Nodal Displacements Research Articles

Computational nodal displacement analysis of acetabulum fossa for injection molded cemented polyethylene acetabular liner

The acetabular liner (AL) is one of the key components that determine the functionality and durability of the total hip joint replacement (THR) device. The performance of Ultra high molecular weight polyethylene (UHMWPE)-based AL depends critically on the manufacturing route and its properties, which are evaluated pre-clinically using a host of experimental and computational analyses. The conventional manufacturing of an AL involves multiple stages, including extrusion/compression molding followed by machining, which is time/cost intensive and leads to material loss. In such a scenario, injection molding is a promising alternative, yet its feasbility remains unexplored for the manufacturing of AL for THA applications. Against this backdrop, the two-fold objectives of this work are to report our recent efforts to establish the efficacy of the injection molding of new generation UHMWPE biomaterial; HU (60 wt% HDPE- 40 wt% UHMWPE blend) for manufacturing AL prototype and to present the key biomechanical response analysis of this prototype, in silico. A range of manufacturing relevant material properties, as well as customized mold design to manufacture HU-based AL with external design features, are discussed. Such guidelines are particularly relevant to mold polymeric parts with a higher thickness (>8 mm). As part of the pre-clinical validation of AL with new design features, a less explored in silico approach to assess biomechanical micro-strain in the acetabulum fossa is presented, and the results are analysed in accordance with the mechanostat theory. The outcomes revealed that for a 100 kg subject weight, average micro-strain in the remodelling region was 1132, while it was determined as 723 for a 55 kg subject weight. Such results highlight the influence of subject weight on micro-strain generation and distribution in the acetabulum fossa. The von Mises stress in AL also increased with subject weight from 17 MPa in a subject weight of 55 kg to 28 MPa in a subject weight of 100 kg. Taken together, this work demonstrates the feasibility and competence of this new generation biomaterial in terms of implant manufacturing via injection molding with a clinically desired biomechanical response.

Global sensitivity analysis of parameters based on sPCE: The case study of a concrete face rockfill dam in northwest China.

To effectively identify the key material parameters of different zones of concrete face rockfill dams and improve the efficiency of parameter optimization, a global sensitivity analysis method of parameters based on sparse polynomial chaotic expansion (sPCE) is proposed in this paper. The latin hypercube sampling method is used to select multiple groups of material parameters, and then finite element method is used to calculate the displacement of dam characteristic nodes in dam body. On this basis, the displacement is expanded by sPCE, and the polynomial basis function is reconstructed by orthogonal matching pursuit to improve the construction and analysis efficiency of the proxy model. According to the chaos coefficients, Sobol' indices are calculated to evaluate the influence of the material parameters and their interaction on different displacements of the dam. The results show that the sPCE model can accurately simulate dam displacement and its statistical characteristics with a relatively small sample size. The sensitivity of the same parameter has spatial variability, and under the influence of parameter levels and spatial distribution of different materials, the parameter sensitivity ranking of different zones has certain differences. The proposed method provides a new reference to sensitivity analysis and uncertainty analysis for practical engineering.

Research on Scheme Design and Structure Optimization of High Speed and High Precision Vehicle Contour Detection Platform

To improve road traffic safety and reduce the incidence of unauthorized vehicle modifications, overloading, and oversizing, this study proposed a detection platform structure to support the quick measurement of key dimensions and high-precision reconstruction of local features using the automotive contour size system. Using CAE technology, static and dynamic characteristic analysis was performed on the platform structure, and optimization design was conducted on the column frame and gantry frame separately. The optimized column frame's first-order modal frequency increased by 243% from 5.16 Hz to 17.65 Hz, with an average optimization ratio of 187% for the first six modal frequencies. For the optimized gantry frame, the average optimization ratio of the key node displacement exceeded 46%, with the first-order modal frequency increasing by 13% from 36.43 Hz to 41.33 Hz, and an average optimization ratio of 21% for the first six modal frequencies. Ultimately, the optimized detection platform's key nodes achieved an average optimization ratio of nearly 60%, with the first-order modal frequency increasing by 240% from 5.16 Hz to 17.57 Hz and an average optimization ratio of 185% for the first six modal frequencies. The optimized platform structure provides robust support for the high-speed and high-precision vehicle contour detection system's implementation.

Synergistic integration of deep neural networks and finite element method with applications of nonlinear large deformation biomechanics

Patient-specific finite element analysis (FEA) holds great promise in advancing the prognosis of cardiovascular diseases by providing detailed biomechanical insights such as high-fidelity stress and deformation on a patient-specific basis. Albeit feasible, FEA that incorporates three-dimensional, complex patient-specific geometry can be time-consuming and unsuitable for time-sensitive clinical applications. To mitigate this challenge, machine learning (ML) models, e.g., deep neural networks (DNNs), have been increasingly utilized as potential alternatives to finite element method (FEM) for biomechanical analysis. So far, efforts have been made in two main directions: (1) learning the input-to-output mapping of traditional FEM solvers and replacing FEM with data-driven ML surrogate models; (2) solving equilibrium equations using physics-informed loss functions of neural networks. While these two existing strategies have shown improved performance in terms of speed or scalability, ML models have not yet provided practical advantages over traditional FEM due to generalization issues. This has led us to the question: instead of abandoning or replacing the traditional FEM framework that can reliably solve biomechanical problems, can we integrate FEM and DNNs to enhance performance?In this study, we propose a synergistic integration of DNNs and FEM to overcome their individual limitations. Using biomechanical analysis of the human aorta as the test bed, we demonstrated two novel integrative strategies in forward and inverse problems. For the forward problem, we developed DNNs with state-of-the-art architectures to predict a nodal displacement field, and this initial DNN solution was then updated by a FEM-based refinement process, yielding a fast and accurate computing framework. For the inverse problem of heterogeneous material parameter identification, our method employs DNN as a regularizer of the spatial distribution of material parameters, aiding the optimizer in locating the optimal solution. In our demonstrative examples, despite that the DNN-only forward models yielded small displacement errors in most test cases; stress errors were considerably large, and for some test cases, the peak stress errors were greater than 50%. Our DNN-FEM integration eliminated these non-negligible errors in DNN-only models and was magnitudes faster than the FEM-only approach. Additionally, compared to FEM-only inverse method with errors greater than 50%, our DNN-FEM inverse approach significantly improved the parameter identification accuracy and reduced the errors to less than 1%.

Investigation on Buckling Performance of Prefabricated Light Steel Frame Materials under the Action of Random Defects during Construction.

This investigation proposes an analytical approach for analyzing the impact of random defects on light steel frame materials. The addition of random defects for the overall and the component units was achieved by integrating Matlab R2022a and Ansys R19.0 finite element software. Nonlinear analysis was conducted to calculate ultimate load factors and nodal ultimate displacements of the materials under various random defects at each stage of construction. A two-factor analysis was employed to investigate the effects of random defects on the calculation results during different construction stages. The investigation reveals that the response of the light steel frame materials to initial defects is more pronounced during the construction stage. Moreover, the construction stage is the main factor that affects the ultimate load factor and nodal ultimate displacement, compared with random defects. The influence of different random defects on structural displacements varies significantly. The displacement development of the light steel frame materials under the influence of component unit defects tends to be more rapid than that of the overall defects. However, their buckling critical loads are essentially similar.

Shaking table tests on the seismic response of truss structure with air spring‐FPS three‐dimensional isolation bearing

AbstractTo investigate the isolation effect of the friction pendulum system (FPS) horizontal isolation bearing and the air spring‐FPS three‐dimensional (3D) isolation bearing in truss structure and the influences of long‐period ground motions on the isolated structure, the shaking table test of a truss structure is designed and conducted. The ordinary ground motions, near‐fault pulse‐like ground motions, and far‐field long‐period ground motions with different spectral characteristics are chosen to input into the structure, and the dynamic characteristics such as the acceleration, displacement, and stress of the truss are compared and analyzed. The horizontal frequency of the horizontal isolated structure is reduced after the horizontal isolation, while the vertical frequency is further reduced after 3D isolation. The isolation effect of nodal acceleration, nodal displacement, and member stress under 3D isolation is better than that of horizontal isolation, especially the vertical seismic response is significantly improved. When the ground motion intensity is small, the FPS horizontal isolation bearing slides little, and reducing the seismic response of the structure is limited. There is a better isolation effect of 3D isolated structure. The vertical frequency ratio Rf under ordinary ground motions is far away from 1 and close to 3, basically avoiding the main energy segment of the ground motions. The advantage of 3D isolated structure is relatively reduced under long‐period ground motions, which lies in the rich low‐frequency components of long‐period ground motions which are difficult to isolate effectively.

Application of the NSGA-II Algorithm and Kriging Model to Optimise the Process Parameters for the Improvement of the Quality of Fresnel Lenses.

The Fresnel lens is an optical system consisting of a series of concentric diamond grooves. One surface of the lens is smooth, while the other is engraved with concentric circles of increasing size. Optical interference, diffraction, and sensitivity to the angle of incidence are used to design the microstructure on the lens surface. The imaging of the optical surface depends on its curvature. By reducing the thickness of the lens, light can still be focused at the same focal point as with a thicker lens. Previously, lenses, including Fresnel lenses, were made of glass due to material limitations. However, the traditional grinding and polishing methods for making Fresnel lenses were not only time-consuming, but also labour-intensive. As a result, costs were high. Later, a thermal pressing process using metal moulds was invented. However, the high surface tension of glass caused some detailed parts to be deformed during the pressing process, resulting in unsatisfactory Fresnel lens performance. In addition, the complex manufacturing process and unstable processing accuracy hindered mass production. This resulted in high prices and limited applications for Fresnel lenses. These factors prevented the widespread use of early Fresnel lenses. In contrast, polymer materials offer advantages, such as low density, light weight, high strength-to-weight ratios, and corrosion resistance. They are also cost effective and available in a wide range of grades. Polymer materials have gradually replaced optical glass and other materials in the manufacture of micro-optical lenses and other miniaturised devices. Therefore, this study focuses on investigating the manufacturing parameters of Fresnel lenses in the injection moulding process. We compare the quality of products obtained by two-stage injection moulding, injection compression moulding, and IMD (in-mould decoration) techniques. The results show that the optimal method is IMD, which reduces the nodal displacement on the Fresnel lens surface and improves the transmission performance. To achieve this, we first establish a Kriging model to correlate the process parameters with optimisation objectives, mapping the design parameters and optimisation objectives. Based on the Kriging model, we integrate the NSGA-II algorithm with the predictive model to obtain the Pareto optimal solutions. By analysing the Pareto frontier, we identify the best process parameters. Finally, it is determined that the average nodal displacement on the Fresnel surface is 0.393 mm, at a holding pressure of 320.35 MPa and a melt temperature of 251.40 °C. Combined with IMD technology, product testing shows a transmittance of 95.43% and an optimisation rate of 59.64%.

Design Optimization of Truss Structures Using a Graph Neural Network-Based Surrogate Model

One of the primary objectives of truss structure design optimization is to minimize the total weight by determining the optimal sizes of the truss members while ensuring structural stability and integrity against external loads. Trusses consist of pin joints connected by straight members, analogous to vertices and edges in a mathematical graph. This characteristic motivates the idea of representing truss joints and members as graph vertices and edges. In this study, a Graph Neural Network (GNN) is employed to exploit the benefits of graph representation and develop a GNN-based surrogate model integrated with a Particle Swarm Optimization (PSO) algorithm to approximate nodal displacements of trusses during the design optimization process. This approach enables the determination of the optimal cross-sectional areas of the truss members with fewer finite element model (FEM) analyses. The validity and effectiveness of the GNN-based optimization technique are assessed by comparing its results with those of a conventional FEM-based design optimization of three truss structures: a 10-bar planar truss, a 72-bar space truss, and a 200-bar planar truss. The results demonstrate the superiority of the GNN-based optimization, which can achieve the optimal solutions without violating constraints and at a faster rate, particularly for complex truss structures like the 200-bar planar truss problem.

Geometric learning for computational mechanics, Part III: Physics-constrained response surface of geometrically nonlinear shells

This paper presents a graph-manifold iterative algorithm to predict the configurations of geometrically exact shells subjected to external loading. The finite element solutions are first stored in a weighted graph where each graph node stores the nodal displacement and nodal director. This collection of solutions is embedded onto a low-dimensional latent space through a graph isomorphism encoder. This graph embedding step reduces the dimensionality of the nonlinear data and makes it easier for the response surface to be constructed. The decoder, in return, converts an element in the latent space back to a weighted graph that represents a finite element solution. As such, the deformed configuration of the shell can be obtained by decoding the predictions in the latent space without running extra finite element simulations. For engineering applications where the shell is often subjected to concentrated loads or a local portion of the shell structure is of particular interest, we use the solutions stored in a graph to reconstruct a smooth manifold where the balance laws are enforced to control the curvature of the shell. The resultant computer algorithm enjoys both the speed of the nonlinear dimensional reduced solver and the fidelity of the solutions at locations where it matters.

A Unit-Load Approach for Reliability-Based Design Optimization of Linear Structures under Random Loads and Boundary Conditions

The low order Taylor’s series expansion was employed in this study to estimate the reliability indices of the failure criteria for reliability-based design optimization of a linear static structure subjected to random loads and boundary conditions. By taking the advantage of the linear superposition principle, only a few analyses of the structure subjected to unit-loads are needed through the entire optimization process to produce acceptable results. Two structural examples are presented in this study to illustrate the effectiveness of the proposed approach for reliability-based design optimization: one deals with a truss structure subjected to random multiple point constraints, and the other conducts shape design optimization of a plane stress problem subjected to random point loads. Both examples were formulated and solved by the finite element method. The first example used the penalty method to reformulate the multiple point constraints as external loads, while the second example introduced an approach to propagate the uncertainty linearly from the nodal displacement vector to the nodal von Mises stress vector. The final designs obtained from the reliability-based design optimization were validated through Monte Carlo simulation. This validation process was completed with only four unit-load analyses for the first example and two for the second example.

Application of the Substructure Method in Direct Calculation of Hull Structures

To study the application of the substructure method in the direct calculation of ship hull structures, this paper uses MSC. PATRAN software for modeling. Based on the full ship model, the structure components corresponding to the units before the tail structure are deleted to form a finite element submodel that only contains the tail structure. The constraints and various design loads under each working condition are retained on the units of the tail section model. The node displacement at the “boundary” of the submodel obtained from the full ship model is used as its constraint condition to study the correctness and applicability of the finite element analysis of the tail section structure based on full ship FEA (Finite Element Analysis). Comparing the deformation and stress results of the stern of the main hull obtained from the full ship model and the tail section model, it is shown that the substructure method is correct and feasible.

The contact condition of viscoelastic contact problems

Three contact states in the viscoelastic contact deformation area between the rigid cylinder and the viscoelastic foundation are discussed based on various material mechanical properties between the rigid and the viscoelastic materials. On this basis, the relationship model between the node displacements and the node stresses in the boundary contact area is established, and the corresponding numerical analytical algorithms are used to simulate the viscoelastic contact deformation. Meanwhile, an example is given to simulate the contact deformation of the conveyor belt.

Multi-objective topology optimization design of truss structures based on evidence theory under limited information

Uncertainty information is often limited and has great discreteness, how to ensure the structure having good robustness under epistemic uncertainty becomes a technical problem for multi-objective topology optimization design. In this paper, the evidence theory is used to quantify the epistemic uncertainty; the plausibility measurement expressing upper limit of failure probability is applied to be a new evaluation of reliability. The total weight and the reliability of structures are taken as the optimization objectives, and a multi-objective robust topology optimization design model based on evidence theory is proposed. Parallelization technique based differential evolution for multi-objective optimization (DEMO) is preferable to search above robust Pareto front due to its merits of non-requirement of any gradient and superior mechanisms of non-dominate strategy. In order to verify the effectiveness of the proposed method, the multi-objective reliability optimization of a wood truss structure is implemented with considering elastic modulus and structural load as uncertain variables. According to the optimization results, six same truss specimens are made for the static random loading test. The failure probability of the truss structure is judged by the stress and node displacement obtained from the test, so as to verify the feasibility of the reliability optimization method based on evidence theory. The experimental results also indicate that the proposed method can avoid the deviation of the optimization results caused by the fluctuation of epistemic uncertainty, and provide a new method for designers to make the optimization results robust even when the data information is insufficient and the cognitive level is limited.

Comparative Analysis of the Mechanical Performance of Timber Flat Truss Typologies for Different Strength Classes via Optimization Algorithm

This study aimed to compare the mechanical performance of two truss typologies, the modified Fan and Howe trusses, using five different wood species: Cambará-rosa, Cupiúba, Angelim-pedra, Garapa, and Jatobá. The spans considered were 6, 9, 12, and 15 m, and the optimization algorithm used was a swarm intelligence algorithm to minimize the structure weight. The study found that wood is a sustainable and viable option for truss constructions, with different wood species having distinct mechanical properties that must be considered when designing the structure. The Howe truss typology presented lower results for the objective function in comparison with the modified Fan truss. The distribution of normal forces in the truss correlated with the characteristic strength in compression and tensile of the species. The Howe truss typology presented a larger amplitude in relation to the modified Fan truss typology in most of the conditions adopted for the Ultimate Limit State constraints. Constraints associated with deflection in the immediate condition were observed to limit the optimization process. The study used the matrix analysis method to evaluate internal efforts and nodal displacements in the trusses. In summary, this study demonstrates the importance of considering all constraints when dimensioning timber structures and the benefits of using different wood species for sustainable construction.

A new lateral force estimator for intelligent tires based on three-dimensional ring model

Tire lateral forces have an important impact on vehicle stability and vehicle ride comfort. In general, the intelligent tire lateral forces are estimated using lateral nodal displacements or data-based methods, which are easily disturbed by measurement noises and lack accuracy in the case of small test numbers. This paper proposes a model-based lateral force estimator for intelligent tires, which includes a mathematical model, a signal processing algorithm, and a Kalman Filter. As the core of the lateral force estimator, the mathematical model is proposed based on the three-dimensional ring model and can describe the analytical relationship between the lateral acceleration signal and the lateral force. Based on the mathematical model, the optimal state observation of the intelligent tire lateral forces is realized using a Kalman Filter. The performance of the lateral force estimator is validated through lateral slip tests under two different vertical loads performed on the MTS Flat Trac III test bench. The results show that the lateral force estimator’s average normalized root mean square (NRMS) error is 5.83%. Compared with the data-based and model-based methods in previous studies, this model-based lateral force estimator provides an efficient method to estimate the intelligent tire lateral forces using fewer tests.

Electromechanical coupling enriched finite element method for dynamic characteristic of piezoelectric materials structures

To improve the application depth and breadth of intelligent unit devices made from piezoelectric structures, their kinetic properties shall be urgently studied. Herein, based on the basic equations and boundary conditions of piezoelectric materials, an interpolation coverage function of node displacement was introduced into the displacement shape function and potential shape function of coupling electromechanical finite element method (FEM), forming an electromechanical coupling enriched FEM generalized shape function. Together with the variation principle, an electromechanical coupling enriched FEM was put forward, and its motion equations were deduced. Next, the free vibration and transient response of the piezoelectric structure were analyzed, and the accuracy and validity of the new method were validated with numerical cases. Results show this method has high application prospects for analyzing the dynamic properties of piezoelectric material structures.

Improved radial basis functions mesh deformation based on parallel point selection strategy and incremental LDLT decomposition

Mesh deformation based on radial basis functions (RBF) is widely used in CFD simulations with moving boundaries due to its simplicity, generality, and robustness, among which selecting an optimum reduced set from all boundary nodes is a crucial data reduction step for fewer support points. However, there remains a great need to reduce the computation costs for large-scale mesh. To solve this issue, a parallel point selection strategy and incremental LDLT decomposition method are proposed. With all boundary nodes distributed to all processors on average, the displacements of boundary nodes are calculated at local ones, and the nodes with local maximum error in divergent processors are added to the support point set in parallel. Besides, the matrix decomposition and solution of weight coefficients are conducted with full uses of corresponding results in the last iteration in incremental LDLT decomposition method. The calculation costs of conventional methods and proposed approaches are theoretically analyzed and compared. Three typical test cases of different mesh scales and motion modes are carried out for validation, including the rigid motion of a NACA0012 airfoil, the bending of an ONERA M6 wing, and the pitch of a missile. The results demonstrate that the proposed approaches greatly improve the efficiency of the data reduction procedure compared with the conventional methods, with similar deformation effect represented by support point distribution and deformed mesh quality.

Non-Destructive Damage Evaluation Based on Static Response for Beam-like Structures Considering Shear Deformation

Shear deformation plays an important role in certain structures, and neglecting shear deformation can affect the accuracy of structural response. This paper proposes a non-destructive damage evaluation method that considers shear deformation, based on static response, for identifying corrosion in beam-like structures. The influence of shear deformation on nodal displacement for simply supported beams with different cross-sections was analyzed. The results indicate that even small errors yield inaccurate identification results when neglecting shear deformation. To solve this problem, analytical displacements of the structure were determined based on the Timoshenko beam theory, and the objective function was established. Additionally, the damage identification results were obtained by minimizing the objective function using the interior point method. Several progressively complex examples were used to demonstrate the effectiveness of the proposed method in identifying damage in beam-like structures.

A Three-Stage Prediction Method for Track Displacement During Shield Tunneling

Track displacement is an important factor of track irregularity. Existing researches related with track displacement prediction generally ignore the influence from underground construction engineering such as shield tunneling, resulting in inaccurate estimation of track displacement. To fill this gap, we propose a three-stage framework to predict the track displacement when the shield tunnel under crosses the existing tunnel. Firstly, by considering the curved shield tunneling, a three-dimensional model is constructed to estimate the total ground displacement during the whole tunneling process. Furthermore, the soil-tunnel interaction model is established to estimate the deformation of the existing tunnel. To tackle the issue of unknown node displacements, cubic splines are used to interpolate the unknown values of tunnel displacements. On this basis, the direct stiffness method is used to estimate the track displacement and to calculate the track irregularity. Finally, the effectiveness of the proposed method is verified and the prediction performance on the track irregularity is evaluated using a real engineering case and the finite element simulation. The main contributions of this article lie in the modeling of the curved scenario for the estimation of the ground loss, as well as the combination of cubic splines and direct stiffness method, which improve the accuracy of the track displacement estimation during shield tunneling.

Construction of a reduced-order model based on tensor decomposition and its application to airbag deployment simulations

We present a construction method for reduced-order models (ROMs) to explore alternatives to numerical simulations. The proposed method can efficiently construct ROMs for non-linear problems with contact and impact behaviors by using tensor decomposition for factorizing multidimensional data and Akima-spline interpolation without tuning any parameters. First, we construct learning tensor data of nodal displacements or accelerations using finite element analysis with some representative parameter sets. Second, the data are decomposed into a set of mode matrices and one small core tensor using Tucker decomposition. Third, Akima-spline interpolation is applied to the mode matrices to predict values within the data range. Finally, the time history responses with new parameter sets are generated by multiplying the expanded mode matrices and small core tensor. The performance of the proposed method is studied by constructing ROMs for airbag impact simulations based on limited learning data. The proposed ROMs can accurately predict airbag deployment behavior even for new parameter sets using the Akima-spline interpolation scheme. Furthermore, an extremely high data compression ratio (more than 1000) and efficient predictions of the response surfaces and Pareto frontier (2000 times faster than that of full finite element analyses using all parameter sets) can be realized.