Structural Mechanics Problems Research Articles

Various estimates of Representative Volume Element sizes based on a statistical analysis of the apparent behavior of random linear composites

This article aims at proposing various estimates of the size of the Representative Volume Element (RVE) of random linear elastic matrix–inclusion composites. These estimates are derived from the computation of the apparent behavior of finite size volume elements (VE) by a new procedure presented in [18] by Salmi et al. (2012) and briefly recalled. Two different points of view to define an RVE are considered: the RVE is defined as being the smallest VE required either to evaluate numerically the considered effective property of the composite by appropriate statistical averaging of apparent ones, or to be allowed to replace any instance of the heterogeneous material by a unique homogeneous equivalent one in structural mechanics problems. In order to introduce the fluctuations of the apparent properties within such definitions of the RVE size, we first study the statistics of the apparent properties. Then, relying on the results of this statistical study, several proposals of RVE criteria are presented and applied to random linear elastic fiber–matrix composites for several contrasts and inclusion (or pore) volume fractions.

An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics

The energy-momentum method, a space---time discretization strategy for elastic problems in nonlinear solid, structural, and multibody mechanics relies critically on a discrete derivative operation that defines an approximation of the internal forces that guarantees the discrete conservation of energy and momenta. In the case of nonlinear elastodynamics, the formulation for general hyperelastic materials is due to Simo and Gonzalez, dating back to the mid-nineties. In this work we show that there are actually infinite second order energy-momentum methods for elastodynamics, all of them deriving from a modified midpoint integrator by an appropriate redefinition of the stress tensor at equilibrium. Such stress tensors can be interpreted as the solutions to local convex projections, whose precise definitions lead to different methods. The mathematical requirements of such projections are identified. Based on this geometrical interpretation several conserving methods are examined.

Recovery of equilibrium on star patches from conforming finite elements with a linear basis

SUMMARY A previous technique for recovering equilibrated stresses from compatible finite element models of structural mechanics problems is extended to cover those cases where the partitioned loads applied to star patches are not initially balanced, regarding rotational equilibrium. The residual moments are removed by adding a suitable corrective stress field to the compatible one before deriving the fictitious body forces. Corrective stress fields are determined by solving another set of local problems based on subdomains that each contain elements forming a neighbourhood of a loaded kernel element. The conditions for the existence of a solution of these problems are studied for simplicial elements. The parameters that control the extended technique are assessed from numerical tests on a variety of two-dimensional linear elastic problems based on constant strain elements for the compatible solutions. These tests are presented in the context of computing bounds on quantities of interest such as total strain energy and local reactions and displacements. Copyright © 2011 John Wiley & Sons, Ltd.

Hybrid asymptotic-direct approach to the problem of finite vibrations of a curved layered strip

A multi-stage approach for the mathematical modeling in the field of nonlinear problems of mechanics of thin-walled structures is the subject of the present paper. A combination of the asymptotic, direct, and numerical methods for consistent and efficient analysis of problems of structural mechanics is presented on the example of plane problem of finite vibrations of a thin curved strip with material inhomogeneity. The method of asymptotic splitting allows for a consistent dimensional reduction of the original two-dimensional continuous problem as the thickness is small: the leading-order solution of the full system of equations of the theory of elasticity results in a one-dimensional formulation of the reduced theory and a problem in the cross-section. The direct approach to a material line extends the results to the geometrically nonlinear range. The appropriate finite element formulation allows for practical applications of the theory; with the numerical solution of the reduced problem, we restore the distributions of stresses, strains, and displacements over the thickness. Numerically and analytically investigated convergence of the solutions of various problems in the original (two-dimensional) and reduced (one-dimensional) models as the thickness tends to zero justifies the analytical conclusion that the curvature and variation of the material properties over the thickness do not require special treatment for classical Kirchhoff’s rods. Further terms of the asymptotic expansion lead to a model with shear and extension, in which curvature appears in a nontrivial way.

Laplacian matrices of product graphs: applications in structural mechanics

Graph products have been studied extensively in the recent decade and applied to many problems in structural mechanics, including configuration processing, parallel computing, and optimal analysis of structures. In this paper, a general theorem is proved for the formation of the Laplacian matrices of product graphs. Using this theorem, exact relationships are derived for eigensolution of Laplacian matrices of product graphs. For the cases where no exact formula is available for calculating the eigenvalues of these matrices, some explanations are provided for the approximate approaches. Applications of the Laplacian matrices of product graphs in structural mechanics are illustrated via some examples.

Finite element analysis of the Girkmann problem using the modern hp-version and the classical h-version

We perform finite element analysis of the so called Girkmann problem in structural mechanics. The problem involves an axially symmetric spherical shell stiffened with a foot ring and is approached (1) by using the axisymmetric formulation of linear elasticity theory and (2) by using a dimensionally reduced shell-ring model. In the first approach the problem is solved with a fully automatic hp-adaptive finite element solver whereas the classical h-version of the finite element method is used in the second approach. We study the convergence behaviour of the different numerical models and show that accurate stress resultants can be obtained with both models by using effective post-processing formulas.

Guided tail modelling for efficient and accurate reliability estimation of highly safe mechanical systems

Classical tail modelling is based on performing a relatively small number of limit-state calculations through Monte Carlo sampling, and then fitting a generalized Pareto distribution to the tail part of the data. The limit-state calculations that do not belong to the tail part are discarded. To reduce the amount of discarded data, this article proposes an efficient tail modelling procedure based on guiding the limit-state evaluations towards the sampling points that have high chances of yielding limit-state values falling into the tail region. The guidance of the limit-state evaluations is achieved through a procedure that utilizes limit-state approximation and distribution fitting. The accuracy of the proposed method is tested through a mathematical problem and four structural mechanics problems, and it is found that the accuracy of reliability estimations can be significantly increased compared to classical tail modelling techniques for the same number of limit-state function evaluations. In addition, it is also found that the improvement in accuracy can be traded off for reducing the number of limit-state evaluations.

Complementarity framework for non-linear dynamic analysis of skeletal structures with softening plastic hinges

In this paper, we describe an algorithm for the incremental state update of elasto-plastic systems with softening. The algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem. In developing the algorithm, we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance, and hence the ability to compute numerical solutions to practical size problems. For example, the notion of a tangent stiffness matrix arises. Numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm. The numerical examples also raise some interesting questions about multiplicity of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.

Variability response functions for effective material properties

The variability response function ( VRF) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a VRF for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists for the effective material properties of statically determinate structures. The VRF for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the VRF for effective material properties, the method of calculation, and results that can be obtained from it.

Patch based stress recovery for plate structures

In this paper we address the application of recovery procedures in advanced problems in structural mechanics. The attention is focused on the recovery by compatibility in patches procedure (RCP) and shear deformable plate structures. The formulation of RCP procedure is extended to shear deformable plate problems (Reissner---Mindlin theory) and is applied to recover stresses from mixed and hybrid stress finite elements. These elements offer new possibilities, for recovery procedures in general, which deserve to be discussed. A comprehensive investigation on which finite element solution can be used as input for the recovery procedures is given through standard benchmark problems, obtained for several values of the thickness on structured and unstructured meshes. The numerical results confirm the effectiveness of the recovery procedure extended to plates problems.

Shape and topological sensitivity analysis in domains with cracks

The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.

Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics

We study the applicability of the discontinuous Petrov–Galerkin (DPG) variational framework for thin-body problems in structural mechanics. Our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate, local computation of the corresponding ‘optimal’ test functions. In the Timoshenko beam problem, the proposed method is shown to provide the best approximation in an energy-type norm which is equivalent to the L 2-norm for all the unknowns, uniformly with respect to the thickness parameter. The same formulation remains valid also for the asymptotic Euler–Bernoulli solution. As another one-dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations. In particular, we derive a special norm for the test space which leads to a robust method in terms of the shell thickness. Finally, we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp-mesh for resolving the boundary layer.

Various approaches for constructing an ensemble of metamodels using local measures

Metamodels are approximate mathematical models used as surrogates for computationally expensive simulations. Since metamodels are widely used in design space exploration and optimization, there is growing interest in developing techniques to enhance their accuracy. It has been shown that the accuracy of metamodel predictions can be increased by combining individual metamodels in the form of an ensemble. Several efforts were focused on determining the contribution (or weight factor) of a metamodel in the ensemble using global error measures. In addition, prediction variance is also used as a local error measure to determine the weight factors. This paper investigates the efficiency of using local error measures, and also presents the use of the pointwise cross validation error as a local error measure as an alternative to using prediction variance. The effectiveness of ensemble models are tested on several problems with varying dimensionality: five mathematical benchmark problems, two structural mechanics problems and an automobile crash problem. It is found that the spatial ensemble models show better performances than the global ensemble for the low-dimensional problems, while the global ensemble is a more accurate model than the spatial ensembles for the high-dimensional problems. Ensembles based on pointwise cross validation error and prediction variance provide similar accuracy. The ensemble models based on local measures reduce cross validation errors drastically, but their performances are not that impressive in reducing the error evaluated at random test points, because the pointwise cross validation error is not a good surrogate for the error at a point.

The “Arch of equilibration” of Charles Hutton (1772)

This paper is a historical re-visitation of the work of C. Hutton, a mathematician of the XVIII century, who wrote an innovative essay on the shape of arches, whose intrados he identifies with the curve of pressure, that he names “arch of equilibration”. His solution was mostly ignored by his contemporaries and after the middle of the XIX century forgotten. But Hutton’s problem is important, also because it offers the first example of an inverse problem in structural mechanics.

Integrated Local Petrov-Galerkin Sinc Method for Structural Mechanics Problems

An integrated, local Petrov-Galerkin Sinc method is introduced and applied to static structural mechanics problems. The method approximates the highest derivative in the local weak form of the governing equation on a rectangular grid, and the lower derivatives and unknown function are found by numerical indenite integration. We suggest that the essential boundary conditions may be applied by the traditional penalty method or by a method in which the stiness matrix is reduced to eliminate dependent degrees of freedom. We propose and compare the performance of three basis functions in terms of their accuracy and convergence properties for two problems: a one-dimensional tapered bar and a two-dimensional plane-stress elasticity problem. Our results indicate that the present method can provide greater accuracy than the Sinc method based on Interpolation of Highest-Derivative, an integration based collocation method suggested by Li and Wu [Li, C. and Wu, X., Numerical solution of dierential equations using Sinc method based on

Post-buckling analysis of planar elastica using a hybrid numerical strategy

Many structural mechanics problems, such as post-buckling of elastica, involve determining multi-equilibrium states of a nonlinear system. Typically, the stable equilibrium states are found by searching for minima of the potential energy function subjected to some constraints. In this paper, the method of genetic algorithm combined with the quasi-Newton method is applied to search for the multiple minima of the strain energy of various elastica problems. The proposed hybrid methodology is relatively straightforward to implement and is adopted to study post-buckling behaviour of planar elastica modelled as multi-link systems for various boundary conditions and sidewall constraints.

Polarization matrices in anisotropic heterogeneous elasticity

Polarization matrices (or tensors) are generalizations of mathematical objects like the harmonic capacity or the virtual mass tensor. They participate in many asymptotic formulae with broad applications to problems of structural mechanics. In the present paper polarization matrices for anisotropic heterogeneous elastic inclusions are investigated, the ambient anisotropic elastic space is allowed to be inhomogeneous near the inclusion as well. By variational arguments the existence of unique solutions to the corresponding transmission problems is proved. Using results about elliptic problems in domains with a compact complement, polarization matrices can be properly defined in terms of certain coefficients in the asymptotic expansion at infinity of the solution to the homogeneous transmission problem. Representation formulae are derived from which properties like positivity or negativity can be read of directly. Further the behavior of the polarization matrix is investigated under small changes of the interface.

Variance decomposition and global sensitivity for structural systems

This paper applies the Sobol’ decomposition of a function of many random variables to a problem in structural mechanics, namely the collapse of a two story two bay frame under gravity load. Prior to introduction of this example application, the Sobol’ decomposition itself is reviewed and extended to cover the case in which the input random variables have Gaussian distribution. Then, an illustrative example is given for a polynomial function of 3 random variables. In the structural example, the Sobol’ decomposition is used to decompose the variance of the response, the collapse load, into contributions from the individual input variables. This decomposition reveals the relative importance of the individual member yield stresses in determining the collapse load of the frame. In applying the Sobol’ decomposition to this structural problem the following issues are addressed: Calculation of the components of the Sobol’ decomposition by Monte Carlo simulation; the effect of input distribution on the Sobol’ decomposition; convergence of estimates of the Sobol’ decomposition with sample size using various sampling schemes; the possibility of model reduction guided by the results of the Sobol’ decomposition.

A parallel algorithm for direct solution of large sparse linear systems, well suitable to domain decomposition methods

The goal of this paper is to develop a parallel algorithm for the direct solution of large sparse linear systems and integrate it into domain decomposition methods. The computational effort for these linear systems, often encountered in numerical simulation of structural mechanics problems by finite element codes, is very significant in terms of run-time and memory requirements.In this paper, a two-level parallelism is exploited. The exploitation of the lower level of parallelism is based on the development of a parallel direct solver with a nested dissection algorithm and to introduce it into the FETI methods. This direct solver has the advantage of handling zero-energy modes in floating structures automatically and properly. The upper level of parallelism is a coarse-grain parallelism between substructures of FETI. Some numerical tests are carried out to evaluate the performance of the direct solver.

Defect Identification in Electrical Circuits via the Virtual Distortion Method. Part 1: Steady-state Case

Virtual Distortion Method, a numerical technique originally developed and applied to various optimization problems in structural mechanics, is adapted to DC/AC circuit analysis. Electro-mechanical analogies with discrete models of plain truss structures are utilized to introduce and implement the main concepts of the method. Simulation of conductance modifications in circuit elements by the equivalent set of virtual sources is a foundation of numerically effective algorithms enabling fast re-analysis and sensitivity analysis. Inverse problem of defect identification is discussed and solution based on distortion approach is provided.