In this paper, we propose an efficient numerical method to solve initial-boundary value problems of Caputo fractional subdiffusion equations with a class of source terms weakly singular in time. To do this, we firstly derive fractional Volterra integro-differential equations equivalent to the given problems and establish some regularity results for solutions to them with respect to first or higher order time derivatives. These results are often called a weak regularity at the initial time of the solution. To effectively approximate the solution with the initial weak regularity, employing a temporal graded mesh, we propose a new numerical approach combined with some fractional quadrature formulas and Galerkin finite element method. Also, for our semi-discretization and fully discrete schemes, the stability and convergence analysis are discussed in details. Finally, we present a numerical example on a uniform and several graded temporal meshes to illustrate our main results.
Accurate simulation of convection-dominated pollutant transport remains a significant challenge due to the presence of steep concentration gradients and the numerical instabilities associated with standard finite element discretizations. This study presents an adaptive Streamline Upwind/Petrov–Galerkin (SUPG) finite element framework for solving two-dimensional advection–diffusion equations governing atmospheric pollutant dispersion. The proposed methodology combines SUPG stabilization with a residual-based a posteriori error estimator to automatically guide local mesh refinement, thereby improving solution accuracy while reducing unnecessary computational effort. The adaptive algorithm iteratively identifies regions with large discretization errors and selectively refines the mesh to accurately resolve localized pollutant plumes and sharp concentration fronts. The numerical performance of the proposed framework is evaluated through representative single-source and two-source pollutant transport problems. Convergence studies, computational efficiency analysis, and comparisons with uniform mesh refinement demonstrate that the adaptive approach achieves lower numerical errors and reduced computational cost for an equivalent number of degrees of freedom. The results further show that the adaptive strategy effectively captures the interaction of multiple pollutant plumes while maintaining numerical stability under convection-dominated conditions. These findings confirm that the proposed adaptive SUPG framework provides an accurate, robust, and computationally efficient numerical tool for atmospheric pollutant transport simulations and establishes a reliable foundation for extending the methodology to more complex environmental transport problems involving nonlinear processes, heterogeneous media, and time-dependent emission scenarios.
Linear tetrahedral elements for nearly incompressible finite-strain hyperelasticity are often affected by volumetric locking and exhibit oscillatory stress/pressure fields. We propose a stabilized linear tetrahedral u-θ element combining mixed VMS stabilization with face-based strain reconstruction smoothed FEM (FSRS-FEM). Implemented in Abaqus/Standard through a user element, the method is assessed by patch test, Cook’s membrane bending, torsion, and contact analyses. At v 0 = 0.4995, the Cook’s membrane displacement error was approximately 0.42% relative to a refined C3D8H reference. The proposed element required approximately 1.6–2.2 times the CPU time of Abaqus C3D4H.
Laser Powder Bed Fusion (LPBF) is a pivotal additive manufacturing technology whose final quality is determined by the complex interplay of process parameters such as laser power and scanning speed. Relying solely on physical experimentation to control internal defects (e.g., pores and cracks) is both inefficient and costly, making numerical simulation an essential tool for process optimization. Among available computational methods, Smoothed Particle Hydrodynamics (SPH) is particularly suited for modeling melt pool dynamics due to its inherent ability to handle free surfaces and evolving interfaces. However, both conventional SPH and the SPH method with kernel gradient correction (KGC) exhibit significant pressure oscillations in melt pool simulations, which degrade the accuracy of key predictions such as pool morphology and temperature distribution. To mitigate this, the high-order Targeted Essentially Non-Oscillatory (TENO) reconstruction scheme can be incorporated to improve numerical accuracy. Yet, the TENO-SPH approach may still exhibit stability issues when simulating intense melt pool flows. In this work, we develop a modified TENO-SPH framework by integrating KGC for the simulation of melt pool flow in LPBF. Numerical tests confirm that the proposed KGC-TENO-SPH model accurately captures melt pool flow behavior and pressure distribution, thereby providing a reliable tool for optimizing process parameters and controlling internal defects.
Finite Particle Method (FPM) has emerged as a significant meshless method, which innovatively realizes the simultaneous solution of function value and its derivative values by solving a linear system. Accordingly, the computational accuracy in the whole computational domain is enhanced. However, when facing problems with higher-order partial differential governing equations, the original FPM usually needs to solve higher-order linear equations for each particle, whose computational complexity will be unacceptable for large-scale engineering problems. In this paper, a fast reduced-order method for high-order FPM equations is proposed. First, the original FPM equations are rewritten to reduce the order of the linear system. Second, matrix decoupling is further performed on the reduced-order system and a fast computation scheme is proposed. Finally, the particle consistency and computational efficiency of the proposed method are analyzed, and its superiorities are verified based on several typical engineering cases. Compared with the traditional FPM, the proposed fast reduced-order FPM could significantly improve the computational efficiency while maintaining its second-order accuracy.
Cardiovascular diseases seriously threaten human life and health, among which congenital heart disease is a common congenital defect. Aortic and pulmonary septal defects (APSD), as a rare congenital cardiac vascular malformation, has a low incidence but serious clinical consequences. Surgery is an important treatment for this disease. At present, surgical methods such as direct ligation, patch method, cut patch method, and catheter occlusion method have been formed. At present, the surgical mortality of this disease is still high, and how to reduce the surgical mortality has become an urgent problem to be solved. Based on the CT image data of the patient, before the surgical treatment of APSD, the postoperative vascular structure of the patient with different surgical plans was simulated by using the numerical simulation method, and the postoperative aorta and pulmonary artery parameters under the four surgical plans were quantified. In the first surgical plan, both the aorta and pulmonary arteries were ligated directly. The second surgical plan is to ligate the aorta directly and repair the pulmonary artery with artificial patch. In the third surgical plan, the aorta was repaired with the pulmonary artery wall, and the pulmonary artery was sutured directly. In the last surgical plan, the aorta is repaired with the pulmonary artery wall and the pulmonary artery is repaired with an artificial patch. The results showed that the postoperative results of different surgical plans were significantly different. The maximum percentage change of pulmonary artery volume after the four surgical plans ranged from 59.90% to 106.67%. The simulation results of the patient in this paper showed that the patient achieved the best effect by undergoing the fourth surgical plan, with a volume change of 98.41-110.52% for the aorta and 83.61-107.99% for the pulmonary artery. The simulated postoperative vascular structure method provided in this paper can quantitatively evaluate different surgical plans for APSD before surgery. It provides a quantitative contrast approach for doctors to select the appropriate surgical plan before operation.
A matrix coefficient formula of damping is developed to solve stiff problems. This is a matrix coefficient formula that has L-stability, explicitness and 2nd-order accuracy. It possesses damping to suppress or eliminate the negligible stiff components of the problems under analysis. It corresponds to the 2-step backward differentiation formula in both performance and property. However, it can possess a noniterative and an iterative solution procedure. An iterative procedure of the formula will perform the same as the 2-step backward differentiation formula. In addition, its noniterative procedure can still perform well as a 2-step backward differentiation formula for general nonlinear problems. However, it may perform slightly worse than the 2-step backward differentiation formula for nonstiff highly nonlinear problems. Hence, the most significant improvement of the proposed formula in contrast to the 2-step backward differentiation formula is that it can have a noniterative process for each step and thus it has no convergent problem per step. As a result, it is an efficient formula to solve stiff problems with negligible fast components. In fact, it is numerically affirmed that it can save many computational efforts for such stiff problems of large systems of ODEs.
With the saturation of above-ground spatial resources, megacities are increasingly shifting their development focus to underground space. However, new underground construction inevitably disturbs adjacent structures. To ensure underground engineering safety, isolation piles are often employed in practice to mitigate such disturbances. It is therefore necessary to conduct a full-lifecycle assessment of construction disturbances and the performance enhancement effects of isolation piles. This study proposes a projection-pursuit-based weighting method for multi-indicator performance data derived from the existing tunnel-deep foundation pit system under construction disturbance, and uses it to assess the enhancement effect of isolation piles on system performance. Refined spatiotemporal coupled numerical simulations for scenarios with and without isolation piles are conducted using Plaxis 2D, and a multi-indicator performance dataset is established in accordance with relevant engineering codes. On this basis, a projection-pursuit-based weighting scheme is adopted to integrate multiple performance indicators into a composite performance measure. To solve the associated projection-direction optimization problem, a hybrid genetic algorithm incorporating elitist preservation, adaptive operator control, and real-coded SBX-Gaussian operators is employed. The results show that the installation of isolation piles markedly improves the comprehensive performance of the existing tunnel-deep foundation pit system throughout the construction process. Compared with a basic GA, the adopted hybrid design provides a more stable and effective solution strategy for the present projection-pursuit-based weighting task. The proposed method offers a practical data-driven tool for comparing mitigation scenarios and supporting system-level interpretation of construction disturbance in underground engineering. The present results are obtained for a case-derived problem under a 2D numerical modeling framework, with field validation mainly available for the excavation stage.
A reduced-order isogeometric scaled boundary finite element method (IG-SBFEM) is developed for solving steady-state heat conduction problems by exploiting cyclic symmetry. First, the IG-SBFEM formulation is established in a symmetry-adapted reference coordinate system. It is then demonstrated that the eigenvalue and coefficient matrices in the IG-SBFEM equations possess a block-circulant structure, which can be transformed into block-diagonal form using an orthogonal basis. Consequently, the eigenvalue problem and system equations can be decomposed into a series of independent subproblems with a smaller solution scale, and computational cost can effectively be reduced. Furthermore, a Lagrange-multiplier-based strategy is proposed to effectively accommodate temperature constraints that do not exhibit cyclic symmetry. Several numerical examples are provided to highlight the accuracy, efficiency, and applicability of the proposed method.
This paper presents an application of differential quadrature method (DQM) based on modified cubic B-splines for obtaining the numerical solution of generalized Black–Scholes equation. The generalized Black–Scholes equation is transformed to a time progressing nondegenerate equation by logarithmic transformation and the transformed model is spatially discretized using differential quadrature method. The time integration of the corresponding ordinary differential equation (ODE) system is executed with SSPRK54 and generalized trapezoidal formula (GTF). Several test problems are examined for the validation of theoretical results. The numerical experiments are compared and computational stability of the system is also ensured.
Given the characteristics of large deformation, strong nonlinearity, and free-surface evolution exhibited by bulk coal during coal mining operations — such as scraper conveying and free-fall deposition — conventional mesh-based numerical methods are prone to mesh distortion, difficulties in boundary tracking, and limited predictive accuracy. To address these issues, this study employs the smoothed particle hydrodynamics (SPH) method and establishes a continuum mechanics-based numerical model for bulk coal by incorporating the Drucker–Prager yield criterion and an elastoplastic constitutive relationship. First, direct shear tests are conducted to obtain key parameters, including the friction angle, shear modulus, and cohesion, for coal particles with different particle-size conditions. Experimental rigs and corresponding SPH numerical models are then developed for two representative scenarios, namely scraper conveying and free-fall coal piling, to systematically investigate the effects of coal-flow height, scraper spacing, and particle size on particle transport behavior, stress-field distribution, and free-surface morphology evolution. Comparisons with experimental results indicate that the SPH simulations agree well with observations in terms of motion patterns, free-surface profiles, and overall macroscopic response trends, demonstrating that the proposed approach can effectively capture the flow mechanisms of bulk coal particles under boundary constraints and yielding effects. The findings provide a theoretical basis and practical reference for dynamic stability assessment and structural parameter optimization of underground bulk-material conveying processes.
To investigate effects of vertical earthquake on seismic performance of multi-story underground subway stations, a series of shaking table tests and pushover analyses conducted by ABAQUS were carried out. Dynamic responses of central column induced by vertical earthquake were focused, especially the variation of axial force, story drift, and the relationship between responses and bearing capacity. Results from shaking table tests show that vertical ground motion increases central column axial force by up to 26.4% and raises the dynamic axial force proportion borne by central columns to 23.5%. And vertical excitation has negligible effect on column story drift, but large story heights result in drift exceeding 1/200 at 0.333 g horizontal acceleration. Results from pushover analyses show that under high axial compression ratio, column drift capacity decreased up to nearly 50%.
To support product design and performance prediction, this study proposes CSDA–IVF, a material-update framework applicable to finite-strain inelastic analysis. CSDA–IVF applies the complex-step derivative approximation to tensor differentiation within an incremental variational formulation, requiring only the Helmholtz free energy and a dissipation potential to compute internal variables, stresses, and consistent tangent moduli, without analytical tensor derivatives. First derivatives attain machine precision, while generalized complex-step second derivatives achieve fourth-order truncation accuracy. An Abaqus/UMAT implementation for finite-strain viscoelasticity in uniaxial tension and Cook’s bending demonstrates accuracy, Newton convergence, and stability, indicating applicability to advanced material modeling.
Existing studies on the mechanism of isolation piles in controlling the displacement of adjacent tunnels have predominantly concentrated on the ultra-deep foundation pit excavation stage, with an inadequate systematic investigation into the entire construction process. A 29m-level ultra-deep foundation pit project adjacent to a tunnel in Shanghai is taken as a case study in this paper. A coupled analysis model of foundation pit-isolation pile-tunnel is established by PLAXIS 2D finite element software, and for the first time systematically explores the displacement control mechanism and influence law of key parameters of isolation piles in the dual stages of excavation and building surcharge of ultra-deep foundation pits. Research findings indicate that during the whole foundation pit construction process, the tunnel presents a deformation pattern characterized by horizontal stretching and vertical compression. Based on the distribution characteristics of pile shaft bending moments, isolation piles can be divided into upper and lower segments with the inflection points serving as the boundary, and the inflection point has the characteristic of surcharge-induced upward shift under the condition of ultra-deep foundation pits, and the distinct action mechanisms of the upper and lower segments of isolation piles in the excavation and surcharge stages of ultra-deep foundation pits are revealed The optimal design criteria of isolation piles for ultra-deep foundation pits are proposed, with the pile length being 2.0 times the foundation pit excavation depth. The embedded isolation piles have a strengthening effect in the surcharge stage on the control of the horizontal displacement of tunnels in ultra-deep foundation pits. A reasonable pile spacing can form a continuous soil arching effect, with the optimal value being 1.2 times the pile diameter. The research results fill the research gap of the control mechanism of isolation piles in the building surcharge stage of ultra-deep foundation pits, and provide exclusive theoretical support for the design of isolation piles in tunnel projects adjacent to ultra-deep foundation pits.
This paper develops a Taylor collocation method (TCM) for solving systems of twodimensional Volterra integral equations with proportional delays. The proposed method constructs explicit formulas for the approximate solution directly, thereby avoiding transformation into large algebraic systems. A convergence analysis is presented to establish the reliability of the scheme. Several numerical examples are provided to illustrate the efficiency and accuracy of the algorithm. The results confirm that Taylor-based collocation techniques offer a powerful and practical tool for tackling multi-dimensional integral equations with delay effects.
A novel method to facilitate direct application of standard one-dimensional finite element for nonprismatic structures is presented here. The method relies on so-called internal elements behaving as substructures generated along particular meshed lines. Strain energy ratio due to assembly of the internal elements is then used to modify the stiffness of the standard element. Numerical examples show the capability of the method to improve the prediction of displacement field and natural frequency provided by the standard element in case of modeling the nonprismatic structures.
This work focuses on evaluating the hydrodynamic pressures exerted on structures such as dams, ports, and offshore structures. The near-infinite liquid reservoir is discretized by coupling finite elements with periodic infinite elements. A reformulation of infinite elements was performed to overcome their dependence on excitation frequencies and reservoir liquid compressibility. For this purpose, infinite elements are introduced into a two-dimensional finite element for a fluid calculation program. In this study, both the containment structure and reservoir bottom are assumed to be infinitely rigid. The liquid in the reservoir is described by the pressure variable. Considered stresses are horizontal components of harmonic accelerations exerted on the reservoir by the monolithic structure. The performance and efficiency of the proposed approach were tested under the influences of several parameters, including liquid compressibility, excitation frequency, truncation boundary position, reservoir bottom inclinations, and fluid-structure interface inclinations. The results of this work are presented and compared with analytical solutions available in the literature and with existing numerical models.
The survivability of military vehicles depends significantly on the ballistic resistance of fuel tanks, which remain one of the most vulnerable components under high-velocity impact. Hybrid fiber-reinforced composites such as Kevlar, Carbon-fiber-reinforced polymer (CFRP), Glass-fiber reinforced polymer (GFRP), and kenaf offer an alternative to traditional metallic fuel tanks due to their superior strength-to-weight ratio and energy dissipation characteristics. This study aims to evaluate the ballistic response of stiffened cylindrical composite fuel tanks fabricated with various hybrid laminate configurations. Explicit Dynamic analysis was performed to simulate the impact of ballast on cylindrical composite fuel tanks made of Kevlar/Kenaf/GFRP, Kevlar/Kenaf/CFRP, and Kevlar/GFRP/CFRP laminate configurations. Each of these laminates was examined with three different orientations, namely, 0(degrees)/45(degrees)/90(degrees), 45(degrees)/90(degrees)/45(degrees), and 90(degrees)/45(degrees)/90(degrees). The results indicated that the Kevlar/GFRP/CFRP laminate with 0(degrees)/45(degrees)/90(degrees) orientation exhibited the most balanced ballistic performance, showing up to 12.9% reduction in deformation and over 42% reduction in residual velocity with enhanced energy absorption ( similar to 70J). Thus, integrating performance-informed stiffeners with hybrid composite laminates provides an effective strategy for developing lightweight, high-performance fuel tanks for defense applications.
