Computational Fluid Dynamics€¦ · Computational Fluid Dynamics Dan Meiron Dale Pullin David Hill...

PSAAP: Predictive Science Academic Alliance Program

Computational Fluid Dynamics

Dan MeironDale Pullin David Hill

Julian Cummings Manuel Lombardini

2/PSAAP: Predictive Science Academic Alliance Program

Overview

• Hypervelocity impact• Overview of Caltech VTF Capability

– AMROC– Solids/fluids solvers

• Eulerian approach to fluid-solid continuum equations– Conservative formulation– Constraints and degeneracy– Phase change– Challenges

• Simulation of plasmas• Progress to date


Hypervelocity impact

Hypervelocity Launcher V

Debris Cloud

V~3-10 Km/SD~1-2mmL/D~ 1-3

Target

θ

Plasma jets

Ejecta


Objectives

• Explore the application of Eulerian approaches to high velocity impact

• Develop numerical methods for– Materials with strength– Damage and fracture– Phase change – melting and vaporization

• Integrate into Eulerian VTF simulation framework• Simulation of high velocity impact experiments• Uncertainty quantification


The Virtual Test Facility

Concept:• Use a level-set-based approach to couple Lagrangian solid

mechanics solvers to Eulerian fluid mechanics solvers• Use distance information to consider geometrically complex

boundary conditions in a ghost fluid method for Cartesian fluid solvers

• Use block-structured mesh adaptation to mitigate boundary approximation errors

• Eulerian-Lagrangian inter-solver communication library synchronizes the boundary data exchange between coupled solver modules

• Implement all components for distributed memory systems with non-blocking MPI communication routines

Infrastructure for fluid-structure interaction simulation of shock- and detonation-driven solid material deformation


Structured hyperbolic AMR with Amroc

• Generic implementation of Berger-Colella SAMR algorithm

• Refined subgrids overlay coarser ones

• Computational decoupling of subgrids by using ghost cells

• Refinement in space and time • Block-based data structures• Cluster-algorithm necessary• Efficient cache-reuse /

vectorization possible• Explicit finite volume scheme

only for single rectangular grid necessary


Fluid solvers

• AMROC encapsulates dynamic mesh adaptation and parallelization to the fluid solver developer

• WENO-TCD scheme with optional LES and chemical reaction capability (D.Hill, C.Pantano)

• Extended Clawpack with full and one-step chemistry (R.Deiterding)

LES cylindrical Richtmyer-Meshkov instability

Hydrogen and ethylene detonation structure

simulations


Solid Dynamics: ARES

• General dynamic plasticity and fracture code• Parallel explicit dynamics, scalable communications• Scalable unstructured parallel meshing• Thermomechanical coupling• Constitutive modeling library

– Single and polycrystal plasticity– Ab initio informed EoS– Shock physics, artificial viscosity

• Nonsmooth contact/friction• Brittle dynamic fracture• Ductile fracture and spall• Shear banding and localization• Scalable parallel fragmentation

ARES simulations of ballistic impact (top) and penetration

(bottom) of a steel plate


Solid - fluid coupling in the VTF

• Ghost fluid method– Incorporate complex moving

boundary/ interfaces into a Cartesian solver (extension of work by R.Fedkiw and T.Aslam)

– Implicit boundary representation via distance function j, normal n=rj / |rj|

– Treat an interface as a moving rigid wall

• Closest Point Transform (CPT)– For each face/edge/vertex;

Find distance, closest point to that primitive for the scan-converted points


Example of solid fluid coupling in the VTF

Water hammer impacting on fracturing shell


Extension to Mie-Gruneisen EOS


2D shock-bubble interaction with end wall reflection

• Single fluid Mie-Gruneisen EOS:

• Aluminum, copper etc (under certain conditions)

• Miller-Puckett algorithm for EOS• Integrated into WENO-TCD within AMROC


Proposed Eulerian approach for hypervelocity impact

• Utilize an Eulerian mixed cell approach– Solid phase– Liquid phase– Vapor phase (including ionization; plasma)– All can be accommodated via a hyperbolic characteristic formulation

supplemented with constraint equations• Develop a new generalized solid mechanics patch solver

– Based on formulation of Miller and Colella• Develop multiple phase formulation • Integrate these solvers into the AMR formulation of the VTF


Nonlinear Elasticity • Advantages of a numerical implementation in conservative form:

– Conserves total mass, momentum, energy– Correct wave-speeds– Correct speed of discontinuities (e.g. shocks in gas-dynamics)

• Gavrilyuk et al (JCP 2008) demonstrate errors when conservation form is not used (1D elastic simulation)

• Early formulation for finite-volume Miller & Colella (JCP 2001)• Characteristic structure

– Primitive variables– Conserved variables

• Neo-Hookean model of Blatz and Ko (JFM ‘62)– Directly calculated all wave-speeds– Real wave speeds– Fits directly in Miller-Colella framework and VTF


Conservation Form: Inverse Deformation Tensor

In an approach similar to Miller&Colella (2001 JCP), the equations are written in terms of mass, momentum, total energy and the Inverse of the deformation tensor,

The material derivative of the material coordinate frame,

may be differentiated with respect to the spatial coordinate frameto produce the evolution equation for the inverse of the deformation tensor.

implied sum over j.


Conservation form for materials with strength

Source terms (not shown here) maybe added to limit numerical error in theform of a non-zero curl for the inverse deformation tensor.


Blatz and Ko Model for Nonlinear Elasticity (Porous Rubber)

The Blatz and Ko model is a two-parameter model with a simple form for the stress te

where

and where

Direct calculation shows that the reduced system is diagonal !


Verification: Linear Modes

The behavior of small amplitude disturbances when evolved in fully nonlinear code is compared with linear theory

Three physical modes with five speeds:•Convective mode (speed u)•Shear mode (u+c2, u-c2)•Compression mode (u+c1, u-c1)

Compressive speed is faster than shear: c1 >

convective shear compression


Verification – linear modes

convective shear compression

As a simple verification test, 1D linear moves were evolved in the 2D nonlinear code with periodic boundary conditions. Each image below shows the evolution of a given mode at three equal-spaced times.


2D Cylinder example

An initial compressed cylinder with density ratio 1.5: f11=f22=sqrt(1.5)

Both directions of the solver are exercised by using problem with radial symmetry.

In late time, small grid-scale disturbancesDevelop in the fine features of the solutionNo explicit attempt has been made to Eliminate curl errors in the inverseDisplacement tensor, and this may be The source of these disturbances.


Some initial shock-contact interactions

• A shock traveling 3x the shear-wave speed

• Interacts with a density jump (ratio 2).

• A large irregular disturbance can be seen between the reflected shock and the shocked second material.

• Significant curl errors are present in this region.

Transmitted shock

Reflected shock


Some initial simulations with CTH

• CTH – Eulerian hydro code developed by Sandia– Used for high rate deformation calculations– Allows for materials strength– Full suite of materials models– Cell based AMR– Uses artificial viscosity to regularize shocks

• Initial (very preliminary) simulation– Considered impact of Ta spheres on Ta target– Ta sphere – 0.5 cm – velocity 5 km/sec– Ta target – 3 cm thickness

• Will be used with Dakota framework to develop UQ for this simulation approach


Initial conditions


Impact at 12 microsecs


Simulation of phase change

• Hypervelocity impact will generate pressures and temperatures sufficient to drive phase transitions

• Several approaches to simulation will be explored– Multiphase Riemann solvers– Use of phase field approach

to track phase boundaries• Advantage of latter approach

is that mixture EOS is not required

• Level set ideas can be utilized in both approaches

Phase diagram for Al

Phase change in lead on lead impact


Implementation

• Several separate Euler solvers simultaneously - one per phase– Each has own patch solver– Coexist within AMROC – If Lagrangian, particle-based solvers are employed, can utilize

presently available Lagrangian-Eulerian capability• Compute phase evolution via appropriate boundary conditions;

thermodynamic processes at phase boundary– Melting (initial impact), re-solidification, arbitrary phase transitions

including vapor, vacuum and mixed phase regions– Tracking of free boundaries and collisions (with appropriate BCs)

• Phase Transition computation via two possible approaches– Phase boundary tracked by level set function and ghost fluid method

• CPT used with GFM modifies cells in mixed phase region– Phase field approach

• Use of free energy functional with phenomological kinetics• Phase evolution via integration of hyperbolic-parabolic PDE


Plasma Simulation

• Experiments in hypervelocity impact; two distinct phases of plasma activity – initial light “flash” (plasma formation and jetting)– time-dependent line emission spectra (plasma equilibration)

• Improved understanding of the material properties for high energy density (HED) states is desired– Look to MD and electronic structure calculations for guidance

• Existing multiphase Eulerian simulation to be extended to allow for vapor phase then ionization transition to plasma


Plasma Simulation• Model of plasma phase requires multiple inputs from fine-scale MD

simulations and ab initio calculations– Thermodynamic properties of plasma constituents in HED state – Viscosity coefficient (inferred from equilibrium fluctuation level)– Thermal and electrical conductivities (using eFF model)

• Realistic plasma initial conditions for hydrodynamic plasma calculations– Molecular dynamics using ReaxFF force fields– Trained with quantum mechanics data

• Plasma model outputs (validation)– Spatial and temporal distributions for all constituents and fluid flows– Evolution of atom and ion line densities and hydrodynamic flow fields– Predicted atomic line emission spectra for plasma

• Plasma model will be developed in three stages– Standalone hydrodynamics Euler solver– Plasma fluid solver integrated into AMROC (hyperbolic system)– Plasma solver coupled to multi-phase impact model via VTF


Challenges for Eulerian approach

• Control of “curl” errors• Develop multiphase GFM approach

– Level set approach has been used previously to simulate phase change problems

• But never for a problem with this level of complexity– Research required for proper phase boundary nucleation and

motion• EOS and constitutive relations for different phases

required– Materials group

• Phase change physics in Eulerian VTF solver• Fracture and fragmentation• Incorporate and couple plasma simulation codes to

multiphase impact model via VTF– New solver class required for AMROC with some similarity to

existing Lagrangian-Eulerian capability


Milestones and progress for this year(CFD contributions)

• Integration of new strength, EoS and transport models into the VTF.– Work on Mie-Gruneisen EOS– Extension of Eulerian solvers to solid mechanics

• Full-system Ta/Ta ballistic runs for UQ analysis, including verification (nonlinear sensitivity analysis) and validation (in coordination with full system experiments) runs.– Initial simulations using Sandia CTH

• Implementation of massively parallel optimization algorithms within the VTF framework that support of UQ analysis and make effective utilization of NNSA petascale computing resources.– Work in progress in collaboration with UQ group


Back up


Aside: Density & Deformation Tensor

Conservation of mass requires that the determinate of the deformation tensor is related to the density by

At first glance it appears that the conserved vector of state

is redundant, by including both density and the inverse deformation tensor, but the explicit inclusion of density is equivalent to a material derivative equation forthe initial (perhaps spatially varying) density.


Eigensystem (1D)

Discontinuity capturing Eulerian numerical methods require knowledge of Eigensystem (characteritics and wavespeeds) for the 1D system.

The decomposition in primitive variables

is related by a transformation to the decompositionin the conserved variables

where


Eigensystem: Primitive Variables

The block-structure of the matrix allowsfor simple analysis. The eigensystemProblem may be viewed as

Two cases:

where

and

Note, we dropped the equationsFor density and energy as their Eigenvalues/vectors are trivial.



The case

Corresponds to the null spaces of A_12 and A_21, but

Assuming non-zero density, implies the eigenvectors are in the null space of A_12

Waves in the null space travel at the convective velocity ‘u’.

The A_12 matrix

In inviscid gas-dynamics, only the top rowof this matrix is non-zero as the stress-tensorIs diagonal. But we shall see that our model ofNon-linear elasticity this matrix is non-signular



The case

The system is equivalent tosolving a 3x3 eigensystem

And then computing the rest of the eigenvector

Notice that if then we have

so the eigenvalues are the multiple speeds of sound (transverse and longitudinal w


Blatz and Ko Model for Nonlinear Elasticity (Porous Rubber)

The Blatz and Ko model is a two-parameter model with a simple form for the stress tewhere

and where

Direct calculation shows that the reduced system is diagonal !

and the eigenvalues are positive


Eigensystem:Blatz and Ko

Recall that the eigensystem for the conserved system maybe computed from The eigensystem for the equations in primitive form by using

In particular the right eigenvectors for the Jacobian of the conserved systemare a transformation of the eigenvectors ‘R’ of the primitive system

For the Blatz and Ko model this becomes

where, the gammas are the eigenvalues of the reduced system.

Computational Fluid Dynamics€¦ · Computational Fluid Dynamics Dan Meiron Dale Pullin David Hill...

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