Mathematics

Research

28 pages, 6168 KB

Open AccessFeature PaperArticle

A Comprehensive Integral-Form Framework for the Stress-Driven Non-Local Model: The Role of Convolution Kernel, Regularization and Boundary Effects

by Luciano Feo, Giuseppe Lovisi and Rosa Penna

Mathematics 2026, 14(5), 872; https://doi.org/10.3390/math14050872 - 4 Mar 2026

Viewed by 401

Abstract

This manuscript presents a study of the Stress-Driven integral Model (SDM) for the bending response of Bernoulli–Euler nanobeams. Unlike conventional approaches that reformulate the nonlocal integral problem into an equivalent differential form, a direct numerical strategy is developed to solve the integral equation. [...] Read more.

This manuscript presents a study of the Stress-Driven integral Model (SDM) for the bending response of Bernoulli–Euler nanobeams. Unlike conventional approaches that reformulate the nonlocal integral problem into an equivalent differential form, a direct numerical strategy is developed to solve the integral equation. The proposed framework enables a systematic comparison of six different convolution kernels (Helmholtz, Gaussian, Lorentzian, triangular, Bessel and hyperbolic cosine), highlighting how their mathematical properties influence the structural response. To address issues related to long-range interactions and the potential ill-posedness of the integral operator, an adaptive regularization procedure based on the Morozov discrepancy principle is introduced. Furthermore, a local clipping and renormalization technique is proposed to properly account for boundary effects while preserving the weighted averaging property of the kernels. Validation against available analytical solutions for the Helmholtz kernel demonstrates high accuracy, with errors below 1%. The results show that the nonlocal parameter significantly affects structural rigidity depending on the kernel shape and that the proposed approach ensures consistent convergence to the local solution as the nonlocal parameter tends to zero. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Graphical abstract

22 pages, 4060 KB

Open AccessArticle

High-Performance Concrete Strength Regression Based on Machine Learning with Feature Contribution Visualization

by Lei Zhen, Chang Qu, Man-Lai Tang and Junping Yin

Mathematics 2025, 13(24), 3965; https://doi.org/10.3390/math13243965 - 12 Dec 2025

Viewed by 732

Abstract

Concrete compressive strength is a fundamental indicator of the mechanical properties of High-Performance Concrete (HPC) with multiple components. Traditionally, it is measured through laboratory tests, which are time-consuming and resource-intensive. Therefore, this study develops a machine learning-based regression framework to predict compressive strength, [...] Read more.

Concrete compressive strength is a fundamental indicator of the mechanical properties of High-Performance Concrete (HPC) with multiple components. Traditionally, it is measured through laboratory tests, which are time-consuming and resource-intensive. Therefore, this study develops a machine learning-based regression framework to predict compressive strength, aiming to reduce experimental costs and resource usage. Under three different data preprocessing strategies—raw data, standard score, and Box–Cox transformation—a selected set of high-performance ensemble models demonstrates excellent predictive capacity, with both the coefficient of determination (

R^{2}

) and explained variance score (EVS) exceeding 90% across all datasets, indicating high accuracy in compressive strength prediction. In particular, stacking ensemble (

R^{2}

-

0.920

,

EVS

-

0.920

), XGBoost regression (

R^{2}

-

0.920

,

EVS

-

0.920

), and HistGradientBoosting regression (

R^{2}

-

0.913

,

EVS

-

0.914

) based on Box–Cox transformation data show strong generalization capability and stability. Additionally, tree-based and boosting methods demonstrate high effectiveness in capturing complex feature interactions. Furthermore, this study presents an analytical workflow that enhances feature interpretability through visualization techniques—including Partial Dependence Plots (PDP), Individual Conditional Expectation (ICE), and SHapley Additive exPlanations (SHAP). These methods clarify the contribution of each feature and quantify the direction and magnitude of its impact on predictions. Overall, this approach supports automated concrete quality control, optimized mixture proportioning, and more sustainable construction practices. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Figure 1

27 pages, 3589 KB

Open AccessArticle

Damage and Failure Modeling of Composite Material Structures Using the Pam-Crash Code

by Eduardo Martin-Santos, Lucia G. Barbu and Pablo Cruz

Mathematics 2024, 12(23), 3847; https://doi.org/10.3390/math12233847 - 6 Dec 2024

Cited by 2 | Viewed by 2474

Abstract

Simulating composite material structures requires complex constitutive models, which normally require fine meshes to obtain an accurate prediction of their behavior. Pam-Crash software has been used for several years in the automotive industry and has been proved to be an efficient tool for [...] Read more.

Simulating composite material structures requires complex constitutive models, which normally require fine meshes to obtain an accurate prediction of their behavior. Pam-Crash software has been used for several years in the automotive industry and has been proved to be an efficient tool for simulating metallic structures, returning good correlations in a fast computational time. However, constitutive models for composite materials in Pam-Crash present some difficulties: some materials are not able to be suitably modeled and the predictive results depend on the mesh refinement. This work proposes a solution for predicting the progressive damage of composite materials in Pam-Crash, which scales the energy dissipated by the damage mechanisms and checks the viability of modeling the material behavior, taking into account the recommended size of finite elements in the automotive industry. The proposed solution is applied for the simulation of Open Hole specimens to evaluate the ultimate strength consistency. After this, it is applied for the simulation of Compact Tension specimens to check the consistency of crack propagation behavior. By considering the target size of the finite elements in the material card definition, the predictions demonstrate great improvement in the equivalence in results between different mesh refinements. Finally, the solution is applied to simulate impact tests on large structures. Good correlations with experimental data are obtained in fast computational times, making this methodology a candidate for application in composite-related automotive simulations. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Figure 1

23 pages, 4074 KB

Open AccessArticle

Efficient Structural Damage Detection with Minimal Input Data: Leveraging Fewer Sensors and Addressing Model Uncertainties

by Fredi Alegría, Eladio Martínez, Claudia Cortés-García, Quirino Estrada, Andrés Blanco-Ortega and Mario Ponce-Silva

Mathematics 2024, 12(21), 3362; https://doi.org/10.3390/math12213362 - 26 Oct 2024

Viewed by 2538

Abstract

In the field of structural damage detection through vibration measurements, most existing methods demand extensive data collection, including vibration readings at multiple levels, strain data, temperature measurements, and numerous vibration modes. These requirements result in high costs and complex instrumentation processes. Additionally, many [...] Read more.

In the field of structural damage detection through vibration measurements, most existing methods demand extensive data collection, including vibration readings at multiple levels, strain data, temperature measurements, and numerous vibration modes. These requirements result in high costs and complex instrumentation processes. Additionally, many approaches fail to account for model uncertainties, leading to significant discrepancies between the actual structure and its numerical reference model, thus compromising the accuracy of damage identification. This study introduces an innovative computational method aimed at minimizing data requirements, reducing instrumentation costs, and functioning with fewer vibration modes. By utilizing information from a single vibration sensor and at least three vibration modes, the method avoids the need for higher-mode excitation, which typically demands specialized equipment. The approach also incorporates model uncertainties related to geometry and mass distribution, improving the accuracy of damage detection. The computational method was validated on a steel frame structure under various damage conditions, categorized as single or multiple damage. The results indicate up to 100% accuracy in locating damage and up to 80% accuracy in estimating its severity. These findings demonstrate the method’s potential for detecting structural damage with limited data and at a significantly lower cost compared to conventional techniques. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Figure 1

19 pages, 419 KB

Open AccessArticle

Rayleigh Waves in a Thermoelastic Half-Space Coated by a Maxwell–Cattaneo Thermoelastic Layer

by Stan Chiriţă and Ciro D’Apice

Mathematics 2024, 12(18), 2885; https://doi.org/10.3390/math12182885 - 16 Sep 2024

Cited by 5 | Viewed by 1610

Abstract

This paper investigates the propagation of in-plane surface waves in a coated thermoelastic half-space. First, it investigates a special case where the surface layer is described by the Maxwell–Cattaneo thermoelastic approach, while the half-space is filled by a thermoelastic material described by the [...] Read more.

This paper investigates the propagation of in-plane surface waves in a coated thermoelastic half-space. First, it investigates a special case where the surface layer is described by the Maxwell–Cattaneo thermoelastic approach, while the half-space is filled by a thermoelastic material described by the classical Fourier law for the heat flux. The contact between the layer and the half-space is assumed to be welded, i.e., the displacements and the temperature, as well as the stresses and the heat flux are continuous through the interface of the layer and the half-space. The boundary and continuity conditions of the problem are formulated and then the exact dispersion relation of the surface waves is established. An illustrative numerical simulation is presented for the case of an aluminum thermoelastic layer coating a thermoelastic copper half-space, highlighting important aspects regarding the propagation of Rayleigh waves in such structures. The exact effective boundary conditions at the interface are also established replacing the entire effect of the layer on the half-space. The general case of the problem is also investigated when both the surface layer and the half-space are described by the Maxwell–Cattaneo thermoelasticity theory. This study helps to further understand the propagation characteristics of elastic waves in layered structures with thermal effects described by the Maxwell–Cattaneo approach. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Figure 1

27 pages, 552 KB

Open AccessFeature PaperArticle

Thermostatistics, Information, Subjectivity: Why Is This Association So Disturbing?

by Didier Lairez

Mathematics 2024, 12(10), 1498; https://doi.org/10.3390/math12101498 - 11 May 2024

Cited by 2 | Viewed by 1878

Abstract

Although information theory resolves the inconsistencies (known in the form of famous enigmas) of the traditional approach of thermostatistics, its place in the corresponding literature is not what it deserves. This article supports the idea that this is mainly due to epistemological rather [...] Read more.

Although information theory resolves the inconsistencies (known in the form of famous enigmas) of the traditional approach of thermostatistics, its place in the corresponding literature is not what it deserves. This article supports the idea that this is mainly due to epistemological rather than scientific reasons: the subjectivity introduced into physics is perceived as a problem. Here is an attempt to expose and clarify where exactly this subjectivity lies: in the representation of reality and in probabilistic inference, two aspects that have been integrated into the practice of science for a long time and which should no longer frighten anyone but have become explicit with information theory. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Figure 1

Review

Jump to: Research, Other

42 pages, 3989 KB

Open AccessFeature PaperReview

Numerical Analysis of Damage in Composites: From Intra-Layer to Delamination and Data-Assisted Methods

by Alireza Taherzadeh-Fard, Alejandro Cornejo, Sergio Jiménez and Lucia G. Barbu

Mathematics 2025, 13(10), 1578; https://doi.org/10.3390/math13101578 - 10 May 2025

Cited by 8 | Viewed by 4472

Abstract

The simulation of damage in composite materials is an important research area that impacts different engineering applications from aerospace structures to renewable energy systems. This review provides a comprehensive analysis of current damage modeling approaches, including intra-layer and inter-layer failures. Various numerical strategies, [...] Read more.

The simulation of damage in composite materials is an important research area that impacts different engineering applications from aerospace structures to renewable energy systems. This review provides a comprehensive analysis of current damage modeling approaches, including intra-layer and inter-layer failures. Various numerical strategies, such as continuum damage mechanics (CDM), cohesive zone models (CZM), extended finite element methods (XFEM), phase-field models (PFM), and peridynamics (PD), are examined to assess their efficiency in predicting crack initiation, propagation, and interaction. Additionally, the role of data-assisted (driven) techniques, such as machine learning, in enhancing predictive capabilities is explored. This review highlights the strengths and limitations of each approach, underscoring the need for further advancements in computational efficiency, multiscale modeling, and integration with experimental data. The findings serve as a foundation for future research into optimizing damage prediction techniques to improve the reliability and durability of composite structures. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Figure 1

Other

Jump to: Research, Review

51 pages, 972 KB

Open AccessSystematic Review

Variational Mechanics for Mining Infrastructure Design: A Systematic Review from Hamilton’s Principle to Physics-Constrained Optimization and Digital Twins

by Luis Rojas, Yuniel Martinez, Alex Paz, Alvaro Peña and José Garcia

Mathematics 2026, 14(4), 689; https://doi.org/10.3390/math14040689 - 15 Feb 2026

Viewed by 355

Abstract

This article presents a systematic synthesis of variationally grounded approaches for the design and optimization of mining structural infrastructure. This study is motivated by the critical need to ensure stiffness, reliability, and operational availability under severe loading, mass constraints, and aggressive environmental conditions. [...] Read more.

This article presents a systematic synthesis of variationally grounded approaches for the design and optimization of mining structural infrastructure. This study is motivated by the critical need to ensure stiffness, reliability, and operational availability under severe loading, mass constraints, and aggressive environmental conditions. Methodologically, the study situates structural modeling and synthesis within the continuity of the principle of stationary action. It demonstrates that, in the quasi-static regime, structural equilibrium is obtained as the stationarity of the total potential energy; consequently, the finite element method (FEM) arises naturally as a Ritz–Galerkin approximation of this underlying variational statement. On this basis, topology optimization is interpreted as a physics-constrained optimization problem wherein the design is posed as an outer optimality level acting over an energetically defined state. It is worth noting that SIMP-based formulations require explicit regularization to define the effective problem being solved. Emphasis is placed on the traceability between physical assumptions, discretization choices, regularization, and the resulting structural interpretations. The core contribution of this paper is a systematic literature review that consolidates evidence across variational mechanics, FEM-based optimization, and industrial applications, identifying recurrent methodological patterns and gaps that currently limit transfer to mining practice. Furthermore, a fully specified illustrative case is included to demonstrate reporting discipline and methodological consistency, rather than as a validation of a new optimization method. The conclusions highlight that a variational reading provides a coherent theoretical backbone for structural analysis, synthesis, simulation, and physics-based digital twins, while also clarifying the extensions required for industrial deployment, such as stability constraints, manufacturability, and multiphysics coupling within Mining 4.0 workflows. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Figure 1

40 pages, 2903 KB

Open AccessSystematic Review

Physics-Informed Neural Networks for the Structural Analysis and Monitoring of Railway Bridges: A Systematic Review

by Yuniel Martinez, Luis Rojas, Alvaro Peña, Matías Valenzuela and Jose Garcia

Mathematics 2025, 13(10), 1571; https://doi.org/10.3390/math13101571 - 10 May 2025

Cited by 13 | Viewed by 10219

Abstract

Physics-informed neural networks (PINNs) offer a mesh-free approach to solving partial differential equations (PDEs) with embedded physical constraints. Although PINNs have gained traction in various engineering fields, their adoption for railway bridge analysis remains under-explored. To address this gap, a systematic review was [...] Read more.

Physics-informed neural networks (PINNs) offer a mesh-free approach to solving partial differential equations (PDEs) with embedded physical constraints. Although PINNs have gained traction in various engineering fields, their adoption for railway bridge analysis remains under-explored. To address this gap, a systematic review was conducted across Scopus and Web of Science (2020–2025), filtering records by relevance, journal impact, and language. From an initial pool, 120 articles were selected and categorised into nine thematic clusters that encompass computational frameworks, hybrid integration with conventional solvers, and domain decomposition strategies. Through natural language processing (NLP) and trend mapping, this review evidences a growing but fragmented research landscape. PINNs demonstrate promising capabilities in load distribution modelling, structural health monitoring, and failure prediction, particularly under dynamic train loads on multi-span bridges. However, methodological gaps persist in large-scale simulations, plasticity modelling, and experimental validation. Future work should focus on scalable PINN architectures, refined modelling of inelastic behaviours, and real-time data assimilation, ensuring robustness and generalisability through interdisciplinary collaboration. Full article

(This article belongs to the Special Issue Advanced Computational Mechanics)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Advanced Computational Mechanics

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (9 papers)

Research

Review

Other

Further Information

Guidelines

MDPI Initiatives

Follow MDPI