Structural Mechanics Problems Research Articles

Direct integration methods in dynamic problems of structural mechanics

A method of direct integration with respect to time is proposed to solve dynamic problems of structural mechanics and comparison with the well-known methods is presented.

Bayesian inference method for model validation and confidence extrapolation

This paper presents a Bayesian-hypothesis-testing-based methodology for model validation and confidence extrapolation under uncertainty, using limited test data. An explicit expression of the Bayes factor is derived for the interval hypothesis testing. The interval method is compared with the Bayesian point null hypothesis testing approach. The Bayesian network with Markov Chain Monte Carlo simulation and Gibbs sampling is explored for extrapolating the inference from the validated domain at the component level to the untested domain at the system level. The effect of the number of experiments on the confidence in the model validation decision is investigated. The probabilities of Type I and Type II errors in decision-making during the model validation and confidence extrapolation are quantified. The proposed methodologies are applied to a structural mechanics problem. Numerical results demonstrate that the Bayesian methodology provides a quantitative approach to facilitate rational decisions in model validation and confidence extrapolation under uncertainty.

Recovery of equilibrium on star patches using a partition of unity technique

AbstractA technique for recovering equilibrated element stresses is developed for finite element models of structural mechanics problems. The data for the method consist of the prescribed loading and the stress and displacement fields resulting from a conventional compatible finite element model. Local problems are defined for each star of elements, via the introduction of fictitious body forces and strains. These problems can be solved independently for equilibrium in a process that can be easily parallelized. The quality of the solutions is assessed, for two‐dimensional linear elastic problems, by using them to compute bounds of the error of the finite element solutions, in terms of both global and local quantities of interest. Copyright © 2009 John Wiley & Sons, Ltd.

Group‐theoretic exploitations of symmetry in computational solid and structural mechanics

AbstractThe use of group theory in simplifying the study of problems involving symmetry is a well‐established approach in various branches of physics and chemistry, and major applications in these areas date back more than 70 years. Within the engineering disciplines, the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group‐theoretic methods over the past 40 years. This paper reviews the advances made in the application of group theory in areas such as bifurcation analysis, vibration analysis and finite element analysis, and summarizes the various implementation procedures currently available. Illustrative examples of typical solution procedures are drawn from recent work of the author. It is shown how the group‐theoretic approach, through the characteristic vector‐space decomposition, enables considerable simplifications and reductions in computational effort to be achieved. In many cases, group‐theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained, before any actual calculations are carried out. Copyright © 2009 John Wiley & Sons, Ltd.

Reliability computation with local polynomial chaos approximations

AbstractReliability estimation for structural mechanics problems involving random fields is discussed. While in most of the literature on stochastic finite element methods to date global approximations with Hermite polynomials are considered, the benefits of local approximations are investigated in this paper. Local polynomial approximations in the vicinity of the point of most probable failure are introduced. An adaptive algorithm is proposed, that allows for efficient and accurate computations of failure probabilities.

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.

Daubechies wavelet beam and plate finite elements

In the last few years, wavelets analysis application has called the attention of researchers in a wide variety of practical problems, particularly for the numerical solutions of partial differential equations using different methods such as finite differences, semi-discrete techniques or finite element method. In some mathematical models in mechanics of continuous media, the solutions may have local singularities and it is necessary to approximate with interpolatory functions having good properties or capacities to efficiently localize those non-regular zones. Due to their excellent properties of orthogonality and minimum compact support, Daubechies wavelets can be useful and convenient, providing guaranty of convergence and accuracy of the approximation in a wide variety of situations. In this work, we show the feasibility of a hybrid scheme using Daubechies wavelet functions and the finite element method to obtain numerical solutions of some problems in structural mechanics. Following this scheme, the formulations of an Euler–Bernoulli beam element and a Mindlin–Reisner plate element are derived. The accuracy of this approach is investigated in some numerical test cases.

Numerical homogenization of foam‐like structures based on the FE2‐approach

AbstractThe computation of foam–like structures is still a topic of research. There are two basic approaches: the microscopic model where the foam–like structure is entirely resolved by a discretization (e.g. with Timoshenko beams) on a micro level, and the macroscopic approach which is based on a higher order continuum theory. A combination of both of them is the FE2‐approach where the mechanical parameters of the macroscopic scale are obtained by solving a Dirichlet boundary value problem for a representative microstructure at each integration point.In this contribution, we present a two–dimensional geometrically nonlinear FE2‐framework of first order (classical continuum theories on both scales) where the microstructures are discretized by continuum finite elements based on the p‐version.The p‐version elements have turned out to be highly efficient for many problems in structural mechanics. Further, a continuum–based approach affords two additional advantages: the formulation of geometrical and material nonlinearities is easier, and there is no problem when dealing with thicker beam–like structures.In our numerical example we will investigate a simple macroscopic shear test. Both the macroscopic load displacement behavior and the evolving anisotropy of the microstructures will be discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Non‐conforming FEM‐BEM coupling for wave propagation phenomena

AbstractIn this paper, a method for coupling finite and boundary element discretizations is presented which allows for non–conforming interface discretizations. This method is applicable (but not restricted ) to static and dynamic problems of acoustics and structural mechanics. The main idea is to employ discrete Dirichlet–to–Neumann maps at each time step within the framework of a Lagrange multiplier domain decomposition method. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

High‐dimensional model representation for structural reliability analysis

AbstractThis paper presents a new computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry. The method involves high‐dimensional model representation (HDMR) that facilitates lower‐dimensional approximation of the original high‐dimensional implicit limit state/performance function, response surface generation of HDMR component functions, and Monte Carlo simulation. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher‐order variable correlations are weak, allowing the physical model to be captured by the first few lower‐order terms. Once the approximate form of the original implicit limit state/performance function is defined, the failure probability can be obtained by statistical simulation. Results of nine numerical examples involving mathematical functions and structural mechanics problems indicate that the proposed method provides accurate and computationally efficient estimates of the probability of failure. Copyright © 2008 John Wiley & Sons, Ltd.

Troubleshooting and rectifying structural mechanics problems – applied mechanics in industry

This paper outlines the general process of troubleshooting and rectifying unexpected structural mechanics problems in industrial plant and infrastructure. Typically the process includes the combination and correlation of site measurements (strain, vibration) and computational simulations (finite element analysis, computational fluid dynamics) to identify root cause sources and guide redesign and rectification means. Details of typical site installations are outlined, including mining machinery, gas pipelines, railway lines, manufacturing plant and ships. Four case studies are included, ranging from resin manufacturing tubular agitators suffering vortex induced resonance, ore grinding mills needing better access manhole design, mine dump trucks, and ship shafting issues.

An Approximate Solution for Boundary Value Problems in Structural Engineering and Fluid Mechanics

Variational iteration method (VIM) is applied to solve linear and nonlinear boundary value problems with particular significance in structural engineering and fluid mechanics. These problems are used as mathematical models in viscoelastic and inelastic flows, deformation of beams, and plate deflection theory. Comparison is made between the exact solutions and the results of the variational iteration method (VIM). The results reveal that this method is very effective and simple, and that it yields the exact solutions. It was shown that this method can be used effectively for solving linear and nonlinear boundary value problems.

Combined Axial and Bending Stiffness in Interval Finite-Element Methods

This paper presents a new approach for the treatment of parameter uncertainty for linear static structural mechanics problems. Parameter uncertainties are introduced as interval numbers. Interval arithmetic is applied to finite-element method to analyze the structural responses due to uncertain loading, axial and bending stiffness. However, a naive use of interval arithmetic in the formulation of finite-element method, i.e., replacing the deterministic parameters with corresponding interval ones, will result in meaningless wide results due to the so-called dependency problem. In the present approach, an element-by-element technique is used to reduce the overestimation and compatibility conditions are ensured by a penalty method. With the newly developed overestimation control, most sources of overestimation are eliminated and a very sharp enclosure for the system response is obtained. A number of numerical examples are introduced.

An approach to coupled thermal hydraulics and structural mechanics problems for the MEGAPIE spallation source target

AbstractThis paper describes some novel heat transfer and stress analysis problems that arose in the context of demonstrating the cooling principle of a liquid–metal neutron spallation source target. The design studies using computational fluid dynamics (CFD) resulted in an optimum flow configuration being defined for the coolant circulation. Simultaneously, stresses in the structural components were examined using finite element method (FEM) techniques. Central to the coupled CFD/FEM analysis strategy was the construction of an interface program, which enabled different specialist groups to carry out the thermal hydraulics and structural mechanics analyses using different commercial or in‐house software packages within a common project, and with fully consistent model data. For the studies reported here, the commercial CFD code CFX‐4 and the commercial structural analysis codes ABAQUS and ANSYS have been employed. Results for steady‐state operation of the target show that the critical lower target components are adequately cooled, and that stresses and displacements are well within tolerances. Transient analysis of a beamtrip and power‐up event is also described, showing that no stress crisis occurs. Copyright © 2007 John Wiley & Sons, Ltd.

Stiffness loss of laminated composite plates with distributed damage by the modified local Green’s function method

Abstract The modified local Green’s function method (MLGFM) is an integral method which has been applied in many structural and continuum mechanics problems. It uses the finite element method as an auxiliary tool to automatically develop the Green’s functions projections, which correspond to the fundamental solutions of the boundary element method. The same numerical approach techniques of the FEM and the BEM are employed. The advantages of this method are high convergence and precision, especially for tractions and fluxes on the boundaries. The aim of this paper is to show the MLGFM capability to solve laminated composite plates with generalized transverse cracks. The damage effect is considered as an additional load vector and no modification is necessary in the stiffness matrix of the problem. Some examples are presented and the results are compared within the ones chosen from the referred literature.

Second-Order Cone and Semidefinite Representations of Material Failure Criteria

A class of important problems in structural mechanics leads to optimization problems with linear objective functions and constraints consisting in (a) linear equalities and (b) inequalities imposed by the material strength, the so-called failure criteria. It is shown that a wide variety of failure criteria can be represented as systems of either second-order cone or semidefinite constraints, giving rise to respective optimization problems.

Tri-diagonal and penta-diagonal block matrices for efficient eigensolutions of problems in structural mechanics

Eigenvalues and eigenvectors have many applications in structural mechanics and combinatorial optimization. In this paper, a set of matrices of special forms is studied for which the calculation of eigenvalues can be performed much easier than with the existing general methods. First tri-diagonal matrices are presented and then the relationships for calculating their eigenvalues are extended to the evaluation of the eigenvalues of block tri-diagonal matrices. Block penta-diagonal matrices are also studied in this paper. The eigensolution of different problems of structural mechanics is performed to show the simplicity of using the present formulations.

Multi-disciplinary Multi-objective Topology Optimization of Electromagnetics and Structural Mechanics (For Case of Optimal Dielectric Resonator Antenna Designs)

The designs of on-board electronic devices used in automobiles need to satisfy not only electronic requirements but also physical requirements such maintaining sufficient stiffness under loads. The development of multi-disciplinary optimization methods that also handle multi-physics issues can facilitate a systematic approach to the design of such devices. In this study, we propose a new multi-disciplinary topology optimization method that handles multi-physics phenomena, namely, the coupling of electromagnetic wave propagation problems with static structural mechanics problems. The Finite Difference-Time Domain method is used to solve the electromagnetic wave propagation problem, while the static structural mechanics problem is solved using the Finite Element Method. New relaxation schemes for the design domain and design variable settings are proposed to deal with the use of the different numerical methods. The integral reflection electric energy and the mean compliance are formulated as objective functions to be minimized in an optimization problem. A multi-objective function is also formulated based on a weighted sum formulation to deal with the two objective functions. The optimization algorithm is constructed based on sequential linear program-ming. A design example of a multiple-band dielectric resonator antenna is provided to show the usefulness of the proposed method.

On using deterministic FEA software to solve problems in stochastic structural mechanics

Over the last three decades there has been an outstanding growth in the development of deterministic finite element codes with extensive analysis capabilities. Extension of such deterministic codes to solve problems in stochastic mechanics is of much interest to the academic research community and industry. In this paper we discuss some of the issues involved in integrating fully grown third-party deterministic finite element codes with stochastic projection schemes. The objective of this study is to lay the foundation for development of an easy-to-use general-purpose stochastic finite element software for carrying out probabilistic analysis of large-scale engineering systems. We present a brief introduction to stochastic reduced basis projection schemes and the steps involved in coupling them with a typical deterministic finite element software. We demonstrate with the help of a number of case studies how a coupled framework can be used for solving problems in probabilistic mechanics.

Incorporation of linear multipoint constraints in domain-decomposition-based iterative solvers – Part II: Blending FETI-DP and mortar methods and assembling floating substructures

Two approaches are presented for extending the FETI-DP domain-decomposition-based iterative method to the solution of problems with Linear Multipoint Constraints (LMPCs). Both cases of arbitrary LMPCs and LMPCs generated by the mortar method for enforcing a weak continuity of the solution across non-matching finite element interfaces are considered. In the first approach, the LMPCs are addressed during the preconditioning step but enforced only at convergence. In the second approach, the LMPCs are enforced at each iteration through the solution of an auxiliary coarse problem. Both extended FETI-DP solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems. The obtained results reveal that both solvers are numerically scalable and complement each other from the memory and CPU performance viewpoints.