ABSTRACT This paper addresses the interpolation of noisy data, a problem in which high‐order polynomial methods frequently exhibit ill‐conditioning and the Runge phenomenon. To mitigate these issues, a hybrid mixed‐type method (MTM) is first developed by augmenting classical collocation equations with Galerkin‐derived moment equations and optimizing a characteristic length parameter within the polynomial representation. Subsequently, the study develops a boundary shape function interpolation method (BSFIM) for two‐dimensional interpolation over rectangular and arbitrary planar domains. In the BSFIM, a family of two‐parameter boundary shape functions is systematically derived such that each basis function automatically and exactly satisfies the measured boundary data. This boundary‐conforming and term‐wise separable basis function design yields a low‐dimensional, and well‐conditioned approximation space that allows the extraction of highly accurate interpolants from substantially fewer interior data points than those required in traditional polynomial or radial basis function (RBF) methods. Extensive numerical examples, including problems on irregular domains, discontinuous scattered data, and Stokes‐flow‐based pressure profile construction, demonstrate that the BSFIM can achieve interpolation errors below the noise level and exhibit robust performance under conditions involving high measurement noise.

ABSTRACT In time‐marching dynamical simulations, treatment of contact forces in deformable bodies represented by finite element meshes requires a compromise between simulation fidelity and computational costs. External forces directly evaluated at the mesh nodes offer better computational performance at the cost of modelling fidelity. Alternatively, externally applied span‐wise member loads can be distributed to the mesh nodes through the element's shape functions. The shape functions enable the development of nodal forces and torques that produce consistent deformations to a load applied along the span of the element. For deformable bodies, represented using finite element methods, which undergo large gross translational and angular motions in time‐marching simulations, constant reorientation of the shape function matrices is required for each evaluation of force distribution. This can be computationally expensive for larger and stiffer systems. In this work, a compact and orientation‐agnostic ‘equivalent’ formulation for expressing nodal loads is presented. The formulation is compared against the consistent formulation for nodal forces of an arbitrarily oriented Euler‐Bernoulli beam. Isolated benchmarking tests and sample application tests indicate nearly identical force distributions in time‐marching simulations for the consistent and equivalent formulations. Evaluated computational performance metrics reveal that the equivalent formulation results in run‐time performance gains. However, the significance of the gains is proportional to the fraction of computational workload associated with the force distribution.

ABSTRACT A mesh that conforms to the geometry is needed for the traditional Finite Element Method (FEM), which is often difficult to generate for complex geometry. It is easier to generate a uniform Cartesian background mesh in which the geometry is embedded. The immersed boundary finite element method (IBFEM) uses a background mesh to approximate the solution while using equations to accurately represent the geometry for finite element analysis. For shell‐like structures, a surface representing the mid‐surface of the shell is utilized, which passes through the 3D elements of the background mesh. In this article, Higher‐Order Shear Deformation Theories (HSDT) are modified for use with immersed boundary shell elements to model laminated composite shells. The rotations and slopes are expressed as derivatives of the displacement field approximated over 3D elements of a background mesh. This requires the displacement field to be tangent‐continuous. Therefore, quadratic or higher‐order B‐spline shape functions that ensure a tangent‐continuous displacement field are utilized to formulate the element. Only three degrees of freedom per node are needed for the immersed boundary shell elements. The shell element presented here is formulated assuming that the strains are infinitesimal. For validating the elements and the modified higher order shear deformation theories, several commonly used laminated composite benchmark problems are used to assess its accuracy. The results obtained using this approach are compared with analytical solutions or with solutions from 3D FEA and isogeometric analysis (IGA) for validation.

ABSTRACT In this article, we propose a novel numerical framework for the non‐isothermal Cahn–Hilliard–Navier–Stokes two‐phase flow system, which couples the incompressible Navier–Stokes equations, the Cahn–Hilliard phase‐field equation, and the heat transport equation to capture temperature‐dependent two‐phase flow dynamics. The proposed scheme achieves three major advances: (i) unconditional energy stability through a combined scalar auxiliary variable (SAV) and zero‐energy‐contribution (ZEC) approach, (ii) linearity and full decoupling of all variables while using a second‐order temporal discretization, and (iii) efficient implementation via discontinuous Galerkin (DG) spatial discretization together with a second‐order projection method for the Navier–Stokes equations. We rigorously prove the unconditional energy stability of the scheme and present key details of its decoupled implementation. Extensive 2D and 3D simulations, including droplet deformation, bubble coalescence, and interfacial instabilities in stratified binary fluids, are presented to demonstrate the accuracy, efficiency, and robustness of the proposed numerical method, thereby confirming its effectiveness for energy‐stable simulation of non‐isothermal two‐phase incompressible flows.

ABSTRACT In this study, a generalized peridynamic model incorporating Seth–Hill bond‐strain measures is proposed to capture mixed‐mode fracture behaviors. We begin with the reformulation of the model introduced by Tupek and Radovitzky within the ordinary state‐based peridynamic (OSB‐PD) framework, where we demonstrate that the three‐dimensional shape tensor state satisfies an integral identity equivalent to the fourth‐order symmetric identity tensor. Based on this identity, the shape tensor state tailored for two‐dimensional problems is constructed, enabling the derivation of the corresponding scalar force state based on Seth–Hill bond‐strain measures for linear elastic materials. This generalized model avoids unphysical material interpenetration and enables the decomposition of the scalar force state over the classic model. Moreover, a nonlocal work‐conjugate stress tensor is developed for the first time by employing the reformulated scalar force state based on the principle of work conjugacy and the integral identity of the shape tensor state. Finally, the maximum principal stress and Drucker–Prager failure criteria are incorporated into the generalized OSB‐PD framework to enable the simulation of mixed‐mode brittle fracture. The accuracy and robustness of the proposed model are validated through several benchmark cases, demonstrating accurate stress evaluation and failure prediction. Notably, the model successfully captures complex crack coalescence patterns in rock subjected to uniaxial compression, underscoring its effectiveness in depicting mixed‐mode fracture processes.

ABSTRACT In this paper, we develop a Numerical Model Reduction (NMR) framework for multi‐scale modeling of electro‐chemically coupled ion transport. Upon introducing the governing equations and employing Variationally Consistent Homogenization, a two‐scale model, consisting of a macro‐scale and a sub‐scale part, is obtained. Instead of solving for the computationally expensive FE 2 simulation, where the macro‐scale and sub‐scale problems are solved in a nested fashion, we exploit NMR by training a surrogate model that replaces the sub‐scale finite element simulations. The surrogate model is trained by performing Proper Orthogonal Decomposition on snapshots of the primary fields. Each macro‐scale quadrature point is no longer occupied by a Representative Volume Element simulation; instead, it is replaced by a surrogate model that consists of a system of Ordinary Differential Equations. In this way, a computationally efficient solution scheme for solving two‐scale problems is obtained.

ABSTRACT Spurious oscillations are a recurring challenge in numerical simulations of advection‐dominated transport, often degrading stability and predictive accuracy. Artificial viscosity is commonly employed to mitigate these effects, but its coefficient is usually tuned empirically, limiting reproducibility and scalability. This study introduces a predictive framework in which the viscosity coefficient is derived analytically from discretization parameters through a closed‐form law obtained via offline optimization guided by a smoothness metric. The methodology is demonstrated for grain aeration, a coupled heat‐moisture transport problem of high practical relevance. The mathematical model was solved using finite differences with the Leith scheme, known for enhanced robustness under realistic aeration conditions. Verification based on apparent order‐of‐convergence analysis of the discretization error confirmed that the stabilized formulation recovered second‐order accuracy, while the unstabilized model exhibited order degradation. Validation against experimental data showed accuracy comparable to manual calibration but with greater stability. Smoothness analysis revealed oscillations only in the energy balance, with mass‐balance equations remaining naturally smooth. Once trained, the predictive law added negligible computational cost (3.5 s per run vs. 162.1 s for mesh refinement ‐ a well‐known technique for reducing oscillations in numerical solutions). The approach eliminates empirical tuning, ensures convergence under experimental conditions, and achieves substantial computational savings for coupled transport problems.

ABSTRACT Group theory has profoundly advanced physics and chemistry in systems with symmetries. Yet its use in structural engineering applications has not yet been fully explored beyond the aesthetics of symmetric designs. This work addresses two significant gaps that have limited the broader adoption of group‐theoretic methods in structural vibration analysis and clarifies their implications for structural design when multiple eigenvalues arise. First, a problem‐independent approach is presented with detailed derivations for constructing group representations for symmetric structures directly from the ‐invariance for structural vibration analysis. This method applies effectively to both dihedral groups and the higher‐order Platonic groups, including tetrahedral (), octahedral (), and icosahedral () symmetries. The method used in this work scales well with structural complexity and enables both explicit and canonical block diagonalizations. Second, this work provides a comprehensive guide to applying finite‐point‐group representations in structural vibration analysis and proves that the dimensions of irreducible representations determine eigenfrequency multiplicities. Although no optimization is performed in this work, this theoretical result has direct implications for structural optimization, resolving longstanding misconceptions about the coalescence of eigenvalues by showing that symmetry is the origin of repeated eigenfrequencies. The theoretical developments are validated on truss structures with dihedral and higher‐order symmetries, accurately predicting their eigenfrequency distributions.

ABSTRACT Neural networks have the potential in approximating the solutions of the Navier–Stokes equations. To address the high computational complexity of high‐order derivatives based on automatic differentiation and the high memory requirements of deep neural network training, a novel Deep Collocation Method (DCM) that combines reversible neural network (RevNet) and spectral differentiation is proposed. The memory‐efficient RevNet is employed to approximate the solutions, while Chebyshev differentiation matrices are derived to rapidly and accurately compute the spatiotemporal derivatives of the approximate solutions, making the computational complexity of high‐order derivatives comparable to that of the forward pass, and significantly reducing memory requirements. A unified spectral discretization scheme is applied in both temporal and spatial dimensions, enabling seamless integration of the neural network and the spatiotemporal Chebyshev differentiation matrices. It effectively incorporates global spatiotemporal information into the residual computation of the governing equations at each collocation point, enabling the neural network to actively account for nonlocal effects of the flow field. Validation results demonstrate that DCM performs well various benchmark flow problems and enables efficient training of deep models even on memory‐constrained devices.