Unit M4 - Mathematical notes 4: Convex analysis · 2018-09-13 · Convex analysis Contents M4.1...

Hyperplasticity Convex analysis M4.1

© G.T. Houlsby, 2017-18

Unit M4 - Mathematical notes 4:

Convex analysis

Contents

M4.1 Introduction 1

M4.2 Some terminology of sets 2

M4.3 Convex sets and functions 4

M4.4 Subdifferentials and subgradients 5

M4.5 Functions defined for convex sets 7

M4.6 Legendre-Fenchel transformation 9

M4.7 The support function 9

M4.8 Further results in convex analysis 12

M4.9 Summary of results for plasticity theory 13

M4.10 Some special functions 14

M4.1 Introduction The terminology of convex analysis allows a number of the issues relating to standard materials to be expressed in a succinct manner. In particular, through the definition of the subdifferential, it allows a rigorous treatment of functions with singularities of various sorts. These arise, for instance, in the treatment of the yield function. A brief summary of some basic concepts of convex analysis is given here. The terminology is based chiefly on that of Han and Reddy (1999). A more detailed introduction to the subject is given by Rockafellar (1970). No attempt is made to provide rigorous, comprehensive definitions here, and for a fuller treatment reference should be made to the above texts. Although it is currently used by only a minority of those studying plasticity, it seems likely that in time convex analysis will become the standard paradigm for expressing plasticity theory.


© G.T. Houlsby, 2017-18

M4.2 Some terminology of sets

We use brackets to indicate a set, so that 5.3,1,0 is simply a set

consisting of the real numbers 0, 1 and 3.5. We use to denote the null (empty) set.

The notation xCx is used to denote the set of values of x that satisfy

the condition xC . Thus a closed set containing an interval of real numbers,

which is denoted by , , is defined by:

bxaxba :, (M4.1)

where we use : to indicate a definition, and the meaning of the contents of the final bracket is “x, such that bxa ”. Similarly an open interval is denoted by:

bxaxba :, (M4.2)

We prefer this notation to the form ba, used in some texts. Mixed intervals

are indicated by:

bxaxba :, (M4.3)

bxaxba :, (M4.4)

In the following C is a subset in a normed vector space V (in simple terms a space in which a measure of distance is defined), usually with the dimension

of nR (with n finite), but possibly infinite dimensional. The notation , is

used for an inner product, or more generally the action of a linear operator on a function. The space V is the space dual to V under the inner product

xx*, , so that Vx and Vx * . More generally V is termed the

topological dual space of V (the space of linear functionals on V).

We extend the concept of an interval of values to define, for Vx and Vy , the closed interval:

101:, yxyx (M4.5)

The open and mixed intervals are similarly defined.

The supremum of a set C of real numbers is the largest value of the set, and the infimum is the lowest value of the set. Thus

xCxCC ,:sup (M4.6)


© G.T. Houlsby, 2017-18

xCxCC ,:inf (M4.7)

where Cx has the meaning “for all x belonging to C”.

Summation and multiplication of sets

The operation of summation of two sets, illustrated in Figure M4.1(a), is defined by:

22112121 ,: CxCxxxCC (M4.8)

The operation of scalar multiplication of a set, illustrated in Figure M4.1(b) is defined by:

CxxC : (M4.9)

It is also convenient to define the operation of multiplication of a set C by a set S of scalars:

CxSxSC ,: (M4.10)

Interior, boundary and closure

The definitions of the interior and boundary of a set are intuitively simple concepts, but their formal definitions depend first on the definition of

distance. In nR the Euclidian distance is defined as:

21,:, yxyxyxyxd (M4.11)

and we define the open ball of radius r and centred at ox as:

rxxdxrxB oo ,:, (M4.12)

C1 + C

2

C2

C1

x2

x1

C

C

x2

x1

(a) (b)

Figure M4.1: (a) summation of sets, (b) scalar multiple of a set


© G.T. Houlsby, 2017-18

The interior of C is then defined as:

CxBxC ,,0:int (M4.13)

where this means that there exists some (possibly very small) so that an open ball of radius is entirely contained in C. The closure of C is defined as the intersection of all sets obtained by adding a ball of non-zero radius to C:

0,0:cl BCC (M4.14)

and finally the boundary of C is that part of the closure of C which is not interior:

CCC int\cl:bdy (M4.15)

A little careful thought reveals that the formal definitions of the interior and boundary correspond to the simple intuitive understanding of these concepts.

M4.3 Convex sets and functions A set C is convex if and only if:

Cyx 1 , Cyx , , 10 (M4.16)

where for instance Cyx , has the meaning “for all x and y belonging to C”.

Simple examples of non-convex and convex sets in two-dimensional space are illustrated in Figure M4.2. A function f whose domain is a convex subset C of V and range is real or is convex if and only if:

yfxfyxf 11 , Cyx , , 10 (M4.17)

This is illustrated for a function of a single variable in Figure M4.3. Convexity requires that NQNP for all N between X and Y. This property has to be true

for all pairs of X,Y within the domain of the function. A function is strictly convex if can be replaced by < in (M4.17) for all yx .

yfxfyxf 11 , Cyx , , 10 , yx

(M4.18)

The effective domain of a function xfdom is defined as that part of the

domain for which the function is not , thus xfVxxf :dom .

The epigraph of a function whose domain is a subset S in V is defined as:

xfRSxxxf ,,,epi (M4.19)


© G.T. Houlsby, 2017-18

A simple geometric interpretation is that the epigraph is the set of points “above” the graph of the function.

A function is convex if and only if its epigraph is a convex set.

M4.4 Subdifferentials and subgradients The concept of the subdifferential of a convex function is a generalisation of the concept of a differentiation. It allows the process of differentiation to be extended to convex functions that are not smooth (i.e. continuous and differentiable in the conventional sense to any required degree). If V is a

X

Y

Non-convex

X

Y

Convex

1 -

Figure M4.2: Non-convex and convex functions

x

z

Q

P

YNX

(1-)

z = f(x)

x + (1-)y

= x + (1-)y

Figure M4.3: Graph of a convex function of one variable


© G.T. Houlsby, 2017-18

vector space and V is its dual under the inner product , , then Vx * is

said to be a subgradient of the function xf , Vx , if and only if

xyxxfyf *, , y .

The subdifferential, denoted by xf is the subset of V consisting of all

the vectors *x satisfying the definition of the subgradient:

yxyxxfyfVxxf ,*,*: (M4.20)

For a function of one variable the subdifferential is the set of the slopes of lines passing through a point on the graph of the function, but lying entirely on or below the graph. The concept is illustrated in Figure M4.4.

The concept of the subdifferential allows us to define “derivatives” of non-

differentiable functions. For example the subdifferential of xw is the

modified signum function, which we define as function:

0,1

0,1,1

0,1

x

x

x

xwxS (M4.21)

Thus at a point x, xf may be a set consisting of a single number equal to

xf , or a set of numbers, or (in the case of a non-convex function) may be

x

w

w = f(x)

P

Figure M4.4: Subgradients of a function at a non-smooth point


© G.T. Houlsby, 2017-18

empty. For a smooth function xfxf .

For functions of more than one variable we use the notation yxfx , to

indicate the subdifferential with respect to the variable x .

M4.5 Functions defined for convex sets The domain of a function xfdom is the set of values of x for which xf

is defined. The range of xf is the set of values xfxxfy dom, .

The indicator function of a set C is a function defined by:

Cx

CxxC ,

,0I (M4.22)

So that the indicator function is simply zero for any x that is a member the set, and elsewhere. The indicator of a convex set is a convex function. Although the indicator appears at first sight to be a rather curious function, it proves to have many applications. In particular it plays an important role in plasticity in that it is closely related to the yield function. (Note that in some other branches of mathematics the indicator function is defined as having the value +1 for any x that is a member the set, and zero elsewhere).

The normal cone xCN of a convex set C, is the set defined by:

CyyxxVxxC ,0*,*:N (M4.23)

It is straightforward to show that 0xCN if Cx int (the point is in

the interior of the set), and that xCN can be identified geometrically with

the cone of normals to C at x if Cx bdy (the point is on the boundary of the

set), and further that xCN is empty if Cx (the point is outside the set).

Furthermore, the subdifferential of an indicator function of any convex set is the normal cone of that set: xx CC NI .

Another important function defined for a convex set is the gauge function or Minkowski function, defined for a set C as:

CxxC 0inf: (M4.24)

In other words xC is the smallest non-negative factor by which the set

can be scaled and x is a member of the scaled set. The meaning is most easily understood for sets which contain the origin (which proves to be the case for all sets of interest in standard materials). In the following we shall therefore assume that C is convex and contains the origin. It is straightforward to see in


© G.T. Houlsby, 2017-18

this case that 1 xC for any point on the boundary of the set, is less than

unity for a point inside the set and greater than unity for a point outside the set. At the origin 00 C .

In the context of plasticity, it is immediately obvious that the gauge may be related to the conventional yield function. If the set C is the set of (generalised) stresses that are accessible for any given state of the internal

variables (the elastic region), then the yield function is a function conventionally taken as zero at the boundary of this set (the yield surface), negative within and positive without. One possible expression for the yield function would therefore be 1 Cy . Other functions could of course

be chosen as the yield function, but this is perhaps the most rational choice, so we follow Han and Reddy (1999) in calling this the canonical yield function. To emphasise the case where the yield surface is written in this way we shall give it the special notation 1 Cy .

The gauge function is always homogeneous of order one in its argument x, so that xx CC . (In the language of convex analysis such functions

are simply referred to as positively homogeneous.) The canonical yield function is therefore conveniently written in the form of a positively homogeneous function of the (generalised) stresses, minus unity. It is also clear that the gauge of the set of accessible generalised stresses contains exactly the same information as the yield function, canonical or otherwise, and there are may be benefits from specifying the gauge rather than the yield function.

It is useful to note that at the boundary of C, the normal cone can also be written 0,xxx CCC IN . This proves to be a convenient

form, allowing the normal cone to be expressed in terms of the subdifferential of the gauge function, and therefore (in plasticity of standard materials) of the canonical yield function.

It is straightforward to see that the definition (M4.24) can be inverted. Given a positively homogeneous function xf one can define a set C, such

that xf is the gauge function of C:

1 xfxC (M4.25)

which has the property that xfxC . It is worth noting too that the

indicator function of a set containing the origin can always be expressed in the following form, and this proves to be useful in the application of convex analysis to plasticity:


© G.T. Houlsby, 2017-18

x

xx

C

CC

1,

0, 1

I

II (M4.26)

The function x0,I is simply zero for all non-positive values of x, and

for positive values.

We also note the subdifferential:

xx

x

xx

CC

C

CC

1,

1,

N

I

IN

(27)

M4.6 Legendre-Fenchel transformation The Legendre-Fenchel transformation (often simply called the Fenchel dual, or conjugate function) is a generalisation of the concept of the Legendre transformation. If xf is a convex function defined for all Vx , its Legendre-

Fenchel transformation is ** xf , where Vx * , defined by:

xfxxxfVx

*,sup** (M4.28)

where Vx

sup means the supremum for any Vx .

It is straightforward to show the Fenchel dual is the generalisation of the Legendre transform. We use the notation that if xfx * and ** xf is the

Fenchel dual of xf then ** xfx .

Some useful Fenchel duals, together with their subdifferentials are given in Table M4.1. The dual of the sum of two functions involves the process of infimal convolution:

ygyxfxgfVy

inf: (M4.29)

Table M4.1 also includes some special functions which are defined in section <TBA>.

M4.7 The support function The support function is also defined for a convex set. For a convex set C in V, if Vx * , then the support function is defined by:

CxxxxC *,sup* (M4.30)


© G.T. Houlsby, 2017-18

Note that although C is a set of values of the variable x, the argument of the support function is the variable *x conjugate to x.

It can be shown that the support function is the Fenchel dual of the indicator function. The support function is always homogeneous of order one in *x , i.e. it is positively homogeneous.

It follows that any homogeneous order one function defines a set in the dual space. In hyperplasticity one can observe that the dissipation function is indeed homogeneous and order one in the internal variable rates. It can thus be interpreted as a support function, and the set it defines in the dual space of generalised stresses is the set of accessible generalised stress states (the elastic region). The Fenchel dual of the dissipation function is the indicator

Function Subdifferential Fenchel dual Subdifferential of dual

xf xf ** xf ** xf

x0I x0N 0 0

xaI xaN ax a

x0,I x0,N *,0 xI *,0 xN

x1,1I x1,1N *x *xS

x1,0I x1,0N *x *xH

x1,I x1,N *pos x *xH

1 0 *0 xI *0 xN

x 1 *1 xI *1 xN

22x x 2*2x *x

nx , 1n 1nnx 1

*1

nn

n

xn

11

*n

n

x

xexp xexp **log* xxx *log x

xgxf xgxf *** xgf

xaf xfa **1

xfa

**1

xfa

axf axfa axf ** axfa

**1

Table M4.1: Some Fenchel duals and their subdifferentials


© G.T. Houlsby, 2017-18

function for this set of accessible states, which is of course zero throughout the set.

Equation (M4.30) can be inverted to obtain the set C from the support function. If ** xf is a homogeneous first order function in *x , then the set

defined by solving the system of inequalities:

*,***, xxfxxxC (M4.31)

satisfies the condition that *** xfxC . Application of (M4.31), with

** xf as the dissipation function, allows the set of accessible (generalised)

stresses to be derived from the dissipation function in a systematic manner, hence the elastic region can be derived from the dissipation function.

The subdifferential of the support function defines a set called the maximal responsive map (see Han and Reddy (1999), although we depart from their notation here):

** xx CC (M4.32)

The normal cone and the maximal responsive map are inverse in the sense that:

xNxxx CC ** (M4.33)

It also follows (see Lemma 4.2 of Han and Reddy (1999)), that C is simply related to the support function by the subdifferential at the origin, i.e. the maximal responsive map at the origin, thus:

00 CCC (M4.34)

Both the gauge and support functions are positively homogeneous. Defining the domain of the support function *dom xS C , it can be

shown (see Han and Reddy, 1999) that:

*

*,sup

*0 x

xxx

CSxC

(M4.35)

and xC is called the polar of *xC , written CC . The process is

symmetric so that we have CC and, defining the domain of the gauge

function xG C dom :

x

xxx

CGxC

*,sup*

0 (M4.36)

Further we have the following inequality:


© G.T. Houlsby, 2017-18

GxSxxxxx CC ,*,*,* (M4.37)

and, the equality holds for *xx C :

Sxxxxxxx CCC *,*,*,* (M4.38)

Application of (M4.35), together with 1 Cy , allows the

canonical yield function to be determined directly from the dissipation function.

M4.8 Further results in convex analysis Whilst the above are the most important results needed, it is worth noting some further relationships between convex sets and functions.

The polar C of a convex set C is defined as:

1** xxC C (M4.39)

in other words it is the set for which the support function of C is the gauge.

Alternatively one may write:

1*,,* xxCxxC (M4.40)

The polar f of a positively homogeneous function convex function f is

another positively homogeneous function defined by:

xxfxxxf ,*,0inf* (M4.41)

If f is finite everywhere and positive except at the origin then this can be

written:

xf

xxxf

x

*,sup*

0 (M4.42)

The above results can be generalised to other than positively homogeneous functions. The polar f of a non-negative function convex

function f which is zero at the origin is defined by:

xxfxxxf ,1*,0inf:* (M4.43)

But note that when *, xx has dimension this must be replaced by:

xxfcxxxf ,*,0inf:* (M4.44)

Where c is a suitable normalising constant with the same dimensions as

*, xx . Equation (M4.43) reduces to (M4.35) for positively homogeneous


© G.T. Houlsby, 2017-18

functions, as the infimum in equation (M4.37) in that case occurs as x , and the 1 becomes negligible.

Finally the polar of the dual of f is the obverse g of a function f, defined

by:

10inf: xfxg (M4.45)

and the operation:

xfxf 1: (M4.46)

is called right scalar multiplication (note that xIxf 00 if xf

and xfxf 0 if xf ).

The indicator and the gauge of a convex set are obverse to each other.

Note in passing that quadratic functions 2axxf are self-obverse so that

2axxfxg .

M4.9 Summary of results for plasticity theory In summary we have the following concepts from convex analysis which

are of relevance in plasticity theory:

A convex set C in V.

The indicator function xCI of the set.

The gauge function xC of the set.

The support function *xC which is the Fenchel dual of the

indicator, and is also the polar of the gauge function.

The normal cone xx CC IN which is a set in V which is the

subdifferential of the indicator function.

The maximal responsive map, which is the subdifferential of the support function ** xx CC

The set C is the subdifferential of the support function at the origin 00 CCC .

The relationships between these quantities are illustrated in Figure 5 for the case of the simple one-dimensional set baC , .

The roles of these concepts in hyperplasticity are explored in more detail below, but Table M4.2 gives the correspondences between some concepts in conventional plasticity theory and in the convex analytical approach.


© G.T. Houlsby, 2017-18

M4.10 Some special functions We have already introduced the modified signum function (M4.21), which we define by:

x

x

x*

x

x*

xNxIx CC *

xIC

xC

Subdifferential Subdifferential

Inverse

Fenchel dual

Polars

** xxx CC

*xC

Gauge

function

Indicator

function

Normal

cone

Support

function

Maximal

responsive map

(0,b)

(-a,0) (b,0)

(0,-a)

1

Convex set baC ,

(-a,1) (b,1)

(b,0)(-a,0)

1a

1

b

*xIC

Indicator

of polar

(-1/a,0) (1/b,0)

Obverse Obverse

Fenchel dual

Polar set baC 1,1

x*

1

Figure M4.5: Relationships between functions of a convex set in one dimension


© G.T. Houlsby, 2017-18

0,1

0,1,1

0,1

x

x

x

xS (M4.21)bis

Closely related is the modified Heaviside step function:

0,1

0,1,0

0,0

12

1

x

x

x

xx SH (M4.47)

It is also useful to define a closely related function:

0,1

0,1,

0,

x

x

x

xH (M4.48)

which can also be written as xx ,01 NH , is useful because it is

the subdifferential of the positive values of x, defined as:

Conventional plasticity theory

Convex analytical approach to (hyper)plasticity

Elastic region. A convex set in (generalised) stress space.

Yield surface. The indicator function or (for some purposes) the gauge function, or equivalently the canonical yield function.

Plastic potential and flow rule.

Gauge function (or equivalently the canonical yield function) and the normal cone.

Plastic work. Support function (equal to the dissipation function). NB for models in which energy can be stored through plastic straining this is not equal to the plastic work.

(No equivalents in conventional theory).

The indicator of the elastic region and the support function (dissipation) are Fenchel duals.

The gauge function of the elastic region and the support function (dissipation) are polars.

Table M4.2: Correspondences between conventional plasticity theory and the convex analytical approach


© G.T. Houlsby, 2017-18

0,

0,pos ,0 xx

xxx I (M4.49)

Note that a careful distinction is needed between xpos , the absolute value

x and the Macaulay bracket x . Note that all of these functions are convex.

Unit M4 - Mathematical notes 4: Convex analysis · 2018-09-13 · Convex analysis Contents M4.1...

Documents

Transcript of Unit M4 - Mathematical notes 4: Convex analysis · 2018-09-13 · Convex analysis Contents M4.1...