Rectangular Mesh Research Articles

Optimal weighted upwind finite volume method for convection–diffusion equations in 2D

This paper develops and analyzes the optimal weighted upwind finite volume method (OWUFVM) for convection–diffusion equations on nonuniform rectangular meshes. The dual meshes in this method are associated with the local Peclet numbers, which are determined by the convection and diffusion coefficients and the mesh size. We first prove the stability and H 1 -norm error estimate. Then we prove a weak error estimate, which plays a key role in deriving the optimal second-order error estimate in the L 2 norm. The L 2 -norm error estimate is the most important result of this paper. We also briefly examine the discrete maximum principle. Numerical experiments are presented to illustrate the theoretical analysis.

The new method for hiatal hernia laparoscopic alloplasty

The aim — to analyze the immediate results of the first experience of a new laparoscopic mesh repair method for the hiatal hernia.Materials and methods. Five patients with an age ranging from 38 to 70 years, and a body mass index ranging from 28 to 35 kg/m2, undergone laparoscopic mesh repair and Nissen fundoplication using new method called «internal buttress of posterior crural repair». The average area of the hernia was 9.8 cm2 (from 8 to 16 cm2). In all cases, there was a thinning of the cruras and the upper part of the border of the hiatal. Two pieces of rectangular mesh Parietene ProGrip (Covidien, USA), measuring from 0.8 ´ 2.5 cm to 1.0 ´ 3.0 cm, were fixed by their hooks to the inner surface of the crura. Then, cruroraphy together with the both pieces of the mesh was done using a self­gripping non­absorbable continuous suture with V­loc 2­0 (Covidien, USA).Results and discussion. The mean duration of repair was 20 minutes (range 15 to 35 minutes). The mesh implants were not shifted when flashing and at the same time their position could be corrected. The rupture of the crura and the bleeding from the bite points were not observed. When tightening the thread, cruras approached to each other well. There were no cases of dysphagia.Conclusions. The new method of the internal buttres of posterior crural repair using self­gripping lightweight polyester mesh (Parietene ProGrip, Covidien, USA) demonstrated its technical feasibility and safety. The new method combines the advantages of positioning of a mesh posteriorly to the crura and the comfortability of posterior crural repair. More cases collection with assessment of long­term results and comparing with other methods is needed.

Range-Restricted Surface Interpolation Using Rational Bi-Cubic Spline Functions With 12 Parameters

This paper discusses the constraint data interpolation or range restricted interpolation for surface data arranges on rectangular meshes that lie above or below an arbitrary plane and between two arbitrary planes by using partially blended rational bi-cubic spline function with 12 parameters. Common research in range restricted surface interpolation is to construct the constrained surface lie above linear plane. However, in this paper, we consider the constraint surfaces up to degree three (cubic). To construct the constrained surface with shape preserving properties, i.e., the resulting surface will lie below or above single planes or between two respective planes, the data dependent sufficient conditions are derived on four parameters; meanwhile, the remaining eight parameters are free parameters to change the shape of the interpolating surface locally. The proposed scheme is tested with various types of data test, including some well-known functions. From the numerical results, we found that the proposed scheme is easy to use, locally control via free parameters, and require less computation compared with some existing schemes as well as visually pleasant for visualization. Furthermore, based on root mean square error (RMSE) and coefficient of determination (R 2 ), the proposed scheme is better than existing scheme with the value of R 2 achieved is in between 0.9701 (97.01%) and 0.9954 (99.54%). This is quite good for range restricted surface data interpolation since we can explain at least 97.01% of the variance in the interpolation by using the proposed scheme. Furthermore, the proposed scheme requires less CPU time (in seconds) compared with the existing scheme.

Development of Bead Modelling for Distortion Analysis Induced by Wire Arc Additive Manufacturing using FEM and Experiment

In this research, Wire Arc Additive Manufacturing is modelled and simulated to determine the most suitable bead modelling strategy. This analysis is aimed to predict distortion by means of thermomechanical Finite Element Method (FEM). The product model with wire as feedstock on plate as substrate and process simulation are designed in form of multi-layered beads and single string using MSC Marc/Mentat. This research begins with finding suitable WAAM parameters which takes into account the bead quality. This is done by using robotic welding system with 01.2mm filler wire (AWS A5.28 : ER80SNi1), shielding gas (80% Ar/ 20% CO2) and 6mm-thick low carbon steel as base plate. Further, modelling as well as simulation are to be conducted with regards to bead spreading of each layers. Two different geometrical modelling regarding the weld bead are modelled which are arc and rectangular shape. Equivalent material properties from database and previous researches are implemented into simulation to ensure a realistic resemblance. It is shown that bead modelling with rectangular shape exhibits faster computational time with less error percentage on distortion result compared to arc shape. Moreover, by using the rectangular shape, the element and meshing are much easier to be designed rather than arc shape bead.

Finite-element analysis of top-casing electric source method for imaging hydraulically active fracture zones

Imaging hydraulically active fracture zones (HAFZ) is of paramount importance to subsurface resource extraction, geologic storage, and hazardous waste disposal. We have developed advanced 3D finite-element (FE) electrical imaging algorithms for HAFZ in the presence of a steel-cased well. The algorithms use tetrahedral FE meshes in the simulation domain and coarse rectangular finite-difference meshes in the imaging domain. This heterogeneous dual-mesh approach is well suited to modeling the multiscale earth model due to steel-cased wells. We find that the algorithms accurately and efficiently simulate surface electric field measurements over a 3D HAFZ at depth when one end point of a surface electric source is connected to a wellhead. For brevity, this configuration is called the top-casing electric source method. By replacing a hollow cased well with a solid prism, we improve our computational efficiency without affecting the solution accuracy. The sensitivity of the top-casing source method to HAFZ highly depends on the continuity of a steel-cased well because it makes currents preferentially flow to HAFZ. The sensitivity also depends on conductivity structures around the well because they control current leaking from the steel-cased well. We find that the method can image a localized HAFZ and detect changes in its width and height. The imaging results are improved when a volume of the imaging domain is constrained from geomechanical perspectives. A primary advantage of the method is the fact that the sources and receivers are placed on the surface, thus not interrupting the well operation.

Development of [formula omitted]-inverse model by using generalized polynomial chaos

We present a hp-inverse model to estimate a smooth, non-negative source function from a limited number of observations for a two-dimensional linear source inversion problem. A standard least-square inverse model is formulated by using a set of Gaussian radial basis functions (GRBF) on a rectangular mesh system with a uniform grid space. Here, the choice of the mesh system is modeled as a random variable and the generalized polynomial chaos (gPC) expansion is used to represent the random mesh system. It is shown that the convolution of gPC and GRBF provides hierarchical basis functions for the linear source inverse model with the hp-refinement capability. We propose a mixed l1 and l2 regularization to exploit the hierarchical nature of the basis functions to find a sparse solution. The hp-inverse model has an advantage over the standard least-square inverse model when the number of data is limited. It is shown that the hp-inverse model provides a good estimate of the source function even when the number of unknown parameters (m) is much larger the number of data (n), e.g., m∕n>40.

Computing the quasipotential for nongradient SDEs in 3D

Nongradient SDEs with small white noise often arise when modeling biological and ecological time-irreversible processes. If the governing SDE were gradient, the maximum likelihood transition paths, transition rates, expected exit times, and the invariant probability distribution would be given in terms of its potential function. The quasipotential plays a similar role for nongradient SDEs. Unfortunately, the quasipotential is the solution of a functional minimization problem that can be obtained analytically only in some special cases. We propose a Dijkstra-like solver for computing the quasipotential on regular rectangular meshes in 3D. This solver results from a promotion and an upgrade of the previously introduced ordered line integral method with the midpoint quadrature rule for 2D SDEs. The key innovations that have allowed us to keep the CPU times reasonable while maintaining good accuracy are (i) a new hierarchical update strategy, (ii) the use of Karush–Kuhn–Tucker theory for rejecting unnecessary simplex updates, and (iii) pruning the number of admissible simplexes and a fast search for them. An extensive numerical study is conducted on a series of linear and nonlinear examples where the quasipotential is analytically available or can be found at transition states by other methods. In particular, the proposed solver is applied to Tao's examples where the transition states are hyperbolic periodic orbits, and to a genetic switch model by Lv et al. (2014) [21]. The C source code implementing the proposed algorithm is available at M. Cameron's web page.

Dispersion and Stability Condition of Seismic Wave Simulation in TTI Media

For seismic waveform simulation in tilted transversely isotropic (TTI) media, we derive explicitly the numerical dispersion relation and the stability condition for the computation of a 2D pseudo-acoustic wave equation. The numerical dispersion relation indicates that the number of sampling points per wavelength has the greatest influence on the dispersion, while the anisotropic parameters of the TTI media and the mesh rotation angle have little influence on the dispersion. Given an appropriate spatial sampling, the stability condition is for the selection of the time step for the implementation of the TTI wave equation. We partition a numerical model using quadrangle grids in Cartesian coordinates, and map it to a computing model in which any non-rectangular meshes in Cartesian coordinates become rectangular meshes. Then we reformulate the pseudo-acoustic wave equation for the TTI media accordingly in the computational space. We implement seismic waveform simulation using the second-order finite-difference method straightforwardly, and show examples with a desirable accuracy using a model with non-rectangular meshes in Cartesian coordinates along a curved surface and fluctuating interfaces in the TTI media.

A Numerical Study of the Effects of Nano-Biological Scaffolds on Inflammatory Vascular Wall.

This research investigated the effect of different nano-biological scaffolds on blood vessel walls by numerical analysis. The dynamics behavior of fluids and the mechanical property of vessel wall were studied when the blood vessel wall was in the state of inflammation. Employing straight and bent blood vessels as 3D models, this research simulated the blood flow in the inflammatory blood vessels and put forward the mathematical method to analyze mechanical characteristics of the vessel walls under the condition of the bi-directional coupling. At the same time, the impact of the nano-biological scaffolds was explored. The results showed that the thrombus in the inflammatory blood vessels significantly influenced the dynamics behavior of blood fluids. The blood velocity in the rectangular mesh nano-biomaterials was greatly decreased. It concluded that the numerical method was feasible to estimate signal feedbacks. Simulating the fluid dynamics behavior in the inflammatory blood vessels, the mechanical characteristics of the vessel wall under the condition of the two-way coupling was consistent with the medical facts. Moreover, the clogging degree had significantly influence on the inflammation of vessel wall. These findings facilitated health care workers to employ proper nano-biological scaffolds in the thrombosis treatment.

A parametric morphing method for generating structured meshes for marine free surface flow applications with plane symmetry

Abstract This article introduces a method for morphing a parametric, rectangular and structured 3D template mesh such that it embodies: (1) a given (half) hull surface properly aligned with a pre-defined air-water interface zone, (2) a high-quality boundary layer region near the hull, and (3) a smooth transition of the mesh to the template mesh structure away from the hull. The performance of the proposed method is successfully validated on three widely studied benchmark cases, namely the parabolic Wigley hull, the Series 60 hull and the DTMB 5415 combatant hull. The proposed automated mesh generation method is useful in shape optimization for computational fluid dynamics, among others. Highlights A new method for creating structured meshes for marine CFD applications is presented. The proposed method is parametric and thus fully programmable. The method includes an automatic resolution of CAD geometry errors. The resulting meshes are strictly orthogonal near the inter-phase zone and the hull. The main application of the proposed method is in CFD-based shape optimization.

An upwind finite volume method for convection-diffusion equations on rectangular mesh

In this paper, we present an upwind finite volume method to solve the convection-diffusion equations with Dirichlet boundary on rectangular mesh. By utilizing the technique of element-by-element analysis, the stability of the method has been proved and the H1-norm error estimate is presented. Furthermore, we provide the proofs of the maximum principle and L∞-norm error estimate. Finally, some numerical experiments are provided to confirm our theoretical results.

An extended family of nonconforming quasi-Wilson elements for solving elasticity problem

This contribution (part two) focuses on numerical implementation and efficiency aspects of an extended family of nonconforming quasi-Wilson elements type. Such a class of nonconforming elements has been introduced recently in Achchab et al. (2015). Here, based on a rectangular mesh, it is used for the approximate solution of a planar elasticity problem. It is shown that this family passes the patch-test, and that by suitably adding some orthogonality conditions, on a general class of enrichment functions, we can derive higher order consistency error estimates. Our general theoretical results, see Theorems 4.1 and 4.2, unify, simplify and extend a number of existing works on the improvement of the order of consistency error. Numerical experiments are carried out to demonstrate that our method is optimal for various Lamé parameter μ, shear modulus λ and locking free when the Poisson parameter ν approaches close to 0.5.

Lowest-Order Weak Galerkin Finite Element Methods for Linear Elasticity on Rectangular and Brick Meshes

This paper investigates lowest-order weak Galerkin finite element methods for solving linear elasticity problems on rectangular and brick meshes. Specifically, constant vectors are used in element interiors and on element interfaces respectively for approximating displacement. For these constant basis functions, their discrete weak gradients are calculated in the local Raviart–Thomas spaces $$ RT_{[0]}^d $$ ( $$ d=2 $$ or 3), whereas their discrete weak divergences are calculated as elementwise constants. Discrete weak strains are calculated accordingly. Then these quantities are used to develop finite element schemes in both strain-div and grad-div formulations, on both rectangular and brick meshes. A theoretical analysis supported by numerical experiments in both 2-dim and 3-dim reveal that the methods are locking-free and have optimal 1st order convergence in displacement, stress, and dilation (divergence of displacement), when the exact solution has full regularity. The methods can also capture low-regularity solutions very well. Strategies for efficient implementation including Schur complement are presented. Extension to quadrilateral and hexahedral meshes, in both theoretical analysis and numerical experiments, is also examined.

An 𝜀-uniform method for a class of singularly perturbed parabolic problems with Robin boundary conditions having boundary turning point

A class of singularly perturbed parabolic differential equations with Robin type boundary condition having boundary turning point is propounded on a rectangular domain in the [Formula: see text]-[Formula: see text] plane. A numerical method comprising a standard upwind finite difference scheme is formulated on a rectangular piecewise uniform fitted mesh [Formula: see text] and it is proved to be [Formula: see text]-uniform. Furthermore, it is shown that the errors are bounded in the supremum norm by [Formula: see text], where [Formula: see text] is a constant independent of [Formula: see text], [Formula: see text] and the perturbation parameter [Formula: see text]. Numerical results are given to illustrate the analytical results.

Coordinate transformation aided finite element method for contour detection of breast tumors in microwave imaging.

A coordinate transformation based finite element method is presented for detecting contours of breast cancer tissues in the context of microwave imaging. The geometry and location of a single or multiple cancerous tumors inside a breast tissue are identified by constructing an inverse problem based on the genetic optimization algorithm. The expensive forward solution phase during repeated cost function evaluations is accelerated by placing a transformation medium inside the finite element mesh. This allows using a uniform or rectangular mesh even when irregular tumor geometries are considered. Therefore, the computational load is reduced. The algorithm is implemented in MATLAB using parallel programming tools. A realistic numerical breast phantom repository, which is available in the University of Wisconsin-Madison webpage, is used to test the algorithm. The performance of the algorithm is also tested by utilizing the Monte Carlo approach, where the inverse problem is run several times and the statistical parameters of the runs are determined. Various simulation results are demonstrated.

Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation

In this paper, the local discontinuous Galerkin method is developed to solve the two-dimensional Camassa–Holm equation in rectangular meshes. The idea of LDG methods is to suitably rewrite a higher-order partial differential equations into a first-order system, then apply the discontinuous Galerkin method to the system. A key ingredient for the success of such methods is the correct design of interface numerical fluxes. The energy stability for general solutions of the method is proved. Comparing with the Camassa–Holm equation in one-dimensional case, there are more auxiliary variables which are introduced to handle high-order derivative terms. The proof of the stability is more complicated. The resulting scheme is high-order accuracy and flexible for arbitrary h and p adaptivity. Different types of numerical simulations are provided to illustrate the accuracy and stability of the method.

Parameter-uniform fractional step hybrid numerical scheme for 2D singularly perturbed parabolic convection–diffusion problems

This article focuses on developing and analyzing an efficient numerical scheme for solving two-dimensional singularly perturbed parabolic convection–diffusion initial-boundary value problems exhibiting a regular boundary layer. For approximating the time derivative, we use the Peaceman–Rachford alternating direction implicit method on uniform mesh and for the spatial discretization, a hybrid finite difference scheme is proposed on a special rectangular mesh which is tensor-product of piecewise-uniform Shishkin meshes in the spatial directions. We prove that the numerical scheme converges uniformly with respect to the perturbation parameter $$\varepsilon $$ and also attains almost second-order spatial accuracy in the discrete supremum norm. Finally, numerical results are presented to validate the theoretical results. In addition to this, numerical experiments are conducted to demonstrate the effect of the time-dependent boundary conditions in the order of convergence numerically by introducing the classical evaluation of the boundary data; and also the improvement in the spatial order of accuracy of the present method by considering the Bakhvalov–Shishkin mesh in the spatial directions.

Spatial Variability in Stability of Aggregates and Organic Matter of an Oxisol

Soil preparation may break its structure, destabilize the aggregates, and cause the loss of organic matter (OM). The study of spatial variability of soil attributes is an important indicator of soil physical quality. The aim of this study was to describe the spatial variability of the stability of aggregates and organic matter in Oxisol (Yellow Latosol) under different management systems.We collected simple samplings of soil in the eastern mesoregion of Maranhão, Brazil. Experimental areas with two distinct management systems were studied: conventional tillage and no-tillage. In each experimental area, we fitted a rectangular mesh of 50 points with 40m of spacing and 0.00 to 0.20 mof depth. The response variables were: weighted mean diameter (WMD); geometric mean diameter (GMD); percentage of aggregates (on classes of size between 1-2 mm and above 2 mm); and organic matter (OM). The no-tillage management showed high values of WMD, GMD, class of aggregates and OM. Maps of WMD and GMD were spatially correlated to OM map at no-tillage management. Soil properties had a spatial-dependent structure. The management system influenced the stability of aggregates and the amount of organic matter.

An Ordered Line Integral Method for computing the quasi-potential in the case of variable anisotropic diffusion

Nongradient stochastic differential equations (SDEs) with position-dependent and anisotropic diffusion are often used in biological modeling. The quasi-potential is a crucial function in the Large Deviation Theory that allows one to estimate transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths. Unfortunately, the quasi-potential can rarely be found analytically. It is defined as the solution to a certain action minimization problem. In this work, the recently introduced Ordered Line Integral Method (OLIM) is extended for computing the quasi-potential for 2D SDEs with anisotropic and position-dependent diffusion scaled by a small parameter on a regular rectangular mesh. The presented solver employs the dynamical programming principle. At each step, a local action minimization problem is solved using straight line path segments and the midpoint quadrature rule. The solver is tested on two examples where analytic formulas for the quasi-potential are available. The dependence of the computational error on the mesh size, the update factor K (a key parameter of OLIMs), as well as the degree and the orientation of anisotropy is established. The effect of anisotropy on the quasi-potential and the maximum likelihood paths is demonstrated on the Maier–Stein model. The proposed solver is applied to find the quasi-potential and the maximum likelihood transition paths in a model of the genetic switch in Lambda Phage between the lysogenic state where the phage reproduces inside the infected cell without killing it, and the lytic state where the phage destroys the infected cell.

A Non-Equilibrium Sediment Transport Model for Dam Break Flow over Moveable Bed Based on Non-Uniform Rectangular Mesh

The use of multiple-level non-uniform rectangular mesh in coupled flow and sediment transport modeling is preferred to achieve high accuracy in important region without increasing computational cost greatly. Here, a robust coupled hydrodynamic and non-equilibrium sediment transport model is developed on non-uniform rectangular mesh to simulate dam break flow over movable beds. The enhanced shallow water and sediment transport equations are adopted to consider the mass and momentum exchange between the flow phase and sediment phase. The flux at the interface is calculated by the positivity preserving central upwind scheme, which belongs to Godunov-type Riemann-problem-solver-free central schemes and is less expensive than other popular Riemann solvers while still capable of tracking wet/dry fronts accurately. The nonnegative water depth reconstruction method is used to achieve second-order accuracy in space. The model was first verified against two laboratory experiments of dam break flow over irregular fixed bed. Then the quantitative performance of the model was further investigated by comparing the computational results with measurement data of dam break flow over movable bed. The good agreements between the measurements and the numerical simulations are found for the flow depth, velocity and bed changes.