Beam Model Research Articles

Dosimetric comparison of M6 CyberKnife plans optimized with Precision and RayStation 12A treatment planning systems.

Treatment planning for CyberKnife (CK) (Accuray, USA) can be performed with Precision (Accuray, USA) or RayStation (RS) (RaySearch Laboratories, Sweden) treatment planning systems (TPS). RaySearch recently released a new version of the CK module in RS 12A. The objective of the study was to compare plan quality between RS 12A and Precision. Fifty nine plans were optimized with both TPS and compared; 39 were for brain metastases and 20 were for vertebral metastases. To avoid bias in plan comparison, Precision plans were recomputed in RS with the dose algorithm and beam model of RS, and then compared to RS plans. The comparison was divided into 3 parts in order to reflect the potential of RS and the differences with Precision, in terms of technical aspects of delivery efficiency and dose distribution. We compared the dose to the target and to the organs at risk (OAR), the conformity index (CI), the gradient, as well as the number of monitor units (MU), and the number of beams and nodes. Finally, a global plan quality index (PQI) was calculated. RS plans showed an equivalent target coverage for brain metastases but worse for vertebrae. OAR sparing was improved in RS but with a lower CI compared to Precision. Using an appropriate planning methodology in RS, plans with comparable quality to Precision could be obtained, but at the cost of a longer optimization time. The PQI obtained with RS was better than Precision, except for some brain cases. RS is an adequate alternative for CK planning as it is possible to obtain plan quality comparable to Precision. However, the optimization time is longer compared to Precision and more attention must be paid to the choice of the initial conditions in terms of the number of beams and nodes.

Shear capacity model of RC beams based on MCMC approach and reliability analysis

The research question revolves around accurately predicting the shear capacity of RC beams and enhancing the reliability of such prediction models. This study aims to refine the accuracy of the RC beam shear capacity model. To augment the safety margin, the enhanced model underwent further calibration through a reliability analysis. The shear strength models, rooted in Bayesian theory’s Markov Chain Monte Carlo (MCMC) method and the Modified Compressive Field Theory (MCFT), were introduced. A research database was curated from test data of 782 RC beams sets under shear stress. From this database, six mechanism models were juxtaposed with the MCMC models for comparison. To meet the desired reliability indexes ([Formula: see text] = 3.2, 3.7, 4.0) in these models, resistance factors ([Formula: see text]) were incorporated. Both the Monte Carlo and First Order Second Moment (FOSM) methods were employed for the reliability analysis, considering inherent model errors and the probability distribution of primary variables. The effect of different elements on the target reliability index was assessed using a tornado diagram. Findings showed that the refined shear model displayed minimal variance when benchmarked against models from existing literature and national codes, yielding a Vtest/ Vcal ratio close to 1.0. The shear components aligned with established mechanical mechanism. Depending on various operational conditions and load combinations, resistance factors fluctuated from 0.377 to 0.516 in the RC beams without stirrup, while resistance factors fluctuated from 0.479 to 0.621 in the RC beams with stirrup. Key determinants, such as the inherent uncertainty in computing model, the live loads, and the cylinder compression strength, significantly influenced the reliability index.

Using a Piecewise Linear Spring to Approximate an Essentially Nonlinear Spring: Design and Validation

This study develops a procedure for designing a piecewise linear spring (PLS) to approximate an essentially nonlinear spring (ENS). The PLS is constructed with a cantilever beam constrained by a pair of single- or double-stop blocks. The design begins by determining the restoring force of the desired ENS using the equivalent stiffness which quantifies the characteristics of a cubic polynomial. Then, based on the force-deflection model of a cantilever beam with an overhang, the configuration parameters for single- and double-stop blocks are determined through a least squares optimization. The numerical simulation demonstrates that the PLS with double-stop blocks approximates the desired ENS behaviors better in terms of the restoring force, potential energy, and instantaneous frequency transition. An experiment apparatus with four tunable stop blocks is developed to validate the numerical simulation results. The static experimental tests verify the accuracy of the analytical model. The dynamic experimental tests show that within the achievable range of displacement, the PLSs behave similarly to the ENS. However, the maximum displacement is smaller than the designed one due to an insufficient exciting force. To address this issue, the desired displacement range is reduced by half. With the redesigned PLSs, the improved experimental results are obtained.

Framework for Regional Seismic Risk Assessments of Groups of Tall Buildings

ABSTRACTA novel probabilistic Monte Carlo‐based framework to conduct regional seismic risk assessments using simplified continuous models is proposed. The hazard at rock outcrop is defined by response spectral ordinates, which are simulated to account for spatial correlation and correlation across different periods simultaneously. For sites on firm and soft soils, a simplified site response analysis using a one‐dimensional continuous non‐uniform shear beam model is used to transform the hazard at rock outcrops to the hazard at all sites of interest. Uncertainty in the soil properties at each site is explicitly considered. Response spectra at each site are computed at the principal orientations of each building using recently proposed directionality models that permit the estimation of the seismic hazard at specific orientations. A one‐dimensional continuous coupled shear‐flexural beam model is used to simulate building dynamic properties accounting for modeling uncertainty and to obtain building responses for each building. The parameters of each model only require information on the building height and the lateral resisting system. All relevant uncertainties associated with each module of the framework are explicitly incorporated and propagated. Finally, a case study of tall buildings in San Francisco subjected to a magnitude 7.0 earthquake on the Hayward Fault is presented to illustrate how the framework can be implemented.

Study of the well-posedness and decay rates for Rao–Nakra sandwich beam models subject to a single internal infinite memory and Dirichlet–Neumann boundary conditions

Study of the well-posedness and decay rates for Rao–Nakra sandwich beam models subject to a single internal infinite memory and Dirichlet–Neumann boundary conditions

Integrated Dynamic Analysis of Thin-Walled Beams: Coupled Bidirectional Bending, Torsion, and Axial Vibrations Under Axial Loads

This paper proposes a beam model integrating the Timoshenko beam theory with Vlasov beam theory to capture the coupled behavior of bidirectional bending, torsion, and axial vibration in thin-walled beams subjected to axial loads. Our model incorporates the effects of shear deformation, rotational inertia, and axial loads, offering a comprehensive approach to complex dynamic behaviors. By utilizing Hamilton’s principle, we derived a complete set of coupled dynamic equations and boundary conditions. The highlight of this model is its capacity to accurately predict the dynamic response of thin-walled beams under multifaceted loading conditions, surpassing traditional models by integrating coupled axial vibrations. This research significantly advances the understanding of the dynamic behavior of thin-walled beams, providing a precise analytical tool for structural design and safety assessment. The robustness and accuracy of the proposed model were validated through extensive theoretical analysis and empirical validation, equipping engineers with critical insights to optimize the design of engineering structures subjected to complex dynamic loads.

The Behavior of Elbow Elements at Pure Bending Applications Compared to Beam and Shell Element Models

Abstract This paper studies the response of ELBOW31 and ELBOW31B element types under pure bending conditions, using shell and beam element models for benchmarking. Various model lengths are evaluated, showing that a model length of six pipe diameters exhibits a hardening effect when total strain exceeds 3.5%, though a strain up to 1% is deemed sufficient for pipeline design. The study examines the effects of ovality modes and boundary conditions such as NOWARP and NOOVAL on the bending response. ELBOW31 with one or two ovality modes yields accurate results, while additional ovality modes or zero ovality mode can lead to overprediction of the elastic bending moment capacity. The introduction of the NOWARP condition enhances the accuracy of the ELBOW31 model, while the NOOVAL condition alone produces unrealistic results. The simplified ELBOW31B model shows good agreement with the ELBOW31-NOWARP model but similarly overpredicts the bending moment when zero ovality mode is used. The study also finds that Poisson's ratio and model length have no significant impact on the bending response when no restrictions are applied. Additional analyses, as presented in Appendices A and B, highlight the importance of D/t ratios in pipeline performance. A D/t ratio of 20 offers a stiffer response with reduced ovalization, while a D/t ratio of 50 results in greater flexibility and increased ovalization. These findings provide valuable insights for the selection of element types, boundary conditions, and D/t ratios in robust pipeline design.

General Solution for Longitudinal Response of Shield Tunnel Considering the Effects of Joints and Soil Shear Resistance

ABSTRACTThis study introduces a general solution for assessing the longitudinal response of shield tunnels, incorporating the combined effects of joints and soil shear resistance. The analysis employs the Timoshenko beam spring model atop a Vlasov foundation, subjected to arbitrary loads and various boundary conditions. Governing equations and relevant boundary conditions are derived using a variational formulation. Validation is conducted against existing simplified analytical solutions and finite element method simulations, showing the efficacy of the proposed solution. Comparative analyses of different models are undertaken based on the potential energy considerations. Additionally, a parametric study explores influential factors such as subgrade shear and reaction coefficients, as well as joint stiffnesses. Findings highlight the significance of accounting for the soil shear resistance and caution against underestimation when using equivalent continuous beam models. The proposed solution improves the prediction accuracy for the longitudinal response of shield tunnels and offers several benefits including compatibility with other analytical models, computation efficiency, and versatility in considering diverse loads and boundary conditions.

Commissioning and implementation of a pencil-beam algorithm with a Lorentz correction as a secondary dose calculation algorithm for an Elekta Unity 1.5T MR linear accelerator.

To commission a beam model in ClearCalc (Radformation Inc.) for use as a secondary dose calculation algorithm and to implement its use into an adaptive workflow for an MR-linear accelerator. A beam model was developed using commissioning data for an Elekta Unity MR-linear accelerator and entered into ClearCalc. The beam model consisted of absolute dose calculation settings, output factors, percent depth-dose (PDD) curves, mutli-leaf collimator (MLC) transmission and dose leaf gap error, and cryostat corrections. Beam profiles were hard-coded by the manufacturer into the beam model and were compared with Monaco-derived profiles. The beam model was tested by comparing point doses in a homogenous phantom obtained through measurements using an ionization chamber in water, Monaco, and ClearCalc for various field sizes, source-surface distances (SSDs), and point locations. Additional testing including point dose verification for test plans using a heterogeneous phantom and patient plans. Post clinical implementation, performance of ClearCalc was evaluated for the first 41 patients treated, which included 215 adaptive plans. PDDs generated using ClearCalc fell within 1.2% of measurements. Field profile comparison between ClearCalc and Monaco showed an average pass rate of 98% using a 3%/3mm gamma criteria. Measured cryostat corrections used in the beam model showed a maximum deviation from unity of 1.4%. Point dose and field monitor units (MUs) comparisons in a homogenous phantom (N=22), heterogeneous phantoms (N=22), and patient plans (N=57) all passed with a threshold of 5%/5MU. Clinically, ClearCalc was implemented as a physics check post adaptive planning completed prior to beam delivery. Point dose and field MUs showed good agreement at a 5%/5MU threshold for prostate stereotactic body radiation therapy (SBRT), pelvic lymph nodes, rectum, and prostate and lymph node plans. This work demonstrated commissioning and clinical implementation of ClearCalc into an adaptive planning workflow. No primary or adaptive plan failures were reported with proper beam model testing.

Development of a rotary piezoelectric energy harvester using magnetic excitation inspired by the fan blade

This work proposes a new rotary piezoelectric energy harvester using magnetic excitation inspired by the fan blade. The configuration and operating principle of the harvester are introduced. Then, the equivalent nonlinear model of the piezoelectric beam is established based on the Euler–Bernoulli theory and the Rayleigh–Ritz method. Finite element simulation is used to obtain the vibration performance of the piezoelectric beam, and the first order natural frequency is obtained as 22.059 Hz. A prototype of the proposed harvester is developed, and a series of experiments are carried out. The effect of magnet deflection angle, magnet mass, and the number of magnets on the output performance of the harvester is studied in detail by experiments. The experimental results proved that the harvester obtained a relatively better output performance when the deflection angle of the drive magnet is 30°. In addition, the harvester generated the maximum output voltage when the rotary speed is 165 rpm, which is consistent with the simulation result. The harvester achieved an average power of 43.5 mW when the resistance was 130 kΩ under the rotary speed of 165 rpm. The output power can satisfy the power consumption of low-power electronic devices, such as LEDs, calculators, and electronic meters.

Beams on elastic foundation: A variable reduction approach for nonlinear contact problems

Beams on elastic foundations are applied to a vast number of engineering problems. Several elastic foundation models are available, from the simplest Winkler element with one parameter to complex models with more parameters and nonlinear characteristics. Analytical and numerical approaches have been developed in the literature for the solution of this problem, often specialized for a particular application. In this paper, a novel numerical approach that can be applied to any combination of beam and foundation models is presented. The method is based on independent meshes for the beam and for the foundation. The independent discretization of the foundation opens the possibility to model any kind of foundation behaviour, including nonlinearities, discontinuities and space-dependent properties. The two meshes are then connected by a variable reduction approach, formulated by standard finite element procedures. Such an approach allows to refine the discretization of the foundation without affecting the dimension of the solving system, i.e. with a limited effect on the computational time. Additionally, a relevant advantage of the presented method is that, contrary to most approaches described in the literature, gaps between the beam and the foundation can be straightforwardly included by an energy-based formulation. Examples of applications to linear, nonlinear, and foundation with gaps are reported in the paper. This innovative approach not only simplifies the modelling process but also offers significant computational advantages, making it a versatile and efficient tool for a wide range of engineering applications involving beam-foundation interactions.

Relative-kinematic formulation of geometrically exact beam dynamics based on Lie group variational integrators

Geometrically exact beam models can accurately describe the statics and dynamics of elastic, slender beams undergoing large deformations. This work introduces a variational integrator for such models based on a relative-kinematic formulation, in which positions and velocities of nodes in the discretized beam are expressed relative to the respective preceding nodes in the kinematic chain. The resulting model is especially suited for applications in robotics and control, since it has a minimum number of states and possesses the same structure as rigid mechanical systems, simplifying the combined modeling of rigid–flexible systems. Moreover, this approach allows to specifically select the modeled deformation modes, avoiding numerical issues associated with stiff, high-frequency modes and greatly increasing numerical efficiency. The proposed model is derived fully variationally in the discrete mechanics framework and under consistent consideration of the underlying Lie group structure, which translates into beneficial numerical properties. We provide a detailed treatment of the inclusion of dissipation and propose two solver strategies for time integration, including a linear-time solver based on recursive rigid-body dynamics algorithms. The model is validated and analyzed in detailed simulation studies underlining its efficiency for the simulation of stiff and slender beams, where the straightforward, but exact reduction to a Kirchhoff beam in the presented frame leads to a substantial speed-up of the simulations.

Criteria for mode shape tracking in Micropolar-Cosserat periodic panels

This study communicates the dispersion nature and motion of propagating waves on the periodic panels resulting from the variation of the Micropolar-Cosserat (MC) parameters such as characteristic length-scale and Cosserat shear modulus. The non-dimensionalization of the system determines the independent parameters for the MC beam model in order to examine the motion of the micro-rotational as well as flexural wave modes. A periodic boundary condition based on Bloch-Floquet’s theorem is employed on the unit cell to maintain the periodicity and assess the eigenvalue domain within the transfer matrix approach. A significant part of this enlightening theoretical comprehension regarding veering, locking, and coupling is elucidated by tracking the mode shapes within the Modal Assurance Criteria (MAC), which is the prime novelty of this research. Periodically pass-band and partial bandwidth are also examined to build up confidence concerning the complex and real wave modes, respectively. A slight variation of MC parameters can dramatically alter the emergence of veering, locking, and coupling phenomena, even 1% only. The band gap (BG) calculated through the two-dimensional (2-D) Finite Element Analysis (FEM) corroborates well with the reduced one-dimensional (1-D) MC periodic panels.

Numerical Model Development of Reinforced Concrete Beam for Structural Health Monitoring

Reinforced concrete (RC) structures are used in various engineering projects of all sizes and shapes. In order to maintain its safety and serviceability, it is necessary to evaluate the durability of an existing structure by conducting structural health monitoring. Thus, the main problem of this study is the lack of inspection, and no predictive maintenance is performed in the existing structure because the conventional method of structure monitoring needs a high cost to execute. Therefore, it will cause structural deterioration and lead to structural failure. In this study, a numerical model of RC beam was developed by using Finite Element Software. The research objective is to validate the real experimental work by using numerical model of RC beam in Marc Mentat software and investigates the potential of the numerical model development for structural health monitoring. In order to achieve the objectives, Marc Mentat software was used to develop a numerical model of RC beam, which validates the results based on the previous experimental work. Next, the qualitative data were collected through an interview session with several experienced Engineers and Project Managers to investigate the potential of the developed numerical model. The collected data were analyzed by executing content analysis and represented in the form of a matrix table. It is found that the numerical analysis produces the same tendencies as previous experimental results. So, the numerical model could validate the experimental work with no significant difference. Moreover, all the respondents agreed that this numerical approach has the potential to be implemented in the structural health monitoring work. This study suggested an alternative method of monitoring the structural conditions. This numerical method promises precise data in real-time with a minimum cost and time compared to the conventional method.

Modeling and analysis of an inextensible beam with inertial and geometric nonlinearities

Modeling and analysis of an inextensible beam with inertial and geometric nonlinearities

Commissioning of a commercial treatment planning system for scanned carbon-ion radiotherapy.

To commission the RayStation (RS) TPS (treatment planning system) for scanned CIRT (carbon-ion radiotherapy) utilizing pencil beam algorithms (PBv4.2). The beam model commissioning entailed employing 1D single beams and 2D monoenergetic fields to validate spot profiles with films, assess beam range using Peakfinder measurements, and evaluate fragment spectra through dose-averaged linear energy transfer (LETd) calculations. 3D dose distributions were verified in homogeneous phantoms for both absorbed and relative biological effectiveness (RBE)-weighted doses, and further assessed in double wedge and anthropomorphic phantoms for absorbed dose only. Finally, RBE-weighted dose verification and patient-specific quality assurance were conducted using 58 beams from 20 clinically treated patient plans. The results demonstrated good agreement in absolute dose distribution between TPS calculations and measurements, with mean dose discrepancies within 3%. However, deviations were slightly higher (>1%) for the cases involving the range shifter (RaShi) compared to those without the RaShi (<1%). Beam range, depth dose distribution, and lateral profiles of spread-out Bragg peaks (SOBPs) closely matched between RS TPS calculations and measurements. Some discrepancies (less than 0.5mm) were observed at field edges and in penumbra regions due to limitations in simulating asymmetrical spots, but within clinical tolerance. After model tuning, RBE-weighted dose calculations in RS TPS were in agreement with those from the clinically used TPS, except for variations exceeding 3% observed at energies exceeding 408.07 MeV/u, primarily attributed to fragment spectra differences. Overall, this study validated the RS TPS for calculating absorbed doses against measurements and RBE-weighted doses against a clinically used TPS. The results suggested that the RS TPS could be utilized for CIRT treatment planning, except for energies exceeding 408.07 MeV/u.

3D gamma analysis between treatment plans for nominally beam‐matched medical linear accelerators using PyMedPhys

AbstractBeam‐matched linear accelerators (linacs) enable flexible patient scheduling and efficient treatment delivery in the event of unexpected machine downtime. The purpose of this study was to test the feasibility of 3D gamma index as an additional metric beyond standard measurement‐based comparisons for more efficient evaluation of treatment plans between linacs with nominally matched beam models to ensure safe patient transfer. Seventeen 3D conformal radiotherapy (3DCRT) plans and thirty‐six volumetric‐modulated radiation therapy (VMAT) plans for different disease sites were selected from the original linac. An in‐house script was used to automatically create new plans for the target linac and calculate dose using parameters of the original plans. 3D gamma analysis was performed to compare plan dose distributions between the target and original linacs using PyMedPhys. The 2%/2 mm gamma pass (γ≤1) rate was &gt;99.99% for all 3DCRT plans. The median 1%/1 mm pass rate was 99.86% but two cases failed (&lt; 90%). For VMAT plans, the median and minimum 2%/2 mm gamma pass rates were 99.43% and 93.81%. For 1%/1 mm, the median pass rate was 92.02% but ten cases failed. The results indicated using 3D gamma index can enhance the confidence and add an extra layer for safe patient transfer.

Test rig for validating the integrated motion measurement of flexible beams

AbstractThe fundamental principle of integrated motion measurement (IMM) is integrated navigation with inertial sensors combined with, for example, a satellite navigation receiver. Accordingly, IMM makes use of the specific advantages of complementary sensors and blends them in a filter algorithm. To meet requirements in advanced motion measurements, for instance, for structural health monitoring or for structural control purposes, the approach of a single rigid body like in classical navigation no longer holds for IMM. As a solution, additional degrees of freedom (DOFs) for the moving structure are introduced, which represent deformations and allow the navigated body to be treated as a flexible structure. To cover a variety of flexible deformation shapes, a sufficient number of distributed inertial sensors is required. To restrict accumulating errors of these sensors, additional structural measurements for aiding are necessary. The signals of the inertial sensors and the aiding measurements are fused by an extended Kalman filter (EKF) to obtain an optimal estimation of the usual navigation states, extended by the deformation variables. The contribution presents the preparation of the experimental validation of an IMM system for a flexible structure, which represents an idealization of a wing or rotor blade by a movable beam. The mechanical and electrical setup of a test rig is described. Furthermore, the simulation of the test configuration, that is the model of a test beam with a variety of distributed sensors for generating artificial measurements as a test reference, is discussed. Finally, for an optimal sensor placement, the two methods of effective independence and maximization of modal energy are compared and experimentally tested for different amounts of additional flexible DOFs.

Semi-analytical model and vibrational energy recovery efficiency of piezoelectric beams with quasi-periodic additional acoustic black holes

Semi-analytical model and vibrational energy recovery efficiency of piezoelectric beams with quasi-periodic additional acoustic black holes

Modeling of a sandwich beam with viscoelastic core partially supported by elastic foundation under moving load

Modeling of a sandwich beam with viscoelastic core partially supported by elastic foundation under moving load