Physics

We propose a novel method for the direct numerical simulation of interfacial flows involving large density contrasts, using a Volume-of-Fluid method. We employ the conservative formulation of the incompressible Navier-Stokes equations for immiscible fluids in order to ensure consistency between the discrete transport of mass and momentum in both fluids. This strategy is implemented on a uniform 3D Cartesian grid with a staggered configuration of primitive variables, wherein a geometrical reconstruction based mass advection is carried out on a grid twice as fine as that for the momentum. The implementation is in the spirit of Rudman (1998) [41], coupled with the extension of the direction-split time integration scheme of Weymouth & Yue (2010) [46] to that of conservative momentum transport. The resulting numerical method ensures discrete consistency between the mass and momentum propagation, while simultaneously enforcing conservative numerical transport to arbitrary levels of precision in 3D. We present several quantitative comparisons with benchmarks from existing literature in order to establish the accuracy of the method, and henceforth demonstrate its stability and robustness in the context of a complex turbulent interfacial flow configuration involving a falling raindrop in air.

We put forth a long short-term memory (LSTM) nudging framework for the enhancement of reduced order models (ROMs) of fluid flows utilizing noisy measurements for air traffic improvements. Toward emerging applications of digital twins in aviation, the proposed approach allows for constructing a realtime predictive tool for wake-vortex transport and decay systems. We build on the fact that in realistic application, there are uncertainties in initial and boundary conditions, model parameters, as well as measurements. Moreover, conventional nonlinear ROMs based on Galerkin projection (GROMs) suffer from imperfection and solution instabilities, especially for advection-dominated flows with slow decay in the Kolmogorov width. In the presented LSTM nudging (LSTM-N) approach, we fuse forecasts from a combination of imperfect GROM and uncertain state estimates, with sparse Eulerian sensor measurements to provide more reliable predictions in a dynamical data assimilation framework. We illustrate our concept by solving a two-dimensional vorticity transport equation. We investigate the effects of measurements noise and state estimate uncertainty on the performance of the LSTM-N behavior. We also demonstrate that it can sufficiently handle different levels of temporal and spatial measurement sparsity, and offer a huge potential in developing next-generation digital twin technologies.

Motivated by the inadequacy of conducting atomistic simulations of crack propagation using static boundary conditions that do not reflect the movement of the crack tip, we extend Sinclair's flexible boundary condition algorithm [Philos. Mag. 31, 647-671 (1975)] and propose a numerical-continuation-enhanced flexible boundary (NCFlex) scheme, enabling full solution paths for cracks to be computed with pseudo-arclength continuation, and present a method for incorporating more detailed far-field information into the model for next to no additional computational cost. The new algorithms are ideally suited to study details of lattice trapping barriers to brittle fracture and can be incorporated into density functional theory and multiscale quantum/classical QM/MM calculations. We demonstrate our approach for Mode III fracture with a 2D toy model and mploy it to conduct a 3D study of Mode I fracture of silicon using realistic interatomic potentials, highlighting the superiority of the new approach over employing a corresponding static boundary condition. In particular, the inclusion of numerical continuation enables converged results to be obtained with realistic model systems containing a few thousand atoms, with very few iterations required to compute each new solution. We also introduce a method to estimate the lattice trapping range of admissible stress intensity factors K − <K< K + very cheaply and demonstrate its utility on both the toy and realistic model systems.

In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the ADI method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on GPU, GPU+OpenMP, and Hybrid (GPU+OpenMP+MPI) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, CUDA, and CUDA+OpenMP implementations.

The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space, allowing a characterization of differential operators while providing structure which may be used to impose biases related to symmetry, isotropy, and conservation form. We demonstrate the effectiveness of our framework for a variety of physics, including local and nonlocal diffusion processes and single and multiphase flows. For the flow physics we demonstrate this approach leads to a learned operator that generalizes to system characteristics not included in the training sets, such as variable particle sizes, densities, and concentration.

Based on our recently proposed plane wave framework, we theoretically study the localized-extended transition in the one dimensional incommensurate systems with cosine type of potentials, which are in close connection to many recent experiments in the ultracold atom and photonic crystal. We formulate a propagator based scattering picture for the transition at the ground state and single particle mobility edge, in which the deeper connection between the incommensurate potentials, eigenstate compositions and transition mechanism is revealed. We further show that there exists a upper limit of localization length for all localized eigenstates, leading to an fundamental difference to the Anderson localization. Numerical calculations are presented alongside the analysis to justify our statements. The theoretical analysis and numerical methods can also be generalized to systems in higher dimensions, with different potentials or beyond the single particle regime, which would benefit the future studies in the related fields.

With this contribution, we give a complete and comprehensive framework for modeling the dynamics of complex mechanical structures as port-Hamiltonian systems. This is motivated by research on the potential of lightweight construction using active load-bearing elements integrated into the structure. Such adaptive structures are of high complexity and very heterogeneous in nature. Port-Hamiltonian systems theory provides a promising approach for their modeling and control. Subsystem dynamics can be formulated in a domain-independent way and interconnected by means of power flows. The modular approach is also suitable for robust decentralized control schemes. Starting from a distributed-parameter port-Hamiltonian formulation of beam dynamics, we show the application of an existing structure-preserving mixed finite element method to arrive at finite-dimensional approximations. In contrast to the modeling of single bodies with a single boundary, we consider complex structures composed of many simple elements interconnected at the boundary. This is analogous to the usual way of modeling civil engineering structures which has not been transferred to port-Hamiltonian systems before. A block diagram representation of the interconnected systems is used to generate coupling constraints which leads to differential algebraic equations of index one. After the elimination of algebraic constraints, systems in input-state-output(ISO) port-Hamiltonian form are obtained. Port-Hamiltonian system models for the considered class of systems can also be constructed from the mass and stiffness matrices obtained via conventional finite element methods. We show how this relates to the presented approach and discuss the differences, promoting a better understanding across engineering disciplines.

We present a robust, highly accurate, and efficient positivity- and boundedness-preserving diffuse interface method for the simulations of compressible gas-liquid two-phase flows with the five-equation model by Allaire et al. using high-order finite difference weighted compact nonlinear scheme (WCNS) in the explicit form. The equation of states of gas and liquid are given by the ideal gas and stiffened gas laws respectively. Under a mild assumption on the relative magnitude between the ratios of specific heats of the gas and liquid, we can construct limiting procedures for the fifth order incremental-stencil WCNS (WCNS-IS) with the first order Harten-Lax-van Leer contact (HLLC) flux such that positive partial densities and squared speed of sound can be ensured in the solutions, together with bounded volume fractions and mass fractions. The limiting procedures are discretely conservative for all conservative equations in the five-equation model and can also be easily extended for any other conservative finite difference or finite volume scheme. Numerical tests with liquid water and air are reported to demonstrate the robustness and high accuracy of the WCNS-IS with the positivity- and boundedness-preserving limiters even under extreme conditions.

The main result in this paper is a provably entropy stable shock capturing approach for the high order entropy stable DGSEM based on a hybrid blending with a subcell low order variant. Since it is possible to rewrite a high order SBP operator into an equivalent conservative finite volume form, we were able to design a low order scheme directly with the LGL nodes that is compatible to the discrete entropy analysis used for the proof of the entropy stable DGSEM. Furthermore, we present a hybrid low order/high order discretisation where it is possible to seamlessly blend between the two approaches, while still being provably entropy stable. We are able to extend the approach to three spatial dimensions on unstructured curvilinear hexahedral meshes. We validate our theoretical findings and demonstrate convergence order for smooth problems, conservation of the primary quantities and discrete entropy stability for an arbitrary blending on curvilinear grids. In practical simulations, we connect the blending factor to a local troubled element indicator that provides the control of the amount of low order dissipation injected into the high order scheme. We modified an existing shock indicator, which is based on the modal polynomial representation, to our provably stable hybrid scheme. The aim is to reduce the impact of the parameters as good as possible. We describe our indicator in detail and demonstrate its robustness in combination with the hybrid scheme, as it is possible to compute all the different test cases without changing the indicator. The test cases include e.g. the double Mach reflection setup, forward and backward facing steps with shock Mach numbers up to 100. The proposed approach is relatively straight forward to implement in an existing entropy stable DGSEM code as only modifications local to an element are necessary.

In this paper, a high order quasi-conservative discontinuous Galerkin (DG) method using the non-oscillatory kinetic flux is proposed for the 5-equation model of compressible multi-component flows with Mie-Grüneisen equation of state. The method mainly consists of three steps: firstly, the DG method with the non-oscillatory kinetic flux is used to solve the conservative equations of the model; secondly, inspired by Abgrall's idea, we derive a DG scheme for the volume fraction equation which can avoid the unphysical oscillations near the material interfaces; finally, a multi-resolution WENO limiter and a maximum-principle-satisfying limiter are employed to ensure oscillation-free near the discontinuities, and preserve the physical bounds for the volume fraction, respectively. Numerical tests show that the method can achieve high order for smooth solutions and keep non-oscillatory at discontinuities. Moreover, the velocity and pressure are oscillation-free at the interface and the volume fraction can stay in the interval [0,1].