Structural Mechanics Problems Research Articles

Parallel finite element solver PARFES for the structural analysis in NUMA architecture

This paper considers the implementation of PARFES – the direct supernodal method for solving systems of linear equations with symmetric matrices resulting from the finite element method applied to problems of structural and solid mechanics. Unlike previous publications describing the implementation of PARFES for multicore SMP computers, this paper focuses on the implementation of the solver on NUMA computers, where storing data in near memory of the corresponding processor is crucial for achieving high performance, which leads to significant changes in the basic solver algorithms.

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.

Bulking of physically nonlinear plates under the action of dynamic shearing loads

The study of the stability of plates under shear under the action of dynamic loads is one of the important problems of structural mechanics. The plates are widely used in construction, mechanical engineering, shipbuilding and aircraft building. The paper presents a method for calculating plates for shear buckling, taking into account the physical nonlinearity of the material. A plate is considered under the action of a shearing dynamic load along the edges. The calculation is based on the Kirchhoff - Love hypotheses and the hypothesis of a non-linear elastic body. The plate material is assumed to be physically nonlinear. The deformation diagram is approximated as a cubic polynomial. The deflection of the plate points is determined in the form of Vlasov - Kantorovich expansions. Basic non-linear differential equations are derived using the energy method. Lagrange’s equations are used to obtain the resolving equations for plate buckling. On the basis of the developed technique, a calculation was made for the stability of a physically nonlinear square plate under the action of a shear dynamic load. The edges of the plate are hinged. The finite system of nonlinear differential equations is integrated numerically by the Runge - Kutta method. Based on the results of calculations, plots of the dependence of the relative value of the deflection of the central point of the plate on the dynamic coefficient Kd (with and without taking into account the physical nonlinearity of the material) are plotted. The influence of the degree of physical nonlinearity of the material, the parameter of the rate of change of the shear load on the criteria for the dynamic stability of a square plate is studied.

Elasticity problems of beams on reaction-driven nonlocal foundation

A challenging task in nonlocal continuum mechanics consists in formulating constitutive relations leading to well-posed structural problems. Several strategies have been adopted to overcome issues inherent applicability of Eringen’s pure nonlocal theory to nanostructures, such as local/nonlocal mixtures of elasticity and integral models involving modified averaging kernels. These strategies can be applied to the ill-posed problem of flexure of a beam on Wieghardt nonlocal foundation without considering any fictitious boundary forces of constitutive type. A consistent formulation of nonlocal elastic foundation underlying a Bernoulli–Euler beam is thus conceived in the present paper by requiring that transverse displacements are convex combination of reaction-driven local and nonlocal phases governed by Winkler and Wieghardt laws, respectively. The proposed integral mixture is proven to be equivalent to a more convenient differential problem, equipped with nonlocal boundary conditions, which can be effectively exploited to solve nonlocal problems of beams resting on mixture reaction-driven continuous foundation. Effectiveness of the developed nonlocal approach is illustrated by analytically solving simple elasto-static problems of structural mechanics.

A framework based on physics-informed neural networks and extreme learning for the analysis of composite structures

This paper presents a novel approach for solving direct problems in linear elasticity involving plate and shell structures. The method relies upon a combination of Physics-Informed Neural Networks and Extreme Learning Machine. A subdomain decomposition method is proposed as a viable mean for studying structures composed by multiple plate/shell elements, as well as improving the solution in domains composed by one single element. Sensitivity studies are presented to gather insight into the effects of different network configurations and sets of hyperparameters. Within the framework presented here, direct problems can be solved with or without available sampled data. In addition, the approach can be extended to the solution of inverse problems. The results are compared with exact elasticity solutions and finite element calculations, illustrating the potential of the approach as an effective mean for addressing a wide class of problems in structural mechanics.

Corrugated shells: An algorithm for generating double-curvature geometric surfaces for structural analysis

Analysis of corrugated shell structures is an interesting problem in Structural Mechanics, which has many practical applications in Civil Engineering and Architecture. Thanks to corrugation, these structures have a remarkable feature: the wavy (undulated) shape in their edge provides significant enhancements in their structural behaviour, increasing the bending stiffness at the edge and allowing for a non-negligible reduction of its thickness. Moreover, looking at the non-linear behaviour, domes corrugation plays a relevant role in instability phenomena, such as the influence of imperfections and increasing resistance to snap-through.A problem in the study of such kind of shells is the definition of mathematical and geometrical model and the construction of a suitable mesh to perform FE analyses. The aim of this paper is to find an automated way to generate a double-curvature geometric surface that can be used both in static and in non-linear stability analyses of such corrugated shell structures. A method to generate a NURBS surface, suitable for a parametric FE analysis from a geometrical model expressed in a parametric form, is proposed and applied to a shell inspired by the well-known dome designed by Pier Luigi Nervi in 1959 for the roof of the Palasport Flaminio in Rome.

A simple and efficient stochastic meshfree method for linear eigenvalue problems in structural mechanics

A novel and simple stochastic meshfree method is proposed in the present study for the stochastic eigenvalue analysis of problems in structural mechanics. Young’s modulus is modelled as homogeneous random field. Both truncated normal and lognormal distribution characteristics are used. The current work proposes to use Monte Carlo simulation (MCS) on a modified system of equations. Here, the stochastic stiffness matrix is approximated using Taylor series expansion as in perturbation methods. However, the method does not make any approximations for eigensolutions, which are also unknown functions of random variables. This enables the numerical integration of stiffness matrix and its derivatives outside the simulation loop. Unlike the direct MCS on system of equations, this simple but previously unexplored modification proposed, can result in accurate and more efficient evaluation of stochastic eigensolutions. A few numerical examples are solved using the proposed method; buckling of columns, buckling of thin plates and free vibration of beams. The probabilistic characteristics evaluated using the proposed method are found to be matching well with those obtained from direct MCS.

Topology optimization for the elastic field using the lattice Boltzmann method

The lattice Boltzmann method (LBM) has often been employed for numerical analysis of fluid dynamics problems and is described as an algorithm based on a simple fully explicit method. When LBM can be applied to structural mechanics problems, there is a possibility that topology optimization can be performed efficiently for non-linear elastic fields and the topology optimization of fluid-structure interaction problems can be performed easily with the single calculation method. We propose a topology optimization method for isotropic linear elastic body problems using LBM in this paper, as a first step to realize this. First, the analysis approach of the isotropic linear elastic field using LBM is constructed with incorporating the expansion technique of the governing equations for applying to a topology optimization framework. Next, based on the density approach, the optimization problem is formulated, and by using the adjoint lattice Boltzmann method (ALBM), the design sensitivity is derived. Finally, the validity of the proposed method is demonstrated with several numerical examples.

FINITE ELEMENT MODELING OF THE STRESS-STRAIN STATE OF PAVE STRUCTURES UNDER THE ACTION OF TRANSPORT LOADS

The problem of mathematical modeling of the stress-strain state of inhomogeneous structures of pavements of roads and bridges throughout the entire life cycle is considered in order to increase their strength and durability. A classical approach to the consideration of the stress-strain state of a certain section of a structure by analyzing an infinitesimal element of this system is described. The problem is reduced to a partial differential equation, which is solved using approximate methods. To solve the problem, an approach is used to discretize the continuous medium of an object through the calculation of a certain number of elements of finite sizes and the study of the stress-strain state of such elements, called finite elements (FE). This approach to solving structural mechanics problems is called the finite element method (FEM). The nature of the distribution of tensile and shear stress zones under the wheels of a heavy-duty vehicle is studied in order to analyze the influence of the superposition of stress fields depending on the center distances and distances between the wheels, as well as the presence of cracks. Note that the delamination of the pavement between the first and second layers leads to a noticeable increase in both contiguous and normal tensile stresses, which is the most dangerous for asphalt concrete materials. Therefore, when modeling stress fields in real structures, the corresponding values of design parameters, characteristics of mechanical properties, sizes and locations of delamination zones should be taken into account. As a result of the calculations, the most unfavorable combinations of these characteristics should be determined. KEY WORDS: FINITE ELEMENT METHOD, STRESS-STRAIN STATE, ROAD STRUCTURE, STRESS FIELD.

Solution of structural mechanics problems by machine learning

Solution of structural mechanics problems by machine learning

NONLINEAR STRUCTURAL ANALYSIS BASED ON THE MODIFIED SEQUENTIAL LOAD METHOD

The article discusses ways to improve the accuracy of solving problems of nonlinear structural mechanics. It is shown that the combination of the method of sequential loading and the Newton-Kantorovich method can improve the accuracy of the solution and reduce the complexity of obtaining results. The solution of the given linear equations can be obtained by numerical and approximate methods known in the literature.

Cracking Patterns of Brittle Hemispherical Domes: an Experimental Study

Crack formation in hemispherical domes is a distinguished problem in structural mechanics. The safety of cracked domes has a long track record; the evolution of the cracking pattern received less attention. Here, we report displacement-controlled loading tests of brittle hemispherical dome specimens, including the evolution of the meridional cracking pattern. The 27 investigated specimens, 20 cm in diameter, were prepared in 3D printed molds, and their material is one of the three mixtures of gypsum and cement. We find that neither the (limited) tensile strength nor the exact value of the thickness significantly affects the statistical description of the cracking pattern, i.e., the cracking phenomenon is robust. The maximal number of the meridional cracks never exceeds seven before the fragments’ disintegration (collapse). We find that the size distribution of the fragments exhibits a lognormal distribution. The evolution is reflected in the load-displacement diagrams recorded in the test, too, as significant drops in the force are accompanied by an emergence of one or more new cracks, reflecting the brittle nature of the phenomenon. A simple, stochastic fragmentation model, in which a segment is fragmented at either in the middle or at the fourth point, fairly recovers the observed size distribution.

A heuristic swarm-based optimization method using multi-variate normal distributions with self-adaptive variance matrices

In this paper a heuristic swarm-based optimization method developed based on stochastic search and concepts from statistics was presented for optimal design of problems in structural mechanics. Although in recent years many heuristic optimization methods have been developed, none of them can be considered as the most effective method for solving all types of optimization problems. In general, heuristic optimization methods are formulated to follow specific patterns and often include some random parameters. It has been shown that adding stochastic parameters and concepts from statistics to such algorithms can improve their performance and increase the chance of finding global optimal points. Based upon same notion, we have developed a new swarm-based optimization algorithm in which multi-variate normal distributions with self-adaptive variance matrices (MNSV) are used to generate superior solutions in each iteration. The self-adaptive variance matrix in each iteration was calculated using the variance of the best points from solution history. Such a choice improved both exploration and exploitation of the algorithm and led to a trade-off between them from beginning towards the end of the solution. Computational complexity of MNSV algorithm was comparable to that of other swarm-based optimization algorithms and performance of the algorithm was tested through optimal solution of several benchmarks from structural and mechanical engineering. While MNSV algorithm was simple in terms of both formulation and implementation, it was effective in finding optimal solutions and often converged using a smaller number of iterations when compared to other algorithms.

A non‐incremental solution procedure for elastoplastic problems in structural mechanics

AbstractIn this contribution, we present a non‐incremental solution procedure for the efficient treatment of elastoplastic problems. To this end, all time history data are decoupled into space and time, and the solution is obtained using the fixed‐point algorithm.

A displacement-controlled arc-length solution scheme

Tracing load-displacement paths in structural mechanics problems is complicated in the presence of critical points of instability where conventional load- or displacement control fails. To deal with this, arc-length methods have been developed since the 1970s, where control is taken over increments of load at these critical points, to allow full transit of the load-displacement path. However, despite their wide use and incorporation into commercial finite element software, conventional arc-length methods still struggle to cope with non-zero displacement constraints. In this paper we present a new displacement-controlled arc-length method that overcomes these shortcomings through a novel scheme of constraints on displacements and reaction forces. The new method is presented in a variety of serving suggestions, and is validated here on six very challenging problems involving truss and continuum finite elements. Despite this paper’s focus on structural mechanics, the new procedure can be applied to any problems that involve nonhomogeneous Dirichlet constraints and challenging equilibrium paths.

Application of the ADINA Software for Tasks of Educational and Technical Practice

Development of the designing process in civil engineering brings the possibility for more complex analysis of the structures. Computational software can be helpful for regular design but also for identification and solving of more specified problems. Finite Element Method (FEM) became one of the most used numerical methods. Numerous commercial programs are based on this method. However, these programs are more complex and require skilled designer to fully utilize their potential. This article brings the lead to apply the FEM method implemented in commercial software ADINA to solve the problem of structural mechanics. It is necessary to connect the theoretical approach of FEM modeling and experience from real case problems. Skilled designer can calibrate the numerical model to obtain relevant outputs. However, certain cases still require a detailed and individual approach during designing that is verified by the in-situ monitoring and observation. Lead of the student to the flexible treatment of the design is one of the important tasks of the educational process. This involves the basic knowledge about appropriate analysis of the problem for effective utilization of the computational software for particular applications even for more complex solutions.

Displacement-function modeling of thermo-mechanical behavior of fiber-reinforced composite structures

A new modeling scheme has been developed for the analysis of thermo-mechanical stresses in structural problems of fiber-reinforced composite materials with mixed mode of physical conditions. In contrast with conventional approaches, the coupled thermal and mechanical problems of structural mechanics are integrated here using a single scalar function of space variables, which is defined in terms of the displacement components of plane elasticity. The displacement-function is governed by a single partial-differential equation of equilibrium that takes into account the two orthogonal gradients of the resulting non-uniform temperature field through two different versions of the formulation. The temperature-dependent elastic field is determined using a two-step solution scheme that utilizes the two versions of displacement-function formulations in two sequential steps. The scheme is developed in such a flexible fashion that even the solution of the corresponding isotropic problems can readily be obtained with thermal and/or mechanical loading as a limiting case of the formulation. The application of the proposed modeling scheme is demonstrated through the solution of a number of problems of structural mechanics, namely plates and beams subjected to thermal as well as thermo-mechanical loadings with both uniform and mixed mode of physical conditions. An attempt is also made to verify the soundness and accuracy of the resulting thermoelastic fields by comparing them with the corresponding analytical as well as computational solutions available in the literature. Moreover, the correspondence between the versions of the displacement-function formulation is also verified by solving identical problems using different versions or different combinations of the versions of the formulation.

Iterative Polynomial Dimensional Decomposition approach towards solution of structural mechanics problems with material randomness

One of the major difficulties in solving stochastic mechanics problems is the curse of dimensionality, where an exponential increase in the dimension of the problem is encountered with the increase in the number of random variables and/or order of expansion considered in any approximation. A prominent method in addressing the curse of dimensionality is ANOVA dimension Decomposition (ADD), which represents a mathematical function with multiple lower variate functions. These lower variate functions are represented using orthogonal polynomials, which yields Polynomial Dimensional Decomposition (PDD). In recent articles, the authors proposed an Iterative Polynomial Chaos (ImPC) based method for the solution of structural mechanics problems, where computational efficacy of ImPC was demonstrated against Polynomial Chaos (PC). In ImPC, the problems are solved iteratively using smaller sizes of PC expansions. Thus, it reduces the curse of dimensionality of PC expansion. The PDD reduces the size of the system matrix by considering a fewer number of random variables at a time, while ImPC can be considered to solve each components of PDD iteratively so that a converged solution can be achieved without increasing the order of expansion, which is termed as iterative PDD in the present study. Thus, the overall convergence can be achieved with a lesser size of the system matrix, which enables to perform analyses with a lesser computational facility. Further, the stiffness matrix size can be reduced by considering the random field at Gauss points instead of the mid point. Numerical studies with both Gaussian and non-Gaussian random field of Young’s modulus are conducted, and computational efficiency of the iterative PDD is compared with that of PDD, ImPC, and first order perturbation method. The iterative PDD is observed to be computationally less demanding and exhibits reduced dimensional curse.

On a cellular developmental method for layout optimization via the two-point topological derivative

This paper introduces a two-point topological derivative for elasticity and a cellular developmental method for layout optimization of structures. The two-point derivative measures the change on a functional due to the addition of a thin ligament between parts of the domain. It contrasts with the existing single-point topological derivatives that measure the change of a functional due to the subtraction of material. Based on the two-point topological derivative, a cellular developmental method is presented, which builds the layout of a structure in a sequence of cellular divisions and cellular dynamics developmental cycles. The methodology introduced in this work can easily account for member sizes and angle constraints among members of the frame, and the optimization can be performed in the free space or in a bounded domain. Benchmark problems are solved to demonstrate the capability of the cellular developmental method to solve optimization problems in structural mechanics.

Parallel finite element solver for multi-core computers with shared memory

A direct finite element solver is considered for symmetric sparse matrices arising from the application of the finite element method to problems of structural mechanics and solid mechanics. The solver is based on the block Cholesky method generalized to indefinite matrices. A distinctive feature of the proposed method from other known approaches is the original algorithm for parallelizing the factorization procedure, as well as assembling the matrix of a system of linear and linearized algebraic equations, based on the analysis of the adjacency graph for the finite elements of the design model. All the proposed parallelization algorithms use dynamic mapping of computational tasks to threads and are based on a general idea that uses a dependency vector that controls the execution of computational tasks in a parallel region. This approach is relatively simple and at the same time demonstrates high efficiency. It was developed for solving nonlinear static and dynamic problems of structural mechanics, requiring multiple assembly and factorization of a sparse matrix, but can be successfully applied in other areas of computational mathematics.