In practical engineering, a multi-input-multi-output (MIMO) structure generally features a significant number of correlated input parameters and output responses. Sensitivity analysis is usually adopted to select key parameters for improving the computational efficiency of structural analysis and design processes. Traditional sensitivity analysis methods based on probabilistic models for MIMO structures may not reliably and efficiently derive the sensitivity indexes of correlated input parameters with limited samples. To solve the above problems, a novel sensitivity analysis method for MIMO structures considering non-probabilistic correlations is proposed to estimate the influence of uncertainties and correlations among the parameters on the responses in a unified framework. Firstly, a multidimensional parallelepiped (MP) model is employed to quantify the uncertainties and non-probabilistic correlations among the parameters. A new non-probabilistic variance propagation equation based on the MP model is then proposed to derive the non-probabilistic variances of output responses. The non-probabilistic independent, correlated, and total sensitivity indexes of each parameter for multi-input-single-output (MISO) structures are defined according to the non-probabilistic variance contribution rates. A dimensional normalization method and a vector projection method are then adopted to extend the non-probabilistic sensitivity indexes of each parameter for MIMO structures with correlations. Two numerical examples and an experimental example are exemplified to verify the proficiency and efficiency of the currently proposed method.

Kolmogorov–Arnold networks (KANs) as an alternative to multi-layer perceptrons (MLPs) are a recent development demonstrating strong potential for data-driven modeling. This work applies KANs as the backbone of a neural ordinary differential equation (ODE) framework, generalizing their use to the time-dependent and temporal grid-sensitive cases often seen in dynamical systems and scientific machine learning applications. The proposed KAN-ODEs retain the flexible dynamical system modeling framework of Neural ODEs while leveraging the many benefits of KANs compared to MLPs, including higher accuracy and faster neural scaling, stronger interpretability and generalizability, and lower parameter counts. First, we quantitatively demonstrated these improvements in a comprehensive study of the classical Lotka–Volterra predator–prey model. We then showcased the KAN-ODE framework’s ability to learn symbolic source terms and complete solution profiles in higher-complexity and data-lean scenarios including wave propagation and shock formation, the complex Schrödinger equation, and the Allen–Cahn phase separation equation. The successful training of KAN-ODEs, and their improved performance compared to traditional Neural ODEs, implies significant potential in leveraging this novel network architecture in myriad scientific machine learning applications for discovering hidden physics and predicting dynamic evolution.

Fluid–structure-interaction (FSI) phenomena are widely concerned in engineering practice and challenge current numerical methods. In this article, the finite element method is strongly coupled with the multi-material arbitrary Lagrangian–Eulerian (MMALE) method to develop a monolithic FSI method named the immersed multi-material arbitrary Lagrangian–Eulerian finite element method (IALEFEM). By immersing the finite elements in the MMALE computational grid, the fluid–solid interface is directly tracked by the element boundary with accurate normal directions. The fluid–structure-interaction is implicitly implemented by assembling the nodal variables and updating the Lagrangian momentum equation on the MMALE grid. Combining the advantages of both MMALE and FEM with the immersed boundary method, the IALEFEM is effective for solving complicated FSI problems with multi-material fluid flow. A slip fluid–structure-interaction method is also proposed to enhance the computational accuracy in simulating FSI problems with significantly different velocity fields. The accuracy and effectiveness of the IALEFEM are verified and validated by several benchmark numerical examples including the shock-cylinder obstacle interaction, flexible panel deformation induced by shock wave, dam break problem with large structural deformation, water entry of a wedge, fragmentation of a cylinder shell induced by blast, response of elastic plate subjected to spherical near-field explosion and structural damage of open-frame building under blast loading.

The identification of geomaterial properties is costly but pivotal for engineering design. A wide range of approaches perform well with sufficiently measured data but their performance is problematic for sparse data. To address this issue, this study proposes an active learning based multi-fidelity residual Gaussian process (AL-MR-GP) modelling framework. A low-fidelity (LF) prediction model is first trained using extensive LF data collected from worldwide sites to generate preliminary estimations. Subsequent training employs active learning to prioritize high-fidelity data from the specific site of interest with larger information gain for calibrating the LF model to make ultimate predictions. The compression index of clays is selected as an example to examine the capability of the proposed framework. The results indicate that using the same number of site-specific datasets, the compression index of clays can be well captured by AL-MR-GP, exhibiting superior accuracy and reliability than models without incorporating multi-fidelity data or active learning. Based on unified LF data, the proposed framework becomes data-efficient for the model development of three sites and is significantly competitive in extrapolation, compared with site-specific models even with active learning. These promising characteristics indicate substantial potential to be extended to broader applications in geotechnical engineering.

This paper presents a dynamic formulation for the simulation of nearly incompressible structures using a mixed finite element method with equal-order interpolation pairs. Specifically, the nodal unknowns are the displacement and the volumetric strain component, something that makes possible the reconstruction of the complete stain at the integration point level and thus enables the use of strain-driven constitutive laws. Furthermore, we also discuss the resulting eigenvalue problem and how it can be applied for the modal analysis of linear elastic solids. The article puts special emphasis on the stabilisation technique used, which becomes crucial in the resolution of the generalised eigenvalue problem. In particular, we prove that using a variational multiscale method assuming the sub-grid scales to lie in the finite element space orthogonal to that of the approximation, namely the Orthogonal Sub-Grid Scales (OSGS), results in a convenient linear and symmetric generalised eigenvalue problem. The correctness, convergence and performance of the method are proven by solving a series of two- and three-dimensional examples.

Tailored Functionally Graded Materials (FGMs) offer the ability to design and engineer materials with specific properties at a changing volume fraction and are widely used in various fields such as aerospace, biomedical engineering, etc. The precise control of physical properties and the connectivity of microstructural sequences are two main challenges in multiscale problems. This paper constructs a novel optimization model for generating FGMs under customized performance, employing an implicit field representation governed by tensor product B-splines. The cross-sectional profile aligns with a microstructure, and thus varying heights correspond to a sequence of microstructures. Fine-tuning the implicit field on connectable FGMs is achieved by optimizing specific properties and addressing the 2-norm problem under connectivity constraints. Therefore, these FGMs can serve as fundamental units for bottom-up multiscale infills. Additionally, we also develop a new model to investigate a more universally applicable concurrent topology optimization method without initial input restrictions. These multiscale optimization results demonstrate excellent performance under tested working conditions.

The parametric probability-box model, often abbreviated as “p-box” is frequently used to characterize epistemic uncertainties. However, structural reliability analysis with p-box uncertainties can often be computationally intensive. This paper presents an efficient method to accurately compute the bounds of failure probabilities within this context. The method’s key innovation lies in its ability to achieve high efficiency with only a single round of model evaluations. First, the Fractional Exponential Moments-based Maximum Entropy Method (FEM-MEM) with Bivariate Dimension Reduction Method (BDRM) is employed for precise reliability assessment, where a single-round of model evaluations are carried out. Subsequently, Bayesian Updating (BU) is applied to adjust the weights obtained from BDRM in response to changes in input variables, while ensuring the invariance of integration points. Following this adjustment, the FEM-MEM is once again employed to compute the failure probability after the information change, without necessitating additional model evaluations. To compute the bounds of failure probabilities, a Kriging model is employed to construct a surrogate relationship between the interval variables and failure probabilities. The accuracy and efficiency of the proposed method are demonstrated through numerical examples, with comparisons made against pertinent double-loop method.

Most scientific machine learning (SciML) applications of neural networks involve hundreds to thousands of parameters, and hence, uncertainty quantification for such models is plagued by the curse of dimensionality. Using physical applications, we show that L0 sparsification prior to Stein variational gradient descent (L0+SVGD) is a more robust and efficient means of uncertainty quantification, in terms of computational cost and performance than the direct application of SGVD or projected SGVD methods. Specifically, L0+SVGD demonstrates superior resilience to noise, the ability to perform well in extrapolated regions, and a faster convergence rate to an optimal solution.

As with many tasks in engineering, structural design frequently involves navigating complex and computationally expensive problems. A prime example is the weight optimization of laminated composite materials, which to this day remains a formidable task, due to an exponentially large configuration space and non-linear constraints. The rapidly developing field of quantum computation may offer novel approaches for addressing these intricate problems. However, before applying any quantum algorithm to a given problem, it must be translated into a form that is compatible with the underlying operations on a quantum computer. Our work specifically targets stacking sequence retrieval with lamination parameters, which is typically the second phase in a common bi-level optimization procedure for minimizing the weight of composite structures. To adapt stacking sequence retrieval for quantum computational methods, we map the possible stacking sequences onto a quantum state space. We further derive a linear operator, the Hamiltonian, within this state space that encapsulates the loss function inherent to the stacking sequence retrieval problem. Additionally, we demonstrate the incorporation of manufacturing constraints on stacking sequences as penalty terms in the Hamiltonian. This quantum representation is suitable for a variety of classical and quantum algorithms for finding the ground state of a quantum Hamiltonian. For a practical demonstration, we performed numerical state-vector simulations of two variational quantum algorithms and additionally chose a classical tensor network algorithm, the DMRG algorithm, to numerically validate our approach. For the DMRG algorithm, we derived a matrix product operator representation of the loss function Hamiltonian and the penalty terms. Although this work primarily concentrates on quantum computation, the application of tensor network algorithms presents a novel quantum-inspired approach for stacking sequence retrieval.

Macroscopic descriptions of both natural and engineered materials usually include a number of phenomenological parameters that have to be estimated from experiments or large-scale microscopic simulations. When dealing with advanced complex materials, these descriptions are sometimes not a priori available or not even known. Using sparsity-promoting techniques one can extract macroscopic dynamic models directly from particle-based simulations. In this work, we showcase such an approach on a simple fluid and test its robustness. We introduce a novel measure for automatic macroscopic model selection that combines stability and accuracy of a model. Using this measure and employing only a few physics-based assumptions, we are able to infer both the mass continuity equation and an equation for the conservation of linear momentum. Moreover, the extracted phenomenological and non-phenomenological parameters agree well with their numerically measured values and the well-known semi-empirical estimates. The presented model selection framework can be applied to simulations or experimental data of more complex systems, described in general by a rich set of coupled nonlinear macroscopic equations.