Complex Mechanical Problems Research Articles

A Kronecker product accelerated efficient sparse Gaussian Process (E-SGP) for flow emulation

In this paper, we introduce an efficient sparse Gaussian process (E-SGP) for the surrogate modelling of fluid mechanics. This novel Bayesian machine learning algorithm enables efficient model training using databases of different structures. It extends the approximated sparse GP algorithm by combining the concept of efficient GP (E-GP) and variational energy free sparse Gaussian processes (VEF-SGP). The developed E-SGP approach leverages the arbitrariness of inducing points and the monotonically increasing nature of the objective function with respect to the number of inducing points in VEF-SGP. By specifying the inducing points on the orthogonal grid/input subspace and using the Kronecker product, E-SGP significantly enhances computational efficiency without imposing constraints on the covariance matrix or increasing the number of parameters that need to be optimised during training.The E-SGP algorithm outperforms E-GP not only in terms of scalability but also in model quality, as evidenced by mean standardised logarithmic loss (MSLL). The computational complexity of E-GP grows cubically with an increasing structured training database, while E-SGP maintains computational efficiency as the model resolution (i.e., the number of inducing points) remains the same. The examples show that E-SGP produces more accurate predictions compared to E-GP when model resolutions are similar. In the application to a partially structured database, E-GP benefits from more training data but comes with higher computational demands, while E-SGP achieves a comparable level of accuracy but is more computationally efficient, making E-SGP a potentially preferable choice for fluid mechanics. Furthermore, E-SGP can produce more reasonable estimates of model uncertainty, whilst E-GP is more likely to produce over-confident predictions. In the application to partially structured databases, the performance of E-SGP is also compared to the structured Gaussian processes latent variable model (SGPLVM). In this case, E-SGP and SGPLVM both exhibit robust performance. However, E-SGP demonstrates a slight advantage over SGPLVM, particularly in terms of MSLL, indicating that E-SGP is better at producing trustworthy and accurate estimates of the uncertainty in the predictions. . While SGPLVM also improves computational efficiency by exploiting the Kronecker product and a small number of inducing points, E-SGP's orthogonal grid specification of inducing points further enhances its scalability and robustness, making it a superior choice for complex fluid mechanics problems with partially structured data.

Spatio-temporal physics-informed neural networks to solve boundary value problems for classical and gradient-enhanced continua

Recent advances have prominently highlighted physics informed neural networks (PINNs) as an efficient methodology for solving partial differential equations (PDEs). The present paper proposes a proof of concept exploring the use of PINNs as an alternative to finite element (FE) solvers in both classical and gradient-enhanced solid mechanics. To this end, spatio-temporal PINNs are designed to represent continuous solutions of boundary value problems within spatio-temporal space. These PINNs directly incorporate the equilibrium and constitutive equations in their differential and rate forms, bypassing the requirement for incremental implementation. This simplifies application of PINNs to solve complex mechanical problems, particularly those involved in the context of gradient-enhanced continua. Moreover, traditional meshing is no longer required as it is replaced by a point cloud, making it possible to overcome meshing drawbacks. The results of this investigation prove the effectiveness of the proposed methodology, especially with regards to non-monotonic loading conditions and irreversible plastic deformation. Compared to classical FE approaches, the proposed spatio-temporal PINNs are more readily applied to complex problems, which are tackled in their raw form. This is especially true for gradient-enhanced continuum problems, where there is no need to introduce additional degrees of freedom as in classical FE approaches. However, PINNs training generally requires more computation time, a challenge that can be mitigated by employing the concept of transfer learning as shown in this paper. This concept, which is very useful when performing parametric studies, involves applying knowledge grained from solving one problem to another different but related one. The use of PINNs as mechanical solvers is shown to be highly promising in the forthcoming era, where advancements in GPU technology can further enhance their performance in terms of computation time.

Analytic analysis of free vibration problem of the plate with a rectangular cutout using symplectic superposition method combined with domain decomposition technique

Plates with rectangular cutouts is widely seen in the field of engineering structures. Therefore, it is crucial to examine analytical solutions for free vibration (FV) of these structures. Despite the existence of approximate/numerical methods, analytical solutions are lacking in the literature. In this study, we employ the symplectic superposition method to effectively analyze the FV problems encountered in plates with rectangular cutouts while integrating the domain decomposition technique. To address the issue of irregular geometry, the rectangular cutout plate is divided into multiple sub-plates. By dividing the problem into multiple sub-problems and solving them separately using variable separation and symplectic eigen expansion, we obtain analytical solutions. Finally, we combine the sub-problem solutions to resolve the initial issue. This solution method can be considered a logical, analytical, and systematic approach as it starts with the fundamental governing equation and is derived without assuming forms of solutions. The study presents a comprehensive set of numerical results that include mode shapes (MSs) and natural frequencies (NFs). The results are rigorously validated using the finite element method (FEM) and relevant literature. The symplectic superposition method demonstrates excellent convergence and precise accuracy, making it suitable for analytically modeling more complex mechanical problems of plates.

ANALISIS MEKANIKA LAGRANGE PADA BENDA TEGAR DENGAN PENDEKATAN DAN PENERAPANNYA PADA SISTEM FISIKA

The Lagrange method is a method used to solve complex mechanical problems by utilizing kinetic and potential energy. This research examines the application of the Lagrange method to rigid body systems, where the position and orientation of the body are determined by general coordinates. This article uses a literature study approach from various relevant sources. The research results show that the Lagrange method is effective in solving mechanical problems in rigid body systems that cannot be solved using Newton's method.

Numerical Analysis of Residual Stresses Effect in Multi-materials

Background: In this study, the effect of mechanical and physical properties of the metal and temperature is highlighted. The purpose of this study is to analyze the effect of these stresses on multi-materials. This work aimed to conduct a three-dimensional numerical study by the finite element method on the levels and distributions of the stresses in the Al2O3/NI/HAYNES multi-materials. These stresses of thermal and mechanical origin are generally detrimental to the service life of multi-material. Methods: The use of numerical resolution by finite element method is the most suitable for complex mechanical problems. It allows a more in-depth analysis of all points of the structure. In this study, a fundamental tool was constructed to resolve the mechanical behavior of materials subjected to complex solicitations. Therefore, the ABAQUS calculation code version 6.14 was used to analyze the residual stresses. Results: By interpreting the results in terms of stress variation, we identified the areas at risk; in particular, the nature of a joint effect plays a decisive role in the assembly of the right material in its mechanical resistance. This nature is defined in terms of stiffness (Young's modulus) and differential expansion (coefficient of thermal expansion). Conclusion: The presence of strong residual stresses can constitute a risk of damage to several materials. The difference between the coefficients of thermal expansion of the two materials (metal and ceramic) linked together induces, in these two constituents, normal internal stresses. This difference determines the level and distribution of these constraints. Moreover, the sign of this difference determines the state of the normal stresses.

Estimation of stress-strain state of a gas pipeline section with different types of defects

This work is aimed at assessing the stress-strain state of a gas pipeline section with different types of defects. The purpose of the calculations is to determine the reliability of the pressurized structure and to assess its serviceability in future operation. The paper presents an assessment of the “dent” type defects and a non-through surface fracture, which were obtained as a result of impact loading. The calculations were performed in the ANSYS mathematical modelling software package, which is a multipurpose package for solving complex physics and mechanics problems based on the finite element method. A model of the gas pipeline section was created in the vicinity of which the defect is located. Preliminary equivalent stresses at normal operation of the object were calculated in this section, as a result of which the place was found, which has the highest load at normal operation. In this zone, defects are applied programmatically. In the process of research, it is revealed that at given geometric parameters, the considered defects do not pose a threat to further operation of the object. The considered assessment method is well suited for objects of different complexity and size, so the scope of application is vast - from machine parts to huge structures.

Extended Applications of the δ-SPH Model for the Numerical Study of Fluid–Soil Interactions

The fluid–soil interactions play a significant role in coastal and ocean engineering applications. However, there are still some complex mechanical problems with large deformations of water–soil interfaces to be solved. As a particle-based Lagrangian method, Smoothed Particle Hydrodynamics (SPH) is good at solving multiphase problems with large deformations of boundaries or interfaces. Therefore, in this work, the [Formula: see text]-SPH method is extended for the simulation of fluid–soil interacting problems. First, based on the weakly compressible assumption, the water is modeled as a viscous fluid while the soil is considered as a material with elastic–perfectly plastic behaviors. The [Formula: see text]-SPH method is implemented on the two phases separately, while the stress diffusive term only acts on the soil. The seepage force is introduced to model the interaction between two phases. After that, several numerical test cases with small to large interface deformations are presented. It is shown that the fluid–soil interacting model based on the [Formula: see text]-SPH model gives satisfying results compared with experimental data. Finally, the model is further extended for the simulation of vertical or oblique water jet scouring problems which demonstrates the potential applications of the SPH model for complex engineering problems.

Carbon nanotubes at the Graphene/hBN interface abnormally enhance its fracture toughness

Different from Van der Waals heterojunction, defects often appear at the interface of the planar heterostructures due to the low-dimensional structure characteristics. These defects lead to complex mechanical problems, and seriously affect application of planar heterostructures in flexible electronics devices. The most commonly used method to improve the toughness is to reduce/eliminate the interfacial structural defects. Using a defective Graphene/hBN interface, we show surprisingly that Carbon Nanotubes (CNTs) are able to enhance the toughness at the interface. It is found that interlayer stress transfer, bonding strain energies and energy release rate of the defective Graphene/hBN interface with CNTs are higher than those of free-defect Graphene/hBN interface, which is in strong contrast to the general view that interfacial defects reduce toughness. By combining the analysis of atomic mechanics and fracture mechanics theory, we find that the abnormal enhancement in interfacial toughness comes from the stress field localization arising from sp2 + sp3 hybrid state and out-of-plane deformation at the interface. In addition, the stress field at the distal end of the simulated models gradually decays with the CNTs height, resulting in the stress fields to be controlled at the interface. This abnormal mechanism provides a new dimension for enhancing toughness in two-dimensional heterostructures.

A DEM bonded particle model compatible with stress/strain constitutive relations

The Discrete Element Method (DEM), also known as distinct element method, is mainly used to study divided media such as granular materials. Moreover, DEM is also widely used for modeling non-linear discontinuous phenomena in rock materials and ceramics such as fragmentation, crack initiation, crack propagation or diffuse cracking. These phenomena are often studied with the Bonded Particle Model (BPM) thanks to the ability of the bond to break in order to simulate a discontinuity in the material. Among the literature, many bonded particle models exist and the choice of the most suitable model remains difficult. Furthermore, these models require fastidious calibration steps because the input bond parameters do not correspond to the whole mechanical response of the DEM assembly. This article introduces a new approach that enables the direct usage of stress–strain laws without calibration steps. Here, these laws directly come from already implemented constitutive laws available in a finite element code. The ability of the proposed software architecture to deal with complex material behaviors (anisotropy, damage and plasticity) is investigated. The obtained results are quite close to the expected ones. This work demonstrates the ability of the proposed approach to deal with such complex mechanical problems which are very challenging to tackle with standard DEM-BPM models.

Segmentation of unsteady cavitation flow fields based on multivariate spatiotemporal hierarchical clustering

Clustering applied to unsteady flow fields can simplify flow field data and partition the flow field into regions of interest. Unfortunately, these areas are often unexplored when applied to complex fluid mechanics problems because multivariate data are difficult to express, and the relationships between flow field snapshots in a time series are difficult to preserve. In this paper, we use joint principal component analysis (JPCA) and fusion principal component analysis (FPCA) to process multivariate data to obtain the static and dynamic characteristics of the cavitation flow field. Based on the static characteristics of the flow field, we use the K-means algorithm and cohesive hierarchical clustering to obtain static flow field segmentation at different levels. Based on the dynamic characteristics of the flow field, we use the proposed time series K-means (TK-means) algorithm and cohesive hierarchical clustering to obtain dynamic flow field segmentation at different levels. The results show that JPCA or FPCA is effective in expressing multivariate features. Static flow field segmentation can obtain time-invariant, physically related structures of unsteady flow. Dynamic flow field segmentation can obtain time-varying, physically related structures of unsteady flow.

A peridynamic-informed neural network for continuum elastic displacement characterization

Physics-informed neural networks have attracted considerable attention for solving scientific and engineering problems because of their low data dependence and physical outputs. The most popular way to embed the laws of physics into neural networks is to apply partial differential equations. However, many complex mechanical problems, such as displacements near the phase interface in heterogeneous plates, are difficult to solve using differential equations expressed by higher-order derivatives. In this work, peridynamic (PD)-informed neural networks are established to characterize the displacement of homogeneous and especially heterogeneous elastic plates. The networks are directly constrained by the integro-differential equations of bond-based PD, thus avoiding higher-order derivatives. The networks are trained without labeled data, and the convergence of the training process is accelerated by a gradual-refinement sampling method. Four simulation experiments show that the displacements of two-dimensional plates with homogeneous or heterogeneous structures are globally approximated by the proposed network with small deviations compare to those obtained with peridynamic numerical solutions. Further investigations show that sampling refinement can significantly increase the prediction accuracy of the networks for complex heterogeneous plates.

Dimensionality reduction through convolutional autoencoders for fracture patterns prediction

Dimensionality reduction methods construct a low dimensional or reduced space. Complex structural mechanics problems can be approximated on a low dimensional space with a Reduced-Order Model or surrogate model. For instance, Principal Components Analysis can compute a reduced basis from a database of full-order snapshots. Principal Components Analysis, however, does not reconstruct low dimensional spaces efficiently for highly nonlinear problems, mainly because it is a linear dimensionality reduction method. In this paper, an original approach based on an autoencoder neural network is developed to construct a nonlinear Reduced-Order Model for a highly nonlinear brittle fracture problem. We show on a set of simulations how the autoencoder can be efficient for dimensionality reduction or compression of highly nonlinear data. A complete deep learning framework is proposed to predict crack propagation patterns directly from the loading conditions. Finally, we validate our nonlinear Reduced-Order Model with data sets generated for two problems using proportional and non-proportional loading conditions. In the first problem, the crack always initiates at the same location but propagates in various directions, which may even vary during the simulation if loading conditions are non-proportional. In the second problem, the crack may additionally initiate at various locations. Results for the two problems evaluate the capabilities of the proposed approach.

A novel method to predict nanofilm morphology on arbitrary-topographical substrate

As physical properties of nanofilms are strongly influenced by their adhered substrates and presented morphologies, it is of vital importance to determine their attaching morphologies on substrates. However, the complex topography of substrate in the practical applications hinders the prediction of attached nanofilm morphology. Previous theoretical methods fail to analyze the complex topography, and molecular dynamics (MD) simulations are limited to small time and size scales. In order to predict the morphology of a nanofilm with an arbitrary size on a substrate with an arbitrary topography, a novel universal method is proposed by combining the mechanical analysis, discrete cosine transform, and multi-resolution analysis together, named multi-resolution discrete cosine transform (MRDCT) method. In this method, the attaching analysis for a complex-topographical substrate is transformed level by level into a sequence of simple mechanical analyses for a 2D-cosine-topographical substrate, of which a semi-analytical model is derived in advance based on the principle of minimum potential energy. Taking the advantages of the pre-derived mechanical model and the multi-level analysis, the MRDCT method has an extremely high efficiency, only about 10−4 of time cost of MD simulations, and moreover, it is not limited by the size scale of nanofilms. The results from both MD simulations and experiments highlight the competency of this method for various complex-topographical substrates. Furthermore, a general conformation criterion in terms of the substrate roughness is derived analytically based on the MRDCT method to assess whether the substrate can be fully conformed by the nanofilm. The proposed MRDCT method can not only be used in the mechanical analysis of film/substrate systems but also provide a new paradigm to solve other complex mechanical problems.

Analysis of 2D contact problems under cyclic loads using IGABEM with Bézier decomposition

Non-uniform rational B-splines (NURBS) are a convenient way to integrate CAD software and analysis codes, saving time from the operator and allowing efficient solution schemes that can be employed in the analysis of complex mechanical problems. In this paper, the Isogeometric Boundary Element Method coupled with Bézier extraction of NURBS and conventional BEM are used for analysis of 2D contact problems under cyclic loads. A node-pair approach is used for the analysis of the slip/stick state. Furthermore, the extent of the contact area is continuously updated to account for the nonlinear geometrical behaviour of the problem. The Newton–Raphson’s method is used for solving the non-linear system. A comparison to analytical results is presented to assess the performance and efficiency of the proposed formulation. Both BEM and IGABEM show good agreement with the exact solution when it is available. On most examples, they are equivalent with some advantage for IGABEM, though the former is slightly more accurate in some situations. This is probably due to the smoothness of NURBS not being able to describe sharp edges on tractions. As expected, IGABEM incurs in higher computational cost due to the basis being more complex than conventional Lagrangian polynomials.

Prediction of stiffness degradation based on machine learning: Axial elastic modulus of [0m /90n ]s composite laminates

This paper, for the first time, proposed an axial elastic modulus degradation prediction method of [0m/90n]s cross-ply laminates using a machine learning (ML) model. The data set of the ML model is established based on the published experiments and a small amount of finite element analysis (FEA) results. The effect of data size on the accuracy of ML prediction is also discussed. The proposed ML model focuses on the process of translating a mechanical problem of damage into a non-linear regression problem of ML, and the mapping between the input and output data, which is hopefully considered for some complex mechanical problems of composites. Meanwhile, the ML method also provides accurate and efficient solution for the engineering practice.

Experiment and Simulation Study on the Dynamic Response of RC Slab under Impact Loading

Reinforced concrete (RC) slab is an important component in civil construction and protection engineering, and its dynamic response under impact loading is a complex mechanical problem, especially for two or multiple continuous impact loads. In this paper, a series of drop hammer impact tests were carried out to investigate the dynamic response of RC slabs with two successive impacts. The time history of impact force and the failure characteristic of the slab surface were recorded. Moreover, four influence factors, including slab thickness, reinforcement ratio, impact location, and drop hammer height have been discussed. Besides, a 3D numerical model based on the finite element method (FEM) was established to expand the research of constrained force, deflection, and vertical stress of an RC slab. The results show that increasing the slab thickness and reinforcement ratio can improve the impact resistance of an RC slab. The impact point location and drop hammer height have a great influence on the dynamic response of the RC slab. In addition, the RC slab will have more obvious damage under the second impact, but the dynamic response becomes weaker. It may be because of the local damage in the concrete caused by the first impact that would weaken the propagation of vibration.

Development of time-space adaptive smoothed particle hydrodynamics method with Runge-Kutta Chebyshev scheme

Smoothed particle hydrodynamics (SPH) method is a Lagrangian particle method that has been widely used for solving complex mechanics problems. To reduce the SPH computational cost, the adaptive SPH (ASPH) with time-varying particle distribution has been proposed using the particle splitting and merging techniques. However, the particle splitting in the adaptive SPH results in a decreased time step due to the reduced particle spacing and thus requires an increased amount of computing time. In the present work, an explicit Runge-Kutta Chebyshev time stepping scheme is implemented within the adaptive SPH, by including the stabilizing substeps into the Runge-Kutta integration method with the use of the Chebyshev polynomials. The number of stabilizing substeps in a single SPH time step can be adaptive after particle splitting while the Runge-Kutta integration time step remains the same. Therefore, the proposed SPH algorithm is adaptive in both time and space domains. Representative example problems are simulated using the developed time-space adaptive SPH method. It is demonstrated that, as compared to the standard SPH and the conventional adaptive SPH with varying particle distribution only, the presented time-space ASPH method exhibits much higher efficiency without compromising accuracy and the superiority in terms of stability.

Optimal 3D arm strategies for maximizing twist rotation during somersault of a rigid-body model

Looking for new arm strategies for better twisting performances during a backward somersault is of interest for the acrobatic sports community while being a complex mechanical problem due to the nonlinearity of the dynamics involved. As the pursued solutions are not intuitive, computer simulation is a relevant tool to explore a wider variety of techniques. Simulations of twisting somersaults have mainly been realized with planar arm motions. The aim of this study was to explore the outcomes of using 3D techniques, with the demonstration that increasing the fidelity of the model does not increase the level of control complexity on the real system. Optimal control was used to maximize twists in a backward straight somersault with both types of models. A multistart approach was used to find large sets of near-optimal solutions. The robustness of these solutions was then assessed by modeling kinematic noise during motion execution. The possibility of using quaternions for representing orientations in this numerical optimization problem was discussed. Optimized solutions showed that 3D techniques generated about two additional twists compared to 2D techniques. The robustness analysis revealed clusters of highly twisting and stable 3D solutions. This study demonstrates the superiority of 3D solutions for twisting in backward somersault, a result that can help acrobatic sports athletes to improve their twisting performance.

A novel element differential method for solid mechanical problems using isoparametric triangular and tetrahedral elements

A novel strong form numerical method, Element Differential Method (EDM), is developed to solve geometrically complex mechanics problems based on triangular or tetrahedral meshes. The discretization of the structure under investigation has been based on Lagrange isoparametric quadrilateral or hexahedral elements while applying EDM. In this paper, a new family of isoparametric triangular and tetrahedral elements with a central node is proposed for EDM. A set of shape functions with analytical expressions for their first and second order partial derivatives is constructed for these triangular and tetrahedral elements, respectively. Moreover, a new element collocation scheme is proposed to establish a system of equations directly from the governing differential equations for internal nodes and traction-equilibrium equations for nodes on edges of an element. In this collocation scheme, no variational principles or virtual energy principles are required to set up the solution scheme, while no integration is needed when forming the coefficients of the system of equations. Numerical examples including standard patch tests and more practical problems are given to demonstrate the correctness of the constructed elements and the efficiency of the proposed element collocation method.

Origin of Unexpectedly Simple Oscillatory Responses in the Excited-State Dynamics of Disordered Molecular Aggregates.

Unraveling the many facets of coherent and incoherent exciton motion in an ensemble of chromophores is an inherently complex quantum mechanical problem that has triggered a vivid debate on the role of quantum effects in molecular materials and biophysical systems. Here the dynamics of a statistical ensemble of molecular aggregates consisting of identical chromophores is investigated within a new theoretical framework. Taking account of intrinsic properties of the system, the Hamiltonian of the aggregate is partitioned into two mutually commuting vibrational and vibronic operators. This representation paves the way for an analysis that reveals the role of static disorder in ensembles of aggregates. Using analytical methods, it is demonstrated that after a critical time τD ≃ 2π/σ (σ being the dispersion of the disorder) any dynamic variable of the aggregate exhibits purely vibrational dynamics. This result is confirmed by exact numerical calculations of the time-dependent site populations of the aggregate. These findings may be useful for the interpretation of recent femtosecond spectroscopic experiments on molecular aggregates.