International Journal for Numerical Methods in Engineering最新文献

This article focuses mainly on the dimensionality reduction for the two-grid Crank-Nicolson (CN) mixed finite element (MFE) (TGCNMFE) model for the temporal fractional-order and spatial fourth-order derivatives sine-Gordon (TFOSFDOSG) equation with actual applied background. To do so, we first split the TFOSFDOSG equation into two nonlinear equations with second-order derivatives by introducing an auxiliary function, and construct the TGCNMFE model through using the CN technique discretizing the time derivatives, and the TGCNMFE method discretizing the space variables. The TGCNMFE model is made up of a nonlinear system on coarser meshes together with a linear system on adequately fine meshes, and can be easy to calculate. Thereafter, most importantly, we create a novel TGCNMFE reduced-dimension (TGCNMFERD) model by means of proper orthogonal decomposition, lowering the dimension of the unknown coefficient vectors of the TGCNMFE solutions, keeping the TGMFE basis functions unchanged, namely keeping the precision unvaried. The main contribution of this article is to theoretically establish the existence, stability, and error estimates of the TGCNMFERD solutions, and in simulation, to confirm the rightness for the obtained theoretical results and the benefit of the TGCNMFERD model through two numerical examples.

Multiscale structural optimization aims to improve structural performance through the simultaneous design of structural layout and local material properties. Through this expanded design space, the complex interactions between microscale and macroscale physics offer new opportunities to improve structural performance at the cost of modeling and computational complexity. In this work, a second-order homogenization model is incorporated into multiscale structural optimization to capture length-scale effects induced by spatially varying microarchitecture. A comprehensive study of the influence of length-scale effects on structural optimization is performed. The approach uses a mixed finite element scheme to analyze the higher-order continuum, recovering the second-order gradients and the higher-order stresses induced by the second-order material model. A second-order homogenization scheme is defined through the method of multiscale virtual power (MMVP). Based on this model, several second-order microstructural material design models are compared, including a solid isotropic material with penalization (SIMP) model, polynomial interpolation, and a neural network (NN) surrogate model. The NN model, trained via finite element analysis of the second-order homogenization problem, is applied in several multiscale structural optimization examples using linear elastic materials. The performance of structures optimized for minimum compliance is compared for several first-order and second-order material models to inform the selection of second-order design models and demonstrate the practical effects of length scale in an optimization context.

We propose a new reduced-order modeling (ROM) strategy for tackling parametrized Darcy-flow systems in which the constraint is given by mass conservation. Our approach employs classical neural-network architectures and supervised learning, but it is constructed in such a way that the resulting ROM is guaranteed to satisfy the linear constraints exactly. The procedure is based on a splitting of the solution into a particular solution satisfying the constraint and a homogenous solution. The homogeneous solution is approximated by mapping a suitable potential function, generated by a neural-network model, onto the kernel of the constraint operator. For the particular solution, instead, we propose an efficient spanning-tree algorithm. Starting from this paradigm, we present three approaches that follow this methodology, obtained by exploring different choices of the potential spaces: derived either via proper orthogonal decomposition (POD) or using the properties of a differential complex. To demonstrate the effectiveness of the proposed strategies and to emphasize their advantages over neural-network regression approaches, we present a series of numerical experiments, ranging from mixed-dimensional problems to nonlinear systems.

This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections, augment linear dimensionality reduction with a nonlinear correction term in the reconstruction map to overcome approximation accuracy limitations of purely linear approaches. While feature map-based approaches typically learn a least squares optimal polynomial correction term, we generalize this approach by learning an optimal nonlinear correction from a user-defined reproducing kernel Hilbert space. Our approach allows one to impose arbitrary nonlinear structure on the correction term, including polynomial structure, and includes feature map and radial basis function-based corrections as special cases. Furthermore, our method has relatively low training cost and has monotonically decreasing error as the latent space dimension increases. We compare our approach to proper orthogonal decomposition and several recent QM approaches on data from several example problems.

The explicit moving particle simulation (EMPS) method, due to its pressure calculation through the equation of state, has high computational efficiency and broad prospects for engineering applications. However, the conventional EMPS method has issues of numerical instability such as high-frequency oscillations and discontinuous distributions of pressure. In this paper, to improve the accuracy and stability of numerical simulations, an enhanced δ_MPS-EMPS method was proposed, in which a new enhanced equation of state and a diffusion term of particle number density were developed. The new enhanced equation of state includes the terms of the velocity divergence and the particle number density. It describes the relationship between pressure and fluid density through the changes in the velocity field and the particle number density field, resulting in a smooth pressure. It is worth noting that the artificial bulk modulus K in this equation is accurately calibrated using the specified pressure field calculated by the MPS method at the first time step. Additionally, the particle number density is corrected by introducing a diffusion term into the mass conservation equation. This approach can reduce high-frequency pressure oscillations by filtering out non-physical density fluctuations. The proposed method was evaluated and validated through simulations of four benchmark cases, including a hydrostatic pressure tank, the deformation of an elliptical droplet, the Taylor-Green vortex flow, the 2D water dam-break flow, and the evolution of a square patch of fluid. The simulation results proved the effectiveness and robustness of the enhanced δ_MPS-EMPS method.