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finite point method for the Simulation of Solidification and Heat Transfer in Continuous Casting Mold

Tsinghua Science and Technology 2004年第5期9卷 570-573页

作者：张雷康进武沈厚发黄天佑 Department of Mechanical Engineering Tsinghua University Beijing 100084 China

A 2-D finite point meshless model was used to simulate the heat transfer and solidification of steel in continuous casting molds to illustrate its use in metallurgy. The latent heat of the pure metal was treated using the temperature recovery method and the latent heat of the alloy was treated using an appar- ent heat capacity method. The model was validated by calculating the classical Stefan moving boundary problem. Analysis of the solid shell growth and temperature distribution of a billet in a mold shows that the solution by the finite point meshless model is quite reasonable, which indicates that the model has potential in metallurgical engineering applications.

关键词： finite point method meshless solidification continuous casting mold

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DISCRETE MAXIMUM PRINCIPLE AND CONVERGENCE OF POISSON PROBLEM FOR THE finite point method

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Journal of Computational Mathematics 2017年第3期35卷 245-264页

作者： Guixia Lv Shunkai Sun Longjun Shen Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics P. O. Box 8009-26 Beijing 100088 China Institute of Applied Physics and Computational Mathematics P. O. Box 8009 Beijing 100088 China

This paper makes some mathematical analyses for the finite point method based on directional difference. By virtue of the explicit expressions of numerical formulae using only five neighboring points for computing first-order and second-order directional differ- entials, a new methodology is presented to discretize the Laplacian operator defined on 2D scattered point distributions. Some sufficient conditions with very weak limitations are obtained, under which the resulted schemes are positive schemes. As a consequence, the discrete maximum principle is proved, and the first order convergent result of O（h） is achieved for the nodal solutions defined on scattered point distributions, which can be raised up to O（h2） on uniform point distributions.

关键词： finite point method Directional difference Meshless Discrete maximum prin-ciple Convergence analysis.

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THEORETICAL ANALYSES ON DISCRETE FORMULAE OF DIRECTIONAL DIFFERENTIALS IN THE finite point method

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Journal of Computational Mathematics 2022年第1期40卷 1-25页

作者： Guixia Lv Longjun Shen Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsP.0.Box 8009-26Beijing 100088China

For the five-point discrete formulae of directional derivatives in the finite point method,overcoming the challenge resulted from scattered point sets and making full use of the explicit expressions and accuracy of the formulae,this paper obtains a number of theoretical results:(1)a concise expression with definite meaning of the complicated directional difference coefficient matrix is presented,which characterizes the correlation between coefficients and the connection between coefficients and scattered geometric characteristics;(2)various expressions of the discriminant function for the solvability of numerical differentials along with the estimation of its lower bound are given,which are the bases for selecting neighboring points and making analysis;(3)the estimations of combinatorial elements and of each element in the directional difference coefficient matrix are put out,which exclude the existence of ***,the theoretical analysis results are verified by numerical *** results of this paper have strong regularity,which lay the foundation for further research on the finite point method for solving partial differential equations.

关键词： finite point method finite difference Scattered point distribution Discrete directional differentials Theoretical analysis

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Relations between Directional Derivatives for Smooth Functions in 2D and 3D Spaces

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Acta Mathematicae Applicatae Sinica 2016年第4期32卷 899-912页

作者： Gui-xia LU Long-jun SHEN Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics P. O. Box 8009-26 Beijing 100088 China

In this paper, relations between directional derivatives are considered for smooth functions both in 2D and 3D spaces. These relations are established in the form of linear combinations of directional derivatives with their coefficients having simple form and structural regularity. By them, expressions based on directional derivatives for some typical differential operators are derived. This builds up a solid mathematical foundation for further study on numerical computation by the finite point method based on directional difference.

关键词： 2D space 3D space directional derivatives finite point method

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Extremum of second-order directional derivatives

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Applied Mathematics(A Journal of Chinese Universities) 2011年第4期26卷 379-389页

作者： LU Gui-xia WU Hao SHEN Long-jun National Key Laboratory of Science and Technology on Computational Physics Institute of Applied Physics and Computational Mathematics Beijing 100088 China.

In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.

关键词： finite point method second-order directional derivative extremum.

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