The use of a Monte Carlo method to calculate the average solid angle subtended by a detector to source in a non-parallel plane

In a previous scientific note, a short computer program for a personal computer was described that calculated the average solid angle subtended by a circular or rectangular detector window to a circular or rectangular source in a parallel plane by a Monte Carlo method. This note describes the development of the program for conditions where the detector window is not parallel to the plane of the source, and in particular looks at a method of analysing orientations that relates straightforwardly to practical measurement.     

A Monte Carlo method to calculate the average solid angle subtended by a detector to an ellipsoidal surface or solid source at any position and orientation

This note discusses and implements an algorithm for modelling a uniform distribution over a scalene ellipsoidal surface or in a solid so that a Monte Carlo method, suitable for a personal computer, can be used to calculate the average solid angle subtended by a right cylinder or rectangular cuboid detector to the ellipsoidal surface or solid at any position and orientation. The distributions over the surface include 2π and 4π surface sources. The algorithm is developed for truncated ellipsoidal solid and surface sources (where the planes of truncation are restricted to those normal to one of the semi-axes). Plane (window) detectors can be modelled by setting the detector height to zero. The discussion includes the estimation of confidence limits. The current performance of personal computers makes it realistic to achieve accuracies of typically better than 0.05%.     

A Monte Carlo method to calculate the average solid angle subtended by a right cylinder to a source that is circular or rectangular, plane or thick, at any position and orientation

This note describes the development of a short program for a personal computer to calculate the solid angle subtended by a right cylinder detector to a circular or rectangular, plane or thick source at any position and orientation to the cylinder. The program also calculates the number of hits on the cylinder side and on each end, and the average pathlength through the detector volume (assuming no scattering or absorption). The current performance of personal computers makes it realistic to model the order of 10(9) simulations of radiation emission and achieve accuracies of solid angle estimates typically better than 0.03%.     

Measurement of propagation time dispersion in a scintillator

One contribution to the time resolution of a scintillation detector is the signal time spread due to path length variations of the detected photons from a point source. In an experimental study, a rectangular scintillator was excited by means of a fast pulsed ultraviolet laser at different positions along its longitudinal axis. Timing measurements with a photomultiplier tube in a detection plane displaced from the scintillator end face showed a correlation between signal time and tube position indicating only a small distortion of photon angles during transmission. The data is in good agreement with a Monte Carlo simulation used to compute the average photon angle with respect to the detection plane and the average propagation time. Limitations on timing performance that arise from propagation time dispersion are expected for long and thin scintillators used in future particle identification systems.     

Light diffusion through a turbid parallelepiped

Solutions of the diffusion approximation to the radiative transport equation are derived for a turbid (rectangular) parallelepiped using the method of image sources and applying extrapolated boundary conditions. The derived solutions are compared with Monte Carlo simulations in the steady-state and time domains. It is found that the diffusion theory is in good agreement with Monte Carlo simulations provided that the light is detected sufficiently far from the incident beam. Applications of the derived solutions, including the determination of the optical properties of the turbid parallelepiped, are discussed.     

An efficient solid angle sampling technique using bounding volume approach

Abstract

Solid angle sampling is an important variance‐reduced technique when solving global illumination problems by Monte Carlo methods. In this paper, we present an efficient solid angle sampling technique where a paraboloidal luminaire is taken as a case study. The efficiency of our technique is due to the fact that we employ a tight bounding volume to approximate the solid angle. Our technique includes three processes. The construction process builds a bounding volume. It is a convex, frustum‐like polyhedron with a compromise between the tightness and the vertex numbers. The projection process approximates the solid angle as a convex spherical polygon on a unit hemisphere. Finally, the triangulation process triangulates the convex spherical polygon into spherical triangles for stratified sampling. We analyzed our technique in Monte Carlo direct lighting and Monte Carlo path tracing rendering algorithms. The results show that our technique provides up to 90% sampling efficiency. The significance of the proposed technique is that solid angle sampling, from being not possible for the paraboloidal luminaire, is now feasible. In addition, this technique is efficient for sampling, and it is also applicable to other types of luminaires, such as cylindrical and conic luminaires.     

Crossed source-detector geometry for a novel spray diagnostic: Monte Carlo simulation and analytical results

Sprays and other industrially relevant turbid media can be quantitatively characterized by light scattering. However, current optical diagnostic techniques generate errors in the intermediate scattering regime where the average number of light scattering is too great for the single scattering to be assumed, but too few for the diffusion approximation to be applied. Within this transitional single-to-multiple scattering regime, we consider a novel crossed source-detector geometry that allows the intensity of single scattering to be measured separately from the higher scattering orders. We verify Monte Carlo calculations that include the imperfections of the experiment against analytical results. We show quantitatively the influence of the detector numerical aperture and the angle between the source and the detector on the relative intensity of the scattering orders in the intermediate single-to-multiple scattering regime. Monte Carlo and analytical calculations of double light-scattering intensity are made with small particles that exhibit isotropic scattering. The agreement between Monte Carlo and analytical techniques validates use of the Monte Carlo approach in the intermediate scattering regime. Monte Carlo calculations are then performed for typical parameters of sprays and aerosols with anisotropic (Mie) scattering in the intermediate single-to-multiple scattering regime.     

Estimation of flux distributions with Monte Carlo functional expansion tallies

Monte Carlo methods provide a powerful technique for estimating the average radiation flux in a volume (or across a surface) in cases where analytical solutions may not be possible. Unfortunately, Monte Carlo simulations typically provide only integral results and do not offer any further details about the distribution of the flux with respect to space, angle, time or energy. In the functional expansion tally (FET) a Monte Carlo simulation is used to estimate the functional expansion coefficients for flux distributions with respect to an orthogonal set of basis functions. The expansion coefficients are then used in post-processing to reconstruct a series approximation to the true distribution. Discrete event FET estimators are derived and their application in estimating radiation flux or current distributions is demonstrated. Sources of uncertainty in the FET are quantified and estimators for the statistical and truncation errors are derived. Numerical results are presented to support the theoretical development.     

Analysis of electrostatic MEMS using meshless local Petrov–Galerkin (MLPG) method

We analyze electrostatic deformations of rectangular, annular circular, solid circular, and elliptic micro-electromechanical systems (MEMS) by modeling them as elastic membranes. The nonlinear Poisson equation governing their deformations is solved numerically by the meshless local Petrov–Galerkin (MLPG) method. A local symmetric augmented weak formulation of the problem is introduced, and essential boundary conditions are enforced by introducing a set of Lagrange multipliers. The trial functions are constructed by using the moving least-squares approximation, and the test functions are chosen from a space of functions different from the space of trial solutions. The resulting nonlinear system of equations is solved by using the pseudoarclength continuation method. Presently computed values of the pull-in voltage and the maximum pull-in deflection for the rectangular and the circular MEMS are found to agree very well with those available in the literature; results for the elliptic MEMS are new.     

Light diffusion model for determination of optical properties of rectangular parallelepiped highly scattering media

A method to determine the absorption and reduced scattering coefficients of rectangular parallelepiped highly scattering media from frequency-domain photon migration measurements is presented. An analytical model for photon diffusion propagation in the rectangular parallelepiped media is established using the method of images and extrapolated boundary conditions. This present technique has simplicity, accuracy, and rapid computability as compared with the Monte Carlo or finite element methods. The theoretical predictions are verified with experimental measurements using a white polyacetal resin, and the errors introduced by using the slab geometry for the optical property determination are identified.     

A Monte Carlo based development of a cavity theory for solid state detectors irradiated in electron beams

Recent Monte Carlo simulations have shown that the assumption in the small cavity theory (and the extension of the small cavity theory by Spencer-Attix) that the cavity does not perturb the electron fluence is seriously flawed. For depths beyond dmax not only is there a significant difference between the energy spectra in the medium and in the solid cavity material but there is also a significant difference in the number of low-energy electrons which cannot travel across the solid cavity and hence deposit their dose in it (i.e. stopper electrons whose residual range is less than the cavity thickness). The number of these low-energy electrons that are not able to travel across the solid state cavity increases with depth and effective thickness of the detector. This also invalidates the assumption in the small cavity theory that most of the dose deposited in a small cavity is delivered by crossers. Based on Monte Carlo simulations, a new cavity theory for solid state detectors irradiated in electron beams has been proposed as: Dmed(p) = meanDdet(p) x s(med,det)S-A x gamma(p)c x S(T) where Dmed(p) is the dose to the medium at point p. MeanDdet(p) is the average detector dose to the same point, s(med,det)S-A is the Spencer-Attix mass collision stopping power ratio of the medium to the detector material, gamma(P)c is the electron fluence perturbation correction factor and S(T) is a stopper-to-crosser correction factor to correct for the dependence of the stopper-to-crosser ratio on depth and the effective cavity size. Monte Carlo simulations have been computed for all the terms in this equation. The new cavity theory has been tested against the Spencer-Attix cavity equation as the small cavity limiting case and also Monte Carlo simulations.     

Stochastic plastic analysis of a shell

A method is proposed for the inclusion of stochastic parameters, using a Monte Carlo simulation technique, in the lower bound plastic analysis of a structural continuum by non-linear mathematical programming. The particular example chosen is a circular, cylindrical concrete silo subject to loading by its contents.     

Analytical differential cross section of electron elastically scattered by solid targets in the energy range up to 100 keV

In this paper, we propose a new approach to determine the adjustable parameters of differential elastic cross-section of Bentabet et al. approximation. The resultant approximation is an analytical expression that can be used to study the interaction of electron beams with solid targets. This approximation is applied for the energy range up to 100 keV, to calculate some quantities such as: the mean penetration depths, the mean number of wide angle collisions and the backscattering coefficient (BSC) of Al, Cu, Ag and Au semi-infinite solid targets. BSC was calculated by using both the Monte Carlo method and the Vicanek and Urbassek analytical model. The obtained results are compared with the experiment and good agreement is remarked.     

A Comparison of Various Patterns of Three‐Dimensional Strain Rosettes

Abstract: The strain tensor, principal strains and precision of the estimates of these values are derived for a range of different layouts of three‐dimensional strain rosettes. These values are based on the Monte Carlo technique applied to experimental work which was carried out on transducers tested in different laboratories. The estimates of precision are determined theoretically and compared with results based on experimental findings. A new design of a miniature tri‐rectangular tetrahedral rosette was manufactured and tested. Results suggest that this transducer does not perform as well as the rectangular patterns.     

氡绝对测量装置的小立体角计算方法研究

为计算确定氡绝对测量装置的小立体角,研究了准直孔半径、氡源半径到准直器距离等相关尺寸参数的测量方法与结果,并基于这些尺寸参数的精确测量,建立了Geant4模型,确定冷凝氡源的探测立体角;采用文献报道的参考方法对模拟计算结果进行了验证评估,结果表明模拟方法是可靠的,该装置在典型探测条件下的冷凝氡源探测的小立体角为0.015 83 Sr,相对标准不确定度仅为6.6×10-6 Sr.     

Calculation of shape factors by means of the equal energy partitioning method and its modification

Abstract

The present study develops a method called the energy equal partitioning method. This method divides all the solid angles of a radiating source properly so that the energy carried by the beams within each small range of the solid angle are equal. The fraction of energy beams arriving at the wall surface is then examined to obtain the shape factor. This method can be efficiently applied to complex geometries. Comparing the rate of convergence and the cost of computing time shows that the shape factor obtained by the present method is better than that obtained by the Monte Carlo method. The influence of the geometric configuration on the accuracy of the shape factor is also investigated. The shape factor obtained by this method is modified by using the original shape factor as a weighting factor. An iteration procedure is developed. The resulting shape factors satisfy the reciprocity law and energy conservation law and the degree of closeness to the exact shape factors is found to be better.     

Plane elasticity problems by barycentric rational interpolation collocation method and a regular domain method

Barycentric rational interpolation collocation method (BRICM) for solving plane elasticity problems with high accuracy is presented. The plane elasticity problems on a circular or rectangular domain can be solved directly by BRICM. Embedded the irregular domain into a regular (circular or rectangular) domain, the governing equations of plane elasticity on regular domain are discretized by the differentiation matrices based on barycentric rational interpolation to form a system of algebraic equations. Discrete boundary conditions are obtained using barycentric rational interpolation. The irregular boundary conditions are imposed by the additional method to form an over-constraint linear system of algebraic equations. Numerical experiments are presented to illustrate the efficiency and high computing precision of proposed method.     

基于FVM数值分析的海工混凝土结构耐久可靠度Monte Carlo模拟

为研究海工混凝土结构耐久性设计及寿命预测的可靠度方法,分析了耐久性设计与寿命预测的半理论、半经验解析模型方法,讨论了截面形状等条件对氯离子扩散传输的影响。该文研究应用有限体积法(FVM)分析求解混凝土中氯离子含量和蒙特卡罗(Monte Carlo)模拟求解耐久失效概率,分析海工混凝土结构耐久可靠度。这种方法可简称FVM-MC方法。在FVM-MC方法中,首先采用FVM方法求解混凝土中的氯离子含量,然后采用Monte Carlo方法对失效概率进行模拟求解。验证分析表明,FVM-MC方法分析模拟精度高,是海工混凝土结构可靠度分析的一种可靠方法。计算结果表明,截面形状等对混凝土结构的耐久可靠度具有显著的影响,采用Fick第二定律解析解的半理论、半经验模型,由于没有考虑截面形状效应,增加了矩形截面构件的耐久失效风险;圆形截面具有近似一维的扩散传输特点,在条件相同情况下,其构件耐久可靠度显著地高于方形或矩形截面的构件。     

Neutron shielding verification measurements and simulations for a 235-MeV proton therapy center

The neutron shielding at the Massachusetts General Hospital's 235-MeV proton therapy facility was investigated with measurements, analytical calculations, and realistic three-dimensional Monte Carlo simulations. In 37 of 40 cases studied, the analytical calculations predicted higher neutron dose equivalent rates outside the shielding than the measured, typically by more than a factor of 10, and in some cases more than 100. Monte Carlo predictions of dose equivalent at three locations are, on average, 1.1 times the measured values. Except at one location, all of the analytical model predictions and Monte Carlo simulations overestimate neutron dose equivalent.     

Neutral points in an atmosphere-ocean system. 1: Upwelling light field

We have developed a Monte Carlo code that utilizes the complete Stokes vector to examine the structure of the degree of linear polarization in the complete observable solid angle at any level in an atmosphere-ocean system. By performing these calculations we are able to compute the positions of neutral points in the upwelling light above and beneath the ocean surface. The locations of these points in a single-scatter calculation and a Monte Carlo treatment are shown for various conditions. The presence of aerosols in the atmosphere and hydrosols in the ocean was found to have an effect on the location of these neutral points.