply this property to prove by the energy method that, regardless of the Mach number, our implicit schemes on a nonuniform rectangular mesh (without any conditions on the mesh steps) are stable with respect to the initial data and the right-hand side uniformly in time and the relaxation parameter.

New 25-point stencils with optimal accuracy for 2-D heat transfer problems. Comparison with the quadratic isogeometric elements

A new approach for the increase in the order of accuracy of high order elements used for the time dependent heat equation and for the time independent Poisson equation has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error. By a simple modification of the coefficients of 25-point stencils, the new approach exceeds the accuracy of the quadratic isogeometric elements by four orders for the heat equation and by twelve orders for the Poisson equation. Despite the significant increase in accuracy, the computational costs of the new technique are the same as those for the conventional quadratic isogeometric elements on a given mesh. The numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new approach is much more accurate than the conventional isogeometric elements at the same number of degrees of freedom. Hybrid methods that combine the new stencils with the conventional isogeometric and finite elements and can be applied to irregular domains are also presented.

Cell-centered Lagrangian Lax-Wendroff HLL hybrid scheme in cylindrical geometry

Lagrangian hydrodynamics as described by the Euler equations is treated by an improved version of the basic predictor-corrector Lax-Wendroff method that also has added HLL-type dissipative fluxes, including both artificial viscosity and energy dissipation. The method in Cartesian geometry is enhanced by a different weighting of the conservative variables in the predictor and a new treatment of material interfaces. The basic method is extended to cylindrical r,z geometry, where it satisfies the geometric conservation law and keeps exact spherical symmetry on equiangular polar meshes. The added viscosity does not preserve symmetry, so in order to achieve that we have added a symmetry correction. Numerical hydrodynamic tests, including the Noh, Sedov, spherical Sod, free expansion and Kidder problems show a reasonable performance of the method. In addition, the spherical Noh problem was simulated on an initially rectangular mesh with a very good result regarding its symmetry.

Numerical simulation of CO2 chemical absorption in a gas-liquid bubble column using the space-time CESE method

Abstract In this study, numerical simulation of the removal process of CO2 using aqueous solutions of piperazine (PZ) in a bubble column has been done. The main focus of this work is to investigate the applicability of the space-time conservation element and solution element (CESE) method for the numerical solution of the coupling of hydrodynamics, mass transfer and chemical reactions. The reactive absorption process has been investigated by means of experiments and numerical simulation. A two-dimensional computational fluid dynamics (CFD) model in the Eulerian framework has been used to describe the gas-liquid system. To obtain the numerical solution, the CESE method based on a uniform rectangular mesh has been implemented by developing a computer code. Experimental study and CFD simulations have been carried out at different conditions of CO2 partial pressure (20 and 36 kPa) and solution concentration (0.1, 0.3 and 0.5 M). The simulation results were in reasonable agreement of 5.226 % and 4.575 % with the experimental values for gas holdup and mass transfer flux, respectively. The comparisons prove that the CESE method can be considered as a proper numerical method for the simulation of complex multiphase flows due to its simplicity, accuracy and low computational costs.

Modeling of Complex Geological Body and Computation of Geomagnetic Anomaly

Modeling plays an important role in engineering exploration. However, the traditional rectangular mesh cannot meet the requirement because the actual shape of the model may be very complex. Therefore, this paper chooses the Delaunay grid for modeling and applies this method to magnetic numerical simulation. At the same time, we make use of the advantages of the Delaunay grid and choose to use grid encryption technology in the boundary and complex area of the model. Then, the formula for calculating the magnetic anomaly of tetrahedron is deduced in detail, and the magnetic anomaly with surface fluctuation is calculated. For a synthetic data model, Delaunay grids with different densities are used to model and calculate surface anomalies. The two results are compared with the analytical solution of the model. The results show that the method has high accuracy. Then, the method is applied to the actual geological body modeling in the Jinchuan mining area. According to local needs, a three-dimensional model with inhomogeneous grid density is established and the surface magnetic field of the model is calculated. Finally, the simulation data are compared with the measured data. The results show that the Delaunay grid modeling method has strong applicability.

A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations

For two-dimensional (2D) time fractional diffusion equations, we construct a numerical method based on a local discontinuous Galerkin (LDG) method in space and a finite difference scheme in time. We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable. Numerical results indicate the effectiveness and accuracy of the method and confirm the analysis.

Traveltime computation and OBN seismic record simulation for rugged seafloor in VTI medium

ABSTRACT To improve the computational accuracy of seismic traveltime for complex structures in inhomogeneous media, a mesh generation method that is adaptive to a rugged seafloor or interface is presented. A hybrid mesh discrete velocity model consisting of conventional rectangular mesh and irregular quadrilateral mesh is also proposed. The local traveltime equations for the hybrid mesh are derived from the linear traveltime interpolation (LTI) method and are shown to be stable. The relationship between the group velocity and group angle is converted into that between group velocity and interpolated point coordinates, extending the LTI ray tracing algorithm to multiwave simulation in vertical transversely isotropic (VTI) medium by multi-stage partitioning. Computations of the first arrival, reflection, multiple reflection, multiple transmission conversion, and multiple reflection conversion seismic waves in the VTI medium with rugged seafloor and complex structural interfaces are realised. The results show that the proposed method can adapt to the rugged seafloor and velocity interfaces of complex structures, resulting in higher computational accuracy of the traveltime and ray paths for ocean bottom node (OBN) seismic wave simulation. The accuracy of the proposed method is verified by comparison with the finite-difference method. This method will be useful for modelling of seismic waves, wavefield identification, and study of seismic wave propagation in OBN seismic observations.

Effective Resistance of Finite Two-Dimensional Grids Based on Infinity Mirror Technique

Conventional numerical circuit analysis tools typically scale superlinearly with the number of nodes. With the rapid increase in nodes in modern VLSI systems, alternative methods are required. The effective resistance is an important characteristic of electrical systems, which is used to simplify the circuit analysis process. An infinite resistive rectangular mesh is commonly assumed in the analysis of grid structures to determine the effective resistance of a grid. The assumption of infinity provides a useful approximation when a large grid is analyzed far from the boundaries. If however the grid is analyzed in close proximity to a boundary or if the grid dimensions are small, the assumption of infinity may lead to significant error. To address this issue, the infinity mirror technique is proposed to determine the effective resistance of a two-dimensional structure, where one or both dimensions are finite. The method exhibits good agreement with nodal analysis, achieving an error below 1% in case studies. The proposed expressions enhance the speed of static grid analysis by several orders of magnitude by replacing computationally expensive nodal analysis with an equivalent reduced grid analysis. A 1,400 fold speedup is achieved in the analysis of 100 nodes within a $10^{3}\times 10^{4}$ grid.

DETERMINATION OF THE EFFECTIVE PERMITTIVITY OF A HETEROGENEOUS MATERIAL

Purpose. To develop a two-dimensional numerical-field model for determining the effective permittivity of a multicomponent material represented by a system of homogeneous volumes with known physical characteristics. Methodology . The model is based on the solution by the finite element method of an electrostatic problem with the subsequent determination of the energy contained in the volume under consideration. Than we have compared this result with the energy of a flat capacitor with a rectangular cross-section of the plates and determined the effective permittivity of test material. We also have used Rayleigh, Odelevsky and Lichtenecker models and the model with a perpendicular arrangement of layers relative to the main electric flux. Results. Based on the developed field model, the effective permittivities for dry, wet and transformer oil-soaked insulating papers of various grades, including taking into account ash, are determined. We have proved that a macroscopically homogeneous multicomponent material is well approximated by uniformly spaced cylindrical volumes with a substance of different nature in a matrix of another substance. We have showed a significant error of the layer model and the Rayleigh model relative to the proposed model. We have showed the equivalence of models with the location of inclusions in the nodes of a rectangular and parallelogram mesh. Originality. For the first time we have proposed wet paper models with an asymmetric arrangement of a cylindrical volume of water with a circular and segment cross-section in a cylindrical pore. For the first time we have proposed models of insulating paper with evenly spaced cylindrical inclusions of different volumes. Practical value. The proposed model allows to calculate the effective permittivity of an inhomogeneous material with a given accuracy without restricting the shape of the components. Based on the proposed field model, it is possible to determine the Lichtenecker index, which allows to calculate the effective permittivity for any ratio of the volumes of the components of a heterogeneous material.

Solution of the Fracture Detection Problem by Machine Learning Methods

Inverse problems of fracture exploration seismology are solved using machine learning methods. A single fracture of fixed size and subvertical orientation is considered in the two-dimensional case. The spatial position and the inclination angle of the fracture are determined using a neural network. The training set consists of solutions of direct problems produced by the grid-characteristic method on regular rectangular meshes in the form of synthetic seismograms obtained by measuring the vertical velocity on the surface of the medium.

The Conformal Finite-Difference Time-Domain Simulation of GPR Wave Propagation in Complex Geoelectric Structures

The finite-difference time-domain (FDTD) method adopts the most popular numerical model simulating ground penetrating radar (GPR) wave propagation in an underground structure. However, a staircase approximation method is usually adopted to simulate the curved boundary of an irregular object in the FDTD and symplectic partitioned Runge-Kutta (SPRK) methods. The approximate processing of rectangular mesh parameters will result in calculation errors and virtual surface waves for irregular targets of an underground structure. In this paper, we examine transverse mode (TM) electromagnetic waves with numerical models of electromagnetic wave propagation in geoelectric structures with conformal finite-difference time-domain (CFDTD) method technology in which the effective dielectric parameters are used to accurately simulate the dielectric surface and to absorb waves at the edges of the grid. The third orders of the transmission boundary are used in this paper. Additionally, three complex geocentric models of inclined layered media, spherical media, and three-layered pavement model with structural damages are set up for simulation calculations, then we carry out the actual radar wave detection in a laboratory as the fourth numerical example. Comparison of simulated reflectance waveform of FDTD, symplectic partitioned Runge-Kutta (SPRK), and CFDTD methods shows that at least 50% of the virtual waves can be reduced by using the proposed algorithm. Wiggle diagrams of FDTD and CFDTD methods show that much of the virtual waves have been reduced, and the radar image is clearer than before. This provides a method for the detection of complex geoelectric and layered structures in actual engineering.