Gradient Recovery Based Two-Grid Finite Element Method for Parabolic Integro-Differential Optimal Control Problems

In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.

Exploring Taiwan’s China Landscape Painting Aesthetic Preferences Through Evaluation Grid Method and the Continuous Fuzzy Kano Model

The primary objective of this study is to apply the Evaluation Grid Method(EGM)and the continuous fuzzy Kano quality model to explore the cognitive preferences of Taiwan China residents regarding the beauty of Taiwan’s China landscape paintings.The aim is to contribute to the development of social and cultural art and promote the widespread appeal of art products.Through a literature review,consultations with aesthetic experts,and the application of Miryoku Engineering’s EGM,this paper consolidates the factors that contribute to the attractiveness of painting art products among Taiwan China residents,taking into account various aesthetic qualities.Simultaneously,the paper introduces the use of the triangular fuzzy golden ratio scale semantics,specifically the equal-ratio aesthetic scale semantics,as a replacement for the traditional subjective consciousness model.Departing from the traditional discrete Kano model that employs the mode as the standard for evaluating quality,this study applies triangular fuzzy numbers to the continuous Kano quality model to analyze the diverse preferences and evaluation standards of the public.The hope is that this research methodology will not only deepen Taiwan China residents’understanding and aesthetic literacy of painting art but also serve as a reference for the popularization of art products.

Application of Choquet Integral-Importance-Performance Analysis and TOPSIS Methods in Approaching the Preference Factors of Calligraphy and Seal Engraving Imagery

Classical Chinese characters,presented through calligraphy,seal engraving,or painting,can exhibit different aesthetics and essences of Chinese characters,making them the most important asset of the Chinese people.Calligraphy and seal engraving,as two closely related systems in traditional Chinese art,have developed through the ages.Due to changes in lifestyle and advancements in modern technology,their original functions of daily writing and verification have gradually diminished.Instead,they have increasingly played a significant role in commercial art.This study utilizes the Evaluation Grid Method(EGM)and the Analytic Hierarchy Process(AHP)to research the key preference factors in the application of calligraphy and seal engraving imagery.Different from the traditional 5-point equal interval semantic questionnaire,this study employs a non-equal interval semantic questionnaire with a golden ratio scale,distinguishing the importance ratio of adjacent semantic meanings and highlighting the weighted emphasis on visual aesthetics.Additionally,the study uses Importance-Performance Analysis(IPA)and Technique for Order Preference by Similarity to Ideal Solution(TOPSIS)to obtain the key preference sequence of calligraphy and seal engraving culture.Plus,the Choquet integral comprehensive evaluation is used as a reference for IPA comparison.It is hoped that this study can provide cultural imagery references and research methods,injecting further creativity into industrial design.

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

“GRIDS”教学法在工程热力学教学中的创新与实践

针对工程热力学课程传统教学课堂沉闷、教学效率低、专业拓展少、课时吃紧、思政入耳不入心和学生创新能力培养不足等问题,依托省级线上线下混合式一流课程,开展基于“GRIDS”教学法的教学创新与实践。混合式教学模式让线下教学融入更多高阶知识点、针对性案例,让课程内容更具弹性;“GRIDS”每一个字母代表一种教学方法;分组案例汇报拓展学科前沿,培养学生团队合作能力;雨课堂随堂测验;4种思政范式让思政“铭于心而践于行”;讨论式互动培养学生批判性学习思维;学习卡片帮助学生把握重难点。各教学方法的形成性评价是过程化考核依据。教学方法形成合力,将教学痛点逐个击破,形成闭环,并持续改进。

Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.

Thermal Comfort Measurement and Landscape Reconstruction Strategy of Street Space Based on Decomposition Grid Method:A Case Study of Yangmeizhu Street

As an important public space in the city,street space is the second residence of people.Taking Yangmeizhu Street as the research object,this study established a street microclimate model in Beijing through microclimate measurement and simulation,and then calculated thermal comfort index of the street and evaluated thermal comfort of the street space and distribution model.The evaluation method used grid method to decompose and study and made several 10m×10m grids for inner space of the street,so as to compare deeply difference of microclimate environment in the street and analyze quantitatively relevance between street space elements(street pavement,street greening,building shadow coverage)and thermal comfort(physiological equivalent temperature PET)in the data.Finally,adaptive strategy of microclimate on street level was put forward to improve the environmental quality of public space in the old city by means of landscape design,aiming at creating a resilient and humanized street space and improving adaptability of microclimate and thermal comfort for living.

Resources calculation of cobalt-rich crusts with the grid subdivision and integral method

On the basis of three geological models and several orebody boundaries, a method of grid subdivision and integral has been proposed to calculate and evaluate the resources of cobalt-rich crusts on the seamounts in the central Pacific Ocean. The formulas of this method are deduced and the interface of program module is designed. The method is carried out in the software ＂Auto mapping system of submarine topography and geomorphology MBChart＂. This method and program will possibly become a potential tool to calculate the resources of seamounts and determine the target diggings for China＇ s next Five-year Plan.

Analysis and management strategies for the low voltage problems of rural power grid based on the distribution network flow calculation method

For a long time, because of the lack of investment capital and enough attentions, the overall constructions of rural power grid were far behind than the urban power grid in Chongqing Jiangbei Power Company. The low voltage problems were highlighted in the rural power grid due to the characteristics of rural power grid. Using the distribution network flow calculation method, we evaluated the low voltage problems of the rural power grid which belongs to Chongqing Jiangbei Power Company. In addition, we collected the data of distribution transformers in electricity consumption peak period. Some practical management strategies were proposed by the analysis and evaluation of potential and appeared low voltage problems.

A Two-Grid Technique for the Penalty Method of the Steady Navier-Stokes Equations

A two grid technique for solving the steady incompressible Navier Stokes equations in a penalty method was presented and the convergence of numerical solutions was analyzed. If a coarse size H and a fine size h satisfy H=O(h 13-s )(s=0(n=2);s=12(n=3), where n is a space dimension), this method has the same convergence accuracy as the usual finite element method. But the two grid method can save a lot of computation time for its brief calculation. Moreover, a numerical test was couducted in order to verify the correctness of above theoretical analysis.

A scheme to treat the singularity in global seismic wavefield simulation using pseudospectral method with staggered grids

The pseudospectral method has been applied to the simulation of seismic wave propagation in 2-D global Earth model. When a whole Earth model is considered, the center of the Earth is included in the model and then singularity arises at the center of the Earth where r=0 since the 1/r term appears in the wave equations. In this paper, we extended the global seismic wavefield simulation algorithm for regular grid mesh to staggered grid configuration and developed a scheme to solve the numerical problems associated with the above singularity for a 2-D global Earth model defined on staggered grid using pseudospectral method. This scheme uses a coordinate transformation at the center of the model, in which the field variables at the center are calculated in Cartesian coordinates from the values on the grids around the center. It allows wave propagation through the center and hence the wavefield at the center can be stably calculated. Validity and accuracy of the scheme was tested by compared with the discrete wavenumber method. This scheme could also be suitable for other numerical methods or models parameterized in cylindrical or spherical coordinates when singularity arises at the center of the model.

TWO-GRID METHOD FOR CHARACTERISTICS FINITE-ELEMENT SOLUTION OF 2D NONLINEAR CONVECTION-DOMINATED DIFFUSION PROBLEM

For two-dimension nonlinear convection diffusion equation, a two-grid method of characteristics finite-element solution was constructed. In this method the nonlinear iterations is only to execute on the coarse grid and the fine-grid solution can be obtained in a single linear step. For the nonlinear convection-dominated diffusion equation, this method can not only stabilize the numerical oscillation but also accelerate the convergence and improve the computational efficiency. The error analysis demonstrates if the mesh sizes between coarse-grid and fine-grid satisfy the certain relationship, the two-grid solution and the characteristics finite-element solution have the same order of accuracy. The numerical is more efficient than that of characteristics example confirms that the two-grid method finite-element method.

ALGEBRAIC MULTI-GRID METHOD IN TWO-DIMENSION ELECTRICALLY LARGE PROBLEMS

In this paper the algebraic multi-grid principle is applied to the multilevel moment method, which makes the new multilevel method easier to implement and more adaptive to structure. Moreover, the error spectrum is analyzed, and the reason why conjugate gradient iteration is not a good relaxation scheme for multi-grid algorithm is explored. The numerical results show that our algebraic block Gauss Seidel multi-grid algorithm is very effective.

Two-grid partition of unity method for second order elliptic problems

A two-grid partition of unity method for second order elliptic problems is proposed and analyzed. The standard two-grid method is a local and parallel method usually leading to a discontinuous solution in the entire computational domain. Partition of unity method is employed to glue all the local solutions together to get the global continuous one, which is optimal in HI-norm. Furthermore, it is shown that the L^2 error can be improved by using the coarse grid correction. Numerical experiments are reported to support the theoretical results.

Application of Multi-Grid Method to the Computation of Electromagnetic Scattering Problems

The multi-grid method has been known as an efficient iterative method for the linear systems and nonlinear systems that arise from finite difference approximations for partial differential equations. In this paper, the multigrid method is extended to the application of solving integral equations which appear in electromagnetic scattering problems. The diakoptic theory is used for this purpose. Compared with other methods, the numerical results show that the multigrid method is powerful to solve electromagnetic scattering problems and can be used to compute electromagnetic scattering problems with electrically large bodies and complex structures.

Solving Schrodinger Equation with Non-Uniform Grids by Scale Transformation Method

A new scale transformation method is used in solving the Schrodinger equation. With it, the uniform grids in the discretization in conventional metho d are changed into non-uniform grids. Consequently, in some cases, the computing quantity will be greatly reduced at keeping the required accuracy. The calcul ation of the quantized inversion layer in MOS structure is used to demonstrate t he efficiency of the new method.

An explicit finite volume element method for solving characteristic level set equation on triangular grids

Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.

A Finite Volume Method with Unstructured Triangular Grids for Numerical Modeling of Tidal Current

The finite volume method （FVM） has many advantages in 2-D shallow water numerical simulation. In this study, the finite volume method is used with unstructured triangular grids to simulate the tidal currents. The Roe scheme is applied in the calculation of the intercell numerical flux, and the MUSCL method is introduced to improve its accuracy. The time integral is a two-step scheme of forecast and revision. For the verification of the present method, the Stoker＇s problem is calculated and the result is compared with the mathematically analytic solutions. The comparison indicates that the method is feasible. A sea area of a port is used as an example to test the method established here. The result shows that the present computational method is satisfactory, and it could be applied to the engineering fields.

HYBRID CARTESIAN GRID/GRIDLESS METHOD FOR CALCULATING VISCOUS FLOWS OVER MULTI-ELEMENT AIRFOILS

A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leaves only small region adjacent to the aerodynamic bodies to be filled with the cloud of points used in the gridless methods,which results in a better combination of the computational efficiency of the Cartesian grid and the flexibility of the gridless method in handling complex geometries.The clouds of points in the local gridless region are implemented in an anisotropic way according to the features of the thin boundary layer of the viscous flows over the airfoils,and the clouds of points at the vicinity of the interface between the grid and the gridless regions are also controlled by using an adaptive refinement technique during the generation of the unstructured Cartesian grid.An implementation of the resulting hybrid method is presented for solving two-dimensional compressible Navier-Stokes(NS)equations.The simulations of the viscous flows over a RAE2822airfoil or a two-element airfoil are successfully carried out,and the obtained results agree well with the available experimental data.

基于2.5D有限元的高密度电法不同装置勘探效果研究

高密度电法是一种适用性广泛的电阻率勘探方法,拥有多种电极排列装置选择。不同装置由于电极分布方式的差异,在针对不同地质目标和测区环境时往往表现出明显不同的探测能力。为探究各排列装置的特点及其适用情况,推导了2.5D电位的变分问题,采用Delaunay三角化算法实现了非结构化网格剖分,进而实现了有限元正演模拟。结合实践,对常见地质情况进行建模,分别使用温纳α、温纳β、施伦贝谢尔、偶极-偶极4种装置开展正反演对比研究。研究发现:(1)在探测未知区域单个异常体时,温纳β、施伦贝谢尔2种排列的效果更好;(2)在探测相邻较近的多个异常体时,应优先采用具有更高水平分辨率的施伦贝谢尔和偶极-偶极排列;(3)对于低阻破碎带的勘探,温纳β和偶极-偶极排列更为适用;(4)当地层具有较好分层界限时,4种排列均可体现良好的分层效果。因此,高密度电法勘探实践中应综合考虑不同排列装置的特点,根据具体情况选用合适的多种排列方式进行测量,并对采集数据进行综合对比分析,以获得更为准确的解释结果。