Numerical Analysis Applications in Improving Engineering Models

Abstract

Numerical analysis serves as the fundamental pillar of modern engineering modeling, providing the mathematical and computational tools necessary to transform complex physical equations into practical numerical solutions. This study reviews contemporary applications of numerical analysis in enhancing engineering models, focusing on methodological advancements in solving partial differential equations, data-driven model order reduction techniques, adaptive mesh methods, and machine learning-based surrogate models. The study employs a synthetic analytical approach that combines a review of recent literature with an analysis of empirical results from various numerical methods, addressing hybrid models that integrate traditional techniques with artificial intelligence. The analysis demonstrates that combining traditional numerical analysis methods with artificial intelligence and reduced-order models has revolutionized the ability of engineers to simulate complex multiphysics systems with unprecedented accuracy and computational efficiency; hybrid models have achieved computational speedups ranging from 72 to 522 times compare to traditional methods. Furthermore, the study addresses challenges related to computational cost and solution sensitivity to initial conditions, examining cutting-edge innovative solutions such as physics-informed neural networks and causality-based learning. The study concludes that future trends in this field are shifting towards the development of hybrid models that combine analytical accuracy with computational speed, with an increasing emphasis on generalization capabilities beyond the training data. This opens up broad prospects for digital engineering and digital twin applications across the industrial, energy, and infrastructure sectors. The study offers a comprehensive perspective on the development of engineering models using advanced numerical analysis techniques and identifies promising future research directions in this vital field.

Introduction

Numerical analysis is an essential tool in engineering because it provides approximate solutions to complex mathematical problems that cannot be solved efficiently using traditional analytical methods. Techniques such as the Finite Element Method (FEM) and Finite Difference Method (FDM) are widely used in structural analysis, fluid dynamics, heat transfer, and other engineering applications. However, these methods can require substantial computational resources, particularly when dealing with complex geometries, multiphysics systems, and repeated simulations for design optimization.

This study reviews recent advancements in numerical analysis and its integration with machine learning to improve computational efficiency, accuracy, and predictive capability. It focuses on four major approaches: adaptive mesh refinement, Model Order Reduction (MOR), data-driven surrogate models, and Physics-Informed Neural Networks (PINNs).

Adaptive mesh refinement improves computational efficiency by concentrating resources in regions where solutions change rapidly. Model Order Reduction simplifies high-dimensional systems into lower-dimensional representations, while surrogate models approximate computationally expensive simulations and enable faster predictions. PINNs combine neural networks with governing physical equations to solve complex engineering problems while incorporating physical constraints into the learning process.

The literature review highlights several promising developments. Reduced-order models combined with adaptive meshing can decrease the number of computational degrees of freedom. Kriging-based surrogate models have demonstrated substantial reductions in simulation time for structural analysis and model updating. Meanwhile, advanced physics-informed methods, including multi-region learning architectures and Physics-Informed Extreme Learning Machines (PIELMs), have achieved significant computational speedups while maintaining accuracy in certain applications. However, PINNs still face challenges involving discontinuities, shock waves, oscillatory solutions, and complex multi-domain problems.

The study identifies a research gap in the lack of unified comparative frameworks for evaluating traditional numerical methods and machine learning-based approaches across diverse engineering applications. It therefore aims to systematically compare their strengths, limitations, computational costs, accuracy, and generalization capabilities.

In conclusion, integrating numerical analysis with machine learning, surrogate modeling, adaptive meshing, and model reduction offers significant potential for faster and more efficient engineering simulations. The most suitable technique depends on the problem's complexity, required accuracy, computational resources, and ability to generalize to new conditions. The study provides a foundation for selecting appropriate numerical and data-driven methods for engineering design, optimization, and real-time modeling.

Conclusion

1) Localized Grid Adaptation: Hierarchical partitioning of the parameter space, combined with anisotropic grid adaptation, achieved a 40% to 70% reduction in the number of degrees of freedom while maintaining accuracy. 2) Surrogate Models: The Kriging model, integrated with a genetic algorithm, proved effective for updating engineering models, achieving a computational speedup of up to 72 times with an average relative error of less than 2%. 3) Physics-Informed Neural Networks (PINNs): The RBF-PIELM model achieved a computational speedup of over 350 times compared to traditional PINNs, while the MRT-PIELM model demonstrated a speedup ranging from 114 to 522 times for multi-domain problems. 4) Handling Discontinuities: A hybrid approach—combining artificial viscosity with loss function reweighting—exhibited the best performance in addressing shock waves.

References

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Copyright

Copyright © 2026 Gadriah Jamaah Ali Madi . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.