Equilibrium Finite Element Formulations - J. P. Moitinho de Almeida - ebook

Equilibrium Finite Element Formulations ebook

J. P. Moitinho de Almeida

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A comprehensive treatment of the theory and practice of equilibrium finite element analysis in the context of solid and structural mechanics Equilibrium Finite Element Formulations is an up to date exposition on hybrid equilibrium finite elements, which are based on the direct approximation of the stress fields. The focus is on their derivation and on the advantages that strong forms of equilibrium can have, either when used independently or together with the more conventional displacement based elements. These elements solve two important problems of concern to computational structural mechanics: a rational basis for error estimation, which leads to bounds on quantities of interest that are vital for verification of the output and provision of outputs immediately useful to the engineer for structural design and assessment. Key features: * Unique in its coverage of equilibrium - an essential reference work for those seeking solutions that are strongly equilibrated. The approach is not widely known, and should be of benefit to structural design and assessment. * Thorough explanations of the formulations for: 2D and 3D continua, thick and thin bending of plates and potential problems; covering mainly linear aspects of behaviour, but also with some excursions into non-linearity. * Highly relevant to the verification of numerical solutions, the basis for obtaining bounds of the errors is explained in detail. * Simple illustrative examples are given, together with their physical interpretations. * The most relevant issues regarding the computational implementation of this approach are presented. When strong equilibrium and finite elements are to be combined, the book is a must-have reference for postgraduate students, researchers in software development or numerical analysis, and industrial practitioners who want to keep up to date with progress in simulation tools.

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Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

List of Symbols

Chapter 1: Introduction

1.1 Prerequisites

1.2 What Is Meant by Equilibrium? Weak to Strong Forms

1.3 What Do We Gain From Strong Forms of Equilibrium?

1.4 What Paths Have Been Followed to Achieve Strong Forms of Equilibrium?

1.5 Industrial Perspectives

1.6 The Structure of the Book

References

Chapter 2: Basic Concepts Illustrated by Simple Examples

2.1 Symmetric Bi-Material Strip

2.2 Kirchhoff Plate With a Line Load

References

Chapter 3: Equilibrium in Other Finite Element Formulations

3.1 Conforming Formulations and Nodal Equilibrium

3.2 Pian's Hybrid Formulation

3.3 Mixed Stress Formulations

3.4 Variants of the Displacement Based Formulations With Stronger Forms of Equilibrium

3.5 Trefftz Formulations

3.6 Formulations Based on the Approximation of a Stress Potential

3.7 The Symmetric Bi-Material Strip Revisited

References

Chapter 4: Formulation of Hybrid Equilibrium Elements

4.1 Approximation of the Stresses

4.2 Approximation of the Boundary Displacements

4.3 Assembling the Approximations

4.4 Enforcement of Equilibrium at the Boundaries of the Elements

4.5 Enforcement of Compatibility

4.6 Governing System

4.7 Existence and Uniqueness of the Solution

4.8 Elements for Specific Types of Problem

4.9 The Case of Geometries With a Non-Linear Mapping

4.10 Compatibility Defaults

4.11 The Dimension of the System of Equations

References

Chapter 5: Analysis of the Kinematic Stability of Hybrid Equilibrium Elements

5.1 Algebraic and Duality Concepts Related to Spurious Kinematic Modes

5.2 Spurious Kinematic Modes in Models of 2D Continua

5.3 Spurious Kinematic Modes in Models of 3D Continua

5.4 Spurious Kinematic Modes in Models of Reissner–Mindlin Plates

5.5 The Stability of Plates Modelled With Kirchhoff Elements

5.6 The Stability of Models for Potential Problems

5.7 How Do We Obtain a Stable Mesh for General Structural Models?

References

Chapter 6: Practical Aspects of the Kinematic Stability of Hybrid Equilibrium Elements

6.1 Identification of Rigid Body and Spurious Kinematic Modes

6.2 Blocking the Spurious Modes

6.3 An Illustration of the Procedures to Remove Spurious Modes

6.4 How Do We Recognize Admissible Loads?

6.5 Quasi-Simplicial Hybrid Elements Created by Hierarchical Mesh Refinement

6.6 Non-Simplicial Hybrid Elements

6.7 A Cautionary Tale of ‘Near Misses’

References

Chapter 7: A Variational Basis of the Hybrid Equilibrium Formulation

7.1 Potential Energy and Complementary Potential Energy

7.2 Hybrid Complementary Potential Energy

7.3 Properties of the Generalized Complementary Energy

7.4 The Babuška–Brezzi Condition and Hybrid Equilibrium Elements

References

Chapter 8: Recovery of Complementary Solutions

8.1 General Features of Partition of Unity Functions

8.2 Recovery of Compatibility From an Equilibrated Solution

8.3 Recovery of Equilibrium From a Compatible Solution

8.4 Numerical Examples

8.5 Extensions of the Recovery Procedures

References

Chapter 9: Dual Analyses for Error Estimation & Adaptivity

9.1 Global Error Bounds

9.2 Estimation of the Error Distribution and Global Mesh Adaptation

9.3 Obtaining Local Quantities of Interest

9.4 Bounding the Error of Local Outputs

9.5 Local Outputs for the Kirchhoff Plate With a Line Load

9.6 Estimation of the Error Distribution and Mesh Adaptation for Local Quantities

9.7 Adaptivity for Multiple Loads and Multiple Outputs

References

Chapter 10: Dynamic Analyses

10.1 Toupin's Principle for Elastodynamics

10.2 Derivation of the Equilibrium Finite Element Equations

10.3 Analysis in the Frequency Domain

10.4 Analysis in the Time Domain

10.5 No Direct Bounds of the Eigenfrequencies?

10.6 Example

References

Chapter 11: Non-Linear Analyses

11.1 Elastic Contact

11.2 Material Non-Linearity

11.3 Limit Analysis

11.4 Geometric Non-Linearity

References

A: Fundamental Equations of Structural Mechanics

A.1 The General Elastostatic Problem

A.2 Compatibility of Strains

A.3 General Elastodynamic Problem

References

B: Computer Programs for Equilibrium Finite Element Formulations

B.1 Auxiliary Programs

B.2 Structure of the Programs

References

Subject Index

End User License Agreement

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cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Chapter 1: Introduction

Figure 1.1 Robert Hooke (1676): ‘As hangs the flexible line, so, but inverted will stand the rigid arch’ (Heyman, 1982). The chain adapts its shape so that internal tensions balance the concentrated vertical forces, and transfer them to the ground/supports. The reflection of the shape of the chain is a thrust line, the trajectory of the compressive stress resultants. Source: Adapted from the hanging chain, Robert Hooke (1676), requoted from Heyman 1982.

Figure 1.2 Statically indeterminate truss and a possible system of force transmission: (

a

) and (

b

) for external loads, from joints to ground; (c) for internal forces that are self-balanced.

Figure 1.3 Determination of a displacement of a hyperstatic beam using a statically admissible virtual moment distribution.

Chapter 2: Basic Concepts Illustrated by Simple Examples

Figure 2.1 Bi-material strip and symmetry simplification.

Figure 2.2 Bi-material strip: stress distributions obtained for . The values for the colours are: for ; for ; for , with blue representing the minimum value, red the maximum and green the central value.

Figure 2.3 Bi-material strip: strain distributions obtained for . The values for the colours are: for ; for ; for .

Figure 2.4 Bi-material strip: strain energy of the solutions as a function of , for .

Figure 2.5 Simply supported plate of thickness . Plan and isometric views.

Figure 2.6 Simply supported plate: some moment distributions. The ranges of the colours are: for ; for ; for .

Figure 2.7 Simply supported plate: error of the strain energy of the solutions as a function of the degree of the approximation functions used.

Chapter 3: Equilibrium in Other Finite Element Formulations

Figure 3.1 Fraeijs de Veubeke's equilibrated triangle.

Figure 3.2 Patches of Fraeijs de Veubeke's equilibrated triangles. When the loading is restricted to the external sides, the patch on the left is stable, whereas the other two will not always achieve equilibrium.

Figure 3.3 Admissible and inadmissible loadings for an unstable mesh of Fraeijs de Veubeke's equilibrated triangles.

Figure 3.5 Bi-material strip: Deformed shapes of the bi-material strip. Obtained from models based on cubic approximations on a mesh of eight elements, .

Figure 3.4 Bi-material strip: stress distributions for the bi-material strip. Obtained from models based on cubic approximations on a mesh of eight elements, . The ranges of the colours are: for ; for ; for .

Chapter 4: Formulation of Hybrid Equilibrium Elements

Figure 4.1 Projection of a basis of linear self-equilibrated stress distributions for two dimensional problems on the boundary of a square region with the lower left corner at the origin of the reference frame. Note that each distribution is in equilibrium, that is, it has zero resultant forces and moments, in the absence of body loads.

Figure 4.2 Mesh for a two dimensional problem.

Figure 4.3 Two dimensional problem: Illustration of the spurious kinematic mode and an alternative stable mesh.

Figure 4.4 Square cantilever. Definition of the boundary conditions, material properties, mesh, reference frames and components of the stress tensor either in the or in the reference frame.

Figure 4.5 Square cantilever. Distributions of the stress components in the two frames considered. The range of values for the colour representation are: for ; for ; for ; for , and ; with blue representing the minimum value, red the maximum and green the central value.

Figure 4.6 Square cantilever. Element tractions and stress trajectories. The width of each trajectory varies proportionally to the corresponding principal stress, from − 4 to 4, with blue representing the minimum value, red the maximum and green the central value.

Figure 4.7 Square cantilever. Boundary displacements, on the left with a non-zero spurious mode amplitude, on the right with the spurious mode blocked.

Figure 4.8 A finite element model of the plate problem.

Figure 4.9 Illustrations of the shear stresses and warping functions on the cross-section of a prismatic bar caused by the action of different stress resultants. A mesh with 522 elements, with approximations of degree 4 was used to obtain these values.

Figure 4.10 Possible configurations of the side

ABC

.

Chapter 5: Analysis of the Kinematic Stability of Hybrid Equilibrium Elements

Figure 5.1 Notation used for a generic triangular element and definition of the side coordinates. The orthogonal reference frame used for side 3 is also illustrated. Note that this frame is different from the oblique frame used at each vertex.

Figure 5.2 Covariant and contravariant base vectors on the sides adjacent to corner 3. The illustrated tractions include all components, whereas the displacements are only those of a spurious mode for which .

Figure 5.3 Illustration of the spurious kinematic modes associated with the top corner for to 4.

Figure 5.4 Illustrations of dependent spurious modes when .

Figure 5.5 A pair of triangular elements, with a rigid body displacement of element .

Figure 5.6 Illustration of the spurious kinematic modes of degree 0, 1 and 2 that can be transmitted between a pair of elements. Non-degenerate and degenerate case, the latter is illustrated when either the rigid body movement of the common interface or the external sides are blocked.

Figure 5.7 Simplicial neighbourhoods of vertices, and their links.

Figure 5.8 Star patches of elements of degree 0.

Figure 5.9 Open star patches of elements of degree 1.

Figure 5.10 Open star patches of elements of degree 2.

Figure 5.11 Examples of unstable closed star patches of degree 2.

Figure 5.12 An external element may buttress a star patch.

Figure 5.13 A hexagonal star patch extracted from a tessellation.

Figure 5.14 A uniform mesh of isosceles elements, with its spurious mode of degree 2.

Figure 5.15 Notation used for the neighbourhood of an edge of a tetrahedral element presented in different views, including three first angle orthographic projections.

Figure 5.16 Oblique reference frame for a normal section along an edge where is the dihedral angle; at a face the direction of the displacement associated with a spurious kinematic mode of an edge is parallel to the normal of the opposite face.

Figure 5.17 Face signature functions, , for different approximation degrees.

Figure 5.18 The three spurious modes associated with the edge shown in bold, when .

Figure 5.19 Local numbering of vertices and edges of a face/interface.

Figure 5.20 Normalized functions , .

Figure 5.21 A primary spurious mode of degree which can be transmitted to a neighbouring element via the face containing point , as defined in Figure 5.19.

Figure 5.22 Examples of the possible cases of pairs of tetrahedral elements with coplanar faces.

Figure 5.23 Patch of tetrahedral elements centred on edge 2–3.

Figure 5.24 Star patches of tetrahedral elements.

Figure 5.25 Oblique components of moment and rotation on sides adjacent to a corner.

Figure 5.26 A pseudo-rigid body rotation for degree .

Figure 5.27 Hexagonal and octagonal patches with .

Figure 5.28 Pseudo-rigid body modes for a Kirchhoff element when degree . The initial position of the sides is marked in light grey and the arrow represents the unit vertex displacement.

Chapter 6: Practical Aspects of the Kinematic Stability of Hybrid Equilibrium Elements

Figure 6.1 Rotated square: Definition of the problem and finite element mesh.

Figure 6.2 Rotated square with approximations of degree 3: Spurious mode, polluted and clean boundary displacements. Notice that the displacements in part (c) are not actually continuous at the vertices, although they appear to be so.

Figure 6.3 Admissible and inadmissible discontinuous tractions on a patch of 2D elements where a spurious kinematic mode is present.

Figure 6.4 Hierarchical refinement of a mesh of triangular elements.

Figure 6.5 Shallow 3-pinned arch.

Figure 6.6 Rotated square: definition of the problem and simple finite element meshes with 4 and 5 elements.

Figure 6.7 Horizontal load applied to the left of ; strain energy of the solution for the simple meshes, as a function of the position of , using elements of degrees 1, 3 and 6.

Figure 6.8 Vertical load applied to the left of ; strain energy of the solution for the simple meshes, as a function of the position of , using elements of degrees 1, 3 and 6.

Figure 6.9 Horizontal load applied across the whole diagonal of the plate; strain energy of the solution for the simple meshes, as a function of the position of , using elements of degrees 1, 3 and 6.

Figure 6.10 Rotated square: action of a partial horizontal load. Convergence curves in terms of the strain energy, and corresponding convergence rates, , for a uniform refinement of the simple finite element meshes. The initial mesh has 4 elements, except for the ‘Partial @0’ solution, where coincides with the diagonal; in this case the initial mesh has five elements.

Chapter 8: Recovery of Complementary Solutions

Figure 8.1 A partition of unity function over a closed star patch.

Figure 8.2 A partition of unity function applied to a 1-dimensional patch.

Figure 8.3 Star patches in the recovery of compatibility.

Figure 8.4 The initial position of an element, its boundary displacements and the element-wise displacement field. is chosen to minimize the integral on the boundary of the square of .

Figure 8.5 A 1-dimensional example. The mesh is uniform, with all elements of length .

Figure 8.6 Plot of axial displacements of patch .

Figure 8.7 Star patches in the recovery of equilibrium

Figure 8.8 Neighbourhoods of element

Figure 8.9 Rotational displacements over a closed star patch in 2D.

Figure 8.10 Decomposition of vertex forces in a closed star patch.

Figure 8.12 Decomposition of vertex forces in an open star patch.

Figure 8.11 Maxwell diagram with target force vectors.

Figure 8.13 Hierarchical shape functions and their dual traction functions for or 2.

Figure 8.14 Variation of tangential tractions in a closed star patch.

Figure 8.15 Square plate. Definition of the boundary conditions, material properties, mesh, reference frames and components of the stress tensor either in the or in the reference frame.

Figure 8.16 Boundary displacements of the equilibrated solution in which the spurious mode has been removed. Notice how the displacements at the vertices, amplified five times in the detail, are not exactly continuous.

Figure 8.20 Compatible displacements: as recovered from the equilibrated solution; as obtained from the compatible model; and their difference. The maximum amplitudes of the displacements are 2.23924 and 2.27220. The maximum amplitude of the difference, which does not occur where the displacement is maximum, is 0.043216. Different scales are applied to the arrows in order to make the small differences visible.

Figure 8.17 Boundary displacements of an equilibrated solution affected by the spurious mode. In this Figure the detailed displacements are shown with the same scale as the whole mesh.

Figure 8.18 The first row shows the element-wise displacement fields obtained from the boundary displacements in Figure 8.16 and 8.17, that is, unaffected and affected by the spurious mode. In the second row we show the displacement field corresponding to the difference between these two solutions, which is a rigid body motion for each element. Different scale factors are used for each plot, in order to better visualize them.

Figure 8.19 Compatible displacements on the star patches, recovered from the equilibrated solution that is free from spurious modes.

Figure 8.21 Stresses obtained from the compatible solution.

Figure 8.23 (a) Equilibrated stresses recovered from the compatible model (b) Stresses obtained from the equilibrated model.

Figure 8.22 Equilibrated stresses on the star patches, recovered from the compatible solution.

Chapter 9: Dual Analyses for Error Estimation & Adaptivity

Figure 9.1 Different problems considered for the computation of errors.

Figure 9.2 Convergence of the error bound for a uniform mesh refinement as a function of the number of elements . Approximations of degree 1 to 4 are used, from top to bottom, for the three problems previously considered.

Figure 9.3 ‘Exact’ energy error for a uniform mesh refinement as a function of the number of degrees of freedom of the minimal system. Approximations of degree 1 to 4 are used, from top to bottom, for the three problems previously considered. Solid and dashed lines are used for the equilibrated and for the compatible solutions.

Figure 9.4 Convergence of the error bound for an adaptive mesh refinement as a function of the number of elements . Approximations of degree 1 to 4 are used, from top to bottom, for the three problems previously considered. The dashed line corresponds to an approximation of degree 4 where the lower convergence at non-regular points is also considered, together with a smaller reduction of the error in each step.

Figure 9.5 Adaptive process—first meshes with more than 500 elements for the mixed problem. The pair of numbers in ‘Dofs’ indicates the dimensions of the minimal systems, as defined in Section 4.11, for the equilibrated and the compatible systems respectively.

Figure 9.6 Setting the amplitude of the vertical virtual load equal to the average vertical displacement in that region is computed. When the imposed normal gap displacement that varies linearly leads to the moment due to the normal stress around the tip of the opening. A unit value for the vertical displacement of the support, , produces the vertical reaction force.

Figure 9.7 Corner displacement of the simply supported plate. Contour lines of the output and of the corresponding error bound, for arbitrary compatible solutions.

Chapter 10: Dynamic Analyses

Figure 10.1 Domain and boundary velocities for the incorrect eigenmode with frequency 0.02 of mesh

1 offset

, with linear stress approximations. The maximum amplitudes of the velocities in the domain and on the boundary are 1.01058 and 283.557 respectively.

Figure 10.2 Relative errors in the first three frequencies for mesh

1

. Notice that for the values presented there is only one case where the numerical frequencies are lower than the reference values; it is marked with a circle and a dotted line is used for the segment joining values with different signs.

Figure 10.3 Domain and boundary velocities for the first six modes of mesh

1

with quartic stress approximations. The corresponding eigenfrequencies are given in Table 10.1.

Figure 10.4 Response curves for mesh

1

, subjected to a harmonic linear load on the top.

Chapter 11: Non-Linear Analyses

Figure 11.1 Contact problem with elastic body and rigid obstacle.

Figure 11.2 Stress-strain relations for plasticity defined in the stress space.

Figure 11.3 Stress-strain relations for plasticity defined in the strain space.

Figure 11.5 (a) Location of the points where plastic strains develop for the ultimate load on the mesh with 412 elements, with a cubic approximation of the moments and 91 control points; distribution of moments in (b) and in (c) for quadratic moments and 45 control points.

Figure 11.4 Meshes used for limit analysis of the square plate simply supported on two adjacent sides, subject to a uniformly distributed load.

Figure 11.6 Large displacements of a body, with small strains. The undeformed configuration marked with the dashed line considers just the effect of the rigid body translation, , while the dotted line correspond to the effect of .

Figure 11.7 Large displacements of a body, with small strains. The position vectors of a point in the initial, rotated and final configurations, , and .

Figure 11.8 Large displacements of a body, with small strains. When the rigid body translation is removed, the strain inducing displacement, , is equal to the difference between the position vector of the deformed configuration, , and the rotated undeformed position vector, .

Figure 11.9 Axially loaded beam with imperfection. Initial geometry and finite element mesh.

Figure 11.10 Axially loaded beam with imperfection. Horizontal () and vertical () displacement curves of the point at the centre of the free end. The vertical direct Cartesian stress () for different load factors is also represented, with the elements subjected to their rigid body displacements. The ranges of the contours are for ; for ; and for .

A: Fundamental Equations of Structural Mechanics

Figure A.1 Positive Cartesian components of stress in 2D.

Figure A.2 Positive Cartesian components of stress in 3D.

Figure A.3 Reference frame used for the beam and shear stresses.

Figure A.4 Positive Cartesian stress resultants for plate bending, acting on an infinitesimal element and projected onto a general boundary. The positive plate rotations are also illustrated.

B: Computer Programs for Equilibrium Finite Element Formulations

Figure B.1 Contents of a simple bidimensional mesh file (

Simple2D.msh

) and the corresponding mesh.

Figure B.2 Contents of a simple tridimensional mesh file (

Simple3D.msh

) and the corresponding mesh.

Figure B.3 A scalar field defined on the mesh of Figure B.1, using a $NodeData file section.

Figure B.4 Problem definition for the mesh of of Figure B.1.

Figure B.5 For the shared side we have that, in general, . A change of coordinates is necessary in order to guarantee that, for example, . We opt to define all functions in terms of , from which the coordinates in the frame of the element are obtained.

List of Tables

Chapter 2: Basic Concepts Illustrated by Simple Examples

Table 2.1 Simply supported plate: Strain energy (

U

) of the compatible solutions, scaled by , as a function of the degree of the approximation (

d

) and of the number of variables (dof)

Table 2.2 Simply supported plate: Complementary strain energy (

U

c

) of the equilibrated solutions, scaled by , as a function of the degree of the approximation (

d

) and of the number of variables (dof).

Chapter 3: Equilibrium in Other Finite Element Formulations

Table 3.1 Bi-material strip: Average values of the horizontal and vertical displacements at the top right corner. Reference values are and

Table 3.4 Bi-material strip: Average values of the tangential stress at the left hand end of the interface, for both materials. Reference value is 0

Table 3.5 Bi-material strip: Relative error in strain energy and relative strain energy of the error

Table 3.3 Bi-material strip: Average values of the vertical direct stress at the left hand end of the interface, for both materials. Reference value is 0.12547

Chapter 4: Formulation of Hybrid Equilibrium Elements

Table 4.1 Ratio between the dimension of the equilibrated and the compatible systems for two dimensional continua

Table 4.2 Ratio between the dimension of the equilibrated and the compatible systems for three dimensional continua

Chapter 5: Analysis of the Kinematic Stability of Hybrid Equilibrium Elements

Table 5.1 Number of malignant modes transmitted between a pair of elements of general degree (Maunder

et al

., 2016)

Chapter 8: Recovery of Complementary Solutions

Table 8.1 Magnitude of the stress components in Figures 8.21 to 8.23

Chapter 9: Dual Analyses for Error Estimation & Adaptivity

Table 9.1 Energies of equilibrium finite element solutions

Table 9.4 Energies of recovered (via partition of unity) compatible solutions

Table 9.2 Energies of recovered (via partition of unity) equilibrium solutions

Table 9.7 Mixed problem. Bounds of the energy error, differences of the energies and ‘exact’ errors for selected solutions

Table 9.5 Force driven problem. Bounds of the energy error, differences of the energies and ‘exact’ errors for selected solutions

Table 9.3 Energies of compatible finite element solutions

Table 9.6 Displacement driven problem. Bounds of the energy error, differences of the energies and ‘exact’ errors for selected solutions

Table 9.8 Convergence data () for the force driven problem with uniform meshes

Table 9.9 Convergence rates for the adaptive process

Table 9.10 Simply supported plate: estimated values of the average displacement on the loaded side and bounds of their errors. Both equilibrated solutions consider only a linear twisting moment. Both compatible solutions are polynomials of degree . The exact value assumed for is 1.68851

Table 9.11 Simply supported plate: estimated values of the average displacement on the free side and bounds of their errors. Both compatible solutions consider only a bilinear transverse displacement. Both equilibrated solutions are polynomials of degree . The exact value assumed for is 1.51226

Table 9.12 Simply supported plate: estimated values of the average displacement on the free side and bounds of their errors. All solutions use polynomial approximations of degree . The exact value assumed for is 1.51226

Chapter 10: Dynamic Analyses

Table 10.1 The first six eigenfrequencies of the equilibrium finite element models

Table 10.2 The first six eigenfrequencies of the compatible finite element models

Chapter 11: Non-Linear Analyses

Table 11.1 Limit loads for quadratic stress approximations and Nielsen's yield criterion

B: Computer Programs for Equilibrium Finite Element Formulations

Table B.1 Main element

types

in

gmsh

Equilibrium Finite Element Formulations

 

 

J. P. Moitinho de Almeida

Department of Civil Engineering, Architecture and Georesources, Instituto Superior Técnico, University of Lisbon, Portugal

 

 

Edward A.W. Maunder

College of Engineering, Mathematics and Physical Sciences, University of Exeter, United Kingdom

 

 

 

 

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Library of Congress Cataloging-in-Publication Data

Names: Almeida, J. P. Moitinho de (Jose Paulo), 1958- author. | Maunder, E. A. W., author.

Title: Equilibrium finite element formulations / J. P. Moitinho de Almeida, Edward A. W. Maunder.

Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2016. | Includes bibliographical references and index.

Identifiers: LCCN 2016039762 (print) | LCCN 2016045455 (ebook) | ISBN 9781118424155 (cloth) | ISBN 9781118926215 (pdf) | ISBN 9781118926208 (epub)

Subjects: LCSH: Finite element method. | Equilibrium. | Structural analysis (Engineering)–Mathematics.

Classification: LCC TA347.F5 A394 2016 (print) | LCC TA347.F5 (ebook) | DDC 518/.25–dc23

LC record available at https://lccn.loc.gov/2016039762

A catalogue record for this book is available from the British Library.

Cover Design: Wiley

Cover Credits: borzaya/Gettyimages

 

 

To

Bruce M. Irons

Baudouin M. Fraeijs de Veubeke

John C. de C. Henderson

For their inspiration to think differently.

Preface

This book is the result of many years of joint work and fun with equilibrium finite element formulations. Although they are often regarded as the ugly duckling of computational mechanics, we know that they have characteristics that are particularly attractive. It thus became our mission to spread the word that a ‘strongly equilibrated finite element’ is not a contradiction.

It all started a long time ago, in the late 1960s, when Edward, then a PhD student in civil engineering at Imperial College, attended a lecture by Fraeijs de Veubeke.1 In retrospect he now has some questions that might have been interesting at the time, but he also admits that the recognition of the practical relevance of the equilibrium formulations for continua that were presented only really came later, while doing structural design of reinforced concrete using stress fields obtained from displacement based finite elements.

Edward was also fortunate to have John Henderson as supervisor and later as a friend, colleague and father figure. John was a polymath with a firm belief in the benefits of a proper mathematical foundation to structural analysis, from vector spaces to algebraic topology. He encouraged the transfer of knowledge gained from the analysis of aircraft structures to the analysis of civil engineering structures, using concepts of static-kinematic duality in the pursuit of equilibrium via flexibility methods.

Zé Paulo's discovery of how to impose strong forms equilibrium, and as consequence the discovery of Fraeijs de Veubeke's and John Henderson's work, happened in 1985, also at Imperial College. His objective was to figure out the characteristics of the solutions of simple elastoplastic models which could enforce either equilibrium or compatibility, leading to interesting complementarities in the results.

Bruce Iron's book Techniques of Finite Elements, published in 1980, with its clear, imaginative and friendly style, focused on ‘enabling the understanding of mathematical and physical concepts, because effective trouble-shooting is best achieved with such harmony’, also had a very strong influence on both authors.

The stars were thus aligned for equilibrium when we first met in the office of David Lloyd Smith at Imperial College, sometime in 1992.2 After many papers, projects and conferences, where stress fields and spurious kinematic modes were dissected to exhaustion, we decided to collect our ideas in a book.

The resulting text provides a comprehensive presentation of an equilibrium formulation of the finite element method, principally with application to 2D, 3D and plate flexure problems in structural mechanics, when strains can be assumed to be infinitesimally small. Equal weight is given to the construction of stress fields that strongly satisfy equilibrium within and between elements, as well as displacements on element boundaries that are ‘broken’ or discontinuous at vertices of 2D plate elements or edges of solid elements.

We present up-to-date developments which enable dual analyses of models to be undertaken, either as a means of verification or as an alternative source of output that may be more directly useful to design engineers.

The book starts with a simple introduction of the concepts involved, followed by a historical introduction of equilibrium in the context of finite element analysis, and comparison with other formulations. We discuss the details of the equilibrium formulation in the context of modelling linear elastic static and dynamic behaviour with particular emphasis on the associated problems of spurious kinematic modes. A more mathematical justification of the formulation is included, where we propose a relevant functional to be used in the variational analysis of the saddle point problem, and attempt to explain its significance in engineering terms to a non-mathematician. We then proceed to present methods to recover complementary conforming and equilibrating solutions from each other, and show how the dual nature of such solutions enables bounds to be enumerated on global or local quantities of interest. The text concludes by opening routes to extending the formulation in order to simulate various forms of non-linear behaviour.

We make particular effort to explain the more mathematical concepts in straightforward terms which we hope will be understandable by the intended readership, namely senior undergraduates of engineering and applied mathematics, graduate researchers and practising engineers with an interest in verification, duality and safe structural design.

The topics are illustrated with a range of numerical examples which have been carefully designed to be simple, but of just sufficient complexity to highlight particular features.

The book contains two appendices: the first to summarize the fundamental equations of structural mechanics, and the second to serve as a companion to the computer programs that were developed in the course of writing the book. These programs are available on request to the first author and we intend to publish them, when they are more mature, under an open source licence.

The time from initial thoughts, in 2009, to the formal proposal, submitted in April 2011 and accepted in March 2012, was almost as long as that to complete the text. As usual, longer than we had anticipated.

We thank, first and foremost, our families for their patience and moral support. Our gratitude also extends to all the colleagues and friends who, directly or indirectly, and in many different ways, have helped us in developing our ideas before or during the writing of this book: in particular to Orlando Pereira and Pierre Beckers who read the manuscript, as well as to Angus Ramsay, Antonio Huerta, Bassam Izzuddin, Bill Harvey, Carlos Tiago, David Lloyd Smith, Eduardo Arantes e Oliveira, João Teixeira de Freitas, John Robinson, Luiz Fernando Martha, Pedro Diez, Philippe Bouillard and Pierre Ladevèze.

We do not forget to thank all the people in the editorial and production teams at Wiley, for their patience and advice throughout the preparation of this book. A special mention should be made regarding the helpful suggestions by the copy editor, Chris Cartwright.

Our recognition obviously includes all those who we have forgotten—sorry for that.

We hope you enjoy this book as much as we enjoyed writing it.

Lisbon and Exeter, March 2016 Zé Paulo and Edward

 

1

 Probably in the Department of Aeronautics, where he also attended memorable lectures given by Kelsey, of Argyris and Kelsey fame!

2

 Following the publication of ‘An alternative approach to the formulation of hybrid equilibrium finite elements’.

List of Symbols

The general meaning of the most commonly used symbols is given in this list. Other meanings may be specified in the text.

Nullspace of

Area of a face

or

(Generalized) body force vector or component

Equilibrium/compatibility matrix for the mixed formulations

or

Curvature vector or component

or

Differential compatibility or equilibrium operator

Equilibrium/compatibility matrix for the hybrid formulations

Degree of a given polynomial approximation

Rigid body displacement

Young's modulus

or

(Generalized) strain vector or component

Bound of the error of a pair of solutions

Element flexibility matrix

Material flexibility matrix

Boundary of the problem

or

Shear strain vector or component

Element stiffness matrix

Material stiffness matrix

Signature function for an edge (

) or a face (

)

Area coordinate on a face

Lagrange multiplier function

Length of a side

Link of vertex

Local output

or

Distributed bending moment vector or component

Mobility matrix

Boundary normal operator

Number of parameters of a given approximation

Simplicial neighbourhood of vertex

Energy norm (may be defined for a displacement, strain, or a stress field)

Poisson's ratio

Domain of the problem

Eigenfrequency

Internal angle at a vertex of a triangle or dihedral angle at an edge of a tetrahedron

Total potential energy

Total complementary energy

Generalized complementary energy

Partition of unity function

or

Interpolation matrix or function

or

Distributed shear force vector or component

Mass density

Stress approximation matrix

or

(Generalized) stress vector or component

Stress approximation parameters

Chapter 1Introduction

1.1 Prerequisites

A very concise description of this book is that it

presents a methodology to predict and explain the distribution of forces and deflections that develop within a loaded structure.

For a layman who is unfamiliar with structural analysis, this description requires further explanations of many important points, namely: What is a structure? What are the forces within the structure? What are loads? Why and how do they get distributed?

We will not try to address these questions. For their answers a basic book on structural analysis, for example, Coates et al. (1988); Hibbeler (2008) or Marti (2013) will provide the necessary knowledge on the concepts used to describe structural behaviour—equilibrium, compatibility and constitutive relations—as well as the variables involved—forces, displacements, stresses and strains.

For all but the simplest problems, the mathematical equations used to describe the relations between these structural variables cannot be solved in a closed form. Of the various techniques that are used to obtain approximate solutions of these equations we will focus our attention on the application of a particular technique, the finite element method (FEM). Though it is possible to gain an understanding of FEM concepts solely from the information that will be presented in this text, it is more convenient to start with a basic book on finite element procedures, for example, Fish and Belytschko (2007).

We will therefore assume that the reader has a basic knowledge of the problems of structural analysis, namely of the fundamental equations of solid mechanics and, at least, some understanding of the procedures involved in the application of the FEM, most probably using a conventional displacement based formulation.

Such a reader, probably with an engineering background in the context of aeronautical, civil or mechanical engineering applications, given a title which includes ‘equilibrium’ and ‘finite elements’ might rather wonder: ‘Why another book? The finite element method is well known and it provides solutions that satisfy equilibrium. Doesn't it?’ The fact is that in most cases it doesn't, since only an approximate form of equilibrium is achieved by displacement based finite element formulations.

Our text presents a way to obtain solutions that are different from the ‘usual’ ones, because they exactly satisfy equilibrium. Nevertheless, since they normally omit the strict enforcement of compatibility conditions, it is not possible to say a priori which will be better. They just fail in different ways.

We believe that exploiting the complementarity of the two approaches allows for an interpretation of the results that is more profound than what is possible with a single type of analysis, naturally providing the tools for the assessment of their quality.

That, in the end, is our goal. Explaining in detail how equilibrated solutions can be obtained is just a step towards it.

1.2 What Is Meant by Equilibrium? Weak to Strong Forms

We expect the reader to understand what is meant by a free body, and being in a state of equilibrium, that is, the forces and their moments sum to zero. However, although checks on equilibrium at the global or overall level of a structure, for example, as represented by its finite element model, are commonly undertaken, deeper investigations into local levels of equilibrium become more problematic.

In FEM there are various shades of meaning, and perhaps expectation, when considering local equilibrium. The concept of a free body normally starts at the level of an infinitesimal element in a continuum (i.e. strong equilibrium between body forces and stresses), which is itself a mathematical abstraction—since we ignore the microscopic structure of the material.

Then the concept moves to the level of a single finite element, and then it may move back to another mathematical abstraction—a node of an element, where we invoke the concept of nodal forces (i.e. corresponding to a weak form of equilibrium between statically equivalent forces).

It is relevant here to note that the concept of a nodal force may not be explicitly mentioned in texts on finite elements, and we are aware of commercially available software where nodal forces are not available to the user, but only stress contours and tables of stresses at particular points!

In practice some confusion exists, and engineers may be unaware of the ‘subtle’ distinctions between these different levels of equilibrium of free bodies, and their significance to the analysis of a finite element model. We frequently hear of engineers who look blank when advised that local equilibrium is usually violated—they appear to have a firm conviction that equilibrium is being satisfied in all necessary aspects. Their first response might be: ‘Does it matter if there are local violations?’

An appropriate reply might be: ‘It all depends on your needs and how well you know the distribution of the loads.’ This is a matter of judgement, but we would advise that engineers, when faced with many uncertainties, can proceed with more confidence knowing that their analysis provides complete equilibrium. Local violations can be regarded as residual loads that are equilibrated by the errors in stress, and such loads are made orthogonal to the displacements allowed by a conforming model. By refining the model, the solutions converge, even when residual loads persist.

Our starting point is the fact that conventional finite element analyses ‘provide solutions that equilibrate the equivalent nodal forces’, where the adjective equivalent plays a central role that is often disregarded in the more basic introductions to the FE method.

Effectively there is equilibrium of equivalent nodal forces in the solutions provided by most FE programs. We will discuss in detail what that means and we will conclude that, in most cases, there are no nodal forces as such. Energetically consistent nodal forces are defined, which are required to produce the same work as the real forces and stresses for all displacements considered. But, in general, this is not sufficient to guarantee equilibrium in a strong, or pointwise, sense.

This happens because only a finite subset of the possible displacements can be included in a given model and the solution space is generally infinite, therefore equilibrium is imposed on an average, or weak form. Generally

the solutions provided by displacement based FE models do not enforce the equilibrium conditions at every point of the domain and/or its boundary.

Our objective is to present in this book a methodology whose models produce solutions that strongly verify all equilibrium conditions. As always there is a drawback for every new approach. In this case the gain in terms of equilibrium will imply a loss in terms of compatibility, which will only be imposed in a weak form.

We will not pretend that these equilibrium formulations are always better than their displacement based counterparts, as each formulation locally enforces one set of conditions, while imposing a weak form of the other.

1.3 What Do We Gain From Strong Forms of Equilibrium?

The complementary nature of these formulations is, in our opinion, the strongest reason for considering solutions obtained from both approaches. It does not matter which one is considered first, as the different approximate solutions that they produce are complementary, in the sense that they satisfy complementary equations in a strong and in a weak form.

As we will show, this complementarity can be used in a natural way to assess the quality of the solutions, and to drive a mesh adaptation process, deciding where it is important to have more, or fewer elements. From a practical point of view it is also relevant to point out that equilibrium solutions have the advantage of being immediately usable as a safe basis for design of ductile structures, when the Static Theorem of Limit Analysis can be invoked (fib, 2013; Marti, 2013; Nielsen and Hoang, 2010).

In particular, equilibrium solutions give us a more rational way of accounting for stress concentrations, especially when they arise due to mathematical singularities where the structural geometry has been simplified, for example, at re-entrant corners. Displacement models tend to pollute local equilibria while seeking infinite values of stress, while equilibrium models continue to provide a statically admissible field in the neighbourhood of a mathematical singularity. Such a stress field represents a redistribution of stress near the singularity, and in this respect it is similar to the behaviour of the real structure, which may adapt itself by local yielding while maintaining stress equilibrium.

On a historical note we recall that, before the introduction of computer techniques, methods requiring strong equilibrium as their starting point formed the basis for most structural design procedures.

Considering the design of arches and their thrust lines, Figure 1.1, the analysis of statically indeterminate trusses and frames by Maxwell-Mohr methods, Figures 1.2 and 1.3, and also in variational methods such as the Trefftz method, we find that most ‘historical’ methods of analysis, in essence, sought to explain how the transmission of forces through the structure may be achieved. The reason for this can be explained by the fact that material resistance is intuitively related to the level of stress within the structure—displacements do not appear to be as important—leading to the application, implicit or explicit, of the Static Theorem of Limit Analysis, already mentioned.

Figure 1.1 Robert Hooke (1676): ‘As hangs the flexible line, so, but inverted will stand the rigid arch’ (Heyman, 1982). The chain adapts its shape so that internal tensions balance the concentrated vertical forces, and transfer them to the ground/supports. The reflection of the shape of the chain is a thrust line, the trajectory of the compressive stress resultants. Source: Adapted from the hanging chain, Robert Hooke (1676), requoted from Heyman 1982.

Figure 1.2 Statically indeterminate truss and a possible system of force transmission: (a) and (b) for external loads, from joints to ground; (c) for internal forces that are self-balanced.

Figure 1.3 Determination of a displacement of a hyperstatic beam using a statically admissible virtual moment distribution.

Furthermore, force, or flexibility, methods can lead to better conditioned systems of equations, and fewer of them (Argyris and Kelsey, 1960; Henderson and Maunder, 1969). These were critical factors when solutions were calculated with manual or semi-manual methods (Kurrer, 2008).

1.4 What Paths Have Been Followed to Achieve Strong Forms of Equilibrium?

Generalizing the approach that is used for the analysis of frame structures using the force method, the obvious solution for the analysis of continua is to combine approximations that verify equilibrium in a strong sense so that the generalized relative displacements corresponding to the hyperstatic forces are zero. In the 1960s and 70s, as a consequence of the historical relevance given to equilibrium in technical culture, there was considerable effort devoted to obtaining such finite element methods (Argyris and Kelsey, 1960; Fraeijs de Veubeke, 1965; Gallagher, 1975; Robinson, 1973).

This approach was practically abandoned, an exception being the work of Kaveh (2004, 2014), mainly due to the difficulty in setting up such approximations, and to the superior computational efficiency of direct stiffness assembly procedures (Przemieniecki, 1985). We will only briefly address this approach in this book.

In the 1960s Fraeijs de Veubeke et al. proposed and developed finite element formulations providing solutions that verify equilibrium in the strong sense using two different approaches: to work with a stress potential, for example the Airy stress function, and to assemble elements where an internally equilibrated stress approximation is assumed in such a way that their boundary tractions are in equilibrium, that is, are codiffusive.

It appears that these equilibrium models did not find favour—maybe due to their relative complexity in implementation, and difficulties encountered in the application of boundary conditions. In any event, the belief in displacement models seems to have grown, together with the idea that the quality of solutions could be judged entirely by post-processing procedures such as those proposed by Zienkiewicz & Zhu in the late 1980s and early 1990s (Zienkiewicz and Zhu, 1987).

However, the value of having complementary solutions for error estimation was still recognized, and a lot of research has been expended into a variety of ways to recover enhanced solutions from displacement models, for example, the superconvergent patch recovery (SPR) methods (Zienkiewicz and Zhu, 1987, 1992) and the error in constitutive relation from P. Ladeveze et al. (Ladevèze and Leguillon, 1983; Ladevèze and Maunder, 1996). These recovery methods, with the exception of the approach proposed by Ladevèze, only lead to a better approximation of strong equilibrium.

Since the 1990s there has been a renewed interest and a renaissance in FEM, with hybrid formulations that directly enforce a strong form of equilibrium (Almeida and Freitas, 1991).

1.5 Industrial Perspectives

Two different perspectives are particularly relevant when considering the industrial application of analyses based on equilibrium finite elements:

verification and validation in ‘simulation governance’ (Szabó and Actis, 2012);

the design and/or assessment of structures, explicitly or implicitly based on the Static Theorem of Limit Analysis.

In this Section we briefly discuss how these aspects can be considered.

1.5.1 Simulation Governance

With the ever wider reliance by industry on finite element analyses to justify compliance with codes of practice or statutory requirements, it is recognized that more formal verification and validation procedures should be adopted.

General procedures have been defined in a rather pithy, but memorable way, to address two questions (Roache, 1998):

‘Am I solving the equations right?’

‘Am I solving the right equations?’

The ‘equations’ refer to the mathematical model which is assumed to describe the physical behaviour; in our case this corresponds to the equations of elasticity. Then (i) concerns verifying that the solution matches the mathematical model. It may fail to do so because of the intrinsic inaccuracy of the numerical model (for us the finite element approximations that are assumed), combined with other numerical aspects, for example, numerical instabilities and ill-conditioning that may be present.

Point (ii) questions the validity of the mathematical model to adequately represent physical reality, for example, are potential non-linearities in behaviour properly accounted for; do the boundary conditions and loads reflect actual conditions; do the constitutive relations properly match those of the real materials?

Thus two complementary approaches to finite element modelling that satisfy compatibility or equilibrium, and which can deliver opposite bounds to quantities of interest, are clearly advantageous, both for the peace of mind of the engineer, and as a means of providing evidence to satisfy statutory requirements on quality control.

1.5.2 Equilibrium in Structural Design and Assessment

Finite element models can evolve as the design of a structure or a device develops, but with different and evolving aims, particularly in the civil engineering context where structures tend to be ‘one-off’ as opposed to the mass produced artefacts of mechanical engineering. For the design of one-off civil engineering structures, high accuracy in the output of quantities of interest is not normally required, since the variability of materials, construction processes, and loading regimes (including the troublesome question of usually indeterminate residual stresses) do not generally justify this. However, a strong sense of equilibrium is important from the simplest initial models at early stages of design to the refined models necessary to justify the final design.

Wittingly, or unwittingly, designers rely on what Heyman (1995) has termed ‘the master safe theorem’, that is, if any equilibrium state can be found, that is, one for which a set of internal forces is in equilibrium with the external loads, and, further, for which every internal portion of the structure satisfies a strength criterion, then the structure is safe.

It is interesting to note that Wren must have had a rather similar basis for his designs: ‘The design must be regulated by the art of staticks, or invention of the centres of gravity, and the duly poising of all parts to equiponderate; without which, a fine design will fail and prove abortive. Hence I conclude, that all designs must, in the first place, be brought to this test, or rejected.’ (Addis, 2007).

In pithier terms, Ed Wilson quotes: ‘equilibrium is essential, compatibility is optional’ and then emphasizes that stresses in conforming elements do not strongly satisfy equilibrium, so that mesh design needs to be considered in order to achieve acceptable levels of stress (Wilson, 2000).

Of course these quotations leave open what is meant by ‘portion’ or ‘part’, but at the smallest practical level we can take this to mean infinitesimal parts of a continuum, and internal forces to mean stresses. The conventional conforming finite element model only enforces the weaker equilibrium of nodal forces, and so an element serves as a portion, but then we need a strength criterion! A useful and attractive feature of equilibrium finite elements is to transfer their interactions from the mathematical concept of nodal forces to the engineering concept of tractions on interfaces, backed up by fully equilibrated internal stress fields.

These are immediately in a form amenable to comparison with strength criteria. It may be noted that, even when the non-elastic properties of the structural material do not strictly justify the use of the master safe theorem, equilibrium is a first line of defence!

Concepts of equilibrium, and their use in design/assessment stages, are embodied in codes of practice for design, but with some warnings when conventional conforming finite element models are used, for example, Eurocodes EN 1990 (Basis of structural design), EN 1992 (Design of concrete structures), and fib Model Code for Concrete Structures 2010.

EN 1990 (1.5.6.2) Global analysis is defined as the ‘determination, in a structure, of a consistent set of either internal forces and moments, or stresses, that are in equilibrium with a particular defined set of actions on the structure, and depend on geometrical, structural and material properties.’

EN 1992: brief reference to the use of finite element methods is made in Section 5 Structural analysis, 5.1.1 General requirements: ‘However, for certain particular elements, the methods of analysis used (e.g. finite element analysis) gives stresses, strains and displacements rather than internal forces and moments. Special methods are required to use these results to obtain appropriate verification.’

The fib Model Code for Concrete Structures 2010 allows the use of the theory of plasticity in design, and this includes the ‘lower bound (static) theorem’. Verification of designs may be assisted by numerical simulations, including the finite element method: ‘In the case of the most widely used stiffness method, the shape of the displacement field is assumed and equilibrium is satisfied only in integral sense. The internal stresses are lower, compared with an exact solution. The approximations introduced by the finite element formulation only, can be a significant source of errors in numerical analysis.’