[en] Highlights: • An indirect ALE DG model on moving triangular meshes with mesh topological adaptability is developed. • The best features of the Lagrangian description, DG method and adaptive topology optimization strategy are incorporated. • The present scheme can handle hydrodynamics problems which involve complex deformations, large distortions and strong shock. • The new indirect ALE DG scheme is more accurate and stable than the conventional Lagrangian DG scheme. In this paper, we present a novel cell-centered indirect Arbitrary-Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme on moving unstructured triangular meshes with mesh topological adaptability, aiming to deal with the strong distortions and large deformation flow problems. The scheme combines the explicit time marching Lagrangian DG methodology with the adaptive mesh topology optimization technique. The scheme consists of the following three steps. Firstly, we utilize the Runge-Kutta DG method to solve the compressible Euler equation in Lagrangian framework, and employ a nodal solver to obtain the nodal velocity and numerical fluxes across element boundaries. The physical variable and nodal position are updated in this step. Secondly, the adaptive mesh topology optimization technique, which includes the mesh refinement, edge collapse operation and mesh regularization, is implemented to eliminate the highly distorted elements and improve the mesh quality. Thirdly, the conservative remapping algorithm is employed, which can maintain the conservative interpolation of the Lagrangian solution onto the remeshed grid. The present indirect ALE DG scheme can ensure the high quality of the mesh by optimizing the topology connectivity, so that the present scheme can successfully simulate complex vortical flow problems for a sufficient simulation time. Due to the inherent Lagrangian nature, the present scheme can naturally track the multi-material flow interface, rather than using algorithms with interface reconstruction or diffuse interfaces. The scheme is validated with several benchmark flow problems. It is demonstrated that the present indirect ALE DG scheme with topological adaptability can accurately simulate flow problems with large fluid deformations and distortions. It can achieve remarkable improvements compared with the conventional Lagrangian DG method with fixed topological connectivity.

[en] Highlights: • A staggered-projection Godunov-type scheme is proposed for the full Baer-Nunziato (BN) two-phase model. • The continuity of Riemann invariants across porosity jumps is fully taken into account. • The generalized Riemann problem (GRP) solver is applied to achieve the second order accuracy. When describing the deflagration-to-detonation transition in solid granular explosives mixed with gaseous products of combustion, a well-developed two-phase mixture model is the compressible Baer-Nunziato (BN) model of flows containing solid and gas phases. As this model is numerically simulated by a conservative Godunov-type scheme, spurious oscillations are likely to generate from porosity interfaces, and may result from the average process of conservative variables that violates the continuity of Riemann invariants across porosity interfaces. In order to reduce numerical oscillations, this paper proposes a staggered-projection Godunov-type scheme over a fixed gas-solid staggered grid, by enforcing that compaction waves with porosity jumps are always inside gaseous grid cells and other discontinuities appear at gaseous cell interfaces. The scheme is based on a standard Godunov scheme for the Baer-Nunziato model on gaseous cells and guarantees the continuity of the Riemann invariants associated with the compaction waves across porosity jumps. While porosity interfaces are moving, a projection process fully takes into account the continuity of associated Riemann invariants and ensures that porosity jumps remain inside gaseous cells. Furthermore, the generalized Riemann problem (GRP) solver is applied, not only to achieve second-order accuracy, reduce numerical oscillations, but guarantees the well-balanced property of the resulting scheme as well.

[en] We propose an effective and light-weight learning algorithm, Symplectic Taylor Neural Networks (Taylor-nets), to conduct continuous, long-term predictions of a complex Hamiltonian dynamic system based on sparse, short-term observations. At the heart of our algorithm is a novel neural network architecture consisting of two sub-networks. Both are embedded with terms in the form of Taylor series expansion designed with symmetric structure. The key mechanism underpinning our infrastructure is the strong expressiveness and special symmetric property of the Taylor series expansion, which naturally accommodate the numerical fitting process of the gradients of the Hamiltonian with respect to the generalized coordinates as well as preserve its symplectic structure. We further incorporate a fourth-order symplectic integrator in conjunction with neural ODEs' framework into our Taylor-net architecture to learn the continuous-time evolution of the target systems while simultaneously preserving their symplectic structures. We demonstrated the efficacy of our Taylor-net in predicting a broad spectrum of Hamiltonian dynamic systems, including the pendulum, the Lotka–Volterra, the Kepler, and the Hénon–Heiles systems. Our model exhibits unique computational merits by outperforming previous methods to a great extent regarding the prediction accuracy, the convergence rate, and the robustness despite using extremely small training data with a short training period (6000 times shorter than the predicting period), small sample sizes, and no intermediate data to train the networks.

[en] Highlights: • Novel strategy to sample from multi-modal distributions. • Creative method to calculate the unknown integration constant. • A whole scheme for realizing the sampling of zero-variance importance function and getting the integration constant. A general purpose strategy is proposed to realize asymptotically the zero-variance importance sampling. The unknown integration constant can also be calculated simultaneously. This strategy can sample efficiently from multi-dimensional zero-variance importance function which is multi-modal by particular Markov Chain random walk. Sampling from this kind of distribution has been a challenge for a long time. Moreover, by using the probability density function reconstruction method, the unknown integration constant can be estimated. This feature is absent in traditional Markov Chain Monte Carlo method. Some multi-dimensional integrals are analyzed carefully. The results show this strategy is efficient.

[en] Highlights: • The Path Integral (PI) method is extended to solve SDEs with combined noises and corresponding governing equation. • We derive and verify the short-time transition probability density function (PDF), which is used to obtain the PI solution. • The path integral solutions have higher accuracy than the Monte Carlo solutions at the tail region of the PDF. • The PI solutions are modified to analyze the first-passage problem. We study the first-passage problem for a process governed by a stochastic differential equation (SDE) driven simultaneously by both parametric Gaussian and Lévy white noises. We extend the path integral (PI) method to solve the SDE with this combined noise input and the corresponding fractional Fokker-Planck-Kolmogorov equations. Then, the PI solutions are modified to analyze the first-passage problem. Finally, numerical examples based on Monte Carlo simulations verify the extension of the PI method and the modification of the PI solutions. The detailed effects of the system parameters on the first-passage problem are analyzed.

[en] Highlights: • A task-based parallel Eulerian-Lagrangian framework. • Asynchronous parallel multi-physics for reacting flows with spray. • Parallel load balanced Eulerian-Lagrangian simulation of combustors with swirl injectors. • Excellent parallel efficiency for applications with large particle count and highly non-uniform distribution. An asynchronous task-based Eulerian-Lagrangian approach for efficient parallel multi-physics simulations that can scale for arbitrary large number of particles and non-uniformly distributed particles is presented. The parallel methodology is based on a task-based partitioning of the multi-physics problem, where each single-physics problem is considered as a task and carried out using its own set of processes. This allows the two problems to solve their governing equations concurrently; therefore, hiding the computational cost incurred of solving an additional physical solver. Applications to complex breakup mechanism leading to highly dynamic computational loading and three-dimensional swirl combustion chamber with reacting flow/spray with extremely uneven particle distribution demonstrate the improved parallel efficiency and great potential of the presented approach.

[en] Highlights: • Multi-dimensional numerical scheme based on high-order finite elements. • Fully implicit formulation including the self-consistent electromagnetic fields. • Numerical energy conservation proved theoretically and on a test problem. • Non-local transport is captured by the model and demonstrated on a physical problem. Detailed description of the transport processes in plasma is crucial for many disciplines. When the mean-free-path of the electrons is comparable or exceeds a characteristic length scale of the plasma profile, non-local behavior can be observed. Predictions of the diffusion theory are not valid and non-local electric and magnetic fields are generated. Kinetic modeling of these phenomena on time scales several orders of magnitude longer than the electron–electron collision time has proven to be cumbersome due to prohibitive requirements on the time step and violation of the conservation laws in the classical explicit Vlasov–Fokker–Planck methods. Therefore, a multi-dimensional conservative implicit Vlasov–Fokker–Planck–Maxwell method is proposed, where the distribution function is approximated by a truncated Cartesian tensor expansion. The electric and magnetic fields are modeled self-consistently, describing the generation process and emergence of non-locality in detail. Mixed finite elements are employed in space and the velocity dimension is discretized by staggered finite differences. Conservation properties are proved theoretically and the overall features are benchmarked on a series of physically representative problems. The second order convergence in velocity and the spatial order proportional to the polynomial order of the finite elements is shown. Further possible extensions of the method are discussed.

[en] Highlights: • Formulate a new first-order hyperbolic system for the Navier-Stokes equations. • Develop efficient and accurate reconstructed discontinuous Galerkin methods. • Present a number of test cases for both steady and time accurate flow problems. • Achieve the designed order of accuracy for the primary variables and their gradients. • Provide an attractive and viable alternative for solving the Navier-Stokes equations. A new first-order hyperbolic system (FOHS) is formulated for the compressible Navier-Stokes equations. The resulting hyperbolic Navier-Stokes system (HNS), termed HNS20G in this paper, introduces the gradients of density, velocity, and temperature as auxiliary variables. Efficient, accurate, compact and robust reconstructed discontinuous Galerkin (rDG) methods are developed for solving this new HNS system. The newly introduced variables are recycled to obtain the gradients of the primary variables. The gradients of these gradient variables are reconstructed based on a newly developed variational formulation in order to obtain a higher order polynomial solution for these primary variables without increasing the number of degrees of freedom. The implicit backward Euler method is used to integrate solution in time for steady flow problems, while the third-order explicit first stage singly diagonally Runge-Kutta (ESDIRK) time marching method is implemented for advancing solutions in time for unsteady flows. The flux Jacobian matrices are obtained with an automatic differentiation toolkit TAPENADE. The approximate system of linear equations is solved with either symmetric Gauss-Seidel (SGS) method or general minimum residual (GMRES) algorithm with a lower-upper symmetric Gauss-Seidel (LU-SGS) preconditioner. A number of test cases are presented to assess accuracy and performance of the newly developed HNS+rDG methods for both steady and unsteady compressible viscous flows. Numerical experiments demonstrate that the developed HNS+rDG methods are able to achieve the designed order of accuracy for both primary variables and the their gradients, and provide an attractive and viable alternative for solving the compressible Navier-Stokes equations.

[en] Highlights: • A novel Lubrication Dynamics method that can efficiently simulate dense non-colloidal suspensions is proposed. • The proposed model conserves both the linear and angular momentum of the particle system. • A semi-implicit splitting integration scheme to solve for the particles translational and rotational velocities is presented. • Based on accuracy and stability, the semi-implicit scheme is shown to be faster than the explicit Velocity-Verlet scheme. In this paper, a novel semi-implicit lubrication dynamics method that can efficiently simulate dense non-colloidal suspensions is proposed. To reduce the computational cost in the presented methodology, inter-particle lubrication-based forces and torques alone are considered together with a short-range repulsion to enforce finite inter-particle separation due to surface roughness, Brownian forces or other excluded volume effects. Given that the lubrication forces are singular, i.e. scaling inversely with the inter-particle gap, the strategy to expedite the calculations is severely compromised if explicit integration schemes are used, especially at high concentrations. To overcome this issue, an efficient semi-implicit splitting integration scheme to solve for the particles translational and rotational velocities is presented. To validate the proposed methodology, a suspension under simple shear test is simulated in three dimensions and its rheology is compared against benchmark results. To demonstrate the stability/speed-up in the calculations, performance of the proposed semi-implicit scheme is compared against a classical explicit Velocity-Verlet scheme. The predicted viscometric functions for a non-colloidal suspension with a Newtonian matrix are in excellent agreement with the reference data from the literature. Moreover, the presented semi-implicit algorithm is found to be significantly faster than the classical lubrication dynamics methods with Velocity-Verlet integration schemes.

[en] Simulation of two-phase gas-liquid flows is a challenging problem in terms of predicting the interface position and appropriately coupling the phases. Stability restrictions induced by surface tension of the liquid phase may increase the level of difficulty of the simulation. The Arbitrary Lagrangian-Eulerian (ALE) method along with an interface tracking technique is an approach for a precise prediction of the interface position. The restrictions in the simulation due to surface tension necessitate an implicitly coupled solution algorithm. In this research, the interface kinematic condition equation is discretized in a new coupled form, including an interface displacement variable as well as the interface velocity components. Furthermore, an implicitly discretized formulation of interface curvature is implemented in the interface normal force balance to facilitate the complete coupling of the interface displacement movement to the hydrodynamic behaviour of the flow. Finally, the governing equations of both phases as well as complete set of interface equations are solved simultaneously in a system of linearized algebraic equations. A partially coupled interface tracking (PCIT) method and a fully coupled interface tracking method (FCIT) are developed and evaluated in predictions of a backward-facing step flow, of liquid falling films with and without interaction with a gas phase flow, of an oscillating drop, and of a rising bubble. The results show that in the viscocapillary regime, the FCIT method keeps its stability in a wide range of CFL numbers, whereas the PCIT method is stable only for CFL ≤1. When the surface tension is ignored in a backward-facing step flow, the PCIT method also remains stable for higher CFL number due to the coupled formulation of interface displacement and slope. The present results are in excellent agreement with previous numerical and experimental work results reported in the literature.