• Abstract

In this paper, we utilize Physics-Informed Neural Networks (PINNs) without any labeled data to solve the fourth-order partial differential equation governing the bending of thin plates. We meticulously formulate and apply our framework to the bending problem of elastic thin plates. Our findings indicate that defining each solution variable through an independent network, meaning without shared network parameters, yields superior performance compared to a single-network model with multiple outputs.

During the training process, we discovered that more accurate results were achieved by adjusting the network architecture to strictly satisfy the boundary conditions, rather than incorporating them as part of the loss function. Remarkably, the PINNs are capable of obtaining relatively good results without the use of any labeled data during training, irrespective of whether soft or hard constraints are applied. In the end, the neural network's predictions for plate deflection, stress, strain, bending, and shear force all have errors less than 5\% compared to the analytical solutions. In the algorithm implementation phase, we leverage the Python library DeepXDE, which facilitates the training of PINNs by providing an efficient and expedited process.

• General Summary

• Abstract

We have proposed a novel framework of troubled-cell indicator (TCI) using K-means clustering on uniform meshes in [SIAM J. Sci. Comput., 43 (2021), pp. A3009--A3031]. Based on this framework, we develop TCIs on two typical types of non-uniform meshes, i.e., triangular meshes and $h$-adaptive rectangular meshes. The TCIs are composed of two parts: one is to create the stencils for troubled-cell indication that are composed of computational cells in a local region, and the other is to detect the troubled cells stencil by stencil. Compared with the uniform meshes, the creation of stencils for non-uniform meshes is no longer trivial. We develop new stencil creation approaches specifically tailored to triangular meshes and $h$-adaptive rectangular meshes, respectively. Another contribution of this work is a new classification criterion in the troubled-cell indication part which is used to determine if there exist troubled cells in a stencil. It contains only one parameter, which leads to a much easier implementation of the TCIs. Numerical results show that the TCIs not only can capture the shocks precisely and produce nonoscillatory solutions, but also work well with multiple indication variables and in a TCI-based $h$-adaptive scheme. These results demonstrate the accuracy and robustness of the TCIs on non-uniform meshes.

• General Summary

• Abstract

A nonconforming $P_2$ finite element  is constructed by enriching the conforming $P_2$ finite element space with seven $P_2$ nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming $P_2$ finite element, combined with the discontinuous $P_1$ finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently such a mixed finite element method produces optimal-order convergent solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.

• General Summary

• Abstract

The local characteristic fields of compressible Euler equations can be decomposed into linearly degenerate and genuinely nonlinear, respectively. The former is associated with a contact discontinuity (the slip lines for multi-dimensional cases) and the latter is associated with shocks and rarefaction waves (Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009). The weighted essentially non-oscillatory (WENO) scheme can capture shock waves well but its resolution for short waves still needs to be improved, while one kind of new scheme constructed by the boundary variation diminishing (BVD) principle with the THINC (Tangent of Hyperbola for INterface Capturing) reconstruction can resolve the contact discontinuities with higher resolution but it may generate spurious numerical phenomena in some special cases. Combining the physical characteristics and the advantages of WENO and BVD schemes, this paper proposes a hybrid reconstruction method, in which the classical WENO scheme is used to discretize the genuinely nonlinear fields and the higher resolution BVD scheme is used to discretize the linearly degenerate field(s). Numerical experiments show that the hybrid method can preserve high accuracy near both contact discontinuities and short waves as the used higher resolution scheme while effectively avoiding spurious numerical phenomena (such as oscillations and overshoots). Compared with the BVD scheme, the hybrid method also improves computational efficiency, since it uses a more efficient WENO scheme in genuinely nonlinear fields.

• General Summary

• Abstract

In this paper, a first-order penalty finite element method for the 2D/3D unsteady incompressible thermomicropolar fluid (UITF) equations is considered, which combines the advantage of the penalty method, first-order backward Euler scheme, and implicit or explicit iteration for the nonlinear terms to get a decoupled temporal evolution procedure, which only needs to solve a series of elliptic subproblems. Theoretically, the stability analysis and optimal error estimates of the temporal semi-discrete method are deduced. Furthermore, the classical MINI element pairs are adopted in concrete spatial discrete, and the feasibility of the method is  verified  by numerical experiments.

• General Summary