Lagrangian Mesh Research Articles

Solution of Two-Dimensional Vorticity Equation on a Lagrangian Mesh

Anovel,vorticity-basedsolutionmethodologyhasbeendevelopedtocomputeunsteadye owpastbodies.Vorticity is evolved on a set of points, and the vorticity in the remainder of the e eld is determined by linear interpolation. Interpolationisaccomplished byDelaunaytriangularizationofthepointsin the e eld. Triangulationofthevorticity e eldprovidesabasistointegratethevorticitytocomputethevelocity.Nodalconnectivityfromthetriangularization also provides a list of the neighboring points that are used to construct a second-order least-squares e t of the vorticity. First- and second-order spatial derivatives can then be computed based on this polynomial e t. Surface vorticity on the body is computed to satisfy the no-slip boundary condition and is introduced into the e ow via diffusion. A diffusion transport velocity was derived to account for spatial movementof the vorticity dueto viscous diffusion. The points are advected by the sum of the induced velocity (computed from the Biot ‐Savart integral ) and the diffusion velocity. The remaining diffusion term includes a form of the Laplacian and is computed directly. This solution scheme was found to be stable as applied to the problem of impulsively started e ow about a circular cylinder and e at plate. Comparisons with experimental and the Blasius boundary-layer solution for a e at plate were used to demonstrate the effectiveness of this method.

Roadmap for crashworthiness finite element simulation of roadside safety structures

The subject of this investigation to develop an accurate simulation of truck impacting a strong-post W-beam guardrail. Detailed methods for system simulation are proposed and three major issues which involve the use of springs to simulate component crashworthiness behavior is investigated. Rail to blockout bolt connection, soil–post dynamic interaction, and the effect of guardrail ends are modeled and simulated. Soil–post interaction is modeled using both Lagrangian and Eulerian meshes; results using the two methods are presented. The present paper provides a roadmap for simulation of highway safety structures.

Solution of two-dimensional vorticity equation on a Lagrangian mesh

Solution of two-dimensional vorticity equation on a Lagrangian mesh

Validated crash simulation of the most common guardrail system in the USA

The subject of this investigation is the development of an accurate simulation of a truck impacting a commonly found strong post/w-beam guardrail system. Detailed methods for system simulation are proposed herein and several major issues, namely. the use of springs to simulate component crashworthiness behaviour, are investigated. Rail to block-out bolt connection, soil-post-dynamic interaction, and effect of ends of guardrail are modelled and simulated. Soil-post interaction is modelled using both Lagrangian and Eulerian meshes and the results using the two methods are presented herein. Both qualitative and quantitative validation of the crash simulation is presented and discussed. The present paper provides a roadmap for the simulation of highway safety structures.

Two-dimensional Lagrangian method for elastic-plastic flow

The present study uses Schulz's two-dimensional radiation hydrodynamic code as a framework. Three new features are added to the basic code. First, finite difference equations for deviatoric stresses and strains, shear modulus and yield strengths are written, i.e. compatibly with Schulz's special formulations. Material properties (i.e. shear modulus and yield strength) are dependent on pressure, compression and equivalent plastic strain (i.e. Steinberg-Guinan model). Second, modifications to the artificial viscosity are made so that the acceleration from the artificial viscosity will project onto the unit vector in the direction of local acceleration. This projection eliminates ‘false heating’ and helps maintain a stable computational grid. Also a very sharp shock front location will be obtained through this new method. Third, a semi-Eulerian method that allows the material flow through the Lagrangian mesh is used for treating the Lagrangian sliding interfaces.

Parallel Transient Dynamics Simulations: Algorithms for Contact Detection and Smoothed Particle Hydrodynamics

Transient dynamics simulations are commonly used to model phenomena such as car crashes, underwater explosions, and the response of shipping containers to high-speed impacts. Physical objects in such a simulation are typically represented by Lagrangian meshes because the meshes can move and deform with the objects as they undergo stress. Fluids (gasoline, water) or fluid-like materials (soil) in the simulation can be modeled using the techniques of smoothed particle hydrodynamics. Implementing a hybrid mesh/particle model on a massively parallel computer poses several difficult challenges. One challenge is to simultaneously parallelize and load-balance both the mesh and particle portions of the computation. A second challenge is to efficiently detect the contacts that occur within the deforming mesh and between mesh elements and particles as the simulation proceeds. These contacts impart forces to the mesh elements and particles which must be computed at each timestep to accurately capture the physics of interest. In this paper we describe new parallel algorithms for smoothed particle hydrodynamics and contact detection which turn out to have several key features in common. Additionally, we describe how to join the new algorithms with traditional parallel finite element techniques to create an integrated particle/mesh transient dynamics simulation. Our approach to this problem differs from previous work in that we use three different parallel decompositions, a static one for the finite element analysis and dynamic ones for particles and for contact detection. We have implemented our ideas in a parallel version of the transient dynamics code PRONTO-3D and present results for the code running on the Pentium-based Intel Teraflop machine at Sandia.

Integrated numerical simulation of indirect laser-driven implosion for ICF

A NIF-type hohlraum target (Lindl 1995; CEA 1995) is analyzed in an integrated calculation that uses last revision (4.2) of MULTI2D code (Ramis et al. 1992). The numerical simulation includes: Lagrangian mesh, tracking of laser beams, inverse bremsstrahlung absorption, 3D radiation transport, radiation-induced hydrodynamics of casing and window panes, and radiatively driven capsule implosion. Electron heat conduction (Spitzer with flux limiter), tabulated Equations of State (Sesame), and LTE opacities are also included. Detailed results on hydrodynamic evolution of the system, compression symmetry, and energy balance are also provided.

Steady-state finite-element analysis of cold forward extrusion

Finite-element analysis of metal forming processes is treated predominantly by the Updated Lagrangian formulation in which the mesh moves and deforms in space, in accordance with the deformation history of the material. However, the steady-state deformation characteristics of some metal forming processes such as extrusion, drawing and rolling, can be analysed advantageously using an Eulerian formulation in which the finite-element mesh is fixed in space. Extension of the latter to include work hardening material response can be accomplished through the use of a combined Eulerian–Lagrangian formulation. Such extension is based on the utilisation of an Updated Lagrangian temporary mesh, to calculate the strain and stress fields, coupled with a mathematical scheme to interpolate the strain and stress tensors into the Eulerian mesh, in order to determine the new distributions at the fixed mesh positions. An example of steady-state cold forward rod extrusion is analysed using both the Updated Lagrangian and the combined Eulerian–Lagrangian formulations. Comparisons are made between the theoretical finite-element predictions and experimental data obtained through the utilisation of visioplasticity and micro-hardness techniques.

Computation of Unsteady Separated Flow Fields Using Anisotropic Vorticity Elements

A novel computational methodology to compute two-dimensional unsteady separated flow fields using a vorticity based formulation is presented. Unlike traditional vortex methods, the elements used in this method are designed to take advantage of the natural anisotropy of most external flows. These vortex elements are disjoint and of compact support. The vorticity is uniform over rectangular elements whose initial thickness is set by a diffusion length scale. The elements are a mathematical construction which enables the vorticity of the flow to be created and followed numerically, and the Biot-Savart integral to be performed. This integral specifies the associated velocity field. Since the vorticity of a single element is of finite extent, the velocity associated with an element is given by a nonsingular expression. Viscous diffusion effects are modeled using random walk and the advection term is computed by transporting the vorticity elements with the local velocity field. Consequently, this Lagrangian mesh continually evolves through time. Since pressure does not explicitly appear in the formulation, surface pressures are computed using a stagnation enthalpy formulation. These elements are used to compute vorticity production, accumulation, transport and viscous diffusion mechanisms for unsteady separated flow fields past a pitching airfoil. Dynamic stall vortex initiation and development were examined and compared with existing experimental data. Surface pressure data and integrated force coefficient data were found to be in excellent agreement with experimental data. Effects of geometry were provided with baseline calculations of the unsteady flow past an impulsively started cylinder. Both qualitative and quantitative comparisons with experimental data for equivalent test conditions establish the applicability of this approach to depict unsteady separated flow fields.

Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems

The material point method is an evolution of particle-in-cell methods which utilize two meshes, one a material or Lagrangian mesh defined over material of the body under consideration, and the second a spatial or Eulerian mesh defined over the computational domain. Although meshes are used, they have none of the negative aspects normally associated with conventional Eulerian or Lagrangian approaches. The advantages of both the Eulerian and Lagrangian methods are achieved by using the appropriate frame for each aspect of the computation, with a mapping between the two meshes that is performed at each step in the loading process. The numerical dissipation normally displayed by an Eulerian method because of advection is avoided by using a Lagrangian step; the mesh distortion associated with the Lagrangian method is prevented by mapping to a user-controlled mesh. Furthermore, explicit material points can be tracked through the process of deformation, thereby alleviating the need to map history variables. As a consequence, problems which have caused severe numerical difficulties with conventional methods are handled fairly routinely. Examples of such problems are the upsetting of billets and the Taylor problem of cylinders impacting a rigid wall. Numerical solutions to these problems are obtained with the material point method and where possible comparisons with experimental data and existing numerical solutions are presented.

A New Scheme for Multidimensional Line Transfer. III. A Two-dimensional Lagrangian Variable Tensor Method with Discontinuous Finite-Element SN Transport

We describe a new code{emdash}ALTAIR{emdash}that has been developed to solve the time-dependent non-LTE problem for a multilevel atom in two-dimensional axisymmetric geometry on a Lagrangian mesh of arbitrary complexity. The method and results are presented here for a two-level atom. The extension to a multilevel atom is made by the equivalent two-level atom (ETLA) methods we have recently described in Paper II of this series. The source function of the line formed by the two-level atom is iterated to self-consistency with the radiation field, which is obtained from a system of coupled moment equations that employs an Eddington tensor closure using the double-splitting method described in Paper I of this series. The Eddington tensor itself is derived from a formal solution of the photon transport equation, which is based on a new discontinuous finite-element technique; the tensor moment system uses a continuous representation. The Eddington tensor is updated in an outer iteration loop, within which the double-splitting iteration is used to find the self-consistent source function for a given tensor. The resulting method shows very rapid convergence of the scattering iteration, even for optically thick regions with scattering albedo near unity. In addition, the spatial structure of the solution ismore » rather insensitive to the shape of the spatial zones; the calculation of problems with zone aspect ratios of 1000 to 1 causes no difficulty. Application to several benchmark calculations, including radiative transfer in a nonorthogonal mesh, is discussed. We find that for two-dimensional problems of astrophysical interest, many angle rays are required to ensure an accurate solution. {copyright} {ital 1996 The American Astronomical Society.}« less

Computation of incipient separation via solution of the vorticity equation on a lagrangian mesh

A novel vorticity based solution methodology has been developed to compute unsteady flow, particularly separation. The vorticity of the flow is determined on a set of points and the vorticity in the field is computed by linear interpolation. This is accomplished by the construction of a connected set of triangular elements formed by Delaunay triangularization of the points in the field. Triangulation of the vorticity field enables computation of first and second order spatial derivatives of the vorticity field using a least squares formulation. An effective diffusion transport velocity was developed to account for spatial movement of the vorticity due to viscosity. Diffusion velocity and direct calculation of the Laplacian allows for a deterministic solution of viscous diffusion. The points are then advected at both the induced velocity (computed from the Biot-Savart integral) and the diffusion velocity. This solution scheme was found to be stable as it was applied to the problem of impulsively started flow about a NACA 0012 airfoil. The phenomena of incipient separation as well as initial development of the unsteady wake was examined.

Softened Lagrangian hydrodynamics for cosmology

view Abstract Citations (95) References (12) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Softened Lagrangian Hydrodynamics for Cosmology Gnedin, Nickolay Y. Abstract A new approach to cosmological hydrodynamics is discussed that is based on a moving, quasi-Lagrangian mesh. The softened Lagrangian hydrodynamics (SLH) method utilizes a high-resolution Lagrangian hydrodynamic code combined with a low-resolution Eulerian solver to deal with severe mesh distortions. Most of the volume of a simulation is treated with the Lagrangian code and only in sites where the Lagrangian approach fails, due to mesh distortions, does the Eulerian part of the code step in. This approach utilizes a high-resolution gravity solver without use of TREE or P3M methods; Poisson's equation is solved on the moving baryonic mesh using a simple relaxation technique. The dark matter is included by means of the cloud-in-cell method on the Lagrangian mesh. All three components of the cosmological code--gravity, dark matter, and baryons--are thus treated selft-consistently with exactly the same resolution. The computer code based on the SLH approach is described in detail, and comparison with existing Eulerian and smooth particle hydrodynamics (SPH) codes is presented. For most purposes the SLH approach turns out to be the intermediate between Eulerian and SPH codes, but it outperforms both of these approaches in resolving caustics. Thus, it may turn out to be a valuable tool to study galaxy formation. Publication: The Astrophysical Journal Supplement Series Pub Date: April 1995 DOI: 10.1086/192141 Bibcode: 1995ApJS...97..231G Keywords: Astronomical Models; Computational Grids; Computerized Simulation; Cosmology; Euler-Lagrange Equation; Hydrodynamics; Lagrangian Function; Mathematical Models; Baryons; Dark Matter; Galactic Evolution; Gravitation; Missing Mass (Astrophysics); Poisson Equation; Astrophysics; COSMOLOGY: THEORY; HYDRODYNAMICS; METHODS: NUMERICAL full text sources ADS |

On the pseudomaterial approach for the analysis of transient forming processes

AbstractA fixed‐mesh method for the analysis of transient forming processes is presented. The mesh covers material regions and zones through which the material may flow. These last zones are identified by a pseudomaterial with relatively small physical parameters. During time processing, the interface between both materials is followed by an arbitrary Lagrangian mesh. This technique appears to be suitable for the treatment of moving surfaces with sharp corners. A particular boundary condition for the Navier‐Stokes equations is also introduced in order to model a porous wall.

A hybrid method for determining material properties from instrumented micro-indentation experiments

The impact code EPIC was employed to study the relationship between the applied force and the penetration depth in a micrometer-scale indentation experiment with oxygen free high conductivity (OFHC) copper. EPIC is an elastic-plastic finite element code that uses a Lagrangian formulation and triangular mesh, which can accommodate large deformation without the need to remesh during the computation process. By fitting the force-penetration curves for a triangular indenter with second degree polynomials, it was demonstrated that the fit changed with two material constants in the constitutive equation. A systematic procedure for determining the material constants is described that is based on matching either the slope or the curvature of the force penetration depth curves from numerical simulation and experiments. It is concluded that material constants can be determined from indentation data obtained using pyramidal or spherical indenters as well as a flat-ended indenter.

RHALE: A MMALE shock physics code written in C++

This paper describes RHALE, a multi-material arbitrary Lagrangian-Eulerian (MMALE) shock physics code. RHALE is the successor to CTH, Sandia's 3-D Eulerian shock physics code, and will be capable of solving problems that CTH cannot adequately address. We discuss the new Lagrangian capabilities of RHALE, which include arbitrary mesh connectivity, superior artificial viscosity, and improved equations of state. We also discuss some of the issues we have encountered in the choice of an axisymmetric element technology and our resolution of these issues. We discuss the MMALE algorithms that have been extended for arbitrary grids in both two and three dimensions and present the results of calculations that are of interest to the hypervelocity impact community. The MMALE addition to RHALE provides the accuracy of a Lagrangian code while allowing a calculation to proceed under very large distortions. Coupling an arbitrary quadrilateral or hexahedral grid to the MMALE solution facilitates modeling of complex shapes with a minimum number of computational cells. RHALE allows regions of a problem to be modeled with Lagrangian, Eulerian or ALE meshes. In addition, regions can switch from Lagrangian to ALE to Eulerian based on user input or mesh distortion. For ALE meshes, new node locations are determined with equipotential schemes. Element quantities are advected with donor, van Leer, or Super-B algorithms. Nodal quantities are advected with the second order SHALE or HIS algorithms. Currently, material interfaces are determined with the SLIC algorithm; however, both two and three dimensional versions of Youngs' interface tracker are being investigated. To facilitate the development of such a lengthy code, we choose to write in the C++ programming language. We feel that object-oriented programming techniques are superior to conventional programming techniques. However, we discuss a few of the efficiency problems we have encountered using these techniques and how we have addressed these problems.

On adaptive-grid computations of variable stars

We show that the use of an implicit adaptive-grid technique is an efficient and up-to-date approach for the calculations of radial oscillations in variable stars. We chose as an illustrative example the radiative envelope of an RR Lyrae variable. For the hydrostatic initial model we compare the Lagrangian ratioed zoning with an adaptive-grid rezoning. We show that the adaptive-grid yields an optimal distribution of the mesh points in the sense that the relevant physical features, the hydrogen and first and second helium ionization zones, are well resolved. For the hydrodynamical evolution we present the full-amplitude model for both the Lagrangian and adaptive-grid computations. We perform a detailed comparison and show that the adaptive-grid method yields limit cycle solutions that are substantially improved over the Lagrangian grid model. This is due to the fact that the Lagrangian mesh sweeps through the ionization zones twice during one oscillation period, whereas the adaptive-mesh resolves them and tracks them continuously. The results are, in particular, smooth radial velocity and light curves. Beyond a physically better defined solution we also observe larger time steps for the convergence towards the limit cycle and for the evolution during one period.

Diapirism and topography

SUMMARY A new numerical technique using markers and a deformable Lagrangian mesh is used to study the deformation of the surface above a rising diapir. This method allows modelling of a free-surface boundary in a self-consistent way. The method makes it possible to investigate many different problems with non-regular geometry. The codes were verified through comparison with analytical solutions for the initial stages and with other numerical codes for mature stages. Our simulations led to the following conclusions. (1) The growth rate of the diapir is more strongly influenced by the viscosity contrast than the wavelengths. This is due to the strong growth rate difference during the initial stage. (2) The maximum elevation above the diapir linearly increases with increasing wavelength and is approximately the same for different viscosity contrasts. (3) The elevation will be considerably higher for a low-viscosity diapir at a given depth than for an isoviscous diapir. Comparisons for different thicknesses of the buoyant layer showed that the highest topography is produced when the two layers have equal thickness. We showed the dependence of the topographic behaviour on the parameter R (the ratio of the density difference between two layers to the density of the upper layer). It was found that topography behaves linearly up to R = 0.15. It indicates that a posteriori estimation of topography for traditional calculations using the free-slip boundary condition works well within this limit.

Inclusion of evolutionary damage measures in Eulerian wavecodes

Most continuum descriptions of damage evolution generally require history-dependent material variables. The Lagrangian formulation of the continuum equations is the natural coordinate system for tracking material-history quantities. In numerical simulations of dynamic events such as in penetration and perforation of target plates by projectiles, the Lagrangian mesh can become severely compressed and distorted which effectively terminates advancing the solution in time. On the other hand, the Eulerian formulation, with its fixed coordinate system, does not suffer from mesh distortion. However, Eulerian descriptions usually follow only what crosses cell boundaries, and instead of computing the time history of material particles, they describe the average instantaneous state of a material in a computational zone. This paper describes the inclusion and evaluation of the history of equivalent plastic strain, which is a representative measure of damage, as an internal state variable within an Eulerian numerical framework. The formulation and method of advection of the equivalent plastic strain are described, and the results for two example problems are discussed and comparisons are made with the results of Lagrangian calculations.

A grey transport acceleration method for time-dependent radiative transfer problems

A new iterative method for solving the time-dependent multifrequency radiative transfer equations is described. The method is applicable to semi-implicit time discretizations that generate a linear steady-state multifrequency transport problem with pseudo-scattering within each time step. The standard “lambda” iteration method is shown to often converge slowly for such problems, and the new grey transport acceleration (GTA) method; based on accelerating the lambda method by employing a grey, or frequency-independent transport equation, is developed. The GTA method is shown, theoretically by an iterative Fourier analysis, and experimentally by numerical calculations, to converge significantly faster than the lambda method. In addition, the GTA method is conceptually simple to implement for general differencing schemes, on either Eulerian or Lagrangian meshes.