Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying...Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.展开更多
In this study, finite element analysis （FEA） has been used to investigate the effects of different Laval nozzle throat sizes on supersonic molecular beam. The simulations indicate the Mach numbers of the molecular s...In this study, finite element analysis （FEA） has been used to investigate the effects of different Laval nozzle throat sizes on supersonic molecular beam. The simulations indicate the Mach numbers of the molecular stream peak at different positions along the center axis of the beam, which correspond to local minimums of the molecular densities. With the increase of the throat diameter, the first peak of the Mach number increases first and then decreases, while that of the molecular number density increases gradually. Moreover, both first peaks shift progressively away from the throat. At the last part, we discuss the possible applications of our FEA approach to solve some crucial problems met in modern transportations.展开更多
Purpose: To improve the accuracy in megavoltage photon beam dose calculation in CBCT-based radiation treatment (RT) plans, using a kilovoltage cone-beam computed tomography (CBCT)-to-density-step (CBCT-SF) function. M...Purpose: To improve the accuracy in megavoltage photon beam dose calculation in CBCT-based radiation treatment (RT) plans, using a kilovoltage cone-beam computed tomography (CBCT)-to-density-step (CBCT-SF) function. Materials and Methods: The CBCT-SF table is constructed from differential histograms of the voxel values of CBCT and Fan-beam CT (FBCT). From the CBCT histograms, frequency peaks representing air, lung, soft tissue and bone are observed and their widths in CT numbers are assigned to the lower and higher bounds of the steps in the CBCT-SF. The CBCT-SF is entered into a planning system as an alternative to the clinical CT-to-density table. The CT image sets studied in this work consist of FBCT and CBCT scans of three patients: a prostate cancer patient, a lung cancer patient and a head and neck patient;and of a humanoid phantom at sections of the pelvis, the thorax and the head. Deformable image registration is used to map the patient FBCT scans to the corresponding CBCT images to minimize anatomical variations. Three-dimensional conformal radiotherapy (3D-CRT) and intensity-modulated radiotherapy (IMRT) plans are made on the FBCT image sets of the patients and the phantom. The plans are recalculated on the CBCT scans using both the conventional CT-to-density table and the CBCT-SF. Dose calculations on the CBCT images and FBCT images are compared using dose differences, distance to agreement (DTA), Gamma analyses and dose volume histogram (DVH) analyses. Results: The results show that IMRT plans optimized using CBCT scans and FBCT scans agree dosimetrically within 1% when the CBCT-SF is used for the CBCT-based plans, including thoracic IMRT plan. In contrast, up to 5% dose difference is observed between IMRT plans optimized on FBCT scans and CBCT scans for thoracic cases if conventional CT-to-density table is used on CBCT images. Conclusions: The simple stepwise mapping of the CBCT numbers to density using the CBCT-SF resolves the inaccuracies in dose calculations previously reported in CBCT-based RT plans. CBCT-SF can be used in Image-Guided adaptive radiotherapy planning.展开更多
The model and analysis of the cantilever beam adhesion problem under the action of electrostatic force are given. Owing to the nonlinearity of electrostatic force, the analytical solution for this kind of problem is...The model and analysis of the cantilever beam adhesion problem under the action of electrostatic force are given. Owing to the nonlinearity of electrostatic force, the analytical solution for this kind of problem is not available. In this paper, a systematic method of generating polynomials which are the exact beam solutions of the loads with di?erent distributions is provided. The polynomials are used to approximate the beam displacement due to electrostatic force. The equilibrium equation o?ers an answer to how the beam deforms but no information about the unstuck length. The derivative of the functional with respect to the unstuck length o?ers such information. But to compute the functional it is necessary to know the beam deformation. So the problem is iteratively solved until the results are converged. Galerkin and Newton-Raphson methods are used to solve this nonlinear problem. The e?ects of dielectric layer thickness and electrostatic voltage on the cantilever beam stiction are studied. The method provided in this paper exhibits good convergence. For the adhesion problem of cantilever beam without electrostatic voltage, the analytical solution is available and is also exactly matched by the computational results given by the method presented in this paper.展开更多
基金Supported by the National Natural Science Foundation of China(51276017)
文摘Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.
基金financially supported by the Science Foundation for International Cooperation of Sichuan Province (2014HH0016)the Fundamental Research Funds for the Central Universities (SWJTU2014: A0920502051113-10000)National Magnetic Confinement Fusion Science Program (2011GB112001)
文摘In this study, finite element analysis （FEA） has been used to investigate the effects of different Laval nozzle throat sizes on supersonic molecular beam. The simulations indicate the Mach numbers of the molecular stream peak at different positions along the center axis of the beam, which correspond to local minimums of the molecular densities. With the increase of the throat diameter, the first peak of the Mach number increases first and then decreases, while that of the molecular number density increases gradually. Moreover, both first peaks shift progressively away from the throat. At the last part, we discuss the possible applications of our FEA approach to solve some crucial problems met in modern transportations.
文摘Purpose: To improve the accuracy in megavoltage photon beam dose calculation in CBCT-based radiation treatment (RT) plans, using a kilovoltage cone-beam computed tomography (CBCT)-to-density-step (CBCT-SF) function. Materials and Methods: The CBCT-SF table is constructed from differential histograms of the voxel values of CBCT and Fan-beam CT (FBCT). From the CBCT histograms, frequency peaks representing air, lung, soft tissue and bone are observed and their widths in CT numbers are assigned to the lower and higher bounds of the steps in the CBCT-SF. The CBCT-SF is entered into a planning system as an alternative to the clinical CT-to-density table. The CT image sets studied in this work consist of FBCT and CBCT scans of three patients: a prostate cancer patient, a lung cancer patient and a head and neck patient;and of a humanoid phantom at sections of the pelvis, the thorax and the head. Deformable image registration is used to map the patient FBCT scans to the corresponding CBCT images to minimize anatomical variations. Three-dimensional conformal radiotherapy (3D-CRT) and intensity-modulated radiotherapy (IMRT) plans are made on the FBCT image sets of the patients and the phantom. The plans are recalculated on the CBCT scans using both the conventional CT-to-density table and the CBCT-SF. Dose calculations on the CBCT images and FBCT images are compared using dose differences, distance to agreement (DTA), Gamma analyses and dose volume histogram (DVH) analyses. Results: The results show that IMRT plans optimized using CBCT scans and FBCT scans agree dosimetrically within 1% when the CBCT-SF is used for the CBCT-based plans, including thoracic IMRT plan. In contrast, up to 5% dose difference is observed between IMRT plans optimized on FBCT scans and CBCT scans for thoracic cases if conventional CT-to-density table is used on CBCT images. Conclusions: The simple stepwise mapping of the CBCT numbers to density using the CBCT-SF resolves the inaccuracies in dose calculations previously reported in CBCT-based RT plans. CBCT-SF can be used in Image-Guided adaptive radiotherapy planning.
文摘The model and analysis of the cantilever beam adhesion problem under the action of electrostatic force are given. Owing to the nonlinearity of electrostatic force, the analytical solution for this kind of problem is not available. In this paper, a systematic method of generating polynomials which are the exact beam solutions of the loads with di?erent distributions is provided. The polynomials are used to approximate the beam displacement due to electrostatic force. The equilibrium equation o?ers an answer to how the beam deforms but no information about the unstuck length. The derivative of the functional with respect to the unstuck length o?ers such information. But to compute the functional it is necessary to know the beam deformation. So the problem is iteratively solved until the results are converged. Galerkin and Newton-Raphson methods are used to solve this nonlinear problem. The e?ects of dielectric layer thickness and electrostatic voltage on the cantilever beam stiction are studied. The method provided in this paper exhibits good convergence. For the adhesion problem of cantilever beam without electrostatic voltage, the analytical solution is available and is also exactly matched by the computational results given by the method presented in this paper.
