Neural-network methods are widely used for partial differential equations (PDEs). Within this framework, gradient-enhanced physics-informed neural networks (gPINNs) incorporate first-order residual derivatives, while fixed directional weighting may unbalance spatial and temporal residual-gradient contributions. Accordingly, an adaptive gradient-balanced gPINN (AGB-gPINN) is proposed using warm-up calibration, dimensionless relative states, and a bounded fixed-sum directional budget without trainable weighting parameters. For Burgers, median relative L 2 error and PDE-MSE over nine seeds decrease by 41.9% and 57.0% relative to fixed-weight gPINN; Allen–Cahn shows smaller gains and lower variability. These results support more balanced and stable gPINN training.
This paper presents a discrete fracture-matrix (DFM) model to simulate coupled fluid flow and geomechanics in fractured rock masses. Aiming at computational efficiency and accuracy, the fractures are explicitly represented by lower-dimensional zero thickness elements. The proposed numerical model is comprised of a matrix-fracture flow model, a matrix deformation model and a stress-strain constitutive model for the fracture deformations. Darcy's law is used to describe the fluid flow in both rock matrix and fractures. Variations of fracture aperture and permeability induced by fracture deformations are considered. A finite element code (COMSOL) is employed to numerically solve the coupled governing equations. To examine the efficiency and accuracy of the proposed model, three simulation cases are conducted in this work. The results obtained are validated against the corresponding analytical solution and published numerical results. The numerical model is then applied to study the coupled hydro-mechanical behavior of fractured reservoir bank slope. The simulation results show that the fractures in the slope result in greater fluid drainage and contribute significantly to deformations.
This study investigates the seismic behavior of T-shaped, L-shaped, and C-shaped buildings with rectangular overhead water tanks, considering the influence of varying water levels (25%, 50%, and 75%) on structural integrity during seismic events. The analysis highlights that T-shaped and L-shaped buildings experience higher seismic forces, including increased water pressure, base shear, and sloshing amplitudes, making them more vulnerable to seismic activity. In contrast, the C-shaped building demonstrates superior seismic stability, exhibiting lower water pressure, base shear, and minimal sloshing, resulting in reduced story drift and dislocation. STAAD study reveals that the C-shaped structure experiences the lowest displacement (22.884 inches), axial forces (46.10[Formula: see text]kN), and shear forces (444.062[Formula: see text]kN in Fy and 453.231[Formula: see text]kN in Fz). Additionally, the Response Surface Methodology (RSM) optimization identifies the C-shaped building as the most desirable design with a desirability value of 0.939, outperforming the T-shaped (0.926) and L-shaped (0.865) buildings. The study emphasizes that incorporating overhead water tanks significantly impacts the seismic performance of structures, with the C-shaped building being the optimal design for minimizing seismic risks.
Level set method has long been renowned for its ease of use and it can be easily extended to higher dimensions. Level set method is deemed as a sharp interface method and uses the signed distance function as the kernel. However, level set method faces a couple of challenges, including the discretization of the curvature-based interfacial tension and the regularization of the Heaviside function, which is used to indicate phases. In this paper, an improved level set method is presented. The key improvements lie in the regularization of the Heaviside function and the use of the interfacial force model. Given the complexity of the curvature-based interfacial tension model, the potential-form model, which is often employed in phase field method and is much simpler in form, is utilized. The improved level set model is then coupled with the Navier–Stokes equation for two-phase flow simulations. The model was tested in two cases, one about droplet impact and the other about the spurious flow in a static droplet. The performance of the two interfacial force models was compared.
Shape optimization problems with Robin boundary conditions have attracted considerable interest and remain a challenging task from a numerical point of view. In this paper, we focus on establishing a mathematical analysis of this class of inverse problems and on constructing both deterministic and heuristic methods for their numerical approximation. As a first step, we derive a constrained optimization problem from the considered ill-posed problem, then we show the optimal solution existence and uniqueness. Using the Lagrangian functional, we compute the first-order optimality conditions as well as the shape gradient and Hessian. For all our proposed numerical methods, we employ the finite element method to discretize the equations. The deterministic minimization algorithms considered are the conjugate gradient and the quasi-Newton techniques, for which we establish the convergence analysis. As heuristic methods, we adopt the differential evolution and genetic algorithms. In addition, we develop a hybrid minimization approach, a quasi-Newton method guided by differential evolution. At the end, we conduct several numerical tests to demonstrate the validity of our proposed methods and provide a comparative assessment of their performance.
