Modular Formulation Research Articles

Consistent crystal plasticity kinematics and linearization for the implicit finite element method

Purpose – The purpose of this paper is to describe several novel techniques for implementing a crystal plasticity (CP) material model in a large deformation, implicit finite element framework. Design/methodology/approach – Starting from the key kinematic assumptions of CP, the presentation develops the necessary CP correction terms to several common objective stress rates and the consistent linearization of the stress update algorithm. Connections to models for slip system hardening are isolated from these processes. Findings – A kinematically consistent implementation is found to require a correction to the stress update to include plastic vorticity developed by slip deformation in polycrystals. A simpler, more direct form for the algorithmic tangent is described. Several numerical examples demonstrate the capabilities and computational efficiency of the formulation. Research limitations/implications – The implementation assumes isotropic slip system hardening. With simple modifications, the described approach extends readily to anisotropic coupled or uncoupled hardening functions. Practical implications – The modular formulation and implementation support streamlined development of new models for slip system hardening without modifications of the stress update and algorithmic tangent computations. This implementation is available in the open-source code WARP3D. Originality/value – In the process of developing the CP formulation, this work realized the need for corrections to the Green-Naghdi and Jaumann objective stress rates to account properly for non-zero plastic vorticity. The paper describes fully the consistent linearization of the stress update algorithm and details a new scheme to implement the model with improved efficiency.

High-speed impact of electromagnetically sensitive fabric and induced projectile spin

This work deals with the dynamic contact of a rigid body with a deformable electromagnetically sensitive fabric structure, represented by a network model. Of particular interest are the electromagnetically induced forces generated on the fabric, which are proportional to the external electric field (E EXT ) and the velocity crossed with the external magnetic field (v × B EXT ). These forces transmit reactions to the rigid contacting object, which can induce rotational motion. Modeling and simulation of this effect can be useful in ballistic shielding applications, because the rotation of an incoming, ogival, projectile allows it to be more easily impeded. A modular formulation for the deformation of impacted fabric structures, represented by a network model, is developed in this paper, characterized by (1) stretching of interconnected yarn networks, described by simple constitutive relations, including yarn damage, (2) interaction with impacting objects, incorporating contact with friction and (3) electromagnetic sensitivity and actuation, demonstrating how the Lorentz force can be harnessed to break symmetric deformation patterns in order to induce spin onto an incoming object, whether that object is electromagnetically sensitive or not.

Whole Core Transport Calculation Employing Hexagonal Modular Ray Tracing and CMFD Formulation

A whole core transport module for hexagonal cores is developed and implemented in the DeCART code. The module consists of a whole core ray tracing kernel for solving the two-dimensional (2-D) method of characteristics (MOC) equation and a coarse-mesh finite difference (CMFD) kernel to accelerate the MOC transport iteration. Whole core ray tracing is realized by incorporating a hexagonal-assembly-based modular ray tracing scheme. The complete path linking constraint forthe modular ray is achieved by adjusting the ray angle and the ray spacing in the range of [0, 30°], and the complete reflection constraint at the problem boundary is satisfied by defining the corresponding reflection angles at the reflection surfaces. The hexagonal CMFD kernel employs unstructured nodes that can treat the irregular-shaped gap cells as well as the regular hexagonal cells. Some features such as cell ray approximation and modified cycle ray scheme are employed to reduce the memory requirements for the segment information and the boundary angular fluxes, respectively. The solution accuracy and execution performance of the hexagonal module are examined for the C5G7 hexagonal variation problems that are established by modifying the original C5G7MOX 3-D extension benchmark. The CMFD kernel shows a significant speedup of 60 in the 2-D core problem. The cell ray approximation does not violate the original solution accuracy when using the default ray spacing of the DeCART code. The modified cycle ray scheme shows its superiority over the simple core ray sweeping scheme in terms of the memory requirement and the original cycle ray scheme in terms of the computing time. Compared with the Monte Carlo solutions, the DeCART solution agrees to within 40 pcm for the eigenvalue and 2% for the pin power distribution.

A multi-objective fuzzy graph approach for modular formulation considering end-of-life issues

Increased awareness of the negative environmental impact caused by electronic waste has driven electronics manufacturers to re-engineer their product design process and include product end-of-life considerations into their design criteria. Design for the Environment (DfE), as a possible solution, lacks an implementation framework. To address this problem, a fuzzy graph based modular product design methodology is developed to implement DfE strategies in product modular formulation considering multiple product life cycle objectives guided by DfE. A fuzzy connected graph approach is used to present the product structure, whereby fuzzy relationship values are determined by applying Analytic Hierarchy Process (AHP) to life cycle environmental objectives along with other functional and production concerns. Based on the fuzzy connected graph, an optimal modular formulation is searched using the graph-based clustering algorithm to identify the best module configuration. An example is provided to illustrate the methodology and the algorithm presented in this paper.

Impedance as a Tool for Investigating Aging in Lithium-Ion Porous Electrodes: I. Physically Based Electrochemical Model

Electrochemical impedance spectroscopy is potentially a powerful diagnostic tool for the investigation of the effects of aging in porous electrodes. A physically based three-electrode model was developed for a composite porous electrode with three porous separators and a reference electrode between a current collector and a plane electrode. Two effects of aging were modeled for this particular electrode chemistry, namely, a resistive corrosion layer on the current collector and a contact resistance between the electronic conductor and the active material of the porous electrode. The derivation of an analytical solution for the impedances between each pair of electrodes in this model yielded a computationally fast, versatile, and modular formulation. The solution was used to study the impact of selected components of the physical model on the impedance spectrum of the porous electrode for a physically relevant base case. Approximating the active material particles as spherical or flake-shaped particles, lognormally or Dirac distributed in size, revealed that the distribution has a negligible impact while the shape makes a noticeable difference. The main aging-related parameters were shown to have quite distinct effects on the impedance spectrum, which is essential for the regression of experimental data and the study of aging hypotheses.

Modular and Recursive Kinematics and Dynamics for Parallel Manipulators

Constrained multibody systems typically feature multiple closed kinematic loops that con- strain the relative motions and forces within the system. Typically, such systems possess far more ar- ticulated degrees-of-freedom (within the chains) than overall end-effector degrees-of-freedom. Thus, actuation of a subset of the articulations creates mixture of active and passive joints within the chain. The presence of such passive joints interferes with the effective modular formulation of the dynamic equations-of-motion in terms of a minimal set of actuator coordinates as well the subsequent recursive solution for both forward and inverse dynamics applications. Thus, in this paper, we examine the development of modular and recursive formulations of equations-of-motion in terms of a minimal set of actuated-joint-coordinates for an exactly-actuated parallel manipulators. The 3 RRR planar parallel manipulator, selected to serve as a case-study, is an illustrative example of a multi-loop, multi-degree-of-freedom system with mixtures of ac- tive/passive joints. The concept of decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism inherent in parallel mechanisms to develop a dynamics formulation that is both recursive and modular. An algorithmic approach to the development of both forward and inverse dynamics is highlighted. The presented simulation studies highlight the overall good nu- merical behavior of the developed formulation, both in terms of accuracy and lack of formulation stiffness.

A modular formulation for flexible multibody systems including nonlinear finite elements

A formulation for flexible multi body systems (MBS) is Investigated, where rigid MBS substructures are coupled with flexible bodies described by a nonlinear finite element (FE) approach Several aspects that turned out to be crucial for the presented approach are discussed The system describing equations are given m differential algebraic form (DAE), where many sophisticated solvers exist In this paper the performance of several solvers is investigated regarding their suitability for the application to the usually highly stiff DAE The substructures are connected With each other by nonlinear algebraic constraint equations Further, partial derivatives of the constraints are required, which often leads to extensive algebraic trans-formations Handcoding of analytically determined derivatives is compared to an approach utilizing algorithmic differentiation

Recursive Kinematics and Inverse Dynamics for a Planar 3R Parallel Manipulator

Abstract We focus on the development of modular and recursive formulations for the inverse dynamics of parallel architecture manipulators in this paper. The modular formulation of mathematical models is attractive especially when existing sub-models may be assembled to create different topologies, e.g., cooperative robotic systems. Recursive algorithms are desirable from the viewpoint of simplicity and uniformity of computation. However, the prominent features of parallel architecture manipulators-the multiple closed kinematic loops, varying locations of actuation together with mixtures of active and passive joints-have traditionally hindered the formulation of modular and recursive algorithms. In this paper, the concept of the decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism of the robots of interest to develop an inverse dynamics algorithm which is both recursive and modular. The various formulation stages in this process are highlighted using the illustrative example of a 3R Planar Parallel Manipulator.

Micro–macro transition for anisotropic, frictional granular packings

From the structure of a static granular solid, we derive the fabric and the stiffness tensor in average over those pairs of interacting particles with contact within the averaging volume. Starting from a linear expansion of the interaction potential around static equilibrium, stress and elastic strain can be derived from the principles of virtual displacement and virtual stress-change, respectively. Our approach includes both normal and tangential forces separately, in a new modular formulation starting from single contacts. The results are applied to a discrete particle simulation, and the findings include a relation between fabric and coordination number that is almost unaffected by the presence of friction, a different qualitative behavior of fabric and stiffness components, and only three independent entries to the stiffness matrix in its eigen-system. More general, anisotropy evolves directed against the direction of compression, and exponentially fast up to a certain maximal (limit) magnitude––a constitutive model for this behavior is proposed; in the critical state shear regime, the anisotropy is considerably smaller.

A connector-based approach to the modular formulation problem for a mechanical product

To respond to marketing competition, companies strive to provide a large variety of products to meet consumers’ requirements. With more choices in products, such variety brings the company closer to the consumers. However, manufacturers are facing a new challenge. The assembly and partitioning of components and subassembly are becoming increasingly more complicated, which cause not only a proliferation of subassemblies but also increases the difficulty in manufacturing. To solve this problem, modular design is a key issue. This study presents a methodology for the formulation of modules on the basis of the connector concept. Based on the information entry in the assembly network sequence proposed by Tseng and Li [12], a modular partition based on the connector concept is explored. Moreover, the problem of assigning components after partitioning and the evaluation of the partitioning results are discussed. In this study, the definition and representation of connectors are described first. Then, the algorithm for the formulation of modules is discussed. Finally, a case study with a detailed analysis is provided to illustrate the application of the proposed methodology.

The liquefaction cycle and the role of drainage in liquefaction

In this work the liquefaction cycle is introduced as a framework to describe the coupled processes that take place in fluid-saturated granular media that lead to liquefaction. The modular formulation of liquefaction makes it possible to test the various processes that contribute to liquefaction separately, and to assemble different formulations of the relevant physics into a numerical model for liquefaction. This view on liquefaction is used here to assess the role of drainage in liquefaction. We present a simple scale analysis of the role of drainage. A numerical implementation of the liquefaction cycle shows, however, that the scale analysis is deceptive for the case when strong spatial variations in the permeability inhibit fluid migration. As an illustration the numerical model is used to quantify the imprint of a low-permeability layer on the liquefaction behavior.

On the spatial formulation of anisotropic multiplicative elasto-plasticity

This contribution aims at a thermodynamically consistent and modular formulation of anisotropic multiplicative elasto-plasticity. Embedded in the well-established framework of non-standard dissipative materials, we adopt as a key ingredient the introduction of additional symmetric arguments––typically structural tensors––into the relevant scalar-valued isotropic tensor functions. On this basis, the fundamental covariance relation leads us to a formulation which allows itself to be set up directly in terms of spatial arguments and thus guarantees a convenient implementation within a finite element setting.

Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials

The paper presents a modular formulation and computational implementation of a class of anisotropic plasticity models at finite strains based on incremental minimization principles. The modular kinematic setting consists of a constitutive model in the logarithmic strain space that is framed by a purely geometric pre- and postprocessing. On the theoretical side, the point of departure is an a priori six-dimensional approach to finite plasticity based on the notion of a plastic metric. In a first step, a geometric preprocessor defines a total and a plastic logarithmic strain measure obtained from the current and the plastic metrics, respectively. In a second step, these strains enter in an additive format a constitutive model of anisotropic plasticity that may have a structure identical to the geometrically linear theory. The model defines the stresses and consistent tangents work-conjugate to the logarithmic strain measure. In a third step these objects of the logarithmic space are then mapped back to nominal, Lagrangian or Eulerian objects by a geometric postprocessor. This geometric three-step-approach defines a broad class of anisotropic models of finite plasticity directly related to counterparts of the geometrically linear theory. It is specified to a model problem of anisotropic metal plasticity. On the computational side we develop an incremental variational formulation of the above outlined constitutive structure where a quasi-hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables. It is shown that this minimization problem is exclusively restricted to the logarithmic strain space in a structure identical to the small-strain theory. The minimization problem determines the internal state of the material for finite increments of time. We develop a discrete formulation in terms of just one scalar parameter for the amount of incremental flow. The existence of the incremental stress potential provides a natural basis for the definition of the geometric postprocessor based on function evaluations. Furthermore, the global initial-boundary-value-problem of the elastic–plastic solid appears in the incremental setting as an energy minimization problem. Numerical examples show that the results obtained are surprisingly close to those obtained by a reference framework of multiplicative plasticity.

A model for turbulent binary breakup of dispersed fluid particles

Accurate predictions of particle size distributions, and therefore of the underlying processes of fluid particle breakup and coalescence are of vital importance in process design, but reliable procedures are still lacking. The present paper aims at developing a modular formulation for the turbulent particle breakup process. The model is to be included in a population balance model which is formulated such as to facilitate the direct future implementation into a full multifluid CFD model. The breakup process is described without introducing adjustable parameters. The current model is a further development of an existing model by Luo and Svendsen (AIChE J. 42 (5) (1996) 1225), which has been expanded and refined, and where an inherent weakness regarding the breakup rate for small particles and small daughter particle fragments are removed. A new criterion regarding the kinetic energy density of the colliding turbulent eddy causing breakup has been introduced. This new criterion is a novel concept describing the breakup process. The details are thoroughly discussed together with possible further modifications. The results from the new model are encouraging because the breakup rate is greatly reduced when the dispersed fluid particles are reduced in size. Further, the response to changes in system variables is reasonable and the distribution of daughter sizes vary in a reasonable way for the different collision possibilities.

Modular Simulation in Multibody System Dynamics

The dynamic analysis of complex engineering systems likeautomobiles is often relieved by a modular approach to make it treatableby a team of engineers. The modular decomposition is based onengineering intuition of corresponding engineering disciplines. In thispaper, a modular formulation of multibody systems is proposed which isbased on the block representation of a multibody system withcorresponding input and output quantities. Advantages of this modularapproach range from independent and parallel modeling of subsystems overthe easy exchange of the resulting modules to the use of differentsoftware for each module. However, the modular simulation of the globalsystem by coupling of simulators may result in an unstable integrationif an algebraic loop exists between the subsystems. This numericalphenomenon is analyzed and a method of simulator coupling whichguarantees stability for general systems including algebraic loops isintroduced. Numerical results of the modular simulation of aslider-crank mechanism are presented.

Electric dipole interactions of chain molecules in nematics: The analysis of segmental ordering in α, ω-dibromoalkanes

The residual electric dipole interactions recently detected in the deuterium NMR spectra of α,ω-dibromoalkanes dissolved in nematic liquid crystalline solvents [J. Phys. Chem. 96, 8176 (1992)] are analyzed in the context of the modular formulation of flexible molecule ordering. The chord model realization of this formulation, previously applied to the study of n-alkanes, is minimally extended to include the additional coupling of the dipole carrying segments of the solute molecules to the dipolar bias produced by the solvent. The ensuing simple model successfully reproduces the observed segmental order parameter profiles of the solute over the entire nematic temperature range of the solvents. Three types of solvents are considered (Phase V, 5CB, 5OCB) differing in their molecular dipole moment constitutions. It is found that in all solvent cases the orientational ordering is suppressed at the dipole moment carrying ends of the solute molecules whereas the ordering in the main body could either be reduced or elevated depending on the configuration of the dipole moments on the solvent molecules. Quantitative results for the effects of the dipolar bias on the potential of mean torque are presented and the mechanism of electric dipole driven ordering in these solvent–solute systems is discussed.

OPEN STRINGS AS Z2-ORBIFOLDS OF CLOSED STRINGS

We present a modular invariant formulation of the open string in terms of the closed string. Chan–Paton factors are understood as multiplicities which arise from a factorization of the closed string. This interpretation of the Chan–Paton factors suggests that the SO (2D/2) open string is consistent to all orders of the loop expansion. We show that the open string may be viewed as a Z2-orbifold of the closed string. Relations are found between various string theories which seem to reinforce an earlier suggestion by Freund that all string theories are derivable from the D = 26 orientable closed bosonic string.

Molecular Ordering in Nematics: The Modular Formulation of the Potential of Mean Torque

Abstract The general framework for the description of orientational ordering of (flexible) molecules in the nematic phase in terms of ordering properties of submolecular units in a modular fashion is presented. Various choices of the elementary submolecular units (segments) leading to different representations of the potential of mean torque are discussed. Transformations among such representations are studied in order to clarify the physical significance of intersegmental correlation terms. It is shown that for n-alkane solutes and chains pendant to nematogenic cores in neat materials, the uncorrelated bond representation (each C‒C bond segment individually interacts with the nematic mean field) used in most of the previous studies of this type gives a poor description of chain ordering. The introduction of orientational correlations among bond segments (particularly nearest neighbor correlations) improves the quality of fits of the order parameter profiles and explains the difficiency of previous models...

N-Hexane proton dipolar couplings and the rotational isomeric state approximation

The three-state rotational isomeric state (RIS) model of conformer statistics is used to analyse the 16 independent dipole coupling constants measured in a recent proton NMR study of n-hexane in a nematic liquid crystal solvent by Gochin et al. The orientational ordering of the hexane molecule is treated in the context of the modular formulation of the potential of mean torque developed recently by Photinos et al. This formulation gives a very accurate description of alkane solute orientational order and conformer probabilities in the nematic solvent. Consequently, substantially more accurate calculated dipolar couplings are obtained than in previous calculations and, moreover, this is achieved without the need to resort to unconventionally high values of the trans-gauche energy difference E g in the RIS model; an optimized fit to the data wherein the location of the gauche dihedral angle RIS state φg and E g are varied yields the values φg = 112° and E g = 0·67 kcal mol-1. The sensitivity of the dipolar ...