ABSTRACT Characterizing heterogeneous inclusions in hyperelastic materials is critical for diverse fields ranging from soft robotics to medical diagnostics. However, this inverse problem remains challenging as traditional engineering approaches struggle to resolve sharp material interface and strong nonlinearity, while data‐driven approaches typically require prior geometric assumptions or accurate stress measurements. To address these limitations, we propose a robust framework based on physics‐informed neural networks equipped with residual‐based adaptive refinement (AR‐PINN). By dynamically appending collocation points according to governing‐equation residuals, our method autonomously resolves both the complex inclusion topologies and the relative shear modulus distributions using only sparse Green–Lagrange strain measurements, requiring no prior knowledge regarding the shape or size of inclusions. Extensive numerical studies demonstrate accurate identification and remarkable robustness under large nonlinear deformation, including uniaxial stretching up to , as well as resistance to measurement noise levels up to 30%. These results firmly establish AR‐PINN as a data‐efficient, assumption‐free, and practical tool for nondestructive characterization and inverse design of heterogeneous soft materials in experimental settings.
ABSTRACT This paper presents a novel B‐spline‐parameterized topology description function (TDF) method to tackle the challenging problem of stress‐constrained topology optimization under sweep‐frequency harmonic excitations. The proposed method employs B‐spline coefficients as design variables, enabling a smooth and implicit representation of evolving topologies. A two‐layer aggregation strategy with a dynamically adjusted correction coefficient is introduced to condense the spatial‐frequency peak stress into a single, smooth, and differentiable global stress constraint. A rigorous error analysis of the introduced two‐layer aggregation strategy is provided. The sensitivities of the global spatial‐frequency stress constraint and the volume objective with respect to the B‐spline coefficients are derived in detail. Furthermore, the model reduction technique based on the modified Gram‐Schmidt orthonormalization is introduced to reduce the cost of repeated harmonic‐response analyses at multiple frequencies. Finally, three numerical examples, including an L‐shaped beam, a U‐shaped beam, and a corbel structure, are investigated to validate the proposed approach. The results show that the developed method effectively controls local stress concentrations under sweep‐frequency harmonic vibrations, providing a robust tool for designing lightweight structures subjected to variable‐frequency dynamic loads.
ABSTRACT Three‐dimensional phase‐field fracture simulations are computationally demanding, as the geometric complexity and topological evolution of crack surfaces in 3D space require a vast number of degrees of freedom to achieve sufficient accuracy. Yet, most existing spatial adaptive schemes rely on rigid refinement patterns and empirical control parameters, which hinder local flexibility and effective adaptive mesh control. To address these limitations, a self‐correcting global–local adaptive meshing method (SC‐GL‐AMM) is developed. A normalized nodal density field is first defined to provide a quantitative indicator of local resolution demand and guide adaptive refinement, enabling efficient and targeted mesh control. Subsequently, global crack sampling is conducted through filtering and clustering to capture 3D fracture features and perform Voronoi partitioning. Within each subdomain, a hexagonal close‐packed node seeding algorithm is applied, supporting parallel processing and adaptively adjusting the distribution of mesh nodes according to the local nodal density field. Constrained Delaunay tetrahedralization is then employed to construct the adaptive mesh, ensuring geometric conformity and maintaining topological consistency across refined regions. Furthermore, a self‐correcting framework is introduced, which is utilized to dynamically update the adaptive criterion based on the evolving solution‐dependent indicators, enhancing robustness for complex 3D fracture scenarios. Comparative and ablation studies are conducted to assess the roles and effects of key algorithmic components. Numerical examples further demonstrate that the proposed method generates high‐quality meshes with flexible adaptation, achieving up to a sixfold speed‐up compared with the conventional tetrahedral subdivision strategy. This strategy thus constitutes a significant enhancement in adaptive mesh control, providing an efficient tool for large‐scale three‐dimensional phase‐field fracture simulations.
ABSTRACT Thin‐shell structures subjected to cyclic loading exhibit strong coupling among in‐plane and out‐of‐plane deformation modes, which renders fatigue life assessment particularly challenging. Existing peridynamic (PD) fatigue formulations predominantly account for fatigue damage driven by axial bond deformation and are therefore not well suited for thin‐shell applications. To address this deficiency, this study introduces a microbeam bond‐based PD shell fatigue model. By employing the bond energy release rate range as a unified scalar fatigue‐driving parameter, the contributions of axial, bending, and torsional deformation modes are inherently incorporated without the need for empirical weighting factors. Building on this framework, phase‐dependent evolution equations are developed separately for the crack initiation and crack propagation stages. The capability of the proposed approach to predict crack trajectories and fatigue life is demonstrated through several numerical examples, with particular emphasis on a cylindrical shell benchmark containing circumferential and inclined cracks, in which local bending and torsional deformations at the crack tip are strongly coupled. Moreover, comparison with a conventional axial deformation‐driven PD fatigue model shows that neglecting bending and torsion leads to a substantial underestimation of the crack growth rate. These findings underscore the critical importance of accounting for coupled axial, bending, and torsional deformations and confirm the predictive capability and suitability of the proposed model for fatigue analysis of thin‐shell structures subjected to multiaxial loading.
ABSTRACT In this paper, an interpolative mapping mortar method (IM‐mortar) is proposed for nonmatching finite element meshes. Displacement continuity is enforced weakly using a Lagrange multiplier field. An intermediate frame is introduced, on which the interface displacement field is reconstructed from the adjoining bodies through interpolative mapping. All interface integrals are then evaluated on the frame mesh, avoiding direct integration over element‐to‐element overlap regions. The proposed method satisfies patch tests to machine precision for the regular configurations considered, whereas exact satisfaction of the patch test is not achieved for the distorted three‐dimensional configuration. Nevertheless, approximately first‐order convergence is observed under mesh refinement. The accuracy and computational performance of the proposed method are further examined through a series of numerical examples.
ABSTRACT Mixed convection nanofluid flowing through stretching/shrinking permeable sheet has significant applications in numerous thermal devices such as heat exchangers, microfluidics, cooling of electronic devices, and materials processing. Nevertheless, the combined role of convective boundary conditions, thermo‐diffusion, viscous dissipation, and flow stability in these configurations is still not well understood. In this work, the mixed convection flow with heat and mass transfer of water‐based AA7075 nanofluid over an exponentially stretching/shrinking permeable surface is investigated. The buoyancy, external forces, and Soret effects are taken into consideration to incorporate the coupling between momentum, temperature, and concentration fields along with convective boundary conditions. The boundary layer equations are reduced to similarity equations and solved numerically by the MATLAB solver bvp4c. Nonunique solutions are found in specific ranges of suction/injection, stretching/shrinking, and mixed convection parameters. Linear temporal stability analysis is carried out by using an eigenvalue method to determine the physically feasible solution. In addition, response surface methodology is used to investigate the interaction of parameters and optimize the heat and mass transfer. The findings reveal that the nanoparticle volume fraction, viscous dissipation, buoyancy forces, and thermo‐diffusion have significant effects on the flow behavior rates of heat and mass transfer. The present numerical, stability, and optimization framework provides useful insight for the design and operation of advanced nanofluid‐based thermal systems.
ABSTRACT Soft material modeling is challenging due, among others, to the substantial structural changes these materials undergo. In this context, meshless methods offer a promising alternative to overcome the limitations of established mesh‐based approximations such as finite elements. However, they introduce new challenges. One of these is the incorporation of surface effects, such as cohesive surface behavior or deformation induced by surface tension, which are important in soft materials such as hydrogels and biological tissues but remain difficult to account for in meshfree settings. In this work, we present and benchmark an enhanced local maximum‐entropy (max‐ent) framework for surface modeling, thereby extending max‐ent to problems simultaneously involving surface tension and large deformation. We propose alternatives to identify boundary orientation in the presence of diffuse boundaries, consistently using 3D surface max‐ent interpolants, and to prevent cross‐coupling errors in irregular domains. We assess the performance of the framework by means of surface‐tension‐driven benchmarks involving strong volumetric contraction, near‐incompressibility, and complex geometries. The results show that the proposed framework yields accurate and robust predictions under severe structural changes, making it a promising tool for the modeling of polymers and soft biological systems in applications involving surface tension, membrane permeability, traction and pressure loading, and interfacial chemical interactions.
ABSTRACT Quantum annealing is emerging as a practical tool for large‐scale combinatorial optimization. We map continuum displacements and element densities to binary variables, turning both deformation analysis and topology optimization into quadratic unconstrained binary optimization (QUBO) problems that run on today's annealers. Two formulations are compared: one derived from the stiffness equation and another from the minimum total potential energy. Benchmarks show the energy‐based QUBO matches finite‐element displacements and stresses more closely and avoids the rapid term‐count growth that slows the stiffness‐based form. The numerical demonstrations reported here include deformation analysis of a 100‐element plate model and topology optimization of cantilever beams discretized with up to 60 triangular elements or 32 rectangular elements. These examples are intended as proof‐of‐concept demonstrations under current hardware and encoding limitations rather than as large‐scale industrial benchmarks. Leveraging this model, a single QUBO embedding density variables–implemented with combinatorial random‐number sums and a quadratic volume penalty–yields two‐dimensional beam layouts that are quantitatively compared with those obtained using the classical Optimality Criteria (OC) method.
ABSTRACT Flexoelectricity denotes a universal electromechanical coupling in dielectric materials, whereby strain gradients break local inversion symmetry and induce electric polarization. The resulting boundary value problem (BVP) is governed by a coupled multi‐physics system of high‐order and mixed‐order partial differential equations with nonstandard boundary conditions. Within this context, physics‐informed neural networks (PINNs) offer an attractive mesh‐free computational framework, providing ‐continuous approximations of the solution fields and enabling the direct evaluation of high‐order derivatives via automatic differentiation. However, we recognize that the distinctive differential structure of flexoelectric BVPs severely degrades the learning performance of naive PINN formulations. The direct use of primary field variables in strong‐form residuals requires repeated evaluation of high‐order derivatives, leading to unstable gradient propagation and degraded training efficiency. More critically, the problem exhibits a mixed‐order structure across the coupled fields, which introduces conflicting gradients within a shared network and results in an ill‐conditioned optimization landscape. In this work, we propose a reformulated PINN architecture for flexoelectric BVPs, retaining the strong‐form and data‐free manner. By incorporating pseudo outputs and auxiliary losses, the proposed architecture recasts the original high‐order system into a lower‐order reformulation; more importantly, it reduces the imbalance of differential orders across coupled fields within each loss component. This reformulation improves the compatibility between the differential structure of the governing equations and the PINN representation, thereby alleviating the associated optimization difficulty and enhancing training efficiency and accuracy. Numerical investigations on representative flexoelectric benchmark problems demonstrate strong performance as a promising numerical approach for flexoelectric BVPs and clear advantages over naive formulations. Beyond flexoelectricity, the present work provides new insights into the treatment of high‐ and/or mixed‐order coupled partial differential systems within the PINN framework.
ABSTRACT The effective Young's modulus of periodic lattice structures is contingent not solely on topology and relative density, but also on the number of unit cells incorporated within the computational domain or specimen. This issue assumes particular significance in uniaxial tension, for which representativeness criteria are less well‐defined than in compression. The present work proposes a convergence‐based numerical framework for the quantitative description of finite effects and the correction of the effective Young's modulus determined for non‐representative domains. The analysis was conducted for six periodic topologies, including three rod structures (BCC, Fluorite, Kelvin) and three TPMS‐type structures (Fischer–Koch, Gyroid shell, Schwarz shell). The linear‐elastic FEM simulations were performed for various relative densities and for cubic domains of N × N × N unit cells, with N from 1 to 12. It has been demonstrated that the convergence of the effective Young's modulus to the asymptotic value can be adequately described by an exponential model governed by a pair of parameters: the convergence length τ , which sets the decay rate, and the normalized amplitude α , which sets the magnitude of the initial deviation. Together they determine the minimum number of cells required for a given error level. On this basis, representativeness error maps and correction factor maps C ( N , ρ rel ) were developed. These maps constitute a calibrated correction framework that allows the response of an underrepresented domain to be rescaled toward the asymptotic stiffness within the calibrated topology–density range, rather than an independent predictive model. A comparison with experimental tensile tests shows that the corrected predictions move closer to the measurements for most configurations. Direct simulations of elongated domains reveal, however, that a correction calibrated on isotropically scaled domains captures only a small part of the stiffness change produced by extending a specimen along the loading axis alone, so the framework is presented as a correction for isotropically scaled domains. At lower relative densities geometric deviations and manufacturing inaccuracies additionally contribute to the remaining discrepancy.
ABSTRACT Fluid‐filled phase‐field fracture simulations require robust, scalable solvers that can handle strongly nonlinear, non‐smooth mechanics and tightly coupled flow on locally refined meshes. In this work, we develop an adaptive finite element framework for quasi‐static, fluid‐filled phase‐field fractures that combines semi‐smooth Newton methods, fixed‐stress iterative coupling, and matrix‐free geometric multigrid preconditioning. The geomechanics subproblem, with displacement and phase field as primary variables, is solved by a semi‐smooth combined Newton method based on a primal‐dual active set formulation. The resulting linear systems are treated with the generalized minimal residual method and matrix‐free geometric multigrid preconditioners on locally refined meshes, with a local smoothing approach and careful treatment in the active set updates. The pressure equation is coupled to the geomechanics system via a fixed‐stress iterative scheme which iterates until both subproblem residuals are sufficiently minimized, and each flow subproblem is likewise preconditioned by matrix‐free geometric multigrid on the same adaptive meshes. In addition, we introduce an enhanced fracture‐width computation, inspired by and extending existing phase‐field aperture formulas, to obtain more accurate and mesh‐robust width fields consistent with the regularized fracture profile. The overall computational framework is demonstrated on a set of two‐ and three‐dimensional benchmark problems, highlighting its robustness, efficiency, and accuracy for simulating fluid‐filled phase‐field fractures with local mesh refinement.
ABSTRACT Engineering materials frequently exhibit considerable disparities between their tensile and compressive moduli. However, there are some difficulties in existing topology optimization methods for material structures with large disparities between their tensile and compressive moduli. To address this issue, a parallel‐computing based topology optimization method for bi‐modulus material structures with large property differences and deformation control is proposed. The main challenges are twofold: (1) the large discrepancy between tensile and compressive moduli usually leads to convergence difficulties within finite element analysis (FEA); (2) the iterative finite element solution procedure imposes a heavy computational burden during the topology optimization process. For the first challenge, new four‐node membrane element formulas with rotational DOFs, which can generate a diagonally dominant stiffness matrix, and an FEA modified model algorithm are proposed, and the form of a smooth elastic modulus matrix related to principal strain states is introduced. These measures ensure the diagonal dominance of the element stiffness matrix, and importantly relieve the FEA difficulty of bi‐modulus material structures with large differences in tensile and compressive moduli, thereby overcoming convergence issues and improving the accuracy of iterative calculations for displacements and strains. To improve computational efficiency, an algorithm based on the Gershgorin circle theorem is proposed to estimate the largest eigenvalue of a two‐level preconditioner matrix integrating diagonal and Chebyshev preconditioners. Subsequently, a novel diagonal‐Chebyshev bi‐level preconditioned conjugate gradient iterative method based on Graphics Processing Units is developed, which significantly reduces the computation time of bi‐modulus material structural optimization. Finally, two numerical examples are presented. Results demonstrate that the proposed method can generate reasonable optimal topologies for the aforementioned bi‐modulus material structures, further verifying the feasibility and superior computational efficiency of the proposed optimization methodology.
Plastic failure of the projectile is a key phenomenon in high-velocity impact events, which leads to shear sharpening for penetration and, in another way, induces the decrease of kinetic energy for protection. Thus, studying the plastic failure of the projectile is full of interest for impact engineering. The transient impact process involves complex physical origins, and this topic has a challenge in science and technology. In this work, a new physics-based numerical model of flat-ended projectile impacting semi-infinite metal target plate at a high velocity is proposed. The plastic deformation and failure of the projectile and the resultant mass loss, the pit-opening performance of the target plate as well as the energy distribution are theoretically calculated, and thus the impact response of both the projectile and target plate is revealed. With timing, the length, upsetting deformation and failure-induced mass loss of the projectile, the pit-opening depth of the target plate and the projectile-target contact force are quantitatively given. Correspondingly, the panoramic view of plastic failure behavior is quantitatively drawn. The numerical results show that the failure-induced mass loss of the projectile increases with the increase of initial impact velocity of the projectile that significantly dissipates the kinetic energy accompanying the plastic deformation of the projectile and pit-opening of the target plate. This numerical model is based on the intrinsic physical mechanisms and offers an effective computational method for quantitatively analyzing the high-velocity impact events.
ABSTRACT The phase‐field model is a developing diffusion method for the brittle crack problem in which the fourth‐order model with higher regularity improves the convergence. As well known, the main problem of phase‐field is the extremely high computational cost. In order to reduce the computational burden and accelerate the convergence, we propose an adaptive method on the continuous quadrilateral spline element to simulate the fourth‐order phase‐field model for the brittle fracture. The quadrilateral element QS‐12 has 12 degrees of freedom (DOFs), the function values and two partial derivatives at each vertex. The simplified stiffness matrix computations by the Bézier coefficients further reduce the calculation costs. Hanging nodes of refined hierarchical meshes are handled by the field variable transformation. Then the indicator for the adaptive process is chosen as the phase‐field variable, and hierarchical quadrilateral mesh or the quadtree mesh is used for the local refinement. Meanwhile, a staggered scheme with a hybrid formulation is applied to solve the phase and displacement field equations. Several benchmarks show the accuracy and robustness of the proposed method. Finally, we show the ability to calculate the complex crack behavior by the compression test of a rock‐like plate with double inclined flaws and a plate with random initial nucleation sites.
Limit analysis and yield design provide a well-defined mathematical framework for upscaling the strength properties of heterogeneous materials. These techniques can be incorporated into an FFT-based computational micromechanics framework to evaluate the strength of heterogeneous materials, based on images of their microstructure. However, the currently existing approaches are limited to rather specific forms of strength criteria, and their convergence for cohesive frictional materials was questioned. In the present contribution, we abstract and reformulate an existing kinematic upper-bound approach that relies on an Augmented Lagrangian and on Uzawa's algorithm, and link it to the traditional and damped ADMM. We show that the strength constraint may be enforced by means of an orthogonal projection operator, and provide the closed-form formulation of the projector for two cases: the Drucker-Prager and the Mohr-Coulomb strength criteria. We also discuss the link between the choice of the discretization and the bounding status of the solution. Finally, building upon the ADMM reformulation of the problem, we ascertain the sublinear convergence of the FFT-based strength homogenization method for a large class of convex strength criteria, which includes cohesive-frictional materials, provided that a projector is available, either analytically or algorithmically. A first set of numerical studies, performed on a toy example, highlights the role of the solver parameters and the discretization on the effectiveness and accuracy of the approach. The obtained FFT-based strength homogenization results are compared to state-of-the-art Finite-Element Limit-Analysis for validation and to discuss the bounding status of the solution. Then, computational experiments performed on industrial scale, potentially porous microstructures, show that FFT-based strength homogenization enables the analysis of the macroscopic strength anisotropy induced by the local geometric anisotropy of the microstructure using reasonable computational resources.
An online adaptive reduced order model (ROM) of the discontinuous Galerkin (DG) method is developed for predicting unsteady scale-resolved flow simulation. The least-squares Petrov-Galerkin (LSPG) projection is chosen as the baseline ROM framework, along with typical hyperreduction techniques for acceleration. Since LSPG requires multiplication operations of the Jacobian and the basis, a Jacobian-free approach is proposed for forming the low-dimensional ROM system, to keep consistent with the Jacobian-free strategy of the original implicit DG method. Then, a comprehensive online adaptation algorithm of the LSPG model is developed by updating the basis and sampling elements with snapshots generated by the full-order DG solver in an efficient way. The key idea for the adaptation is to update the ROM with the most recent flow information to predict the unseen features. Given a set of parameters, the proposed algorithm firstly runs the full-order DG solver for a short period, secondly generates the initial basis and reduced mesh, and finally runs the adaptive ROM for future-state predictions, which enables the model to possess predictive capability. Several benchmark cases have been conducted for verification and comparison to static ROMs. The chosen cases involve typical challenges, that is, transportation and discontinuity, for static ROMs, including the isentropic vortex convection, the Sod shock tube, the Kelvin-Helmholtz instability, and the two-dimensional Riemann problem. The results demonstrate that the adaptive ROM is able to effectively address the above challenges encountered by its static counterpart from a predictive perspective while achieving reasonable acceleration.
ABSTRACT The accurate prediction of displacement and stress fields in pressure vessels is essential for the safe and reliable design of these structures, particularly when dealing with nonlinear behavior such as that of hyperelastic functionally graded materials (FGMs). Conventional finite element method (FEM) approaches, while robust, often require extensive meshing and may encounter challenges in graded or strongly nonlinear domains. In this work, we present a Deep Energy Method (DEM) framework, based on physics‐informed neural networks, to analyze the axisymmetric deformation of ellipsoidal vessels subjected to internal and external pressure loading. The formulation employs automatic differentiation and a hyperelastic constitutive model to evaluate the strain‐energy functional, with pressure boundary conditions incorporated through natural work terms. Both homogeneous and exponentially graded materials are studied, enabling direct assessment of how material gradation influences the structural response. Three neural network architectures with varying depth and width are examined to evaluate the effect of model complexity on accuracy and convergence. The DEM solution is obtained by minimizing the total potential energy, comprising the internal strain energy and pressure work, to train a neural network representation of the displacement field. Convergence analyses confirm stable optimization for all cases, with graded vessels showing fast and smooth loss reduction due to the stabilizing effect of material gradation. DEM predictions of displacement agree closely with FEM benchmarks in both expansion‐type (internal pressure) and contraction‐type (external pressure) scenarios, accurately capturing global deformation modes as well as local responses near the inner and outer boundaries. To further ensure numerical robustness, a Monte Carlo sampling strategy is also employed for a selected test case, yielding displacement and von Mises stress fields fully consistent with structured‐grid DEM and FEM solutions. von Mises stress fields derived from DEM match FEM references for both homogeneous and graded vessels. While stress concentrations appear near loaded boundaries in homogeneous materials, gradation leads to smoother stress transitions, and DEM successfully reproduces these patterns. Qualitative comparisons and magnitude‐wise agreement across all network architectures demonstrate the method's capability to reflect nonlinear constitutive behavior and spatial material variation. The findings highlight several advantages of DEM: it provides smooth mesh‐free solutions, naturally accommodates material gradation, and embeds the variational energy principle directly within the learning process. The method shows strong potential for extension to more complex geometries and nonlinear multiphysics problems, although sensitivity to optimization parameters and training cost remain important considerations. It also offers a powerful and reliable alternative to FEM for the nonlinear analysis of homogeneous and functionally graded pressure vessels. The method not only reproduces FEM accuracy but also provides additional flexibility and robustness across different network architectures and sampling schemes, positioning DEM as a promising tool for next‐generation computational mechanics.
This paper presents a generalized shape function (gSF) approach for multi-material topology optimization that utilizes a compact design space to produce optimized configurations featuring a large number of materials. Building upon 1D (linear), 2D (bilinear), and 3D (trilinear) shape functions, generalized nD (n-linear) shape functions are conceptualized to map the multi-material simplex domain. Natural coordinates of these shape functions are considered the design variables used to determine the material densities. These densities are mathematically proven to satisfy the essential barycentric properties, guaranteeing a physically valid material interpolation space. Furthermore, we demonstrate that applying density filtering directly to the natural coordinates is mathematically equivalent to filtering the densities themselves, and that the tailored projection scheme preserves these vital barycentric properties in the projected states. The versatility, efficacy, and success of the gSF approach are demonstrated across various 2D and 3D stiff-structure (SS) and compliant-mechanism (CM) design problems. Strain energy is minimized for SS, whereas a multicriteria objective is minimized for CMs with given volume constraints. Sensitivity analysis is performed using the adjoint variable method, and the optimization problem is solved using the method of moving asymptotes. Results for SSs and CMs in 2D and 3D, respectively, up to 24 and 15 different materials, are presented. Objective history plots indicate smooth convergence. The proposed approach removes practical restrictions on the number of candidate materials, offering excellent scalability for large-scale engineering applications.
ABSTRACT Lattice systems are indispensable for modeling and analyzing physical phenomena in materials with discrete or heterogeneous micro‐ or meso‐structures. However, the computational requirements for practical engineering applications of lattice systems remain high. The quasicontinuum (QC) method addresses this by reducing the system of equations using a finite element mesh interpolation, rather than considering all nodes of a fully resolved lattice. Nevertheless, interfaces between separate phases in heterogeneous materials, such as concrete, require fine meshes throughout the domain, diminishing the effectiveness of QC. Enrichment strategies originally introduced in the extended finite element method (XFEM) can also account for material interfaces in discrete systems using nonconforming meshes, thereby resolving this issue. In a previous study, we applied Heaviside enrichment to investigate concrete mesostructures with the extended QC method, achieving a tenfold reduction in the number of unknowns while maintaining similar accuracy compared to discretizations with fully resolved interfaces. In the present study, we employ the meshless local maximum entropy (LME) interpolation, which transitions seamlessly from widespread meshfree to linear basis functions. Additionally, we combine LME interpolation with Heaviside enrichment and systematically investigate the role of the locality parameter and its optimization in heterogeneous lattices. This combination of optimized LME basis functions with Heaviside enrichment leads to an order‐of‐magnitude improvement in displacement accuracy while using the same number of degrees of freedom (DOF) compared to QC with linear interpolation. Moreover, we identify optimized distributions of the LME locality parameter and propose simple, non‐optimized rules that deliver comparable accuracy at a fraction of the computational cost. Results from three numerical examples show that the optimal locality‐parameter fields are nonuniform near interfaces and can be approximated by simple pattern‐based rules that retain much of the benefit of full optimization.
ABSTRACT Peridynamics (PD), similarly to some other nonlocal continuum theories, exhibits truncated interaction horizons near free surfaces, cracks, and voids in bounded domains. This loss of neighbors causes artificial surface effects and inconsistencies in the derivative and energy operators that persist under discretization refinement, limiting PD accuracy and robustness. First, the analysis and numerical isolation of the effects of horizon truncation on these operators are carried out, showing how it induces surface softening in bond‐based and stiffening/softening in state‐based PD formulations. Then, a purely PD‐based surface correction is proposed to restore full‐horizon behavior while preserving the original horizon radius and bond topology. Each node is assigned a scalar influence weight, and every bond is scaled by the average of the endpoint weights to maintain symmetry. The nodal influence weights are computed in a preprocessing optimization step enforcing agreement between truncated‐ and full‐horizon derivative and energy operators. Unlike existing approaches, the proposed method does not rely on ghost particles, variable horizons, or reference solutions from classical continuum mechanics (CCM). Benchmarks in one, two, and three dimensions, including a dynamic brittle fracture test, show that the optimized nodal influence weights reduce boundary‐induced surface effects, improve energy consistency, and accelerate convergence of PD solutions. Analytical evaluation of the calibration targets over full‐horizons also improves interior volume integration accuracy. The proposed scheme is simple to implement in existing PD codes, incurs only a modest preprocessing cost, performs robustly for flat and curved boundaries on both regular and irregular point sets, produces reliable crack paths, and applies to both bond‐based and ordinary state‐based PD formulations. An open‐source Python implementation, perifit , is provided.
