## Computational Physics

The integrated computational approach for predicting the behavior of the demonstration and discovery systems will consist of combining macro- and micro-continuum representations. Our two-fold strategy recognizes that only a relatively small part of the material will generally be instantaneously exposed to shocks and/or undergoing large plastic flow, phase change, and reaction. The rest of the material may be adequately described by macroscopic constitutive models, based on homogenization of the complex but slowly-varying microstructure. However, the shocked and/or reactive regions will generally need a more fundamentally based simulation since no reliable and universal constitutive model exists. We will perform all the computations at the continuum level, since current experimental data does not provide evidence for considering molecular/atomistics scales. Our approach will take advantage of the instantaneous localization knowledge of those regions where full simulations are necessary. In short, we propose to solve phase-averaged *macro-continuum* equations with constitutive laws provided by results obtained from full simulations of the multi-phase mixture using *micro-continuum* equations. We will refer to codes that solve macro- and micro-problems as M&m codes.

### The Demonstration System — Ni-Al.

During the last several decades nickel aluminides have been extensively studied as candidates for a variety of high temperature structural applications due to their excellent high temperature oxidation resistance. Under shock-loading, radically modified structures, metastable phases, and novel compounds may be formed with very different properties. In the case of Ni-Al the ductility may be improved by a critical reduction in grain size. The intermetallic Ni-Al system was one of the first composites investigated using the so-called combustion synthesis method. Members of our team demonstrated the reaction sensitivity of mechanically activated Ni-Al mixtures. As such, this system is relatively well understood and many parameters that are critical for model development and validation are available.

### The Discovery System — c-BN.

After using the Ni-Al system for V&V/UQ of the computational model, we aim to apply the model for the design of novel materials. In particular, we will explore conditions for synthesis of phases/compounds and provide predictions of non-equilibrium structures that will form under shock wave-processing. Specifically, we plan to identify conditions under which we can synthesize cubic boron nitride (c-BN). c-BN is not found in nature and therefore can only be produced synthetically. The hardness of c-BN is inferior only to diamond, but its thermal and chemical stability is superior. Because of these properties, c-BN surpasses diamond in high temperature mechanical applications. Presently, synthesis of c-BN employs the same methods used to synthesize diamond from graphite: it is produced by treating hexagonal boron nitride (h-BN) at high pressure (between 5 and 18 GPa) and high temperature (between 2300 and 3800 K). Our approach is different but still requires high pressure, i.e. shock wave conditions.

### Macro-continuum Modeling

For the macro-continuum modeling on centimeter scales, we will solve the balance equations for mass, momenta, and energy in Eulerian form using a Wavelet Adaptive Multiscale Representation (WAMR) solver. The Eulerian formulation is more suited for problems with large motions, as in impact/shock events. However, the constitutive model update requires information about the reference state, i.e. the Lagrangian description. To accomplish this, for computational convenience one evolves the inverse of the deformation gradient. Such an equation is equivalent to Maxwell’s compatibility criterion used in the analysis of shocks in the Lagrangian formulation.

**Wavelet adaptive multiresolution representation (WAMR)** is a numerical method for the solution of partial differential equations in multidimensional domain. It is ideally suited for handling problems developing localized structures, which might occur intermittently anywhere in the computational domain, or change their locations and scales in space and time. The main advantage of the WAMR algorithm is that it requires a far fewer number of grid points than other algorithms when applied to problems with a large range of spatial scales. Taking advantage of wavelet features, WAMR performs an adaptive compression of the solution, and exhibits exponential convergence on the unevenly spaced collocation points. The wavelet amplitudes provide a direct measure of the local approximation error at each collocation point, so that the WAMR algorithm produces automatically verified solutions. The WAMR algorithm has been implemented in Fortran90 and uses MPI standards to take advantage of large-scale computational resources.

### Micro-continuum Modeling

At the micro-continuum level, we will also solve the set of unsteady equations that describe the conservation of mass, momenta, and energy for each material component and phase. However, here we will use their material forms and solve them employing micro-continuum constitutive equations using a Parallel Generalized Finite Element (PGFem3D) solver.

**PGFem3D** is a highly scalable finite element library for computing the multiscale response of heterogeneous materials subjected to chemo-thermo-mechanical loads. PGFem3D uses a vertically coupled multiscale framework to bridge many length-scales for prediction of inelastic material behavior in the 3D finite strain setting. PGFem3D has been used to simulate elasto-viscoplasticity in polycrystalline microstructures, multiscale effects of particle debonding in reinforced elastomers, and multiscale failure of heterogeneous interfaces. Recently, the high performance library was used in a hierarchically parallel and fully-coupled multiscale solver for inhomogeneous interfaces. PGFem3D is used in the C-SWARM to compute the multiscale constitutive response of the heterogeneous reactive materials through the generalized computational theory of homogenization (GCTH). GCTH provides the accurate homogenized macroscopic response (e.g., stress, heat flux, etc.) while capturing physical processes at the microscale in sufficient detail. Using the detailed microscale information to compute the macroscale response, PGFem3D provides physical insight into various physical mechanisms at play in heterogeneous composites.

### Principal Investigators

### Karel Matouš

**Associate Professor,
University of Notre Dame,
Director, Lead Team Member**

### Gretar Tryggvason

**Department Head and
Charles A. Miller, Jr.
Distinguished Professor,
Johns Hopkins Department of
Mechanical Engineering**

### Samuel Paolucci

**Professor Emeritus,
University of Notre Dame**