Lagrangian Research Articles

Selection of the numerical formulation for modeling the effect of tool cutting edge microgeometries in machining of Ti6Al4V titanium alloy

The Finite-element method (FEM) is often used to simulate the metal machining process. Currently, several formulations are used in metal cutting simulation, such as Lagrangian (LAG), Eulerian (EUL), Arbitrary Lagrangian–Eulerian (ALE), and Coupled Eulerian–Lagrangian (CEL). The selection of the numerical formulation that better reproduces the material separation in machining to form the chip is a critical issue. This is quite important when the ratio between the uncut chip thickness and the cutting edge microgeometry, often represented by a cutting edge radius, is very low. In this study, orthogonal cutting of Ti6Al4V titanium alloy using different tool edge microgeometries is investigated using LAG and CEL approaches. In the case of LAG approach, two types of cutting models were develop: one using a sacrificial layer (hereby called LAG-SL), and another without sacrificial layer (hereby called LAG-nSL). The cutting models have included a constitutive model (both plasticity and damage) considering the effects of strain, strain rate, temperature, and state of stress, which have proved to be accurate enough to represent the mechanical behavior of the Ti6Al4V alloy in machining. Comprehensive comparisons between CEL and LAG-based cutting models and experimental results are carried out in terms of chip morphology, forces, and residual stresses. When compared to the experimental results, LAG-nSL model gives the best predictions of the maximum chip compression ratio (CCRm) with an error of 9.5%, cutting force (12.8%) and thrust force (9.3%) than the other two models. In the case of the of residual stress profiles, LAG-SL offers good predictions of the maximum compressive residual stress by a preference ratio of 75%. Although CEL yields the worst predictions of chip morphology and forces, it is preferred from the perspective of the thickness of the layer affected by residual stress.

Comparison of Machining Simulations of Aerospace Alloy Al6061-T6 Using Lagrangian and Smoothed Particle Hydrodynamics Techniques

This research focuses on the study of the simulation capabilities of the lagrangian (LAG) model and Smoothed Particle Hydrodynamics (SPH) model for the orthogonal dry machining of aluminum alloy Al6061-T6. A three-dimensional finite element model was developed and verified using experimental data from the published literature. The numerical models were developed using lagrangian boundary conditions via finite element modeling in ABAQUS/Explicit 6.14. The cutting simulations were carried out at low and medium cutting speeds. Johnson–Cook material constitutive law and Johnson–Cook damage law were used in both models. The numerical methodologies are compared based on cutting forces, chip morphology, shear angle, chip separation criterion, and chip thickness. The findings of the present work show that the LAG model is good for predictions regarding cutting forces and chip morphology, while the SPH model is good for predictions regarding the shear angle and chip thickness. The difference between results generated by both models mainly occurred due to the friction coefficient. The comparative study shown here offers a guidance approach for various numerical models for appropriate parameter analysis.

Numerical investigations on residual stresses in orthogonal cutting of Ti-6A1-4V

Titanium is widely employed in aerospace space, where the surface integrity plays an important role in part quality. This paper presents the numerical and experimental results of residual stress when machining Titanium based on a CEL (Coupled Eulerian-Lagrangian) method. The performance of proposed model is compared with LAG (Lagrangian) and ALE (Arbitrary Lagrangian-Eulerian) methods regarding segment chip formation and residual stress. The correctness of simulation results is validated by orthogonal cutting experiments of Ti-6Al-4V as well as the experimental data given in recent published literature. The effects of uncut chip thickness, cutting velocity and edge radius on residual stress are discussed through simulation results.

Lagrangian and Eulerian accelerations in turbulent stratified shear flows

The Lagrangian (LA) and Eulerian Acceleration (EA) properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from $Ri=0$, corresponding to unstratified shear flow, to $Ri=1$, corresponding to strongly stratified shear flow. The probability density functions (pdfs) of both LA and EA have a stretched-exponential shape and they show a strong and similar influence on the Richardson number. The extreme values of the EA are stronger than those observed for the LA. Geometrical statistics explain that the magnitude of the EA is larger than its Lagrangian counterpart due to the mutual cancellation of the Eulerian and convective acceleration, as both vectors statistically show an anti-parallel preference. A wavelet-based scale-dependent decomposition of the LA and EA is performed. The tails of the acceleration pdfs grow heavier for smaller scales of turbulent motion. Hence the flatness increases with decreasing scale, indicating stronger intermittency at smaller scales. The joint pdfs of the LA and EA indicate a trend to stronger correlations with increasing Richardson number and at larger scales of the turbulent motion. A consideration of the terms in the Navier--Stokes equation shows that the LA is mainly determined by the pressure-gradient term, while the EA is dominated by the nonlinear convection term.

Dynamic vulnerability assessment and damage prediction of RC columns subjected to severe impulsive loading

Reinforced concrete (RC) columns are crucial in building structures and they are of higher vulnerability to terrorist threat than any other structural elements. Thus it is of great interest and necessity to achieve a comprehensive understanding of the possible responses of RC columns when exposed to high intensive blast loads. The primary objective of this study is to derive analytical formulas to assess vulnerability of RC columns using an advanced numerical modelling approach. This investigation is necessary as the effect of blast loads would be minimal to the RC structure if the explosive charge is located at the safe standoff distance from the main columns in the building and therefore minimizes the chance of disastrous collapse of the RC columns. In the current research, finite element model is developed for RC columns using LS-DYNA program that includes a comprehensive discussion of the material models, element formulation, boundary condition and loading methods. Numerical model is validated to aid in the study of RC column testing against the explosion field test results. Residual capacity of RC column is selected as damage criteria. Intensive investigations using Arbitrary Lagrangian Eulerian (ALE) methodology are then implemented to evaluate the influence of scaled distance, column dimension, concrete and steel reinforcement properties and axial load index on the vulnerability of RC columns. The generated empirical formulae can be used by the designers to predict a damage degree of new column design when consider explosive loads. With an extensive knowledge on the vulnerability assessment of RC structures under blast explosion, advancement to the convention design of structural elements can be achieved to improve the column survivability, while reducing the lethality of explosive attack and in turn providing a safer environment for the public.

Limit Behavior of Complex Special Lagrangian Equations With Neumann Boundary-Value Conditions

Abstract We consider the following complex special Lagrangian equations with Neumann boundary conditions on a domain $\Omega \subset \mathbb{C}^{n} $: $$\begin{align*}& \left\{\begin{array}{ll} {\sum\limits_{i=1}^n \arctan k \lambda_{i} (H(u_{k,\varepsilon} ))=\Theta_k(z)} & {\textrm{ in } \Omega} \\{D_{\nu} {u_{k,\varepsilon }}=-\varepsilon{u_{k,\varepsilon }}+\varphi(z)} & {\textrm{ on } \partial \Omega}\end{array}\right.. \end{align*}$$We prove uniform global estimates for these equations. As $k\to +\infty $, we prove that $u_{ k,\varepsilon }$ converges to the solution of $J$-type equation with the same Neumann boundary condition on $\Omega $.

Study of segmented chip formation in cutting of high-strength lightweight alloys

Chip segmentation results in fluctuation of the cutting force, deteriorated tool wear and surface finish, thereby plays an important role in the machining process. Although extensive research has been carried out on studying the segmented chip formation, the mechanism of chip segmentation has remained under debate. This paper aims to investigate the mechanism by combining numerical and experimental methods. Finite element (FE) models of the orthogonal cutting process of A2024–T351 aluminum alloy and Ti6Al4V titanium alloy were developed with three numerical formulations: Lagrangian (LAG), arbitrary Lagrangian–Eulerian (ALE), and coupled Eulerian and Lagrangian (CEL). The appropriate model for predicting the segmented chip formation process was selected by systematic comparison. The mechanism of chip segmentation was thoroughly investigated by the selected numerical model. It revealed that the adiabatic shear band (ASB) is generated not only from the chip root but also from the free chip surface. The finding was then validated by observing the microstructure of chips from high-speed dry-cutting tests.

Thermal frequency analysis of FG sandwich structure under variable temperature loading

The thermal eigenvalue responses of the graded sandwich shell structure are evaluated numerically under the variable thermal loadings considering the temperature-dependent properties. The polynomial type rule-based sandwich panel model is derived using higher-order type kinematics considering the shear deformation in the framework of the equivalent single-layer theory. The frequency values are computed through an own home-made computer code (MATLAB environment) prepared using the finite element type higher-order formulation. The sandwich face-sheets and the metal core are discretized via isoparametric quadrilateral Lagrangian element. The model convergence is checked by solving the similar type published numerical examples in the open domain and extended for the comparison of natural frequencies to have the final confirmation of the model accuracy. Also, the influence of each variable structural parameter, i.e. the curvature ratios, core-face thickness ratios, end-support conditions, the power-law indices and sandwich types (symmetrical and unsymmetrical) on the thermal frequencies of FG sandwich curved shell panel model. The solutions are helping to bring out the necessary influence of one or more parameters on the frequencies. The effects of individual and the combined parameters as well as the temperature profiles (uniform, linear and nonlinear) are examined through several numerical examples, which affect the structural strength/stiffness values. The present study may help in designing the future graded structures which are under the influence of the variable temperature loading.

Investigation of process parameters in orthogonal cutting using finite element approaches

The cutting force in orthogonal cutting of steel AISI 1045 was predicted by applying 2D finite element analysis (FEA) using two methods; (i) Lagrangian (LAG) and (ii) Arbitrary Lagrangian Eulerian (ALE). Johnson-Cook (J-C) models were used for defining plastic and failure properties of simulated materials. The predicted force was validated experimentally by using dynamometer. Comparison held between the simulation methods and experimental work in terms of results accuracy, reading stability, and chip morphology. Furthermore, this study adopted new modeling idea to control the excessive distortion of mesh elements along chip separation line by defining nearly zero damage criterion for these elements. The results demonstrated that LAG and ALE methods could predict the cutting force but with different accuracy, as LAG and ALE results deviated from experimental results with minimum error percentage 3.6% and 0.14% respectively. As well, ALE method showed stable force readings and continues smooth chip during simulation, while LAG method showed unstable force readings and discontinuous realistic chip.

Discretization approaches to model orthogonal cutting with Lagrangian, Arbitrary Lagrangian Eulerian, Particle Finite Element method and Smooth Particle Hydrodynamics formulations

With significant technological growth and computational power, it is possible to simulate metal cutting processes with increased complexity. In the modeling phase, a fundamental question about choosing the best discretization approach arises. Lagrangian and Eulerian formulations are the two classical discretization approaches. Alternative methods to these mesh-based approaches are being developed in recent times, such as particle-based and meshless methods. In this work, we employed four discretization approaches: pure Lagrangian (LAG), Arbitrary Lagrange Eulerian (ALE), Particle Finite Element Method (PFEM) and Smooth Particle Hydrodynamics (SPH), to simulate a turning operation of AISI 4140 steel. This paper aims to compare the conventional approaches (LAG and ALE) to newer approaches (PFEM and SPH). Firstly, orthogonal cutting models were benchmarked against a turning experiment presented in the literature, by comparing the obtained cutting and passive forces. Secondly, a detailed comparative study of parameters, such as forces, stresses, temperatures, chip morphology etc. of the four discretization approaches was performed. The study was then extended to negative rake angles to study the effect on the discretization approaches. The work concludes with identifying the advantages and drawbacks of different discretization approaches.

Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X0, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X0, X) of the subdiagram type whenever X0 is nonlinear. It remains unsolved whether rigidity holds when (X0, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X0, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure $$\overline{\omega} : \mathscr{C}(S) \rightarrow S$$ on a uniruled projective manifold $$(X,\,{\cal K})$$ equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve l emanating from a general point x ∈ S, there exists an immersed neighborhood Nl of l which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ⊂ X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ⩾ 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X0 into X which induces an isomorphism on second homology groups. By studying ℂ*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X0 ⊂ X. Under the additional hypothesis that all holomorphic sections in Γ(X0, Tx∣x0) lift to global holomorphic vector fields on X, we prove that the admissible pair (X0, X) is rigid. As examples we check that (X0, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ≅ ℂ2n, n ⩾ 3, and when X0 ⊂ X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

INVESTIGATING DOUBLY CHARGED LEPTONS AT FUTURE ENERGY FRONTIER MUON-PROTON COLLIDERS

We study the doubly charged leptons considered in extended isospin models at muon-proton colliders. We respect the lepton flavor conservation and take into consideration the single production of the doubly charged leptons related to second generation. We give the effective lagrangian describing the doubly charged lepton gauge interactions. We calculate the signal and corresponding background cross sections and analyze the kinematical distributions to obtain the suitable cuts for the discovery. We choose the W-boson hadronic decay channel to get the accessible mass limits and couplings of doubly charged leptons for various muon-proton colliders.

ATTILA 4.0: Lagrangian advective and convective transport of passive tracers within the ECHAM5/MESSy (2.53.0) chemistry–climate model

Abstract. We have extended ATTILA (Atmospheric Tracer Transport in a LAgrangian model), a Lagrangian tracer transport scheme, which is online coupled to the global ECHAM/MESSy Atmospheric Chemistry (EMAC) model, with a combination of newly developed and modified physical routines and new diagnostic and infrastructure submodels. The new physical routines comprise a parameterisation for Lagrangian convection, a formulation of diabatic vertical velocity, and the new grid-point submodel LGTMIX to calculate the mixing of compounds in Lagrangian representation. The new infrastructure routines simplify the transformation between grid-point (GP) and Lagrangian (LG) space in a parallel computing environment. The new submodel LGVFLUX is a useful diagnostic tool to calculate online vertical mass fluxes through horizontal surfaces. The submodel DRADON was extended to account for emissions and changes of 222Rn on Lagrangian parcels. To evaluate the new physical routines, two simulations in free-running mode with prescribed sea surface temperatures were performed with EMAC–ATTILA in T42L47MA resolution from 1950 to 2010. The results show an improvement of the tracer transport into and within the stratosphere when the diabatic vertical velocity is used for vertical advection in ATTILA instead of the standard kinematic vertical velocity. In particular, the age-of-air distribution is more in accordance with observations. The global tropospheric distribution of 222Rn, however, is simulated in agreement with available observations and with the results from EMAC in grid space for both Lagrangian systems. Additional sensitivity studies reveal an effect of inter-parcel mixing on the age of air in the tropopause region and the stratosphere, but there is no significant effect for the troposphere.

Geometrically nonlinear analysis of functionally graded porous beams

In this paper, geometrically non-linear analysis of a functionally graded simple supported beam is investigated with porosity effect. The material properties of the beam are assumed to vary though height direction according to a prescribed power-law distributions with different porosity models. In the nonlinear kinematic model of the beam, the total Lagrangian approach is used within Timoshenko beam theory. In the solution of the nonlinear problem, the finite element method is used in conjunction with the Newton-Raphson method. In the study, the effects of material distribution such as power-law exponents, porosity coefficients, nonlinear effects on the static behavior of functionally graded beams are examined and discussed with porosity effects. The difference between the geometrically linear and nonlinear analysis of functionally graded porous beam is investigated in detail. Also, the effects of the different porosity models on the functionally graded beams are investigated both linear and nonlinear cases.

Static behavior of a laterally loaded guardrail post in sloping ground by LS-DYNA

This study aims to present accurate soil modeling and validation of a single roadside guardrail post as well as a single concrete pile installed near cut slopes or compacted sloping embankment. The conventional Winkler\'s elastic spring model and p-y curve approach for horizontal ground cannot directly be applied to sloping ground where ultimate soil resistance is significantly dependent on ground inclination. In this study, both grid-based 3-D FE model and particle-based SPH (smoothed particle hydrodynamics) model available in LS-DYNA have been adopted to predict the static behavior of a laterally loaded guardrail post. The SPH model has potential to eliminate any artificial soil stiffness due to the deterioration of the node-connected Lagrangian soil mesh. For this purpose, this study comprises two parts. Firstly, only 3-D FE modeling has been tested to show the numerical validity for a single concrete pile in sloping ground using Mohr-Coulomb material. However, this material option cannot be implemented for SPH elements. Nevertheless, Mohr-Coulomb model has been used since this material model requires six input soil data that can be obtained from the comparative papers in literatures. Secondly, this work is extended to compute the lateral resistance of a guardrail post located near the slope using the hybrid approach that combines Lagrange FE elements and SPH elements by the suitable node-merging option provided by LS-DYNA. For this analysis, the FHWA soil material developed for application to road-base soils has been used and also allows the application of SPH element.

Three approaches for modeling residual stresses induced by orthogonal cutting of AISI316L

Different to model the orthogonal cutting process methods can be found in the literature. Each approach has its advantages that make it more appropriate for modeling particular cutting cases than others. In this work, three finite element models of the orthogonal cutting process are developed: (i) Lagrangian (LAG), (ii) Arbitrairy Lagrangian and Eulerian (ALE) and (iii) hybrid model. The purpose of this paper is to improve these models in order to predict machining induced Residual Stresses (RS). A critical assessment of the applicability of each method is presented and their efficiency in the prediction of RS profiles is discussed. The absence of the damage criterion and consequently the absence of suppressing mesh elements makes the ALE model more reliable than the LAG one when predicting the RS curves. However, even though the hybrid model presents similar performances in the numerical prediction, the RS profiles are closer to the experimental results, owing to the accuracy of its input data.

R-Adaptive Reconnection-based Arbitrary Lagrangian Eulerian Method-R-ReALE

R-Adaptive Reconnection-based Arbitrary Lagrangian Eulerian Method-R-ReALE

Mechanism of Emission Reduction in HSDI Diesel Engine: Split Injection

In order to meet the stringent emission standards significant efforts have been imparted to the research and development of cleaner IC engines. Diesel combustion and the formation of pollutants are directly influenced by spatial and temporal distribution of the fuel injected. The development and validation of computational fluid dynamics (CFD) models for diesel engine combustion and emissions is described. The complexity of diesel combustion requires simulation with many complex interacting sub models in order to have a success in improving the performance and to reduce the emissions. In the present work an attempt has been made to develop a multidimensional axe-symmetric model for CI engine combustion and emissions. Later simulations have been carried out using split injection for single, double and three pulses (split injection) for which commercial validation tool FLUENT was used for simulation. The tool solves basic governing equations of fluid flow that is continuity, momentum, species transport and energy equation. Using finite volume method turbulence was modeled by using RNG K-ɛ model. Injection was modeled using La Grangian approach and reaction was modeled using non premixed combustion which considers the effects of turbulence and detailed chemical mechanism into account to model the reaction rates. The specific heats were approximated using piecewise polynomials. Subsequently the simulated results have been validated with the existing experimental values. The peak pressure obtained by simulation for single and double is 10% higher than to that of experimental value. Whereas for triple injections 5% higher than to that of experimental value. For quadruple injection the pressure has been decreased by 10% when compared to triple injection.NOX have been decreased in simulation for single, double and triple injections by 15%, 28% and 20%.For quadruple injection NOX were reduced in quadruple injection by 20% to that of triple injection. The simulated value of soot for single, double and triple injections are 12%, 22% and 12% lesser than the experimental values. For quadruple injection the soot levels were almost negligible. The simulated heat release rates for single, double and triple were reduced by 12%, 18% and 11%. For quadruple injection heat release is reduced same as to that of triple injection.

Modelling hollow organs for impact conditions: a simplified case study

This study compares the performances of three numerical approaches [Lagrangian (LAG), arbitrary Lagrangian–Eulerian (ALE) and control volume (CV)] for modelling the response of a short cylindrical pipe representing a portion of the intestines subjected to large and rapid compressions. While not being able to simulate sustained fluid flow, the LAG approach provided similar results as the ALE for moderate levels of compression. However, it was the stiffest approach for larger levels and had numerical issues for extreme compressions. While the ALE did not have these issues, its computing cost was very high, which would be problematic for large models. The CV approach had the lowest computing cost and seemed promising for larger compressions. However, its response was the softest and further investigations are needed to define its dependency to modelling parameters.

An adaptive meshfree method for phase-field models of biomembranes. Part II

Display Omitted HighlightsWe propose a phase-field method to simulate membranes embedded in a viscous fluid.The method is meshfree, Lagrangian, robust, and easily made parallel.Adaptivity for free:...