We present an algorithm for the numerical solution of systems of fully nonlinear PDEs using stochastic coded branching trees. This approach covers functional nonlinearities involving gradient terms of arbitrary orders, and it requires only a boundary condition over space at a given terminal time $T$ instead of Dirichlet or Neumann boundary conditions at all times as in standard solvers. Its implementation relies on Monte Carlo estimation, and uses neural networks that perform a meshfree functional estimation on a space-time domain. The algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.

A novel neural network method is developed for solving systems of conservation laws whose solutions may contain abrupt changes of state, including shock waves and contact discontinuities. In conventional approaches, a low-cost solution patch is usually used as the input to a neural network for predicting the high-fidelity solution patch. With that technique, however, there is no way to distinguish a smeared discontinuity from a smooth solution with large gradient in the input, and the two almost identical inputs correspond to two fundamentally different high-fidelity solution patches in training and predicting. To circumvent this difficulty, we use local patches of two low-cost numerical solutions of the conservation laws in a converging sequence as the input to a neural network. The neural network then makes a correct prediction by identifying whether the solution contains discontinuities or just smooth variations with large gradients, because the former becomes increasingly steep in a converging sequence in the input, and the latter does not. The inputs can be computed from low-cost numerical schemes with coarse resolution, in a local domain of dependence of a space-time location where the prediction is to be made. Despite smeared input solutions, the output provides sharp approximations of solutions containing shock waves and contact discontinuities. The method works effectively not only for regions with discontinuities, but also for smooth regions of the solution. It is efficient to implement, once trained, and has broader applications for different types of differential equations.

In this work, a genuinely two-dimensional HLL-type approximate Riemann solver is proposed for hypo-elastic plastic flow. To consider the effects of wave interaction from both the $x$- and $y$-directions, a corresponding 2D elastic-plastic approximate solver is constructed with elastic-plastic transition embedded. The resultant numerical flux combines one-dimensional numerical flux in the central region of the cell edge and two-dimensional flux in the cell vertex region. The stress is updated separately by using the velocity obtained with the above approximate Riemann solver. Several numerical tests, including genuinely two-dimensional examples, are presented to test the performances of the proposed method. The numerical results demonstrate the credibility of the present 2D approximate Riemann solver.

We study efficient simulation of steady state for multi-dimensional rarefied gas flow, which is modeled by the Boltzmann equation with BGK-type collision term. A nonlinear multigrid solver is proposed to resolve the efficiency issue by the following approaches. The unified framework of numerical regularized moment method is first adopted to derive the high-quality discretization of the underlying problem. A fast sweeping iteration is introduced to solve the derived discrete problem more efficiently than the usual time-integration scheme on a single level grid. Taking it as the smoother, the nonlinear multigrid solver is then established to significantly improve the convergence rate. The OpenMP-based parallelization is applied in the implementation to further accelerate the computation. Numerical experiments for two lid-driven cavity flows and a bottom-heated cavity flow are carried out to investigate the performance of the resulting nonlinear multigrid solver. All results show the wonderful efficiency and robustness of the solver for both first- and second-order spatial discretization.

Constitutive modeling of heterogeneous hyperelastic materials is still a challenge due to their complex and variable microstructures. We propose a multiscale data-driven approach with a hierarchical learning strategy for the discovery of a generic physics-constrained anisotropic constitutive model for the heterogeneous hyperelastic materials. Based on the sparse multiscale experimental data, the constitutive artificial neural networks for hyperelastic component phases containing composite interfaces are established by the particle swarm optimization algorithm. A microscopic finite element coupled constitutive artificial neural networks solver is introduced to obtain the homogenized stress-stretch relation of heterogeneous materials with different microstructures. And a dense stress-stretch relation dataset is generated by training a neural network through the FE results. Further, a generic invariant representation of strain energy function (SEF) is proposed with a parameter set being implicitly expressed by artificial neural networks (SANN), which describes the hyperelastic properties of heterogeneous materials with different microstructures. A convexity constraint is imposed on the SEF to ensure that the multiscale constitutive model is physically relevant, and the $ℓ_1$ regularization combined with thresholding is introduced to the loss function of SANN to improve the interpretability of this model. Finally, the multiscale model is hierarchically trained, cross-validated and tested using the experimental data of cord-rubber composite materials with different microstructures. The proposed multiscale model provides a convenient and general methodology for constitutive modeling of heterogeneous hyperelastic materials.

In this paper, we present a hybrid form of weighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws by applying linear and nonlinear methods for smooth and discontinuous zones individually. To fulfill this algorithm, it is inseparable from the recognition ability of the discontinuity detector adopted. In specific, a troubled-cell indicator is utilized to recognize unsmooth areas such as shock waves and contact discontinuities, while avoiding misjudgments of smooth structures. Some classical detectors are classified into three basic types: derivative combination, smoothness indicators and characteristic decomposition. Meanwhile, a new improved detector is proposed for comparison. Then they are analyzed through identifying a series of waveforms firstly. After that, hybrid schemes using such indicators, as well as different detection variables, are examined with Euler equations, so as to investigate their ability to distinguish practical discontinuities on various levels. Simulation results demonstrate that the proposed algorithm has similar performances to pure WCNS, while it generally saves 50 percent of CPU time for 1D cases and about 40 percent for 2D Euler equations. Current research is in the hope of providing some reference and establishing some standards for judging existing discontinuity detectors and developing novel ones.

The DDFV (Discrete Duality Finite Volume) method is a finite volume scheme mainly dedicated to diffusion problems, with some outstanding properties. This scheme has been found to be one of the most accurate finite volume methods for diffusion problems. In the present paper, we propose a new monotonic extension of DDFV, which can handle discontinuous tensorial diffusion coefficient. Moreover, we compare its performance to a diamond type method with an original interpolation method relying on polynomial reconstructions. Monotonicity is achieved by adapting the method of Gao et al [A finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes] to our schemes. Such a technique does not require the positiveness of the secondary unknowns. We show that the two new methods are second-order accurate and are indeed monotonic on some challenging benchmarks as a Fokker-Planck problem.

In this paper, we discuss the conforming finite element method for a modified interior transmission eigenvalues problem. We present a complete theoretical analysis for the method, including the a priori and a posteriori error estimates. The theoretical analysis is conducted under the assumption of low regularity on the solution. We prove the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms, and we also analyze the reliability of estimators for eigenvalues. Finally, we report numerical experiments to show that our posteriori error estimator is effective and the approximations can reach the optimal convergence order. The numerical results also indicate that the conforming finite element eigenvalues approximate the exact ones from below, and there exists a monotonic relationship between the conforming finite element eigenvalues and the refractive index through numerical experiments.

In this paper, the electromagnetic scattering from overfilled cavities with inhomogeneous anisotropic media is investigated. To solve the scattering problem, a Petrov-Galerkin finite element interface method on non-body-fitted grids is presented. We reduce the infinite domain of scattering to a bounded domain problem by introducing a transparent boundary condition. The level set function is used to capture complex boundary and interface geometry that is not aligned with the mesh. Non-body-fitted grids allow us to save computational costs during mesh generation and significantly reduce the amount of computer memory required. The solution is built by connecting two linear polynomials across the interfaces to satisfy the jump conditions. The proposed method can handle matrix coefficients produced by permittivity and permeability tensors of anisotropic media. The final linear system is sparse, making it more suitable for most iterative methods. Numerical experiments show that the proposed method has good convergence and realizability. Meanwhile, we discover that the absorbing properties of anisotropic media clearly and positively influence the reduction of radar cross section. It has also been demonstrated that the method can achieve second-order accuracy.