The modeling of crack growth in three-dimensional(3D)space poses significant challenges in rock mechanics due to the complex numerical computation involved in simulating crack propagation and interaction in rock mater...The modeling of crack growth in three-dimensional(3D)space poses significant challenges in rock mechanics due to the complex numerical computation involved in simulating crack propagation and interaction in rock materials.In this study,we present a novel approach that introduces a 3D numerical manifold method(3D-NMM)with a geometric kernel to enhance computational efficiency.Specifically,the maximum tensile stress criterion is adopted as a crack growth criterion to achieve strong discontinuous crack growth,and a local crack tracking algorithm and an angle correction technique are incorporated to address minor limitations of the algorithm in a 3D model.The implementation of the program is carried out in Python,using object-oriented programming in two independent modules:a calculation module and a crack module.Furthermore,we propose feasible improvements to enhance the performance of the algorithm.Finally,we demonstrate the feasibility and effectiveness of the enhanced algorithm in the 3D-NMM using four numerical examples.This study establishes the potential of the 3DNMM,combined with the local tracking algorithm,for accurately modeling 3D crack propagation in brittle rock materials.展开更多
A new numerical manifold （NMM） method is derived on the basis of quartic uniform B-spline interpolation. The analysis shows that the new interpolation function possesses higher-order continuity and polynomial consis...A new numerical manifold （NMM） method is derived on the basis of quartic uniform B-spline interpolation. The analysis shows that the new interpolation function possesses higher-order continuity and polynomial consistency compared with the conven- tional NMM. The stiffness matrix of the new element is well-conditioned. The proposed method is applied for the numerical example of thin plate bending. Based on the prin- ciple of minimum potential energy, the manifold matrices and equilibrium equation are deduced. Numerical results reveal that the NMM has high interpolation accuracy and rapid convergence for the global cover function and its higher-order partial derivatives.展开更多
The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts high...The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.42172312 and 52211540395)support from the Institut Universitaire de France(IUF).
文摘The modeling of crack growth in three-dimensional(3D)space poses significant challenges in rock mechanics due to the complex numerical computation involved in simulating crack propagation and interaction in rock materials.In this study,we present a novel approach that introduces a 3D numerical manifold method(3D-NMM)with a geometric kernel to enhance computational efficiency.Specifically,the maximum tensile stress criterion is adopted as a crack growth criterion to achieve strong discontinuous crack growth,and a local crack tracking algorithm and an angle correction technique are incorporated to address minor limitations of the algorithm in a 3D model.The implementation of the program is carried out in Python,using object-oriented programming in two independent modules:a calculation module and a crack module.Furthermore,we propose feasible improvements to enhance the performance of the algorithm.Finally,we demonstrate the feasibility and effectiveness of the enhanced algorithm in the 3D-NMM using four numerical examples.This study establishes the potential of the 3DNMM,combined with the local tracking algorithm,for accurately modeling 3D crack propagation in brittle rock materials.
基金supported by the Fund of National Engineering and Research Center for Highways in Mountain Area(No.gsgzj-2012-05)the Fundamental Research Funds for the Central Universities of China(No.CDJXS12240003)the Scientific Research Foundation of State Key Laboratory of Coal Mine Disaster Dynamics and Control(No.2011DA105287-MS201213)
文摘A new numerical manifold （NMM） method is derived on the basis of quartic uniform B-spline interpolation. The analysis shows that the new interpolation function possesses higher-order continuity and polynomial consistency compared with the conven- tional NMM. The stiffness matrix of the new element is well-conditioned. The proposed method is applied for the numerical example of thin plate bending. Based on the prin- ciple of minimum potential energy, the manifold matrices and equilibrium equation are deduced. Numerical results reveal that the NMM has high interpolation accuracy and rapid convergence for the global cover function and its higher-order partial derivatives.
基金supported by the National Key R&D Program of China (Grant No.2018YFC0407002)the National Natural Science Foundation of China(Grant Nos.11502033 and 51879014)
文摘The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.
