Dynamic fracture in flexoelectric solids under combined electro-thermo-mechanical loading has significant challenges for accurate and efficient numerical modeling. Existing phase-field models have not systematically investigated dynamic fracture in flexoelectric solids under coupled electro-thermo-mechanical loading. In this work, an adaptive isogeometric phase-field model is presented for simulating fracture processes in flexoelectric solids. For spatial discretization, polynomial splines over hierarchical T-meshes (PHT-splines) are employed to represent the unknown field variables, viz., displacement, temperature, electric potential, and phase-field variable. The use of PHT-splines ensures that the resulting approximation has C1 continuity as required by the fourth-order governing differential equations in space, and for stable temporal resolution, we employ the Hilber-Hughes-Taylor (HHT) time integration scheme. Further computational efficiency is improved by local adaptive refinement in the vicinity of the damage process zone, which is driven by the user-defined threshold value on the phase field variable. Five numerical examples of varying complexities are solved to demonstrate the efficacy of the proposed framework. From the systematic numerical study, it can be inferred that the proposed methodology yields accurate results at a fraction of the computational cost. The numerical framework offers a robust and efficient numerical tool for investigating dynamic fracture in flexoelectric material subjected to coupled loading conditions.

Efficient simulation of brittle fracture in heterogeneous materials with uniformly distributed small-scale discontinuities is a significant challenge due to the associated high computational cost and evolving complex fracture patterns. To address this issue, this study integrates a multiscale multi-patch isogeometric analysis with the phase-field method to simulate brittle fracture. In the proposed framework, the computational domain is partitioned into coarse-scale and fine-scale regions. The coarse-scale region is modeled using multiscale multi-patch isogeometric analysis, in which heterogeneous basis functions are constructed numerically to capture the influence of underlying small-scale heterogeneities. The fine-scale region is dynamically constructed within crack-propagation zone using a multi-patch modeling strategy, while the coarse-scale region is maintained in the remaining domain to reduce computational cost. Within the fine-scale region, the displacement and phase-field variables are represented using polynomial splines over hierarchical T-meshes (PHT-splines), and Nitsche’s method is adopted to enforce continuity of field variables across adjacent patches. Furthermore, numerical heterogeneous basis functions are used to ensure consistent coupling between coarse- and fine-scale regions. Numerical studies demonstrate that the proposed method achieves accurate results with low computational cost, making it a reliable and efficient framework for modeling brittle fracture in heterogeneous materials.

This study presents a phase field model for simulating ductile crack growth under small-strain conditions. In this work, the key contribution lies in the introduction of an exponential toughness degradation function that relates fracture toughness to the accumulated plastic strain, while the crack-driving force remains governed by the elastic energy. Unlike many existing ductile phase-field formulations, this approach preserves the linearity of the AT2 phase-field subproblem, thereby simplifying numerical implementation. An isogeometric analysis framework is adopted for describing the unknown fields and the geometries. To accommodate complex geometries, the computational domain is constructed using a multi-patch modeling strategy. Continuity of both displacement and phase-field variables across the patch interfaces is weakly enforced through Nitsche’s method. Locally refined non-uniform rational B-splines (LR NURBS) are employed to maintain an exact geometric representation and enable targeted refinement near damage zones, effectively reducing the number of active degrees of freedom and computational cost. The accuracy and efficiency of the proposed multi-patch isogeometric analysis-based ductile phase-field framework are compared against the numerical and experimental results reported in the literature.

By integrating the phase-field method with additive manufacturing filter, a topology optimization framework is proposed for self-supporting structures in both two and three dimensions. With phase field as design variable and minimizing the structural compliance as a primary objective function, the design of self-supporting structures is solved as an optimization problem. The evolution of phase field variable is governed by Allen-Cahn equation, which effectively redefines the interface between the solid and void phases, while the volume constraint is en forced through Lagrange multiplier. In addition, an additive manufacturing filter is incorporated to eliminate unprintable geometries from the optimization space, thereby ensuring self-supportability. Several examples are presented to demonstrate the robustness of the proposed framework.

Silver doping in PbTe 0.88 Ag 0.12 improves stability, reducing oxidation compared to pure PbTe, while maintaining high thermoelectric performance.

Electroelastic shells are widely used in soft actuators, sensors, and energy harvesters owing to their large electrically induced deformations. However, the accurate simulation of their complex nonlinear multiphysics coupling, including bifurcation and post-buckling responses, remains challenging. This work presents an isogeometric Kirchhoff-Love shell formulation for the nonlinear analysis of electroelastic thin structures undergoing finite deformations. The formulation incorporates geometrically nonlinear kinematics, Maxwell-stress-induced electromechanical coupling, material incompressibility, and initial prestretch. Catmull–Clark subdivision surfaces are employed to ensure the C1 continuity required by Kirchhoff–Love shell theory. Consistent tangent operators are derived analytically, and a static condensation procedure is introduced to satisfy the plane-stress constraint. To trace bifurcation and post-buckling equilibrium paths, a staged Newton–Raphson algorithm with arc-length continuation and eigenmode perturbation is adopted. Numerical examples involving spherical membranes, prestretched circular plates, and toroidal membranes demonstrate the capability of the proposed framework to accurately capture large deformations, symmetry-breaking instabilities, and post-buckling responses under coupled electromechanical loading.

Conventional moving-mesh methods for free surface seepage analysis often suffer from mesh distortion and numerical instability. On the other front, fixed-mesh approaches tend to compromise either on accuracy or computational efficiency, particularly in three-dimensional problems. To circumvent these drawbacks, the present work simulates the three-dimensional steady-state unconfined seepage problems within the framework of adaptive isogeometric analysis, with the goal of attaining efficient and stable numerical results. Truncated hierarchical NURBS (TH-NURBS) is introduced for local refinement, and together with a permeability adjustment scheme, this framework enables precise free surface localization without the need for constant remeshing. The modeling and simulation of complex three-dimensional geometries are accomplished by multiple patches in conjunction with Nitsche's method. The proposed adaptive strategy relies on an a posteriori error estimator to identify and refine critical regions during iteration on the fly, leading to significant gains in computational efficiency without sacrificing numerical accuracy. The robustness of the framework is verified across varied seepage conditions, and numerical results show excellent agreement with reference solutions. It is shown that the proposed framework establishes a basis for future multi-physics coupling and practical engineering applications in seepage analysis.

Lead telluride (PbTe) is a well-known thermoelectric material with a figure of merit (zT) typically exceeding 1 at intermediate temperatures. However, due to its intermetallic properties, the oxidation of PbTe introduces oxygen-related defects. These defects not only destabilize the crystal but also reshape the delicate balance between electrical and thermal conductivity. The present work investigates the influence of Ag incorporation on its thermoelectric transport and oxidation behavior. Unconventional p-type silver-doped PbTe (PbTe1-xAgx, where x = 0, 0.04, 0.08, 0.12) was synthesized. Structural analysis confirmed the successful incorporation of Ag into the lattice. Density functional theory (DFT) calculations revealed that Ag doping induces a narrow bandgap semiconducting to semi-metallic transition, which potentially enhanced the electrical conductivity. Concurrently, phonon calculations indicated intensified phonon scattering, which contributes to a suppressed lattice thermal conductivity. This synergistic effect of electronic band engineering and phonon suppression proved as a highly effective strategy for enhancing thermoelectric performance. As a result, the thermoelectric figure of merit, zT, improved from 1.06 for pristine PbTe to 1.37 at 753 K for the PbTe0.88Ag0.12 composition, representing a 29.2% enhancement. Beyond improving the performance, Ag doping also significantly enhanced the material's operational stability. Comprehensive thermal and structural analyses, supported by time- and temperature-dependent oxidation studies, demonstrated that Ag doping leads to a markedly reduced oxidation rate. This suppression of oxidation effectively stabilizes the material against degradation at both room temperature and intermediate operating temperatures.

To improve bridge deck icing warning, this study investigates two related tasks, namely bridge deck temperature prediction and bridge deck icing-state prediction, within a unified surrogate-modeling framework. Three representative surrogate models, including deep neural network (DNN), Gaussian process (GP), and polynomial chaos expansion (PCE), are established and compared in COMSOL under consistent workflows, datasets, and evaluation metrics. For bridge deck temperature prediction, real monitoring data collected from a bridge in Shandong Province are used. For bridge deck icing-state prediction, both a MATLAB-generated synthetic dataset and a real dataset derived from monitored samples through physical labeling and class balancing are considered. The results show that all three surrogate models can effectively predict bridge deck temperature, among which GP achieves the best overall regression performance, while DNN performs comparably well. For icing-state prediction, DNN exhibits the best overall classification performance on both the synthetic and real datasets. GP and PCE show relatively stronger ability to identify icy samples on the real dataset, but this advantage is accompanied by an increased probability of false alarms. The results indicate that the suitability of surrogate models is task-dependent: GP is more advantageous for bridge deck temperature regression, whereas DNN provides a better balance of accuracy, robustness, and generalization for bridge deck icing-state prediction. This study provides a useful reference for surrogate-model selection in bridge deck icing warning.

This work proposes an adaptive thermomechanical phase-field framework for simulating three-dimensional (3D) thermally-induced fracture in brittle materials. Adaptive mesh refinement is performed via the COMSOL application programming interface (API), using a user-defined phase-field threshold as the refinement indicator. COMSOL’s built-in adaptive meshing produces a locally refined mesh without hanging nodes. A staggered iteration scheme is adopted to ensure computational robustness of the fracture phase-field framework. To further improve computational efficiency in 3D fracture problems, the restarted generalized minimum residual (GMRES) method, combined with smoothed aggregation algebraic multigrid (SAAMG), is adopted as an iterative linear system solver. Through the analysis of three numerical examples, the accuracy and efficiency of the present framework can be validated. The open-source model enables users without programming experience to simulate fractures in a multiphysics environment.

This paper develops an adaptive isogeometric analysis framework to simulate planar thermoelastic problems in isotropic and orthotropic media. Truncated hierarchical non-uniform rational B-spline (TH-NURBS) is employed for exact geometric description and local mesh refinement with its superior properties. The framework integrates multi-patch technology for complex geometric description, and combines Nitsche’s method with the closest projection algorithm to couple non-matching patches. A stress recovery-based error estimator is adopted to guide the local refinement. The proposed framework exhibits high accuracy, as evidenced by four numerical examples. We also compare the error convergence rate of the proposed framework with that of the global refinement method, confirming its advantage in computational efficiency.

For seepage problems with free surfaces, traditional approaches use either moving mesh strategies or fixed mesh strategies. Both these approaches have inherent challenges, on one hand, the moving mesh methods often encounter computational instability caused by mesh distortion, whereas the fixed mesh methods face challenges in balancing computational accuracy and efficiency. To overcome these limitations, this paper presents a twodimensional steady-state unconfined seepage framework based on adaptive multi-patch isogeometric analysis, designed to enable efficient, stable, and precise numerical simulations. The proposed framework incorporates a Zienkiewicz-Zhu error estimator to drive local mesh refinement, and utilizes truncated hierarchical non-uniform rational B-splines (TH-NURBS) for accurate modeling and localized adaptive refinement. The multi-patch technique, integrated with Nitsche's method, is adopted for the simulation of complex geometries. Validation with several numerical examples shows that the framework provides favorable computational accuracy and efficiency, highlighting its potential for application in complex engineering problems.

This paper presents a computationally efficient framework for stress-constrained topology optimization using an integrated level set method that combines inexact adjoint sensitivity analysis with multigrid acceleration. The proposed methodology introduces a novel convergence criterion based on adjoint sensitivity stabilization, replacing traditional force-residual-based tolerances in iterative solvers. A dual-projection strategy is employed: sharp Heaviside functions enable accurate boundary stress evaluation, while smooth Heaviside functions govern material density evolution for stable topology updates. To reduce computational overhead, multigrid-preconditioned iterative solvers are terminated using sensitivity-based convergence criteria, yielding over an 80% reduction in Krylov iterations for adjoint sensitivity computations. The approach tightly couples discrete adjoint formulations with velocity field parameterization, effectively eliminating intermediate density artifacts and ensuring localized stress control. Numerical studies demonstrate triple-quintuple acceleration in optimization convergence relative to conventional methods, while maintaining compliance with stress constraints within 0.5%-3.5% of direct-solver reference solutions. The framework also integrates p-norm stress aggregation to manage local constraint violations and supports large-scale optimization with improved numerical stability. Validation through ANSYS boundary post-processing confirms the accuracy and robustness of the proposed approach. Overall, this work presents a scalable and efficient method for high-fidelity stress-constrained topology optimization, integrating inexact sensitivity analysis, multigrid preconditioning, and specialized projection strategies to achieve accurate results with significantly reduced computational cost.

In this paper, a nonlocal computational framework for simulating thermo-mechanical fatigue fracture by combining an adaptive multi-patch isogeometric analysis (IGA) with a gradient damage model is presented. A novel fatigue degradation function and an energy-limiter theory are introduced to accurately capture material deterioration under cyclic combined thermo-mechanical loading. To handle complex shapes, Nitsche’s method is employed to couple multiple geometric patches while maintaining solution continuity. Computational efficiency is significantly enhanced through truncated hierarchical NURBS for local adaptive mesh refinement and an adaptive cycle jump scheme that skips redundant loading cycles. Numerical experiments, including gear teeth and notched plates, validate that the model precisely predicts crack propagation paths and fatigue life while reducing calculation time by over 70%. Ultimately, the study offers a robust tool for assessing the safety and durability of critical engineering components subjected to harsh environmental conditions.

Mixed-mode behaviors are critical in both quasi-static and dynamic fracture. To investigate dynamic mixed-mode fracture in quasi-brittle materials, we developed a rate-independent Phase-Field Cohesive-Zone Model (PF-CZM). This model incorporates damage-induced anisotropy through a directional decomposition scheme. A modified G-criterion determines the fracture plane's orientation, and a multiscale framework is introduced to stabilize crack orientation once formed. We solved the governing equations using an implicit time integration scheme implemented in Julia, and adaptive mesh refinement (AMR) accelerated computations. Numerical examples confirm the model's length-insensitivity and its flexibility in capturing failure mode transitions. Notably, our work reveals that Y-joint crack structures in dynamic Brazilian split test form from the intersection of a central mode-I crack and corner-generated mode-II cracks. This study marks the first application of a directional decomposition scheme to dynamic fracture, offering novel into-brittle mixed-mode fracture.

Different software platforms may influence the surrogate model’s comparison and evaluation results. To overcome this challenge, this paper develops a unified surrogate modeling framework to predict the structural responses of a steel box girder bridge under moving-load analysis, ensuring consistency in both data generation and model assessment. Numerical simulations are conducted in MIDAS Civil to generate datasets of bridge’s deflection and combined stresses under variations in uniformly distributed and concentrated loads, as well as their simultaneous changes. Based on identical datasets, deep neural network, polynomial chaos expansion, and Gaussian process, are constructed and implemented within COMSOL for a consistent comparison. The models’ performance is quantified using the mean absolute error, root mean square error, and coefficient of determination. Bridge structural responses under moving-load analysis are effectively captured by all three surrogate models. The Gaussian process model performs best for single-parameter variations, while the polynomial chaos expansion model is more robust for multi-parameter cases. Stress responses are more sensitive to prediction errors than deflection. The proposed framework serves as a valuable benchmark for surrogate modeling-based prediction of bridge structural responses.

In addressing seepage problems with free surfaces, conventional moving mesh methods often encounter numerical instability due to mesh distortion, whereas fixed mesh approaches typically fail to deliver high-accuracy solutions with controllable computational cost. To overcome these limitations, this study develops an adaptive isogeometric analysis framework for two-dimensional steady-state unconfined seepage, aiming to achieve efficient and stable numerical solutions. Combining the local refinement capability of truncated hierarchical NURBS with the permeability adjustment strategy, the proposed framework is shown to capture the free surface accurately while avoiding mesh reconstruction required by traditional methods. By introducing an adaptive criterion based on a posteriori error estimator, the system can intelligently identify and refine key regions near the free surface during the iteration process, significantly improving computational efficiency without compromising accuracy. Numerical results demonstrate that the proposed method exhibits excellent robustness under conditions with different seepage properties, and shows exceptionally good agreement with reference solutions. This method also provides a reliable foundation for subsequent multi-physics coupling analysis and seepage simulation in complex engineering scenarios.

High-fidelity analysis of periodic piezoelectric structures is computationally prohibitive due to the complex electromechanical coupling and periodic microstructural heterogeneity. Isogeometric analysis (IGA) using single-patch modeling cannot efficiently handle complex topologies such as honeycombs or lattice structures. To overcome these challenges, this paper proposes an extended multiscale multi-patch isogeometric analysis (EMs-MPIGA) based on the penalty method for two-dimensional periodic piezoelectric structures. The non-uniform rational B-spline (NURBS) basis functions are employed for geometrical description and mechanical simulation of unit cells to obtain the numerical heterogeneity basis functions of piezoelectric problems. The equivalent matrices, including the stiffness, piezoelectric coupling, and dielectric system matrices of unit cells, are constructed using the computed heterogeneity basis functions. Additionally, downscaling computation is applied to obtain the displacement, electric potential, and von Mises stress of the unit cells. The accuracy, reliability, and robustness of the proposed method are validated via several multiscale simulations.