Abstract: Concurrent multiscale methods are essential for the understanding and prediction of behavior of engineering systems when a small-scale event will eventually determine the performance of the entire system. Here, we describe the recently-proposed [1] domain-decomposition-based Schwarz alternating method as a means for concurrent multiscale coupling in finite deformation quasistatic and dynamic solid mechanics. The approach is based on the simple idea that if the solution to a partial differential equation is known in two or more regularly shaped domains comprising a more complex domain, these local solutions can be used to iteratively build a solution for the more complex domain. The proposed approach has a number of advantages over competing multiscale coupling methods, most notably its concurrent nature, its ability to couple non-conformal meshes with different element topologies, and its non-intrusive implementation into existing codes. In this talk, we will first overview our original formulation of the Schwarz alternating method for multiscale coupling in the context of quasistatic solid mechanics problems [1]. We will discuss the method's proven convergence properties, and demonstrate its accuracy, convergence and scalability of the proposed Schwarz variants on several quasistatic solid mechanics examples simulated using the Albany/LCM code. The bulk of the talk will present some recent extensions of the Schwarz alternating formulation to dynamic solid mechanics problems [2]. Our dynamic Schwarz formulation is not based on a space-time discretization like other dynamic Schwarz-like methods; instead, it uses a governing time-stepping algorithm that controls time-integrators within each subdomain. As a result, the method is straight-forward to implement into existing codes (e.g, Albany/LCM), and allows the analyst to use different time-integrators with different time steps within each domain. We demonstrate on several test cases (including bolted-joint problems of interest to production) that coupling using the proposed method introduces no dynamic artifacts that are pervasive in other coupling methods (e.g., spurious wave reflections near domain boundaries), regardless of whether the coupling is done with different mesh resolutions, different element types like hexahedral or tetrahedral elements, or even different time integration schemes, like implicit and explicit. Furthermore, on dynamic problems where energy is conserved, we show that the method is able to preserve the property of energy conservation. REFERENCES [1] A. Mota, I. Tezaur, C. Alleman. “The alternating Schwarz method for concurrent multiscale coupling”, Comput. Meth. Appl. Mech. Engng. 319 (2017) 19-51. [2] A. Mota, I. Tezaur, G. Phlipot. "The Schwarz alternating method for dynamic solid mechanics", in preparation for submission to Comput. Meth. Appl. Mech. Engng.

Abstract: The Exascale Computing Project (ECP, https://exascaleproject.org/) is a U.S. Dept. of Energy effort developing hardware, software infrastructure, and applications for computational platforms capable of performing 10^18 floating point operations per second (one “exaop”). The Exascale Additive Manufacturing Project (ExaAM) is one of the applications selected for development of models that would not be possible on even the largest of today’s computational systems. In addition to ORNL, partners include Lawrence Livermore National Laboratory (LLNL), Los Alamos National Laboratory (LANL), the National Institute for Standards and Technology (NIST), as well as key universities such as Purdue Univ., UCLA, and Penn. State Univ. Since we are both leveraging existing simulation software and also developing new capabilities, we will describe the physics components that comprise our simulation environment and report on progress to date using highly-resolved melt pool simulations to inform part-scale finite element thermomechanics simulations, drive microstructure evolution, and determine constitutive mechanical property relationships based on those microstructures using polycrystal plasticity. The coupling of melt pool dynamics and thermal behavior, microstructure evolution, and microscale mechanical properties provides a unique, high-fidelity model of the process-structure-property relationship for additively manufactured parts. We will report on the numerics, implementation, and performance of the nonlinearly consistent coupling strategy, including convergence behavior, sensitivity to fluid flow fidelity, and challenges in timestepping. The ExaAM team includes James Belak, co-PI (LLNL), Nathan Barton (LLNL), Matt Bement (LANL), Curt Bronkhorst (Univ. of Wisc.), Neil Carlson (LANL), Robert Carson (LLNL), Jean-Luc Fattebert (ORNL), Neil Hodge (LLNL), Zach Jibben (LANL), Brandon Lane (NIST), Lyle Levine (NIST), Chris Newman (LANL), Balasubramaniam Radhakrishnan (ORNL), Matt Rolchigo (LLNL), Stuart Slattery (ORNL), and Steve Wopschall (LLNL). This work was supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration.

Abstract: We introduce a new parallelizable numerical multiscale method for advection-dominated problems as they often occur in engineering and geosciences. State of the art multiscale simulation methods work well in situations in which stationary and elliptic scenarios prevail but are prone to fail when the model involves dominant lower order terms which is common in applications. We suggest to overcome the associated difficulties through a reconstruction of subgrid variations into a modified basis by solving many independent (local) inverse problems that are constructed in a semi-Lagrangian step. Globally the method looks like a Eulerian method with multiscale stabilized basis. The method is extensible to other types of Galerkin methods, higher dimensions, nonlinear problems and can potentially work with real data. We provide examples inspired by tracer transport in climate systems in one and two dimensions and numerically compare our method to standard methods.

Abstract: Galerkin projection provides a formal means to project a differential equation onto a set of preselected basis functions. This may be done for the purpose of formulating a numerical method, as in the case of spectral methods, or formulation of a reduced-order model (ROM) for a complex system. Here, a new method is proposed in which the basis functions used in the projection process are determined from an asymptotic (perturbation) analysis. These asymptotic basis functions (ABF) are obtained from the governing equation itself; therefore, they contain physical information about the system and its dependence on parameters contained within the mathematical formulation. We refer to this as reduced-physics modeling (RPM) as the basis functions are obtained from a physical, i.e.\ model-driven, rather than data-driven, approach. Therefore, the ABF hold the potential to provide an accurate RPM of a system that captures the physical dependence on model parameters and is accurate over a wider range of parameters than possible for traditional ROM methods. This new approach is tailor-made for modeling multiscale problems as the various scales, whether overlapping or distinct in time or space, are formally accounted for in the ABF. A regular-perturbation problem is used to illustrate that projection of the governing equations onto the ABF allows for determination of accurate approximate solutions for values of the ``small'' parameter that are much larger than possible with the asymptotic expansion alone.

Abstract: A mathematical model of the flow of a polyatomic gas containing a combination of the Navier-Stokes-Fourier model (NSF) and the model kinetic equation of polyatomic gases is presented. At the heart of the hybrid components is a unified physical model, as a result of which the NSF model is a strict first approximation of the model kinetic equation. The model allows calculations of flow fields in a wide range of Knudsen numbers (Kn), as well as fields containing regions of high dynamic nonequilibrium. The boundary conditions on a solid surface are set at the kinetic level, which allows, in particular, to formulate the boundary conditions on the surfaces absorbing or emitting gas. The hybrid model was tested. The example of the problem of the shock wave profile shows that up to Mach numbers near 2 the combined model gives smooth solutions even in those cases where the sewing point is in a high gradient region. For the Couette flow, smooth solutions are obtained at M=5, Kn=0.2. As a result of research, a weak and insignificant difference between the kinetic region of the hybrid model and the “pure” kinetic model was established. A model effect was discovered: in the region of high nonequilibrium, there is an almost complete coincidence of the solutions of the kinetic region of the combined model and the “pure” kinetic solution. This work was conducted with the financial support of the Ministry of Education and Science of the Russian Federation, project №9.7170.2017/8.9.