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ALE3D: An Arbitrary Lagrangian-Eulerian Multi-Physics Code

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ALE3D: An Arbitrary
Lagrangian-Eulerian
Multi-Physics Code
Charles Noble, Andrew Anderson, Nathan Barton, Jamie
Bramwell, Arlie Capps, Michael Chang, Jin Chou, David
Dawson, Emily Diana, Timothy Dunn, Douglas Faux, Aaron
Fisher, Patrick Greene, Ines Heinz, Yuliya Kanarska, Saad
Khairallah, Benjamin Liu, Jon Margraf, Albert Nichols, Robert
Nourgaliev, Michael Puso, James Reus, Peter Robinson, Alek
Shestakov, Jerome Solberg, Daniel Taller, Paul Tsuji,
Christopher White, Jeremy White
May 23, 2017
ALE3D: An Arbitrary Lagrangian-Eulerian Multi-Physics Code
Lawrence Livermore National Laboratory ii
Disclaimer
This document was prepared as an account of work sponsored by an agency of the United States
government. Neither the United States government nor Lawrence Livermore National Security,
LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any
legal liability or responsibility for the accuracy, completeness, or usefulness of any information,
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name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its
endorsement, recommendation, or favoring by the United States government or Lawrence
Livermore National Security, LLC. The views and opinions of authors expressed herein do not
necessarily state or reflect those of the United States government or Lawrence Livermore
National Security, LLC, and shall not be used for advertising or product endorsement purposes.
Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security,
LLC, for the U.S. Department of Energy, National Nuclear Security Administration under
Contract DE-AC52-07NA27344.
LLNL-TR-732040
Introduction
ALE3D is a multi-physics numerical simulation software tool utilizing arbitrary-Lagrangian-
Eulerian (ALE) techniques. The code is written to address both two-dimensional (2D plane and
axisymmetric) and three-dimensional (3D) physics and engineering problems using a hybrid
finite element and finite volume formulation to model fluid and elastic-plastic response of
materials on an unstructured grid. As shown in Figure 1, ALE3D is a single code that integrates
many physical phenomena.
Figure 1. ALE3D supports a wide range of applications.
ALE3D supports a wide range of simulation needs. The ALE and mesh relaxation capabilities
broaden the scope of application in comparison to tools restricted to Lagrangian-only or
Eulerian-only approaches, while maintaining accuracy and efficiency for large, multi-physics
and complex geometry simulations. For some applications ALE can deliver accuracy similar to
Eulerian techniques using as few as 1/10th the number of mesh elements and a reduction in
memory requirements.
Figure 2 provides a chart showing that ALE3D has an integrated flexible and extendable
architecture. Beyond its foundation as a hydrodynamics and structures code, ALE3D has multi-
physics capabilities that integrate various packages through an operator splitting approach.
Additional ALE3D features include heat conduction, chemical kinetics, species diffusion,
incompressible flow, a wide range of material models, chemistry models, multi-phase flow, and
magnetohydrodynamics, which can be used in numerous combinations for long (implicit) to
short (explicit) time-scale applications.
ALE3D: An Arbitrary Lagrangian-Eulerian Multi-Physics Code
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Figure 2. ALE3D has an integrated flexible and extendable architecture supporting a
variety of mesh generators, and material models can be pluggable components.
The discretized domain or mesh may consist of arbitrarily connected hexahedral, shell and beam
elements. In 2D simulations, the mesh can comprise of arbitrarily connected quadrilaterals. The
mesh can be constructed from disjointed blocks of elements that interact at the boundaries via
slide surfaces or other types of boundary conditions.
Large mesh distortions can be addressed via mesh relaxation and/or the use of ALE techniques
where material is advected through the mesh. All components of the code participate in
advection and all the mature capabilities operate with slide surfaces. Advection is the process
whereby the mesh is modified to alleviate tangling or to preserve an Eulerian grid. Advection is
implemented in a Lagrange plus remap fashion. For each cycle, after a Lagrangian motion, the
state variables flow from the original mesh to the modified mesh. The mesh is allowed to cross
material boundaries and create multi-material elements. The ALE and mesh relaxation capability
broadens the scope of applications in comparison to tools restricted to Lagrangian or Eulerian
(advection) only approaches, while maintaining accuracy and efficiency for large, multi-physics
and complex geometry simulations.
Slide surfaces are boundaries between disjoint sections of the grid that may or may not be in
contact. They represent either physical contact discontinuities or a discontinuity in the zoning.
The unstructured grid is composed of 3D hexagonal elements that can be arbitrarily connected.
Triangular or prismatic elements are not allowed, but shell and beam structural elements can
couple to 3D elements.
The 2D capability includes explicit and implicit hydrodynamics, thermal diffusion, chemistry,
deflagration, shape generation, advection, and most explicit slide surface features.
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Thermal and chemistry packages are tightly coupled and have been successfully used for long
(implicit) to short (explicit) time-scale applications. The incompressible flow module has been
used in simulations of turbulent thermal convection. A multi-phase flow package (a continuum
representation of solid particulate transport in solids, liquids and gases and impulse load at
impact) is available for simulation of multi-phase fluid, structure interaction. Another package
that solves the magnetohydrodynamics equations (MHD in 3D only) can simulate a variety of
high magnetic field, high velocity, thermal applications.
ALE3D operates on a wide variety of platforms, ranging from laptops to the world’s largest
supercomputers. ALE3D has native implementations for Windows™ and Mac workstations for
smaller scale problem sets, and it is portable to virtually any Unix-based machine with C++/C
and Fortran compilers available. The code will also run in parallel on multi-processor
Windows™ and Mac machines. While most users will be interested in Linux-based versions of
the code, it has also been ported to several other lightweight kernel operating systems.
Operation on massively parallel machines has always been a core requirement for the code.
Parallelization is implemented by decomposing the mesh into computational subdomains.
Message passing is the primary mode for communication between domains. The emphasis on
message passing means that it is possible to run parallel computations on a network of
workstations, as well as utilizing multi-core processors available on most workstations.
Explicit Hydrodynamics
ALE3D’s hydrodynamics capability captures the behavior of solids and fluids and has been
successfully used for long (implicit) to short (explicit) time-scale applications. The explicit
hydrodynamics module was developed to model the behavior of objects undergoing deformation
due to the application of shocks in the kilobar regime. For such problems, the natural time step is
consistent with the Courant time scale which governs the stability of the equations.
The code explicitly conserves mass and momentum. Following DYNA3D (Hallquist, 1982), the
stress gradients and strain rates for the Lagrange step are evaluated by a lowest-order finite-
element method. A diagonal mass matrix is used. For second-order accuracy a staggered space
and time grid is also used. The stress gradient calculation has been modified so that it is
represented by an integral of the shape function over the surface of an element rather than an
integral of the gradient of the shape function over the volume. (The stress in general is
discontinuous at the surface so that the stress gradient is a delta function.) This modification
corrects a problem in which distorted elements generated forces even though the stress field was
constant. Likewise, a more accurate volume calculation is done following J. Dukowicz (JCP)
(1984). Hour-glass modes are damped using the method of Flanagan and Belytschko (1981).
The hydrodynamics is energy based rather than temperature based. Energy conservation is not
explicitly enforced but depends on the accuracy of the time integration. This averts the problem
of converting lost kinetic energy into internal energy and overheating materials. The pressure,
viscosity and strain work evaluations are fully time centered. To integrate –PdV for nonlinear
equations-of-state, a third-order Runge-Kutta method is used. Although the explicit
hydrodynamics module is isentropic except at shocks, a temperature variable exists for each
region, and the temperature Equation of State (EOS) can be evaluated either as part of a
constitutive model or from EOS tables.
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Shocks are treated with a scalar artificial viscosity. The default version simply uses
d
v/v for
compressing elements. Both linear and quadratic terms are available. R. Christensen’s monotonic
artificial viscosity (Christensen, 1990) has also been implemented. This scalar viscosity is
constructed using velocity gradients from which the linear component of the gradient has been
subtracted. The form can be derived from the linearized Riemann problem.
The most commonly used equations-of-state and constitutive models are analytic models using
the Mie-Grüneisen and Steinberg-Guinan forms, respectively (Steinberg, 1980 and 1996).
Several engineering constitutive models are available to facilitate modeling of structures.
Tabular equation-of-state models can be accessed through the Livermore equation-of-state
(LEOS) tables. The majority of the available material models are for isotropic materials and a
von Mises yield condition is applied. The deviatoric part of the stress tensor is integrated through
time, and the Jaumann stress rate is used to satisfy objectivity. A number of micro-structurally
based models have also been added to provide a more accurate representation of damage
evolution mechanisms leading to fracture and to capture anisotropy at both the single crystal and
polycrystal size scales. A few of these latter models do not have time-centered energy
evaluations or use high-order integration for the EOS. For all material models, a Navier-Stokes
viscosity is available to augment the stress tensor.
For most problems, high explosive (HE) detonation is simulated by using a programmed burn
model with a beta burn override. Several options are available for computing HE lighting times.
Reactive flow models for HE detonation are also available. These models use pressure and
volume dependent rate laws to describe the detonation process. The reactive flow models are
zoning dependent, need high resolution, and may have limited usefulness in 3D.
Implicit Mechanics
The implicit mechanics module was developed to model problems that evolve at time scales that
are orders of magnitude greater than the Courant time scale that determines the stability of the
explicit equations. The implicit hydrodynamics is a finite element displacement formulation with
single- or eight-point integration. Single-point integration requires the addition of hourglass
stabilization forces. The formulation solves the non-linear equilibrium equations using a
Newton-Raphson iteration surrounding a linearization of the equations. This linearization is
based on estimates for material properties, etc., at the end of the time step. Convergence of the
non-linear iterations is achieved when corrections to the displacements and the nodal forces are
sufficiently small. Slide surface constraints are supported. The implicit time integration can be
run using a quasi-static approximation, or the inertial terms can be included via a Hilber-Hughes-
Taylor stabilization of the standard Newmark time integration. The code can convert
automatically from implicit to explicit hydrodynamics when the time step is sufficiently small
that the explicit integration is more computationally efficient. The reverse transfer from explicit
to implicit hydrodynamics is also possible.
Slide Surfaces
The interaction between multiple material surfaces is captured with a numerical technique called
slide surfaces. As mentioned above, slide surfaces model contact discontinuities or mesh
discontinuities. Slide surfaces may be either two-sided (master / slave) or single-sided. In two-
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sided surfaces, the nodes of each side interact with the faces of the other side to conserve
momentum and enforce impenetrability. With single-sided surfaces, any node may interact with
any face, barring nearest neighbors. Single-sided slides are useful for surfaces that fold upon
themselves and surfaces consisting of many discrete segments.
There are several options for contact enforcement: the point-on-plane method for explicit or
implicit hydrodynamics, the common-plane method (available only for single-sided surfaces) for
explicit hydrodynamics, and a mortar method for implicit hydrodynamics. In addition, ALE3D
provides an automatic slide surface capability with either the point-on-plane or common-plane
approach.
For the point-on-plane method, slide surface velocities are computed with a momentum
conserving algorithm. Each side of a slide surface is independently accelerated using interpolated
physics quantities from the opposing side. First a node is accelerated as if it were on a free
surface or a surface with a pressure boundary condition. The boundary pressure is the average of
the normal stresses on each side of the surface. The acceleration of the node is decomposed into
normal and tangential components. The normal component for each side is combined with the
interpolated normal component from the opposite side to form a center-of-mass acceleration.
This acceleration is used to integrate the velocity. Unless friction is called for, the tangential
acceleration remains that of the free surface. A distinction is then made between master and slave
surfaces for the final application of continuity boundary conditions. This last step corrects for
any lack of perfect continuity in the normal direction due to truncation errors.
Techniques for ordering nodes on one side of the slide surface with respect to nodes on the other
side and for making projections normal to the slide surface are borrowed from DYNA3D. These
techniques have been modified, however, to make them more robust. The ordering algorithm has
been improved so that nodes are not prone to penetrating the slide surface and getting lost. The
normal projection algorithm uses normals that vary across a slide surface element and capture the
effects of curvature of the surface.
The opposing sides of a slide surface may be separated and come into contact during the course
of a calculation. This is called a void. When voids close, the momenta from the opposing sides
are combined to form the center-of-mass momentum. The collision is inelastic for the first layer
of nodes on each side of the slide surface. Void closing is always calculated from the perspective
of the master side. This is done to avoid miscounting momentum transferred from one side to the
other if nodes on each side of the surface close in different cycles.
By default, the tangential velocities on each side of a slide surface are decoupled. However,
coulomb friction can be applied. In this case, a tangential force is applied that opposes any
relative velocity. This force is proportional to the normal force that is inferred from the
acceleration required to change the normal velocity from its free surface value to its center-of-
mass value. For so-called tied sliding both the normal and tangential accelerations are combined
into center-of-mass accelerations. This enables one to use slide surfaces to affect a zoning
change in the middle of a region.
Special provision is made for intersecting or overlapping slide surfaces. Intersections are
assumed to be orthogonal and errors grow to the extent that this condition is violated.
The common-plane method of contact enforcement checks for contact between each pair of
faces, defined as penetration of a plane constructed midway between the two faces. If both faces
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penetrate the plane, then the intersection of their penetrations projected onto the common plane
is the area of contact, and an opposing force (penalty method) is applied to resist and reverse
penetration. This force is distributed to the nodes of the faces in contact.
The mortar method of contact is a face-on-face type of method and is implemented for use with
implicit hydrodynamics. This method considers the faces on one side of a surface that are
connected to a node, and the faces on the other side that overlap these faces, and enforces that a
weighted sum of the gap volumes of the overlap be zero.
At most one slide surface may be designated as an autocontact surface. The faces of the
autocontact surface are not explicitly specified, but typically consist of all external faces (faces
with a zone on only one side) which are not included in any other slide surface.
FEusion Embedded Mesh Coupling
The FEusion library provides an embedded mesh method that allows the coupling of two
separate, spatially overlapping meshes. The embedded mesh approach can greatly simplify the
pre-processing requirements while potentially avoiding many run-time issues related to tangling
of an ALE mesh conforming around Lagrange objects.
The approach used here, which was developed by Puso et al., uses Lagrange multipliers from a
piecewise constant space on the cut background elements to constrain the jump between
background and foreground velocities. A stabilization scheme penalizes the difference in face-
adjacent Lagrange multipliers tractions. These multipliers are solved for implicitly using a
conjugate gradient (CG) iterative method on the subset of the mesh where they are active, i.e.,
only on the cut background cells.
Advection
ALE3D uses an arbitrary Lagrangian-Eulerian (ALE) algorithm. The algorithm consists of two
distinct steps: (i) a Lagrangian step that updates nodal positions, nodal velocities, and zonal
quantities, and (ii) a remap/advection step that remaps the results of the Lagrangian step onto a
mesh determined by the relaxation method specified by the user.
In the ALE3D implementation of the ALE algorithm, the Lagrangian step is executed, a new
representation of the mesh is created, and then material variables such as mass, momentum, and
energy are advected from the old mesh to the new mesh. The advection of the material variables
is done by computing fluxes of the state variables between the Lagrangian mesh and the new
mesh. The pressure is recalculated after the remap with a call to the EOS routines. If the
materials have strength, the stress deviators, plastic strain, and other constitutive model variables
are also advected.
The advection step comprises a relaxation phase and an advection phase. An “ideal” grid is first
created using an equipotential or condition number grid relaxation algorithm. (A pure Eulerian
option is also available.) The state variables are then remapped onto the new grid by constructing
fluxes between the old grid and the new grid. The fluxes for extensive variables (mass, internal
energy, and momentum) are conservative. For pure zones, a second-order, monotonic algorithm
is used to calculate the fluxes. This technique was pioneered by van Leer (1977). Velocities are
updated by applying the results of momentum conservation. This leads to a loss of kinetic energy
ALE3D: An Arbitrary Lagrangian-Eulerian Multi-Physics Code
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(which is quadratic rather than linear in the velocity), and high-order advection is required to
limit the loss. As a rule, lost kinetic energy is not added to internal energy to explicitly force total
energy conservation. However, kinetic energy lost in the vicinity of a shock can be put back into
the internal energy field. This is useful for the purpose of propagating shocks over long distances
without degradation.
For problems with several material regions, the process of advection can create mixed elements
in which more than one material region resides in an element. While material region interfaces
are currently not tracked explicitly during the advection step, they are inferred from the volume
fractions of each region in neighboring zones. This is done to determine the order in which
regions are moved from one element to another. The ordering process preserves the integrity of
region objects as they move through the mesh. When volume fractions provide no guidance,
regions are moved from one element to another in the order in which they are numbered, so it
can be very important to order regions in such a way that “more important” (i.e., non-
background) materials have lower region numbers. Separate values of the thermodynamic state
variables are maintained for each region in a mixed element, and advection is done using a first-
order upwind method. Regions in mixed elements are allowed to relax towards pressure
equilibrium. The relaxation algorithm is based on a linearized solution to the Riemann problem.
Free surfaces are not relaxed unless tangential relaxation is explicitly set up for a free surface
nodeset. Advection can take place up to, but not across, a slide surface. Ordinarily only the slave
nodes of a two-sided slide surface are relaxed, but master-side relaxation is available (similar to
tangential relaxation). If necessary, and if slide surface nodes are forced to line up across the
slide surface, the slide surface can be deleted during the course of the calculation and thereby
allow for advection after that time.
Thermal Diffusion
Thermal diffusion is the conduction of heat from a hot to a cold temperature location in a solid or
fluid. The thermal diffusion module was originally incorporated to model manufacturing
processes such as casting, forging, rolling, and extrusion. It is implemented in a manner that
allows for application to most engineering heat transfer problems. The thermal diffusion module
has also been coupled to a chemical kinetics module for the purpose of modeling thermal
ignition of high explosives. The existing capabilities include conduction with orthotropic and
temperature dependent properties, phase changes, enclosure thermal radiation, thermal contact
resistance across interfaces, and temperature, flux, thermal radiation, and convection boundary
conditions.
Two modes of coupling between the hydrodynamics and the heat transfer are supported. The first
solves the hydrodynamics and heat transfer consecutively at each step by operator splitting. The
heat transfer step changes energy without changing volume and the hydrodynamics step changes
volume without transferring heat. Cell-centered energy is made consistent with nodal
temperatures by including an adiabatic expansion source term, g (dV/V), and strain heating.
Chemical reactions also contribute a source term. The second mode solves the hydrodynamics
and heat transfer iteratively using the intermediate results of the package until they both
converge.
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The heat conduction equations are solved using a finite element approach. The solution includes
implicit time integration, direct or iterative matrix solution, hexahedral elements with second-
order Gaussian quadrature integration, nonlinear convergence by successive substitution or
Newton-Raphson methods, and variable time step control with sub-cycling of the thermal
diffusion and hydrodynamics.
Heat transfer can occur across a slide surface by perfect thermal contact or by heat conduction
using ‘virtual elements’. Virtual elements are conduction and thermal radiation resistive elements
that are inserted between the slide surfaces. The virtual elements can have zero thickness or their
resistance can be scaled with gap distance. The perfect thermal contact algorithm is formulated
as a penalty method. It maintains symmetry at boundary edges through geometrical and mass
weighting, and scales the penalty function with the magnitude of the diagonal term of the
coefficient matrix.
Chemistry and Chemical Diffusion
ALE3D’s chemistry model is capable of capturing the composition, structure and properties of
chemically reacting materials. The chemical kinetics module was developed to model
decomposition of high explosive materials in thermal environments. A chemical material is a set
of materials that can transform among each other. These materials (or species) define all the
properties of the chemical material. Several models for combining the properties of the
individual species are provided. In principle, any material model that supplies a temperature and
pressure can be used as a species in a chemical material. It is even possible to combine several
models by including a chemical material as a species in another chemical material.
The composition of a chemical material can be modified through a set of user defined chemical
reactions. ALE3D organizes the reactions into groups that act on the same set of species. Only
reactions that involve all of the species present in a chemical material are included in that
material’s full reaction scheme. The change in composition is calculated implicitly with a self-
correcting Newton-Raphson technique. The temperature derivative of the change is also
calculated for use by the thermal module.
The chemical kinetics package is normally run tightly coupled to the thermal module. However,
it can also be run during the hydrodynamics phase. When run during the hydrodynamics phase,
the chemical kinetics package can be used to emulate a variety of reactive flow detonation
propagation models. This emulation is accomplished by including the appropriate set of reaction
mechanisms with the appropriate material models for the reacting species. Both the standard
Lee-Tarver (Lee and Tarver, 1980) and PERMS (Propellant Energetic Response to Mechanical
Stimuli) (Maienschein et al., 1997) models can be implemented this way.
A chemical diffusion model is also available in ALE3D. It allows for diffusion within a chemical
material (not between chemical materials). The species diffusion equations are solved using a
finite-volume approach. The operator is explicit in time and computes fluxes of chemical species
across faces between elements based on the locally determined chemical potential gradient. Both
tracer, where the diffusing species is assumed to be only a small fraction of the total mass of the
system, and non-tracer models are available.
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Incompressible Flow
The incompressible flow package solves the incompressible Navier-Stokes equations and is
useful for simulations involving a fluid that can be approximated with a constant density, such as
low-speed aerodynamics or internal fluid mechanics. As a rule of thumb, flows with a Mach
number less than 0.3 are candidates to be considered incompressible. An obvious exception to
this rule is any flow where the density variations drive the flow, no matter how fast it is going.
The package can be coupled to the thermal package for simulations of natural convection of a
fluid with temperature gradients. The incompressible flow model may also be run simultaneously
with the standard ALE hydro package for coupled fluid-structure problems. The fluid and solid
components are coupled through their boundary conditions. The incompressible flow module
includes explicit, semi-implicit (either the advection or viscous terms implicit) and fully implicit
time integration options. An advection-diffusion solver is also available to model species
concentrations within an incompressible material.
Multiphase Flow
The ALE3D multiphase flow model was developed for simulating mixed materials with separate
velocity fields. This model reuses the species framework developed for the chemical materials.
In principle, any ALE3D material model that supplies a temperature and pressure can be used as
a species in the chemistry and multiphase packages. Often, it is desirable to create a hierarchical
chemical material for use in the multiphase model. An example of this is a multiphase flow of
particles in a fluid flow, each composed of multiple constituents that can react amongst
themselves. This hierarchy of material models enables chemical reactions to be modeled in a
unified manner with multiphase flow.
There are several models for drag and compaction in the multiphase flow package. These
include semi-analytic models where the drag terms can be integrated analytically to those which
are evaluated using backward Euler. For cases where the compaction viscosity is unknown, the
pressure relaxation module defaults to an infinite relaxation rate or pressure equilibration.
Magnetohydrodynamics
ALE3D’s magnetohydrodynamics model is capable of capturing the dynamics of electrically
conducting solids and fluids. The magnetohydrodynamics (MHD) module was developed
primarily for the modeling of coupled electro-thermal-mechanical (ETM) systems that are
inherently 3D in nature. Example applications for this capability include explosively driven
magnetic flux compression generators, induction heating / metal forming and electromagnetic
rail gun systems. The ALE3D MHD module solves the resistive magnetic induction equation
given a collection of specified current and voltage sources. The equation is solved in the
Lagrangian frame using a mixed finite element method employing H(Curl) and H(Div) finite
element basis functions which preserves the solenoidal nature of the magnetic field to machine
precision. Electromagnetic force and resistive Joule heating terms are coupled to the equations of
motion and thermal diffusion in an operator split manner. For problems that require mesh
relaxation, magnetic advection is performed using the method of algebraic constrained transport
that is valid for unstructured hexahedral grids with arbitrary mesh velocities. The advection
method maintains the divergence free nature of the magnetic field and is second-order accurate
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in regions where the solution is sufficiently smooth. For regions in which the magnetic field is
discontinuous (e.g., MHD shocks), the advection step is limited using the method of algebraic
flux correction, which is local extremum diminishing and divergence preserving.
Parallelization and Scaling
ALE3D is parallelized across the problem space using domain decomposition to associate
separate pieces of physical space to individual processors. The implementation uses MPI to
communicate across processor boundaries. Several of the key characteristics of the ALE3D
software infrastructure contribute to the code’s scalability. Field data are contained in simple C
arrays, giving the compiler canonical loop iteration patterns to optimize. Ghost boundary data is
always stored contiguously, removing the need for gather / scatter operations during MPI
communication. Nearly all MPI communication is point-to-point, with only two global
reductions per time step: the duration of the next time step, and a global error status check.
The amount of point-to-point communication required is dependent on the type of calculation
being run. Problems running explicit hydrodynamics require a single communication to collect
the sum of the forces at nodes along domain boundaries, although some optional algorithms (e.g.
monotonic Q) require more communication. Problems running advection require a 20x – 30x
greater amount of communication, both in terms of the number of communication points, and the
amount of data typically sent. A rough breakdown of the steps performed in the advection are:
nodal relaxation; calculation of volume fluxes; identification of mixed elements and interface
reconstruction; advection of element centered variables; and momentum (node centered)
advection. If there are slide surfaces, additional communications are required. The approach
taken in ALE3D is to use a separate decomposition for slide surfaces. Nodes on one side of a
surface (the “master” side) are assigned statically to the various processors in a load-balanced
manner, and then nodes on the other (“slave”) side of the surface that are currently “close” to the
masters on a processor are assigned to the same processor.
The daunting task of fully characterizing the performance of ALE3D’s many packages has never
been done in a systematic way, but we present an overview of the scaling of the Lagrangian
hydrodynamics package that forms the core of ALE3D. Figure 3 shows the weak scaling
behavior of a 5695 Element / process Sedov problem from 256 to 96,000 processes.
ALE3D: An Arbitrary Lagrangian-Eulerian Multi-Physics Code
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Figure 3. Weak scaling results for ALE3D’s Lagrangian mechanics package.
Conclusion
The ALE3D code continues be under active development at LLNL. In addition to new physics
and methodology improvements, the underlying computer science framework is being modified
to ensure excellent performance on the latest generation of High Performance Computing
machines. This Export Controlled and Official Use Only code is available to analysts in the
Department of Defense and associated contractors for work related to national defense. Contact
the authors for additional information.
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Lawrence Livermore National Laboratory 14
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... However, these quoted alternatives would require sophisticated, 58 and likely costly, numerical implementations for updating the stress and other constitutive variables (e.g., the Jaumann rate derived previously 7,25 would not necessarily apply). On the other hand, the assumptions and approximations used in the current framework take advantage of existing, efficient numerical algorithms for explicit dynamic finite element (FE) computations, 21,42,53,54,59 thus justifying their use. ...
... 24,71 In simulations, L corresponds to the minimum size of elements of the grid or mesh, and the time step size ∆t should be smaller than the minimum characteristic time for accuracy and stability (i.e., the Courant-Friedrichs-Lewy condition) if explicit integration methods are used, as is conventional for the momentum balance. 41,42,59 As an example, take L = 1 mm and invoke physical properties of Fe, 24 with µ = µ 0 the limiting case for magnetic diffusion. The longitudinal sound speed is c L ≈ 6 km/s, giving t mech ≈ 0.17 µs. ...
... However, if electrical conductivity Σ is very low, then t mag can be prohibitively small. For this reason, implicit integration methods are commonly used in magnetohydrodynamics host codes 41,42 for stable solution of the magnetic diffusion equation with larger step sizes. ...
... The sharp shock front travels at sufficiently high velocities through the material, as compared to the exposure time of the camera's gate, to induce a motion blurring effect that is detectable in the X-ray radiograph.Four cases, investigating a 1D shock wave propagating through polymethyl methacrylate (PMMA) 29 were studied experimentally and with simulations. The simulations were conducted with the hydrodynamic simulation code, ALE3D (Arbitrary Lagrangian Eulerian Three-Dimensional Analysis) 30 and radiographic simulation code, HADES 31 . The experimental results were collected at DCS@APS using the single stage gas gun at beamline 35-ID-D (Fig. 1). ...
... The shock propagation in the PMMA sample was simulated using LLNL's hydrodynamic simulation code, ALE3D (Arbitrary Lagrangian Eularian Three-Dimensional Analysis) 30 , which specifies the mechanical state of the material, giving stress and density location at every spatial location within a material undergoing deformation, as an input into LLNL's radiographic ray tracing simulation code, HADES 31 ( Supplementary Fig. 2). The inputs into ALE3D include descriptions of the PMMA sample and Al-6061 impactor materials' equation of states, which describe the P-V-T relationship (pressure-volume-temperature) and evolving mechanical state model (i.e. ...
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Time-integrated radiography using MeV Bremsstrahlung X-ray sources is the norm for imaging during system-level testing of components and structures under dynamic condition. One source of error in the analysis of the time-integrated radiography data sets stems from motion blur which smears out sharp interfaces to a greater degree with longer exposure times, which become necessary to provide sufficient signal-to-noise with low X-ray penetration of objects of interest. To quantify motion blur, a 1D shock wave through PMMA was investigated experimentally at The Dynamic Compression Sector at The Advanced Photon Source (DCS@APS) with tapered broadband and 25.46 ± 1.06 keV narrowband X-rays. Four cameras with different exposure times were used for each experiment to compare the effect that exposure time has on motion blur. In addition, our methodology to accurately simulate motion blur in terms of transmission and shape is presented and compared to our experimental results and quantified. There is a high level of agreement between the experimental and simulation results across the range of data sets investigated in this study with a percent difference range of 0.29–1.31% for the four shots. The methodology of this work serves as a steppingstone towards a physically validated model that could be used in conjunction with experimental results to deconvolve physical parameters, densities, and interfaces of interest in a way that would not be possible with experimental results alone.
... Metal AM processes are governed by a complex interplay of multiple physical phenomena, including heat transfer, multi-phase flows, and phase transitions. In metal AM processes, the laser-material interaction introduces rapid localized heating/cooling, large thermal gradients, and com-(such as Arbitrary-Lagrangian Eulerian [26][27][28][29][30][31][32] and front tracking method [33][34][35][36][37]), the interface-capturing methods circumvent mesh motion and re-meshing by capturing the interface changes implicitly by solving a scalar partial differential equation (PDE) in a background mesh. This feature is advantageous, given the large deformations or topological changes of the material interface in metal AM processes, which can be difficult and sometimes impossible for interface-tracking methods using mesh motion and re-meshing schemes. ...
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This paper presents an effective high-fidelity multi-physics model for metal additive manufacturing (AM). Using a mixed interface-capturing/interface-tracking approach, the model integrates level set and variational multiscale formulation for thermal multi-phase flows and explicitly handles the gas-metal interface evolution without mesh motion and re-meshing schemes. We integrate the mixed formulation with an energy-conservative ray tracing-based laser model and a mass-fixing algorithm that accounts for phase transitions. First, we present the mathematical details of the proposed model. Then, we apply the model to simulate the NIST A-AMB2022-01 Benchmark test, emphasizing the prediction of thermal history, laser absorption rate, melt pool dimensions, and pore formation. The results show the model’s strong capability to accurately capture the complex physics of metal AM processes and its potential in simulation-based process optimization.
... As a result of numerous research efforts over the last decade, many numerical methods have been developed to understand the complex multiphysics of metal additive manufacturing. Employing the arbitrary Lagrangian Eulerian (ALE) method, Lawrence Livermore National Laboratory built a solver [10], [11], [12] with which they discovered that "periodic oscillations of the melt pool's morphology existed prior to a transition to chaotic and pore-generating turbulence" [13]. The National University of Singapore at the department of mechanical engineering developed a set of CFD toolkit with volume of fluid technique for free surface tracking (VOF) to simulate not only the directed energy deposition (DED) process but also the multi-layer and multi-track laser powder bed fusion (LPBF) processes [14], [15], [16], [17]. ...
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Porosity defects in laser powder bed fusion processes are critical, particularly when initiated in the high laser energy density regime, known as the keyhole regime. In-situ understanding of melt-pool dynamics is essential, which can be achieved through high energy radiography techniques. However, these techniques are costly and time-consuming, prompting the need for high fidelity numerical models. This paper presents a numerical model aimed at studying the correlation between powder bed fusion process inputs and the formation of deep and narrow vapor depressions (keyholes) thereby revealing interesting aspects that are difficult if not impossible to identify using experimental setups. The model consists of two coupled components: a multi-phase, multi-physics model to solve melt pool dynamics and a high-performance ray tracing-based multi-reflection laser radiation model. The metal free surface predicted by the first model is input into the radiation model, which calculates the effective energy absorbed, resulting in a fully coupled simulation Results show good agreement with experimental benchmark data from dynamic x-ray radiography techniques, with maximum deviations of 11% in average keyhole depths and 2 to 8% in laser absorption time profile and final pores were successfully predicted. The investigation of the impact of maximal temperature spot on keyhole shape and dynamics, revealed distinct keyhole shapes categorized by average maximal temperature spot and uncovering a clear pattern behind pore-keyhole separation, leading to a better understanding of chaotic keyhole mode where pores are generated and pushed into the melt pool. The developed numerical model offers a cost-effective alternative to experimental techniques and can aid in optimizing process parameters to mitigate porosity defects in additive manufacturing.
... Periodic re-meshing reconciles these domains into a single reference frame. Detailed descriptions of ALE formulations can be found in Le Tallec & Mouro (2000), Souli et al. (2000), Legay et al. (2006) and Noble et al. (2017). ...
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Underwater explosions are inherently complex and unique physical phenomena markedly distinct from those occurring above the surface. This distinctiveness is primarily attributed to the relatively incompressible nature of water, which fundamentally alters the propagation and impact of shock waves. The study of underwater explosions is paramount in applications such as underwater demolitions for construction and salvage operations. These applications require a comprehensive understanding in order to mitigate the disturbances’ impact on marine structures and ecosystems. Studying underwater explosions and their mitigation encompasses various disciplines, including fluid mechanics, materials science and structural engineering. The work reviewed in this study contributes significantly to enhancing safety measures in marine structures by providing critical insights into the behaviour of structures under extreme conditions. This includes understanding the behaviour of gas bubbles formed by explosions, the transmission of shock waves through different media and the resultant forces exerted on structures submerged in water. Consequently, this review is meant to aid in designing robust and resilient marine systems capable of withstanding severe loading conditions caused by underwater explosions by providing key engineering considerations. The continuous evolution of this research area is essential for advancing maritime technology, ensuring the safety of undersea operations and protecting marine environments from the adverse effects of extreme subaqueous loadings.
... To understand the evolution of detonation within this device and to optimize the design for a jet-mitigating buffer, we simulate the device using the LLNL hydrocode, ALE3D (short for Arbitrary Lagrangian-Eulerian three-dimensional analysis). 26 As a first pass, a 2D slice of the device was taken and discretized on a grid at a resolution of 100 zones/cm (Fig. 2). The LWG inlet is modeled as an HMX-based HE undergoing programmed burn, where the HE is modeled using a Jones-Wilkins-Lee (JWL) equation of state. ...
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In this work, we detail a novel application of inverse design and advanced manufacturing to rapidly develop and experimentally validate modifications to a shaped charge jet analog. The shaped charge jet analog comprises a copper liner, a high explosive (HE), and a silicone buffer. We apply a genetic algorithm to determine an optimal buffer design that can be placed between the liner and the HE that results in the largest possible change in jet velocity. The use of a genetic algorithm allows for discoveries of unintuitive, complex, yet optimal buffer designs. Experiments using the optimal design verified the effectiveness of the buffer and validated the machine learning approach to hydrodynamic design optimization.
... These filaments can then either be rendered as shapes in an ACIS format, as is done in programs like Cubit 18 and OpenSCAD 19 (shown in the left and right images of Figure 1c) for later meshing, or can be mapped onto a predefined mesh generated as shown in the ALE3D example (Figure 1c center). 20 The G-code rendering program can also be easily modified to account for specific callouts (or "breadcrumbs") in the toolpath instructions such as changes in filament width, filament shape (square vs. cylindrical filaments), or layer height. The component renders can then be formed into meshes for direct import to hydrocodes and simulated for performance comparisons. ...
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Additive manufacturing (or 3D printing) can be used to create complex parts with features that could not otherwise be made via traditional machining or casting techniques. Direct-ink-write is the simplest AM technique whereby a material is extruded out of a nozzle while moving along a prescribed path. The DIW techniques developed at LLNL have been used to create multi-material components with unique geometries and compositional gradients. However, qualifying these parts remains a challenge. New features (geometric or compositional) are introduced at each step of the AM process including material formulation, toolpath design, and part fabrication. Fortunately, AM natively lends itself to fabrication simulation and in situ data collection which can be used to capture these features and create digital twins of the as-manufactured parts. These digital twins can, in turn, be used directly in simulations for part qualification. This presentation will feature ongoing efforts at LLNL to create digital twins of parts at two different stages of the AM process: (1) toolpath design and (2) part fabrication. We will begin by demonstrating a newly developed computational tool that reads machine instructions to generate digital twins of the part as it was meant to be manufactured for import into simulations. We will then demonstrate a different in situ data collection methods which can be used to create digital twins of multi-material AM parts using data collected during the printing process. The work showcased here will enable the evolution of a high-quality AM process that can yield "born qualified" parts.
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Understanding the physical and chemical response of materials to impulsive deformation is crucial for applications ranging from soft robotic locomotion to space exploration to seismology. However, investigating material properties at extreme strain rates remains challenging due to temporal and spatial resolution limitations. Combining high-strain-rate testing with mechanochemistry encodes the molecular-level deformation within the material itself, thus enabling the direct quantification of the material response. Here, we demonstrate a mechanophore-functionalized block copolymer that self-reports energy dissipation mechanisms, such as bond rupture and acoustic wave dissipation, in response to high-strain-rate impacts. A microprojectile accelerated towards the polymer permanently deforms the material at a shallow depth. At intersonic velocities, the polymer reports significant subsurface energy absorption due to shockwave attenuation, a mechanism traditionally considered negligible compared to plasticity and not well explored in polymers. The acoustic wave velocity of the material is directly recovered from the mechanochemically-activated subsurface volume recorded in the material, which is validated by simulations, theory, and acoustic measurements. This integration of mechanochemistry with microballistic testing enables characterization of high-strain-rate mechanical properties and elucidates important insights applicable to nanomaterials, particle-reinforced composites, and biocompatible polymers.
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Understanding the physical and chemical response of materials to impulsive deformation is crucial for applications ranging from soft robotic locomotion 1–3 to space exploration 4,5 to seismology ⁶. However, investigating material properties at extreme strain rates remains challenging due to temporal and spatial resolution limitations. Combining high-strain-rate testing with mechanochemistry uniquely encodes the molecular-level deformation within the material itself, thus enabling the direct quantification of the material response. Here, we demonstrate a mechanophore-functionalized block copolymer that self-reports unique energy dissipation mechanisms, such as bond rupture and acoustic wave dissipation, in response to high-strain-rate impacts. A microprojectile accelerated towards the polymer permanently deforms the material at a shallow depth. At intersonic velocities, the polymer reports significant subsurface energy absorption due to shockwave attenuation, a mechanism traditionally considered negligible compared to plasticity and not well explored in polymers. The acoustic wave velocity of the material is directly recovered from the mechanochemically-activated subsurface volume recorded in the material, which is validated by simulations, theory, and acoustic measurements. This integration of mechanochemistry with microballistic testing enables characterization of high-strain-rate mechanical properties and elucidates new insights applicable to nanomaterials ⁷, particle-reinforced composites ⁸, and biocompatible polymers ⁹.
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A new approach for treating the mechanical interactions of overlapping finite element meshes is presented. Referred to as embedded mesh methods here, these overlapping mesh methods typically include a foreground solid mesh and a background Euler fluid grid or solid mesh. A number of different approaches have been used in previous work to characterize the interactions of the background and foreground meshes at the interface. Lagrange multipliers are well suited to enforce the continuity constraints but care must be taken such that the resulting formulation is stable. Several Lagrange multiplier techniques are examined in this work and applied to coupling solid meshes and fluid-structure interaction type problems. In addition, details regarding implementation in a two-step, multi-material, Arbitrary Lagrangian Eulerian (ALE) code are presented. Example problems demonstrate convergence and applicability to a range of problems. In particular, the fluid-structure interaction examples focus on blast applications.
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Fromm's second-order scheme for integrating the linear convection equation is made monotonic through the inclusion of nonlinear feedback terms. Care is taken to keep the scheme in conservation form. When applied to a quadratic conservation law, the scheme notably yields a monotonic shock profile, with a width of only 112 mesh.
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A classic problem in Lagrangian numerical hydrodynamics is the conservative transfer of quantities from an old,, distorted mesh to a new mesh. The same problem arises whenever the mesh is changed as, for example, in adaptive mesh techniques. This transfer of information is an interpolation process which is frequently called rezoning (or remapping). The general problem of conservative rezoning from one arbitrary mesh to another may be formulated as follows: m/sub k/ = ..integral integral integral../sub V//sub k/rho(r)dV. That is, we compute the mass m/sub k/ of each cell of the new mesh by integrating the known density distribution in the old mesh over the cell volume V/sub k/. A direct intregation is generally prohibitive. We show, however, that is is possible to convert this integral to a surface integral by the appropriate use of the divergence theorem, thus greatly reducing the complexity of the problem. For two-dimensional generall quadrilateral meshes the resulting method is exact and particularly simple.
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An ignition and growth concept is used, within the framework of a one-dimensional Lagrangian hydrodynamic code, to model the shock initiation of heterogeneous solid explosives. The leading shock wave of an initiating pulse is assumed to ignite a small fraction of the explosive at localized heated regions. These ignited regions then grow as material is consumed at their boundaries. The growth rate for a particular material is assumed to have the characteristic pressure dependence of high-pressure laminar burning experiments. Results of the model calculations are in good quantitative agreement with recent manganin pressure gage and particle velocity gage measurements of the buildup of the initiating shock front to detonation for both sustained and short duration pulses in four solid explosives: PBX−9404, TATB, cast TNT, and PETN. The predicted run distances to detonation as functions of shock pressure at various initial densities and the predicted reaction zone lengths of the fully developed detonation waves also correlate well with experimental data for these four solid explosives.
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This report provides a theoretical manual for DYNA3D, a vectorized explicit three-dimensional finite element code for analyzing the large deformation dynamic response of inelastic solids. A contact-impact algorithm that permits gaps and sliding along material interfaces is described. By a specialization of this algorithm, such interfaces can be rigidly tied to admit variable zoning without the need of transition regions. Spatial discretization is achieved by the use of 8-node solid elements, and the equations-of-motion are integrated by the central difference method. DYNA3D is operational on the CRAY-1 and CDC7600 computers.
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The treatment of zero energy modes which arise due to one-point integration of first-order isoparametric finite elements is addressed. A method for precisely isolating these modes for arbitrary geometry is developed. Two hourglass control schemes, viscous and elastic, are presented and examined. In addition, a convenient one-point integration scheme which analytically integrates the element volume and uniform strain modes is presented.
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A model, applicable at high‐strain rate, is presented for the shear modulus and yield strength as functions of equivalent plastic strain, pressure, and internal energy (temperature). The parameters needed to implement the model have been determined for 14 metals. Using this model, hydrodynamic computer simulations have been successful in reproducing measured stress and free‐surface‐velocity–vs–time data for a number of shock‐wave experiments.
Godunov Methods on a Staggered Mesh -An Improved Artificial Viscosity
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Equation of State and Strength Properties of Selected Materials
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Steinberg, D.J., "Equation of State and Strength Properties of Selected Materials," Lawrence Livermore National Laboratory, Report UCRL-MA-106439 (1996).