An inverse problem of elastica of a variable-arclength beam subjected to a concentrated load is investigated. The beam is fixed at one end, and can slide freely over a hinge support at the other end. The inverse probl...An inverse problem of elastica of a variable-arclength beam subjected to a concentrated load is investigated. The beam is fixed at one end, and can slide freely over a hinge support at the other end. The inverse problem is to determine the value of the load when the deflection of the action point of the load is given. Based on the elasitca equations and the elliptic integrals, a set of nonlinear equations for the inverse problem are derived, and an analytical solution by means of iterations and Quasi-Newton method is presented. From the results, the relationship between the loads and deflections of the loading point is obtained.展开更多
A tapered rod mounted at one end (base) and subject to a normal force at the other end (tip) is a fundamental structure of continuum mechanics that occurs widely at all size scales from radio towers to fishing rods to...A tapered rod mounted at one end (base) and subject to a normal force at the other end (tip) is a fundamental structure of continuum mechanics that occurs widely at all size scales from radio towers to fishing rods to micro-electromechanical sensors. Although the bending of a uniform rod is well studied and gives rise to mathematical shapes described by elliptic integrals, no exact closed form solution to the nonlinear differential equations of static equilibrium is known for the deflection of a tapered rod. We report in this paper a comprehensive numerical analysis and experimental test of the exact theory of bending deformation of a tapered rod. Given the rod geometry and elastic modulus, the theory yields virtually all the geometric and physical features that an analyst, experimenter, or instrument designer might want as a function of impressed load, such as the exact curve of deformation (termed the elastica), maximum tip displacement, maximum tip deflection angle, distribution of curvature, and distribution of bending moment. Applied experimentally, the theory permits rapid estimation of the elastic modulus of a rod, which is not easily obtainable by other means. We have tested the theory by photographing the shapes of a set of flexible rods of different lengths and tapers subject to a range of impressed loads and using digital image analysis to extract the coordinates of the elastica curves. The extent of flexure in these experiments far exceeded the range of applicability of approximations that linearize the equations of equilibrium or neglect tapering of the rod. Agreement between the measured deflection curves and the exact theoretical predictions was excellent in all but several cases. In these exceptional cases, the nature of the anomalies provided important information regarding the deviation of the rods from an ideal Euler-Bernoulli cantilever, which thereby permitted us to model the deformation of the rods more accurately.展开更多
In this paper,the extremals of curvature energy actions on non-null Frenet curves in 3-dimensional Anti-de Sitter space are studied.We completely solve the Euler-Lagrange equation by quadratures.By using the Killing f...In this paper,the extremals of curvature energy actions on non-null Frenet curves in 3-dimensional Anti-de Sitter space are studied.We completely solve the Euler-Lagrange equation by quadratures.By using the Killing fields,we obtain existence for closed general-ized elastica fully immersed in Anti-de Sitter space H_1~3.展开更多
In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and...In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and operator splitting techniques,the resulting saddle-point problem is solved by a serial of subproblems.To tackle the nonlinear constraints arising in the model,a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution.We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic,real-world and medical images for image denoising,image inpainting and image zooming problems.展开更多
Snap-through phenomenon widely occurs for elastic systems,where the systems lose stability at critical points.Here snapthrough of an elastica under bilateral displacement control at a material point is studied,by rega...Snap-through phenomenon widely occurs for elastic systems,where the systems lose stability at critical points.Here snapthrough of an elastica under bilateral displacement control at a material point is studied,by regarding the whole elastica as two components,i.e.,pinned-clamped elasticas.Specifically,stiffness-curvature curves of two pinned-clamped elasticas are firstly efficiently located based on the second-order mode,which are used to determine the shapes of two components.Similar transformations are used to assemble two components together to form the whole elastica,which reveals four kinds of shapes.One advantage of this way compared with other methods such as the shooting method is that multiple coexisting solutions can be located accurately.O n the load-deflection curves,four branches correspond to four kinds of shapes and first two branches are symmetrical to the last two branches relative to the original point.For the bilateral displacement control,the critical points can only appear at saddle-node bifurcations,which is different to that for the unilateral displacement control.Specifically,one critical point is found on the first branch and two critical points are found on the secondary branch.In addition,the snap-through among different branches can be well explained with these critical points.展开更多
Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construc...Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construct efficient algo-rithms to obtain the minimizers of original high order functionals.In this paper,we propose a new linearized augmented Lagrangian method for Euler’s elastica image denoising model.We detail the procedures of finding the saddle-points of the aug-mented Lagrangian functional.Instead of solving associated linear systems by FFTor linear iterative methods(e.g.,the Gauss-Seidel method),we adopt a linearized strat-egy to get an iteration sequence so as to reduce computational cost.In addition,we give some simple complexity analysis for the proposed method.Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method,and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.展开更多
In this paper,we propose an algorithm based on augmented Lagrangian method and give a performance comparison for two segmentation models that use the L^(1)-and L^(2)-Euler’s elastica energy respectively as the regula...In this paper,we propose an algorithm based on augmented Lagrangian method and give a performance comparison for two segmentation models that use the L^(1)-and L^(2)-Euler’s elastica energy respectively as the regularization for image seg-mentation.To capture contour curvature more reliably,we develop novel augmented Lagrangian functionals that ensure the segmentation level set function to be signed dis-tance functions,which avoids the reinitialization of segmentation function during the iterative process.With the proposed algorithm and with the same initial contours,we compare the performance of these two high-order segmentation models and numerically verify the different properties of the two models.展开更多
基金The project supported by the National Natural Science Foundation of China(10272011)
文摘An inverse problem of elastica of a variable-arclength beam subjected to a concentrated load is investigated. The beam is fixed at one end, and can slide freely over a hinge support at the other end. The inverse problem is to determine the value of the load when the deflection of the action point of the load is given. Based on the elasitca equations and the elliptic integrals, a set of nonlinear equations for the inverse problem are derived, and an analytical solution by means of iterations and Quasi-Newton method is presented. From the results, the relationship between the loads and deflections of the loading point is obtained.
文摘A tapered rod mounted at one end (base) and subject to a normal force at the other end (tip) is a fundamental structure of continuum mechanics that occurs widely at all size scales from radio towers to fishing rods to micro-electromechanical sensors. Although the bending of a uniform rod is well studied and gives rise to mathematical shapes described by elliptic integrals, no exact closed form solution to the nonlinear differential equations of static equilibrium is known for the deflection of a tapered rod. We report in this paper a comprehensive numerical analysis and experimental test of the exact theory of bending deformation of a tapered rod. Given the rod geometry and elastic modulus, the theory yields virtually all the geometric and physical features that an analyst, experimenter, or instrument designer might want as a function of impressed load, such as the exact curve of deformation (termed the elastica), maximum tip displacement, maximum tip deflection angle, distribution of curvature, and distribution of bending moment. Applied experimentally, the theory permits rapid estimation of the elastic modulus of a rod, which is not easily obtainable by other means. We have tested the theory by photographing the shapes of a set of flexible rods of different lengths and tapers subject to a range of impressed loads and using digital image analysis to extract the coordinates of the elastica curves. The extent of flexure in these experiments far exceeded the range of applicability of approximations that linearize the equations of equilibrium or neglect tapering of the rod. Agreement between the measured deflection curves and the exact theoretical predictions was excellent in all but several cases. In these exceptional cases, the nature of the anomalies provided important information regarding the deviation of the rods from an ideal Euler-Bernoulli cantilever, which thereby permitted us to model the deformation of the rods more accurately.
基金Supported by the NSF of China(10671066,10971066)Supported by the Shanghai Leading Academic Discipline Project(B407)
文摘In this paper,the extremals of curvature energy actions on non-null Frenet curves in 3-dimensional Anti-de Sitter space are studied.We completely solve the Euler-Lagrange equation by quadratures.By using the Killing fields,we obtain existence for closed general-ized elastica fully immersed in Anti-de Sitter space H_1~3.
文摘In this paper,a fast algorithm for Euler’s elastica functional is proposed,in which the Euler’s elastica functional is reformulated as a constrained minimization problem.Combining the augmented Lagrangian method and operator splitting techniques,the resulting saddle-point problem is solved by a serial of subproblems.To tackle the nonlinear constraints arising in the model,a novel fixed-point-based approach is proposed so that all the subproblems either is a linear problem or has a closed-form solution.We show the good performance of our approach in terms of speed and reliability using numerous numerical examples on synthetic,real-world and medical images for image denoising,image inpainting and image zooming problems.
基金supported by the National Natural Science Foundation of China(Grants 91648101 and 11972290)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(Grant CX201811)the Fundamental Research Funds for the Central Universities(Grant 3102018zy012).
文摘Snap-through phenomenon widely occurs for elastic systems,where the systems lose stability at critical points.Here snapthrough of an elastica under bilateral displacement control at a material point is studied,by regarding the whole elastica as two components,i.e.,pinned-clamped elasticas.Specifically,stiffness-curvature curves of two pinned-clamped elasticas are firstly efficiently located based on the second-order mode,which are used to determine the shapes of two components.Similar transformations are used to assemble two components together to form the whole elastica,which reveals four kinds of shapes.One advantage of this way compared with other methods such as the shooting method is that multiple coexisting solutions can be located accurately.O n the load-deflection curves,four branches correspond to four kinds of shapes and first two branches are symmetrical to the last two branches relative to the original point.For the bilateral displacement control,the critical points can only appear at saddle-node bifurcations,which is different to that for the unilateral displacement control.Specifically,one critical point is found on the first branch and two critical points are found on the secondary branch.In addition,the snap-through among different branches can be well explained with these critical points.
基金supported by the NNSF of China grants 11526110,11271069,61362036 and 61461032,the 863 Program of China grant 2015AA01A302the Open Research Fund of Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing(2016WICSIP013)the Youth Foundation of Nanchang Institute of Technology(2014KJ021).
文摘Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construct efficient algo-rithms to obtain the minimizers of original high order functionals.In this paper,we propose a new linearized augmented Lagrangian method for Euler’s elastica image denoising model.We detail the procedures of finding the saddle-points of the aug-mented Lagrangian functional.Instead of solving associated linear systems by FFTor linear iterative methods(e.g.,the Gauss-Seidel method),we adopt a linearized strat-egy to get an iteration sequence so as to reduce computational cost.In addition,we give some simple complexity analysis for the proposed method.Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method,and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.
基金X.C.Tai was supported by the startup grant at Hong Kong Baptist University,grant RG(R)-RC/17-18/02-MATH and FRG2/17-18/033.
文摘In this paper,we propose an algorithm based on augmented Lagrangian method and give a performance comparison for two segmentation models that use the L^(1)-and L^(2)-Euler’s elastica energy respectively as the regularization for image seg-mentation.To capture contour curvature more reliably,we develop novel augmented Lagrangian functionals that ensure the segmentation level set function to be signed dis-tance functions,which avoids the reinitialization of segmentation function during the iterative process.With the proposed algorithm and with the same initial contours,we compare the performance of these two high-order segmentation models and numerically verify the different properties of the two models.
