Letbe a simplex in The integral type Meyer-Komg-Zeller operators are constructed on the simplex T, and the degree of approximation of these operators for L'-functions is obtained.
The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on...The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains.Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands,this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time.This method is initially applied within the framework of the numerical manifold method(NMM)to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem.First,a comprehensive overview of the fundamentals of the NMM and the simplex integration(SI)method is provided in this paper.Subsequently,the NMM framework for solving the Laplace equation is elaborated upon,with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration.Finally,a series of rigorous numerical experiments is conducted,where the proposed method demonstrates improved accuracy and efficiency.In conclusion,this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains,which is a promising approach for increasing accuracy and convergence across various integration-based methods.This groundbreaking achievement has not yet been reported in the publicly available literature.展开更多
Thermal cracking of rocks can significantly affect the durability of underground structures in engineering practices such as geothermal energy extraction,storage of nuclear waste and tunnelling in freezeethaw cycle in...Thermal cracking of rocks can significantly affect the durability of underground structures in engineering practices such as geothermal energy extraction,storage of nuclear waste and tunnelling in freezeethaw cycle induced areas.It is a scenario of strong coupled thermomechanical process involving discontinuity behaviours of rocks.In this context,a numerical model was proposed to investigate the thermal cracking of rocks,in a framework of the continuous-discontinuous element method(CDEM)for efficiently capturing the initiation and propagation of multiple cracks.A simplex integration strategy was adopted to account for the influences of temperature-dependent material properties.Several benchmark tests were considered and the obtained results were compared with analytical solutions and numerical results from the literature.The results show that the fracture degree of the cases when considering temperature-dependent material parameters had 10%differences approximately compared with the cases with constant parameters.展开更多
The three-dimensional numerical manifold method(3D-NMM),which is based on the derivation of Galerkin's variation,is a powerful calculation tool that uses two cover systems.The 3D-NMM can be used to handle continue...The three-dimensional numerical manifold method(3D-NMM),which is based on the derivation of Galerkin's variation,is a powerful calculation tool that uses two cover systems.The 3D-NMM can be used to handle continue-discontinue problems and extend to THM coupling.In this study,we extended the 3D-NMM to simulate both steady-state and transient heat conduction problems.The modelling was carried out using the raster methods(RSM).For the system equation,a variational method was employed to drive the discrete equations,and the crucial boundary conditions were solved using the penalty method.To solve the boundary integral problem,the face integral of scalar fields and two-dimensional simplex integration were used to accurately describe the integral on polygonal boundaries.Several numerical examples were used to verify the results of 3D steady-state and transient heat-conduction problems.The numerical results indicated that the 3D-NMM is effective for handling 3D both steadystate and transient heat conduction problems with high solution accuracy.展开更多
文摘Letbe a simplex in The integral type Meyer-Komg-Zeller operators are constructed on the simplex T, and the degree of approximation of these operators for L'-functions is obtained.
基金supported by the Natural Science Foundation of Shanghai(Grant No.21ZR1468500)the Fundamental Research Funds for the Central Universities(Grant No.22120240299)。
文摘The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains.Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands,this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time.This method is initially applied within the framework of the numerical manifold method(NMM)to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem.First,a comprehensive overview of the fundamentals of the NMM and the simplex integration(SI)method is provided in this paper.Subsequently,the NMM framework for solving the Laplace equation is elaborated upon,with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration.Finally,a series of rigorous numerical experiments is conducted,where the proposed method demonstrates improved accuracy and efficiency.In conclusion,this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains,which is a promising approach for increasing accuracy and convergence across various integration-based methods.This groundbreaking achievement has not yet been reported in the publicly available literature.
基金the financial support from the Natural Science Foundation of Hebei Province(Grant No.E2020050012)the National Natural Science Foundation of China(NSFC)(Grant No.52178324)the National Key Research and Development Project of China,the Ministry of Science and Technology of China(Grant No.2018YFC1505504).
文摘Thermal cracking of rocks can significantly affect the durability of underground structures in engineering practices such as geothermal energy extraction,storage of nuclear waste and tunnelling in freezeethaw cycle induced areas.It is a scenario of strong coupled thermomechanical process involving discontinuity behaviours of rocks.In this context,a numerical model was proposed to investigate the thermal cracking of rocks,in a framework of the continuous-discontinuous element method(CDEM)for efficiently capturing the initiation and propagation of multiple cracks.A simplex integration strategy was adopted to account for the influences of temperature-dependent material properties.Several benchmark tests were considered and the obtained results were compared with analytical solutions and numerical results from the literature.The results show that the fracture degree of the cases when considering temperature-dependent material parameters had 10%differences approximately compared with the cases with constant parameters.
基金supported by the National Natural Science Foundation of China(Grant Nos.42277165,41920104007,and 41731284)the Fundamental Research Funds for the Central Universities,China University of Geosciences(Wuhan)(Grant Nos.CUGCJ1821 and CUGDCJJ202234)the National Overseas Study Fund(Grant No.202106410040)。
文摘The three-dimensional numerical manifold method(3D-NMM),which is based on the derivation of Galerkin's variation,is a powerful calculation tool that uses two cover systems.The 3D-NMM can be used to handle continue-discontinue problems and extend to THM coupling.In this study,we extended the 3D-NMM to simulate both steady-state and transient heat conduction problems.The modelling was carried out using the raster methods(RSM).For the system equation,a variational method was employed to drive the discrete equations,and the crucial boundary conditions were solved using the penalty method.To solve the boundary integral problem,the face integral of scalar fields and two-dimensional simplex integration were used to accurately describe the integral on polygonal boundaries.Several numerical examples were used to verify the results of 3D steady-state and transient heat-conduction problems.The numerical results indicated that the 3D-NMM is effective for handling 3D both steadystate and transient heat conduction problems with high solution accuracy.
