Constitutive modeling lies at the core of mechanics, allowing us to map strains onto stresses for a material in a given mechanical setting. Historically, researchers relied on phenomenological modeling where simple mathematical relationships were derived through experimentation and curve fitting. Recently, to automate the constitutive modeling process, data-driven approaches based on neural networks

The resolution and accuracy of numerical partial differential equation solvers are governed by the mesh density and the order of accuracy of the solver. Anisotropic mesh adaptation combined with a posteriori error estimation is known to be a powerful tool to enhance the efficiency of the solvers. However, in engineering applications involving multiple complex configurations, optimization or uncertainty

In cardiovascular mechanics, reaching consensus in simulation results within a physiologically relevant range of parameters is essential for reproducibility purposes. Although currently available benchmarks contain some of the features that cardiac mechanics models typically include, some important modeling aspects are missing. Therefore, we propose a new set of cardiac benchmark problems and solutions

The cascading reliability evaluation of multi-failure modes of complex system/structure usually needs to repeatedly establish mathematical models with the step-by-step modeling strategy, which weakens the correlation between multi-failure modes. To improve the efficiency and precision of cascading reliability evaluation, a hybrid adaptive coordination surrogate modeling-based improved moving regression

This paper presents a comprehensive framework for spatiotemporal flow field analysis based on complex network theory, emphasizing dimensionality reduction, spatiotemporal feature identification, modeling, and sparsification. The framework first redefines transient flow fields using graph theory and applies clustering techniques to discretize the flow field into different vortex structures, achieving

Kriging is a prevalent surrogate model technique in optimization and data-driven analysis, known for its high accuracy and statistical error estimation. However, training Kriging models often requires extensive global optimization of hyperparameters, posing significant challenges when applying these methods to large-scale datasets. Previous research has mainly focused on expediting the training process

This article presents stochastic augmented Lagrangian multiplier methods to solve contact problems with uncertainties, in which stochastic contact constraints are imposed by weak penalties and stochastic Lagrangian multipliers. The stochastic displacements of original stochastic contact problems are first decomposed into two parts, including contact and non-contact stochastic solutions. Each part is

Currently, due to the complexity of multi-scale topology optimization (MSTO), most of them are optimized based on deterministic conditions, ignoring the influence of uncertain factors on multi-scale structural design optimization. This article aims to develop a novel approach to probabilistic reliability-based topology optimization of multi-scale structure (RBTOM) to address the optimization design

Aircraft design requires extensive aerodynamic data to characterize various flight conditions throughout the aircraft’s flight envelope. Typically, the aerodynamic data is acquired through wind tunnel testing or numerical analysis, which are costly and inevitably entails multiple sources of uncertainty. In the present work, we propose a multi-fidelity Bayesian neural network (MFBNN) framework for multi-source

The growing demand for accurate, efficient, and scalable solutions in computational mechanics highlights the need for advanced operator learning algorithms that can efficiently handle large datasets while providing reliable uncertainty quantification. This paper introduces a novel Gaussian Process (GP) based neural operator for solving parametric differential equations. The approach proposed leverages

Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural networks utilize local convolution in a neighborhood to potentially address these challenges, yet they often suffer from issues such as over-smoothing and over-squashing

We introduce a generative learning framework to model high-dimensional parametric systems using gradient guidance and virtual observations. We consider systems described by Partial Differential Equations (PDEs) discretized with structured or unstructured grids. The framework integrates multi-level information to generate high fidelity time sequences of the system dynamics. We demonstrate the effectiveness

In this work, an efficient thermo-mechanically coupled two-scale finite element (FE)-fast Fourier transform (FFT)-based simulation approach for elasto-viscoplastic polycrystalline materials is proposed. Assuming a separation of scales, the macroscopic and microscopic boundary value problems are solved individually and linked by a scale transition. While the macroscopic boundary value problems are solved

We present Basis-to-Basis (B2B) operator learning, a novel approach for learning operators on Hilbert spaces of functions based on the foundational ideas of function encoders. We decompose the task of learning operators into two parts: learning sets of basis functions for both the input and output spaces and learning a potentially nonlinear mapping between the coefficients of the basis functions. B2B

In this work, we introduce two innovative types of discontinuity-removing physics-informed neural networks (DR-PINNs) aimed at solving variable coefficient elliptic interface problems on curved surfaces, i.e., decoupling DR-PINN and coupling DR-PINN. Initially, by leveraging the level set function associated with the surface, we reframe the surface differential operators as conventional differential

Advancements in manufacturing technologies have enabled material system design optimization across multiple length scales. However, microstructural anomalies (defects) that are present at different scales have not been considered comprehensively enough for systems to be robust to manufacturing variations and uncertainties. Addressing these anomalies through uncertainty quantification and propagation

Accurately modelling the interactions between continuous and discontinuous materials is essential for advancing engineering solutions across a wide range of fields. Owing to fundamental differences in the governing equations of discontinuous and continuous numerical models, distinct solution schemes have been developed, presenting challenges for their coupling. This paper proposes a unified scheme

The machine learning (ML) methods have been applied to numerical solutions to partial differential equations (PDEs) in recent years and achieved great success in PDEs with smooth solutions and in high dimensional PDEs. However, it is still challenging to develop high-precision ML solvers for PDEs with non-smooth solutions. The linear elastic fracture mechanics equation is a typical non-smooth problem

The macroscopic stress formulation for periodic systems in Kohn–Sham density functional theory is critically examined. The identification of the stress through the partial variation of the energy with respect to cell deformation is cast in a strictly large deformation context. The nature of the non-uniqueness in the stress expression which emanates from this variation is extensively discussed. The

Multi-fidelity surrogate models combining dimensionality reduction and an intermediate surrogate in the reduced space allow a cost-effective emulation of simulators with functional outputs. The surrogate is an input–output mapping learned from a limited number of simulator evaluations. This computational efficiency makes surrogates commonly used for many-query tasks. Diverse methods for building them

In engineering applications, the Metropolis-Hastings (M-H) algorithm with high rejection rates is employed to evaluate implicit response functions, making reliability analysis for small failure probabilities with multiple input random variables difficult and inefficient. To address the challenge and estimate highly nonlinear limit state functions in a more efficient and accurate way, a novel reliability

This work introduces an efficient methodology for: (i) predicting dynamic responses across a broad frequency band for large-scale, highly complex structures, and (ii) forecasting their high-frequency response using associated low-frequency information. Structures of interest are characterized by a large number of degrees of freedom (DOFs) and numerous local vibration modes that couple, within the frequency

We present a novel probabilistic approach for generating multi-fidelity data while accounting for errors inherent in both low- and high-fidelity data. In this approach a graph Laplacian constructed from the low-fidelity data is used to define a multivariate Gaussian prior density for the coordinates of the true data points. In addition, few high-fidelity data points are used to construct a conjugate

We study in detail the pressure stabilizing effects of the non-iterated fixed-stress splitting in poromechanical problems which are nearly undrained and incompressible. When applied in conjunction with a spatial discretization which does not satisfy the discrete inf–sup condition, namely a mixed piecewise linear–piecewise constant spatial discretization, the explicit fixed-stress scheme can have a

The mesh flexibility offered by the virtual element method has made it increasingly popular in the context of adaptive remeshing. There exists a healthy literature concerning error estimation and adaptive refinement techniques for virtual elements while the topic of adaptive coarsening (i.e. de-refinement) is in its infancy. The notion of a quasi-optimal mesh is based on the principle of quasi-even

Data-based uncertainty quantification plays a significant role in the design of various patterns of new materials and structures. However, significant challenges remain due to missing data, inherent uncertainties, and incomplete material properties arising from the manufacturing process. In this paper, we quantitatively investigate the uncertainty in the probability of the mechanical response of bio-inspired

The rapid advancement in technology and engineering leads to increasingly complex structural working conditions. Especially, in the field of aeronautics and astronautics, structures are frequently subjected to high temperatures together with external forces, posing great threat to structural health. Consequently, the identification of both mechanical and thermal loads is crucial for structural health

This paper presents a probabilistic approach to represent and quantify model-form uncertainties in the reduced-order modeling of complex systems using operator inference techniques. Such uncertainties can arise in the selection of an appropriate state–space representation, in the projection step that underlies many reduced-order modeling methods, or as a byproduct of considerations made during training

In this paper, we propose several improvements to existing fictitious domain/distributed Lagrange multiplier (FD/DLM) type immersed fluid–structure interaction (FSI) methods targeted for FSI analysis of heart valve dynamics. We utilize the variational multiscale (VMS) method to improve accuracy and robustness on under-resolved grids expected with immersed FSI techniques, as well as for the wide range

Physics-Informed Neural Networks (PINN) are a machine learning tool that can be used to solve direct and inverse problems related to models described by Partial Differential Equations by including in the cost function to minimise during training the residual of the differential operator. This paper proposes an adaptive inverse PINN applied to different transport models, from diffusion to advection–diffusion–reaction

In recent years, the architected structures have attracted extensive attention due to their lightweight feature and excellent mechanical properties. The development of additive manufacturing technologies has expedited the development of the computational design of architected structures. However, the parametric design and simulation of architected structures are full of challenges because of their

In this paper, we introduce a fully-monolithic, implicit finite element method designed for investigating fluid–structure interaction problems within a fully Eulerian framework. Our approach employs a coupled Navier–Stokes Cahn–Hilliard phase-field model, recently developed by Mokbel et al. (2018). This model adeptly addresses significant challenges such as large solid deformations, topology changes

This work introduces a new solution-transfer process for slab-based space–time finite element methods. The new transfer process is based on Hsieh–Clough–Tocher (HCT) splines and satisfies the following requirements: (i) it maintains high-order accuracy up to 4th order, (ii) it preserves a discrete maximum principle, (iii) it asymptotically enforces mass conservation, and (iv) it constructs a smooth

An incremental high-cycle fatigue damage model is combined with topology optimization to design structures subject to non-proportional loads. The optimization aims to minimize the mass under compliance and fatigue constraints. The fatigue model is based on the concept of an evolving endurance surface and a system of ordinary differential equations that model the local fatigue damage evolution. A recent

This paper presents a concurrent optimization method for structural topology and fabrication sequence, aiming at designing for multi-axis additive manufacturing. The proposed method involves two fields: the density field representing the structure, and the time field representing the manufacturing sequence. In addition, angle variables are introduced to represent the designable build directions. The

Enriched finite element methods (e-FEMs) have become a popular choice for modeling problems containing material discontinuities (e.g., multi-phase materials and fracture). The main advantage as compared to the standard finite element method (FEM) remains the versatility in the choice of discretizations, since e-FEMs resolve discontinuities by completely decoupling them from the finite element mesh

This paper presents a novel equilibrium-based approach to the linear homogenization of shear-deformable beams and plates with periodic heterogeneity. The proposed approach leverages the fact that, under equilibrium, the stress resultants and sectional strains in beams and plates vary at most linearly with respect to the axial or in-plane coordinates. Consequently, the displacement fields within a representative

A phase-field monolithic scheme based on the gradient projection method is developed to model crack propagation in brittle materials under cyclic loading. As a type of active set method, the gradient projection method is particularly attractive to enforce the irreversibility condition imposed on the phase-field variables as bound constraints, or box constraints. This method has the advantages of allowing

We develop H(div)-conforming mixed finite element methods for the unsteady Stokes equations modeling single-phase incompressible fluid flow. A projection method in the framework of the incremental pressure correction methodology is applied, where a predictor problem and a corrector problem are sequentially solved, accounting for the viscous effects and incompressibility, respectively. The predictor

Modeling cardiovascular blood flow is central to many applications in biomedical engineering. To accommodate the complexity of the cardiovascular system, in terms of boundary conditions and surrounding vascular tissue, computational fluid dynamics (CFD) often are coupled to reduced circuit and/or solid mechanics models. These allow for realistic simulations of hemodynamics in the heart or the aorta

Geometry projection-based topology optimization has attracted a great deal of attention because it enables the design of structures consisting of a combination of geometric primitives and simplifies the integration with computer-aided design (CAD) systems. While the approach has undergone substantial development under the assumption of linear theory, it remains to be developed for non-linear hyperelastic

Diffusion models have gained attention for their ability to represent complex distributions and incorporate uncertainty, making them ideal for robust predictions in the presence of noisy or incomplete data. In this study, we develop and enhance score-based diffusion models in field reconstruction tasks, where the goal is to estimate complete spatial fields from partial observations. We introduce a

This work presents a matrix-free finite element solver for finite-strain elasticity adopting an hp-multigrid preconditioner. Compared to classical algorithms relying on a global sparse matrix, matrix-free solution strategies significantly reduce memory traffic by repeated evaluation of the finite element integrals.

This study presents an approach to solving an inverse problem through the application of Deep Reinforcement Learning (DRL) coupled with homogenization. The underlying objective is to determine the micro-structural parameters of a composite material, including particle radius, Young’s moduli and Poisson’s ratios in order to achieve a specific target bulk modulus at the macro-scale using DRL. This approach

The original computational framework of elasto-plastic localizing gradient damage, also called as the localizing gradient plasticity (LGP) model, considers that damage in a material causes reduction of yield strength alone. It does not account for the physical process of stiffness-degradation due to damage of ductile materials. Therefore, in this work, a new constitutive model of elasto-plastic localizing

Peridynamics (PD) is a nonlocal continuum mechanics theory over two scales, where the macro-scale material responses are built based on the interaction forces among meso-scale material points. Unlike molecular dynamics, where the pairwise bond forces consist only of normal components along the bonds, the peridynamics bond forces proposed in our recent work can have both normal and tangential components

Euler’s elastica is a classical model of flexible slender structures relevant in many industrial applications. Static equilibrium equations can be derived via a variational principle. The accurate approximation of solutions to this problem can be challenging due to nonlinearity and constraints. We here present two neural network-based approaches for simulating Euler’s elastica. Starting from a data

The Hagstrom–Warburton (HW) boundary operators play an important role in the development of high-order computational schemes for problems in unbounded domains. They have been used on truncating boundaries in the formulation of a sequence of high-order local Absorbing Boundary Conditions (ABCs) and in the Double Absorbing Boundary (DAB) method. These schemes proved to be very accurate, efficient, and

Fast-Fourier Transform (FFT) methods have been widely used in solid mechanics to address complex homogenization problems. However, current FFT-based methods face challenges that limit their applicability to intricate material models or complex mechanical problems. These challenges include the manual implementation of constitutive laws and the use of computationally expensive and complex algorithms

We develop a mixed finite element domain decomposition method on non-matching grids for the Biot system of poroelasticity. A displacement–pressure vector mortar function is introduced on the interfaces and utilized as a Lagrange multiplier to impose weakly continuity of normal stress and normal velocity. The mortar space can be on a coarse scale, resulting in a multiscale approximation. We establish

In the paper we develop the optimal local truncation error method (OLTEM) with the non-diagonal and diagonal mass matrices on unfitted Cartesian meshes for the 3-D time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. 27-point stencils that are similar to those for linear finite elements are used with OLTEM. There are no unknowns for OLTEM on interfaces between

For many important problems it is essential to be able to accurately quantify the statistics of extremes for specific quantities of interest, such as extreme atmospheric weather events or ocean-related quantities. While there are many classical approaches to perform such modeling tasks, recent interest has been increasing in the usage of generative models trained on available data. Despite the sporadic

This work presents a straightforward and efficient isogeometric phase-field framework for predicting creep crack propagation in elasto-plastic materials. In contrast to conventional models that utilize viscous strain energy as the driving force, the proposed approach introduces an asymptotic degradation function for fracture toughness, effectively quantifying material damage resulting from creep strain

We present neural network-based constitutive models for hyperelastic geometrically exact beams. The proposed models are physics-augmented, i.e., formulated to fulfill important mechanical conditions by construction, which improves accuracy and generalization. Strains and curvatures of the beam are used as input for feed-forward neural networks that represent the effective hyperelastic beam potential

Structural lightweight is a core technical requirement for the structural design of aerospace and new energy power equipment structures. For multi-scale variable stiffness design optimization of discrete fiber-reinforced composite laminates, one of the challenges is how to avoid the explosion of design variable combinations caused by the increase in the number of candidate discrete fiber laying angles

Friction Stir Welding (FSW) is a solid-state joining process that has several benefits over conventional welding techniques. A major challenge is its high sensitivity to process parameters such as advancing and rotational speed. Simulations are a key tool for understanding the material flow and temperature evolution and help to find the best processing parameters for a given FSW task. This work proposes

The theory of spectral submanifolds (SSMs) has emerged as a powerful tool for constructing rigorous, low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems. A direct computation of SSMs requires explicit knowledge of nonlinear coefficients in the equations of motion, which limits their applicability to generic finite-element (FE) solvers. Here, we propose a non-intrusive

A seamless integration of neural networks with Isogeometric Analysis (IGA) was first introduced in [1] under the name of Hierarchical Deep-learning Neural Network (HiDeNN) and has systematically evolved into Isogeometric Convolution HiDeNN (in short, C-IGA) [2]. C-IGA achieves higher order approximations without increasing the degree of freedom. Due to the Kronecker delta property of C-IGA shape functions

Motivated by the mismatch between the mechanical performance calculated numerically in topologically optimized designs and that observed in the associated parts fabricated by additive manufacturing (AM) processes, we integrate material characteristics produced via AM processes into topology optimization at low computational cost, by introducing a density-based topology optimization formulation that