Proposed Discretization Method Research Articles

Mesh‐Free Discrete Laplace–Beltrami Operator

AbstractIn this work we propose a new discretization method for the Laplace–Beltrami operator defined on point‐based surfaces. In contrast to the existing point‐based discretization techniques, our approach does not rely on any triangle mesh structure, turning out truly mesh‐free. Based on a combination of Smoothed Particle Hydrodynamics and an optimization procedure to estimate area elements, our discretization method results in accurate solutions while still being robust when facing abrupt changes in the density of points. Moreover, the proposed scheme results in numerically stable discrete operators. The effectiveness of the proposed technique is brought to bear in many practical applications. In particular, we use the eigenstructure of the discrete operator for filtering and shape segmentation. Point‐based surface deformation is another application that can be easily carried out from the proposed discretization method.

Transient three-dimensional contact problems: mortar method. Mixed methods and conserving integration

The present work deals with the development of an energy-momentum conserving method to unilateral contact constraints and is a direct continuation of a previous work (Hesch and Betsch in Comput Mech 2011, doi: 10.1007/s00466-011-0597-2 ) dealing with the NTS method. In this work, we introduce the mortar method and a newly developed segmentation process for the consistent integration of the contact interface. For the application of the energy-momentum approach to mortar constraints, we extend an approach based on a mixed formulation to the segment definition of the mortar constraints. The enhanced numerical stability of the newly proposed discretization method will be shown in several examples.

A discontinuous enrichment method for variable‐coefficient advection–diffusion at high Péclet number

AbstractA discontinuous Galerkin method with Lagrange multipliers is presented for the solution of variable‐coefficient advection–diffusion problems at high Péclet number. In this method, the standard finite element polynomial approximation is enriched within each element with free‐space solutions of a local, constant‐coefficient, homogeneous counterpart of the governing partial differential equation. Hence in the two‐dimensional case, the enrichment functions are exponentials, each exhibiting a sharp gradient in a carefully chosen flow direction. The continuity of the enriched approximation across the element interfaces is enforced weakly by the aforementioned Lagrange multipliers. Numerical results obtained for two benchmark problems demonstrate that elements based on the proposed discretization method are far more competitive for variable‐coefficient advection–diffusion analysis in the high Péclet number regime than their standard Galerkin and stabilized finite element comparables. Copyright © 2010 John Wiley & Sons, Ltd.

데이터의 공간적 분포를 고려한 퍼지 이산화와 특징선택에의 응용

임상 데이터마이닝에서 최적의 특징 집합을 선택하는 것은 주어진 데이터로부터 생성된 모델의 복잡성을 줄일 뿐만 아니라 유용성을 향상시키는 데에 매우 중요하고, 선택된 특징들의 임계값은 질병의 감별진단을 위해 임상 전문가의 결정기준으로 사용된다. 본 논문에서는 데이터의 공간적인 분포, 즉 중첩영역에서 중복 속성값을 포함하는 데이터의 분리성 정도를 평가함으로써 연속형 속성을 가진 데이터에 대한 퍼지 이산화기법을 제안한다. 제안된 방법에서 중복 속성값의 가중치 평균값은 각 특징의 임계값(즉 경계값)을 결정하기 위해서 사용되었고, 러프집합은 전체 특징들 중에서 중요특징들의 집합을 선택하기 위해서 이용하였다. 제안된 방법의 타당성을 검증하기 위해 호흡곤란을 주호소로 내원한 668명의 환자 데이터를 근거로 3가지 이산화방법과 제안된 이산화방법에 대한 실험을 수행하였다. 실험결과, 퍼지분할을 기반으로 한 이산화방법이 하드분할을 기반으로 한 이산화방법에 비해서 평균 분류정확도와 G-mean 성능에서 보다 좋은 결과를 제공함을 확인하였다. In clinical data minig, choosing the optimal subset of features is such important, not only to reduce the computational complexity but also to improve the usefulness of the model constructed from the given data. Moreover the threshold values (i.e., cut-off points) of selected features are used in a clinical decision criteria of experts for differential diagnosis of diseases. In this paper, we propose a fuzzy discretization approach, which is evaluated by measuring the degree of separation of redundant attribute values in overlapping region, based on spatial distribution of data with continuous attributes. The weighted average of the redundant attribute values is then used to determine the threshold value for each feature and rough set theory is utilized to select a subset of relevant features from the overall features. To verify the validity of the proposed method, we compared experimental results, which applied to classification problem using 668 patients with a chief complaint of dyspnea, based on three discretization methods (i.e., equal-width, equal-frequency, and entropy-based) and proposed discretization method. From the experimental results, we confirm that the discretization methods with fuzzy partition give better results in two evaluation measures, average classification accuracy and G-mean, than those with hard partition.

Controller design for nonlinear systems with time delay using model algorithm control (MAC)

This paper presents a digital model algorithm control (MAC) for nonlinear time-delayed systems. The MAC controller is designed based on the sampled-data representation of the dynamic system. A new discretization method using the Taylor series and zero order hold assumption is proposed to discretize the continuous nonlinear time-delayed system. The proposed discretization method can provide accurate time discretization results and convert the discrete-time system into a general nonlinear system, so that the MAC method can be employed. Several simulations are conducted to verify the performance of the proposed method.

Discretization methods with analytical solutions for a convection–reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection–reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [S.K. Godunov, Difference methods for the numerical calculations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), pp. 271–306] and [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method. The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order total variation diminishing methods, see [A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1993), pp. 357–393.]. In this article, we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods. For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method. At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.

On the convergence of a new time window discretization method for the traveling salesman problem with time window constraints

In this paper, we discuss convergence of a time window discretization method for the traveling salesman problem with time window constraints. This recently proposed discretization method leads to a lower bound to the minimization problem. Analysis shows that this method always guarantees convergence to the optimal solution. We also illustrate that the traditional method that gives feasible solutions does not always provide such a guarantee.

Numerical Evaluation of Biocide Treatment against Sulfate Reducing Bacteria in Oilfield Water Pipelines

A simple two-dimensional mathematical model capable of describing the interactions taking place in the bulk fluid and biofilm between sulfate reducing bacteria (SRB), substrates and biocide agents was used for the numerical evaluation of biocide treatment against SRB. The characteristics inherent to thick biofilms formed in large diameter oilfield pipelines were taken into account by assuming that the biocide treatment will remove little of the biofilm and not kill the SRB present in the biofilm. The evaluation model considered SRB disinfection/proliferation, and biocide deactivation within the bulk fluid, as well as many features of biofilm interactions with the bulk fluid such as mass transfer, live SRB detachment and biocide agent deactivation. These interactions are represented by simple coefficients as far as possible. Simulation results, using reported kinetic parameters for SRB, acetate and biocide agents such as chlorine and glutaraldehyde, showed that biocide treatments in oilfield water pipelines are sensitive to the disinfection rate coefficient and the biocide agent concentration as well as to the decay rate coefficient of the biocide agent in the bulk fluid, but are not sensitive to the biocide deactivation rate on the biofilm surface, nor to the SRB concentration in the biofilm. Various extreme conditions of biocide treatments were simulated using the proposed discretization method, which was adapted to take into account the biofilm interactions with the bulk fluid, to investigate the sensitivity of the numerical method and the suitability of the model developed to determine minimum effective biocide concentrations.

A mixed hybrid formulation based on oscillated finite element polynomials for solving Helmholtz problems

A mixed-hybrid-type formulation is proposed for solving Helmholtz problems. This method is based on (a) a local approximation of the solution by oscillated finite element polynomials and (b) the use of Lagrange multipliers to “weakly” enforce the continuity across element boundaries. The computational complexity of the proposed discretization method is determined mainly by the total number of Lagrange multiplier degrees of freedom introduced at the interior edges of the finite element mesh, and the sparsity pattern of the corresponding system matrix. Preliminary numerical results are reported to illustrate the potential of the proposed solution methodology for solving efficiently Helmholtz problems in the mid- and high-frequency regimes.

ENERGY BASED DISCRETIZATION OF AN ADSORPTION COLUMN

A new method for the spatial discretization of complex multi-scale systems described by partial differential equations is presented. This method allows to preserve the global power balance equation and the geometric structure of the system. The modelling of the adsorption column is based on a network approach. The key notions are the energy function and the description of the power transfers within the system and through its boundaries with the help of a power-conserving geometric structure. The proposed discretization method preserves this geometric structure and is thermodynamically consistent.

Digital Control for Takagi-Sugeno Fuzzy System with Multirate Sampling

In this paper, a new dual-rate digital control technique for the Takagi-Sugeno (T-S) fuzzy system is suggested. The proposed method takes account of the stabilizablity of the discrete-time T-S fuzzy system at the fast-rate sampling points. Our main idea is to utilize the lifted control input. The proposed approach is to obtain the dual-rate discrete-time T-S fuzzy system by discretizing the overall dynamics of the T-S fuzzy system with the lifted control, and then to derive the sufficient conditions for the stabilization in the sense of the Lyapunov asymptotic stability for this system. An example is provided for showing the feasibility of the proposed discretization method.

데이터 분포를 고려한 연속 값 속성의 이산화

This paper proposes a new approach that converts continuous-valued attributes to categorical-valued ones considering the distribution of target attributes(classes). In this approach, It can be possible to get optimal interval boundaries by considering the distribution of data itself without any requirements of parameters. For each attributes, the distribution of target attributes is projected to one-dimensional space. And this space is clustered according to the criteria like as the density value of each target attributes and the amount of overlapped areas among each density values of target attributes. Clusters which are made in this ways are based on the probabilities that can predict a target attribute of instances. Therefore it has an interval boundaries that minimize a loss of information of original data. An improved performance of proposed discretization method can be validated using C4.5 algorithm and UCI Machine Learning Data Repository data sets.

Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media

This paper presents a numerical method for simulating flow fields in a stochastic porous medium that satisfies locally the Darcy equation, and has each of its hydraulic parameters represented as one realization of a three-dimensional random field. These are generated by using the Turning Bands method. Our ultimate objective is to obtain statistically meaningful solutions in order to check and extend a series of approximate analytical results previously obtained by a spectral perturbation method (L. W. Gelhar and co-workers). We investigate the computational aspects of the problem in relation with stochastic concepts. The difficulty of the numerical problem arises from the random nature of the hydraulic conductivities, which implies that a very large discretized algebraic system must be solved. Indeed, a preliminary evaluation with the aid of scale analysis suggests that, in order to solve meaningful flow problems, the total number of nodes must be of the order of 106. This is due to the requirement that Δxi ≪ gli ≪ Li, where Δxi is the mesh size, λi is a typical correlation scale of the inputs, and Li is the size of the flow domain (i = 1, 2, 3). The optimum strategy for the solution of such a problem is discussed in relation with supercomputer capabilities. Briefly, the proposed discretization method is the seven-point finite differences scheme, and the proposed solution method is iterative, based on prior approximate factorization of the large coefficient matrix. Preliminary results obtained with grids on the order of one hundred thousand nodes are discussed for the case of steady saturated flow with highly variable, random conductivities.