At present the mechanical model of the interac- tion between a disc cutter and rock mainly concerns indentation experiment, linear cutting experiment and tunnel boring machine （TBM） on-site data. This is not in line...At present the mechanical model of the interac- tion between a disc cutter and rock mainly concerns indentation experiment, linear cutting experiment and tunnel boring machine （TBM） on-site data. This is not in line with the actual rock-breaking movement of the disc cutter and impedes to some extent the research on the rock-breaking mechanism, wear mechanism and design theory. Therefore, our study focuses on the interaction between the slantingly installed disc cutter and rock, developing a model in accordance with the actual rock-breaking movement. Displacement equations are established through an analysis of the velocity vector at the rock-breaking point of the disc cutter blade; the func- tional relationship between the displacement parameters at the rock-breaking point and its rectangular coordinates is established through an analysis of micro-displacement vectors at the rock-breaking point, thus leading to the geometric equations of rock deformation caused by the slantingly installed disc cutter. Considering the basically linear relationship between the cutting force of disc cutters and the rock deformation before and after the leap break of rock, we express the constitutive relations of rock deformation as generalized Hooke＇s law and analyze the effect of the slanting installa- tion angle of disc cutters on the rock-breaking force. This will, as we hope, make groundbreaking contributions to the development of the design theory and installation practice of TBM.展开更多
Product variation reduction is critical to improve process efficiency and product quality, especially for multistage machining process（MMP）. However, due to the variation accumulation and propagation, it becomes qui...Product variation reduction is critical to improve process efficiency and product quality, especially for multistage machining process（MMP）. However, due to the variation accumulation and propagation, it becomes quite difficult to predict and reduce product variation for MMP. While the method of statistical process control can be used to control product quality, it is used mainly to monitor the process change rather than to analyze the cause of product variation. In this paper, based on a differential description of the contact kinematics of locators and part surfaces, and the geometric constraints equation defined by the locating scheme, an improved analytical variation propagation model for MMP is presented. In which the influence of both locator position and machining error on part quality is considered while, in traditional model, it usually focuses on datum error and fixture error. Coordinate transformation theory is used to reflect the generation and transmission laws of error in the establishment of the model. The concept of deviation matrix is heavily applied to establish an explicit mapping between the geometric deviation of part and the process error sources. In each machining stage, the part deviation is formulized as three separated components corresponding to three different kinds of error sources, which can be further applied to fault identification and design optimization for complicated machining process. An example part for MMP is given out to validate the effectiveness of the methodology. The experiment results show that the model prediction and the actual measurement match well. This paper provides a method to predict part deviation under the influence of fixture error, datum error and machining error, and it enriches the way of quality prediction for MMP.展开更多
A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primit...A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.展开更多
The recently developed hard-magnetic soft(HMS)materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields,e.g.,soft ro...The recently developed hard-magnetic soft(HMS)materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields,e.g.,soft robotics,flexible electronics,and biomedicine.Theoretical investigations on large deformations of HMS structures are significant foundations of their applications.This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar deformations of flexible beams.A so-called quaternion beam model is proposed to break the singularity limitation of the existing geometrically exact(GE)beam model.The singularity-free governing equations for the three-dimensional(3D)large deformations of an HMS beam are first derived,and then solved with the Galerkin discretization method and the trustregion-dogleg iterative algorithm.The correctness of this new model and the utilized algorithms is verified by comparing the present results with the previous ones.The superiority of a quaternion beam model in calculating the complicated large deformations of a flexible beam is shown through several benchmark examples.It is found that the purpose of the HMS beam deformation is to eliminate the direction deviation between the residual magnetization and the applied magnetic field.The proposed new model and the revealed mechanism are supposed to be useful for guiding the engineering applications of flexible structures.展开更多
A new method was proposed for quasi-static deployment analysis of deployable space truss structures. The structure is assumed a rigid assembly, whose constraints are classified as three categories:rigid member constra...A new method was proposed for quasi-static deployment analysis of deployable space truss structures. The structure is assumed a rigid assembly, whose constraints are classified as three categories:rigid member constraint, joint-attached kinematic constraint and boundary constraint. And their geometric constraint equations and derivative matrices are formulated. The basis of the null space and M-P inverse of the geometric constraint matrix are employed to determine the solution for quasi-static deployment analysis. The influence introduced by higher terms of constraints is evaluated subsequently. The numerical tests show that the new method is efficient.展开更多
For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy,including the Willmore and the Helfrich flows,we consider a numerical approach.In this study,we construct a s...For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy,including the Willmore and the Helfrich flows,we consider a numerical approach.In this study,we construct a structure-preserving method based on a discrete variational derivative method.Furthermore,to prevent the vertex concentration that may lead to numerical instability,we discretely introduce Deckelnick’s tangential velocity.Here,a modification term is introduced in the process of adding tangential velocity.This modified term enables the method to reproduce the equations’properties while preventing vertex concentration.Numerical experiments demonstrate that the proposed approach captures the equations’properties with high accuracy and avoids the concentration of vertices.展开更多
We study a fourth order curve flow, which is the gradient flow of a functional describing the shapes of human red blood cells. We prove that for any smooth closed initial curve in ?2, the flow has a smooth solution fo...We study a fourth order curve flow, which is the gradient flow of a functional describing the shapes of human red blood cells. We prove that for any smooth closed initial curve in ?2, the flow has a smooth solution for all time and the solution subconverges to a critical point of the functional.展开更多
A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is devel- oped. The connection between the space-time symmetry and energy-momentum conservation laws of the system is...A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is devel- oped. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space- time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles＇ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler-Lagrange （EL） equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.展开更多
Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the eq...Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations （second-order, fourth-order and sixth- order） into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.展开更多
A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrins...A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto the intrinsic Laplacian of the curvature, second order terms related to the curva-ture, first order terms related to anisotropy and by a given external velocity field. Theevolution is numerically stabilized by an asymptotically uniform tangential redistri-bution of grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numer-ically approximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.展开更多
This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow ...This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.展开更多
We present a general framework for a higher-order spline level-set （HLS） method and apply this to biomolecule surfaces construction. Starting from a first order energy functional, we obtain a general level set formu...We present a general framework for a higher-order spline level-set （HLS） method and apply this to biomolecule surfaces construction. Starting from a first order energy functional, we obtain a general level set formulation of geometric partial differential equation, and provide an efficient approach to solving this partial differential equation using a C2 spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. One example of our HLS method is demonstrated their individual atomic coordinates which is the construction of biomolecule and solvated radii as prerequisites. surfaces （an implicit solvation interface） with展开更多
The author studies the L2 gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any λi ≥ 0 and co, the author obtains a lower bound on the lifespan of th...The author studies the L2 gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any λi ≥ 0 and co, the author obtains a lower bound on the lifespan of the smooth solution, which depends only on the concentration of curvature for the initial surface.展开更多
This paper proposes a new neural algorithm to perform the segmentation of an observed scene into regions corresponding to different moving objects byanalyzing a time-varying images sequence. The method consists of a c...This paper proposes a new neural algorithm to perform the segmentation of an observed scene into regions corresponding to different moving objects byanalyzing a time-varying images sequence. The method consists of a classificationstep, where the motion of small patches is characterized through an optimizationapproach, and a segmentation step merging neighboring patches characterized bythe same motion. Classification of motion is performed without optical flow computation, but considering only the spatial and temporal image gradients into anappropriate energy function minimized with a Hopfield-like neural network givingas output directly the 3D motion parameter estimates. Network convergence is accelerated by integrating the quantitative estimation of motion parameters with aqualitative estimate of dominant motion using the geometric theory of differentialequations.展开更多
基金supported by the National Natural Science Foundation of China(51075147)863 Project(2012AA041803)
文摘At present the mechanical model of the interac- tion between a disc cutter and rock mainly concerns indentation experiment, linear cutting experiment and tunnel boring machine （TBM） on-site data. This is not in line with the actual rock-breaking movement of the disc cutter and impedes to some extent the research on the rock-breaking mechanism, wear mechanism and design theory. Therefore, our study focuses on the interaction between the slantingly installed disc cutter and rock, developing a model in accordance with the actual rock-breaking movement. Displacement equations are established through an analysis of the velocity vector at the rock-breaking point of the disc cutter blade; the func- tional relationship between the displacement parameters at the rock-breaking point and its rectangular coordinates is established through an analysis of micro-displacement vectors at the rock-breaking point, thus leading to the geometric equations of rock deformation caused by the slantingly installed disc cutter. Considering the basically linear relationship between the cutting force of disc cutters and the rock deformation before and after the leap break of rock, we express the constitutive relations of rock deformation as generalized Hooke＇s law and analyze the effect of the slanting installa- tion angle of disc cutters on the rock-breaking force. This will, as we hope, make groundbreaking contributions to the development of the design theory and installation practice of TBM.
基金Supported by National Natural Science Foundation of China(Grant Nos.51205286,51275348)
文摘Product variation reduction is critical to improve process efficiency and product quality, especially for multistage machining process（MMP）. However, due to the variation accumulation and propagation, it becomes quite difficult to predict and reduce product variation for MMP. While the method of statistical process control can be used to control product quality, it is used mainly to monitor the process change rather than to analyze the cause of product variation. In this paper, based on a differential description of the contact kinematics of locators and part surfaces, and the geometric constraints equation defined by the locating scheme, an improved analytical variation propagation model for MMP is presented. In which the influence of both locator position and machining error on part quality is considered while, in traditional model, it usually focuses on datum error and fixture error. Coordinate transformation theory is used to reflect the generation and transmission laws of error in the establishment of the model. The concept of deviation matrix is heavily applied to establish an explicit mapping between the geometric deviation of part and the process error sources. In each machining stage, the part deviation is formulized as three separated components corresponding to three different kinds of error sources, which can be further applied to fault identification and design optimization for complicated machining process. An example part for MMP is given out to validate the effectiveness of the methodology. The experiment results show that the model prediction and the actual measurement match well. This paper provides a method to predict part deviation under the influence of fixture error, datum error and machining error, and it enriches the way of quality prediction for MMP.
文摘A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.
基金Project supported by the National Key Research and Development Program of China(No.2018YFA0703200)the National Natural Science Foundation of China(Nos.52205594 and51820105008)+1 种基金the China National Postdoctoral Program for Innovative Talents(No.BX20220118)the China Postdoctoral Science Foundation(No.2021M701306)。
文摘The recently developed hard-magnetic soft(HMS)materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields,e.g.,soft robotics,flexible electronics,and biomedicine.Theoretical investigations on large deformations of HMS structures are significant foundations of their applications.This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar deformations of flexible beams.A so-called quaternion beam model is proposed to break the singularity limitation of the existing geometrically exact(GE)beam model.The singularity-free governing equations for the three-dimensional(3D)large deformations of an HMS beam are first derived,and then solved with the Galerkin discretization method and the trustregion-dogleg iterative algorithm.The correctness of this new model and the utilized algorithms is verified by comparing the present results with the previous ones.The superiority of a quaternion beam model in calculating the complicated large deformations of a flexible beam is shown through several benchmark examples.It is found that the purpose of the HMS beam deformation is to eliminate the direction deviation between the residual magnetization and the applied magnetic field.The proposed new model and the revealed mechanism are supposed to be useful for guiding the engineering applications of flexible structures.
基金National Natural Science Foundation ofChina(No.10 10 2 0 10 )
文摘A new method was proposed for quasi-static deployment analysis of deployable space truss structures. The structure is assumed a rigid assembly, whose constraints are classified as three categories:rigid member constraint, joint-attached kinematic constraint and boundary constraint. And their geometric constraint equations and derivative matrices are formulated. The basis of the null space and M-P inverse of the geometric constraint matrix are employed to determine the solution for quasi-static deployment analysis. The influence introduced by higher terms of constraints is evaluated subsequently. The numerical tests show that the new method is efficient.
基金This work was supported by JSPS KAKENHI Grant Nos.19K14590,21K18301,Japan.
文摘For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy,including the Willmore and the Helfrich flows,we consider a numerical approach.In this study,we construct a structure-preserving method based on a discrete variational derivative method.Furthermore,to prevent the vertex concentration that may lead to numerical instability,we discretely introduce Deckelnick’s tangential velocity.Here,a modification term is introduced in the process of adding tangential velocity.This modified term enables the method to reproduce the equations’properties while preventing vertex concentration.Numerical experiments demonstrate that the proposed approach captures the equations’properties with high accuracy and avoids the concentration of vertices.
基金supported by Postdoctoral Science Foundation of China, National Natural Science Foundationof China (No. 10631020, 10871061)the Grant for PhD Program of Ministry of Education of China
文摘We study a fourth order curve flow, which is the gradient flow of a functional describing the shapes of human red blood cells. We prove that for any smooth closed initial curve in ?2, the flow has a smooth solution for all time and the solution subconverges to a critical point of the functional.
基金This research was supported by the Na- tional Magnetic Confinement Fusion Energy Research Project （Grant Nos. 2015GB111003 and 2014GB124005）, the National Natural Science Foundation of China （Grant Nos. NSFC- 11575185, 11575186, and 11305171）, JSPS-NRF-NSFC A3 Fore- sight Program （Grant No. 11261140328）, the Key Research Pro- gram of Frontier Sciences CAS （QYZDB-SSW-SYS004）, Geo- Algorithmic Plasma Simulator （GAPS） Project, and the National Magnetic Confinement Fusion Energy Research Project （Grant No. 2013GB111002B）.
文摘A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is devel- oped. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space- time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles＇ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler-Lagrange （EL） equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.
基金supported in part by NSFC under the Grant 60773165NSFC Key Project under the Grant 10990013National Key Basic Research Project of China under the Grant 2004CB318000
文摘Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations （second-order, fourth-order and sixth- order） into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.
基金This work was supported by grants:VEGA 1/0269/09,APVV-0351-07,APVV-RPEU-0004-07(K.Mikula and M.Balazovjech)and APVV-0247-06(D.Sevcovic).
文摘A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto the intrinsic Laplacian of the curvature, second order terms related to the curva-ture, first order terms related to anisotropy and by a given external velocity field. Theevolution is numerically stabilized by an asymptotically uniform tangential redistri-bution of grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numer-ically approximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.
基金Partially supported by NSF grants and a Simons fund.
文摘This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.
基金Bajaj is supported in part by NSF of USA under Grant No. CNS-0540033NIH under Grant Nos. P20-RR020647, R01- EB00487, R01-GM074258, R01-GM07308.+2 种基金Xu and Zhang are supported by the National Natural Science Foundation of China under Grant No. 60773165the National Basic Research 973 Program of China under Grant No. 2004CB318000. Zhang is also supported by Beijing Educational Committee Foundation under Grant No. KM200811232009.
文摘We present a general framework for a higher-order spline level-set （HLS） method and apply this to biomolecule surfaces construction. Starting from a first order energy functional, we obtain a general level set formulation of geometric partial differential equation, and provide an efficient approach to solving this partial differential equation using a C2 spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. One example of our HLS method is demonstrated their individual atomic coordinates which is the construction of biomolecule and solvated radii as prerequisites. surfaces （an implicit solvation interface） with
基金Project supported by the National Natural Science Foundation of China(No.11026121)the TrainingProgramme Foundation for the Excellent Talents of Beijing(No.2012D005003000004)
文摘The author studies the L2 gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any λi ≥ 0 and co, the author obtains a lower bound on the lifespan of the smooth solution, which depends only on the concentration of curvature for the initial surface.
文摘This paper proposes a new neural algorithm to perform the segmentation of an observed scene into regions corresponding to different moving objects byanalyzing a time-varying images sequence. The method consists of a classificationstep, where the motion of small patches is characterized through an optimizationapproach, and a segmentation step merging neighboring patches characterized bythe same motion. Classification of motion is performed without optical flow computation, but considering only the spatial and temporal image gradients into anappropriate energy function minimized with a Hopfield-like neural network givingas output directly the 3D motion parameter estimates. Network convergence is accelerated by integrating the quantitative estimation of motion parameters with aqualitative estimate of dominant motion using the geometric theory of differentialequations.
