Three-dimensional Nonlinear Model Research Articles

Physics-informed neural network estimation of material properties in soft tissue nonlinear biomechanical models

AbstractThe development of biophysical models for clinical applications is rapidly advancing in the research community, thanks to their predictive nature and their ability to assist the interpretation of clinical data. However, high-resolution and accurate multi-physics computational models are computationally expensive and their personalisation involves fine calibration of a large number of parameters, which may be space-dependent, challenging their clinical translation. In this work, we propose a new approach, which relies on the combination of physics-informed neural networks (PINNs) with three-dimensional soft tissue nonlinear biomechanical models, capable of reconstructing displacement fields and estimating heterogeneous patient-specific biophysical properties and secondary variables such as stresses and strains. The proposed learning algorithm encodes information from a limited amount of displacement and, in some cases, strain data, that can be routinely acquired in the clinical setting, and combines it with the physics of the problem, represented by a mathematical model based on partial differential equations, to regularise the problem and improve its convergence properties. Several benchmarks are presented to show the accuracy and robustness of the proposed method with respect to noise and model uncertainty and its great potential to enable the effective identification of patient-specific, heterogeneous physical properties, e.g. tissue stiffness properties. In particular, we demonstrate the capability of PINNs to detect the presence, location and severity of scar tissue, which is beneficial to develop personalised simulation models for disease diagnosis, especially for cardiac applications.

Dimension-reduced mathematical modeling of self-shaping wooden composite bilayers

ABSTRACT Swelling and shrinkage of wood provides challenges when used as construction material. On the positive side, it allows for manufacturing curved wood elements by self-shaping upon drying. We present a two-dimensional model for simulating the moisture-induced bending of wooden bilayers and more general composite plates with multiple, periodically distributed material phases. The model is derived in a mathematically rigorous way from a fully three-dimensional nonlinear hyperelasticity model of the plate and is computationally cheap to handle. We focus specifically on the scenario of free deformations. While the derived model is capable to describe general isometric bending deformations, it has the property that free deformations in equilibrium are given by uniaxial bending deformations that can be computed without the need for solving the plate equations via finite element simulations. The hereby gained computational efficiency enables extensive parameter studies to be conducted regarding the design of wooden composite plates with complex composite geometry. We validate the model for homogeneous wooden bilayers and obtain good agreement between simulations and experimental data. Then we use the model to predict the bending behavior of periodically perforated bilayers. The comparison to other existing models and implications for practical use are discussed.

Dynamic response of a permafrost railway subgrade with 3D train-track-subgrade-ground model simulations

Studying train-induced response characteristics is essential for safely operating permafrost railway subgrades. A three-dimensional thermal-mechanical coupling nonlinear dynamic model of train-track-subgrade-ground relationships was established to analyse the train-induced dynamic stress, acceleration and stress path characteristics of a permafrost railway subgrade, and field monitoring data were used to verify this model. The differences between the 2D and 3D models are also discussed, along with the seasonal changes, train speed, axle load, and train type affecting permafrost subgrades. The main results are as follows. First, the vibration load significantly impacts the subgrade 6 m below the sleeper, producing distinct vertical dynamic stress waves due to the wheels and bogies. Dynamic compression stress dominates the subgrade and is influenced by the train structure, speed, and sleeper spacing. While the 2D model tends to underestimate the dynamic stress in shallower layers, it concurs with the 3D model in deeper subgrade dynamics within a 10% margin of error. Then, the principal stress axis of the subgrade soil rotates synchronously with train movements, exhibiting regular stress paths in the YZ plane (longitudinal section) with depth-dependent variations in the stress cycles and deviatoric stress. Finally, predominantly originating from sleeper-induced vibrations, the subgrade vibration acceleration varies with the train speed, sleeper spacing, and season and is most pronounced in the vertical direction. This study provides theoretical guidance for the vibration response of permafrost subgrades on the Qinghai-Tibet Railway (QTR).

Anisotropic characteristics and creep model for thin-layered rock under true triaxial compression

The failure phenomenon of thin-layered rock tunnels not only exhibits asymmetric spatial characteristics, but also significant time-dependent characteristics under high in-situ stress, which is attributed to the time-dependent fracture of thin-layered rocks. This paper conducted a series of true triaxial creep compression tests on typical thin-layered rock siliceous slate with acoustic emission technique to reveal its anisotropic time-dependent fracture characteristics. The anisotropic long-term strength, creep fracturing process, and fracture orientation characteristics of thin-layered rocks under different loading angles (β, ω) and intermediate principal stress were summarized. A three-dimensional (3D) non-linear visco-plastic creep model for thin-layered rock was developed to simulate its anisotropic creep behavior. The time-dependent fracturing of rocks during true triaxial creep loading is reflected through the change of equivalent strain based on an improved Euler iteration method. By constructing the plastic potential function and overstress index related to loading angles and stress state, the anisotropic time-dependent fracturing process and propagation of thin-layered rocks under different loading angles and intermediate principal stress are expounded. The model was validated experimentally to show it can reflect the long-term strength and creep deformation characteristics of thin-layered rocks under true triaxial compression.

Fatigue crack growth and microstructural analysis of rail steel specimens under periodic overloads

The present paper aims to study the effects of periodic overloads on the fatigue crack growth and microstructure of railway rail specimens. Numerical analysis and experimental tests were exploited to examine the crack path in specimens. A three-dimensional nonlinear boundary element model was used to predict the crack path in specimens. The effect of several parameters on the fatigue life of the rail steel specimens were evaluated using numerical model and experimental tests. The numerical and experimental test results revealed that the fatigue life curve had an optimal overload ratio at which the fatigue life was maximized. The number of fixed cycles was kept fixed between the overloads, measuring the effect of the overload ratio on the fatigue crack growth. Also, the fractographic analysis showed that by increasing the stretching speed of the rail steel specimens, the amount of plastic deformation is reduced and the brittleness of the material is increased.

Numerical investigation of seismic amplification characteristics in loess ridge region of Xiji, northwest China.

The seismic effects on sloped terrain, which are of paramount importance for engineering design and earthquake risk mitigation, have always been a central focus of earthquake engineering research. In this study, generalized geometric models of loess ridges at varying heights were created, and a three-dimensional nonlinear numerical model was established using FLAC3D. Seismic ground motion time histories at different frequencies and actual earthquake ground motion records were input into the model to analyze the peak acceleration amplification effects experienced by the surface of loess ridges when subjected to SV waves. The study's outcomes reveal that seismic amplification on the slopes of loess ridges is characterized by non-linearity with respect to slope height. Instead, it exhibits rhythmic variations, with the rate of change in these rhythms increasing in correspondence with the frequency of seismic motion and the height of the slope. Under low-intensity seismic motion, a linear increase in acceleration amplification is observed at the ridge's crest concerning the height of the loess ridge. However, under high-intensity seismic motion, the relationship between amplification and slope height becomes less significant. Typically, the peak acceleration at the ridge's crest is reported to be 1.5 to 2.5 times that observed at the slope's base. The amplification effect at the ridge's crest is more pronounced in the low-frequency and high-frequency segments when compared to the mid-frequency range. Conversely, significant amplification is observed in the high-frequency range in the lower sections of the slope near the base. It is further noted that the amplification effect at the ridge's crest displays distinct behavior at different frequencies, characterized by narrow frequency bands of maximum amplification, with peak amplification factors exceeding 10 in some cases. These research findings have practical significance and provide valuable references for engineering construction and seismic risk mitigation planning in loess regions.

Investigation of the displacement-based seismic performance of geogrid earth-retaining walls using three-dimensional finite element modeling

This study evaluates the earthquake-induced movement of geogrid earth-retaining (GER) walls. A thorough investigation was conducted on a GER wall model, utilizing a comprehensive finite element (FE) analysis. This research focuses on investigating and designing hollow prefabricated concrete panels and conventional gravity-type stone masonry GER walls. It also displays comparative studies such as the displacement of the wall, deflection of the wall, lateral pressure of the wall, settlement of the backfill reinforcement, vertical pressure of the backfill, lateral pressure of the backfill, vertical settlement of the foundation, and settlements of soil layers across the height and acceleration of the walls of the GER walls. The FE simulations used a three-dimensional (3D) nonlinear dynamic FE model of full-scale GER walls. The seismic performance of models has also been examined in terms of wall height. It was found that the seismic motion significantly impacts the height of the GER walls. In addition, the validity of the proposed study model was assessed by comparing it to the conventional reinforcement concrete and gravity-type GRE wall and ASSHTO guidelines using finite element (FE) simulation results. Based on the findings, the hollow prefabricated concrete panels were the most practical alternative due to their lower deflection and displacement. Based on the observation, it was also found that the hollow prefabricated GER wall is the most viable option, as the settlement and lateral pressure in the former type are high.

A three-dimensional micropolar beam model with application to the finite deformation analysis of hard-magnetic soft beams

The main purpose of this contribution is to develop a three-dimensional (3D) nonlinear beam model based on the micropolar continuum theory. To do so, a kinematic model based on the deformation of three directors and accounting for the micro-rotation tensor of the micropolar theory is introduced. One of the main characteristics of the present beam model is that 3D constitutive equations without any modification can be directly used in the formulation. Furthermore, it is known that a body couple field is induced in hard-magnetic soft materials (HMSMs) when subjected to external magnetic fluxes. Therefore, the stress tensor in HMSMs is asymmetric, in general. Since the asymmetry of stress is one of the main features of the micropolar theory, the present formulation can be used for analyzing the deformation of beams made of HMSMs. Accordingly, the virtual external work of the present model is formulated so that it accounts for the contribution from uniform or constant-gradient external magnetic fluxes on the beam. Moreover, a Total Lagrangian (TL) nonlinear finite element (FE) formulation to provide numerical solutions of the related problems is developed. Several numerical examples are solved to investigate the capability of the developed formulation. It is shown that the present formulation can model the size-dependent behavior of beam-like structures if the material length-scale parameter of the micropolar constitutive model is comparable to the thickness of the beam. Moreover, the proposed model can successfully predict the finite deformation of 3D beams made of HMSMs subjected to magnetic loading.

Three-Dimensional Nonlinear Strength Model of Soil Considering Tenso–Shear Coupling

Three-Dimensional Nonlinear Strength Model of Soil Considering Tenso–Shear Coupling

A study on the nonlinear relationship between market, subsidy, and income of photovoltaic enterprises based on chaos theory

With the annual promotion of the international “dual carbon” goals, countries attach great importance to the development and innovation of clean energy. The United States, Japan, and China have all created many policies for the research and market development of photovoltaic energy. This article incorporates market dynamic regulation capability into a two-dimensional system of government subsidy policies and photovoltaic revenue, constructs a three-dimensional dynamic nonlinear model based on market dynamic regulation capability, government subsidies, and enterprise revenue, and numerically simulates and analyzes the impact of parameter and initial value changes in the equation on enterprise revenue. The market dynamic regulation capability is obtained from Chaotic attractors and dynamic evolution graphs of the nonlinear evolution between government subsidies and corporate profits in different scenarios. Research has shown that: (1) Rapidly improving the dynamic regulation ability of the market cannot continuously increase the revenue of the photovoltaic industry; (2) The changes in market dynamics affect the dependence of enterprises on government subsidies; (3) The demand for government subsidies by enterprises gradually decreases with the increase of their own profits.

Seismic Response Compression of Various MSE Walls Based on 3D Modeling

This study evaluates the earthquake-induced movement of mechanically stabilized earth (MSE) walls. A thorough investigation was conducted on an MSE wall model, utilizing a comprehensive finite element (FE) analysis. This research focuses on investigating and designing MSE walls made of reinforcement concrete and hollow precast concrete panels. It also involves comparative studies such as on the vertical pressure of the wall, horizontal pressure of the wall, lateral pressure of the wall, settlement of the wall, settlement of the backfill reinforcement, vertical pressure of the backfill, horizontal pressure of the backfill, lateral pressure of the backfill, vertical settlement of the foundation, and settlements of soil layers across the height of the MSE walls. The FE simulations used a three-dimensional (3D) nonlinear dynamic FE model of full-scale MSE walls. The seismic performance of MSE walls has also been examined in terms of wall height. It was found that the seismic motion significantly impacts the height of the walls. In addition, the validity of the proposed study model was assessed by comparing it to the reinforcement concrete wall and ASSHTO guidelines using finite element (FE) simulation results. Based on the findings, the hollow prefabricated MSE wall was the most practical alternative due to its lower displacement and settlement. The specifics of the modeling approach used in this study and the lessons learned serve as benchmarks for future comparable lines of inquiry and practitioners, especially as the computational power of desktop computers continues to rise.

Identification of spectrum of secondary acoustic-gravity waves in the middle and upper atmosphere in a high-resolution numerical model

Considerable attention has recently been paid to the study of so-called “secondary” acoustic-gravity waves (AGWs) that arise as a result of instability and nonlinear interactions of “primary” wave modes propagating from atmospheric sources, among themselves, and with the mean flow. In this paper, for the first time, the horizontal spatial spectra of primary and secondary AGWs are separated at fixed altitude levels in the middle and upper atmosphere at different time moments, which are simulated using a three-dimensional nonlinear high-resolution model AtmoSym. It is shown that in a short time after switching on the plane wave source at the lower boundary of the model, the spectrum consists of a peak corresponding to primary AGW and quasi-white noise generated by random atmospheric disturbances and the numerical model. Later, secondary peaks appear in the spectra at horizontal wave numbers, which are multiples of the wave numbers of primary AGW. The proposed separation of the spectra of primary and secondary AGWs makes it possible to estimate the relative contribution of secondary AGW at different altitudes, at different times, and with different stability of background temperature and wind profiles in the atmosphere.

Выделение спектра вторичных акустико-гравитационных волн в средней и верхней атмосфере в численной модели высокого разрешения

Considerable attention has recently been paid to the study of so-called “secondary” acoustic-gravity waves (AGWs) that arise as a result of instability and nonlinear interactions of “primary” wave modes propagating from atmospheric sources, among themselves, and with the mean flow. In this paper, for the first time, the horizontal spatial spectra of primary and secondary AGWs are separated at fixed altitude levels in the middle and upper atmosphere at different time moments, which are simulated using a three-dimensional nonlinear high-resolution model AtmoSym. It is shown that in a short time after switching on the plane wave source at the lower boundary of the model, the spectrum consists of a peak corresponding to primary AGW and quasi-white noise generated by random atmospheric disturbances and the numerical model. Later, secondary peaks appear in the spectra at horizontal wave numbers, which are multiples of the wave numbers of primary AGW. The proposed separation of the spectra of primary and secondary AGWs makes it possible to estimate the relative contribution of secondary AGW at different altitudes, at different times, and with different stability of background temperature and wind profiles in the atmosphere.

Effects of fling step on the seismic response of reinforced concrete arch bridges

Many arch bridges have been built in near-fault regions around the world and may be affected by fling step, which is the permanent ground displacement caused by the long-period pulse within the frequency content of an earthquake. This study investigates the effect of fling step on reinforced concrete deck-type arch bridges with short to medium spans. Three-dimensional non-linear models of three existing reinforced concrete arch bridges in Iran, with span lengths of 23, 35 and 60 m, were developed. The structures were subjected to seven near-fault earthquakes in longitudinal and transverse directions, once with the fling-step pulse included and once with it removed. The results showed that fling step has a noticeable effect on the seismic response indicators investigated, but may increase or decrease the seismic demand. The maximum curvature in the presence of fling step in relation to that without fling step ranged between 0.54 and 2.06. Unseating in the longitudinal direction was most sensitive to the fling step, and showed a more than 200% increase in some cases. Effects of the fling step were correlated with the ratio of the period of the fundamental mode of vibration of the structure to that of the fling-step pulse.

Innovative scheme for RC building progressive collapse prevention

Progressive collapse of reinforced concrete (RC) building due to column failure can cause human casualties and severe irreparable damage or total collapse of the building. Different methods have been presented to mitigated progressive collapse of RC buildings; however, most of them are installed externally after construction and are costly, demanding, and require manufacturing mitigating elements such as steel cables, steel devices, or composites. This study presents an innovative mitigating scheme to improve the progressive collapse resistance of such buildings. The proposed scheme applies during construction and comprises of adding short wasted leftover steel bars from the construction site with special configuration in the slab around potential failed interior column. To examine the effectiveness of the proposed scheme, three-dimensional nonlinear fiber element-based model was developed, validated, and utilized for the analysis of such buildings. The findings of the study show that the proposed scheme improved progressive collapse behavior of the building with a percentage increase in total dissipated energy up to 241.8%. The scheme is eco-efficient and cost effective as it requires minimum amount of material, saves energy required to produce new steel bars, and can serve as an effective solution to mitigate progressive collapse of RC buildings subjected to column failure.

Three-dimensional impact-angle-constrained fixed-time cooperative guidance algorithm with adjustable impact time

To address the problem wherein multiple missiles attack a target simultaneously in a three-dimensional space at an adjustable impact time and expected impact angles, this paper investigates a leader-following fixed-time cooperative guidance algorithm based on the three-dimensional nonlinear cooperative guidance model. First, in the leader's line-of-sight direction, the fixed-time convergence theory is introduced, and an impact time-controlled guidance algorithm is proposed so that the leader can accurately attack the target at any prespecified impact time. Then, in the line-of-sight direction of the followers, by utilizing the designed consensus error function and the fixed-time convergence theory, a time consensus tracking algorithm is investigated, which makes full use of the information interaction between missiles, and can achieve a rapid convergence of the time-to-go of the leader and followers within a fixed time. Next, in the normal and lateral directions to the line-of-sight, an innovative three-dimensional impact-angle-constrained guidance algorithm is designed by utilizing integral global sliding mode control technique, enabling the flexibility of guidance performance design. Meanwhile, during the design process of the impact-angle-constrained guidance algorithm, it eliminates the reaching phase of the traditional terminal sliding mode, lessens the energy consumption, and avoids the singularity phenomenon directly. In addition, the stability of the proposed algorithms is rigorously proved. Finally, numerical simulation demonstrates the effectiveness and superiority of the proposed guidance algorithm.

Effect of Lumbar Fusion and Pelvic Fixation Rigidity on Hip Joint Stress: A Finite Element Analysis.

This study compared hip stress among different types of lumbopelvic fusion based on finite element (FE) analysis. We believe that the number and placement of S2 alar iliac (AI) screws and whether the screws loosen likely influence hip joint stress in the FE model. Spinopelvic fixation has been shown to increase the risk of progression for hip joint osteoarthritis. The biomechanical mechanism is not well understood. We hypothesize that the rigid pelvic fixation may induce stress at adjacent joints. A three-dimensional nonlinear FE model was constructed from the L4 vertebra to the femoral bone. From the intact model, we made four fusion models, each with different lower vertebrae instrumentation: (1) intact, (2) L4-S1 fusion, (3) L4-S2 AI screw fixation, (4) L4-S2 AI screw fixation with S2 AI screw loosening, and (5) L4-S1 and dual sacral AI screw fixation. A compressive load of 400N was applied vertically to the L4 vertebra, followed by an additional 10Nm bending moment about different axes to simulate either flexion, extension, left lateral bending, or right axial rotation. The distal femoral bone was completely restrained. The von Mises stress and angular motion were analyzed across the hip joints within each fusion construct model. Hip joint cartilage stress and range of motion increased for all postures as pelvic fixation became more rigid. The dual sacral AI screw fixation model increased stress and angular motion at the hip joint more than intact model. Our results suggest that more rigid fixation of the pelvis induces additional stress on the hip joint, which may precipitate or accelerate adjacent joint disease. Dual sacral AI fixation led to the highest stress while loosening of S2 AI decreased stress on the hip joint. This study illustrates that more rigid fixation among lumbosacral fusion constructs increases biomechanical stress on the hip joints.

Performance of parallel double pipelines under non-uniform seismic excitation

This paper investigates the structural safety of parallel double-buried pipelines under non-uniform seismic excitation considering the soil-pipeline interaction. A three-dimensional nonlinear numerical model has been established to study the effects of seismic wave propagation and ground motion intensity, as well as the pipeline internal pressure and spacing between the pipelines on their structural safety. The dynamic response of the parallel double buried pipelines was evaluated in terms of the acceleration amplification, settlement of ground surface, and strain, displacement, and elliptical deformation of the pipeline. The results demonstrate that the seismic response of buried pipelines under non-uniform excitation has obvious nonlinear characteristics, especially when the loading intensity is high. It is found that the internal pressure has virtually no effect on the site acceleration response, but it amplifies the settlement of the ground surface. Meawhile, it inhibits the elliptic deformation of the pipeline. Furthermore, the variation of the elliptic deformation of parallel buried pipelines is found to be the same for different pipeline spacings considered in this study. On the other hand, the pipelines spacing has a significant influence on their seismic response as the interaction between the two pipelines gradually decreases with the increase in the spacing.

Study by finite element modeling of hydrogen atom diffusion and distribution at a dent on existing pipelines for hydrogen transport

Pipelines, especially existing natural gas pipelines, provide an efficient and economical means for hydrogen transport, accelerating realization of a full-scale hydrogen economy. However, the existing aged pipelines usually contain dents, which serve as effective traps to accumulate hydrogen atoms, potentially causing hydrogen embrittlement and pipeline failures. To date, there has been a strong interest in development techniques for assessment of hydrogen atom distribution at a dent on pipelines. In this work, a three-dimensional nonlinear finite element-based model was developed to investigate the hydrogen atom diffusion and distribution at a constrained dent on an X52 steel pipe. Parametric effects, including dent depth and internal pressure, were determined. The presence of a dent on pipelines drastically changes the local stress and strain distributions, facilitating diffusion and accumulation of hydrogen atoms at the dent. The minimum hydrogen atom concentration at the crystalline lattice sites, (i.e., 3.8 mol/m3) is located on the outer surface of the dent center, and the maximum hydrogen atom concentration (i.e., 12.1 mol/m3) is found on both sides of the dent with 103 mm from the dent center in the circumferential direction. The hydrogen atom concentration at trap sites at the dent is much lower than that at the lattice sites. The diffusion and accumulation of hydrogen atoms at the dent area is affected by two competitive mechanisms, i.e., hydrostatic stress and plastic strain, depending on the initial concentration of hydrogen atoms entering the interior of the pipe steel. At a low initial concentration such as 0.001 mol/m3, the equivalent plastic strain dominates the distribution of hydrogen atoms at the dent, and the hydrogen atom concentration at the lattice sites is much lower than the concentration at the trap sites. At high initial hydrogen atom concentrations such as 0.1 and 1 mol/m3, the hydrostatic stress is the dominant factor to determine the distribution of hydrogen atoms, which mainly accumulate at the lattice sites. With an increased depth of the dent, both the von Mises stress and plastic strain concentrations increase, and so is the hydrogen atom concentration. The increase of internal pressure from 5 MPa to 15 MPa only slightly elevates the von Mises stress, equivalent plastic strain and the hydrogen atom concentration.

Three-dimensional nonlinear model of rock creep under freeze-thaw cycles.

In areas with large differences between day and night temperature, the freeze-thaw cycle and frost heaving force in rock mass generate cracks within the rock, which seriously threatens the stability and safety of geotechnical engineering structures and surrounding buildings. This problem can be solved by developing a reasonable model that accurately represents the rock creep behavior. In this study, we developed a nonlinear viscoelastic-plastic creep damage model by introducing material parameters and a damage factor while connecting an elastomer, a viscosity elastomer, a Kelvin element, and a viscoelastic-plastic element in series. One- and three-dimensional creep equations were derived, and triaxial creep data were used to determine the model parameters and to validate the model. The results showed that the nonlinear viscoelastic-plastic creep damage model can accurately describe rock deformation in three creep stages under freeze-thaw cycles. In addition, the model can describe the time-dependent strain in the third stage. Parameters G1, G2, and η20' decrease exponentially with the increase in the number of freeze-thaw cycles while parameter λ increases exponentially. These results provide a theoretical basis for studying the deformation behavior and long-term stability of geotechnical engineering structures in areas with large diurnal temperature differences.