Analysis of the quench propagation along Nb3Sn Rutherford cables with the THELMA code. Part II: Model predictions and comparison with experimental results

To improve the technology of the new generation of accelerator magnets, prototypes are being manufactured and tested in several laboratories. In parallel, many numerical analyses are being carried out to predict the magnets behaviour and interpret the experimental results. This paper focuses on the quench propagation velocity, which is a crucial parameter as regards the energy dissipation along the magnet conductor. The THELMA code, originally developed for cable-in-conduit conductors for fusion magnets, has been used to study such quench propagation. To this purpose, new code modules have been added to describe the Rutherford cable geometry, the material non-linear thermal properties and to describe the thermal conduction problem in transient regime. THELMA can describe the Rutherford cable at the strand level, modelling both the electrical and thermal contact resistances between strands and enabling the analysis of the effects of local hot spots and quench heaters. This paper describes the model application to a sample of Short Model Coil tested at CERN: a comparison is made between the experimental results and the model prediction, showing a good agreement. A comparison is also made with the prediction of the most common analytical models, which give large inaccuracies when dealing with low n-index cables like Nb₃Sn cables.     

Analysis of the quench propagation along Nb3Sn Rutherford cables with the THELMA code. Part I: Geometric and thermal models

The paper describes the new lumped thermal model recently implemented in THELMA code for the coupled electromagnetic–thermal analysis of superconducting cables. A new geometrical model is also presented, which describes the Rutherford cables used for the accelerator magnets. A first validation of these models has been given by the analysis of the quench longitudinal propagation velocity in the Nb₃Sn prototype coil SMC3, built and tested in the frame of the EUCARD project for the development of high field magnets for LHC machine. This paper shows in detail the models, while their application to the quench propagation analysis is presented in a companion paper.     

Discrete cohesive zone model for mixed-mode fracture using finite element analysis

The discrete cohesive zone model (DCZM) is implemented using the finite element (FE) method to simulate fracture initiation and subsequent growth when material non-linear effects are significant. Different from the widely used continuum cohesive zone model (CCZM) where the cohesive zone model is implemented within continuum type elements and the cohesive law is applied at each integral point, DCZM uses rod type elements and applies the cohesive law as the rod internal force vs. nodal separation (or rod elongation). These rod elements have the provision of being represented as spring type elements and this is what is considered in the present paper. A series of 1D interface elements was placed between node pairs along the intended fracture path to simulate fracture initiation and growth. Dummy nodes were introduced within the interface element to extract information regarding the mesh size and the crack path orientation. To illustrate the DCZM, three popular fracture test configurations were examined. For pure mode I, the double cantilever beam configuration, using both uniform and biased meshes were analyzed and the results show that the DCZM is not sensitive to the mesh size. Results also show that DCZM is not sensitive to the loading increment, either. Next, the end notched flexure for pure mode II and, the mixed-mode bending were studied to further investigate the approach. No convergence difficulty was encountered during the crack growth analyses. Therefore, the proposed DCZM approach is a simple but promising tool in analyzing very general two-dimensional crack growth problems. This approach has been implemented in the commercial FEA software ABAQUS^® using a user defined subroutine and should be very useful in performing structural integrity analysis of cracked structures by engineers using ABAQUS^®.     

An embedded cohesive crack model for finite element analysis of brickwork masonry fracture

This paper presents a numerical procedure for fracture of brickwork masonry based on the strong discontinuity approach. The model is an extension of the cohesive model prepared by the authors for concrete, and takes into account the anisotropy of the material. A simple central-force model is used for the stress versus crack opening curve. The additional degrees of freedom defining the crack opening are determined at the crack level, thus avoiding the need of performing a static condensation at the element level. The need for a tracking algorithm is avoided by using a consistent procedure for the selection of the separated nodes. Such a model is then implemented into a commercial code by means of a user subroutine, consequently being contrasted with experimental results. Fracture properties of masonry are independently measured for two directions on the composed masonry, and then input in the numerical model. This numerical procedure accurately predicts the experimental mixed-mode fracture records for different orientations of the brick layers on masonry panels.     

An object-oriented finite element framework for multiphysics phase field simulations

The phase field approach is a powerful and popular method for modeling microstructure evolution. In this work, advanced numerical tools are used to create a framework that facilitates rapid model development. This framework, called MARMOT, is based on Idaho National Laboratory’s finite element Multiphysics Object-Oriented Simulation Environment. In MARMOT, the system of phase field partial differential equations (PDEs) are solved simultaneously together with PDEs describing additional physics, such as solid mechanics and heat conduction, using the Jacobian-Free Newton Krylov Method. An object-oriented architecture is created by taking advantage of commonalities in the phase field PDEs to facilitate development of new models with very little effort. In addition, MARMOT provides access to mesh and time step adaptivity, reducing the cost for performing simulations with large disparities in both spatial and temporal scales. In this work, phase separation simulations are used to show the numerical performance of MARMOT. Deformation-induced grain growth and void growth simulations are also included to demonstrate the muliphysics capability.     

Simulation and evaluation of tablet-coating burst based on finite element method

The objective of this study was to simulate and evaluate the burst behavior of coated tablets. Three-dimensional finite element models of tablet-coating were established using software ANSYS. Swelling pressure of cores was measured by a self-made device and applied at the internal surface of the models. Mechanical properties of the polymer film were determined using a texture analyzer and applied as material properties of the models. The resulted finite element models were validated by experimental data. The validated models were used to assess the factors those influenced burst behavior and predict the coating burst behavior. The simulation results of coating burst and failure location were strongly matched with the experimental data. It was found that internal swelling pressure, inside corner radius and corner thickness were three main factors controlling the stress distribution and burst behavior. Based on the linear relationship between the internal pressure and the maximum principle stress on coating, burst pressure of coatings was calculated and used to predict the burst behavior. This study demonstrated that burst behavior of coated tablets could be simulated and evaluated by finite element method.     

热压工艺热-化学-应力三维数值模型及有限元分析

建立了复合材料热压工艺的三维热-化学-应力耦合数学模型。该模型考虑了整个工艺周期过程中的热-化学应变、材料的黏弹性效应、各向异性及玻璃化转变温度与固化度的关系。其中热-化学模型可采用完全耦合的形式求解,而应力模型则可采用单向耦合的形式求解。对AS4/3501-6层合平板的热压工艺过程用有限元方法进行了数值模拟,得到的层合板翘曲度与实验结果相符。计算结果表明,层合板厚度减小或长度增大,都会使翘曲变形增大,而工艺压强对翘曲变形影响很小。层合板的翘曲变形随着模具与层合板热膨胀系数差距的变小而减小。       

A finite element model of skeletal muscles

The present paper surveys recent developments in constitutive and computational modelling of skeletal muscles, concerning mainly the generalization to two- and three-dimensional (2D, 3D) continuum deformation analysis of typical one-dimensional (1D) Hill-type muscle models. Extending our previous work in the field and recent contributions by other authors, we describe a constitutive model for skeletal muscles that incorporates all the features of the 3 typical elements (parallel elastic, series elastic and contractile elements) in Hill's muscle model. In particular the proposed incompressible transversely isotropic model incorporates: a multiplicative split of the fibre stretch into contractile and (series) elastic stretches; the possibility of energy storage in the series elastic element; the dependence of the contractile stress on the strain rate; the governing equation of activation dynamics, so that general histories of neural stimulation may be taken as input data. The resulting 2D or 3D constitutive equations are implemented as user subroutines in the large deformation finite element software package ABAQUS. Simple numerical tests are presented and discussed, as well as an example that involves passive or active deformations of a pelvic floor muscle using shell finite elements.     

An augmented finite element method for modeling arbitrary discontinuities in composite materials

An augmented finite element method (“A-FEM”) is presented that is a variant of the method of Hansbo and Hansbo (Comput Methods Appl Mech Eng, 193: 3523–3540, 2004), which can fully account for arbitrary discontinuities that traverse the interior of elements. Like the method of Hansbo and Hansbo, the A-FEM preserves elemental locality, because element augmentation is implemented within single elements and involves nodal information from the modified element only. The A-FEM offers the additional convenience that the augmentation is implemented via separable mathematical elements that employ standard finite element nodal interpolation only. Thus, the formulation is fully compatible with standard commercial finite element packages and can be incorporated as a user element without access to the source code. Because possible discontinuities include both elastic heterogeneity and cracks, the A-FEM is ideally suited to modeling damage evolution in structural or biological materials with complex morphology. Elements of a multi-scale approach to analyzing damage mechanisms in laminated or woven textile composites are used to validate the A-FEM and illustrate its possible uses. Key capabilities of the formulation include the use of meshes that need not conform to the surfaces of heterogeneities; the ability to apply the augmented element recursively, enabling modeling of multiple discontinuities arising on different, possibly intersecting surfaces within an element; and the ease with which cohesive zone models of nonlinear fracture can be incorporated.     

A finite element model for contact analysis of multiple Cosserat bodies

The objective of this paper is to develop a finite element model for multi-body contact analysis of Cosserat materials. Based on the parametric virtual work principle, a quadratic programming method is developed for finite element analysis of contact problems. The contact problem with friction between two Cosserat bodies is treated in the same way as in plastic analysis. The penalty factors, that are normally introduced into the algorithm for contact analysis, have a direct influence on accuracy of solution. There is no available rule for choosing a reasonable value of these factors for simulation of contact problems of Cosserat materials, and they are therefore cancelled through a special technique so that the numerical results can be of high accuracy. Compared with the conventional work on Cosserat elasticity, the newly developed model is on the contact analysis of the Cosserat materials and is seldom found in the existing literatures. Four examples are computed to illustrate the validity and importance of the model developed.     

An accelerated pseudo-genetic algorithm for dynamic finite element model updating

An effective accelerated pseudo-genetic algorithm (APGA), which combines an adaptive pseudo-genetic algorithm (P-GA) with an accelerated random search (ARS) method, is proposed to update finite element (FE) models in the presence of measured data. The algorithm explores the higher probability of converging to a global solution provided by genetic algorithms and the accelerated hill-climbing ability given by ARS. The objective of the optimization problem is to minimize the difference between measured and numerical FE vibration data. The effectiveness of the approach is first tested on mathematical benchmark functions. The best version of APGA is then applied to a simulated beam structure to test the applicability of the new approach for FE model updating. Finally, the algorithm is applied to update two real structures using measured modal data. The application of this new algorithm obtains results that correlate well with experiments in reduced time.     

A finite element model of skeletal muscles

The present paper surveys recent developments in constitutive and computational modelling of skeletal muscles, concerning mainly the generalization to two- and three-dimensional (2D, 3D) continuum deformation analysis of typical one-dimensional (1D) Hill-type muscle models. Extending our previous work in the field and recent contributions by other authors, we describe a constitutive model for skeletal muscles that incorporates all the features of the 3 typical elements (parallel elastic, series elastic and contractile elements) in Hill's muscle model. In particular the proposed incompressible transversely isotropic model incorporates: a multiplicative split of the fibre stretch into contractile and (series) elastic stretches; the possibility of energy storage in the series elastic element; the dependence of the contractile stress on the strain rate; the governing equation of activation dynamics, so that general histories of neural stimulation may be taken as input data. The resulting 2D or 3D constitutive equations are implemented as user subroutines in the large deformation finite element software package ABAQUS. Simple numerical tests are presented and discussed, as well as an example that involves passive or active deformations of a pelvic floor muscle using shell finite elements.     

有限元简化模型应用于超声换能器模拟分析

依据有限元方法的基本物理思想,在某些不需要计算辐射声场的准确声学参数和波束特性的工程应用方面,对流体模型进行充分简化,提出了简化模型处理的有效方法,利用该方法对超声换能器进行模拟分析,并进行了样品的制作和测试,实测结果与模型简化分析处理的结果基本一致。可以证明,用该方法进行换能器的优化设计是可行和高效的。     

基于FEPG有限元分析系统的树脂充模过程模拟

为优化复杂预成型体结构的液体成型工艺,基于有限元法/生死节点法模拟了复合材料液体模塑成型过程树脂流动,并针对典型矩形平板、圆板结构、I型加筋壁板充模过程进行了仿真与验证。结果表明:典型矩形平板和圆板结构的充模过程模拟结果与理论解一致性较好,验证了生死节点法跟踪树脂流动前锋的有效性。含有方腔的变厚度圆柱体和正方体三维实体结构的充模过程模拟验证了有限元方法对三维结构的适用性。基于有限元法/生死节点法的液体充模过程模拟方法对于复杂求解区域具有更好适应性,可用于复杂实体结构的液体模塑成型工艺过程树脂流动规律预测、指导模具设计及工艺优化。      

An implicit finite element material model for energetic particulate composite materials

The predisposition of energetic particulate composite materials, or high explosives (HE), to ignition by bulk heating (cookoff) poses serious safety problems. Because unexpected initiation of HE must be a major consideration in any activity involving employment of the material, its behaviour under a variety of conditions is of much interest. The formulation of a numerical constitutive model that can be employed in an implicit finite element code to predict the mechanical and ignition behaviour of HE is presented. The capability of the developed material model is then demonstrated through its implementation in the ABAQUS finite element code to simulate the response of HE test configurations. The simulated response is found to compare favourably with the physical test results, in the cases where test data exist. Copyright © 2000 John Wiley & Sons, Ltd.     

On the model building for transmission line cables: a Bayesian approach

This work is aimed at building models to predict the bending vibrations of stranded cables used in high-voltage transmission lines. The present approach encompasses model calibration, validation and selection based on a statistical framework. Model calibration is tackled using a Bayesian framework and the Delayed Rejection Adaptive Metropolis (DRAM) sampling algorithm is employed to explore the posterior probability of the unknown model parameters. Two model classes are proposed to predict the bending vibrations of a typical high-voltage stranded cable. Both model classes account for the aerodynamic damping with the surrounding medium and the bending stiffness of the cable. The difference between the two relies on the damping model chosen to quantify the energy dissipation due to friction among the constituent wires of the cable. Model ranking is rigorously quantified by means of a Bayesian model class selection approach, in which both the data-fitting capability and complexity of each model class are simultaneously taken into account. Experimental tests are performed on a laboratory span with a typical high-voltage stranded cable. The measured frequency response functions are the observable quantities employed in the Bayesian model updating for the two model classes proposed. Both model classes provide comparable and accurate predictions for the cable’s frequency response functions within the range [5, 25] Hz, with the fractional derivative-based model class providing the most accurate predictions. Nonetheless, both model classes failed to accurately reproduce the measured cable’s dynamic response within the frequency range [25, 30] Hz.     

An Abaqus implementation of the extended finite element method

In this paper, we introduce an implementation of the extended finite element method for fracture problems within the finite element software ABAQUS^TM. User subroutine (UEL) in Abaqus is used to enable the incorporation of extended finite element capabilities. We provide details on the data input format together with the proposed user element subroutine, which constitutes the core of the finite element analysis; however, pre-processing tools that are necessary for an X-FEM implementation, but not directly related to Abaqus, are not provided. In addition to problems in linear elastic fracture mechanics, non-linear frictional contact analyses are also realized. Several numerical examples in fracture mechanics are presented to demonstrate the benefits of the proposed implementation.     

Simulation of static recrystallization in non-isothermal annealing using a coupled cellular automata and finite element model

A coupled cellular automata and finite element model has been proposed to evaluate static recrystallization kinetics during non-isothermal annealing of cold deformed low carbon steels. The effects of various factors including heating rate, annealing temperature, and initial microstructures have been considered in the model to accurately predict the static recrystallization kinetics and the final microstructures. In order to examine the employed algorithm, the predicted results have been compared with the experimental observations and a good agreement was found between the two sets of results.     

A simple model describing the non-linear biaxial tensile behaviour of PVC-coated polyester fabrics for use in finite element analysis

A simple model based on experimental observations of the yarn-parallel biaxial extension of PVC-coated polyester fabric cruciform specimens is proposed. In situ loading conditions are considered. The material behaviour is assumed to be plane stress orthotropic for a particular load ratio, while the elastic properties can vary with the load ratio in order to represent the complex interaction between warp and fill yarns. A linear relationship is experimentally found between elastic moduli and normalized load ratios for a wide range of PVC-coated polyester (Type I to Type IV). Two new parameters corresponding to the moduli variations are introduced to complement the existing plane stress orthotropic model. Theoretical results show that only five biaxial tests are required to accurately describe the material response with the proposed material model. Finally, the model was integrated in a commercial finite element software. It is shown that the proposed material model significantly increases the accuracy of the finite element predictions compared to the standard orthotropic linear material model with almost identical computation times.