A multi-material Eulerian formulation for the efficient solution of impact and penetration problems

An Eulerian formulation is necessary for the accurate solution of contact and impact problems involving penetration and fracture. The Eulerian mesh is fixed in space, thereby eliminating all the problems associated with a distorted mesh that are commonly encountered with a Lagrangian formulation. Since the material flows through the mesh, additional data is necessary in an Eulerian formulation to describe the current contents of an element and additional calculations must be performed to update the data. The additional calculations, which account for the material transport between the elements, are usually much more expensive than the Lagrangian terms in the calculation. As a consequence, Eulerian calculations have been restricted to hypervelocity impacts, which cannot be solved in any other manner. This paper discusses strategies for restructuring the transport calculations so that the Eulerian formulation may be applied to a broader range of problems in science and engineering. Example calculations, performed on a workstation, are presented to demonstrate the efficiency of the proposed strategies.This research was supported by the National Science Foundation Division of Design and manufacturing Systems grant DDM 90-09545.     

Coupling of fully Eulerian and arbitrary Lagrangian–Eulerian methods for fluid-structure interaction computations

We present a specific application of the fluid-solid interface-tracking/interface-capturing technique (FSITICT) for solving fluid-structure interaction. Specifically, in the FSITICT, we choose as interface-tracking technique the arbitrary Lagrangian–Eulerian method and as interface-capturing technique the fully Eulerian approach, leading to the Eulerian-arbitrary Lagrangian–Eulerian (EALE) technique. Using this approach, the domain is partitioned into two sub-domains in which the different methods are used for the numerical solution. The discretization is based on a monolithic solver in which finite differences are used for temporal integration and a Galerkin finite element method for spatial discretization. The nonlinear problem is treated with Newton’s method. The method combines advantages of both sub-frameworks, which is demonstrated with the help of some benchmarks.     

Eulerian formulation using stabilized finite element method for large deformation solid dynamics

This paper describes an Eulerian formulation for large deformation solid dynamics. In the present Eulerian approach, an advective equation is solved using the Stream‐Upwind/Petrov–Galerkin finite element method. The Eulerian finite element method is applied to path‐dependent solid analyses such as impact bar and ductile necking problems. These computational results using the Eulerian finite element method are compared with the results obtained from using the Lagrangian finite element method and an Eulerian formulation based on a finite difference method. Copyright © 2007 John Wiley & Sons, Ltd.     

Families of continuous spin tensors and applications in continuum mechanics

A new method is proposed for generating families of continuous spin tensors associated with families of corotational rates of second-order tensors using isotropic tensor functions of the same tensor arguments and different forms of continuous antisymmetric scalar spin functions of scalar arguments. Tensor functions are represented in terms of eigenprojections of a symmetric tensor S, which is one of the arguments of these functions. Each member of the generated family is represented as the sum of some basic spin tensor associated with the basic corotational tensor rate and the above-mentioned tensor function, whose structure is matched to the structure of the tensor function required to construct the twirl tensor of the triad of orthonormal eigenvectors of the tensor S (but this twirl tensor itself does not belong to the family of continuous spin tensors). The developed method is used in continuum mechanics to generate two families of continuous spin tensors associated with two families of objective corotational rates: Lagrangian and Eulerian. In these families, isotropic tensor functions are constructed using Lagrangian and Eulerian tensor arguments of the kinematic type, respectively. It is shown that if the same scalar spin function is used in deriving tensor functions of Lagrangian and Eulerian tensor arguments, then the corotational tensor rates associated with the generated spin tensors are objective (Lagrangian and Eulerian) counterparts of each other. It is shown that the spin tensors associated with the classical Eulerian corotational tensor rates (Zaremba–Jaumann, Green–Naghdi, d-rate) and their Lagrangian counterparts (including material rate) belong to the generated families of continuous spin tensors. It is also shown that both of these families of continuous spin tensors are subfamilies of the families of material spin tensors derived by Xiao et al. (J Elast 52:1–41, 1998). It is noted that the twirl tensors of the Lagrangian and Eulerian triads associated with the Gurtin–Spear corotational rates of tensors belong to the families of material spin tensors but do not belong to the families of continuous spin tensors. The final section gives expressions of continuous spin tensors from families associated with the families of Lagrangian and Eulerian corotational tensor rates which are appropriate for applications.     

A three‐invariant hardening plasticity for numerical simulation of powder forming processes via the arbitrary Lagrangian–Eulerian FE model

In this paper, a three‐invariant cap plasticity model with an isotropic hardening rule is presented for numerical simulation of powder compaction processes. A general form is developed for single‐cap plasticity which can be compared with some common double‐surface plasticity models proposed for powders in literature. The constitutive elasto‐plastic matrix and its components are derived based on the definition of yield surface, hardening parameter and non‐linear elastic behaviour, as function of relative density of powder. Different aspects of the new single plasticity are illustrated by generating the classical plasticity models as special cases of the proposed model. The procedure for determination of powder parameters is described by fitting the model to reproduce data from triaxial compression and confining pressure experiments. The three‐invariant cap plasticity is performed within the framework of an arbitrary Lagrangian–Eulerian formulation, in order to predict the non‐uniform relative density distribution during large deformation of powder die pressing. In ALE formulation, the reference configuration is used for describing the motion, instead of material configuration in Lagrangian, and spatial configuration in Eulerian formulation. This formulation introduces some convective terms in the finite element equations and consists of two phases. Each time step is analysed according to Lagrangian phase until required convergence is attained. Then, the Eulerian phase is applied to keep mesh configuration regular. Because of relative displacement between mesh and material, all dependent variables such as stress and strain are converted through the Eulerian phase. Finally, the numerical schemes are examined for efficiency and accuracy in the modelling of a rotational flanged component, an automotive component, a conical shaped‐charge liner and a connecting‐rod. Copyright © 2005 John Wiley & Sons, Ltd.     

A comparison of modelling methods for polydispersed wet‐steam flow

Spontaneous nucleation of water droplets in moist air or steam may result in droplet spectra which are complex in shape and which span a broad range of sizes. This is particularly true if the flow is transonic or supersonic with shock waves present, or if an already droplet‐laden flow re‐expands to give secondary or tertiary nucleations. Computation of such flows requires careful modelling of the size distributions if two‐phase behaviour is to be accurately predicted. In this paper, three methods are presented for treating size distributions and growth of the liquid phase in condensing steam: a mixed Eulerian–Lagrangian method, a fully Eulerian method, and a method based on moments of the droplet spectra. These are compared by computing condensing flow within a one‐dimensional supersonic nozzle under conditions that yield very different types of size spectra. Copyright © 2003 John Wiley & Sons, Ltd.     

A hybrid variational‐collocation immersed method for fluid‐structure interaction using unstructured T‐splines

We present a hybrid variational‐collocation, immersed, and fully‐implicit formulation for fluid‐structure interaction (FSI) using unstructured T‐splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational‐collocation method. We use T‐splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T‐splines offer us two main advantages against non‐uniform rational B‐splines: they can be locally refined and they are unstructured. The generalized‐α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T‐junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 John Wiley & Sons, Ltd.     

Full Eulerian simulations of biconcave neo-Hookean particles in a Poiseuille flow

For a given initial configuration of a multi-component geometry represented by voxel-based data on a fixed Cartesian mesh, a full Eulerian finite difference method facilitates solution of dynamic interaction problems between Newtonian fluid and hyperelastic material. The solid volume fraction, and the left Cauchy–Green deformation tensor are temporally updated on the Eulerian frame, respectively, to distinguish the fluid and solid phases, and to describe the solid deformation. The simulation method is applied to two- and three-dimensional motions of two biconcave neo-Hookean particles in a Poiseuille flow. Similar to the numerical study on the red blood cell motion in a circular pipe (Gong et al. in J Biomech Eng 131:074504, 2009), in which Skalak’s constitutive laws of the membrane are considered, the deformation, the relative position and orientation of a pair of particles are strongly dependent upon the initial configuration. The increase in the apparent viscosity is dependent upon the developed arrangement of the particles. The present Eulerian approach is demonstrated that it has the potential to be easily extended to larger system problems involving a large number of particles of complicated geometries.     

Eulerian elasto‐visco‐plastic formulations for residual stress prediction

A stabilized, Galerkin finite element formulation for modeling the elasto‐visco‐plastic response of quasi‐steady‐state processes, such as welding, laser surfacing, rolling and extrusion, is presented in an Eulerian frame. The mixed formulation consists of four field variables, such as velocity, stress, deformation gradient and internal variable, which is used to describe the evolution of the material's resistance to plastic flow. The streamline upwind Petrov–Galerkin method is used to eliminate spurious oscillations, which may be caused by the convection‐type of stress, deformation gradient and internal variable evolution equations. A progressive solution strategy is introduced to improve the convergence of the Newton–Raphson solution procedure. Two two‐dimensional numerical examples are implemented to verify the accuracy of the Eulerian formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

CTH: A three-dimensional shock wave physics code

CTH is a software system under development at Sandia National Laboratories Albuquerque to model multidimensional, multi-material, large deformation, strong shock wave physics. One-dimensional recti-linear, cylindrical, and spherical meshes; two-dimensional rectangular, and cylindrical meshes; and three-dimensional rectangular meshes are currently available. A two-step Eulerian solution scheme is used with these meshes. The first step is a Lagrangian step in which the cells distort to follow the material motion. The second step is a remesh step where the distorted cells are mapped back to the Eulerian mesh.

CTH has several models that are useful for simulating strong shock, large deformation events. Both tabular and analytic equations of state are available. CTH can model elastic-plastic behavior, high explosive detonation, fracture, and motion of fragments smaller than a computational cell. The elastic-plastic model is elastic-perfectly plastic with thermal softening. A programmed burn model is available for modelling high explosive detonation. The Jones-Wilkins-Lee equation of state is available for modelling high explosive reaction products. Fracture can be initiated based on pressure or principle stress. A special model is available for moving fragments smaller than a computational cell with statistically the correct velocity. This model is very useful for analyzing fragmentation experiments and experiments with witness plates.

CTH has been carefully designed to minimize the dispersion present in Eulerian codes. It has a high-resolution interface tracker that prevents breakup and distortion of material interfaces. It uses second order convection schemes to flux all quantities between cells.

This paper describes the models, and novel features of CTH. Special emphasis will be placed on the features that are novel to CTH or are not direct generalizations of two-dimensional models. Another paper by Trucano and McGlaun (1989) describes several hypervelocity impact calculations using CTH.     

Computational and experimental studies on bed dynamics of a gas–solid fluidized bed using Geldart-A particle: A comparison

The bed dynamics of a two-dimensional gas–solid fluidized bed is studied experimentally and computationally using Geldart-A particles. Commercial software ANSYS FLUENT 13 is used for computational studies. Unsteady behavior of gas–solid fluidized bed is simulated by using the Eulerian–Eulerian model coupled with the kinetic theory of granular flow. The two-equation standard k?? model is used to describe the turbulent quantities. The simulation predictions are compared with experimentally observed data on volume fraction, bed pressure drop and bed expansion ratio. The results of simulations are found to be in close agreement with the experimental observations, implying that computational fluid dynamics (CFD) can be used for the design of an efficient bench-scale catalytic fluidized bed reactor.     

An immersed boundary element method for level‐set based topology optimization

In this paper, we propose a new BEM for level‐set based topology optimization. In the proposed BEM, the nodal coordinates of the boundary element are replaced with the nodal level‐set function and the nodal coordinates of the Eulerian mesh that maintains the level‐set function. Because this replacement causes the nodal coordinates of the boundary element to disappear, the boundary element mesh appears to be immersed in the Eulerian mesh. Therefore, we call the proposed BEM an immersed BEM. The relationship between the nodal coordinates of the boundary element and the nodal level‐set function of the Eulerian mesh is clearly represented, and therefore, the sensitivities with respect to the nodal level‐set function are strictly derived in the immersed BEM. Furthermore, the immersed BEM completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization. By using the immersed BEM, we construct a concrete topology optimization method for solving the minimum compliance problem. We provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results. Copyright © 2012 John Wiley & Sons, Ltd.     

Source terms in Eulerian–Lagrangian contaminant transport simulation

Standard Eulerian treatment of source terms in Eulerian–Lagrangian numerical simulations results in poor performance at higher Courant numbers. To regain the customary high accuracy of Eulerian–Lagrangian methods under these conditions, a Lagrangian treatment of source terms is needed. It is also important to include the effects of fluid sources as well as contaminant sources. A new Lagrangian source formulation is presented, which has been implemented in a finite element simulator for contaminant transport in rivers and estuaries. Test problems demonstrate the high accuracy of the technique under a range of conditions, and its applicability to general multi‐dimensional problems and unstructured grids. Copyright © 2004 John Wiley & Sons, Ltd.     

Thermo‐elasto‐plastic finite element analysis of quasi‐state processes in Eulerian reference frames

An Eulerian finite element formulation for quasi‐state one way coupled thermo‐elasto‐plastic systems is presented. The formulation is suitable for modeling material processes such as welding and laser surfacing. In an Eulerian frame, the solution field of a quasi‐state process becomes steady state for the heat transfer problem and static for the stress problem. A mixed small deformation displacement elasto‐plastic formulation is proposed. The formulation accounts for temperature dependent material properties and exhibits a robust convergence. Streamline upwind Petrov–Galerkin (SUPG) is used to remove spurious oscillations. Smoothing functions are introduced to relax the non‐differentiable evolution equations and allow for the use of gradient (stiffness) solution scheme via the Newton–Raphson method. A 3‐dimensional simulation of a laser surfacing process is presented to exemplify the formulation. Results from the Eulerian formulation are in good agreement with results from the conventional Lagrangian formulation. However, the Eulerian formulation is approximately 15 times faster than the Lagrangian. Copyright © 2001 John Wiley & Sons, Ltd.     

绝对节点坐标列式流体单元建模方法及在液体晃动问题中的研究

计算流体力学的建模方法主要采用欧拉描述,而多体系统动力学的建模方法主要采用拉格朗日描述。与欧拉描述关注于流过空间固定点或固定体积上的流场状态不同,绝对节点坐标列式流体单元采用拉格朗日描述,能够跟踪流体物质点,建立流体与多体系统的统一描述。该文在绝对节点坐标列式流体单元方法基础上,提出和完善单元建模理论,并使用绝对节点坐标列式流体单元实现了对流体系统的建模,并首次将绝对节点坐标列式流体单元应用于液体晃动分析,初步验证了理论的正确性和可行性。仿真结果表明,在单元数量较少情况下,绝对节点坐标列式流体单元可以满足晃动计算需求。     

Finite element analysis of internal flows with heat transfer

This paper presents a finite element-based model for the prediction of 2-D and 3-D internal flow problems. The Eulerian velocity correction method is used which can render a fast finite element code comparable with the finite difference methods. Nine different models for turbulent flows are incorporated in the code. A modified wall function approach for solving the energy equation with high Reynolds number models is presented for the first time. This is an extension of the wall function approach of Benim and Zinser and the method is insensitive to initial approximation. The performance of the nine turbulent models is evaluated by solving flow through pipes. The code is used to predict various internal flows such as flow in the diffuser and flow in a ribbed channel. The same Eulerian velocity correction method is extended to predict the 3-D laminar flows in various ducts. The steady state results have been compared with benchmark solutions and the agreement appears to be good.     

Modeling mass loading effects in industrial cyclones by a combined Eulerian–Lagrangian approach

Cyclone separation is studied by means of numerical simulations. While the gas flow is modeled by a modified Reynolds stress (RS) model, the behavior of the particles is pictured by a combined Eulerian–Lagrangian approach. A mono-disperse Eulerian particle phase is utilized to account for inter-particle collisions, while the effects of fractional separation and particle-wall collisions are considered by poly-disperse Lagrangian particles. The above particle models interact in two ways. On the one hand, the Lagrangian particles determine the local mean diameter of the substitute Eulerian particle class. On the other hand, especially in regions of high particle concentration, the Eulerian particle phase exerts an additional collisional force onto the Lagrangian particle trajectories. An industrial cyclone is chosen as a test case and the numerical results are evaluated with respect to pressure drop as well as to global and fractional separation efficiency. In this context the influence of the cyclone’s mass loading and wall roughness is highlighted. Simulations indicate that the separation efficiency improves with increasing mass loading until an excess loading is reached while at the same time the pressure drop is reduced. Furthermore, it can be shown that rough walls lead to a reduction of separation efficiency while simultaneously the pressure drop decreases. The simulations results are compared with both an analytic theory of Muschelknautz [Die Berechnung von Zykonabscheidern für Gase. Chem Ing Techn 44, (1+2):63–71, 1972] as well as with real plant measurements.