The cubic B-splines taken as trial function, the large deflection of a circular plate with arbitrarily variable thickness,as well as the buckling load, have been calculated by the method of point collocation. The supp...The cubic B-splines taken as trial function, the large deflection of a circular plate with arbitrarily variable thickness,as well as the buckling load, have been calculated by the method of point collocation. The support can be elastic. Loads imposed can be polynomial distributed loads, uniformly distributed radial forces or moments along the edge respectively or their combinations. Convergent solutions can still be obtained by this method under the load whose value is in great excess of normal one. Under the action of the uniformly distributed loads, linear solutions of circular plates with linearly or quadratically variable thickness are compared with those obtained by the parameter method. Buckling of a circular plate with identical thickness beyond critical thrust is compared with those obtained by the power series method.展开更多
In this paper, the bicubic splines in product form are used to construct the multi-field functions for bending moments, twisting moment and transverse displacement of the plate on elastic foundation. The multivariable...In this paper, the bicubic splines in product form are used to construct the multi-field functions for bending moments, twisting moment and transverse displacement of the plate on elastic foundation. The multivariable spline element equations are derived, based on the mixed variational principle. The analysis and calculations of bending, vibration and stability of the plates on elastic foundation are presented in the paper. Because the field functions of plate on elastic foundation are assumed independently, the precision of the field variables of bending moments and displacement is high.展开更多
In order to achieve refined precipitation grid data with high accuracy and high spatial resolution,hourly precipitation grid dataset with 1 km spatial resolution in Anhui Province from May to September (in the rainy s...In order to achieve refined precipitation grid data with high accuracy and high spatial resolution,hourly precipitation grid dataset with 1 km spatial resolution in Anhui Province from May to September (in the rainy season) in 2016 and from August 2017 to July 2018 was established based on thin plate smoothing spline (TPS),which meets the needs of climate change research and meteorological disaster risk assessment.The interpolation errors were then analyzed.The grid precipitation product obtained based on thin plate smoothing spline,CLDAS-FAST and CLDAS-FRT were evaluated.The results show that the interpolated values of hourly precipitation by thin plate smoothing spline are close to the observed values in the rainy season of 2016.The errors are generally below 0.9 mm/h.However,the errors of precipitation in the mountainous areas of eastern Huaibei and western Anhui are above 1.2 mm/h.On monthly scale,the errors in June and July are the largest,and the proportion of absolute values of the errors ≥2 mm/h is up to 2.0% in June and 2.2% in July.The errors in September are the smallest,and the proportion of absolute values of the errors ≥2 mm/h is only 0.6%.The root mean square (RMSE) is only 0.37 mm/h,and the correlation coefficient (COR) is 0.93.The interpolation accuracy of CLDAS-FRT is the highest,with the smallest RMSE (0.65 mm/h) and mean error ( ME =0.01 mm/h),and the largest COR (0.81).The accuracy of the precipitation product obtained by thin plate smoothing spline interpolation is close to that of CLDAS-FAST.Its RMSE is up to 0.80 mm/h,and its ME is only -0.01 mm/h.Its COR is 0.73,but its bias (BIAS) is up to 1.06 mm/h.展开更多
The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh disto...The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh distortions. In a previous work, we constructed an 8-node quadrilateral spline element L8 using the triangular area coordinates and the B- net method, which can be insensitive to mesh distortions and possess the second order completeness in the Cartesian co- ordinates. In this paper, a thin plate spline element is devel- oped based on the spline element L8 and the refined tech- nique. Numerical examples show that the present element indeed possesses higher accuracy than the DKQ element for distorted meshes.展开更多
Common tools based on landmarks in medical image elastic registration are Thin Plate Spline (TPS) and Compact Support Radial Basis Function (CSRBF). TPS forces the corresponding landmarks to exactly match each other a...Common tools based on landmarks in medical image elastic registration are Thin Plate Spline (TPS) and Compact Support Radial Basis Function (CSRBF). TPS forces the corresponding landmarks to exactly match each other and minimizes the bending energy of the whole image. However, in real application, such scheme would deform the image globally when deformation is only local. CSRBF needs manually determine the support size, although its deformation is limited local. Therefore, to limit the effect of the deformation, new Compact Support Thin Plate Spline algorithm (CSTPS) is approached, analyzed and applied. Such new approach gains optimal mutual information, which shows its registration result satisfactory. The experiments also show it can apply in both local and global elastic registration.展开更多
Cubic B-spline taken as trial function, the nonlinear bending of a circular sandwich plate was calculated by the method of point collocation. The support could be elastic. A sandwich plate was assumed to be Reissner m...Cubic B-spline taken as trial function, the nonlinear bending of a circular sandwich plate was calculated by the method of point collocation. The support could be elastic. A sandwich plate was assumed to be Reissner model. The formulae were developed for the calculation of a circular sandwich plate subjected to polynomial distributed loads, uniformly distributed moments, radial pressure or radial prestress along the edge and their combination. Buckling load was calculated for the first time by nonlinear theory. Under action of uniformly distributed loads, results were compared with that obtained by the power series method. Excellences of the program written by the spline collocation method are wide convergent range, high precision and universal.展开更多
THz Radar Cross Section(RCS)measurement setup based on THz Time Domain Spectroscopy(TDS)is built to provide large scaled targets test ability in recent years.As calibrations,the metal plates and dihedrons are used in ...THz Radar Cross Section(RCS)measurement setup based on THz Time Domain Spectroscopy(TDS)is built to provide large scaled targets test ability in recent years.As calibrations,the metal plates and dihedrons are used in our experiments.The measurements are performed in a monostatic terahertz time-domain setup.The author proposed time domain and frequency domain calibration methods for angular RCS of calibrations,comparing the measurements with the theory to verify the ability of the time domain measurement setup.展开更多
In the view of Reissner's and Kirchhoff's theories, respectively, we formulate the isotropicalized governing equations for the anisotropic plates, and give the proof of the equivalence relation between these t...In the view of Reissner's and Kirchhoff's theories, respectively, we formulate the isotropicalized governing equations for the anisotropic plates, and give the proof of the equivalence relation between these two plate-models for the simply-supported rectangular orthotropic plates. The well-known fundamental solutions of the isotrqpic plates are utlized for the spline integral equation analysis of anisotropic plates.Even with sparse meshes the satisfactory results can be obtained. The analysis of plates on two-parameter elastic foundation is so simple as the common case that only a few terms should be added to the formulas of fictitious loads.展开更多
In this paper, Von Karman's set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection bein...In this paper, Von Karman's set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection being taken as a perturbation parameter. These sets of linear equations are solved by the spline finite-point (SFP) method and by the spline finite element (SFE) method. The solutions for rectangular plates having any length-to-width ratios under a uniformly distributed load and with various boundary conditions are presented, and the analytical formulas for displacements and deflections are given in the paper. The computer programs are worked out by ourselves. Comparison of the results with those in other papers indicates that the results of this paper are satisfactorily better.展开更多
The tectonic evolution history of the South China Sea(SCS) is important for understanding the interaction between the Pacific Tectonic Domain and the Tethyan Tectonic Domain,as well as the regional tectonics and geody...The tectonic evolution history of the South China Sea(SCS) is important for understanding the interaction between the Pacific Tectonic Domain and the Tethyan Tectonic Domain,as well as the regional tectonics and geodynamics during the multi-plate convergence in the Cenozoic.Several Cenozoic basins formed in the northern margin of the SCS,which preserve the sedimentary tectonic records of the opening of the SCS.Due to the spatial non-uniformity among different basins,a systematic study on the various basins in the northern margin of the SCS constituting the Northern Cenozoic Basin Group(NCBG) is essential.Here we present results from a detailed evaluation of the spatial-temporal migration of the boundary faults and primary unconformities to unravel the mechanism of formation of the NCBG.The NCBG is composed of the Beibu Gulf Basin(BBGB),Qiongdongnan Basin(QDNB),Pearl River Mouth Basin(PRMB) and Taixinan Basin(TXNB).Based on seismic profiles and gravity-magnetic anomalies,we confirm that the NE-striking onshore boundary faults propagated into the northern margin of the SCS.Combining the fault slip rate,fault combination and a comparison of the unconformities in different basins,we identify NE-striking rift composed of two-stage rifting events in the NCBG:an early-stage rifting(from the Paleocene to the Early Oligocene) and a late-stage rifting(from the Late Eocene to the beginning of the Miocene).Spatially only the late-stage faults occurs in the western part of the NCBG(the BBGB,the QDNB and the western PRMB),but the early-stage rifting is distributed in the whole NCBG.Temporally,the early-stage rifting can be subdivided into three phases which show an eastward migration,resulting in the same trend of the primary unconformities and peak faulting within the NCBG.The late-stage rifting is subdivided into two phases,which took place simultaneously in different basins.The first and second phase of the early-stage rifting is related to back-arc extension of the Pacific subduction retreat system.The third phase of the earlystage rifting resulted from the joint effect of slab-pull force due to southward subduction of the proto-SCS and the back-arc extension of the Pacific subduction retreat system.In addition,the first phase of the late-stage faulting corresponds with the combined effect of the post-collision extension along the Red River Fault and slab-pull force of the proto-SCS subduction.The second phase of the late-stage faulting fits well with the sinistral faulting of the Red River Fault in response to the Indochina Block escape tectonics and the slab-pull force of the proto-SCS.展开更多
The spline finite strip method （FSM） is one of the most popular numerical methods for analyzing prismatic structures. Efficacy and convergence of the method have been demonstrated in previous studies by comparing on...The spline finite strip method （FSM） is one of the most popular numerical methods for analyzing prismatic structures. Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems. To date, no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such, in this paper, the mathematical exact solutions of spline finite strips in the plate analysis are derived using a unitary transformation approach （abbreviated as the U-transformation method herein）. These exact solutions are presented for the first time in open literature. Unlike the conventional spline FSM which involves assembly of the global matrix equation and its numerical solution, the U-transformation method decouples the global matrix equation into the one involving only two unknowns, thus rendering the exact solutions of the spline finite strip to be derived explicitly. By taking Taylor＇s series expansion of the exact solution, error terms and convergence rates are also derived explicitly and compared directly with other numerical methods. In this regard, the spline FSM converges at the same rate as a non-conforming finite element, yet involving a smaller number of unknowns compared to the latter. The convergence rate is also found superior to the conventional finite difference method.展开更多
This paper addresses the regularity and finite dimensionality of the global attractor for the plate equation on the unbounded domain. The existence of the attractor in the phase space has been established in an earlie...This paper addresses the regularity and finite dimensionality of the global attractor for the plate equation on the unbounded domain. The existence of the attractor in the phase space has been established in an earlier work of the author. It is shown that the attractor is actually a bounded set of the phase space and has finite fractal dimensionality.展开更多
A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth...A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains.In the solution of differential equations,various derivatives of the unknown function are denoted as new functions.Then,the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals.Therefore,the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative.During the application of the proposed method,boundary conditions can be automatically included in the integration operations,and relevant matrices can be assured to exhibit perfect sparse patterns.As examples,we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes.By comparing the solutions obtained by the proposed method with the exact solutions,the new multiresolution method is found to have a convergence rate of fifth order.The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method(FEM)with tens of thousands of elements.In addition,because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order,we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.展开更多
For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-...For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-resistances R<sub>01,23</sub>, R<sub>12,30</sub> with the sheet resistance without any other parameters. If the domain has one hole van der Pauw’s equation becomes an inequality with upper and lower bounds, the envelopes. This was conjectured by Szymański et al. in 2013, and only recently it was proven by Miyoshi et al. with elaborate mathematical tools. The present article gives new proofs closer to physical intuition and partly with simpler mathematics. It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also given.展开更多
文摘The cubic B-splines taken as trial function, the large deflection of a circular plate with arbitrarily variable thickness,as well as the buckling load, have been calculated by the method of point collocation. The support can be elastic. Loads imposed can be polynomial distributed loads, uniformly distributed radial forces or moments along the edge respectively or their combinations. Convergent solutions can still be obtained by this method under the load whose value is in great excess of normal one. Under the action of the uniformly distributed loads, linear solutions of circular plates with linearly or quadratically variable thickness are compared with those obtained by the parameter method. Buckling of a circular plate with identical thickness beyond critical thrust is compared with those obtained by the power series method.
文摘In this paper, the bicubic splines in product form are used to construct the multi-field functions for bending moments, twisting moment and transverse displacement of the plate on elastic foundation. The multivariable spline element equations are derived, based on the mixed variational principle. The analysis and calculations of bending, vibration and stability of the plates on elastic foundation are presented in the paper. Because the field functions of plate on elastic foundation are assumed independently, the precision of the field variables of bending moments and displacement is high.
基金Supported by New Technology Integration Project of Anhui Meteorological Bureau(AHXJ201704)Project for Masters and Doctors of Anhui Meteorological Bureau(RC201701)
文摘In order to achieve refined precipitation grid data with high accuracy and high spatial resolution,hourly precipitation grid dataset with 1 km spatial resolution in Anhui Province from May to September (in the rainy season) in 2016 and from August 2017 to July 2018 was established based on thin plate smoothing spline (TPS),which meets the needs of climate change research and meteorological disaster risk assessment.The interpolation errors were then analyzed.The grid precipitation product obtained based on thin plate smoothing spline,CLDAS-FAST and CLDAS-FRT were evaluated.The results show that the interpolated values of hourly precipitation by thin plate smoothing spline are close to the observed values in the rainy season of 2016.The errors are generally below 0.9 mm/h.However,the errors of precipitation in the mountainous areas of eastern Huaibei and western Anhui are above 1.2 mm/h.On monthly scale,the errors in June and July are the largest,and the proportion of absolute values of the errors ≥2 mm/h is up to 2.0% in June and 2.2% in July.The errors in September are the smallest,and the proportion of absolute values of the errors ≥2 mm/h is only 0.6%.The root mean square (RMSE) is only 0.37 mm/h,and the correlation coefficient (COR) is 0.93.The interpolation accuracy of CLDAS-FRT is the highest,with the smallest RMSE (0.65 mm/h) and mean error ( ME =0.01 mm/h),and the largest COR (0.81).The accuracy of the precipitation product obtained by thin plate smoothing spline interpolation is close to that of CLDAS-FAST.Its RMSE is up to 0.80 mm/h,and its ME is only -0.01 mm/h.Its COR is 0.73,but its bias (BIAS) is up to 1.06 mm/h.
基金supported by the National Natural Science Foundation of China(11001037,11102037,11290143)the Fundamental Research Funds for the Central Universities(DUT13LK07)
文摘The quadrilateral discrete Kirchhoff thin plate bending element DKQ is based on the isoparametric element Q8, however, the accuracy of the isoparametric quadrilateral elements will drop significantly due to mesh distortions. In a previous work, we constructed an 8-node quadrilateral spline element L8 using the triangular area coordinates and the B- net method, which can be insensitive to mesh distortions and possess the second order completeness in the Cartesian co- ordinates. In this paper, a thin plate spline element is devel- oped based on the spline element L8 and the refined tech- nique. Numerical examples show that the present element indeed possesses higher accuracy than the DKQ element for distorted meshes.
基金the National Natural Science Foundation of China (No.60572101) the Natural Science Foundation of Guangdong Province (No.31789).
文摘Common tools based on landmarks in medical image elastic registration are Thin Plate Spline (TPS) and Compact Support Radial Basis Function (CSRBF). TPS forces the corresponding landmarks to exactly match each other and minimizes the bending energy of the whole image. However, in real application, such scheme would deform the image globally when deformation is only local. CSRBF needs manually determine the support size, although its deformation is limited local. Therefore, to limit the effect of the deformation, new Compact Support Thin Plate Spline algorithm (CSTPS) is approached, analyzed and applied. Such new approach gains optimal mutual information, which shows its registration result satisfactory. The experiments also show it can apply in both local and global elastic registration.
文摘Cubic B-spline taken as trial function, the nonlinear bending of a circular sandwich plate was calculated by the method of point collocation. The support could be elastic. A sandwich plate was assumed to be Reissner model. The formulae were developed for the calculation of a circular sandwich plate subjected to polynomial distributed loads, uniformly distributed moments, radial pressure or radial prestress along the edge and their combination. Buckling load was calculated for the first time by nonlinear theory. Under action of uniformly distributed loads, results were compared with that obtained by the power series method. Excellences of the program written by the spline collocation method are wide convergent range, high precision and universal.
基金The Science and Technology Commission of Shanghai Municipality under Grant(16ZR1435000)
文摘THz Radar Cross Section(RCS)measurement setup based on THz Time Domain Spectroscopy(TDS)is built to provide large scaled targets test ability in recent years.As calibrations,the metal plates and dihedrons are used in our experiments.The measurements are performed in a monostatic terahertz time-domain setup.The author proposed time domain and frequency domain calibration methods for angular RCS of calibrations,comparing the measurements with the theory to verify the ability of the time domain measurement setup.
文摘In the view of Reissner's and Kirchhoff's theories, respectively, we formulate the isotropicalized governing equations for the anisotropic plates, and give the proof of the equivalence relation between these two plate-models for the simply-supported rectangular orthotropic plates. The well-known fundamental solutions of the isotrqpic plates are utlized for the spline integral equation analysis of anisotropic plates.Even with sparse meshes the satisfactory results can be obtained. The analysis of plates on two-parameter elastic foundation is so simple as the common case that only a few terms should be added to the formulas of fictitious loads.
文摘In this paper, Von Karman's set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection being taken as a perturbation parameter. These sets of linear equations are solved by the spline finite-point (SFP) method and by the spline finite element (SFE) method. The solutions for rectangular plates having any length-to-width ratios under a uniformly distributed load and with various boundary conditions are presented, and the analytical formulas for displacements and deflections are given in the paper. The computer programs are worked out by ourselves. Comparison of the results with those in other papers indicates that the results of this paper are satisfactorily better.
基金This research was funded by National Program on Global Change and Air-Sea Interaction,SOA(No.GASI-GEOGE-01)National Key Research and Development Program of China(2017YFC0601401 and 2016YFC0601002)+2 种基金Qingdao National Laboratory for Marine Science and Technology(2016ASKJ13,2017ASKJ02)the financially support from the Aoshan Talents Program Supported by Qingdao National Laboratory for Marine Science and Technology to Prof.Sanzhong Li(No.2015ASTP-0S10)the Taishan Scholar Program to Prof.Sanzhong Li
文摘The tectonic evolution history of the South China Sea(SCS) is important for understanding the interaction between the Pacific Tectonic Domain and the Tethyan Tectonic Domain,as well as the regional tectonics and geodynamics during the multi-plate convergence in the Cenozoic.Several Cenozoic basins formed in the northern margin of the SCS,which preserve the sedimentary tectonic records of the opening of the SCS.Due to the spatial non-uniformity among different basins,a systematic study on the various basins in the northern margin of the SCS constituting the Northern Cenozoic Basin Group(NCBG) is essential.Here we present results from a detailed evaluation of the spatial-temporal migration of the boundary faults and primary unconformities to unravel the mechanism of formation of the NCBG.The NCBG is composed of the Beibu Gulf Basin(BBGB),Qiongdongnan Basin(QDNB),Pearl River Mouth Basin(PRMB) and Taixinan Basin(TXNB).Based on seismic profiles and gravity-magnetic anomalies,we confirm that the NE-striking onshore boundary faults propagated into the northern margin of the SCS.Combining the fault slip rate,fault combination and a comparison of the unconformities in different basins,we identify NE-striking rift composed of two-stage rifting events in the NCBG:an early-stage rifting(from the Paleocene to the Early Oligocene) and a late-stage rifting(from the Late Eocene to the beginning of the Miocene).Spatially only the late-stage faults occurs in the western part of the NCBG(the BBGB,the QDNB and the western PRMB),but the early-stage rifting is distributed in the whole NCBG.Temporally,the early-stage rifting can be subdivided into three phases which show an eastward migration,resulting in the same trend of the primary unconformities and peak faulting within the NCBG.The late-stage rifting is subdivided into two phases,which took place simultaneously in different basins.The first and second phase of the early-stage rifting is related to back-arc extension of the Pacific subduction retreat system.The third phase of the earlystage rifting resulted from the joint effect of slab-pull force due to southward subduction of the proto-SCS and the back-arc extension of the Pacific subduction retreat system.In addition,the first phase of the late-stage faulting corresponds with the combined effect of the post-collision extension along the Red River Fault and slab-pull force of the proto-SCS subduction.The second phase of the late-stage faulting fits well with the sinistral faulting of the Red River Fault in response to the Indochina Block escape tectonics and the slab-pull force of the proto-SCS.
基金Project supported by the Division Research Grant from City University of Hong Kong(No.DRG 13/08-09)
文摘The spline finite strip method （FSM） is one of the most popular numerical methods for analyzing prismatic structures. Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems. To date, no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such, in this paper, the mathematical exact solutions of spline finite strips in the plate analysis are derived using a unitary transformation approach （abbreviated as the U-transformation method herein）. These exact solutions are presented for the first time in open literature. Unlike the conventional spline FSM which involves assembly of the global matrix equation and its numerical solution, the U-transformation method decouples the global matrix equation into the one involving only two unknowns, thus rendering the exact solutions of the spline finite strip to be derived explicitly. By taking Taylor＇s series expansion of the exact solution, error terms and convergence rates are also derived explicitly and compared directly with other numerical methods. In this regard, the spline FSM converges at the same rate as a non-conforming finite element, yet involving a smaller number of unknowns compared to the latter. The convergence rate is also found superior to the conventional finite difference method.
基金Project Supported by the Scientific Research Fund of Zhejiang Provincial Education Department(No.Y200804289)the Natural Science Foundation of Ningbo City(No.2010A610102)the K.C.Wong Magna Fund in Ningbo University
文摘This paper addresses the regularity and finite dimensionality of the global attractor for the plate equation on the unbounded domain. The existence of the attractor in the phase space has been established in an earlier work of the author. It is shown that the attractor is actually a bounded set of the phase space and has finite fractal dimensionality.
基金Project supported by the National Natural Science Foundation of China(No.11925204)the 111 Project(No.B14044)。
文摘A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains.In the solution of differential equations,various derivatives of the unknown function are denoted as new functions.Then,the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals.Therefore,the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative.During the application of the proposed method,boundary conditions can be automatically included in the integration operations,and relevant matrices can be assured to exhibit perfect sparse patterns.As examples,we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes.By comparing the solutions obtained by the proposed method with the exact solutions,the new multiresolution method is found to have a convergence rate of fifth order.The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method(FEM)with tens of thousands of elements.In addition,because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order,we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.
文摘For plane singly-connected domains with insulating boundary and four point-sized contacts, C<sub>0</sub> …C<sub>3</sub>, van der Pauw derived a famous equation relating the two trans-resistances R<sub>01,23</sub>, R<sub>12,30</sub> with the sheet resistance without any other parameters. If the domain has one hole van der Pauw’s equation becomes an inequality with upper and lower bounds, the envelopes. This was conjectured by Szymański et al. in 2013, and only recently it was proven by Miyoshi et al. with elaborate mathematical tools. The present article gives new proofs closer to physical intuition and partly with simpler mathematics. It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also given.
