Atom tracking technology enhanced with innovative algorithms has been implemented in this study,utilizing a comprehensive suite of controllers and software independently developed domestically.Leveraging an on-board f...Atom tracking technology enhanced with innovative algorithms has been implemented in this study,utilizing a comprehensive suite of controllers and software independently developed domestically.Leveraging an on-board field-programmable gate array(FPGA)with a core frequency of 100 MHz,our system facilitates reading and writing operations across 16 channels,performing discrete incremental proportional-integral-derivative(PID)calculations within 3.4 microseconds.Building upon this foundation,gradient and extremum algorithms are further integrated,incorporating circular and spiral scanning modes with a horizontal movement accuracy of 0.38 pm.This integration enhances the real-time performance and significantly increases the accuracy of atom tracking.Atom tracking achieves an equivalent precision of at least 142 pm on a highly oriented pyrolytic graphite(HOPG)surface under room temperature atmospheric conditions.Through applying computer vision and image processing algorithms,atom tracking can be used when scanning a large area.The techniques primarily consist of two algorithms:the region of interest(ROI)-based feature matching algorithm,which achieves 97.92%accuracy,and the feature description-based matching algorithm,with an impressive 99.99%accuracy.Both implementation approaches have been tested for scanner drift measurements,and these technologies are scalable and applicable in various domains of scanning probe microscopy with broad application prospects in the field of nanoengineering.展开更多
Adaptive mesh refinement(AMR)is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy.Accurate treatment on AMR hierarchies requires accurate prolongation of the so...Adaptive mesh refinement(AMR)is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy.Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh.For scalar variables,suitably high-order finite volume WENO methods can carry out such a prolongation.However,classes of PDEs,such as computational electrodynamics(CED)and magnetohydrodynamics(MHD),require that vector fields preserve a divergence constraint.The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh.As a result,the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate.In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact.Extension to higher orders using analytically exact methods is very challenging.To overcome that challenge,a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces,where the vector field components are collocated.This approach is almost divergence constraint-preserving,therefore,we call it WENO-ADP.To make it exactly divergence constraint-preserving,a touch-up procedure is developed that is based on a constrained least squares(CLSQ)method for restoring the divergence constraint up to machine accuracy.With the touch-up,it is called WENO-ADPT.It is shown that refinement ratios of two and higher can be accommodated.An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods,where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares.We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields,where the divergence of the vector field has to match a charge density and its higher moments.We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.展开更多
Several important PDE systems,like magnetohydrodynamics and computational electrodynamics,are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-pre...Several important PDE systems,like magnetohydrodynamics and computational electrodynamics,are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion.Recently,new classes of PDE systems have emerged for hyperelasticity,compressible multiphase flows,so-called firstorder reductions of the Einstein field equations,or a novel first-order hyperbolic reformulation of Schrödinger’s equation,to name a few,where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field.We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume(FV)WENO-like schemes for PDEs that support a curl-preserving involution.(Some insights into discontinuous Galerkin(DG)schemes are also drawn,though that is not the prime focus of this paper.)This is done for two-and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction.The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented.In two dimensions,a von Neumann analysis of structure-preserving WENOlike schemes that mimetically satisfy the curl constraints,is also presented.It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems.Numerical results are also presented to show that the edge-centered curl-preserving(ECCP)schemes meet their design accuracy.This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy.By its very design,this paper is,therefore,intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.展开更多
基金Project supported by the National Science Fund for Distinguished Young Scholars(Grant No.T2125014)the Special Fund for Research on National Major Research Instruments of the National Natural Science Foundation of China(Grant No.11927808)the CAS Key Technology Research and Development Team Project(Grant No.GJJSTD20200005)。
文摘Atom tracking technology enhanced with innovative algorithms has been implemented in this study,utilizing a comprehensive suite of controllers and software independently developed domestically.Leveraging an on-board field-programmable gate array(FPGA)with a core frequency of 100 MHz,our system facilitates reading and writing operations across 16 channels,performing discrete incremental proportional-integral-derivative(PID)calculations within 3.4 microseconds.Building upon this foundation,gradient and extremum algorithms are further integrated,incorporating circular and spiral scanning modes with a horizontal movement accuracy of 0.38 pm.This integration enhances the real-time performance and significantly increases the accuracy of atom tracking.Atom tracking achieves an equivalent precision of at least 142 pm on a highly oriented pyrolytic graphite(HOPG)surface under room temperature atmospheric conditions.Through applying computer vision and image processing algorithms,atom tracking can be used when scanning a large area.The techniques primarily consist of two algorithms:the region of interest(ROI)-based feature matching algorithm,which achieves 97.92%accuracy,and the feature description-based matching algorithm,with an impressive 99.99%accuracy.Both implementation approaches have been tested for scanner drift measurements,and these technologies are scalable and applicable in various domains of scanning probe microscopy with broad application prospects in the field of nanoengineering.
基金DSB acknowledges support via NSF grants NSF-19-04774,NSF-AST-2009776NASA-2020–1241.
文摘Adaptive mesh refinement(AMR)is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy.Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh.For scalar variables,suitably high-order finite volume WENO methods can carry out such a prolongation.However,classes of PDEs,such as computational electrodynamics(CED)and magnetohydrodynamics(MHD),require that vector fields preserve a divergence constraint.The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh.As a result,the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate.In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact.Extension to higher orders using analytically exact methods is very challenging.To overcome that challenge,a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces,where the vector field components are collocated.This approach is almost divergence constraint-preserving,therefore,we call it WENO-ADP.To make it exactly divergence constraint-preserving,a touch-up procedure is developed that is based on a constrained least squares(CLSQ)method for restoring the divergence constraint up to machine accuracy.With the touch-up,it is called WENO-ADPT.It is shown that refinement ratios of two and higher can be accommodated.An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods,where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares.We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields,where the divergence of the vector field has to match a charge density and its higher moments.We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.
基金Dinshaw S.Balsara acknowledges support via NSF grants NSF-19-04774,NSFAST-2009776 and NASA-2020-1241Michael Dumbser acknowledges the financial support received from the Italian Ministry of Education,University and Research(MIUR)in the frame of the Departments of Excellence Initiative 2018-2022 attributed to DICAM of the University of Trento(grant L.232/2016)and in the frame of the PRIN 2017 project Innovative numerical methods for evolutionary partial differential equations and applications.
文摘Several important PDE systems,like magnetohydrodynamics and computational electrodynamics,are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion.Recently,new classes of PDE systems have emerged for hyperelasticity,compressible multiphase flows,so-called firstorder reductions of the Einstein field equations,or a novel first-order hyperbolic reformulation of Schrödinger’s equation,to name a few,where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field.We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume(FV)WENO-like schemes for PDEs that support a curl-preserving involution.(Some insights into discontinuous Galerkin(DG)schemes are also drawn,though that is not the prime focus of this paper.)This is done for two-and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction.The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented.In two dimensions,a von Neumann analysis of structure-preserving WENOlike schemes that mimetically satisfy the curl constraints,is also presented.It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems.Numerical results are also presented to show that the edge-centered curl-preserving(ECCP)schemes meet their design accuracy.This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy.By its very design,this paper is,therefore,intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.
