Second-generation high-temperature superconducting(HTS)conductors,specifically rare earth-barium-copper-oxide(REBCO)coated conductor(CC)tapes,are promising candidates for high-energy and high-field superconducting app...Second-generation high-temperature superconducting(HTS)conductors,specifically rare earth-barium-copper-oxide(REBCO)coated conductor(CC)tapes,are promising candidates for high-energy and high-field superconducting applications.With respect to epoxy-impregnated REBCO composite magnets that comprise multilayer components,the thermomechanical characteristics of each component differ considerably under extremely low temperatures and strong electromagnetic fields.Traditional numerical models include homogenized orthotropic models,which simplify overall field calculation but miss detailed multi-physics aspects,and full refinement(FR)ones that are thorough but computationally demanding.Herein,we propose an extended multi-scale approach for analyzing the multi-field characteristics of an epoxy-impregnated composite magnet assembled by HTS pancake coils.This approach combines a global homogenization(GH)scheme based on the homogenized electromagnetic T-A model,a method for solving Maxwell's equations for superconducting materials based on the current vector potential T and the magnetic field vector potential A,and a homogenized orthotropic thermoelastic model to assess the electromagnetic and thermoelastic properties at the macroscopic scale.We then identify“dangerous regions”at the macroscopic scale and obtain finer details using a local refinement(LR)scheme to capture the responses of each component material in the HTS composite tapes at the mesoscopic scale.The results of the present GH-LR multi-scale approach agree well with those of the FR scheme and the experimental data in the literature,indicating that the present approach is accurate and efficient.The proposed GH-LR multi-scale approach can serve as a valuable tool for evaluating the risk of failure in large-scale HTS composite magnets.展开更多
Numerical solver using a uniform grid is popular due to its simplicity and low computational cost, but would be unfeasible in the presence of tiny structures in large-scale media. It is necessary to use a nonuniform g...Numerical solver using a uniform grid is popular due to its simplicity and low computational cost, but would be unfeasible in the presence of tiny structures in large-scale media. It is necessary to use a nonuniform grid, where upsampling the wavefield from the coarse grid to the fine grid is essential for reducing artifacts. In this paper, we suggest a local refinement scheme using the Fourier interpolation, which is superior to traditional interpolation methods since it is theoretically exact if the input wavefield is band limited.Traditional interpolation methods would fail at high upsampling ratios(say 50); in contrast, our scheme still works well in the same situations, and the upsampling ratio can be any positive integer. A high upsampling ratio allows us to greatly reduce the computational burden and memory demand in the presence of tiny structures and large-scale models, especially for 3D cases.展开更多
This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-...This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.展开更多
Based on our proposed adaptivity strategy for the vibration of Reissner-Mindlin plate,we develop it to apply for the vibration of Kirchhoff plate.The adaptive algorithm is based on the Geometry-Independent Field appro...Based on our proposed adaptivity strategy for the vibration of Reissner-Mindlin plate,we develop it to apply for the vibration of Kirchhoff plate.The adaptive algorithm is based on the Geometry-Independent Field approximaTion(GIFT),generalized from Iso-Geometric Analysis(IGA),and it can characterize the geometry of the structure with NURBS(Non-Uniform Rational B-Splines),and independently apply PHT-splines(Polynomial splines over Hierarchical T-meshes)to achieve local refinement in the solution field.TheMAC(Modal AssuranceCriterion)is improved to locate unique,as well as multiple,modal correspondence between different meshes,in order to deal with error estimation.Local adaptivity is carried out by sweeping modes from low to high frequency.Numerical examples showthat a proper choice of the spline space in solution field(with GIFT)can deliver better accuracy than using NURBS solution field.In addition,for vibration of heterogeneous Kirchhoff plates,our proposed method indicates that the adaptive local h-refinement achieves a better solution accuracy than the uniform h-refinement.展开更多
In this paper,we propose a refined local learning scheme to reconstruct a high resolution(HR)face image from a low resolution(LR)observation.The contribution of this work is twofold.Firstly,multi-direction gradient fe...In this paper,we propose a refined local learning scheme to reconstruct a high resolution(HR)face image from a low resolution(LR)observation.The contribution of this work is twofold.Firstly,multi-direction gradient features are extracted to search the nearest neighbors for each image patch,then the non-negative matrix factorization(NMF)is used to reduce the complexity in weight calculation,and the initial HR embedding is estimated from the training pairs by preserving local geometry.Secondly,a global reconstruction constraint and post-processing by non-local filtering is incorporated into super-resolution(SR)reconstruction process to reduce the image artifacts and further improve the image visual quality.Experimental results show that the proposed algorithm improves the SR performance both in subjective and objective assessments compared with several existing methods.展开更多
A complete mesh free adaptive algorithm （MFAA）, with solution adaptation and geometric adaptation, is developed to improve the resolution of flow features and to replace traditional global refinement techniques in s...A complete mesh free adaptive algorithm （MFAA）, with solution adaptation and geometric adaptation, is developed to improve the resolution of flow features and to replace traditional global refinement techniques in structured grids. Unnecessary redundant points and elements are avoided by using the mesh free local clouds refinement technology in shock influencing regions and regions near large curvature places on the boundary. Inviscid compressible flows over NACA0012 and RAE2822 airfoils are computed. Finally numerical results validate the accuracy of the above method.展开更多
Three-dimensional forward modeling magnetotellurics （MT） problems. We present a is a challenge for geometrically complex new edge-based finite-element algorithm using an unstructured mesh for accurately and efficien...Three-dimensional forward modeling magnetotellurics （MT） problems. We present a is a challenge for geometrically complex new edge-based finite-element algorithm using an unstructured mesh for accurately and efficiently simulating 3D MT responses. The electric field curl-curl equation in the frequency domain was used to deduce the H （curl） variation weak form of the MT forward problem, the Galerkin rule was used to derive a linear finite-element equation on the linear-edge tetrahedroid space, and, finally, a BI-CGSTAB solver was used to estimate the unknown electric fields. A local mesh refinement technique in the neighbor of the measuring MT stations was used to greatly improve the accuracies of the numerical solutions. Four synthetic models validated the powerful performance of our algorithms. We believe that our method will effectively contribute to processing more complex MT studies.展开更多
Predicting rolling bearing fatigue life requires knowledge of the three-dimensional(3D)stress fields in the roller and raceway near the lubricated contact.Owing to the increasingly severe operating conditions,the effe...Predicting rolling bearing fatigue life requires knowledge of the three-dimensional(3D)stress fields in the roller and raceway near the lubricated contact.Owing to the increasingly severe operating conditions,the effect of localized features such as surface roughness,subsurface inclusions,and even the crystallographic structure of the material becomes important.Achieving such detail requires(locally)extremely dense gridding in simulations,which in 3D is a major challenge.Multigrid techniques have been demonstrated to be capable of solving such problems.In this study,multigrid techniques are shown to further increase the efficiency of the solution by exploiting local grid refinement while maintaining the simplicity of a uniform discretization.This is achieved by employing increasingly finer grids only locally,where the highest resolution is required.Results are presented for dry contact and elastohydrodynamically lubricated contact cases,circular as well as elliptic,with varying crystallographic structure,and with surface roughness.The results show that the developed algorithm is very well suited for detailed analysis,with also excellent prospects for computational diagnostics involving actual material crystallographic structure from electron backscatter diffraction measurements.展开更多
This paper provides a survey of local refinable splines,including hierarchical B-splines,T-splines,polynomial splines over T-meshes,etc.,with a view to applications in geometric modeling and iso-geometric analysis.We ...This paper provides a survey of local refinable splines,including hierarchical B-splines,T-splines,polynomial splines over T-meshes,etc.,with a view to applications in geometric modeling and iso-geometric analysis.We will identify the strengths and weaknesses of these methods and also offer suggestions for their using in geometric modeling and iso-geometric analysis.展开更多
We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral...We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 （Ω）-context along with local discrete Helmholtz-type decompositions of the edge element space.展开更多
A preconditioning method for the finite element stiffness matrix is given in this paper. The triangulation is refined in a subregion; the preconditioning process is composed of resolution of two regular subproblems; t...A preconditioning method for the finite element stiffness matrix is given in this paper. The triangulation is refined in a subregion; the preconditioning process is composed of resolution of two regular subproblems; the condition number of the preconditioned matrix is 0(1 + log H/h), where H and h are mesh sizes of the unrefined and local refined triangulations respectively.展开更多
In order to reach the best numerical properties with the numerical manifold method(NMM),uniform finite element meshes are always favorite while constructing mathematical covers,where all the elements are congruent.In ...In order to reach the best numerical properties with the numerical manifold method(NMM),uniform finite element meshes are always favorite while constructing mathematical covers,where all the elements are congruent.In the presence of steep gradients or strong singularities,in principle,the locally-defined special functions can be added into the NMM space by means of the partition of unity,but they are not available to those complex problems with heterogeneity or nonlinearity,necessitating local refinement on uniform meshes.This is believed to be one of the most important open issues in NMM.In this study multilayer covers are proposed to solve this issue.In addition to the first layer cover which is the global cover and covers the whole problem domain,the second and higher layer covers with smaller elements,called local covers,are used to cover those local regions with steep gradients or strong singularities.The global cover and the local covers have their own partition of unity,and they all participate in the approximation to the solution.Being advantageous over the existing procedures,the proposed approach is easy to deal with any arbitrary-layer hanging nodes with no need to construct super-elements with variable number of edge nodes or introduce the Lagrange multipliers to enforce the continuity between small and big elements.With no limitation to cover layers,meanwhile,the creation of an even error distribution over the whole problem domain is significantly facilitated.Some typical examples with steep gradients or strong singularities are analyzed to demonstrate the capacity of the proposed approach.展开更多
In this study, numerical manifold method(NMM) coupled with non-uniform rational B-splines(NURBS) and T-splines in the context of isogeometric analysis is proposed to allow for the treatments of complex geometries and ...In this study, numerical manifold method(NMM) coupled with non-uniform rational B-splines(NURBS) and T-splines in the context of isogeometric analysis is proposed to allow for the treatments of complex geometries and local refinement. Computational formula for a 9-node NMM based on quadratic B-splines is derived. In order to exactly represent some common free-form shapes such as circles, arcs, and ellipsoids, quadratic non-uniform rational B-splines(NURBS) are introduced into NMM. The coordinate transformation based on the basis function of NURBS is established to enable exact integration for the manifold elements containing those shapes. For the case of crack propagation problems where singular fields around crack tips exist, local refinement technique by the application of T-spline discretizations is incorporated into NMM, which facilitates a truly local refinement without extending the entire row of control points. A local refinement strategy for the 4-node mathematical cover mesh based on T-splines and Lagrange interpolation polynomial is proposed. Results from numerical examples show that the 9-node NMM based on NURBS has higher accuracies. The coordinate transformation based on the NURBS basis function improves the accuracy of NMM by exact integration. The local mesh refinement using T-splines reduces the number of degrees of freedom while maintaining calculation accuracy at the same time.展开更多
CEGAR （Counterexample-guided abstraction refinement）-based slicing is one of the most important techniques in reducing the state space in model checking. However, CEGAR-based slicing repeatedly explores the state sp...CEGAR （Counterexample-guided abstraction refinement）-based slicing is one of the most important techniques in reducing the state space in model checking. However, CEGAR-based slicing repeatedly explores the state space handled previously in case a spurious counterexample is found. Inspired by lazy abstraction, we introduce the concept of lazy slicing which eliminates this repeated computation. Lazy slicing is done on-the-fly, and only up to the precision necessary to rule out spurious counterexamples. It identifies a spurious counterexample by concretizing a path fragment other than the full path, which reduces the cost of spurious counterexample decision significantly. Besides, we present an improved over-approximate slicing method to build a more precise slice model. We also provide the proof of the correctness and the termination of lazy slicing, and implement a prototype model checker to verify safety property. Experimental results show that lazy slicing scales to larger systems than CEGAR-based slicing methods.展开更多
In this work,we propose an efficient multi-mesh adaptive finite element method for simulating the dendritic growth in two-and three-dimensions.The governing equations used are the phase field model,where the regularit...In this work,we propose an efficient multi-mesh adaptive finite element method for simulating the dendritic growth in two-and three-dimensions.The governing equations used are the phase field model,where the regularity behaviors of the relevant dependent variables,namely the thermal field function and the phase field function,can be very different.To enhance the computational efficiency,we approximate these variables on different h-adaptive meshes.The coupled terms in the system are calculated based on the implementation of the multi-mesh h-adaptive algorithm proposed by Li(J.Sci.Comput.,pp.321-341,24(2005)).It is illustrated numerically that the multi-mesh technique is useful in solving phase field models and can save storage and the CPU time significantly.展开更多
A new numerical approach based on a multiblock, multigrid, local refinement method has been developed. The multiblock structure makes grid generation for complex geometries easier,multigrid techniques significantly a...A new numerical approach based on a multiblock, multigrid, local refinement method has been developed. The multiblock structure makes grid generation for complex geometries easier,multigrid techniques significantly accelerate the rate of convergence and the local refinement method provides high spatial resolution of boundary lay6r and separated vortical flows with much reduced computer memory and CPU time. Sample three-dimensional flow computations for complex geometries are presented to illustrate the advantages in term of memory savings and reductions of CPU time in obtaining high spatial resolution solutions. High spatial resolution in the inner viscous layer is achieved by using the values of y+ for the first grid centers rom the wall in the order of 1 to 7. The required solution resolution in the dominantly viscous flow region (boundary layer, vortex core, or three-dimensional separated zone) dictates the acceptable spacing of grid cells in that region. It is very desirable to be able to reduce grid spacing locally in the critical region without changing the overall grid. The local refinement technique doubles grid numbers in all three directions for every level of refinement. Multiple levels of local grid refinement with moderate grid stretching can be used to provide fine grid spacing in rapidly changing flow regions such as near the wall, vortex core and separated shear layer. Thus, the method provides the required fine spatial resolution in the dominantly viscous region which is relatively small compared to the entire computational domain. This also avoids the used of excessively large aspect ratios of the grid cells near the rapidly changing now regions thus reduces truncation error and improves robustness. Furthermore, the results of numerical experimentation indicate that the present multigrid local refinement technique provides very effective numerical communications across the fine and coarse grid interfaces because the finer mesh is embedded entirely in the coarse mesh, and the occurrence of numerical instability in the interface is reduced.Other recent progress in the application of the preconditioning method to improve the rate of convergence and the modification of the Baldwin-Lomax turbulence model for accurate prediction of the cross-now separation problems is also discussed, Numerical examples are presented to illustrate that these advances have significantly improved our computational capability. Some future research trends are also indicated.展开更多
We propose an a-posteriori error/smoothness indicator for standard semidiscrete finite volume schemes for systems of conservation laws,based on the numerical production of entropy.This idea extends previous work by th...We propose an a-posteriori error/smoothness indicator for standard semidiscrete finite volume schemes for systems of conservation laws,based on the numerical production of entropy.This idea extends previous work by the first author limited to central finite volume schemes on staggered grids.We prove that the indicator converges to zero with the same rate of the error of the underlying numerical scheme on smooth flows under grid refinement.We construct and test an adaptive scheme for systems of equations in which the mesh is driven by the entropy indicator.The adaptive scheme uses a single nonuniform grid with a variable timestep.We show how to implement a second order scheme on such a space-time non uniform grid,preserving accuracy and conservation properties.We also give an example of a p-adaptive strategy.展开更多
In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining fu...In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.11932008 and 12272156)the Fundamental Research Funds for the Central Universities(No.lzujbky-2022-kb06)+1 种基金the Gansu Science and Technology ProgramLanzhou City’s Scientific Research Funding Subsidy to Lanzhou University of China。
文摘Second-generation high-temperature superconducting(HTS)conductors,specifically rare earth-barium-copper-oxide(REBCO)coated conductor(CC)tapes,are promising candidates for high-energy and high-field superconducting applications.With respect to epoxy-impregnated REBCO composite magnets that comprise multilayer components,the thermomechanical characteristics of each component differ considerably under extremely low temperatures and strong electromagnetic fields.Traditional numerical models include homogenized orthotropic models,which simplify overall field calculation but miss detailed multi-physics aspects,and full refinement(FR)ones that are thorough but computationally demanding.Herein,we propose an extended multi-scale approach for analyzing the multi-field characteristics of an epoxy-impregnated composite magnet assembled by HTS pancake coils.This approach combines a global homogenization(GH)scheme based on the homogenized electromagnetic T-A model,a method for solving Maxwell's equations for superconducting materials based on the current vector potential T and the magnetic field vector potential A,and a homogenized orthotropic thermoelastic model to assess the electromagnetic and thermoelastic properties at the macroscopic scale.We then identify“dangerous regions”at the macroscopic scale and obtain finer details using a local refinement(LR)scheme to capture the responses of each component material in the HTS composite tapes at the mesoscopic scale.The results of the present GH-LR multi-scale approach agree well with those of the FR scheme and the experimental data in the literature,indicating that the present approach is accurate and efficient.The proposed GH-LR multi-scale approach can serve as a valuable tool for evaluating the risk of failure in large-scale HTS composite magnets.
基金supported by the National Natural Science Foundation of China (Grant No.41130418)the National Major Project of China (under grant 2017ZX05008-007)+1 种基金supports from the Youth Innovation Promotion Association CAS (2012054)Foundation for Excellent Member of the Youth Innovation Promotion Association (2016)
文摘Numerical solver using a uniform grid is popular due to its simplicity and low computational cost, but would be unfeasible in the presence of tiny structures in large-scale media. It is necessary to use a nonuniform grid, where upsampling the wavefield from the coarse grid to the fine grid is essential for reducing artifacts. In this paper, we suggest a local refinement scheme using the Fourier interpolation, which is superior to traditional interpolation methods since it is theoretically exact if the input wavefield is band limited.Traditional interpolation methods would fail at high upsampling ratios(say 50); in contrast, our scheme still works well in the same situations, and the upsampling ratio can be any positive integer. A high upsampling ratio allows us to greatly reduce the computational burden and memory demand in the presence of tiny structures and large-scale models, especially for 3D cases.
基金Supported by the National Natural Science Foundation of China(60933008 and 61272300)
文摘This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.
基金This study was funded by Natural Science Foundation of China(Grant No.12102095)Research grant for 100 Talents of Guangxi Plan,The Starting Research Grant for High-Level Talents from Guangxi University,Generalized Isogeometric Analysis with Homogeniztion Theory for Soft Acoustic Metamaterials(AD20159080)+2 种基金Science and Technology Major Project of Guangxi Province(AA18118055)Guangxi Natural Science Foundation(2018JJB160052)Application of Key Technology in Building Construction of Prefabricated Steel Structure(BB30300105).
文摘Based on our proposed adaptivity strategy for the vibration of Reissner-Mindlin plate,we develop it to apply for the vibration of Kirchhoff plate.The adaptive algorithm is based on the Geometry-Independent Field approximaTion(GIFT),generalized from Iso-Geometric Analysis(IGA),and it can characterize the geometry of the structure with NURBS(Non-Uniform Rational B-Splines),and independently apply PHT-splines(Polynomial splines over Hierarchical T-meshes)to achieve local refinement in the solution field.TheMAC(Modal AssuranceCriterion)is improved to locate unique,as well as multiple,modal correspondence between different meshes,in order to deal with error estimation.Local adaptivity is carried out by sweeping modes from low to high frequency.Numerical examples showthat a proper choice of the spline space in solution field(with GIFT)can deliver better accuracy than using NURBS solution field.In addition,for vibration of heterogeneous Kirchhoff plates,our proposed method indicates that the adaptive local h-refinement achieves a better solution accuracy than the uniform h-refinement.
基金the National Natural Science Foundation of China(Nos.61171165 and 60802039)the Natural Science Foundation of Jiangsu(No.BK2010488)+1 种基金the Qing Lan Project of Jiangsu Province"the Six Top Talents"of Jiangsu Province Grant(No.2012DZXX-36)
文摘In this paper,we propose a refined local learning scheme to reconstruct a high resolution(HR)face image from a low resolution(LR)observation.The contribution of this work is twofold.Firstly,multi-direction gradient features are extracted to search the nearest neighbors for each image patch,then the non-negative matrix factorization(NMF)is used to reduce the complexity in weight calculation,and the initial HR embedding is estimated from the training pairs by preserving local geometry.Secondly,a global reconstruction constraint and post-processing by non-local filtering is incorporated into super-resolution(SR)reconstruction process to reduce the image artifacts and further improve the image visual quality.Experimental results show that the proposed algorithm improves the SR performance both in subjective and objective assessments compared with several existing methods.
文摘A complete mesh free adaptive algorithm （MFAA）, with solution adaptation and geometric adaptation, is developed to improve the resolution of flow features and to replace traditional global refinement techniques in structured grids. Unnecessary redundant points and elements are avoided by using the mesh free local clouds refinement technology in shock influencing regions and regions near large curvature places on the boundary. Inviscid compressible flows over NACA0012 and RAE2822 airfoils are computed. Finally numerical results validate the accuracy of the above method.
基金National High Technology Research and Development Program(863 Program)(No.2006AA06Z105,2007AA06Z134)
文摘Three-dimensional forward modeling magnetotellurics （MT） problems. We present a is a challenge for geometrically complex new edge-based finite-element algorithm using an unstructured mesh for accurately and efficiently simulating 3D MT responses. The electric field curl-curl equation in the frequency domain was used to deduce the H （curl） variation weak form of the MT forward problem, the Galerkin rule was used to derive a linear finite-element equation on the linear-edge tetrahedroid space, and, finally, a BI-CGSTAB solver was used to estimate the unknown electric fields. A local mesh refinement technique in the neighbor of the measuring MT stations was used to greatly improve the accuracies of the numerical solutions. Four synthetic models validated the powerful performance of our algorithms. We believe that our method will effectively contribute to processing more complex MT studies.
文摘Predicting rolling bearing fatigue life requires knowledge of the three-dimensional(3D)stress fields in the roller and raceway near the lubricated contact.Owing to the increasingly severe operating conditions,the effect of localized features such as surface roughness,subsurface inclusions,and even the crystallographic structure of the material becomes important.Achieving such detail requires(locally)extremely dense gridding in simulations,which in 3D is a major challenge.Multigrid techniques have been demonstrated to be capable of solving such problems.In this study,multigrid techniques are shown to further increase the efficiency of the solution by exploiting local grid refinement while maintaining the simplicity of a uniform discretization.This is achieved by employing increasingly finer grids only locally,where the highest resolution is required.Results are presented for dry contact and elastohydrodynamically lubricated contact cases,circular as well as elliptic,with varying crystallographic structure,and with surface roughness.The results show that the developed algorithm is very well suited for detailed analysis,with also excellent prospects for computational diagnostics involving actual material crystallographic structure from electron backscatter diffraction measurements.
基金supported by National Natural Science Foundation of China(Grant Nos.11031007 and 60903148)the Chinese Universities Scientific Fund+2 种基金Scientific Research Foundation for the Returned Overseas Chinese ScholarsState Education Ministry,the Chinese Academy of Sciences Startup Scientific Research Foundationthe State Key Development Program for Basic Research of China(973 Program)(Grant No.2011CB302400)
文摘This paper provides a survey of local refinable splines,including hierarchical B-splines,T-splines,polynomial splines over T-meshes,etc.,with a view to applications in geometric modeling and iso-geometric analysis.We will identify the strengths and weaknesses of these methods and also offer suggestions for their using in geometric modeling and iso-geometric analysis.
基金supported in part by China NSF under the grant 60873177by the National Basic Research Project under the grant 2005CB321702
文摘We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 （Ω）-context along with local discrete Helmholtz-type decompositions of the edge element space.
文摘A preconditioning method for the finite element stiffness matrix is given in this paper. The triangulation is refined in a subregion; the preconditioning process is composed of resolution of two regular subproblems; the condition number of the preconditioned matrix is 0(1 + log H/h), where H and h are mesh sizes of the unrefined and local refined triangulations respectively.
基金supported by the National Basic Research Program of China("973"Project)(Grant Nos.2011CB013505&2014CB047100)the National Natural Science Foundation of China(Grant Nos.11572009&51538001)
文摘In order to reach the best numerical properties with the numerical manifold method(NMM),uniform finite element meshes are always favorite while constructing mathematical covers,where all the elements are congruent.In the presence of steep gradients or strong singularities,in principle,the locally-defined special functions can be added into the NMM space by means of the partition of unity,but they are not available to those complex problems with heterogeneity or nonlinearity,necessitating local refinement on uniform meshes.This is believed to be one of the most important open issues in NMM.In this study multilayer covers are proposed to solve this issue.In addition to the first layer cover which is the global cover and covers the whole problem domain,the second and higher layer covers with smaller elements,called local covers,are used to cover those local regions with steep gradients or strong singularities.The global cover and the local covers have their own partition of unity,and they all participate in the approximation to the solution.Being advantageous over the existing procedures,the proposed approach is easy to deal with any arbitrary-layer hanging nodes with no need to construct super-elements with variable number of edge nodes or introduce the Lagrange multipliers to enforce the continuity between small and big elements.With no limitation to cover layers,meanwhile,the creation of an even error distribution over the whole problem domain is significantly facilitated.Some typical examples with steep gradients or strong singularities are analyzed to demonstrate the capacity of the proposed approach.
基金supported by the National Basic Research Program of China("973"Project)(Grant No.2014CB047100)the National Natural Science Foundation of China(Grant No.41372316)
文摘In this study, numerical manifold method(NMM) coupled with non-uniform rational B-splines(NURBS) and T-splines in the context of isogeometric analysis is proposed to allow for the treatments of complex geometries and local refinement. Computational formula for a 9-node NMM based on quadratic B-splines is derived. In order to exactly represent some common free-form shapes such as circles, arcs, and ellipsoids, quadratic non-uniform rational B-splines(NURBS) are introduced into NMM. The coordinate transformation based on the basis function of NURBS is established to enable exact integration for the manifold elements containing those shapes. For the case of crack propagation problems where singular fields around crack tips exist, local refinement technique by the application of T-spline discretizations is incorporated into NMM, which facilitates a truly local refinement without extending the entire row of control points. A local refinement strategy for the 4-node mathematical cover mesh based on T-splines and Lagrange interpolation polynomial is proposed. Results from numerical examples show that the 9-node NMM based on NURBS has higher accuracies. The coordinate transformation based on the NURBS basis function improves the accuracy of NMM by exact integration. The local mesh refinement using T-splines reduces the number of degrees of freedom while maintaining calculation accuracy at the same time.
基金Supported by the National Natural Science Foundation of China under Grant No. 60873038the National Key Technology Research and Development Program of the Ministry of Science and Technology of China under Grant Nos. 2009BAH42B02 and 2012BAH08B02
文摘CEGAR （Counterexample-guided abstraction refinement）-based slicing is one of the most important techniques in reducing the state space in model checking. However, CEGAR-based slicing repeatedly explores the state space handled previously in case a spurious counterexample is found. Inspired by lazy abstraction, we introduce the concept of lazy slicing which eliminates this repeated computation. Lazy slicing is done on-the-fly, and only up to the precision necessary to rule out spurious counterexamples. It identifies a spurious counterexample by concretizing a path fragment other than the full path, which reduces the cost of spurious counterexample decision significantly. Besides, we present an improved over-approximate slicing method to build a more precise slice model. We also provide the proof of the correctness and the termination of lazy slicing, and implement a prototype model checker to verify safety property. Experimental results show that lazy slicing scales to larger systems than CEGAR-based slicing methods.
基金Part of Hu’s research was carried out while visiting Hong Kong Baptist UniversityHis research was also supported by an National Basic Research Program of China under the grant 2005CB32170+2 种基金Li’s research was partially supported by the National Basic Research Programof China under the grant 2005CB321701Foundation forNational ExcellentDoc-toral Dissertation Award of China and the Joint Applied Mathematics Research Institute between Peking University and Hong Kong Baptist UniversityTang’s research was sup-ported by CERG Grants of Hong Kong Research Grant Council and FRG grants of Hong Kong Baptist University.
文摘In this work,we propose an efficient multi-mesh adaptive finite element method for simulating the dendritic growth in two-and three-dimensions.The governing equations used are the phase field model,where the regularity behaviors of the relevant dependent variables,namely the thermal field function and the phase field function,can be very different.To enhance the computational efficiency,we approximate these variables on different h-adaptive meshes.The coupled terms in the system are calculated based on the implementation of the multi-mesh h-adaptive algorithm proposed by Li(J.Sci.Comput.,pp.321-341,24(2005)).It is illustrated numerically that the multi-mesh technique is useful in solving phase field models and can save storage and the CPU time significantly.
文摘A new numerical approach based on a multiblock, multigrid, local refinement method has been developed. The multiblock structure makes grid generation for complex geometries easier,multigrid techniques significantly accelerate the rate of convergence and the local refinement method provides high spatial resolution of boundary lay6r and separated vortical flows with much reduced computer memory and CPU time. Sample three-dimensional flow computations for complex geometries are presented to illustrate the advantages in term of memory savings and reductions of CPU time in obtaining high spatial resolution solutions. High spatial resolution in the inner viscous layer is achieved by using the values of y+ for the first grid centers rom the wall in the order of 1 to 7. The required solution resolution in the dominantly viscous flow region (boundary layer, vortex core, or three-dimensional separated zone) dictates the acceptable spacing of grid cells in that region. It is very desirable to be able to reduce grid spacing locally in the critical region without changing the overall grid. The local refinement technique doubles grid numbers in all three directions for every level of refinement. Multiple levels of local grid refinement with moderate grid stretching can be used to provide fine grid spacing in rapidly changing flow regions such as near the wall, vortex core and separated shear layer. Thus, the method provides the required fine spatial resolution in the dominantly viscous region which is relatively small compared to the entire computational domain. This also avoids the used of excessively large aspect ratios of the grid cells near the rapidly changing now regions thus reduces truncation error and improves robustness. Furthermore, the results of numerical experimentation indicate that the present multigrid local refinement technique provides very effective numerical communications across the fine and coarse grid interfaces because the finer mesh is embedded entirely in the coarse mesh, and the occurrence of numerical instability in the interface is reduced.Other recent progress in the application of the preconditioning method to improve the rate of convergence and the modification of the Baldwin-Lomax turbulence model for accurate prediction of the cross-now separation problems is also discussed, Numerical examples are presented to illustrate that these advances have significantly improved our computational capability. Some future research trends are also indicated.
文摘We propose an a-posteriori error/smoothness indicator for standard semidiscrete finite volume schemes for systems of conservation laws,based on the numerical production of entropy.This idea extends previous work by the first author limited to central finite volume schemes on staggered grids.We prove that the indicator converges to zero with the same rate of the error of the underlying numerical scheme on smooth flows under grid refinement.We construct and test an adaptive scheme for systems of equations in which the mesh is driven by the entropy indicator.The adaptive scheme uses a single nonuniform grid with a variable timestep.We show how to implement a second order scheme on such a space-time non uniform grid,preserving accuracy and conservation properties.We also give an example of a p-adaptive strategy.
基金Supported by the Scientific Research Foundation for the Doctor,Nanjing University of Aeronautics and Astronautics(No.1008-907359)
文摘In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.
