In the past decade,notable progress has been achieved in the development of the generalized finite difference method(GFDM).The underlying principle of GFDM involves dividing the domain into multiple sub-domains.Within...In the past decade,notable progress has been achieved in the development of the generalized finite difference method(GFDM).The underlying principle of GFDM involves dividing the domain into multiple sub-domains.Within each sub-domain,explicit formulas for the necessary partial derivatives of the partial differential equations(PDEs)can be obtained through the application of Taylor series expansion and moving-least square approximation methods.Consequently,the method generates a sparse coefficient matrix,exhibiting a banded structure,making it highly advantageous for large-scale engineering computations.In this study,we present the application of the GFDM to numerically solve inverse Cauchy problems in two-and three-dimensional piezoelectric structures.Through our preliminary numerical experiments,we demonstrate that the proposed GFDMapproach shows great promise for accurately simulating coupled electroelastic equations in inverse problems,even with 3%errors added to the input data.展开更多
The possibility of using a nodal method allowing irregular distribution of nodes in a natural way is one of the main advantages of the generalized finite difference method (GFDM) with regard to the classical finite di...The possibility of using a nodal method allowing irregular distribution of nodes in a natural way is one of the main advantages of the generalized finite difference method (GFDM) with regard to the classical finite difference method. Moreover, this feature has made it one of the most-promising meshless methods because it also allows us to reduce the time-consuming task of mesh generation and the numerical solution of integrals. This characteristic allows us to shape geological features easily whilst maintaining accuracy in the results, which can be a source of great interest when dealing with this kind of problems. Two widespread geophysical investigation methods in civil engineering are the cross-hole method and the seismic refraction method. This paper shows the use of the GFDM to model the aforementioned geophysical investigation tests showing precision in the obtained results when comparing them with experimental data.展开更多
Numerical quadrature is an important ingredient of Galerkin meshless methods. A new numerical quadrature technique, partition of unity quadrature （PUQ）,for Galerkin meshless methods was presented. The technique is b...Numerical quadrature is an important ingredient of Galerkin meshless methods. A new numerical quadrature technique, partition of unity quadrature （PUQ）,for Galerkin meshless methods was presented. The technique is based on finite covering and partition of unity. There is no need to decompose the physical domain into small cell. It possesses remarkable integration accuracy. Using Element-free Galerkin methods as example, Galerkin meshless methods based on PUQ were studied in detail. Meshing is always not required in the procedure of constitution of approximate function or numerical quadrature, so Galerkin meshless methods based on PUQ are “truly” meshless methods.展开更多
In this paper, a meshless method is introduced for NDT computation. Compared with the conventional FEM, it can avoid the onerous mesh generation and updating, only a distribution of points and the description of the b...In this paper, a meshless method is introduced for NDT computation. Compared with the conventional FEM, it can avoid the onerous mesh generation and updating, only a distribution of points and the description of the boundaries are needed. The mathematical background for moving least square approximation employed in the method is given, and the numerical implementation is discussed. Application of the method for MFL computation and comparison with the results from FEM are also presented.展开更多
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 usi...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.展开更多
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 fir...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.展开更多
基金the Natural Science Foundation of Shandong Province of China(Grant No.ZR2022YQ06)the Development Plan of Youth Innovation Team in Colleges and Universities of Shandong Province(Grant No.2022KJ140)the Key Laboratory ofRoad Construction Technology and Equipment(Chang’an University,No.300102253502).
文摘In the past decade,notable progress has been achieved in the development of the generalized finite difference method(GFDM).The underlying principle of GFDM involves dividing the domain into multiple sub-domains.Within each sub-domain,explicit formulas for the necessary partial derivatives of the partial differential equations(PDEs)can be obtained through the application of Taylor series expansion and moving-least square approximation methods.Consequently,the method generates a sparse coefficient matrix,exhibiting a banded structure,making it highly advantageous for large-scale engineering computations.In this study,we present the application of the GFDM to numerically solve inverse Cauchy problems in two-and three-dimensional piezoelectric structures.Through our preliminary numerical experiments,we demonstrate that the proposed GFDMapproach shows great promise for accurately simulating coupled electroelastic equations in inverse problems,even with 3%errors added to the input data.
基金The authors acknowledge the support of the Escuela Tecnica Superior de Ingenieros Industriales(UNED)of Spain,project 2019-IFC02of the Universidad Politecnica de Madrid(UPM)(Research groups 2019).
文摘The possibility of using a nodal method allowing irregular distribution of nodes in a natural way is one of the main advantages of the generalized finite difference method (GFDM) with regard to the classical finite difference method. Moreover, this feature has made it one of the most-promising meshless methods because it also allows us to reduce the time-consuming task of mesh generation and the numerical solution of integrals. This characteristic allows us to shape geological features easily whilst maintaining accuracy in the results, which can be a source of great interest when dealing with this kind of problems. Two widespread geophysical investigation methods in civil engineering are the cross-hole method and the seismic refraction method. This paper shows the use of the GFDM to model the aforementioned geophysical investigation tests showing precision in the obtained results when comparing them with experimental data.
文摘Numerical quadrature is an important ingredient of Galerkin meshless methods. A new numerical quadrature technique, partition of unity quadrature （PUQ）,for Galerkin meshless methods was presented. The technique is based on finite covering and partition of unity. There is no need to decompose the physical domain into small cell. It possesses remarkable integration accuracy. Using Element-free Galerkin methods as example, Galerkin meshless methods based on PUQ were studied in detail. Meshing is always not required in the procedure of constitution of approximate function or numerical quadrature, so Galerkin meshless methods based on PUQ are “truly” meshless methods.
文摘In this paper, a meshless method is introduced for NDT computation. Compared with the conventional FEM, it can avoid the onerous mesh generation and updating, only a distribution of points and the description of the boundaries are needed. The mathematical background for moving least square approximation employed in the method is given, and the numerical implementation is discussed. Application of the method for MFL computation and comparison with the results from FEM are also presented.
基金Supported by the Young Teacher Foundation of the Department of Mechanical Engineering at Tsinghua University and the Iron-Steel Research Conjunct Foundation of the National Natural Science Foundation of China and the Baosteel Co. of China (No. 5017403
文摘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.
基金This project was supported by the National Natural Science Foundation of China （11371066, 11372050）, and the Foundation of National Key Laboratory of Science and Technology Computation Physics.
文摘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.
基金This work is supported by the National Natural Science Foundation of China(52275009,52037002,52207038)the National Double First-classConstruction Special Funds(4316002181)the Fundamental Research Funds for the Central Universities(3216002101A2,3216002209A1).
