The augmented failure probability (AFP) global sensitivity (AFPGS) can effectively quantify the impact of random inputs on structural safety when their distribution parameters are also uncertain. Solving the AFPGS involves a double-loop nested process: the outer loop extracts input samples, while the inner loop focuses on solving for the conditional AFPs under the different input realizations, which requires repeated and costly model evaluations. To address this limitation, this study proposes an efficient sparse-grid-integrated fourth-moment method. Firstly, AFPGS is mathematically reformulated as a weighted sum of absolute differences between unconditional and conditional AFPs at finite integration nodes, which fundamentally eliminates the double-loop nested sampling structure. Secondly, an adaptive Kriging surrogate model combined with sparse grid integration is developed to accurately estimate the first four statistical moments of the performance function. Importantly, the Kriging surrogate model constructed for unconditional AFP estimation is reused for all conditional AFP calculations, making the computational cost of AFPGS analysis equivalent to that of the unconditional AFP estimation. Thirdly, the Yeo–Johnson transformation is introduced to improve the normality of responses at sparse grid nodes, significantly enhancing the accuracy and stability of the fourth-moment method. The effectiveness and superiority of the proposed method are validated through numerical and engineering examples.
This paper presents two high order computational schemes for coupled nonlinear singular differential systems. The schemes are based on higher order Haar wavelet collocation coupled with quasilinearization and the Newton–Raphson method. These methods effectively deal with nonlinearity as well as singularity. The numerical results demonstrate high accuracy with a relatively smaller number of collocation points. Convergence analysis confirms the stability and robustness of the methods. The proposed approach offers an efficient tool for solving nonlinear singular problems arising in astrophysics and fluid dynamics applications.
A novel topology optimization algorithm named EFGM(Adam_FNN) is proposed for orthotropic multi-material structures by coupling neural networks (NN) and element free Galerkin method (EFGM). A fully connected neural network (FNN) is used to perform topology predictions, and the backpropagation technique of the neural network is applied to perform sensitivity analysis. The differences between different neural network-based topology optimization algorithms and the FEM-based iterative solution algorithm are compared. A numerical example is conducted to examine the influence of network depth, learning rate, material categories, and the off-angle of anisotropic materials on topology results. The results indicate that the proposed algorithm can rapidly obtain a relatively clear topology structure. A shallow neural network may lead to intermediate density regions in the topology, and it is recommended that the number of hidden layers exceeds 4. An excessively large network learning rate may cause poorer convergence of the algorithm. It is recommended that the initial learning rate be set between 0.01 and 0.001 and gradually decayed during the optimization process. The compliance can be lowered by adjusting the orthotropic multi-material off-angle, and the reasonable off-angles are 0–30° or 120–150°.
Accurate prediction of transonic surface flow fields over supercritical airfoils is essential for aerodynamic analysis and shape optimization. However, under transonic conditions, these surface flow fields exhibit both large scale smooth variations and shock-induced localized high-gradient structures. Existing deep learning models are prone to spectral bias when learning such multiscale features, leading to preferential convergence in smooth regions and insufficient fitting in localized high-gradient regions. To address this issue, we propose a Multiscale Information Fusion Feature Equalization Operator Network (MIFE-ONet), for surface flow field prediction over supercritical airfoils. In contrast to conventional multi-branch neural operators that mainly fuse branch features at the final stage, MIFE-ONet introduces layer-wise information fusion to progressively couple geometry, angle of attack, and surface-position features. This strengthens the interaction among heterogeneous inputs and improves the prediction accuracy of surface flow fields. A neural tangent kernel (NTK) analysis further shows that unequal residual convergence rates across feature directions contribute to the imbalanced learning of multiscale flow structures. Accordingly, MIFE-ONet employs multiple trunk branches with different scale transformations applied to the same surface coordinates, allowing the model to learn flow field features at different effective spatial scales and improving the prediction balance between smooth regions and localized high-gradient structures. Experimental results show that MIFE-ONet achieves the best performance across four error metrics, with a mean relative error (MRE) of 6.40
A physics-informed neural network (PINN) is developed to identify a closed-form moment–curvature constitutive law for elastoplastic Euler–Bernoulli beams. The fiber-level von Mises response integrated through the thickness is recast as a structural plasticity problem with a single yield surface in moment space. The primary novelty is the data-driven identification of the structural kinematic hardening closure, an emergent property of the dimensional reduction, absent from the local material law, embedded inside the exact algorithmic return mapping, enforcing algorithmic admissibility by construction rather than by penalty terms. A cyclic maturity indicator and architectural constraints ensure Masing-rule consistency between virgin and stabilized branches and positivity of the hardening modulus. Trained weights are extracted to explicit linear algebra, enabling microsecond-scale constitutive evaluation in standard finite element codes. Two boundary value problems validate the 1D model against 2D plane-stress references and closed-form analytical solutions, confirming quantitative agreement in force–displacement response and deflection profiles across cyclic loading–unloading. Computation times are orders of magnitude shorter than the plane-stress reference, demonstrating the practical efficiency of the model reduction.
A growing challenge in research and industrial engineering applications is the need for repeated, systematic analysis of large-scale computational models, for example, patient-specific digital twins of diseased human organs: The analysis requires efficient implementation, data, resource management, and parallelization, possibly on distributed systems. To tackle these challenges and save many researchers from repetitive, time-consuming tasks, we present QUEENS (Quantification of Uncertain Effects in Engineering Systems), an open-source Python framework for composing and managing simulation analyses with arbitrary (physics-based) solvers on distributed computing infrastructures. Besides simulation management capabilities, QUEENS offers a comprehensive collection of efficiently implemented state-of-the-art algorithms ranging from routines for convergence studies and common optimization algorithms to more advanced sampling algorithms for uncertainty quantification and Bayesian inverse analysis. Additionally, we provide our latest cutting-edge research in multi-fidelity uncertainty quantification, efficient multi-fidelity Bayesian inverse analysis, and probabilistic machine learning. QUEENS adopts a Bayesian, probabilistic mindset but equally supports standard deterministic analysis without requiring prior knowledge of probability theory. The modular architecture allows rapid switching between common types of analyses and facilitates building sophisticated hierarchical algorithms. Encouraging natural incremental steps and scaling towards complexity allows researchers to consider the big picture while building towards it through smaller, manageable steps. We aim to create a large scientific community that seeks to collect, extend, and develop methods at the intersection of physics-based simulations, data-driven machine-learning techniques, and sophisticated simulation analytics. The open-source repository is available at https://github.com/queens-py/queens .
Accurate prediction of time‑dependent multi‑responses of surrounding rock during tunnel excavation is critical for engineering safety and rational support design. Traditional numerical simulations are accurate but computationally prohibitive for real‑time use, whereas purely data‑driven models lack physical consistency. To address these issues, a physics‑informed surrogate model integrating Kolmogorov-Arnold Networks (KAN) and Long Short-Term Memory (LSTM) networks is proposed. The model takes seven static mechanical parameters of the rock mass as input and predicts the temporal evolution of deformation, stress, and damage characteristic zones including loosened, plastic, and disturbed zones induced by excavation. To overcome the lack of unified identification criteria, a quantitative determination method based on thermodynamic internal variable theory is introduced. The loosened zone is delineated by plastic shear strain, plastic volumetric strain, and their depth‑correlated Pearson coefficient curves; the disturbed zone boundary is determined using the cumulative energy dissipation density and its depth‑based Pearson coefficient curve. A Physical Knowledge Module (PKM) is embedded, encoding deformation growth, stress relaxation, and damage propagation into a composite loss function that combines data loss and physical loss. The framework is applied to the SJLS tunnel project. Bayesian optimization and Dirichlet sampling are employed for hyperparameter tuning. Results show that the model achieves high prediction accuracy (R2 > 0.99), outperforming standalone KAN, LSTM, and other baselines. The PKM enhances both predictive accuracy and physical consistency. The proposed framework serves as an efficient tool for real‑time prediction of multiple time‑dependent responses and for mechanical parameter inversion based on multi‑source monitoring data. A physics-informed KAN-LSTM framework for predicting tunnel responses is proposed. A numerical criterion for determining surrounding rock damage extents is established. Evolutionary pattern of time-dependent responses of surrounding rock are considered.
Radiative magnetohydrodynamic bioconvective transport of viscoelastic nanofluids in porous media is governed by strongly nonlinear and tightly coupled mechanisms that challenge conventional numerical solvers. Physics-informed neural networks (PINNs) offer a mesh-free alternative by embedding the governing conservation equations directly into the solution framework; however, their predictive capability is highly sensitive to hyperparameter selection, often leading to slow convergence and inconsistent accuracy. To address this limitation, a hybrid Reptile Search Algorithm (RSA)–optimized PINN is developed for the similarity equations describing upper-convected Maxwell nanofluid flow over an axially stretching cylinder in a Darcy–Forchheimer porous medium with thermal radiation and bioconvection effects. The RSA adaptively identifies optimal network hyperparameters, providing stable and consistent convergence and enabling reduction of the physics-based residuals to near machine-precision levels while avoiding premature convergence commonly observed in conventional training strategies. The optimized solutions are validated against a fourth-order collocation boundary-value solver, demonstrating excellent agreement for velocity, temperature, nanoparticle concentration, and motile microorganism distributions. Comparative studies with PSO, GA, and Bayesian-optimized PINNs show that the RSA-based framework yields improved robustness, reduced solution variability, and enhanced numerical stability in handling the highly nonlinear transport system. Statistical assessments using MAE, RMSE, TIC, and ENSE further confirm the accuracy of the approach. The proposed RSA–PINN formulation therefore provides an efficient and reliable computational tool for simulating complex multiphysics heat and mass transfer in non-Darcy porous boundary-layer configurations.
A novel AT1 phase-field framework is introduced for simulating quasi-static anti-plane shear fracture in geometrically linear elastic bodies. In this approach, ξ -based local mesh adaptivity is unified with an algebraically nonlinear strain energy density function to circumvent the physically inconsistent crack-tip singularities inherent to classical linear elastic fracture mechanics. A modified Francfort-Marigo energy functional with Ambrosio-Tortorelli-type regularization is proposed, in which the characteristic length of the damage zone, ξ , is dynamically optimized to enhance computational efficiency and approximation accuracy. The total energy functional–comprising nonlinear strain energy, evolving surface energy, and linear regularization–is minimized via variational principles to yield a coupled system of quasilinear partial differential equations, which are discretized using conforming bilinear finite elements. The formulation is governed by four parameters, including two asymptotically calibrated penalty terms. It is demonstrated that this spatially adaptive strategy is best suited for adaptive mesh simulation of quasi-static crack propagation and outperforms regular mesh refinement strategies, enabling accurate fracture propagation even while maintaining a significantly larger regularization length compared to non-adaptive counterparts.
This paper presents compact and educational Matlab implementations for integrated topology optimization of functionally graded cellular structures in 2D and 3D. The method combines a modified Solid Isotropic Material with Penalization (SIMP) scheme with computational homogenization and a multiple-variable cutting (M-VCUT) level set method to jointly optimize macroscopic topology and microscopic configurations. To avoid repeated homogenization, an offline database is constructed to store homogenized equivalent elastic tensors and queried during optimization. This ensures computational efficiency while enabling continuous microstructural grading. The framework extends the classic 88-line SIMP code to a multiscale setting. Our contributions are: (1) offline generation of homogenized parameters for representative microstructures, (2) a code architecture for integrated optimization for functionally graded cellular structures (top88_2D.m, top88_3D.m) and (3) explicit geometric reconstruction (shapestore_2D.m, shapestore_3D.m) to generate optimized cellular structure. Numerical examples demonstrate the effectiveness of the proposed framework for compliance minimization. The open-source Matlab codes provide a practical and accessible platform for research and education in graded cellular materials and multiscale topology optimization.
The inverse design of cellular metamaterials—finding unit-cell geometries that realize prescribed effective properties—is intrinsically one-to-many, yet conditional generators often collapse to narrow outputs and graded assembly can introduce interface discontinuities. We present Style-IH-GAN, a style-modulated generative adversarial network for triply periodic minimal surface (TPMS)-based cellular materials. A mapping network converts latent noise into style vectors that modulate a property-conditioned generator, while mode-seeking and property-consistency losses encourage geometric alternatives under fixed targets. For graded synthesis, the model emits a spatial field of TPMS descriptors in one batched forward pass, and trilinear interpolation supports continuous inter-cell reconstruction. The predictor is used only as a differentiable training signal and controlled baseline metric; unit-cell fidelity is evaluated by numerical homogenization over 750 generated samples across 15 targets. Compared with a vanilla conditional generative adversarial network, Style-IH-GAN achieves 2.2 × higher shape-parameter diversity and 19
Efficiently estimating extremely small time-dependent failure probability (TDFP) is a challenge for highly reliable engineering structures. To address this challenge, this paper proposes an efficient clustering-based stratified importance sampling (CSIS) method. The proposed CSIS alternately constructs mixture importance sampling (IS) densities and stratifies the input space. It uses samples from mixture IS densities to enable time-dependent performance function to adaptively divide the input space into time-dependent failure stratifications with decreasing probabilities. Then, clustering analysis is conducted on failure samples within each stratification to construct mixture IS densities for subsequent stratifications. After failure stratifications approach the target failure domain, the current mixture IS densities are used to estimate extremely small TDFP. In the proposed CSIS, adaptive stratification reduces the exploration complexity of rare time-dependent failure domain. Meanwhile, the explicitly formulated mixture IS densities, which are constructed on clustering centroids of failure samples in each stratification, significantly reduces TDFP estimation variance by stepwise approaching critical regions contributing most to TDFP. And computational efficiency of CSIS is further enhanced by the adaptive Kriging model of time-dependent performance function. Examples demonstrate that the proposed method achieves significant computational efficiency improvements over existing advanced methods while maintaining accuracy of estimating extremely small TDFPs.
This paper presents an adaptive coupling method that integrates ordinary state-based peridynamics with the quadtree-based scaled boundary finite element method, specifically developed for modeling porous materials. The adaptive mesh partition is automatically achieved based on a newly developed displacement-induced criterion, in which the SBFEM nodes will be automatically transformed into PD points when the relative displacement between them exceeds a certain critical threshold. The inherent compatibility of SBFEM with quadtree meshes is fully exploited to refine the mesh locally in the critical fracture areas only. A precomputed strategy is adopted, where the stiffness matrices of 16 standard quadtree elements are calculated in advance and cached to avoid redundant computations. The validation cases have demonstrated that the present adaptive coupling method can achieve highly accurate predictions of crack paths in porous plates, and the relative errors are less than 0.8
Dynamic Mode Decomposition (DMD) has emerged as a widely used data-driven tool for analyzing complex dynamical systems. While numerous extensions of DMD have been proposed to improve its applicability to nonlinear partial differential equations (PDEs), the standard formulation of DMD has not always been explored to its full potential. In this work, we demonstrate that DMD via ensembling can recover the spectra of the linearized dynamics, corresponding to the Koopman operator, restricted to linear observables of nonlinear PDEs in the neighborhood of their equilibria with good accuracy. By stacking trajectories from multiple initial conditions, the ensemble approach yields a more comprehensive representation of the underlying dynamics and captures richer spatiotemporal structures. In the local regime, the resulting model captures the essential dynamics without the need for nonlinear observable lifting, kernel-based formulations, or deep learning architectures. We investigated the role of the sampling distribution by relating sampling strategies to convergence rates associated with corresponding quadrature rules. We evaluated the ensemble formulation on several systems, including the convection-diffusion equation, Burgers’ equation, Fisher’s equation, the Kuramoto–Sivashinsky equation, as well as the Navier–Stokes dynamics corresponding to flow past a circular cylinder, and compared the learned spectra with their analytical counterparts. The learned spectra show excellent agreement with analytical counterparts, demonstrating that DMD via ensembling can reliably uncover the underlying dynamics of complex dynamical systems.
Hyperthermia is increasingly recognized as an effective adjunctive therapy for cancer treatment, yet its clinical impact depends on accurate and rapid treatment planning tailored to patient anatomy. This work presents a multi-anatomical hyperthermia treatment planning framework designed for both brain and breast tumors, two clinically relevant sites with distinct anatomical and technical challenges. The proposed framework, referred to as the MATLAB-based brain and breast hyperthermia simulator, is based on the finite element method (FEM) and serves as a computationally efficient alternative to commonly used finite-difference time-domain (FDTD) platforms such as Sim4Life. The proposed framework integrates three-dimensional finite-element electromagnetic and thermal modeling with annular phased-array dipole applicators (twelve elements for brain and eight for breast), operating at 915 MHz to enable deep penetration and localized heating. A convex optimization strategy based on Second-Order Cone Programming is employed to determine phase and amplitude excitation coefficients that maximize tumor heating while suppressing unwanted hotspots in healthy tissue. Validation was performed across five tumor locations per anatomy using the Duke head model, segmented breast phantoms, and geometry-matched breast and head phantoms implemented in Sim4Life and COMSOL Multiphysics for FDTD- and FEM-based benchmarking. Quantitative evaluation using European Society of Hyperthermic Oncology recommended specific absorption rate (SAR) and temperature quality indicators demonstrates close agreement with benchmark simulations in terms of SAR focusing, thermal coverage, and peak temperature metrics. Overall, this study introduces a unified and computationally efficient FEM-based framework for brain and breast hyperthermia planning, validated against both FDTD- and FEM-based platforms.
Modeling cross-scale failure processes remains challenging due to the high computational cost associated with size effects and the need to calibrate macro-scale damage models from micro-scale fracture mechanics. In this work, we propose a model-based neural network (MBNN) that serves as a micro-scale informed constitutive surrogate for macro-scale damage simulations. Within an energy-based homogenization framework, the approach links micro-crack evolution to continuum-scale damage variables while embedding mechanics-consistent constitutive laws into the network architecture, resulting in sparse and physically interpretable parameters. Damage evolution is formulated as a joint model-selection and parameter-identification problem, enabled by a differentiable Gumbel-Softmax mechanism that automatically selects appropriate constitutive and damage behaviors from RVE-level fracture data. Compared with conventional deep neural networks, the proposed framework exhibits improved robustness, interpretability, and extrapolation behavior. The MBNN is validated on multiple cross-scale failure benchmarks, demonstrating favorable accuracy and computational efficiency. Applications to femoral bone damage further illustrate its applicability to biomechanical problems. In addition, an extension of the MBNN to progressive damage modeling is investigated and validated using experimental data, demonstrating its ability to capture gradual degradation, thereby extending the framework beyond ideal brittle fracture.
Recent advances in topology optimization (TO) have increasingly incorporated machine learning techniques to enhance computational efficiency. Among them, online learning methods emerge as promising approaches for large-scale TO designs. Unlike off-line methods requiring pre-generated training data, online learning frameworks adaptively train surrogate models during the optimization process itself using data from earlier iterations. Current state-of-the-art approaches typically employ a two-scale localized training paradigm, where machine learning models iteratively learn mappings between coarse-scale sensitivity fields and fine-scale structural responses within localized patches. While this localized strategy improves computational scalability and provides implicit data augmentation, it faces a limitation: the exclusion of global structural information often leads to underdetermined mappings and compromised prediction accuracy, as fine-scale sensitivities inherently depend on structural context beyond individual patches. To address this issue, we propose a multi-grid decomposition strategy and integrate it within the online learning framework. This strategy hierarchically incorporates global structural information into local training patches to enhance mapping uniqueness. Additionally, our framework implements online learning entirely within a single mesh resolution, bypassing costly coarse-scale simulations while maintaining accuracy. We employ Fourier neural operators as our surrogate model of choice, leveraging their superior capability in capturing structural sensitivity patterns directly from spatial information The proposed method is validated through 2D and 3D compliance design examples, demonstrating significant computational savings by reducing high-dimensional numerical simulations. The influence of key hyperparameters on the framework’s performance is also investigated.
Due to the distinct combustion properties of hydrogen compared to conventional fossil fuels, novel burner designs are required. Additive manufacturing techniques enable the realization of complex geometries, which can be obtained through simulation-based design optimization. However, the considerable costs associated with high-fidelity CFD simulations of the governing physical processes in hydrogen burners limits their direct use in optimization. Multi-fidelity surrogate modeling provides an efficient alternative by combining inexpensive low-fidelity with accurate high-fidelity simulation results. We present a modular toolchain for simulation-based shape optimization of combustion devices that integrates (low-fidelity) RANS and (high-fidelity) LESs, which provide the basis for a multi-fidelity surrogate model. Inherently, this approach incorporates time-averaged and transient characteristics of the newly designed burners. Our approach is therefore capable of capturing both, quantities derived from averaged solutions and those influenced by transient dynamics. We apply the multi-fidelity toolchain to the optimization of the internal geometry of a weakly turbulent hydrogen burner nozzle. The results show that purely geometric modifications can significantly influence mixture characteristics through changes in flow features such as recirculation zones and jet-in-crossflow interactions. The optimization toolchain reliably produces designs that meet the desired mixture characteristics in both time-averaged and transient metrics. The results demonstrate the efficacy of this toolchain for the design optimization of burner nozzles (or other combustion devices) and it is suited to make use of the design freedom offered by additive manufacturing.
This paper proposes a hybrid unstructured mesh generation method for turbomachinery simulations, aiming to overcome the limitations of conventional approaches in geometric adaptability, size control, viscous boundary-layer resolution, and automation. A hybrid surface representation framework is first constructed, in which the discrete surface representation serves as the background mesh for size control, while the continuous surface representation defines the geometric boundary for mesh generation. A unified topological data structure is then established to manage the discrete representation, continuous representation, and mesh model in a consistent manner. To handle the complex geometric configurations of turbomachinery, feature recognition and local mesh refinement techniques are developed, and the size function is defined on the optimized background mesh. An efficient size-function smoothing strategy is further proposed to ensure a smooth transition between regions with different mesh densities. The proposed method is validated through numerical simulations of several representative turbomachinery cases, and the predicted results are compared with experimental data as well as with those obtained using the commercial software packages ICEM-CFD and Pointwise. The results demonstrate that the proposed method achieves comparable or superior mesh quality while significantly improving mesh generation efficiency, and the simulation results show good agreement with experimental data.
A new semi-implicit, two-phase, double-point formulation of the Material Point Method (MPM) for soil–water interaction with seepage and free-surface flows under large deformation is presented in this paper. The approach advances the water phase implicitly while keeping the soil phase explicit, enabling stable, efficient time integration in problems that involve rapid seepage and strong free-surface motion. The proposed framework models high-Reynolds-number interphase drag through a non-linear Darcy’s law implemented for the first time within an incremental fractional step MPM formulation without enlarging the implicit solve. This methodology also enhances the numerical stability for fast flows and wave breaking via a hyperelastic constitutive treatment of slightly compressible viscous water, and mitigates spurious oscillations through a new stabilisation approach for the velocity. Robustness of soil–water interface is achieved by combining nodal-based, free-surface detection, suited for higher-order spline functions with smooth porosity–permeability transitions that avoid constitutive divergence at sharp material boundaries. Validation against laboratory benchmark cases reported in the literature, including pure-water dam break, dam-break seepage through a porous barrier, two granular-collapse tsunami experiments, and a dam-break wave over a movable granular bed, shows accurate and stable free-surface evolution, pressure time histories, seepage fronts, and wave-gauge records. Using an advanced critical-state soil model (NorSand) further improves the reproduction of granular flow kinematics. The results demonstrate that the proposed formulation is a reliable and computationally efficient tool for geotechnical hazards involving intense soil–water coupling, seepage, sediment transport and free water.
