Overview - University of Colorado Boulder

1
Overview
1–1
Chapter 1: OVERVIEW
TABLE OF CONTENTS
Page
§1.1
§1.2
§1.3
§1.4
§1.5
§1.6
Scope . . . . . . . . . . . . . . . . . . . . . . . .
Where the Book Fits
. . . . . . . . . . . . . . . . .
§1.2.1
Top Level Classification . . . . . . . . . . . . . .
§1.2.2
Computational Mechanics . . . . . . . . . . . . .
§1.2.3
Statics versus Dynamics . . . . . . . . . . . . . .
§1.2.4
Linear versus Nonlinear
. . . . . . . . . . . . .
§1.2.5
Discretization Methods . . . . . . . . . . . . . .
§1.2.6
FEM Variants . . . . . . . . . . . . . . . . .
The FEM Analysis Process . . . . . . . . . . . . . . . .
§1.3.1
The Physical FEM
. . . . . . . . . . . . . . .
§1.3.2
The Mathematical FEM . . . . . . . . . . . . . .
§1.3.3
Synergy of Physical and Mathematical FEM
. . . . . .
§1.3.4
Streamlined Idealization and Discretization
. . . . . . .
Method Interpretations . . . . . . . . . . . . . . . . .
§1.4.1
Physical Interpretation
. . . . . . . . . . . . . .
§1.4.2
Mathematical Interpretation . . . . . . . . . . . .
The Solution Morass
. . . . . . . . . . . . . . . . . .
Historical Background
. . . . . . . . . . . . . . . . .
§1.6.1
Smooth Nonlinearities
. . . . . . . . . . . . . .
§1.6.2
Rough Nonlinearities
. . . . . . . . . . . . . .
§1.6.3
Hybrid Approach . . . . . . . . . . . . . . . .
§1.6.4
Summary of Present Status
. . . . . . . . . . . .
1–2
1–3
1–3
1–3
1–5
1–7
1–7
1–7
1–8
1–8
1–8
1–9
1–10
1–12
1–12
1–12
1–13
1–14
1–14
1–15
1–16
1–17
1–18
§1.2
WHERE THE BOOK FITS
§1.1. Scope
This is a textbook about nonlinear structural analysis using the Finite Element Method (FEM) as a
discretization tool. It is intended to support a course at the second-year level of graduate studies in
Aerospace, Mechanical, or Civil Engineering. The focus of the book is on geometrically nonlinear
problems as well as structural stability. Basic prerequisite to understanding the material covered
here is an introductory FEM course at the graduate level. Such course typically focuses on linear
problems and assumes working knowledge of matrix algebra, as well as that of structural and solid
mechanics at the undergraduate level.
This Chapter presents an overview of what the book covers. As noted above, it is assumed that the
reader has a solid knowledge of linear finite elements, whence those aspects are glossed over. For
convenient reference, Tables 1.1 and 1.2 gather technical terms often used througout this book. The
first table catalogs basic terminology on mechanical systems, while the second one collects terms
used in stability analysis.
§1.2. Where the Book Fits
This Section outlines where the book material fits within the large scope of Mechanics. In the
ensuing multilevel classification, topics addressed in some depth are emphasized in bold typeface.
§1.2.1. Top Level Classification
Definitions of Mechanics in dictionaries usually state two flavors:
•
The branch of Physics that studies the effect of forces and energy on physical bodies.1
•
The practical application of that science to the design, construction or operation of material
systems or devices, such as machines, vehicles or structures.
These flavors are science and engineering oriented, respectively. But dictionaries are notoriously
outdated. For our objectives it will be convenient to distinguish four major areas:

Theoretical


Applied
Mechanics
(1.1)

 Computational
Experimental
Theoretical mechanics deals with fundamental laws and principles studied for their intrinsic scientific value. Applied mechanics transfers this theoretical knowledge to scientific and engineering
applications, especially as regards the construction of mathematical models of physical phenomena.
Computational mechanics solves specific problems by model-based simulation through numerical
methods implemented on digital computers. Experimental mechanics subjects the knowledge derived from theory, application and simulation to the ultimate test of observation.
Remark 1.1. Paraphrasing an old joke about mathematicians, one may define a computational mechanician
as a person who searches for solutions to given problems, an applied mechanician as a person who searches
for problems that fit given solutions, and a theoretical mechanician as a person who can prove the existence
of problems and solutions. As regards experimentalists, make up your own joke.
1
Here the term “bodies” includes all forms of matter, whether solid, liquid or gaseous; as well as all physical scales, from
subatomic through cosmic.
1–3
Chapter 1: OVERVIEW
Table §1.1. Terminology Pertaining to Mechanical Systems
Term
Definition
System
Configuration
State
Degrees of Freedom
Kinematic DOF
Input
Output
Model
Discrete Model
Continuum Model
Event
Behavior
Reference State
A functionally related set of components regarded as a physical entity.
The relative disposition or arrangement of system components.
The condition of the system as regards its form, structure or constitution.
A set of state variables that uniquely characterizes the state. Abbr.: DOF.
A DOF that is directly linked to the system geometry; e.g., a displacement.
The set of all actions that can influence the state of a system, or component.
The set of all quantities that characterize the state of a system, or component.
A mathematical idealization of a physical system.
A model with a finite # of DOF. Often expressed as vector equations.
A model with an infinite # of DOF. Often expressed as a ODE or PDE.
A change in the state variables produced by an agent.
A pattern of events.
A state of the system adopted as base or origin to measure relative changes.
Often the same as the undeformed state under zero loads.
Motion
The change in system geometry, as measured from a reference state.
Kinematics
The study of system motion independently of force agents.
Kinetics
The study of forces as action agents, and their effect on the system state.
Kinematic Constraint Any condition that restricts the system motion, usually expressed in terms of
kinematic DOF. Also simply called constraint.
Environment
A entity set that does not belong to the system, but can influence its behavior.
Open System
A system that is influenced by entities outside the system (its environment).
Closed System
A system that is not affected by entities outside the system.
Interaction
The mutual effect of a system component(s) on other component(s).
Forces
The action agents through which effects are transmitted between system
components, or between environment entities and system components.
Internal Forces
Forces that act between system components.
External Forces
Forces that act between environment entities and system components.
Constraint Force
A force manifested by removing a constraint while keeping it enforced.
Reaction Force
A constraint force that is an external force.
Conservative Force
A force derivable from a potential; e.g., gravity.
Nonconservative Force A force not derivable from a potential; e.g., friction.
Applied Load
An external force specified as data. Also simply called load.
Reference Load(s)
A set of applied loads taken as reference for application of a load factor.
Load Factor
A scalar, usually denoted by λ, which scales reference loads to get the actual
applied loads. Also called load parameter and load multiplier.
System Response
Values of the DOF, or subset thereof, expressed as function of the load factor,
or of the load level if only one load is applied. Also simply called response.
Equilibrium State
A state in which internal and external forces are in equilibrium.
The associated configuration is called an equilibrium configuration.
Undeformed State
The equilibrium state under zero applied loads, or, equivalently, λ = 0.
The associated configuration is called an undeformed configuration.
Equilibrium Response A system response in which all states are equilibrium states.
State Space
A RCC frame with a DOF subset as axes.
Response Space
A RCC frame with load factor as one axis, and a DOF subset as the other(s).
Response Plot
A display of the system response in response space.
Equilibrium Path
An equilibrium response viewed in response space.
1–4
§1.2
WHERE THE BOOK FITS
Table §1.2. Terminology Pertaining To Stability Analysis
Term
Definition
Perturbation
An externally imposed disturbance of an equilibrium state while actual
loads are kept fixed. It may involve application of forces or motions.
Allowed Perturbation A perturbation that satisfies kinematic constraints. Also called admissible
perturbation, and (in the sense of variational calculus) virtual variation.
Stability
The ability of a system to recover an equilibrium state upon being
disturbed by any of the allowed perturbations.
Instability
The inability of a system to recover an equilibrium state upon being
disturbed by at least one allowed perturbation.
Stable
Qualifier for an equilibrium state, or configuration, at which stability holds.
Unstable
Qualifier for an equilibrium state, or configuration, at which instability occurs.
Neutrally Stable
Qualifier for an equilibrium state, or configuration, at which transition
between stability and instability occurs.
Critical
A qualifier that flags the occurrence of neutral stability. Applicable to state,
configuration, load, and load factor. For example: critical load.
Critical Point
In an equilibrium response plot, a location where a critical state occurs.
Buckling
Name used by structural engineers for the occurrence of a bifurcation point.
Snapping
Name used by structural engineers for the occurrence of a limit point.
Also called snap-through, snap buckling, and snap-through buckling.
Bifurcation Point
A critical point at which two or more equilibrium paths cross.
Limit Point
A critical point at which the load factor is stationary (usually a maxmin).
Primary Path
An equilibrium path that crosses the reference state. A.k.a. fundamental.
Secondary Path
An equilibrium path that is not the primary path.
First Critical Point
Acronym: FCP. The first critical point encountered when traversing the
primary equilibrium path when starting from the reference state.
Critical Load Factor
The value of the load factor λcr at the FCP.
Critical Load
The value of the applied load, or load set, at the FCP.
First Bifurcation Point Acronym: FBP. The FCP of bifurcation type encountered, if any.
First Limit Point
Acronym: FLP. The FCP of limit type encountered, if any.
Prebuckling
Qualifier for structural behavior before a FCP of bifurcation type is reached.
Postbuckling
Qualifier for structural behavior after a FCP of bifurcation type is traversed.
Divergence
Stability loss by growing monotonic motion under nonconservative loading.
Flutter
Stability loss by growing oscillatory motion under nonconservative loading.
Stability Equation(s)
The equation, or set of equations, used to assess stability.
Stability Matrix
A matrix whose determinant set to zero provides the stability equation(s).
Stability Determinant The determinant of the stability matrix.
Linearized
A qualifier that refers to a stability analysis in which stability equations are
linearized about the configuration being considered.
Linearized Prebuckling Acronym: LPB. A linearized stability analysis in which deformations prior
to the FBP are neglected. This allows setting up the stability equations in the
undeformed (reference) geometry. Not applicable if the FCP is of limit type.
Geometrically Exact
A qualifier that refers to an stability analysis in which the exact geometry of
the deflected configuration is used. Contrast with Linearized Prebuckling.
Static Criterion
A stability analysis methodology that factors out the time domain.
Dynamic Criterion
A stability analysis methodology that explicitly includes the time domain.
Tangent Stiffness Test Acronym: TST. The static criterion for stability expressed in terms of the
spectrum (the set of all eigenvalues) of the tangent stiffness matrix.
Singular Stiffness Test Acronym: SST. The static criterion for locating critical points expressed in
terms of the vanishing of the determinant of the tangent stiffness matrix.
1–5
Chapter 1: OVERVIEW
§1.2.2. Computational Mechanics
Branches of computational mechanics can be distinguished according to the physical scale under
study:

Nanomechanics






 Micromechanics
Solids and Structures
Computational Mechanics
(1.2)

Continuum mechanics Fluids




Multiphysics


Systems
Nanomechanics deals with phenomena at the molecular and atomic levels. As such, it is closely
related to particle physics and chemistry. At the atomic scale it transitions to quantum mechanics.
Micromechanics looks primarily at the crystallographic and granular levels of matter. Its main
technological application is the design and fabrication of materials and microdevices.
Continuum mechanics studies bodies at the macroscopic level, using continuum models in which the
microstructure is homogenized by phenomenological averaging. Traditional areas of application
are solid and fluid mechanics. Structural mechanics is a conjoint branch of solid mechanics, since
structures, for obvious reasons, are fabricated with solids. Computational solid mechanics favors a
applied-sciences approach, whereas computational structural mechanics emphasizes technological
applications to the design and analysis of structures.
Computational fluid mechanics deals with problems that involve the equilibrium and motion of
liquid and gases. Well developed subareas are hydrodynamics, aerodynamics, atmospheric physics,
propulsion, and combustion.
Multiphysics is a more recent newcomer.2 This area is meant to include mechanical systems that
transcend the classical boundaries of solid and fluid mechanics. A key example is interaction
between fluids and structures, which has important application subareas such as aeroelasticity and
hydroelasticity. Phase change problems such as ice melting and metal solidification fit into this
category, as do the interaction of control, mechanical and electromagnetic systems.
Finally, system identifies mechanical objects, whether natural or artificial, that perform a distinguishable function. Examples of man-made systems are airplanes, building, bridges, engines, cars,
microchips, radio telescopes, robots, roller skates and garden sprinklers. Biological systems, such
as a whale, amoeba, virus or pine tree are included if studied from the viewpoint of biomechanics.
Ecological, astronomical and cosmological entities also form systems.3
In the progression of (1.2), system is the most general concept. Systems are studied by decomposition: its behavior is that of its components plus the interaction between the components. Components are broken down into subcomponents and so on. As this hierarchical breakdown process
2
This unifying term is in fact missing from most dictionaries, as it was introduced by computational mechanicians in the
1970s. Some multiphysics problems, however, are older. For example, aircraft aeroelasticity emerged in the 1920s.
3
Except that their function may not be clear to us. “What is that breathes fire into the equations and makes a universe
for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions
of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?”
(Stephen Hawking).
1–6
§1.2
WHERE THE BOOK FITS
continues the individual components become simple enough to be treated by individual disciplines,
but interactions may get more complex. Thus there are tradeoff skills in deciding where to stop.4
§1.2.3. Statics versus Dynamics
Continuum mechanics problems may be subdivided according to whether inertial effects are taken
into account or not:

 Statics Time Invariant
(1.3)
Continuum mechanics
Quasi-static

Dynamics
In statics inertial forces are ignored or neglected. Static problems may be subclassified into time
invariant and quasi-static. For the former time need not be considered explicitly; any historical
time-like response-ordering parameter (should one is needed) will do. In quasi-static problems such
as foundation settlement, creep flow, rate-dependent plasticity or fatigue cycling, a more realistic
estimation of time is required but inertial forces are neglected because motions are slow.
In dynamics the time dependence is explicitly considered because the calculation of inertial (and/or
damping) forces requires derivatives respect to actual time to be taken.
§1.2.4. Linear versus Nonlinear
A classification of static problems that is particularly relevant to this book is
Linear
Statics
Nonlinear
(1.4)
Linear static analysis deals with static problems in which the response is linear in the cause-andeffect sense. For example: if the applied forces are doubled, the displacements and internal stresses
also double. Problems outside this domain are classified as nonlinear.
§1.2.5. Discretization Methods
A final classification of computational solid and structural mechanics (CSSM) is based on the
discretization method by which the continuum mathematical model is discretized in space, i.e.,
converted to a discrete model of finite number of degrees of freedom:

Finite Element Method (FEM)



 Boundary Element Method (BEM)
Computational solid and structural 

Finite Difference Method (FDM)
(1.5)
mechanics spatial discretization
Finite Volume Method (FVM)





 Spectral Method
Meshfree Method
For linear problems finite element methods currently dominate the scene, with boundary element
methods posting a strong second choice in selected application areas. For nonlinear problems the
dominance of finite element methods is overwhelming.
4
Thus in breaking down a car engine, say, the decomposition does not usually proceed beyond the components that may
be bought at a automotive shop.
1–7
Chapter 1: OVERVIEW
Classical finite difference methods in solid and structural mechanics have virtually disappeared
from practical use. This statement is not true, however, for fluid mechanics, where finite difference
discretization methods are still important. Finite-volume methods, which focus on conservation
laws, are important in highly nonlinear problems of fluid mechanics. Spectral methods are based
on global transformations, based on eigendecomposition of the governing equations, that map the
physical computational domain to transform spaces where the problem can be efficiently solved.
Recent newcomers to the scene are the meshfree methods. These are finite difference methods on
arbitrary grids constructed using a subset of finite element techniques
§1.2.6. FEM Variants
The term Finite Element Method actually identifies a broad spectrum of techniques that share common features, as outlined in introductory FEM textbooks. Two subclassifications that fit applications
to structural mechanics particularly well are

Displacement


Equilibrium
FEM Formulation

 Mixed
Hybrid
FEM Solution
Stiffness
Flexibility
Mixed (a.k.a. Combined)
(1.6)
Using the classification of (1.1) through (1.5) we can now state the book topic more precisely:
The continuum-model-based simulation of nonlinear static structures discretized by FEM
(1.7)
Of the FEM variants listed in (1.6) emphasis will be placed on the displacement formulation and
stiffness solution. This particular combination is called the Direct Stiffness Method or DSM.
§1.3. The FEM Analysis Process
Processes that use FEM involve carrying out a sequence of steps in some way. Those sequences
take two canonical configurations, depending on (i) the environment in which FEM is used and (ii)
the main objective: model-based simulation of physical systems, or numerical approximation to
mathematical problems. Both are reviewed below to introduce terminology used in the sequel.
§1.3.1. The Physical FEM
A canonical use of FEM is simulation of physical systems. This requires models of such systems.
Consequenty the methodology is often called model-based simulation.
The process is illustrated in Figure 1.1. The centerpiece is the physical system to be modeled.
Accordingly, this configuration is called the Physical FEM. The processes of idealization and
discretization are carried out concurrently to produce the discrete model. The solution step is
handled by an equation solver often customized to FEM, which delivers a discrete solution (or
solutions).
Figure 1.1 also shows an ideal mathematical model. This may be presented as a continuum limit or
“continuification” of the discrete model. For some physical systems, notably those well modeled
by continuum fields, this step is useful. For others, such as complex engineering systems (say, a
1–8
§1.3
Ideal
Mathematical
model
THE FEM ANALYSIS PROCESS
generally
irrelevant
CONTINUIFICATION
SOLUTION
FEM
Physical
system
Discrete
model
IDEALIZATION &
DISCRETIZATION
Discrete
solution
VERIFICATION
solution error
simulation error= modeling + solution error
VALIDATION
Figure 1.1. The Physical FEM. The physical system (left) is the source of the simulation process.
The ideal mathematical model (should one go to the trouble of constructing it) is inessential.
flying aircraft) it makes no sense. Indeed Physical FEM discretizations may be constructed and
adjusted without reference to mathematical models, simply from experimental measurements.
The concept of error arises in the Physical FEM in two ways. These are known as verification
and validation, respectively. Verification is done by replacing the discrete solution into the discrete
model to get the solution error. This error is not generally important. Substitution in the ideal
mathematical model in principle provides the discretization error. This step is rarely useful in
complex engineering systems, however, because there is no reason to expect that the continuum
model exists, and even if it does, that it is more physically relevant than the discrete model.
Validation tries to compare the discrete solution against observation by computing the simulation
error, which combines modeling and solution errors. As the latter is typically unimportant, the
simulation error in practice can be identified with the modeling error. In real-life applications this
error overwhelms the others.5
§1.3.2. The Mathematical FEM
The other canonical way of using FEM focuses on the mathematics. The process steps are illustrated
in Figure 1.2. The spotlight now falls on the mathematical model. This is often an ordinary
differential equation (ODE), or a partial differential equation (PDE) in space and time. A discrete
finite element model is generated from a variational or weak form of the mathematical model.6
This is the discretization step. The FEM equations are solved as described for the Physical FEM.
On the left, Figure 1.2 shows an ideal physical system. This may be presented as a realization of
the mathematical model. Conversely, the mathematical model is said to be an idealization of this
system. E.g., if the mathematical model is the Poisson’s PDE, realizations may be heat conduction
5
“All models are wrong; some are useful” (George Box)
6
The distinction between strong, weak and variational forms is discussed in advanced FEM courses. In the present book
such forms will be largely stated (and used) as recipes.
1–9
Chapter 1: OVERVIEW
VERIFICATION
Mathematical
model
Discretization + solution error
IDEALIZATION
REALIZATION
FEM
SOLUTION
Ideal
physical
system
Discrete
model
IDEALIZATION &
DISCRETIZATION
Discrete
solution
VERIFICATION
solution error
generally irrelevant
Figure 1.2. The Mathematical FEM. The mathematical model (top) is the source of
the simulation process. Discrete model and solution follow from it. The ideal physical
system (should one go to the trouble of exhibiting it) is inessential.
or an electrostatic charge-distribution problem. This step is inessential and may be left out. Indeed
Mathematical FEM discretizations may be constructed without any reference to physics.
The concept of error arises when the discrete solution is substituted in the “model” boxes. This
replacement is generically called verification. As in the Physical FEM, the solution error is the
amount by which the discrete solution fails to satisfy the discrete equations. This error is relatively
unimportant when using computers, and in particular direct linear equation solvers, for the solution
step. More relevant is the discretization error, which is the amount by which the discrete solution
fails to satisfy the mathematical model.7 Replacing into the ideal physical system would in principle
quantify modeling errors. In the Mathematical FEM this is largely irrelevant, however, because the
ideal physical system is merely that: a figment of the imagination.
§1.3.3. Synergy of Physical and Mathematical FEM
The foregoing canonical sequences are not exclusive but complementary. This synergy8 is one of
the reasons behind the power and acceptance of the method. Historically the Physical FEM was the
first one to be developed to model complex physical systems such as aircraft, as narrated in §1.7.
The Mathematical FEM came later and, among other things, provided the necessary theoretical
underpinnings to extend FEM beyond structural analysis.
A glance at the schematics of a commercial jet aircraft makes obvious the reasons behind the Physical
FEM. There is no simple differential equation that captures, at a continuum mechanics level,9 the
structure, avionics, fuel, propulsion, cargo, and passengers eating dinner. There is no reason for
7
This error can be computed in several ways, the details of which are of no importance here.
8
Such interplay is not exactly a new idea: “The men of experiment are like the ant, they only collect and use; the reasoners
resemble spiders, who make cobwebs out of their own substance. But the bee takes the middle course: it gathers its
material from the flowers of the garden and field, but transforms and digests it by a power of its own.” (Francis Bacon).
9
Of course at the (sub)atomic level quantum mechanics works for everything, from landing gears to passengers. But
it would be slightly impractical to represent the aircraft by, say, 106 interacting particles modeled by the Schrödinger
1–10
§1.3 THE FEM ANALYSIS PROCESS
al
atic
hem
Matmodel
FEM
ary
Libr
t
en
pon e
Com
ret
c
s
i
d del
mo
NT
ONE
P
COMEVEL
L
ent
ponns
o
Com
i
t
a
equ
TEM
SYS EL
LEV e
plet
Comution
sol
l
sica
Phy tem
sys
em
Systrete
disc del
mo
Figure 1.3. Combining physical and mathematical modeling through multilevel FEM. Only
two levels (system and component) are shown for simplicity.
despair, however. The time honored divide and conquer strategy, coupled with abstraction, comes
to the rescue.
First, separate the structure out and view the rest as masses and forces. Second, consider the aircraft
structure as built up of substructures (a part of a structure devoted to a specific function): wings,
fuselage, stabilizers, engines, landing gears, and so on.
Take each substructure, and continue to break it down into components: rings, ribs, spars, cover
plates, actuators, etc. Continue through as many levels as necessary. Eventually those components
become sufficiently simple in geometry and connectivity that they can be reasonably well described
by the mathematical models provided, for instance, by Mechanics of Materials or the Theory of
Elasticity. At that point, stop. The component level discrete equations are obtained from a FEM
library based on the mathematical model.
The system model is obtained by going through the reverse process: from component equations
to substructure equations, and from those to the equations of the complete aircraft. This system
assembly process is governed by the classical principles of Newtonian mechanics, which provide
the necessary inter-component “glue.” The multilevel decomposition process is diagramed in
Figure 1.3, in which intermediate levels are omitted for simplicity
Remark 1.2. More intermediate decomposition levels are used in systems such as offshore and ship structures,
which are characterized by a modular fabrication process. In that case multilevel decomposition mimics the
way the system is actually fabricated. The general technique, called superelements, is discussed in Chapter 11.
Remark 1.3. There is no point in practice in going beyond a certain component level while considering the
complete system. The reason is that the level of detail can become overwhelming without adding relevant
equations. More seriously, Truesdell and Toupin correctly note that “Newtonian mechanics, while not appropriate to the
corpuscles making up a body, agrees with experience when applied to the body as a whole, except for certain phenomena
of astronomical scale” [780, p. 228].
1–11
Chapter 1: OVERVIEW
member
support
joint
Physical System
;;
;;
;;
;;
;;
;;
IDEALIZATION
Figure 1.4. The idealization process for a simple structure. The physical system — here a
roof truss — is directly idealized by the mathematical model: a pin-jointed bar assembly. For
this particular structure idealized and discrete models coalesce.
information. Usually that point is reached when uncertainty impedes further progress. Further refinement
of specific components is done by the so-called global-local analysis technique outlined in Chapter 10. This
technique is an instance of multiscale analysis.
§1.3.4. Streamlined Idealization and Discretization
For sufficiently simple structures, passing to a discrete model is carried out in a single idealization
and discretization step, as illustrated for the truss roof structure shown in Figure 1.4. Other levels
are unnecessary in such cases. Of course the truss may be viewed as a substructure of the roof, and
the roof as a a substructure of a building. If so the multilevel process would be more appropriate.
§1.4. Method Interpretations
Just like there are two complementary ways of using the FEM, there are two complementary
interpretations for explaining it, a choice that obviously impacts teaching. One interpretation
stresses the physical significance and is aligned with the Physical FEM. The other focuses on the
mathematical context, and is aligned with the Mathematical FEM. They are outlined next.
§1.4.1. Physical Interpretation
The physical interpretation focuses on the flowchart of Figure 1.1. This interpretation has been
shaped by the discovery and extensive use of the method in the field of structural mechanics. The
historical connection is reflected in the use of structural terms such as “stiffness matrix”, “force
vector” and “degrees of freedom,” a terminology that carries over to non-structural applications.
The basic concept in the physical interpretation is the breakdown (≡ disassembly, tearing, partition,
separation, decomposition) of a complex mechanical system into simpler, disjoint components
called finite elements, or simply elements. The mechanical response of an element is characterized
in terms of a finite number of degrees of freedom. These degrees of freedoms are represented as
the values of the unknown functions as a set of node points. The element response is defined by
1–12
§1.4
METHOD INTERPRETATIONS
algebraic equations constructed from mathematical or experimental arguments. The response of
the original system is considered to be approximated by that of the discrete model constructed by
connecting or assembling the collection of all elements.
The breakdown-assembly concept occurs naturally when an engineer considers many artificial and
natural systems. For example, it is easy and natural to visualize an engine, bridge, aircraft or
skeleton as being fabricated from simpler parts.
As discussed in §1.3.3, the underlying theme is divide and conquer. If the behavior of a system
is too complex, the recipe is to divide it into more manageable subsystems. If these subsystems
are still too complex the subdivision process is continued until the behavior of each subsystem is
simple enough to fit a mathematical model that represents well the knowledge level the analyst
is interested in. In the finite element method such “primitive pieces” are called elements. The
behavior of the total system is that of the individual elements plus their interaction. A key factor in
the initial acceptance of the FEM was that the element interaction could be physically interpreted
and understood in terms that were eminently familiar to structural engineers.
§1.4.2. Mathematical Interpretation
This interpretation is closely aligned with the flowchart of Figure 1.2. The FEM is viewed as
a procedure for obtaining numerical approximations to the solution of boundary value problems
(BVPs) posed over a domain . This domain is replaced by the union ∪ of disjoint subdomains
(e) called finite elements. In general the geometry of is only approximated by that of ∪.(e)
The unknown function (or functions) is locally approximated over each element by an interpolation
formula expressed in terms of values taken by the function(s), and possibly their derivatives, at a
set of node points generally located on the element boundaries. The states of the assumed unknown
function(s) determined by unit node values are called shape functions. The union of shape functions
“patched” over adjacent elements form a trial function basis for which the node values represent the
generalized coordinates. The trial function space may be inserted into the governing equations and
the unknown node values determined by the Ritz method (if the solution extremizes a variational
principle) or by the Galerkin, least-squares or other weighted-residual minimization methods if the
problem cannot be expressed in a standard variational form.
Remark 1.4. In the mathematical interpretation the emphasis is on the concept of local (piecewise) approx-
imation. The concept of element-by-element breakdown and assembly, while convenient in the computer
implementation, is not theoretically necessary. The mathematical interpretation permits a general approach
to the questions of convergence, error bounds, trial and shape function requirements, etc., which the physical
approach leaves unanswered. It also facilitates the application of FEM to classes of problems that are not so
readily amenable to physical visualization as structures; for example electromagnetics and heat conduction.
1–13
Chapter 1: OVERVIEW
§1.5. The Solution Morass
In nonlinear analysis the two FEM interpretations are not equal in importance. Nonlinear analysis
demands a persistent attention to the underlying physics to avoid getting astray as the “real world”
is covered by layer upon layer of mathematics and numerics.
Why is concern for physics of paramount importance? A key component of finite element nonlinear
analysis is the solution of the nonlinear algebraic systems of equations that arise upon discretization.
FACT
The numerical solution of nonlinear systems in “black box” mode is much
more difficult than in the linear case.
The key difficulty is tied to the essentially obscure nature of general nonlinear systems, about
which little can be said in advance. And you can be sure that Murphy’s Law10 works silently in the
background.
A particularly vexing aspect of dealing with nonlinear systems is the solution morass. A determinate
system of 1, 1000, or 1000000 linear equations has, under mild conditions, one and only one
solution. The computer effort to obtain this solution can be estimated fairly accurately if the
sparseness (or denseness) of the coefficient matrix is known. Thus setting up linear equation
solvers as “black-box” stand-alone functions or modules is a perfectly sensible thing to do.
By way of contrast, a system of 1000 cubic polynomial equations has 31000 ≈ 10300 solutions in
the complex plane. This is much, much larger than the number of atoms in the Universe, which is
merely 1050 give or take a few. Suppose just several millions of these are real solutions. Which
ones have physical meaning? And how do you get those without wasting time on the others?
This combinatorial difficulty is overcome by the concept of continuation, which engineers also call
incremental analysis. Briefly speaking, we start the analysis from an easily computable solution
— for example, the linear solution — and then try to follow the behavior of the system as actions
applied to it are changed by small steps called increments. The previous solution is used as a starting
point for the iterative solution-search procedure. The underlying prescription: follow the physics.
This technique is interwined with the concept of response explained in Chapter 2.
Remark 1.5. Not surprisingly, incremental analysis was created by the aerospace engineers that first used
the finite element method for geometrically nonlinear analysis in the late 1950s. Techniques have been
considerably refined since then, but the underlying idea remains the same.
We conclude this overview chapter with a historical perspective on nonlinear finite element methods
in solid and structural mechanics, along with a succint (but no up-to-date) bibliography.
10
If something can go wrong, it will go wrong.
1–14
§1.6 HISTORICAL BACKGROUND
§1.6.
Historical Background
In the history of finite element methods the year 1960 stands out. The name “finite element method” appears
for the first time in the open literature in an article by Clough [138]. And Turner, Dill, Martin and Melosh [788]
publish a pioneering paper in nonlinear structural analysis. The then-five-year-old “direct stiffness method”
(what we now call displacement-assumed finite element method) was applied to
“problems involving nonuniform heating and/or large deflections . . . in a series of linearized steps.
Stiffness matrices are revised at the beginning of each step to account for changes in internal loads,
temperatures, and geometric configuration.”
Several decades and thousands of publications later, computerized nonlinear structural analysis has acquired
full adult rights, but has not developed equally in all areas.
The first fifteen years (1960-1975) were dominated by formulation concerns. For example, not until the
late 1960s were correct finite-deflection incremental forms for displacement models rigorously derived. And
interaction of flow-like constitutive behavior with the spatial discretization (the so called “incompressibility
locking” effects) led to important research into constitutive equations and element formulations.
While the investigators of this period devoted much energy to obtaining correct and implementable nonlinear
finite-element equations, the art of solving such equations in a reliable and efficient manner was understandably
neglected. This helps to explain the dominance of purely incremental methods. Corrective methods of Newton
type did not get much attention until the early 1970s, and then only for geometrically nonlinear problems. At the
time of this writing (mid 1990s), progress in numerical solution techniques has been uneven: well developed
for certain problems, largely a black art in others. To understand the difference, it pays to distinguish between
smooth nonlinearities and rough nonlinearities.
§1.6.1. Smooth Nonlinearities
Problem with smooth nonlinearities are characterized by continuous, path-independent nonlinear relations at
the local level. Some examples:
1.
Finite deflections (geometric nonlinearities). Nonlinear effects arise from strain-displacement equations,
which are well behaved for all strain measures in practical use.
2.
Nonlinear elasticity. Stresses are nonlinear but reversible functions of strains.
3.
Follower forces (e.g., pressure loading). External forces are smooth nonlinear functions of displacements.
A unifying characteristic of this problem is that nonlinearities are of equality type, i.e., reversible, and these
relations are continuous at each point within the structure. Mathematicians call these smooth mappings.
It is important to point out, however, that the overall structural behavior is not necessarily smooth; as witnessed
by the phenomena of buckling, snapping and flutter. But at the local level everything is smooth: nonlinear
strain-displacement equations, nonlinear elasticity law, follower pressures.
Methods for solving this class of problems are highly developed, and have received a great deal of attention
from the mathematical and numerical analysis community. This research has directly benefitted many areas
of structural analysis.
Let us consider finite deflection problems as prototype. Within the finite element community, these were
originally treated by purely incremental (step-by-step) techniques; but anomalies detected in the mid-1960’s
prompted research into consistent linearizations. A good exposition of this early work is given in the book by
Oden [535]. Once formulation questions were settled, investigators had correct forms of the “residual” outof-balance forces and tangent stiffness matrix, and incremental steps began to be augmented with corrective
iterations in the late 1960s. Conventional and modified Newton methods were used in the corrective phase.
These were further extended through restricted step (safeguarded Newton) and, more recently, variants of the
powerful conjugate-gradient and quasi-Newton methods.
1–15
Chapter 1: OVERVIEW
But difficulties in detecting and traversing limit and bifurcation points still remained. Pressing engineering
requirements for post-buckling and post-collapse analyses led to the development of displacement control,
alternating load/displacement control, and finally arclength control. The resultant increment-control methods
have no difficulty in passing limit points. The problem of reliably traversing simple bifurcation points without
guessing imperfections remains a research subject, while passing multiple or clustered bifurcation points
remains a frontier subject. A concerted effort is underway, however, to subsume these final challenges.
These reliable solution methods have been implemented into many special-purpose finite element programs,
and incorporation into general-purpose programs is proceeding steadily.
Remark 1.6. As noted above, incremental methods were the first to be used in nonlinear structural analysis. Among the
pre-1970 contributions along this line we may cite Argyris and coworkers [?,25], Felippa [211], Goldberg and Richards
[317], Marcal, Hibbitt and coworkers [371,482,483], Oden [533], Turner, Martin and coworkers [484,788,789],
Remark 1.7. The earliest applications of Newton methods to finite element nonlinear analysis are by Oden [533], Mallet
and Marcal [479], and Murray and Wilson [521,522]. During the early 1970s Stricklin, Haisler and coworkers at Texas
A&M implemented and evaluated self-corrective, pseudo-force, energy-search and Newton-type methods and presented
extensive comparisons; see Stricklin et. al. [729,732], Tillerson et. al. [761], and Haisler et. al. [342]. Almroth, Brogan,
Bushnell and coworkers at Lockheed began using true and modified Newton methods in the late 1960s for energy-based
finite-difference collapse analysis of shells; see Brogan and Almroth [111], Almroth and Felippa [15], Brush and Almroth
[114], and Bushnell [115,116]. By the late 1970s Newton-like methods enjoyed widespread acceptance for geometrically
nonlinear analysis.
Remark 1.8. The first displacement control strategies for finite element post-buckling and collapse analysis were presented
by Argyris [27] and Felippa [211] in 1966. They were generalized in different directions by Sharifi and Popov [691,692]
(fictitious springs), Bergan et. al. [83,85], (current stiffness parameter), Powell and Simons [612] and Bergan and Simons
[86] (multiple displacement controls). A modification of Newton’s method to traverse bifurcation points was described by
Thurston [760]. Arclength control schemes for structural problems may be found in the following source papers: Wempner
[821], Riks [658], Schmidt [681], Crisfield [164,165], Ramm [630], Felippa [220,222], Fried [293], Park [566,570], Padovan
[558,?], Simo et.al. [694], Yang and McGuire [846], Bathe and Dvorkin [57]. Other articles of particular interest are Bathe
and Cimento [55], Batoz and Dhatt [60], Bushnell [116], Bergan and Simons [86], Geradin et al. [311,312], Meek and Tan
[495], Ramm [631], Riks [659,660,661], Sobel and Thomas [705], Zienkiewicz et al. [854,858,861]. Several conferences
have been devoted exclusively to nonlinear problems in structural mechanics, for example [33,90,54,536,748,749,843].
Finite element textbooks and monographs dealing rather extensively with nonlinear problems are by Oden [534], Bathe [?],
Bushnell [117], White [823] and Zienkiewicz [859].
Remark 1.9. In the mathematical literature the concept of continuation (also called homotopy and imbedding) can be traced
back to the 1930s. A survey of the work up to 1950 is given by Ficken [267]. The use of continuation by parameter
differentiation as a numerical method is attributed to Davidenko [180]. Key papers of this early period are by Freudenstein
and Roth [292], Deist and Sefor [185] and Meyer [505], as well as the survey by Wasserstrom [817]. The early history is
covered by Wacker [809].
Remark 1.10. Arclength continuation methods in the mathematical literature are generally attributed to Haselgrove [354]
and Klopfestein [433] although these papers remained largely unnoticed until the late 1970s. Important contributions to
the mathematical treatment are by Abbott [1], Anselone and Moore [22], Avila [?], Brent [107], Boggs [104], Branin
[105], Broyden [112,113], Cassel [128], Chow et. al. [135], Crandall and Rabinowitz [163], Georg [307,308], Keller
and coworkers [131,183,184,424,427], Matthies and Strang [493], Moore [514,515], Pönish [607,608], Rheinboldt and
coworkers [186,498,652,653], Watson [815] and Werner and Spence [822]. Of these, key contributions in terms of subsequent
influence are [135,425,652]. For surveys and edited proceedings see Allgower [9,10], Byrne and Hall [120], Küpper
[443,444], Rall [627], Wacker [809], and references therein. Textbooks and monographs dealing with nonlinear equation
solving include Chow and Hale [136], Dennis and Schnabel [188], Kubı́ček and Hlaváček [441], Kubı́ček and Marek [442],
Ortega and Rheinboldt [552], Rabinowitz [623], Rall [626], Rheinboldt [656], and Seydel [689]. Of these, the book by
Ortega and Rheinboldt [552] remains a classic and an invaluable source to essentially all mathematically oriented work done
prior to 1970. The book by Seydel [689] contains material on treatment of conventional and Hopf bifurcations not readily
available elsewhere. Nonlinear equation solving is interwined with the larger subject of optimization and mathematical
programming; for the latter the textbooks by Gill, Murray and Wright [315] and Fletcher [276] are highly recommended.
1–16
§1.6 HISTORICAL BACKGROUND
§1.6.2. Rough Nonlinearities
Rough nonlinearities are characterized by discontinuous field relations, usually involving inequality constraints.
Examples: flow-rule plasticity, contact, friction. The local response is nonsmooth.
Solution techniques for these problems are in a less satisfactory state, and case-by-case consideration is called
for. The local and overall responses are generally path-dependent, an attribute that forces the past response
history to be taken into account. Furthermore, contact problems introduce inequality constraints.
The key difficulty is that conventional solution procedures based on Taylor expansions or similar differential
forms may fail, because such Taylor expansions need not exist! An encompassing mathematical treatment
is lacking, and consequently problem-dependent handling is presently the rule. For this class of problems
incremental methods, as opposed to incremental-iterative methods, still dominate.
Remark 1.11. Earliest publications on computational plasticity using finite element methods are by Gallagher et. al. [296],
Argyris [24,26], Marcal [482], Pope [609] and Felippa [211]. By now there is an enormous literature on the numerical
treatment of inelastic processes, especially plasticity and creep. Fortunately the survey by Bushnell [117], although focusing
on plastic buckling, contains over 300 references that collectively embody most of the English-speaking work prior to 1980.
Other important surveys are by Armen [37] and Willam [829]. For contact problems, see Oden et. al. [537], Bathe and
Chaudhary [58], Kikuchi and Oden [429,430], Simo et. al. [694] Stein et. al. [719], Nour-Omid and Wriggers [528], and
references therein.
§1.6.3. Hybrid Approach
What does an analyst do when faced with an unfamiliar nonlinear problem? If the problem falls into the
smooth-nonlinear type, there is no need to panic. Robust and efficient methods are available. Even if the
whizziest methods are not implemented into one’s favorite computer program, there is a wealth of theory and
practice available for trouble-shooting.
But what if the problem include rough nonlinearities? A time-honored general strategy is divide and conquer.
More specifically, two powerful techniques are frequently available: splitting and nesting.
Splitting can be used if the nonlinearities can be separated in an additive form:
Smooth + Rough
This separation is usually done at the force level. Then the smooth-nonlinear term is treated by conventional
techniques whereas the rough-nonlinear term is treated by special techniques. This scheme can be particularly
effective when the rough nonlinearity is localized, for example in contact and impact problems.
Nesting may be used when a simple additive separation is not available. This is best illustrated by an actual
example. In the early 1970s, some authors argued that Newton’s method would be useless for finite-deflection
elastoplasticity, as no unique Jacobian exists in plastic regions on account of loading/unloading switches. The
argument was compelling but turned out to be a false alarm. The problem was eventually solved by “nesting”
geometric nonlinearities within the material nonlinearity, as illustrated in Fig. 1.1.
In the inner equilibrium loop the material law is “frozen”, which makes the highly effective Newton-type methods applicable. The non-conservative material behavior is treated in an outer loop where material properties
and constitutive variables are updated in an incremental or sub-incremental manner.
Another application of nesting comes in the global function approach (also called Rayleigh-Ritz or reducedbasis approach), which is presently pursued by several investigators. The key idea is to try to describe the
overall response behavior by a few parameters, which are amplitudes of globally defined functions. The small
nonlinear system for the global parameters is solved in an inner loop, while an external loop involving residual
calculations over the detailed finite element model is executed occasionally.
Despite its inherent implementation complexity, the global function approach appears cost-effective for smooth,
path-independent nonlinear systems. This is especially so when expensive parametric studies are involved, as
in structural optimization under nonlinear stability constraints.
1–17
Chapter 1: OVERVIEW
Remark 1.12. For geometric-material nesting and subincremental techniques see Bushnell [115,118], and references therein.
The global-function approach in its modern form was presented by Almroth, Stern and Brogan [17] and later pursued by
Noor and coworkers under the name of reduced-basis technique; see [526,527]. For perturbation techniques see the survey
by Gallagher [298].
§1.6.4. Summary of Present Status
Solution techniques for smooth nonlinearities are in a fairly satisfactory state. Although further refinements
in the area of traversing bifurcation points can be expected, incremental-iterative methods implemented with
general increment control appear to be as reliable as an engineer user may reasonably expect.
For rough nonlinearities, case-by-case handling is still necessary in view of the lack of general theories and
implementation procedures. Separation or nesting of nonlinearities, when applicable, can lead to significant gains in efficiency and reliability, but at the cost of programming complexity and problem-dependent
implementations. The only unifying path might be the inclusion of real-time dynamics.
1–18