Mesh Scale Research Articles

Two deterministic approaches to topography evolution

We discuss two deterministic approaches to model free-molecular transport and reaction inside features on patterned wafers during thermal deposition processes. One approach that has been widely used to model chemical vapor deposition processes, which are largely steady state at the equipment scale, is based upon the ballistic transport and reaction model (BTRM). The major computational burden in BTRM based codes is usually computing the geometry-defined matrix of transmission probabilities; each element of which is used to determine what fraction of material leaving one point on the surface goes to each other point. This computation scales quadratically in the size of the discretizations, but is done in parallel. The second approach to modeling transport and reaction in the free-molecular flow regime is based on the kinetic transport and reaction model (KTRM). The KTRM starts from the Boltzmann equation and is particularly appropriate for processes operated under transient conditions at the reactor scale; e.g., atomic layer deposition. The KTRM is computationally expensive; models in three spatial dimensions require the discretization of the three spatial dimensions, three velocity dimensions, and time for transient studies. The spatial mesh scales with the third power of the system size, for the same resolution. The KTRM is implemented in parallel.

Explicit off-line criteria for stable accurate time filtering of strongly unstable spatially extended systems

Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and physical instabilities on both large and small scales. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Here, explicit off-line test criteria for stable accurate discrete filtering are developed for use in the above context and mimic the classical stability analysis for finite difference schemes. First, constant coefficient partial differential equations, which are randomly forced and damped to mimic mesh scale energy spectra in the above problems are developed as off-line filtering test problems. Then mathematical analysis is used to show that under natural suitable hypothesis the time filtering algorithms for general finite difference discrete approximations to an sxs partial differential equation system with suitable observations decompose into much simpler independent s-dimensional filtering problems for each spatial wave number separately; in other test problems, such block diagonal models rigorously provide upper and lower bounds on the filtering algorithm. In this fashion, elementary off-line filtering criteria can be developed for complex spatially extended systems. The theory is illustrated for time filters by using both unstable and implicit difference scheme approximations to the stochastically forced heat equation where the combined effects of filter stability and model error are analyzed through the simpler off-line criteria.

A topologically faithful, tissue-guided, spatially varying meshing strategy for computing patient-specific head models for endoscopic pituitary surgery simulation

This paper presents a method for tessellating tissue boundaries and their interiors, given as input a map consisting of relevant tissue classes of the head, in order to produce anatomical models for finite-element-based simulation of endoscopic pituitary surgery. Our surface meshing method is based on the simplex model, which is initialized by duality from the topologically accurate results of the Marching Cubes algorithm, and which affords explicit control over mesh scale, while using tissue information to adhere to relevant boundaries. Our mesh scale strategy is spatially varying, based on the distance to a central point or linearized surgical path. The tetrahedralization stage also features a spatially varying mesh scale, consistent with that of the surface mesh.

A topologically faithful, tissue-guided, spatially varying meshing strategy for computing patient-specific head models for endoscopic pituitary surgery simulation

This paper presents a method for tessellating tissue boundaries and their interiors, given as input a map consisting of relevant tissue classes of the head, in order to produce anatomical models for finite-element-based simulation of endoscopic pituitary surgery. Our surface meshing method is based on the simplex model, which is initialized by duality from the topologically accurate results of the Marching Cubes algorithm, and which affords explicit control over mesh scale, while using tissue information to adhere to relevant boundaries. Our mesh scale strategy is spatially varying, based on the distance to a central point or linearized surgical path. The tetrahedralization stage also features a spatially varying mesh scale, consistent with that of the surface mesh.

A multi-level discontinuous Galerkin method for solving the stationary Navier–Stokes equations

In this paper, we consider the multi-level discontinuous finite element method for solving the stationary incompressible Navier–Stokes equations. On the coarsest mesh the discrete nonlinear Navier–Stokes equations are solved by using piecewise polynomial functions, which are totally discontinuous across inter-element boundaries and are pointwise divergence free on each element for the velocity and are continuous functions for the pressure. Subsequent approximations are generated on a succession of refined grids by solving the Newton linearized Navier–Stokes equations using piecewise polynomial functions which are similar to that on the coarsest mesh. Finally, the well-posedness and the optimal error estimate for the multi-level discontinuous Galerkin method are provided. The error analysis shows that when the mesh scales k j + 1 = O ( k j 2 ) , h j + 1 = O ( h j 2 ) with j = 0 , 1 , … , J − 1 are chosen, the multi-level finite element method can save a large amount of computational time compared with the one-level finite element method.

Failure analysis of laminated structures by FEM based on nonlinear constitutive relationship

Nonlinear characteristic of matrix material is considered in the Bridging Model constitutive theory for composite materials. The model is consistent in that the laminate nonlinear constitutive equations automatically deteriorate to those of the isotropic matrix material when the fiber content becomes zero or when the fiber becomes the same as that of the matrix. This model is programmed into UGENS subroutine of the commercial finite element code ABAQUS for analyzing the nonlinear response and ultimate strength of composite laminates readily. In the paper, the influence of different load increments, mesh scales, and stiffness discount schemes on the finite element (FE) analyzed results has been studied. Strength envelopes and stress–strain or load–deflection curves of several laminates subjected to in-plane as well as bending loads are obtained. It is shown that the FE results agree well with analytical and experimental data.

Application of Morishita Spread Index in the study of landslides from the Kulekhani watershed, central Nepal

The Kulekhani watershed was severely affected by sedimentation, landslides, and gully erosion in 1993. The sediments produced were ultimately supplied to the reservoir decreasing its gross storage capacity. A statistical tool, called Morishita Spread Index, was applied to detect the landslide distribution pattern in the watershed. Morishita Spread Index was greater than 1 and decreased with increasing mesh scale. Such a trend indicates a cluster-type of distribution pattern. The landslides are concentrated mainly in the gullies and streams, and there is a high probability that landslides will continue to occur there in the future.

Matrix-fracture exchange in a fractured porous medium: stochastic upscaling

We present in this Note a stochastic approach to the matrix-fracture exchange in a heterogeneous fractured porous medium. We introduce an intermediate scale, called the unit-scale, between the local-scale (fracture-scale) and the large-scale characteristic of the reservoir mesh (reservoir block). This paper focuses on the problem of upscaling fluid exchange phenomena from the unit scale to the reservoir mesh or block scale. Simplifying the Darcian flow terms enables us to obtain a probabilistic solution of the dual continuum problem, in continuous time, in the case of a purely random exchange coefficient. This is then used to develop several upscaling approaches to the fluid exchange problem, and to analyze the so-called ‘effective’ exchange coefficient. The results are a first contribution to the more general problem of upscaling multidimensional flow-exchange processes in space and time, in randomly heterogeneous dual continua. To cite this article: M. Kfoury et al., C. R. Mecanique 332 (2004).

Optimal imaging with adaptive mesh refinement in electrical impedance tomography

In non-linear electrical impedance tomography the goodness of fit of the trial images is assessed by the well-established statistical χ2 criterion applied to the measured and predicted datasets. Further selection from the range of images that fit the data is effected by imposing an explicit constraint on the form of the image, such as the minimization of the image gradients. In particular, the logarithm of the image gradients is chosen so that conductive and resistive deviations are treated in the same way. In this paper we introduce the idea of adaptive mesh refinement to the 2D problem so that the local scale of the mesh is always matched to the scale of the image structures. This improves the reconstruction resolution so that the image constraint adopted dominates and is not perturbed by the mesh discretization. The avoidance of unnecessary mesh elements optimizes the speed of reconstruction without degrading the resulting images. Starting with a mesh scale length of the order of the electrode separation it is shown that, for data obtained at presently achievable signal-to-noise ratios of 60 to 80 dB, one or two refinement stages are sufficient to generate high quality images.

EARTHQUAKE GROUND MOTION SIMULATION THROUGH THE 2-D SPECTRAL ELEMENT METHOD

The application of the 2-D Chebyshev spectral element method (SPEM) to engineering seismology problems is reviewed in this paper. The SPEM is a high-order finite element technique which solves the variational formulation of the seismic wave propagation equations. The computational domain is discretised into an unstructured grid composed by irregular quadrilateral elements. This property makes the SPEM particularly suitable to compute numerically accurate solutions of the full wave equations in complex media. The earthquake is simulated following an approach that can be considered "global", that is all the factors influencing the wave propagation — source, crustal heterogeneity, fine details of the near-surface structure, and topography — are taken into account and solved simultaneously. The basic earthquake source is represented by a 2-D point double couple model. Ruptures propagating along fault segments placed on the model plane are simulated as a finite summation of elementary point sources. After a general introduction, the paper first gives an overview of the method; then it concentrates on some methodological topics of interest for practical applications, such as quadrangular mesh generation, source definition and scaling, numerical accuracy and computational efficiency. Limitations and advantages of using a 2-D approach, although sophisticated such as the SPEM, are addressed, as well. The effectiveness of the method is illustrated through two case histories, i.e. the ground shaking prediction in Catania (Sicily, Italy) for a catastrophic earthquake, and the analysis of the ground motion in the presence of a massive structure.

Earthquake Ground Motion Simulation Through the 2-D Spectral Element Method

The application of the 2-D Chebyshev spectral element method (SPEM) to engineering seismology problems is reviewed in this paper. The SPEM is a high-order finite element technique which solves the variational formulation of the seismic wave propagation equations. The computational domain is discretised into an unstructured grid composed by irregular quadrilateral elements. This property makes the SPEM particularly suitable to compute numerically accurate solutions of the full wave equations in complex media. The earthquake is simulated following an approach that can be considered global, that is all the factors influencing the wave propagation — source, crustal heterogeneity, fine details of the near-surface structure, and topography — are taken into account and solved simultaneously. The basic earthquake source is represented by a 2-D point double couple model. Ruptures propagating along fault segments placed on the model plane are simulated as a finite summation of elementary point sources. After a general introduction, the paper first gives an overview of the method; then it concentrates on some methodological topics of interest for practical applications, such as quadrangular mesh generation, source definition and scaling, numerical accuracy and computational efficiency. Limitations and advantages of using a 2-D approach, although sophisticated such as the SPEM, are addressed, as well. The effectiveness of the method is illustrated through two case histories, i.e. the ground shaking prediction in Catania (Sicily, Italy) for a catastrophic earthquake, and the analysis of the ground motion in the presence of a massive structure.

On the use of boundary conditions for variational formulations arising in financial mathematics

The general intractability of derivative security valuation models to present techniques, both analytic and numerical, arguably remains one of the preeminant problem of mathematical finance. It is the focus of this paper to examine the applicability of a promising recent development, namely Radial Basis Functions (RBF), to the problem of option valuation. A Black–Scholes framework is considered for American and European options written on a one and two risky assets. The performance of RBF and Finite-Differencing algorithms are examined with respect to artificial boundary conditions, computational domain, domain decomposition, and mesh scaling.

Renormalization group and perfect operators for stochastic differential equations.

We develop renormalization group (RG) methods for solving partial and stochastic differential equations on coarse meshes. RG transformations are used to calculate the precise effect of small-scale dynamics on the dynamics at the mesh size. The fixed point of these transformations yields a perfect operator: an exact representation of physical observables on the mesh scale with minimal lattice artifacts. We apply the formalism to simple nonlinear models of critical dynamics, and show how the method leads to an improvement in the computational performance of Monte Carlo methods.

Meso/macro-mechanical behaviour of textile reinforcements for thin composites

The simulation of the drawing process for woven reinforcements aims at evaluating the possibility of forming a shape for a given fabric. The mode of deformation during the forming of woven reinforcements is very specific and different from that for metal sheets. Knowledge of fabric behaviour is necessary. In the present work, this is studied experimentally by means of biaxial tests that bring out the influence of the geometric non-linearity, at the mesh scale, on the macroscopic behaviour of the fabric. Models based on bar and yarn theories, are identified from these tests and used in a specific finite-element formulation for fabrics. These elements are efficient since they take into account only the tension stiffness of the yarns (the only one present in a fabric). Three-dimensional simulations of the elementary mesh supplement this study with local information.

EFFECT OF SCALE OF CATCHMENT ELEMENT ON RUNOFF RESPONSE OF THE DISTRIBUTED MODEL

Distributed runoff models have been becoming powerful for describing the naturalrunoff processes. Such kind of model should reflect topographical features of a catchment. Then, it is necessary to examine about the possibility of aggregationas the catchment model. The purpose of this study is to investigate the effect of catchment scale of the element on runoff response characteristics. Runoff simulations in the conditions of the overland flow and the interflow by using two kinds of different mesh sizes of digital elevation model in the Hayakawa River, central Japan, were carried out. Then the concentration times of these runoffs were discussed. The result is as follows; 1) the concentration time depends on the mesh scale, 2) the influence of aggregation caused a difference in case of computation, and 3) the problem is solved by using a model parameter that gets equivalent average concentration time.

Multiple surface crack initiation and growth in glass-ceramics loaded by a sphere

A study is presented in which a finite element method was used to visualize initiation and growth of a crack system observed in Hertzian contact between a sphere and a glass-ceramic half-space with limited plasticity. This method uses a micromechanically based approach to simulate crack evolution, and a unified stress-based fracture criterion to predict the crack initiation and growth. Stress intensity factors (SIFs) at the crack tip are first computed. The effect of plasticity on the SIFs is investigated. A fracture parameter is proposed as a function of material toughness and mesh scale. Visualizations of the crack system that were performed to verify the proposed unified stress-based fracture criterion are reported.

Application of Morishita Spread Index to temporal and spatial distribution characteristics of landslides in Shanxi and Gansu provinces of China

Morshita Spread Index Iδ was applied for the study of temporal and spatial distribution characteristics of landslides in the Shanxi and Gansu provinces of China. For this purpose, the landslides larger than 105 m3 in volume were considered. In the study area, the spatial distribution of Morishita Spread Index Iδ (l) isgreater than 1 and decreases with increasing mesh scale.
 Such a trend indicates cluster distribution of landslides. On the other hand, the temporal distribution of Morishita Spread Index Iδ (t) for the above landslides showed a maximum and a minimum, corresponding to the years with high frequency of landslide occurrence.

FUNDAMENTAL EXAMINATION ON THE SYNTHETIC SEDIMENT ROUTING MODEL IN A WHOLE RIVER SYSTEM

It is required to develop an appropriate sediment routing model for describing behavior of sedimentation in a whole river system. This research deals with a synthetic sediment routing model based on DEM (Digital Elevation Models). This kind of model has several subjects to be examined from the viewpoints on model construction such as computational time interval, tuning hydraulic parameters, effects of computational mesh scale on the results, pseudo channel network made from DEM and so on. Authors have hitherto been examined some of the subjects mentioned above in the mountainous Hayakawa River basin (509km2). In order to apply the model to larger river basin, the Fujikawa River basin (3, 571km2), a new method of automatical tracing of the channel network is adopted. In this paper, difference of channel networks between the conventional method and the new one is examined, and reasonable way of smoothing river bed as an initial situation is searched. The results of some cases of simulation on sedimentation in the basin is also discussed.

DISCHARGE IN A MOUNTAINOUS REGION USING THE RECTANGULAR GRIDS

In run-off analysis, it is impotant to consider the physical processes, for example, the infiltration of rainfall into soil and the storage of water in a storage layer etc, because the effective rainfall is governed by these processes. In this study, in order to consider the physical process of run-off in a river basin, an infiltration flow equation was used for the run-off analysis. In computation, the finite difference method is used, but if the mesh size was large, results of calculation would not agree with the theoretical solutions. So the conservation of water was taken into account in the infiltration flow equation. By using this model, the computation result agreed with the observed one very well even in the large mesh scale grids.

A Study on the Planning of Biotope Network with Dragonfly as Indicator in City Zone

In this study, a biotope network plan adopting drawnfly as indicator living thing in Hirakata city, Osaka Prefecture was done using GIS. Firstly, the condition of dragonfly habitat-pond, river, paddy field, wood, field and weed-for species diversity at 1km mesh scale was analyzed. Then, it was found that the area of pond, wood, field and weed are very critical environmental variables. Secondly, the spatial distribution of dragonfly habitat was studied while the scope of dragonfly behavior for ingestion and breeding was estimated at city scale.On the basis of the above, the policy of biotope network was suggested, for example, biotope improvement and creation for the connection of an ecological core in nature reserve area and patches in man-made landuse area.