力学进展

AI for PDEs in solid mechanics: A review

WANG Yizheng, ZHUANG Xiaoying, TIMON Rabczuk, LIU Yinghua

2025, 55(2): 231-287. doi: 10.6052/1000-0992-24-016

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Abstract:
In recent years, deep learning has become ubiquitous and is empowering various fields. In particular, the combination of artificial intelligence and traditional science (AI for science, AI4Science) has attracted widespread attention. In the field of AI4Science, the use of artificial intelligence algorithms to solve partial differential equations (AI4PDEs) has become the focus of computational mechanics research. The core of AI4PDEs is to fuse data with equations and can solve almost any PDEs. Due to the advantages of AI4PDEs in data fusion, computational efficiency using AI4PDEs is usually increased by tens of thousands of times compared to traditional algorithms. Therefore, this article comprehensively reviews the research on AI4PDEs, summarizes the existing AI4PDEs algorithms and theories, discusses its application in solid mechanics, including forward and inverse problems, and outlines future research directions, especially the foundation model of computational mechanics. Existing algorithms of AI4PDEs include physics-informed neural networks (PINNs), deep energy methods (DEM), operator learning, and (physics-informed neural operator, PINO). AI4PDEs has numerous applications in scientific computing, and this paper focuses on application of AI4PDEs in the forward and inverse problems of solid mechanics. The forward problems include linear elasticity, elasto-plasticity, hyperelasticity, and fracture mechanics; while the inverse problems encompass the identification of material parameters, constitutive laws, defect recognition, and topology optimization. AI4PDEs represents a novel method of scientific simulation, which offers approximate solutions for specific problems by leveraging large datasets and then fine-tunes according to the specific physical equations, avoiding the need to start calculations from scratch as traditional algorithms do. Thus, AI4PDEs is a prototype for the foundation model of computational mechanics in the future, capable of significantly accelerating traditional numerical methods. We believe that utilizing artificial intelligence to empower scientific computing is not only a vital direction for the future of computation but also a dawn of humanity in scientific research, laying the foundation for mankind to reach new heights in scientific development.

Research progress and performance evaluations for self-starting single-solve explicit time integrators

LI Jinze, LIU Yaokun, YU Kaiping, CUI Naigang

2025, 55(2): 288-339. doi: 10.6052/1000-0992-24-030

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Direct time integration methods play a critical role in the numerical computation of large-scale nonlinear dynamic systems, particularly in the field of engineering simulation and design. The self-starting single-solve explicit time integration methods have become essential tools in this domain due to their efficiency and reliability in handling complex nonlinear systems. However, as these algorithms continue to evolve and diversify, their performance varies significantly, underscoring the urgent need for a systematic review and in-depth analysis of their capabilities. This paper first introduces the key performance metrics for evaluating time integration methods, including accuracy, stability, amplitude and phase error, providing a theoretical foundation for readers. It then offers a detailed review of the development of self-starting single-solve explicit time integration methods, systematically tracing the evolution of various algorithms. Finally, the performance of several self-starting single-solve explicit methods is compared in terms of spectral properties, accuracy, stability, and error characteristics, with numerical verification performed using typical examples and engineering structures. The paper highlights two explicit methods that currently exhibit superior performance: The fully explicit GSSE method and the velocity-implicit GSSI method. Both methods are characterized by their self-starting capability, single-solution, explicitness, maximized conditional stability, controllable numerical dissipation (over the full range), and identical second-order accuracy. The primary distinction between the two lies in the computational effort required for damping problems and the size of the conditional stability domain in the presence of damping. The paper also explores future research directions for explicit time integration methods, emphasizing the potential for further optimization and development.

Cellular mechanical intelligence

CHENG Bo, LU Mengnan, JIA Yuanbo, XU Feng

2025, 55(2): 340-377. doi: 10.6052/1000-0992-24-028

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Biological intelligence, which includes features such as perception, memory, learning, problem-solving and decision-making, is widely observed in humans, animals and other higher organisms with nervous systems. Recent studies have shown that single cells also exhibit behaviours that resemble human-like intelligence in their interactions with the microenvironment, such as “multimodal perception”, “problem solving”, “learning and memory”, and “evolutionary adaptation”. Cellular intelligence, as a newly proposed and disruptive theoretical concept, raises fundamental questions, including the principles underlying the emergence of cellular intelligence, the mechanisms by which collective cell behaviour emerges as collective intelligence, and the evolutionary drivers for single cells to evolve into multicellular life forms. As the fields of biomechanics and mechanobiology have advanced, numerous studies have demonstrated the significant influence of the mechanical microenvironment on cellular physiological behaviour. Under mechanical stimulation, even single cells exhibit intelligent behaviours similar to those observed in higher organisms. Based on this, the concept of “cellular mechanical intelligence” is proposed in this paper. We summarise the characteristics of intelligent behaviours in terms of mechanical perception, mechanical decision making, mechanical memory and mechanical learning, with the aim of providing new insights and perspectives on the mechanisms underlying cellular mechanical intelligence and its potential applications, such as in cellular intelligent medicine.

Research advances in wind-train-bridge coupling vibration

HAN Yan, HU Peng, WANG Lidong, LIU Hanyun, CAI Chunsheng

2025, 55(2): 378-418. doi: 10.6052/1000-0992-24-032

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Researching the coupling vibration of wind-train-bridge systems is one of the crucial means to ensure the safe operation of trains on bridges under strong wind conditions. In the last two decades, domestic and foreign scholars have conducted extensive research in this field, accumulating fruitful research results. The study of the coupling vibration of the wind-train-bridge system mainly includes three aspects: the aerodynamic characteristics of the train-bridge system, the calculation of the coupling vibration of the wind-train-bridge system, and the evaluation, prevention, and control of driving safety. Firstly, the primary purpose of studying the aerodynamic characteristics of the train-bridge system is to analyze the aerodynamic interference mechanism of the train-bridge system and accurately obtain the wind loads acting on the train and bridge. Depending on whether the train on the bridge is stationary or moving, it can be divided into two cases, i.e., stationary train and moving train. Secondly, on the basis of obtaining wind loads on the train-bridge system, it is necessary to solve the dynamic response of the wind-train-bridge system to obtain the response time-history curve of the entire process of the train passing through the bridge. This research mainly involves modeling methods for the train-bridge system, solving theories of wind-train-bridge coupling vibration equations, and using efficient calculation methods. Thirdly, based on the obtained dynamic response, the ultimate goal is to evaluate the driving safety of trains on the bridge and propose prevention and control measures. This research mainly includes the evaluation indicators and methods of driving safety and the main preventive and control measures for ensuring the driving safety of trains under strong wind conditions. Finally, in conjunction with the main issues existing in the field of the coupling vibration of wind-train-bridge systems, prospects for future research directions are provided to promote the development of this research field.

Quantum computing: The new focus in computational mechanics

XU Yongchun, KUANG Zengtao, HUANG Qun, YANG Jie, HU Heng

2025, 55(2): 419-430. doi: 10.6052/1000-0992-24-039

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Quantum computing has the potential to exponentially surpass classical computing in terms of computational power, but its practical applications need further expansion. At the same time, computational mechanics offers a wide range of applications, but faces challenges of significant computational power requirements arising from multi-scale, multi-physics, and extreme conditions, among others. Therefore, the complementary development of quantum computing and computational mechanics holds great promise. This paper reviews the current state of quantum computing applications in computational mechanics and discusses future trends in this field.

Stress or Strain?

LI Shuguang

2025, 55(2): 431-453. doi: 10.6052/1000-0992-24-035

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This paper is intended to reconcile the stress-based and strain-based formulations for material failure criteria, where a longstanding and deep division is present. The two approaches do not naturally agree with each other, and they not genuinely complement each other, either. Most popular criteria are stress-based when originally proposed, including the maximum stress, Tresca, von Mises, Raghava-Caddell-Yeh and the Mohr criteria. Their formulations are unique and self-consistent, i.e., capable of reproducing the input data. Their strain-based counterparts, with the maximum strain criterion being considered as the strain-based counterpart of the maximum stress criterion, are neither unique nor necessarily self-consistent. It has been proven that the self-consistent ones reproduce their respective stress-based counterparts identically in effect with a disadvantage of requiring an additional material property to apply, without a single benefit. For the Mohr criterion as a special case, a strain-based counterpart is simply infeasible in general. All undesirable features of strain-based criteria are rooted in a single source: The failure strains can only be measured under a uniaxial stress state, which corresponds to a combined strain state in general, not a uniaxial strain state! Given the arguments presented, the reconciliation proves to be biased completely towards the stress-based side if mathematics, logic and common sense prevail over perception and prejudice.

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2025, Volume 55, Issue 2

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Current Issue 2025, Volume 55, Issue 2

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Current Issue
2025, Volume 55, Issue 2