Nodal Velocities Research Articles

Coupled finite-volume method and smoothed-particle hydrodynamics method for numerical simulation of interactions between inviscid shock waves and structures

In this work, a novel coupled finite-volume method (FVM) and a smoothed-particle-hydrodynamics (SPH) method were developed for the simulation of interactions between inviscid shock waves and structures. In this approach, which considers the particles of a meshless method immersed in an FVM grid, the FVM grid cells are classified into either pure or mixed FVM cells, the latter of which contain SPH particles. A finite-element-method shape function is applied to map the variables from the SPH particles to the FVM cells, and the nodal and cell velocities are then obtained. The interaction of the fluid with the structure is computed using moving reflection boundary conditions at cell interfaces with SPH particles. The interactions of the structure with the fluid are computed from the pressure differences around the SPH particles. The processes for computing the coupled FVM–SPH method are described in detail herein. The validity of the presented coupled FVM–SPH method was verified using a theoretical model of a piston, and the numerical results were found to agree well with the theoretical approximations, indicating the accuracy of the proposed coupled method. The results of the method were then compared with the results of an experiment involving a blast-driven steel plate. Good agreement between the experimental and numerical results was obtained, and the maximum difference was 3.44%, demonstrating the effectiveness of the proposed coupled FVM–SPH method when applied to the interaction of a shock wave with a structure.

Power Plastics: A Hybrid Lagrangian/Eulerian Solver for Mesoscale Inelastic Flows

We present a novel hybrid Lagrangian/Eulerian method for simulating inelastic flows that generates high-quality particle distributions with adaptive volumes. At its core, our approach integrates an updated Lagrangian time discretization of continuum mechanics with the Power Particle-In-Cell geometric representation of deformable materials. As a result, we obtain material points described by optimized density kernels that precisely track the varying particle volumes both spatially and temporally. For efficient CFL-rate simulations, we also propose an implicit time integration for our system using a non-linear Gauss-Seidel solver inspired by X-PBD, viewing Eulerian nodal velocities as primal variables. We demonstrate the versatility of our method with simulations of mesoscale bubbles, sands, liquid, and foams.

An artificial neural network-based machine learning approach to correct coarse-mesh-induced error in computational fluid dynamics modeling of cell culture bioreactor

Computational Fluid Dynamics (CFD) is a valuable tool for studying fluid environments within cell culture bioreactors and optimizing processing parameters, but it can be computationally expensive. This study developed an artificial neural network (ANN)-based machine learning model to predict and correct the coarse-mesh-induced errors in CFD modeling of a spinner flask bioreactor. A baseline ANN model was trained to predict the velocity error function between the coarse and optimized reference mesh results at one rotational speed (90 rpm), demonstrating that the ANN-based approach could correct the coarse-mesh velocity with RMSE values of nodal velocities improved by an average of ∼20% at different rotational speeds. The effect of ANN structure, input data normalization, and training dataset combinations on prediction performance was evaluated. More neurons and hidden layers generated better results but required more computational time for training. The model’s generalization capabilities were further evaluated in case studies of correcting velocity and Kolmogorov length at different fluid viscosity and bioreactor impeller geometry conditions. Results suggested that the ANN model had better generalization in correcting Kolmogorov length than velocity. This research provides insights into using a machine learning approach to enhance CFD modeling in bioreactor applications, contributing to advancing tissue engineering processes.

Spectral wave explicit weak-scattering approach for higher harmonic wave loads and ringing response of flexible monopile

This paper presents an accurate potential decomposition strategy for wave–structure interactions, known as the spectral wave explicit weak-scattering approach, where the entire original problem is divided into incident and scattered components. The incident component is evaluated using spectral models in advance; hence, only the scattered component is to be solved via a higher-order boundary element approach. The proposed coupled model is validated using published experimental and computational fluid dynamics results for wave interactions with a vertical cylinder. The results obtained are consistent with wave elevations and wave loads for regular waves and focused wave groups. A finite-element numerical model is applied to measure the severity of structural responses. Fluid–structure interactions are simulated by converting the obtained pressures into equivalent nodal forces, and new nodal velocities are converted into boundary conditions. The ringing response of a bottom-fixed monopile due to moderately steep focused wave groups is discussed based on various focal positions relative to the monopile, incident wave conditions, and damping ratios. The time–frequency characteristics of the ringing phenomenon are successfully revealed via wavelet analyses.

Coupling effects of strain rate and fatigue damage on wheel-rail rolling contact behaviour: a dynamic finite element simulation

ABSTRACT A comprehensive 3-D explicit finite element (FE) model was established to investigate the frictional wheel-rail rolling contact responses in this study, in which the strain rate effect and initial fatigue damage of wheel/rail materials were taken into account via inputting relevant stress-strain response curves. Four friction exploitation levels were included to examine the dynamic contact responses in terms of contact load, contact patch area, creep characteristic, and stress/strain state. It is found that the nodal velocities on the rail surface at the trailing and leading edges of the contact patch are in opposite directions once the slip occurs. The strain rate hardening effect will significantly increase the Mises stress and suppress the plastic deformation of the wheel and rail at high creep levels, while the Mises stress will decrease with the increase of fatigue damage but the plastic strain will increase correspondingly.

Modeling Fracture in Rate-Dependent Polymer Networks: A Quasicontinuum Approach

Abstract Soft materials, such as rubber and gels, exhibit rate-dependent response where the stiffness, strength, and fracture patterns depend largely on loading rates. Thus, accurate modeling of the mechanical behavior requires accounting for different sources of rate dependence such as the intrinsic viscoelastic behavior of the polymer chains and the dynamic bond breakage and formation mechanism. In this chapter, we extend the QC approach presented in Ghareeb and Elbanna (2020, An Adaptive Quasi-Continuum Approach for Modeling Fracture in Networked Materials: Application to Modeling of Polymer Networks, J. Mech. Phys. Solids, 137, p. 103819) to include rate-dependent behavior of polymer networks. We propose a homogenization rule for the viscous forces in the polymer chains and update the adaptive mesh refinement algorithm to account for dynamic bond breakage. Then, we use nonlinear finite element framework with predictor–corrector scheme to solve for the nodal displacements and velocities. We demonstrate the accuracy of the method by verifying it against fully discrete simulations for different examples of network structures and loading conditions. We further use the method to investigate the effects of the loading rates on the fracture characteristics of networks with different rate-dependent parameters. Finally, We discuss the implications of the extended method for multiscale analysis of fracture in rate-dependent polymer networks.

A new formulation for kinetic dynamic relaxation method based on the Lagrangian interpolation

This research presents a new algorithm for the Dynamic Relaxation (DR) method with kinetic damping by utilizing Lagrangian interpolation functions to derive new iterative kinetic DR equations. This procedure leads to an algorithm that, unlike the ubiquitous DR methods, does not require the calculation of nodal velocities, thereby marching forward only through successive nodal displacement. Also, the power iteration method is used to determine the optimal time step ratio. By utilizing this time step, the restarting analysis phase, which is one of the drawbacks of the kinetic DR strategy, is eliminated. In order to evaluate the performance and efficiency of the proposed method, some truss and frame structures with geometrically nonlinear behavior are analyzed. Results prove that the number of convergence iterations and analysis duration is reduced in the new kinetic DR scheme compared to other conventional DR methods.

Lagrangian interpolation for kinetic dynamic relaxation method with the variable load factor

Common dynamic relaxation (DR) methods use central finite differences to derive fundamental equations. In this study, Lagrangian interpolation functions are utilized to formulate iterative DR relationships with kinetic damping. This procedure leads to an algorithm that, unlike the ubiquitous DR methods, does not require the calculation of nodal velocities thereby marching forward only through successive nodal displacement. Also, the power iteration method is used to determine the optimal time-step ratio. By utilizing this time-step, the restarting analysis phase which is one of the drawbacks of the kinetic DR strategy is eliminated. Furthermore, a new variable load factor is developed based on the concept of minimization of displacement increment function. To evaluate the performance and efficiency of the proposed method, some truss and frame structures with geometrically nonlinear behavior are analyzed. Results prove that the number of convergence iterations and analysis time are reduced in the new kinetic DR scheme in comparison with other conventional DR methods.

АНАЛИЗ ВЛИЯНИЯ КОНСЕРВАТИВНОГО СГЛАЖИВАНИЯ КОНЕЧНО-ЭЛЕМЕНТНОГО РЕШЕНИЯ НА ПОДАВЛЕНИЕ МОД НУЛЕВОЙ ЭНЕРГИИ

The problem of high-speed penetration of a non-deformable cylinder into a steel plate is considered. The defining system of equations is formulated in Lagrange variables in a three-dimensional formulation. The equation of motion is derived from virtual work capacities balance. Kinematic relations are recorded in the metric of the current state. The relations of the flow theory with kinematic and isotropic hardening are used as equations of state. The contact interaction of the cylinder and the plate is modeled by non-penetration conditions. The numerical solution of the problem under given boundary and initial conditions is based on the moment scheme of the finite element method and “cross” type explicit time integration scheme. To discretize the defining system of equations with respect to spatial variables, 8-node isoparametric finite elements with multilinear form functions are used. To suppress the high-frequency oscillations of the numerical solution, the procedure of nodal displacement velocities conservative smoothing is used. The smoothing algorithm is based on the momentum conservation law, focused on finite element grids consisting of blocks that are mutually unambiguously mapped to a unit cube. To analyze the nodal displacement velocities monotonicity, the numerical solution splitting in the directions of the finite element grid lines is used. As the results of computer modeling have shown, the finite elements of the plate are exposed large deformations and rotation angles as a rigid whole during local intense dynamic loading. The conservative smoothing procedure influence on the numerical solution stability is analyzed. It is shown that in the problem under consideration, without applying the conservative smoothing procedure, zero-energy modes develop in the contact zone in the finite-element grid of the plate (an hourglass-type instability) and the collision process cannot be modeled before the cylinder rebounds.

Tensile Fracture Mechanism of Masonry Wallettes Parallel to Bed Joints: A Stochastic Discontinuum Analysis

Nonhomogeneous material characteristics of masonry lead to complex fracture mechanisms, which require substantial analysis regarding the influence of masonry constituents. In this context, this study presents a discontinuum modeling strategy, based on the discrete element method, developed to investigate the tensile fracture mechanism of masonry wallettes parallel to the bed joints considering the inherent variation in the material properties. The applied numerical approach utilizes polyhedral blocks to represent masonry and integrate the equations of motion explicitly to compute nodal velocities for each block in the system. The mechanical interaction between the adjacent blocks is computed at the active contact points, where the contact stresses are calculated and updated based on the implemented contact constitutive models. In this research, different fracture mechanisms of masonry wallettes under tension are explored developing at the unit–mortar interface and/or within the units. The contact properties are determined based on certain statistical variations. Emphasis is given to the influence of the material properties on the fracture mechanism and capacity of the masonry assemblages. The results of the analysis reveal and quantify the importance of the contact properties for unit and unit–mortar interfaces (e.g., tensile strength, cohesion, and friction coefficient) in terms of capacity and corresponding fracture mechanism for masonry wallettes.

A meso-scale discrete element method framework to simulate thermo-mechanical failure of concrete subjected to elevated temperatures

This paper presents mesoscale thermo-mechanical analyses of plain (unreinforced) concrete based on the discrete element method (DEM). The proposed discontinuum modelling strategy represents the aggregates and matrix as a system of deformable polyhedral blocks, interacting along their boundaries. The nodal velocities of each block are calculated via the explicit integration scheme of DEM, and contact stresses are computed based on the relative contact displacements of the adjacent blocks. To better predict the thermo-mechanical behaviour of concrete, fracture energy-based contact constitutive models are implemented by considering temperature dependency at the zone and contact properties. First, the discrete meso models are tested under uniaxial compression loading at room temperature. Then, transient thermo-mechanical tests are performed considering different load levels. The results of the computational models are compared with the macroscopic response quantities of concrete obtained from the available experimental studies in the literature. The results indicate that the developed DEM framework predicts the complex mesoscale thermo-mechanical response history and the typical damage progression observed in concrete. Furthermore, the fracture patterns, crack propagation with temperature, and the differential thermal expansion phenomena are studied in detail.

Simulation of the in-plane structural behavior of unreinforced masonry walls and buildings using DEM

In this study, a novel computational modeling strategy is proposed to estimate the lateral load capacity and behavior of unreinforced masonry (URM) structures. All commonly noted failure mechanisms are captured via the proposed modeling strategy using the discrete element method (DEM) in three-dimensions (3D). Masonry walls are represented as a system of elastic discrete blocks, where the nodal velocities are evaluated by integrating the equations of motion using the central difference method. Then, the mechanical interactions among adjacent blocks are examined utilizing the relative contact displacements and employed in the contact stress calculation. Through this research, a new stress-displacement contact constitutive model is considered and implemented in the commercial software 3DEC, which includes softening stress-displacement behavior for tension, shear, and compression along with the fracture energy concept. The results of the discontinuum models are validated on small- and large-scale experimental studies available in the literature with good agreement. Furthermore, important inferences are made regarding the effect of block size, the number of contact points, and contact stiffness values for robust and accurate simulations of masonry walls.

A three-dimensional dual-scale coupled approach and its application in geotechnics

To study the microscopic and macroscopic behaviour of geotechnical engineering materials, attention is increasingly being paid to the dual-scale coupled computation of continuum and discrete models. However, two-dimensional dual-scale coupling cannot reveal the actual behaviour of materials. Therefore, a three-dimensional dual-scale coupled approach to accomplish the information transmission between discrete-element model and finite-difference model was deduced, and a corresponding coupled program was developed. The point of application of the resultant force on the interface in the discrete-element model was sought by introducing the principle of least squares, and the element nodal forces at the interface in the continuum model were obtained based on the formulation of load transfer from the literature. The velocities acting on the interface in the discrete-element model were calculated by bilinear interpolation according to the nodal velocities of the interface in the continuum model, which achieved the velocity transmission from the continuous zone to the discrete-element zone. The coupling was accomplished through the velocity and force transmissions at the interface between discrete-element model and finite-difference model. Finally, the algorithm and program were validated through an example of dynamic compaction.

A total Lagrangian position-based finite element formulation for free-surface incompressible flows

In this work, we propose a position-based finite element formulation for incompressible Newtonian flows under total Lagrangian description. Such formulation is different from the traditional finite element approach used in fluid dynamics by using current nodal positions as main variable instead of nodal velocities. The variational form of the governing equations is derived by applying the stationary total mechanical energy principle written in terms of current positions. To introduce full incompressibility, we use equal-order mixed finite elements combined to a pressure stabilizing Petrov-Galerkin technique, circumventing Ladyzhenskaya-Babuška-Brezzi conditions. This leads to a method directly applicable to finite strain incompressible flow problems that, if combined to re-meshing techniques, is capable of simulating more complex problems, resulting into a suitable tool for free-surface flows analysis in general. The efficiency and robustness of the proposed approach is demonstrated with numerical studies and comparison of obtained results to numerical and experimental data from literature.

A surface moving mesh method based on equidistribution and alignment

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed based on the current mesh. A selection of two- and three-dimensional examples are presented.

Bearing capacity of embedded foundations using quasi-kinematic limit analysis

Bearing capacity of rectangular foundations, with varying aspect ratios (1 ≤ L/B ≤ 5) and embedded at shallow to medium depths (D/B ≤ 5), has been evaluated by using the three-dimensional cell-based smoothed finite elements quasi-kinematic limit analysis. The results have also been obtained for circular and strip foundations by performing exclusively the axisymmetric and plane strain analyses. The associated optimization problem has been solved by employing the semidefinite programming (SDP). Shape factors (sc, sq and sγ) and depth factors (dc, dq and dγ) are given as a function of soil internal friction angle (ϕ). The shape factors have been expressed as a function of L/B, on the other hand, the depth factors have been given as a function of D/B and L/B. A thorough comparison of the results has been made with the different solutions available in literature. The variations of (i) power dissipation function, (ii) maximum plastic shear strain rates, and (iii) the nodal velocities patterns, have also been examined to interpret the failure mechanism.

The Effect of Solder Joint Microstructure on the Drop Test Failure—A Peridynamic Analysis

We investigate the drop reliability of different solder joint microstructures using a coupled board-level finite-element (FE) analysis and the microscale peridynamic (PD) simulations that can capture fracture in the heterogeneous solder joint. A new PD model for elastic behavior across a material interface is introduced and used in computing the damage and failure in the two-phase microstructure of a solder joint. This new model eliminates oscillations in strains at an interface. The microstructural geometry is digitized from the scanning electron microscopy images of experimentally tested samples. To simulate the drop test conditions, we employ the input acceleration method in the board-level FE model and calculate nodal velocities for the boundary conditions imposed on the microscale PD model. We test two different microstructures of intermetallic component (IMC) (AuSn4) and SAC305: flash and thick Au content. The results show significantly more fracture in the samples with larger Au content compared to the flash samples, which correlate well with experimental observations. The computed results show that the failure mechanism for solder joints with high Au content is a fracture through the AuSn4 IMC as well as along the interface between the inclusions and the matrix. A newly introduced measure, the quasi-damage index, estimates the locations in the microstructure of the solder joint with the highest risk for initiation of cracks and damage.

Realistic estimates of superconducting properties for the cuprates: reciprocal-space diagrammatic expansion combined with variational approach

We propose a systematic approach to the systems of correlated electrons, the so-called -DE-GWF method, based on reciprocal-space (-resolved) diagrammatic expansion of the variational Gutzwiller-type wave function for parametrized models of correlated fermions. The present approach, in contrast to either variational Monte-Carlo (VMC), or the recently developed real-space diagrammatic expansion of the Gutzwiller-type wave function (direct-space DE-GWF technique), is applicable directly in the thermodynamic limit and thus is suitable for describing selected singular features of the wave-vector-dependent quantities. We employ the -DE-GWF method to extract the non-analytic part of the two leading moments of the fermion spectral-density function across the (two-dimensional) Brillouin zone for the Hubbard model and away from the half-filling. Those moments are used to evaluate the nodal quasiparticle velocities and their spectral weights in the correlated superconducting state. The two velocities determined in that manner exhibit scaling with the electron concentration qualitatively different from that obtained earlier for the excited states of the high-Tc cuprates within the projected quasi-particle ansatz, and the results are in a very good quantitative agreement with experimental data if interpreted as those characterizing the spectrum below and above the observed kink. We provide a detailed discussion of the two gaps and two excitation branches (two velocities) appearing naturally within our DE-GWF approach. The two separate sets of characteristics distinguish the renormalized quasiparticle states very close to the Fermi surface from the deeper correlated-state properties. Also, an enhancement of the -dependent magnetic susceptibility is shown to contain a spin-fluctuation contribution within our language. Finally, the -DE-GWF approach is compared to both the VMC and real-space DE-GWF results for the cases of Hubbard and t-J-U models.

Upper-Bound Finite-Element Limit Analysis of Axisymmetric Problems for Mohr-Coulomb Materials Using Semidefinite Programming

An upper-bound formulation for performing finite-element limit analysis by using semidefinite programming (SDP) for an axisymmetric stability problem involving the Mohr-Coulomb yield criterion has been presented. The SDP technique has an advantage in that it can deal with the yield criterion directly in its native form in terms of principal stresses and strains without any smoothing of the parent yield surface. The associated flow rule and plastic power dissipation are expressed entirely in terms of principal plastic strain rates. Nodal velocities and element plastic strain rates were framed as basic governing variables without involving stresses. The solution was obtained by using the SDP solver MOSEK in MATLAB. The problem of finding the bearing capacity of a circular footing was dealt with by choosing a planar domain that was discretized with (1)three-noded constant strain finite elements and (2)six-noded linear strain finite elements. The results were obtained with and without the provision of the velocity discontinuities, and were compared with those reported in literature. It was learned that quite accurate solutions can be obtained with the application of six-noded linear strain triangular elements and using velocity discontinuities along all the elements' interfaces.

Maxis—A rezoning and remapping code in two dimensional cylindrical geometry

This paper presents the new version of our code Maxis (Lin et al., 2011). Maxis is a local rezoning and remapping code in two dimensional cylindrical geometry, which can be employed to address the grid distortion problem of unstructured meshes. The new version of Maxis is mostly programmed in the C language which considerably improves its computational efficiency with respect to the former Matlab version. A new algorithm for determining the intersection of two arbitrary convex polygons is also incorporated into the new version. Some additional linking functions are further provided in the new version for the purpose of combining Maxis and MULTI2D. New version program summaryProgram title: MaxisProgram files doi:http://dx.doi.org/10.17632/nnxvbg9pbz.1Licensing provisions: GPLv3Programming language: C and MatlabSubprograms used: MULTI2D CPC 180 (2009) 977Journal reference of previous version: Zhiwei Lin, Comput. Phys. Commun. 182 (2011) 1361–1376Does the new version supersede the previous version?: YesNature of problem: Although the large distortion of mesh zones moving with fluid velocities causes the shortening of time step and the increasing of computational error, and thus makes it indispensable that the rezoning of distorted zones and the remapping of physical variables are performed, it should be noted that rezoning and remapping can also bring in the loss of the Lagrangian motion information of the original mesh. Therefore the essential problem is how to minimize the loss caused by rezoning and remapping and to improve the mesh quality concurrently.Solution method: We employ the local rezoning and remapping technique to confine the loss within the worse or insignificant regions, that is, only those worse regions determined by the mesh-closure or mesh-union methods are rezoned.Reasons for the new version: Due to the multi loops of the remapping algorithm, the former version of Maxis [1] in the Matlab language is quite inefficient. Moreover, the linking functions for combining Maxis and MULTI2D [2] are not present in the previous version.Summary of revisions: The new version of Maxis is mostly programmed in the C language and its computational efficiency is greatly increased. The remapping algorithm of the new version is based on a new scheme for determining the intersection of two arbitrary convex polygons. Some linking functions are also provided along. The detailed new features of Maxis are as follows.(1) The development of Maxis in CThe former version of Maxis is developed in the Matlab language and is quite inefficient due to the multi loops of the remapping algorithm. Hence we programme the time consuming part of Maxis in the C language because of its high efficiency, that is, the following C functions are developed:closure_of_polygon.c:which can be used to compute mesh closure;new_topology.c:which can be used to compute the topology of the new mesh;cell_remapping.c:which can be used to compute the zone/cell centered physical variables of the new mesh such as mass densities;node_remapping.c:which can be used to compute the node centered physical variables of the new mesh such as nodal velocities;convex_polygon_intersection.c:which can be used to compute the intersection of two arbitrary convex polygons. But the top level of the code Maxis still uses the Matlab language, since Matlab is easy to use and makes it rather convenient to visualize and analyze numerical results. The advantages of the C and Matlab languages can be fully exploited by doing so. The thought of combining two computer languages is also adopted by MULTI2D in which the C and r94 languages are utilized at different computing levels.(2) A new scheme for determining the intersection of two arbitrary convex polygonsBoth cell_remapping.c and node_remapping.c are based on a new algorithm for computing the intersection of two arbitrary convex polygons as follows. Let P1 and P2 be two convex polygons. Then it holds that P1=∩i=1n1li, where li is the half plane determined by the ith edge of P1 and n1 is the number of the edges of P1. Then we can have that P1∩P2=P2∩l1∩l2∩⋯∩ln1.Hence the calculation of the intersection of two convex polygons can be reduced to a series of calculations of the intersection between a convex polygon and a half plane, which can be readily programmed. Such an example is given in Fig. 1.(3) Some linking functionsThe new version of Maxis comprises some linking functions for the purpose of combining Maxis and MULTI2D. For example, the r94 function readremap.r is supplied to import the data of the Maxis remap.out into MULTI2D so that the Lagrangian calculation of an ALE simulation [3,4] can proceed.(4) New Maxis manualA new Maxis manual for detailed operations is provided along.Restrictions: It appears that there is no uniform criterion to distinguish the worse mesh zones from the good ones.Unusual features: The time consuming part of Maxis is developed in the C language, whereas the top level of the code still utilizes the Matlab language, which takes full advantage of both languages.