Dynamical System

.

5PMP Dynamical

Systems:Kinematic Constraints

5–1

Chapter 5: PMP DYNAMICAL SYSTEMS: KINEMATIC CONSTRAINTS 5–2

TABLE OF CONTENTS

Page§5.1. Overview 5–3

§5.2. Analytically-Related Constraint Classification 5–3§5.2.1. Algebraic, Differential or Integral . . . . . . . . . . . 5–4§5.2.2. Stationary or Time Dependent . . . . . . . . . . . 5–4§5.2.3. Equality or Inequality . . . . . . . . . . . . . . 5–5

§5.3. Implementation-Related Constraint Classification 5–5§5.3.1. Single or Multiple Freedoms . . . . . . . . . . . . 5–5§5.3.2. Homogeneous or Nonhomogeneous . . . . . . . . . . 5–5§5.3.3. Linear or Nonlinear . . . . . . . . . . . . . . . 5–7

§5.4. Unified Treatment By Constraint Forces 5–7

§5.5. Holonomic SFC Examples 5–8§5.5.1. Scleronomic, Homogeneous SFC . . . . . . . . . . . 5–8§5.5.2. Rheonomic, Nonhomogeneous SFC . . . . . . . . . 5–9§5.5.3. Pseudo-Nonholonomic SFC . . . . . . . . . . . . 5–10

§5.6. Holonomic, Homogeneous MFC Examples 5–10§5.6.1. Scleronomic, Homogeneous MFC by Master-Slave . . . . 5–10§5.6.2. Rheonomic, Nonhomogeneous MFC by Master-Slave . . . . 5–11

§5.7. Penalty Function Augmentation Methods for MFC 5–12

§5.8. Lagrange Multiplier Adjunction Methods for MFC 5–12

§5. Notes and Bibliography. . . . . . . . . . . . . . . . . . . . . . 5–12

5–2

5–3 §5.2 ANALYTICALLY-RELATED CONSTRAINT CLASSIFICATION

§5.1. Overview

Kinematic constraints are restrictions on the motion of a dynamic system.1 If there are no con-straints, the system is said to be unconstrained or unrestrained. If an unconstrained model representsa structure or solid body, it is also called free-free or floating.2 The PMP example system treatedin the previous Chapter is of this type. Familiar examples: an airplane or bird in flight, a planet,satellite or rocket in orbit.

Constrained systems are more common, however, especially in Civil and Mechanical Engineering.For functional reasons, such as stability, control, safety, or operational usability, motions may berestricted by devices such as building foundations, bridge supports, member joints, failsafe stops,sliding guides, etc. This Chapter explains how to impose some practically important constrainttypes into a PMP dynamical system, once the unconstrained semidiscrete EOM are set up by oneof the methods covered in the previous Chapter. Although the example is admittedly simplistic,most techniques can be extended to the more complex systems treated in later Chapters.

The introduction of kinematic constraints into a dynamical system comprises three stages:

• Idealization: stating physical restrictions on the motion as mathematical expressions.

• Implementation: modifying unconstrained EOM so that the idealized constraints are verified.

• Solution: accounting for constraints in the solution procedure if and as necessary.

The first step results in a mathematical statement that may include displacements and/or velocities(involvement of accelerations is relatively rare), as well as possibly time and associated forces. Insimple cases the idealization may be obvious from inspection. For instance, suppose that pointmass 2 of the PMP example system of the previous Chapter, illustrated in Figure 4.1, can onlymove along x . Then evidently uy2 = 0. Such local or single-DOF constraint forms are often calledspatial boundary conditions in the literature.3

More complicated scenarios lead to constraints that link multiple DOF. These are discussed in [106,Chapters 8–9] for the static case. But the spectrum of kinematic constraints in dynamics is farricher, because they may display time dependence and/or involve temporal state derivatives such asvelocities. Accordingly, constraint implementation techniques span a wider range, and may bringabout additional modeling as well as computational difficulties. To prevent such troubles it helpsto beware of the different types of constraint that may occur in practice. Hence a categorization isin order.

For convenient reference, the classifications outlined in §5.2 and §5.3 are summarized in Tables 5.1and 5.2, respectively.

1 The appropriate definition for constraint, taken from the American Heritage Dictionary, is “something that restricts,limits or regulates.” Synonyms: coercion, control, limitation, prevention, regulation, restriction, restraint, suppression.The qualifier “kinematic” introduces the flavor of physical motion. The term “constraint” is relatively recent; othernames found in the older literature are: side conditions and auxiliary conditions.

2 Qualifier free-free has the connotation of “exempt from external authority, interference or restrictions,” whereas qualifierfloating is used in the sense of “having little or no attachment to a particular place.”

3 In Mathematics, the term “boundary condition” technically applies to situations where the boundary of a region can bereadily identified, as in boundary-value problems. But occasionally the meaning is informally extended to encompassconstraints of local type, whether pertaining to a boundary or not.

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§5.2. Analytically-Related Constraint Classification

In Classical Mechanics4 constraints were primarily classified according to mathematical featuresrelevant to the use of analytical solution techniques. These include: (i) algebraic, differential orintegral form; (ii) time dependence or independence, and (iii) equality or inequality. The followingdefinitions assume that the idealized expression is of equality type since inequality constraints arebeyond the scope of the book; see §5.2.3 for more details.

§5.2.1. Algebraic, Differential or Integral

An equality kinematic constraint is holonomic5 if it can be expressed in the algebraic form

fk(u, t) = 0, (5.1)

in which k denotes a constraint index, u is the vector of state variables (in PMP systems, point-massdisplacements). Holonomic constraints are often associated with conservative systems.

An equality kinematic constraint is nonholonomic if it is expressed as a differential or algebraic-differential form that includes state variable velocities collected in vector u:

fk(u, u, t) = 0, (5.2)

and is not reducible to (5.1) by either integration, or differentiation followed by elimination. [Con-straints originally stated as (5.2) but reducible to (5.1) are sometimes called pseudo-holonomicor semiholonomic.] These are often associated with nonconservative systems (those exhibitingdissipative effects such as friction), as well as rolling contact or robotic motions.

An equality constraint is isoparametric if it is expressed as a definite integral in time; for example

∫ t2

t1

fk(u, u, t) dt = C, (5.3)

where C is a constant. Such conditions are important in contexts that involve conservation of somequantity such as path length or energy, but will not be treated here.

An obvious generalization of (5.2) is an equality differential form that includes accelerations:fk(u, u, u, t) = 0, and is not reducible to either (5.1) or (5.2). But as previously noted such formsare comparatively rare.6

4 “Classical” refers to a period extending roughly from Newton to the advent of digital computers in the early 1950s. Formajor advances in the treatment of kinematic constraints during this period, see Notes and Bibliography.

5 The distinction between holonomic and nonholonomic constraints was first emphasized by Hertz in 1894 [381], and furtherstudied by Ferrers [375], Routh [414] and other scientists of the time. For a historical account of these developments,see [390].

6 One reason for their rarity is that selected accelerations can often be eliminated using the original EOM.

5–4

5–5 §5.3 IMPLEMENTATION-RELATED CONSTRAINT CLASSIFICATION

§5.2.2. Stationary or Time Dependent

Here are two more Victorian-era adjectives for impressing your Facebook friends. The holonomicconstraint (5.1) is called scleronomic if it is stationary (that is, time independent), thus reducing tofk(u) = 0. It is rheonomic otherwise.7 Such a distinction is not relevant for the nonholonomicform (5.2), which depends at least implicitly on time because it contains temporal derivatives.

This classification is vacuous for the static case, in which only the scleronomic holonomic formfk(u) = 0 occurs since both time and velocities are absent.

§5.2.3. Equality or Inequality

Note the qualifier equality used for the definitions in §5.2.1. If the equals sign in (5.1) or (5.2) isreplaced by > or ≥, the expression becomes an inequality constraint. In Mechanics these occurprimarily in contact and impact problems as well as optimization. Those forms are beyond the scopeof this book, and will not be considered further. Computational handling of inequality constraintsnormally demands application-specific tools, such as search algorithms for contact-impact problemsand sequential quadratic programming (SQP) methods in optimization. Those highly specializedtopics are covered in advanced textbooks and monographs.

§5.3. Implementation-Related Constraint Classification

The classification discussed in §5.2 was important in the pre-computer era, where long and labo-rious dynamic calculations were carried out by hand.8 As computational dynamics evolved in thecomputer era, certain constraint features previously ignored or neglected were found to have signif-icant impact on the computer implementation as well as in organizing time-stepping calculations.These are collected in Table 5.2 and summarized in the following subsections.

§5.3.1. Single or Multiple Freedoms

If the constraint involves just one state variable, it is called Single Freedom Constraint or SFC. if itlinks more than one, it is a Multiple Freedom Constraint, or MFC. The distinction was introducedfor static FEM analysis in IFEM [106, Chapters 8–9], and is also important in dynamics.

Suppose that for the example PMP system we specify a holonomic SFC such as uy2 = 0, oruy2 = 2 + 1

2 t2. The accelerations u y2 are computed, and together with uy2 replaced into the matrixEOM. Known data is transferred to the RHS and the modified EOM partitioned to exhibit theremaining, unconstrained DOF. This reduction technique is illustrated with examples in §5.5.

How about uy2 = 4u y2? Some preprocessing is required. It is integrated to yield uy2 = exp(t/4)+Cy2, with Cy2 determined by initial conditions. Then one proceeds as in the previous case. Ifintegration is difficult or impossible, treat as a MFC.

For a holonomic MFC, the three implementation methods described in IFEM [106, Chapter 8-9], and collected in Table 5.3, are still applicable with some extensions illustrated later. For anonholonomic MFC, only the Lagrange multiplier method, used with extreme care, offers hope.

7 Roots come from Greek sclero: hard or solid, and rheo: flowing.8 For example, the mathematician E. W. Brown (1866–1938) spent most of his professional life doing accurate calculations

of the Moon orbit, reaching 1500 perturbation terms. A laptop can reproduce his 1908 tables in a few seconds.

5–5


Table 5.1. Constraint Classification Related to Analytical Issues

Type Definition Comments

Holonomic Algebraic form (5.1)Nonholonomic Differential form (5.2) If not reducible to (5.1)Isoparametric Integral form (5.3) Not treated here

Scleronomic Holonomic form (5.1), with no t dependenceRheonomic Holonomic form (5.1), with explicit t dependence

Equality = separates LHS and RHSInequality ≥ or > separates LHS and RHS Not treated here

Table 5.2. Constraint Classification Related to Implementation Issues

Type Description Comments

Single freedom (SFC) One DOF in (5.1) or (5.2)Multiple freedom (MFC) Multiple DOF in (5.1) or (5.2) Tougher to implement

Homogeneous Homogeneous polynomial in state variables No RHS modificationNonhomogeneous Not meeting previous definition

Linear State variables appear linearlyNonlinear State variables appear nonlinearly Not treated here

Table 5.3. Constraint Implementation Methods

Method Variants Procedure

Elimination Direct Substitute, then partitionMaster-slave Congruential transformation on stiffness & mass

Penalty function Standard (one pass) Inject penalty elements of fixed weightsAugmented Lagrangian Inject penalty elements but iterate on weights

Lagrange multiplier Standard Constraint forces are made system unknownsRegularized Similar to Augmented Lagrangian

§5.3.2. Homogeneous or Nonhomogeneous

A scleronomic holonomic constraint is homogeneous if it is expressible as a homogeneous polyno-mial in the state variables. Otherwise it is nonhomogeneous. For the PMP example system, uy2 = 0is homogeneous, and so is (ux2 − ux1)

n = 0 for any integer exponent n > 0. On the other handux2 = t , ux2 − ux1 = 2, and u2

x2 = 1 + u2x1, are nonhomogeneous. Rheonomic (time dependent)

holonomic constraints, as well as nonholonomic ones, are always considered nonhomogeneous.

5–6

5–7 §5.4 UNIFIED TREATMENT BY CONSTRAINT FORCES

��

f con

Figure 5.1. Remove the bridge (log) and replace its effect on the elephant by constraint forcesfcon on the legs. The elephant stays happy: nothing happens. This illustrates Newton’s Third Lawof action and reaction, whence constraint forces are often called reaction forces by engineers.

The significance of this feature in computer implementation is that, if the constraint is nonhomo-geneous, applied force terms must be modified and converted to effective forces if elimination orpenalty function methods are used. No such modifications are needed in the homogeneous case.

§5.3.3. Linear or Nonlinear

A holonomic constraint is linear if any displacements present in the constraint appear linearly.Otherwise the constraint is nonlinear. The dependence on time may be arbitrary. For instanceux1 − 2uy1 = t2 is linear but (ux2 − ux1)

2 + (uy2 − uy1)2 = 0 is not.

Likewise, a nonholonomic constraint is linear if any displacements and velocities present in theconstraint appear linearly. The dependence on time may be arbitrary.

The presence of nonlinear constraints affects computer implementation if the time marching solutionprocedure iterates at each step, as is often the case with implicit time integrators. If so, suchconstraints must be appropriately linearized.

Nonlinear constraints are not treated in this book.

For convenient reference, Table 5.3 collects constraint implementation methods. Their basic fea-tures are described in IFEM [106, Chapters 3, 8 and 9]. Modifications required for the dynamiccase can be learned from the examples that follow.

§5.4. Unified Treatment By Constraint Forces

Looking at the Tables 5.1 through 5.3 may discourage the reader. There are 12 constraint classeslisted, along with 3 implementation methods with 2 variants each. But there is no cause for alarm.Only a few of the simplest constraints are those overwhelmingly found in practical engineering andphysics problems.

In previous chapters the unifying role of forces in unconstrained dynamic systems has been em-phasized. Their role is equally strong once constraints are introduced. The basic rule is

Any kinematic constraint may be replaced by a system of forces (5.4)

See cartoon in Figure 5.1. Unsurpringly, these forces are called constraint forces. Since this recipeis nothing more than Newton’s Third Law of action and reaction under another name, they are often

5–7


1

x3x3f , u

x2x2f , ux1x1f , u

y3 y3f , u

��

45o

45o 2

3

(1)

(2)(3)

y1y1f , u

y2y2f , u

Figure 5.2. Support conditions for Example worked out in §5.5.1.

called reaction forces by engineers, particularly when they happen at a foundation or support. Froma mathematical standpoint, constraint forces can always be identified with Lagrange multipliers,which explains the power and generality of this methodology.

The examples that follow will illustrate the rule (5.4) in action.

§5.5. Holonomic SFC Examples

We begin our set of examples with the simplest kinematic constraints. These are holonomic, singlefreedom constraints (SFC). Three variations on this theme are considered in this Section.

§5.5.1. Scleronomic, Homogeneous SFC

The floating example truss studied in Chapter 4 (see Figure 4.3) is now supported as shown inFigure 5.2. Node 1 is fixed whereas node 2 can only moved along x . Thus displacements ux1, uy1,and uy2 vanish and so do the associated accelerations ux1, u y1, and u y2. We can state the constraintsas

ux1 = uy1 = uy2 = 0, ux1 = u y1 = u y2 = 0. (5.5)

In the classification covered previously, these are scleronomic (time-independent), holonomic, andhomogeneous SFC. To implement, insert (5.5) into the unsconstrained EOM (4.15) derived in theprevious Chapter to get

0 0 00 0 03 0 00 0 00 4 00 0 4

[ ux2

ux3

u y3

]+

−10 −10 −100 −10 −1010 0 00 0 −50 10 100 −10 15

[ ux2

ux3

uy3

]=

0000

2H(t)H(t)

+

f conx1

f cony10

f cony200

= f ext +f con. (5.6)

in which fcon collects unknown constraint (support reaction) forces. System (5.6) is obtained fromthe matrix EOM (4.15) by striking out columns 1, 2 and 4 of M and K. (Those can be deleted

5–8

5–9 §5.5 HOLONOMIC SFC EXAMPLES

because they multiply zero entries.) Now (5.6) partitions naturally into two subsets. Keeping rows3, 5 and 6 produces the reduced system[ 3 0 0

0 4 00 0 4

] [ ux2

ux3

u y3

]+

[ 10 0 00 10 100 −10 15

] [ ux2

ux3

uy3

]=

[ 02H(t)H(t)

](5.7)

which has no constraint forces. This system provides three linear ODE for three unknowns, andmay be solved once the initial conditions are specified. The other subset consists of rows 1, 2 and4. Because accelerations drop out (note the 3 null rows in the reduced M), we get an algebraicsystem:

f con =[ f con

x1f con

y1f con

y2

]=

[ −10 −10 −100 −10 −100 0 −5

] [ ux2

ux3

uy3

], (5.8)

This is called the reaction recovery system because the constraint (reaction) forces may be computed,if so desired, once ux2, ux3 and uy3 are obtained by solving (5.7) in time. Note that (5.8) is timeindependent, and merely reflects static force equilibrium.

§5.5.2. Rheonomic, Nonhomogeneous SFC

The truss support conditions of the foregoing example are modified by making ux1, uy1, and uy2

specified functions of time, which are denoted by gx1, gy1, and gy2, respectively:

ux1 = gx1(t), uy1 = gy1(t), uy2 = gy2(t), ux1 = gx1(t), u y1 = gy1(t), u y2 = gy2(t). (5.9)

These are rheonomic (time-dependent), holonomic, and nonhomogeneous SFC. As before insert(5.14) into (4.15). But now contributions from columns 1, 2 and 4 do not vanish, and are transferredto the RHS before those columns are removed:

0 0 00 0 03 0 00 0 00 4 00 0 4

[ ux2

ux3

u y3

]+

−10 −10 −100 −10 −10

10 0 00 0 −50 10 100 −10 15

[ ux2

ux3

uy3

]= f ext + f,tra + f con. (5.10)

in which

f ext =

0000

2H(t)H(t)

, f tra =

−5gx1 − 20gx1 − 10gy1

−5gy1 − 10gx1 − 10gy1

10gx1

−3gy2 − 5gy2

10gx1 + 10gy1

10gx1 + 10gy1 + 5gy2

, f con =

f conx1

f cony10

f cony200

,

denote external (applied), transfer and constraint forces, respectively. Only f con carries unknowns.Keeping rows 3, 5 and 6 provides the reduced EOM system[ 3 0 0

0 4 00 0 4

] [ ux2

ux3

u y3

]+

[ 10 0 00 10 100 −10 15

] [ ux2

ux3

uy3

]=

[ 10gx1

2H(t) + 10gx1 + 10gy1

H(t) + 10gx1 + 10gy1 + 5gy2

](5.11)

5–9


This RHS is a known function of time; so (5.11) can be directly solved once the initial conditionsare given. The other subset, which consists of rows 1, 2 and 4, gives the reaction recovery system[ f con

x1f con

y1f con

y2

]=

[ −10 −10 −100 −10 −100 0 −5

] [ ux2

ux3

uy3

]+

[ 5gx1 + 20gx1 + 10gy1

5gy1 + 10gx1 + 10gy1

3gy2 + 5gy2

]. (5.12)

This can be used to recover the constraint (support reaction) forces once ux2, ux3 and uy3 areobtained by solving (?) in time.

§5.5.3. Pseudo-Nonholonomic SFC

The truss support conditions are specified in terms of time-dependent velocities, and time dif-ferentated to get accelerations:

ux1 = 1 + t, u y1 = 2t, uy2 = −3t2, ux1 = 1, u y1 = 2, u y2 = −6t. (5.13)

These look like nonholonomic constraints, but actually can be integrated in time to get displace-ments:

ux1 = t + 12 t2 + Cx1, ux1 = t2 + Cy1, uy2 = −t3 + Cy2. (5.14)

The constants of integration can be determined from the initial conditions. For example supposethat at t = 0 we have ux1(0) = uy1(0) = uy2(0) = 0. Then Cx1 = Cy1 = Cy2 = 0, and we areback to the rheonomic, holonomic SFC case treated in §5.5.2.

§5.6. Holonomic, Homogeneous MFC Examples

The examples in this Section pertain to a more complicated case: a set of holonomic homogeneousconstraints, at least one of which involves multiple freedoms (MFC). The appearance of MFC callsfor more powerful implementation methodologies that straightforward elimination.

§5.6.1. Scleronomic, Homogeneous MFC by Master-Slave

The example truss is subject to three MFC:

uy1 = −ux1, uy2 = ux2, uy3 = −ux3. (5.15)

Interpretation for infinitesimal displacements: the truss rotates about the midpoint of element(3) halfway between nodes 1 and 3. Differentiate (5.23) twice in time to establish accelerationconstraints:

u y1 = −ux1, u y2 = ux2, u y3 = −ux3. (5.16)

To apply the master-slave elimination method, pick 3 slave displacement DOF, say the y componentsuy1, uy2 and uy3, which will be eliminated.9 The x components: ux1, ux2 and ux3 are the masterDOF. Write down the DOF transformation

ux1

uy1

ux2

uy2

ux3

uy3

=

1 0 0−1 0 00 1 00 1 00 0 10 0 −1

[ ux1

ux2

ux3

],

ux1

u y1

ux2

u y2

ux3

u y3

=

1 0 0−1 0 00 1 00 1 00 0 10 0 −1

[ ux1

ux2

ux3

]. (5.17)

9 For details of this method in the static case, see IFEM [106, Chapter 8].

5–10

5–11 §5.6 HOLONOMIC, HOMOGENEOUS MFC EXAMPLES

In compact formu = T u, u = T û (5.18)

Likewise for velocities: û = T û, which may be used for initial conditions as noted below. Substituteu and u in the unconstrained matrix EOM M u + Ku = f and premultiply both sides by TT to getthe transformed EOM

TT M T û + TT K T u = TT f. (5.19)

or

M û + K u = f. (5.20)

in which

M = TT M T =[ 8 0 0

0 6 00 0 10

], K = TT K T =

[ 10 10 0−10 15 5

0 5 5

], f = TT f =

[ 00

2H(t)

]. (5.21)

The transformed stiffness matrix M has rank 2 and nullity 1. The initial conditions u(0) = u0 andu(0) = v0 transform as per u(0) = u0 = T u0 and û(0) = v0 = T v0. Once the reduced matrixEOM (5.26) is solved in time for u = u(t), the slave displacements, velocities and accelerationsmay be recovered from the constraints (5.23); or, equivalently, from the master/slave transformationequations.

§5.6.2. Rheonomic, Nonhomogeneous MFC by Master-Slave

The example truss is subject again to three MFC but now they are nonhomogeneous and rheonomic:

uy1 = −ux1 + g1(t), uy2 = ux2 + g2(t), uy3 = −ux3 + g3(t). (5.22)

Here gi (t), i = 1, 2, 3 are arbitrary but twice differentiable functions of time, which may reduceto constants. If all three functions vanish (5.23) coalesces with the homogeneous case (5.23).Differentiating (5.23) twice in time gives the acceleration constraints:

u y1 = −ux1 + g1(t), u y2 = ux2 + g2(t), u y3 = −ux3 + g3(t). (5.23)

As in the previous example, the slave displacements are uy1, uy2 and uy3, while ux1, ux2 and ux3

are the master DOF. The master-slave transformations may be written in the compact matrix form

u = T u + g, u = T û + g, (5.24)

in which T, u and u are the same as in (5.17), and

g = [ g1 g2 g3 ]T , g = [ g1 g2 g3 ]T , (5.25)

The transformed matrix EOM are

TT M T û + TT K T u = TT (f − Kg) (5.26)

or

M û + K u = f. (5.27)

5–11


in which M anf K are the same as above, but f is different:

f =[ 4g1

−5g2 + 5g3 − 3g2

H(t) − 5g2 + 5g3 + 5g3

]. (5.28)

Examples on nonholonomic MFC will not be given here, since they are non treatable by master-slavemethods unless they integrate to holonomic constraints.

§5.7. Penalty Function Augmentation Methods for MFC

General recommendation: be careful in using penalty function methods to treat dynamic constraints.To justify this statement one needs to resort to variational techniques, which are the topic of nextChapter.

§5.8. Lagrange Multiplier Adjunction Methods for MFC

These is the most powerful technique available for the treatment of constraints. But explanation ofuse is deferred until next Chapter, which introduces variational forms.

Notes and Bibliography

Constraints were not important in early days of particle dynamics since the main application was astronomical:the movement of celestial bodies. Few kinematic constraints appear there except collitions. (Curiously thesame thing appears at the other end of the scale: atomic physics and quantum mechanics.) Collitions werestudied in the XVIII Century, still using particle models. Euler treated isoparametric constraints using thecalculus of variations that he invented. The first investigator that studied constraints in some generality wasLagrange [389], who developed his method of multipliers as a powerful technique that has been heavily usedto date.

In his Principles of Mechanics book, Hertz was the first to emphasize a clear dictinction between holonomicand nonholonomic constraints. The importance of the latter had increased during the XIX Century as theIndustrial Revolution focused attention on rolling motions. But the treatment of those constraints opened upa controversy that persists to the current day, complete with incorrect and misleading statements in even themost reputed textbooks and monographs. This unfortunate saga is well documented in the Hertz biographyby Lutzen [390, Chapter 23].

Within the FEM context, development in this topic have been minimal and tyipcally application dependent.

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