Structural Mechanics Problems Research Articles

Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics

Physics-informed neural network has emerged as a promising approach for solving partial differential equations. However, it is still a challenge for the computation of structural mechanics problems since it involves solving higher-order partial differential equations as the governing equations are fourth-order nonlinear equations. Here we develop a multi-level physics-informed neural network framework where an aggregation model is developed by combining multiple neural networks, with each one involving only first-order or second-order partial differential equations representing different physics information such as geometrical, constitutive, and equilibrium relations of the structure. The proposed framework demonstrates a remarkable advancement over the classical neural networks in terms of the accuracy and computation time. The proposed method holds the potential to become a promising paradigm for structural mechanics computation and facilitate the intelligent computation of digital twin systems.

Recognition of local fiber orientation state in prepreg platelet molded composites via deep learning

This paper presents a novel deep learning approach for reconstruction of local through-the-thickness fiber orientation distribution (FOD) in discontinuous long-fiber, prepreg-platelet molded composites (PPMC). The thermal-residual strains on composite surfaces are used as inputs to train a fully convolutional neural network, named U-Net, to predict spatially varying, local FOD. High fidelity synthetic data was generated via computational simulation of PPMC with stochastic material orientation state and was used for training the deep learning model. The proposed U-Net model allowed for rapid recognition of PPMC morphology by solving the inverse structural mechanics problem of determining the average fiber orientation through the composite thickness based on the provided surface strain measurements. Upon training and validation, the U-Net deep learning model was deployed to rapidly predict complex distributions of the local through-the-thickness FOD in PPMC.

Weak-formulated physics-informed modeling and optimization for heterogeneous digital materials.

Numerical solutions to partial differential equations (PDEs) are instrumental for material structural design where extensive data screening is needed. However, traditional numerical methods demand significant computational resources, highlighting the need for innovative optimization algorithms to streamline design exploration. Direct gradient-based optimization algorithms, while effective, rely on design initialization and require complex, problem-specific sensitivity derivations. The advent of machine learning offers a promising alternative to handling large parameter spaces. To further mitigate data dependency, researchers have developed physics-informed neural networks (PINNs) to learn directly from PDEs. However, the intrinsic continuity requirement of PINNs restricts their application in structural mechanics problems, especially for composite materials. Our work addresses this discontinuity issue by substituting the PDE residual with a weak formulation in the physics-informed training process. The proposed approach is exemplified in modeling digital materials, which are mathematical representations of complex composites that possess extreme structural discontinuity. This article also introduces an interactive process that integrates physics-informed loss with design objectives, eliminating the need for pretrained surrogate models or analytical sensitivity derivations. The results demonstrate that our approach can preserve the physical accuracy in data-free material surrogate modeling but also accelerates the direct optimization process without model pretraining.

Latent Crossover for Data-Driven Multifidelity Topology Design

Abstract Topology optimization is one of the most flexible structural optimization methodologies. However, in exchange for its high level of design freedom, typical topology optimization cannot avoid multimodality, where multiple local optima exist. This study focuses on developing a gradient-free topology optimization framework to avoid being trapped in undesirable local optima. Its core is a data-driven multifidelity topology design (MFTD) method, in which the design candidates generated by solving low-fidelity topology optimization problems are updated through a deep generative model and high-fidelity evaluation. As its key component, the deep generative model compresses the original data into a low-dimensional manifold, i.e., the latent space, and randomly arranges new design candidates over the space. Although the original framework is gradient free, its randomness may lead to convergence variability and premature convergence. Inspired by a popular crossover operation of evolutionary algorithms (EAs), this study merges the data-driven MFTD framework and proposes a new crossover operation called latent crossover. We apply the proposed method to a maximum stress minimization problem in 2D structural mechanics. The results demonstrate that the latent crossover improves convergence stability compared to the original data-driven MFTD method. Furthermore, the optimized designs exhibit performance comparable to or better than that in conventional gradient-based topology optimization using the P-norm measure.

ЭНЕРГЕТИЧЕСКИЕ СВОЙСТВА СИММЕТРИЧНЫХ ДЕФОРМИРУЕМЫХ СИСТЕМ

Energy methods for calculating structures, which have become popular for a century, are based on the Lagrange principle and have the meaning of equality of work of external forces and internal forces. Having proved their effectiveness in the overwhelming majority of problems of structural mechanics, they became the dominant approach in formulating the problems of studying solid deformable systems and gave rise to the main methodology for solving problems. As a result, a situation has arisen that the internal potential energy of a deformed body remains insufficiently studied.
 The paper develops an approach to the study of the symmetric structure at critical levels of strain energy. The criterion of critical levels of strain energy, based on the concepts of "self-stress" ("self-balance") of a deformable body. Limiting values of the structure strain energy may get by varying the reactions and deflections in the nodal points. The extreme values of forces and displacements of the rods are calculated in matrix form from the values of nodal reactions (displacements).
 Methodology for studying the energy properties of a system is shown on the examples of the study of symmetric rod systems without involving the concept of external forces. The technique is based on matrix methods of structural mechanics and the mathematical apparatus of eigenvalue problems. The comparison of structural design and structural analysis solution of structural mechanics tasks by traditional methods and with the proposed methodology is carried out.

Cycle Bases of Graphs for the Analysis of Frames Structures Using Force Method

The formation of suitable cycle bases corresponding to sparse flexibility matrices for the force method of frame analysis has always been an interesting problem in structural mechanics. These cycle bases are needed for the formation of static bases for efficient force method of structural analysis. Similarly, such bases are required in the mesh analysis of other networks. This paper reviews methods for the cycle basis selection by utilizing different embeddings on higher dimensional topological spaces, and using the ideas and concept from this study, graph theory algorithms are developed for efficient computational algorithms for the formation of subminimal, and minimal cycle bases.

A general framework of high-performance machine learning algorithms: application in structural mechanics

Data-driven models utilizing powerful artificial intelligence (AI) algorithms have been implemented over the past two decades in different fields of simulation-based engineering science. Most numerical procedures involve processing data sets developed from physical or numerical experiments to create closed-form formulae to predict the corresponding systems’ mechanical response. Efficient AI methodologies that will allow the development and use of accurate predictive models for solving computational intensive engineering problems remain an open issue. In this research work, high-performance machine learning (ML) algorithms are proposed for modeling structural mechanics-related problems, which are implemented in parallel and distributed computing environments to address extremely computationally demanding problems. Four machine learning algorithms are proposed in this work and their performance is investigated in three different structural engineering problems. According to the parametric investigation of the prediction accuracy, the extreme gradient boosting with extended hyper-parameter optimization (XGBoost-HYT-CV) was found to be more efficient regarding the generalization errors deriving a 4.54% residual error for all test cases considered. Furthermore, a comprehensive statistical analysis of the residual errors and a sensitivity analysis of the predictors concerning the target variable are reported. Overall, the proposed models were found to outperform the existing ML methods, where in one case the residual error was decreased by 3-fold. Furthermore, the proposed algorithms demonstrated the generic characteristic of the proposed ML framework for structural mechanics problems.

Mechanical response of concrete pipe rehabilitated by liner under external load: Analytical solution

AbstractUsing pipe lines to rehabilitate concrete drainage pipes is a popular trenchless technique. In this paper, the concrete pipe and liner are equivalent to a symmetrical structural mechanics problem, and the mechanical solution of the composite structure is obtained using the force method. Subsequently, the analytical solutions are compared with the test and finite element results. The distribution law of axial force, bending moment, and shear force of the composite structure are obtained, and the circumferential strain law of the inner and outer surfaces of the host pipe is analyzed. Lastly, the influence of pipe diameter and liner thickness on the bearing capacity of the host pipe is examined. The results indicate that the axial and shear force distribution of the composite structure remains the same for different pipe diameters, as the distribution diagrams of shear and axial force along the circumferential direction coincide. While the elastic modulus and thickness of the liner have no effect on the internal force distribution of the composite structure, they do directly affect the stress state of the host pipe. Additionally, the bottom support causes abrupt changes in the bending moments and strains of the host pipe.

Fuzzy sets application in the problems of structural mechanics and optimal design

Fuzzy sets application in the problems of structural mechanics and optimal design

A consistent group-theoretic transformation for the block-diagonalization of structural matrices

Symmetry properties of physical systems may be studied through symmetry groups. In recent times, group theory has found application in the study of various problems in structural mechanics, specifically bifurcation, buckling, kinematics and vibration. Computational simplifications are achieved by decomposing the vector space of the problem into smaller subspaces that are independent of each other. When the basis vectors of a subspace are used as the symmetry-adapted variables of that subspace, a smaller problem (associated with a matrix of smaller dimensions) automatically results. However, the same decomposition may be achieved by first obtaining the structural matrix of the system, and then transforming this into a non-overlapping block-diagonal matrix, each independent block being associated with a subspace of the problem. The advantage of this approach is its greater amenability to computer programming, but it does not always give the correct results unless a very specific procedure is followed. The purpose of this contribution is to present a consistent group-theoretic approach for the block diagonalization of structural matrices.

CLOSED-FORM SOLUTION OF A METAL PLATE WITH A CENTRAL CRACK UNDER TENSION

Accurate prediction of stress and displacement in plates with central cracks is vital for engineering design safety. However, the impact of finite plate size on stress and displacement distribution is an important factor that has not been fully addressed in previous studies, highlighting the need for further investigation. Thus, this study presents a closed-form elastic solution for a rectangular plate with a central crack under tension load. The solution uses complex polynomial functions satisfying equilibrium equations and boundary conditions. Numerical finite element models verify the analytical solution's accuracy for plates with varying crack lengths. The presented closed-form solution accounts for the impact of finite plate size on stress and displacement distribution accurately. Developing closed-form solutions for structural mechanics problems enables engineers to optimize designs for safety, efficiency, and cost-effectiveness more accurately.

Historic adjacent concrete buildings strengthened by cable-ties under seismic pounding effects: A stochastic approach considering uncertain input parameters

Cultural Heritage structures include existing old industrial framed reinforced concrete (RC) buildings. The present study deals with a stochastic numerical treatment for the pounding problem concerning the seismic interaction between historic adjacent framed structures strengthened by cable-ties (tension-only bracings) when the input parameters are uncertain. This problem concerns here the unilateral contact between neighbouring structures during earthquakes and is considered as an inequality problem of dynamic structural contact mechanics. The Monte Carlo method is used for treating the uncertainty concerning input parameters. The purpose here is to estimate numerically and to control actively the influence of the cable-ties on the seismic response of the adjacent structures. Finally, in a practical case of two seismically interacting historic framed reinforced concrete (RC) structures, the effectiveness of the proposed methodology is shown.

A locking free virtual element formulation for Timoshenko beams

The virtual element method (VEM) provides new ways of deriving discretization for problems in structural and solid mechanics, starting with the contribution by Beirão da Veiga et al. (2013) for elastic solids. Interestingly, the virtual element method allows also to revisit the construction of different elements which have the same shape as finite elements. This is even true for one-dimensional structures like trusses and beams. Here we study a virtual element development of the Timoshenko beam which surprisingly leads to a straight forward formulation of locking free and in the linear range even exact Timoshenko beam elements. These elements can be easily incorporated into classical finite element codes since they have the same number of unknowns as finite elements for beams. The formulation allows to compute nonlinear structural problems undergoing large deflections and rotations using the formulation provided in Reissner (1972).

Использование разработанного алгоритма проверки совместности системы линейных неравенств для решения оптимизационных задач в строительстве

Optimization methods have been used to solve many problems in construction. Such well-known problems as the transport problem, some problems of structural mechanics, the problem of the optimal location of objects on the construction site, the assignment of the composition of construction teams in the production of construction and installation works, the tasks of technological equipment and a number of others can be reduced to linear programming problems. An algorithm for checking the compatibility of a system of n linear inequalities in the space of real numbers R of dimension d is given. The algorithm is based on the sequential construction of hyperplanes in the space R of dimension (d+2).

Modified replica exchange-based MCMC algorithm for estimation of structural reliability based on particle splitting method

We consider the problem of evaluation of the reliability integral when the limit state functions could have complicating features such as (a) discontinuity/rapid changes in values of GU in the space spanned by U, (b) existence of multiple, possibly unimportant, regions of failure with rapid changes in performance function near one or more of the important regions, and (c) multiple regions of failure with substantial contributions to failure probability. We tackle this problem by using the particle splitting framework and introduce an improved Markov chain Monte Carlo sampler based on replica exchange strategy. This is shown to enhance the capacity of samples to detect and explore important regions of failure. The application of the bootstrap technique to deduce the sampling variance of the estimator of the probability of failure is developed. Also given are a few examples of problems in structural mechanics where limit state functions with the aforementioned difficulties arise. The performance of the proposed method in dealing with these difficulties is compared with those of existing methods.

Structural stress prediction based on deformations using artificial neural networks trained with artificial noise

AbstractArtificial Neural Networks (ANN) have found application for multiple problems in structural mechanics and civil engineering. In the new approach developed here, ANNs are used for the determination of the maximum stress resultants in a structure on the basis of monitored displacements. Initially, a simple supported beam subjected up to two vertical forces is considered. The beam is solved analytically for different combinations of load positions and magnitudes defined by Monte‐Carlo sampling. The resulting beam deflections and maximum stress resultants are employed for the training of feedforward neural networks in order to generate metamodels able to predict the maximum bending moment knowing only a few beam input deflections. Subsequently, the concept is applied to a real beam experiment, where a reinforced concrete beam has been tested. It is shown, that the perfect correlation of the beam displacements used as inputs for the ANN training leads to extrapolations when the metamodel is applied to the measured displacements of the real experiment, which are not perfectly correlated. This is source of wrong predictions when the network is applied to these data. The introduction of artificial noise in the synthetic input data in a controlled way before training helps to overcome extrapolation and to increase repeatability and robustness of the metamodels. Various noise levels are applied to the synthetic data to monitor how the ANN performances change over increasing noise and the results along with the limitations of the approach are illustrated.

О ЕСТЕСТВЕННЫХ ГРАНИЧНЫХ УСЛОВИЯХ В ЗАДАЧЕ О ПОТЕРЕ УСТОЙЧИВОСТИ ПЛАСТИНЫ С ЭЛЛИПТИЧЕСКОЙ ВСТАВКОЙ ПРИ РАСТЯЖЕНИИ

This paper studies the issue of determination and influence of natural boundary conditions in the problem of buckling of a thin plate with an elliptical inclusion under tension. First, the naturalness of the boundary conditions of the “free edge” type for a plate with a hole is proved. Then a plate with welded inclusion is considered and natural boundary conditions are derived. The limit cases are checked for an absolutely soft inclusion and for an absolutely rigid one. It is shown that the first case leads to a problem with a hole and the corresponding natural boundary conditions, and in the second case, to the absence of natural conditions, since a problem with a clamped edge occurs. The authors conclude that the use of additional restrictions will lead to the construction of a basis that will rapid up the convergence of the method. Variational methods are widely used in all fields of mechanics, including in the field of machine, aircraft, and rocket engineering. An exact solution to the problems of elasticity theory and structural mechanics is not always possible to construct, therefore, in practice, great importance is attached to various approximate methods. Among them a special place is occupied by variational methods based on the direct minimization of the corresponding energy of the body and making it possible to build approximate analytical solutions in the form of a functional series. The goal of variational methods is to construct a partial sum of this series, which, with a sufficient number of terms, will be maximum close to the solution. However, the issue of convergence is influenced by many factors, and one of them is the natural boundary conditions, which are derived in this paper for the problem of the stability loss of the plate with the elliptical inclusion under tension.

Analysis of a column with variable, arbitrary shaped cross section, including initial imperfections, geometric and material nonlinearity

In this paper, we propose a general method for solving structural mechanics problems by developing a “functional model”. It is implemented by defining the required geometrical and physical relationships in their general form via analytical and numerical functions. For that purpose, a special programming language Calcpad is used. The obtained data structure is processed by a core engine, which generates the computer code, required for the solution. This model can be used for calculation of different partial cases, after specifying the specific input functions and parameters. In this way, the need for developing a different procedure for each separate case is eliminated. The proposed method is applied for the analysis of a column with variable, arbitrary shaped cross section, accounting for initial imperfections, geometric and material nonlinearity.

Interpolation-based immersed finite element and isogeometric analysis

We introduce a new paradigm for immersed finite element and isogeometric methods based on interpolating function spaces from an unfitted background mesh into Lagrange finite element spaces defined on a foreground mesh that captures the domain geometry but is otherwise subject to minimal constraints on element quality or connectivity. This is a generalization of the concept of Lagrange extraction from the isogeometric analysis literature and also related to certain variants of the finite cell and material point methods, as well as the concept of free-form deformation from design. Crucially, the interpolation may be approximate without sacrificing high-order convergence rates, which distinguishes the present method from existing high-order finite cell, CutFEM, and immersogeometric approaches. The interpolation paradigm also permits non-invasive reuse of existing finite element software for immersed analysis. We analyze the properties of the interpolation-based immersed paradigm for a model problem and implement it on top of the open-source FEniCS finite element software, to apply it to a variety of problems in fluid, solid, and structural mechanics where we demonstrate high-order accuracy and applicability to practical geometries like trimmed spline patches.

Evaluation of global sensitivity analysis methods for computational structural mechanics problems

Abstract The curse of dimensionality confounds the comprehensive evaluation of computational structural mechanics problems. Adequately capturing complex material behavior and interacting physics phenomenon in models can lead to long run times and memory requirements resulting in the need for substantial computational resources to analyze one scenario for a single set of input parameters. The computational requirements are then compounded when considering the number and range of input parameters spanning material properties, loading, boundary conditions, and model geometry that must be evaluated to characterize behavior, identify dominant parameters, perform uncertainty quantification, and optimize performance. To reduce model dimensionality, global sensitivity analysis (GSA) enables the identification of dominant input parameters for a specific structural performance output. However, many distinct types of GSA methods are available, presenting a challenge when selecting the optimal approach for a specific problem. While substantial documentation is available in the literature providing details on the methodology and derivation of GSA methods, application-based case studies focus on fields such as finance, chemistry, and environmental science. To inform the selection and implementation of a GSA method for structural mechanics problems for a nonexpert user, this article investigates five of the most widespread GSA methods with commonly used structural mechanics methods and models of varying dimensionality and complexity. It is concluded that all methods can identify the most dominant parameters, although with significantly different computational costs and quantitative capabilities. Therefore, method selection is dependent on computational resources, information required from the GSA, and available data.