# Under Body Blasts

The main objective of this research project is two-fold: (1) to develop * predictive* High Performance Computational (HPC) models for underbody blast and its effects on personnel and vehicles, and (2) to develop nonlinear Model Order Reduction (MOR) methods that are applicable to these and other HPC models in order to enable

__parametric studies__*. The availability of such models and MOR methods would significantly reduce the number and cost of the real tests that must be performed in order to properly assess threats and improve safety.*

__in a reasonable amount of time__The collapse of a vehicle due to an under body blast is a transient, high-speed, multi-phase fluid-structure interaction problem characterized by ultrahigh compressions, shock waves, large structural displacements and deformations, self-contact, and the initiation and propagation of cracks in the structure. The development of a physics-based, multi-disciplinary, high-fidelity computational model for this problem is a formidable challenge. It requires incorporating in the computations several equations of state and material failure models, capturing the precise effects on the pressure peaks of many factors such as the rate of structural collapse, and accounting for all possible interactions between the air flow, structure, soil, and any other near-by media. It also requires developing a new class of computational methods for the solution of large deformation multi-fluid-structure interaction problems with unprecedented nonlinear stability and accuracy properties.

At AHPCRC, two complementary approaches are being considered for constructing high-fidelity computational models for the under body blast problem. The first approach models the effect of air blast using CONWEP software kernels. It bypasses the physics-based modeling of the fluid subsystem and transforms the problem into essentially a high-speed nonlinear transient structural dynamics problems. The second approach recognizes the under body blast problem as a highly nonlinear, two-way coupled, fluid-structure interaction problem in which shock waves propagate in a multi-material domain involving arbitrary equations of state and large density jumps. The global domain of interest includes a deformable solid subdomain that undergoes topological changes due to crack propagation. This problem is formulated and solved by coupling the state-

of-the-art, massively parallel flow solver AERO-F — and more speficially, its FIVER module — and the highly scalable, nonlinear structural analyzer AERO-S. The key components of the Finite Volume Method with Exact Riemann (FIVER) problems include: (1) the definition of a discrete surrogate material interface, (2) the computation first of a reliable inviscid approximation of the fluid state vector on each side of a discrete material interface via the construction and solution of local, exact, two-phase (JWL/air, JWL/structure, and air/structure) Riemann problems, (3) the algebraic solution of this auxiliary problem when the equation of state allows it, (4) the solution of this two-phase Riemann problem using sparse grid tabulations otherwise, (5) a ghost fluid scheme for approximating next the diffusive and source terms, (6) a systematic procedure for populating the ghost or inactive fluid grid points that guarantees under specified conditions the desired order of spatial accuracy, and (7) an energy conserving algorithm for enforcing the equilibrium transmission condition at a fluid-structure interface and therefore properly communicating with a nonlinear finite element structural analyzer.

To be predictive, the physics-based computational models outlined above are bound to be large-scale. Hence, their feasibility for parametric studies requires first the reduction of their sizes by several orders of magnitude, while maintaining their nonlinear stability and as much as possible their level of accuracy. This in turn requires the development of nonlinear MOR methods for both fluid and structural subsystems. This is another daunting challenge that is also pursued under this research project. More specifically, stable and accurate MOR methods are being developed for each component of the under body blast problem based on Galerkin and Petrov-Galerkin projection methods and local reduced-order bases. These methods are equipped with suitable hyper reduction and pre-computing techniques for maximizing their speed-up. A taste of their potential is illustrated here with preliminary results obtained for their application to the simulation of the response of a generic hull to an air blast loading.

**References** for FIVER and fluid-structure interaction (selected)

- K. Wang, J. Gretarsson, A. Main and C. Farhat, "Computational Algorithms for Tracking Dynamic Fluid-Structure Interfaces in Embedded Boundary Methods",
*International Journal for Numerical Methods in Fluids*, Vol. 70, pp. 515-535 (2012) - C. Farhat, J.-F. Gerbeau and A. Rallu, "FIVER: A Finite Volume Method Based on Exact Two-Phase Riemann Problems and Sparse Grids for Multi-Material Flows with Large Density Jumps",
*Journal of Computational Physics*, Vol. 231, pp. 6360-6379 (2012) - X. Zeng and C. Farhat, "A Systematic Approach for Constructing Higher-Order Immersed Boundary and Ghost Fluid Methods for Fluid and Fluid-Structure Interaction Problems",
*Journal of Computational Physics*, Vol. 231, pp. 2892-2923 (2012) - S. Brogniez, A. Rajasekharan and C. Farhat, "Provably Stable and Time-Accurate Extensions of Runge-Kutta Schemes for CFD Computations on Moving Grids",
*International Journal for Numerical Methods in Fluids*, Vol. 69, pp. 1249-1270 (2012) - K. Wang, A. Rallu, J.-F. Gerbeau and C. Farhat, "Algorithms for Interface Treatment and Load Computation in Embedded Boundary Methods for Fluid and Fluid-Structure Interaction Problems",
*International Journal for Numerical Methods in Fluids*, Vol. 67, pp. 1175-1206 (2011) - C. Farhat, A. Rallu, K. Wang and T. Belytschko, "Robust and Provably Second-Order Explicit-Explicit and Implicit-Explicit Staggered Time-Integrators for Highly Nonlinear Fluid-Structure Interaction Problems",
*International Journal for Numerical Methods in Engineering*, Vol. 84, pp. 73-107 (2010) - C. Farhat, A. Rallu and S. Shankaran, "A Higher-Order Generalized Ghost Fluid Method for the Poor for the Three-Dimensional Two-Phase Flow Computation of Underwater Implosions",
*Journal of Computational Physics*, Vol. 227, pp. 7674-7700 (2008)

**References** for model order reduction method (selected)

- D. Amsallem, M. Zahr and C. Farhat, "Nonlinear Model Order Reduction Based on Local Reduced-Order Bases",
*International Journal for Numerical Methods in Engineering*, International Journal for Numerical Methods in Engineering, Vol. 92, pp. 891–916, 7 (2012) - U. Hetmaniuk, R. Tezaur and C. Farhat, "Review and Assessment of Interpolatory Model Order Reduction Methods for Frequency Response Structural Dynamics and Acoustics Problems",
*International Journal for Numerical Methods in Engineering*, Vol. 90, pp. 1636-1662 (2012) - D. Amsallem and C. Farhat, "Stabilization of Projection-Based Reduced-Order Models",
*International Journal for Numerical Methods in Engineering*, Vol. 91, pp. 343-456 (2012) - D. Amsallem and C. Farhat, "An Online Method for Interpolating Linear Parametric Reduced-Order Models",
*SIAM Journal on Scientific and Statistical Computing*, Vol. 33, pp. 2169-2198 (2011) - K. Carlberg and C. Farhat, "A Low-Cost, Goal-Oriented Compact Proper Orthogonal Decomposition Basis For Model Reduction of Static Systems",
*International Journal for Numerical Methods in Engineering*, Vol. 86, pp. 381-402 (2011) - K. Carlberg, C. Bou-Mosleh and C. Farhat,"Efficient Nonlinear Model Reduction via a Least-Squares Petrov-Galerkin Projection and Compressive Tensor Approximations",
*International Journal for Numerical Methods in Engineering*, Vol. 86, pp. 155-181 (2011) - D. Amsallem, J. Cortial and C. Farhat, "Toward Real-Time CFD-Based Aeroelastic Computations Using a Database of Reduced-Order Information",
*AIAA Journal*, Vol. 48, pp. 2029-2037 (2010) - D. Amsallem, K. Carlberg, J. Cortial and C. Farhat, "A Method for Interpolating on Manifolds Structural Dynamics Reduced-Order Models",
*International Journal for Numerical Methods in Engineering*, Vol 80, pp. 1241 - 1258 (2009) - P. Avery, C. Farhat and U. Hetmaniuk, "A Padé-Based Factorization-Free Algorithm for Identifying the Eigenvalues Missed by a Generalized Symmetric Eigensolver",
*International Journal for Numerical Methods in Engineering*, Vol. 79, pp. 239-252 (2009) - D. Amsallem and C. Farhat, "An Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity",
*AIAA Journal*, Vol. 46, pp. 1803-1813 (2008)