Barna A. Szabó

• PhD
• Professor (Full) at Washington University in St. Louis

Until very recently, computer codes offering nonlinear capabilities were based on the h-version of the finite element method. In this paper we describe an algorithm and its implementation based on the p-version of the finite element method. The functionalities to be discussed are inelastic material behavior based on the deformation theory of plasti...

The formulation of a system of hierarchic models for the simulation of the mechanical response of slender elastic bodies, such as elastic rods, is considered. The present work is concerned with aspects of implementation and numerical examples. We use a finite element formulation based on the principle of minimum potential energy. The displacement f...

This chapter introduces the finite element method as a method by which the exact solution of a mathematical problem, cast in a generalized form, can be approximated. It also introduces the relevant mathematical concepts, terminology and notation in the simplest possible setting. The chapter considers the formulation of a second order ordinary diffe...

This chapter is concerned with finding the macromechanical properties of unidirectional fiber‐matrix laminae from the mechanical properties of its constituents. It considers idealized unidirectional laminae, assuming that the fiber arrangement fits a perfect hexagonal or square pattern. An algorithm for finding the macroscopic thermomechanical prop...

The questions of how to choose an effective finite element discretization scheme, given a set of input data, and how to extract the quantities of interest from the finite element solution and estimate their relative errors, are addressed in this chapter. Preprocessing is concerned with the collection and verification of input data and the formulati...

This chapter is concerned with the formulation, calibration, validation and ranking of mathematical models with reference to a classical problem of mechanical engineering; that of predicting fatigue failure in metallic machine components and structural elements subjected to alternating loads in the high cycle regime. The mathematical model consider...

This chapter discusses the important class of dimensionally reduced models. The starting point is the generalized formulation of the problem of linear elasticity. In order to present the main points in a simple setting, mathematical models for beams are derived from the generalized formulation of the problem of two‐dimensional elasticity. The formu...

In this chapter, the strong forms of boundary value problems are formulated from first principles with reference the problems of heat conduction in solid bodies and elasticity. The formulation of mathematical models for linear problems in heat conduction and elasticity is described in strong and generalized forms. The three‐dimensional analogue of...

This chapter is concerned with the algorithmic aspects of the finite element method. The finite element spaces, standard elements, the corresponding shape functions and mapping functions are described for two‐ and three‐dimensional formulations. Two‐dimensional finite element meshes are comprised of triangular and quadrilateral elements. As in the...

The formulation of nonlinear models is a very large and diverse field. This chapter presents a brief introduction, with emphasis on the algorithmic aspects and examples. Mathematical models of heat conduction often involve radiation heat transfer and the coefficients of heat conduction are typically functions of the temperature. The formulation of...

This chapter considers the functioning of mechanical systems and their imitative representation by mathematical models. It focuses on predicting certain quantities of interest (QoIs), given the geometry, material properties and loads. The usual QoIs in the simulation of mechanical systems are displacements, stresses, strains, stress intensity facto...

The article summarizes the responses to a challenge problem posed on the October 2018 issue of Benchmark (a NAFEMS publication) and gives an example of solution verification.

An end-to-end example of the application of the procedures of verification, validation, and uncertainty quantification (VVUQ) is presented with reference to mathematical models formulated for the prediction of fatigue failure in the high cycle range. A validation metric based on the log likelihood function is defined. It is shown that the functiona...

In this work we propose a stochastic model for estimating the occurrence of crack initiations on the surface of metallic specimens in fatigue problems that can be applied to a general class of geometries. The stochastic model is based on spatial Poisson processes with intensity function that combines stress-life (S-N) curves with averaged effective...

In this work we propose a stochastic model for estimating the occurrence of crack initiations on the surface of metallic specimens in fatigue problems that can be applied to a general class of geometries. The stochastic model is based on spatial Poisson processes with intensity function that combines stress-life (S-N) curves with averaged effective...

In the first part of this chapter the basic algorithmic structure and performance characteristics of the p-version of the finite element method are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the p-version is fully established, and a very substantial amount of engineering exp...

Design rules are stated in the form of a predictor of failure initiation and its allowable value in the context of design of mechanical and structural components subjected to cyclic loading. Calibration and validation procedures are described and illustrated for the predictors with the aid of published experimental data. It is noted that mathematic...

The effects of small notches on the predicted fatigue life of mechanical and structural components are examined. It is shown that the classical results of Neuber, Peterson and others can be interpreted to mean integral averages of stresses or strains over small volumes that depend on the solution of a problem of elasticity. This interpretation perm...

In this work, we present a statistical treatment of stress-life (S-N) data drawn from a collection of records of fatigue experiments that were performed on 75S-T6 aluminum alloys. Our main objective is to predict the fatigue life of materials by providing a systematic approach to model calibration, model selection and model ranking with reference t...

The difference between finite element modeling and numerical simulation is explained. It is concluded that finite element modeling should not be relied upon in mechanical design.

How is it possible that finite element models are full of errors and yet produce credible results?

An algorithm for the computation of the homogenized material properties of unidirectional fiber-reinforced laminae is described. The method of extraction is such that superconvergence is realized. It is shown that the homogenized material properties are substantially independent of the number of RVEs. An algorithm for the determination of the micro...

Simulation governance is discussed from the perspectives of formulation and application of design rules in structural, mechanical and aerospace engineering. The key technical requirements are described in the context of what is variously called validation pyramid, building block method and validation experiment hierarchy and illustrated by an examp...

We study an old mathematical model, developed before the computer era, for analyzing the strength of a stiffened shell roof. The specific problem considered is a textbook example presented in K. Girkmann: Flächentragwerke, 3rd edition, 1954. Here the roof consists of a spherical dome and a stiffening ring of rectangular cross section attached to th...

Mathematical models and their numerical solution must be sufficiently reliable to justify basing engineering decisions on them. The reliability of a numerical solution is established through verification, the reliability of a mathematical model is evaluated by comparing predictions based on the model with the outcome of physical experiments. The co...

At the end of the solution process the finite element solution is stored in the form of data sets that contain the coefficients of the shape functions, the mapping functions and indices that identify the polynomial space associated with each element. Some of the data of interest, such as temperature, displacement, flux, strain, stress, can be compu...

The classical formulation of problems of linear heat conduction and linear elasticity in three dimensions is presented. Essential and natural boundary conditions as well as boundary conditions of convenience (symmetry, antisymmetry and periodicity) are described. The hierarchic view of mathematical models is presented through the formulation of spe...

In many cases it is advantageous to make certain a priori assumptions concerning the mode of deformation of an elastic body and use dimensionally reduced models instead of fully three-dimensional models. Dimensionally reduced models are well suited for structural analysis were the goals of computation are to determine structural stiffness, displace...

A finite element space is characterized by a finite element mesh and the polynomial degrees and mapping functions assigned to the elements. The basis functions are constructed from a polynomial space defined on a standard element. The standard element is mapped to the elements of the mesh by mapping functions. Isoparametric mapping and mapping by t...

In the process of conceptualization one usually starts with simple linear models bearing in mind the restrictive assumptions incorporated in those models. In order to properly formulate mathematical models for the representation of some physical reality, it is necessary to consider the effects of the various assumptions and, when necessary, remove...

Half-Title PageWiley Series PageTitle PageCopyright PageDedication PageTable of ContentsAbout the AuthorsSeries PrefacePreface

The concept of regularity of the exact solution is introduced. Understanding this concept is essential for analysts because proper construction of finite element spaces depends on the regularity of the underlying solution which can be determined from the attributes of the domain, the material properties, the loading and constraint conditions. Furth...

Conceptualization and validation are addressed in the context of fatigue and fracture of mechanical and structural components. The goal is to reduce epistemic uncertainties through systematic examination of alternative models in validation experiments. To support such an effort, it is necessary to design, implement and deploy a system of procedures...

The finite element method is outlined in one-dimensional setting. The classical and generalized formulations are described. The notions of energy space and finite element spaces are introduced. Conventional shape functions based on Lagrange polynomials and hierarchic shape functions based on Legendre polynomials are described. Key theorems are prov...

The generalized formulation for linear heat conduction problems and elasticity are derived in three dimensions. The conditions that permit dimensional reduction and the corresponding essential and natural boundary conditions are described. The main points are illustrated with examples.

The contour integral methodThe energy release rate

Gaussian quadratureGauss–Lobatto quadrature

A mathematical model formulated for the prediction of distortion of airframe components manufactured from 7050-T7451 aluminum plates, caused by residual stresses introduced by the manufacturing process of the plate, was tested in validation experiments. The results indicate that distortion in thin gauge parts is caused primarily by machining-induce...

This paper is concerned with the problem of verification of the numerical accuracy of computed information with particular reference to a model problem in solid mechanics. The basic concepts and procedures are outlined and illustrated by examples. KeywordsVerification-Reliability-Error estimation-Numerical approximation

The global maritime operating environment of U.S. Naval Aviation assets necessitates their prolonged exposure to severe corrosive environments. The resulting corrosion damage on flight critical structural components has a significant adverse impact on fleet readiness and total ownership costs. To address these issues, NAVAIR has initiated a multiye...

a b s t r a c t An overview of the general considerations underlying the selection of mathematical models and the methods used for estimation and control of the associated errors and the requisite technical capabilities are presented. An example that illustrates the main points of the paper is presented.

The crack compliance method is a destructive experimental method used for the estimation of residual stress profiles in thick metal plates. Simplifying assumptions, such as dimensional reduction, are generally used in applications of this method. The question of how the simplifying assumptions affect the estimated residual stresses is addressed in...

A mathematical model was developed to solve a steady free surface flow problem and a rapid drawdown problem in a two‐dimensional porous medium. The same problem was also solved by an analogue device and excellent agreement was found to exist between the two solutions. This paper contains the formulation of the numerical problem from first principle...

The reliability of the solutions of problems of two-dimensional linear elasticity is examined in relation to the solutions of the corresponding three-dimensional problems. It is shown that the solution of a long cylindrical body with free ends can be approximated by the classical plane strain solution plus a correction which is the solution of anot...

Shell-like structures are viewed as fully three-dimensional solid bodies that allow the imposition of restrictions on the transverse variation of displacement vector components in certain regions. An important practical problem is to select a simplified mathematical model for a particular application so that the simplifications do not affect the da...

Distortion and buckling of aluminum aerospace components can be caused by machining-induced residual stress or by residual stress induced earlier in material processing. This stress is characterized through layer removal experiments and measurements of surface location. This stress is correlated to two machining process parameters, which can be cha...

Procedures for verification and validation through adaptive control of the errors of discretization and idealization in finite element analysis (FEA), guided by feedback information, are discussed. A hierarchic framework for the construction of sequences of finite element spaces and working models is outlined and an example is presented.

The basic algorithmic structure and performance characteristics of the p-version of the finite element method are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the p-version is fully established, and a very substantial amount of engineering experience covering linear and nonlin...

The basic algorithmic structure and performance characteristics of the p-version of the finite element method are surveyed with reference to elliptic problems in solid mechanics. For this class of problems, the theoretical basis of the p-version is fully established, and a very substantial amount of engineering experience covering linear and nonlin...

Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal

The development of efficient and reliable methods for the design and analysis of fastened structural connections and bonded joints is among the most important and most challenging problems in aerospace applications because these connections are common sites of failure initiation and there is a complicated nonlinear interaction between the fasteners...

The steps undertaken for a high-order accurate simulation of fluid-structure inter-action problems are presented. Specifically, an interface is designed to couple a parallel spectral/hp element fluid solver NεκT αr with the hp-FEM solid solver StressCheck . The objective is to perform DNS of flows past wings using realistic representation of flows...

The solution to the Laplace operator in three-dimensional domains in the vicinity of straight edges is presented as an asymptotic expansion involving eigenpairs with their coefficients called edge flux intensity functions (EFIFs). The eigenpairs are identical to their two-dimensional counterparts over a plane perpendicular to the edge. Extraction o...

Higher order (p-version) finite element methods have been shown to be clearly superior to low order finite element methods when properly applied. However, realization of the full benefits of p-version finite elements for general 3-D geometries requires the careful construction and control of the mesh. A 2-D elasticity problem with curved boundary i...

The numerical treatment of mechanical contact problems in two dimensions using the p-version of the finite element method (FEM) coupled with minor iterative modification of the mesh is investigated. The method of solution is based on the augmented Lagrangian technique. The accuracy in the contact zone is increased using a special form of the hp-ver...

The definition, essential properties and formulation of hierarchic models for laminated plates and shells are presented. The hierarchic models satisfy three essential requirements: approximability; asymptotic consistency, and optimality of convergence rate. Aspects of implementation are discussed and the performance characteristics are illustrated...

Linear models for estimating the limits of elastic stability and the effects of stress stiffening are formulated, starting from the assumption that in its reference configuration the elastic body is subjected to an initial stress field which satisfies the equations of linear elasticity. When the body is subjected to some kinematically admissible pe...

The problem of finding a nearly optimal distribution of polynomial degrees on a fixed finite element mesh is discussed. An a posteriori error estimator based on the minimum complementary energy principle is proposed which utilizes the displacement vector field computed from the finite element solution. This estimator, designed for p- and hp-extensi...

Background: Magnetic resonance imaging tissue tagging is a relatively recent methodology that describes ventricular systolic function in terms of intramyocardial ventricular deformation. Because the analysis involves the use of many intramyocardial points to describe systolic deformation, it is theoretically more sensitive at describing subtle dif...

This final technical report presents a summary of a three years project in which the mathematical basis of failure criteria for metals and composite materials was investigated. The investigation covered criteria associated with crack formation and structural instability. The main results are: (a) General procedures for the determination of mathemat...

This grant was used for upgrading an SGI Power Challenge L supercomputer and for the acquisition of 3 SGI O2 workstations to serve DoD research projects in three general areas: (1) Research on the p- and hp-versions of the Finite Element Method conducted by the Center for Computational Mechanics. (2) Research on non-linear control systems conducted...

In the p-version of the finite element method the size of the elements is fixed independently of the number of degrees of freedom. Therefore, accurate representation of the curves and surfaces which bound the solution domain, so that the quality of the representation is independent of the number of elements, is very important. Another important req...

Prediction and measurement of residuum shape change inside the prosthesis under various loading conditions is important for prosthesis design and evaluation. Residual limb surface measurements with the prosthesis in situ were used for construction of a finite element model (FEM). These surface measurements were obtained from volumetric computed tom...

This paper presents a new formulation for geometrically non-linear problems and their numerical treatment by the p-version of the finite element method. The formulation is characterized by a weak form based on the spatial representation of the equilibrium equations, a deformed geometry mapped by the displacement field, a natural description for non...

An extension of the p-version of the finite element method, called the space enrichment method, is described and its application to normal contact problems, using the penalty and augmented Lagrangian formulations, is illustrated by examples. The contacting bodies are represented by the equations of the linear theory of elasticity. The numerical sol...

In end-stage pulmonary hypertension (PH), the degree of right ventricular (RV) dysfunction has been considered so severe as to require combined heart-lung transplantation. Nevertheless, left ventricular (LV) and RV hemodynamics return to relatively normal levels after single-lung transplantation (SLT) alone. Accordingly, to test the hypothesis that...

Objectives: To determine nonlinear material properties of passive, diastolic myocardium using magnetic resonance imaging (MRI) tissue-tagging, finite element analysis (FEA) and nonlinear optimization. Background: Alterations in the diastolic material properties of myocardium may pre-date the onset of or exist exclusive of systolic ventricular dy...

This paper presents a new formulation for geometrically nonlinear problems and their numerical treatment by the p-version of the finite element method. The formulation is characterized by a weak form based on the spatial representation of the equilibrium equations, a deformed geometry mapped by the displacement field, a natural description for non-...

The use of finite element analysis (FEA) software products in mechanical and structural design are examined. It is found that current FEA software products are being under-utilized because they do not satisfy the requirements of the design process well. Specifications for future FEA software products are outlined.

The solution of linear elastostatic problems in the neighborhood of singular points is characterized, by a sequence of eigenpairs and their coefficients, called generalized stress intensity factors (GSIFs). For general singular points, as cracks in anisotropic multimaterial interfaces, and V-notches in composite materials, only numerical approximat...