Training data-driven turbulence models with high fidelity Reynolds stress can be impractical and recently such models have been trained with velocity and pressure measurements. For gradient-based optimization, such as training deep learning models, this requires evaluating the sensitivities of the RANS equations. This paper explores the use of an ensemble approximation of the sensitivities of the RANS equations in training data-driven turbulence models with indirect observations. A deep neural network representing the turbulence model is trained using the network’s gradients obtained by backpropagation and the ensemble approximation of the RANS sensitivities. Different ensemble approximations are explored and a method based on explicit projection onto the sample space is presented. As validation, the gradient approximations from the different methods are compared to that from the continuous adjoint equations. The ensemble approximation is then used to learn different turbulence models from velocity observations. In all cases, the learned model predicts improved velocities. However, it was observed that once the sensitivity of the velocity to the underlying model becomes small, the approximate nature of the ensemble gradient hinders further optimization of the underlying model. The benefits and limitations of the ensemble gradient approximation are discussed, in particular as compared to the adjoint equations.

In this paper, we develop a new class of linear time-integration schemes for phase-field models. The newly proposed schemes extend the recently developed energy quadratization technique by introducing extra free parameters to further stabilize the schemes and improve their accuracy. The freshly proposed schemes have several advantages. First of all, they are rather generic such that they apply to most existing phase-field models in the literature. The resulted schemes are also linear in time, which means only a linear system needs to be solved during each time marching step. Thus, it significantly reduces the computational cost. Besides, they are unconditionally energy stable such that a larger time step size is practical. What is more, the solution existence and uniqueness in each time step are guaranteed without any dependence on the time step size. To demonstrate the generality of the proposed schemes, we apply them to several typical examples, including the widely-used molecular beam epitaxy (MBE) model, the Cahn-Hilliard equation, and the diblock copolymer model. Numerical tests reveal that the proposed schemes are accurate and efficient. This new family of linear and unconditionally energy stable schemes provides insights in developing numerical approximations for general phase field models.

The fundamental nature of flow instability in wall bounded flows changes with Mach number. The objectives of this study are two-fold, (i) compute the instability modes in high Mach number Poiseuille flows using linear stability analysis (LSA) and, (ii) perform direct numerical simulations (DNS) of the instability development using a solver based on gas kinetic method (GKM) for the purpose of code validation by comparison against LSA results. The LSA and DNS are performed for the case of Poiseuille flow over a range of Mach numbers – from moderately supersonic to hypersonic speeds. First, LSA is employed to identify the most unstable mode over the range of Mach numbers. We then perform two sets of GKM-DNS to corroborate the LSA results over the Mach number range. In the first set of simulations, the background field is initially perturbed with the most unstable mode identified by LSA and the evolution is monitored. It is shown that GKM-DNS accurately captures the exponential growth in kinetic energy for all Mach numbers. The second set of GKM-DNS simulations is performed by superposing the background pressure field with random initial perturbations. After an initial transient period, the modes predicted by LSA dominate the DNS flow field evolution. The wave-vector and mode shapes of the dominant instability are well replicated by GKM-DNS at each Mach number. These insights in the linear regime of high speed Poiseuille flow and validation of GKM are important for understanding and simulating wall bounded flows.

In this work, a novel and simple phase-field-based lattice Boltzmann (LB) method is proposed for the axisymmetric two-phase electro-hydrodynamic flows. The present LB method is composed of three LB models, which are used to solve the axisymmetric Allen-Cahn equation for the phase field, the axisymmetric Poisson equation for the electric potential, and the axisymmetric Navier-Stokes equations for the flow field. Compared with the previous LB models for the axisymmetric Poisson equation, which can be viewed as the solvers to the convection-diffusion equation, the present model is a genuine solver to the axisymmetric Poisson equation. To test the capacity of the LB method, the deformation of a single leaky or perfect dielectric drop under a uniform electric field is considered, and the effects of electric strength, conductivity ratio, and permittivity ratio are investigated in detail. It is found that the present numerical results are in good agreement with some available theoretical, numerical and/or experimental data.

The topology optimization of a linearized elasticity system with the area (volume) constraint is investigated. A non-ersatz material approach is proposed. By introducing a fixed background domain, the linearized elasticity system is extended into the background domain by a characteristic function. The piecewise constant level set (PCLS) method is applied to represent the original material region and the void region. A quadratic function of PCLS function is proposed to replace the characteristic function. The functional derivative of the objective functional with respect to PCLS function is derived, which is zero in the void region and nonzero in the original material region. A penalty gradient algorithm is proposed. Four numerical experiments of 2D and 3D elastic structures with different boundary conditions are presented, illustrating the validity of the proposed algorithm.

The maximum entropy moment system extends the Euler equation to nonequilibrium gas flows by considering higher order moments such as the heat flux. This paper presents a systematic study of the maximum entropy moment system of Boltzmann equation. We consider a hypothetical one-dimensional gas and study a five-moment model. A numerical algorithm for solving the optimization problem is developed to produce the maximum entropy distribution function from known moments, and the asymptotic behaviour of the system around the singular region known as the Junk’s line, as well as that near the boundary of the realizability domain is analyzed. Furthermore, we study the properties of the system numerically, including the behaviour of the system around the Maxwellian and within the interior of the realizability domain, and properties of its characteristic fields. Our studies show the higher order entropy-based moment models to differ significantly from the Euler equations. Much of this difference comes from the singularity near the Junk’s line, which would be removed if a truncation of the velocity domain is employed.

Considering the fact that the lattice discrete effects are neglected while introducing a body force into the simplified lattice Boltzmann method (SLBM), we propose a consistent forcing scheme in SLBM for incompressible flows with external forces. The lattice discrete effects are considered at the level of distribution functions in the present forcing scheme. Consequently, it is more accurate compared with the original forcing scheme used in SLBM. Through Taylor series expansion and Chapman-Enskog (CE) expansion analysis, the present forcing scheme can be proven to recover the macroscopic Navier-Stokes (N-S) equations. Then, the macroscopic equations are resolved through a fractional step technique. Furthermore, the material derivative term is discretized by the central difference method. To verify the results of the present scheme, we simulate with multiple forms of external force interactions including the space- and time-dependent body forces. Hence, the present forcing scheme overcomes the disadvantages of the original forcing scheme and the body force can be accurately imposed in the present scheme even when a coarse mesh is applied while the original scheme fails. Excellent agreements between the analytical solutions and our numerical results can be observed.

Robust control design for quantum systems with uncertainty is a key task for developing practical quantum technology. In this paper, we apply neural networks to learn the control of a quantum system with uncertainty. By exploiting the auto differentiation function developed for neural network models, our method avoids the manual computation of the gradient of the cost function as required in traditional methods. We implement our method using two algorithms. One uses neural networks to learn both the states and the controls and one uses neural networks to learn only the controls but solve the states by finite difference methods. Both algorithms incorporate the sampling-based learning process into the training of the networks. The performance of the algorithms is evaluated on a practical numerical example, followed by a detailed discussion about the advantage and trade-offs between our method and the other numerical schemes.

This work considers numerical methods for the time-dependent Schrödinger equation of incommensurate systems. By using a plane wave method for spatial discretization, the incommensurate problem is lifted to a higher dimension that results in semidiscrete differential equations with extremely demanding computational cost. We propose several fully discrete time stepping schemes based on the idea of "layer-splitting", which decompose the semidiscrete problem into sub-problems that each corresponds to one of the periodic layers. Then these schemes handle only some periodic systems in the original lower dimension at each time step, which reduces the computational cost significantly and is natural to involve stochastic methods and parallel computing. Both theoretical analysis and numerical experiments are provided to support the reliability and efficiency of the algorithms.

Large-scale reservoir modeling and simulation of gas reservoir flows in fractured porous media is currently an important topic of interest in petroleum engineering. In this paper, the dual-porosity dual-permeability (DPDP) model coupled with the Peng-Robinson equation of state (PR-EoS) is used for the mathematical model of the gas reservoir flow in fractured porous media. We develop and study a parallel and highly scalable reservoir simulator based on an adaptive fully implicit scheme and an inexact Newton type method to solve this dual-continuum mathematical model. In the approach, an explicit-first-step, single-diagonal-coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with adaptive time stepping is proposed for the fully implicit discretization, which is second-order and L-stable. And then we focus on the family of Newton–Krylov methods for the solution of a large sparse nonlinear system of equations arising at each time step. To accelerate the convergence and improve the scalability of the solver, a class of multilevel monolithic additive Schwarz methods is employed for preconditioning. Numerical results on a set of ideal as well as realistic flow problems are used to demonstrate the efficiency and the robustness of the proposed methods. Experiments on a supercomputer with several thousand processors are also carried out to show that the proposed reservoir simulator is highly scalable.

Flash calculation plays significant roles in petroleum and chemical industries. Since Michelsen proposed his milestone studies in 1982, through several decades of development, the current research interests on flash calculation have been shifted from accuracy to efficiency, but the ultimate goal remains the same; that is accurate determination of equilibrium phase amounts and compositions at a given condition. On the other hand, finding the transition route and its related saddle point is often crucial to understand the whole energy landscape of flash models, which would provide new insights for designing numerical algorithms or optimizing existing ones. In this study, an efficient numerical approach is developed by coupling the string method with the exponential time differencing (ETD) scheme to investigate the minimum energy paths and first-order saddle points of VT flash models with Peng-Robinson equation of state. As a promising alternative to the conventional approach, VT flash calculates phase equilibria under a new variable specification of volume and temperature. The Rosenbrock-type ETD scheme is used to reduce the computational difficulty caused by the high stiffness of the model systems. The proposed ETD-String method successfully calculates the minimum energy paths of single-component and two-component VT flash models with strong stiffness. Numerical results also show good feasibility and accuracy in calculation of equilibrium phase amounts and compositions.

We propose a new family of mapped WENO schemes by using several adaptive control functions and a smoothing approximation of the signum function. The proposed schemes admit an extensive permitted range of the parameters in the mapping functions. Consequently, they have the capacity to achieve optimal convergence rates, even near critical points. Particularly, the new schemes with fine-tuned parameters illustrates a significant advantage when solving problems with discontinuities. It produces numerical solutions with high resolution without generating spurious oscillations, especially for long output times.