Modeling and Convexity E-Book

Eduardo Souza de Cursi

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 625

Veröffentlichungsjahr: 2013

Beschreibung

This reference book gives the reader a complete but comprehensive presentation of the foundations of convex analysis and presents applications to significant situations in engineering. The presentation of the theory is self-contained and the proof of all the essential results is given. The examples consider meaningful situations such as the modeling of curvilinear structures, the motion of a mass of people or the solidification of a material. Non convex situations are considered by means of relaxation methods and the connections between probability and convexity are explored and exploited in order to generate numerical algorithms.

Leseprobe

Table of Contents

Introduction

PART 1. MOTIVATION: EXAMPLES AND APPLICATIONS

Chapter 1. Curvilinear Continuous Media

1.1. One-dimensional curvilinear media

1.2. Supple membranes

Chapter 2. Unilateral System Dynamics

2.1. One-dimensional curvilinear media

2.2. Contact dynamics

Chapter 3. A Simplified Model of Fusion/Solidification

3.1. Contact dynamics

Chapter 4. Minimization of a Non-Convex Function

4.1. Probabilities, convexity and global optimization

Chapter 5. Simple Models of Plasticity

5.1. Ideal elastoplasticity

PART 2. THEORETICAL ELEMENTS

Chapter 6. Elements of Set Theory

6.1. Elementary notions and operations on sets

6.2.The axiomof choice

6.3. Zorn’s lemma

Chapter 7. Real Hilbert Spaces

7.1. Scalar product and norm

7.2. Bases anddimensions

7.3. Open sets and closed sets

7.4. Sequences

7.5.Linear functionals

7.6. Complete space

7.7. Orthogonal projection onto a vector subspace

7.8. Riesz’s representation theory

7.9. Weak topology

7.10. Separable spaces: Hilbert bases and series

Chapter 8. Convex Sets

8.1. Hyperplanes

8.2.Convex sets

8.3. Convex hulls

8.4. Orthogonal projection on a convex set

8.5. Separation theorems

8.6. Convex cone

Chapter 9. Functionals on a Hilbert Space

9.1. Basic notions

9.2. Convex functionals

9.3. Semi-continuous functionals

9.4. Affine functionals

9.5. Convexification and LSC regularization

9.6. Conjugate functionals

9.7. Subdifferentiability

Chapter 10. Optimization

10.1.The optimization problem

10.2.Basic notions

10.3. Fundamental results

Chapter 11. Variational Problems

11.1. Fundamental notions

11.2.Zeros of operators

11.3. Variational inequations

11.4. Evolutione quations

Bibliography

Index

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Modélisation et convexité, first published 2008 in France by Hermes Science/ Lavoisier, © LAVOISIER 2008

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2010

The rights of Eduardo Souza de Cursi and Rubens Sampaio to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Cursi, Eduardo Souza de.

[Modélisation et convexité. English]

Modeling and convexity / Eduardo Souza de Cursi, Rubens Sampaio.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-177-3

1. Engineering mathematics. 2. Machinery--Mathematical models. 3. Convex sets. I. Sampaio, Rubens. II. Title

TA342.C87 2010

620.001'51--dc22

2010007879

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-177-3

Introduction

From a mathematical point of view, the first approach leads us to write differential equations or partial differential equations describing the state of (equilibrium or movement) of the system; the second approach leads us to write variational equations, that is to say, variational (or weak) formulations of these particular equations; the third method leads to the study of the variation of the energy functional, that is to say, the differential of an energy functional.

The last two approaches are known as variational methods. It is interesting to note that the apparently alternative idea which consists of writing the conservation of physically significant quantities (e.g. mass, quantity of movement, or energy) for every part of the system is truly a variant of variational approaches.

Modern scientific computation and especially finite elements and finite volumes are based on variational approaches and seem nowadays intrinsically related to the pervasive use of the computer in sciences in general and to that of mechanics in particular. The truth is that variational methods are both modern and old fashioned. It is often the case in the development of science that new scientists revisit and update classic ideas in order to reach new horizons. Paraphrasing a classic quote ascribed to Newton, we can conclude from the history of variational methods that, in order to see further, the best option is to stand on a giant’s shoulder.

One of the greatest scientific challenges has been the lever – studied by humanity for practically 2,000 years before being resolved. Aristotle had already suggested that the workings of the lever could be explained by the analysis of “natural” movements and “non-natural” movements of the system. Thus, he started the revolutionary idea of explaining the equilibrium by analyzing possible movements.

Likewise, the renowned Heron of Alexandria – whose imprint on mechanics teaching is still noticeable nowadays – formulated the saving principle. This principle can be reinterpreted in terms of variations of functionals. In the Middle Ages, Jordanus Nemorarius studied the lever problem introducing a new notion of linearity close to that of virtual work. Other great names in science also contributed to the development of variational methods, such as Torricelli, Fermat, Huyghens, Leibiniz, the Bernouilli family, Euler, Varignon, Maupertuis, Fourier, d’Alembert, Coulomb, Lagrange and Hamilton.

Briefly, we can see two reference points in the history of variational methods – the introduction of differential and integral calculus and the arrival of the computer age. These two events have favored variational approaches and gained significant popularity thanks to the work of the Russian dam builder Boris Galerkin. Right before WWI and after his trip through Europe, he became inspired in building the approximations now named after him. Galerkin’s work was properly mathematized in the work of another Russian man – Sobolev – and it was also resumed by the French school founded by Hadamard, Fréchet, Schwarz, Lions, and Moreau. These scholars came up with syntheses considered classic in the published studies [LAU 72, EKE 74] (see also [ROC 70a]).

Problems comprising constraints (or restrictions) – for which specific care is needed (in relation to reaction forces) – demand special attention in the study of physics, or more recently, economic systems. In this field, variational methods show all their power and flexibility and so they bring systematic procedures as well as varied and adapted numerical methods to the center of the analysis (see, e.g. [PAN 84, FR 01]). In these situations, the theory which establishes variational approaches is the Convex Analysis, the subject of this book. In this field, numerous situations of great practical interest bring us to open and unsolved mathematical problems. This should contribute to researchers’ interest in this field.

This book aims to present the basis of convex analysis and its applications to modeling through examples. Let us note that convex analysis deals not only with convex sets and convex functions but also with non-convex ones, as will be shown in this book.

The authors thank all those who have contributed to the realization and distribution of this book. A special mention goes to CAPES-COFECUB international co-operation project, CNPq, FAPERJ and SBMAC. Eduardo Souza de Cursi adds a special mention dedicated to those who helped him follow the path of his childhood dreams.

Part 1

Motivation: Examples and Applications

Chapter 1

Curvilinear Continuous Media

In certain situations – frequently in practice – movements and efforts connected to three-dimensional solids can be approached from similar fields related to one-dimensional and two-dimensional continuous media. This is the case, for example, of theories about strings, bars, beams, membranes, plates and shells.

These approximations generally rest on geometrical hypotheses. For example, when one of the dimensions of a solid is much bigger than the others, the geometry of this solid is approximated by that of a curve and its movements are described in approximately the same way as those of the same curve; thus we obtain a one-dimensional model.

Likewise, when one of the dimensions is much smaller than the others, we can approximate the geometry of a solid to that of a surface and its movements to deformations of the same surface; the result is a two-dimensional model.

Thus, the descriptions discussed and obtained constitute the type of curvilinear media of great relevance for engineers. The apparent geometrical simplicity of these models is deceitful; they are very rich in terms of applications and theoretical and practical difficulties, from which many remain unsolved to date. The study of curvilinear media brings us to the basis of methods, as we will see in some examples.

Figure 1.1.Tubular structure described by a curve and a straight section

1.1. One-dimensional curvilinear media

Let us consider a three-dimensional solid of “tubular” nature, that is to say, described by a curve and a straight section (Figure 1.1).

The configuration of the solid is

We can approximate the geometry of the solid as if it were that of the curve (x(α) : α ∈ (0,A)), when the biggest right-section dimension S(α) (e.g. the longest diameter or side of a right section S(α)) is considered small in comparison to the length of the curve.

σ is the Cauchy Stress Tensor of internal efforts (that is to say, the Eulerian tensor of internal efforts). In order to use the approximation below efficiently, it is necessary to reduce σ to a resultant force T(α) and a resulting moment M(α) applied to a point x(α), that is to say,

Figure 1.2.Equilibrium of an element of a tubular solid

Internal efforts are represented by T and M. When the solid is submitted to an external load corresponding to a density of the resulting efforts c(α) and resulting moments m(α), the equilibrium of a string element is given by (Figure 1.2):

that is to say,

(1.1)

In order to complete the equations, it is necessary to have, on the one hand, conditions to the limits and, on the other hand, a constitutive law (e.g. a connection between the variation of geometry, i.e., x′, and the internal efforts).

1.1.1. Ideally flexible string

Figure 1.3.Unilateral behavior: T ≥ 0

traction efforts are possible and the medium reacts to the application of compressive force by a geometrical change in order to transform the compression into traction (see Figure 1.3). Thus, T ≥ 0 for all physically feasible configurations.

As seen from Lagrange’s point of view, the configuration of the string is given by a map x : [0, ℓ] → 3, which associates the position-vector x(a) with the particle a ∈ [0, ℓ]. We then have

(1.2)

Equation (1.1) is reduced to

(1.3)

For an elastic string, Hooke’s law is

(1.4)

where K > 0 is the modulus of elasticity and ε is the deformation:

(1.5)

The conditions of the usual limits for string are to fix one side and the force applied to the other:

(1.6)

The configurations of the equilibrium of a string are the solutions to the following.

PROBLEM 1.1. Let K > 0, ℓ > 0, and let c and F be given. Determine (x, T) verifying equations (1.2)–(1.6).

Let us note that equation (1.4) combined with equation (1.5) prove that

(1.7)

Thus, the admissible configurations of a string are the elements of V+.

NOTE 1.1. In a more formal way, V is a Hilbert Space:

This space can be provided with the scalar product:

1.1.1.1. The essential difficulty

and on the other hand,

Thus

Therefore,

Thus

Then x ∉ V+. Thus, x ∈ V+ and x ∉ V+, which is a contradiction. Therefore, no configuration of equilibrium exists for the string.

Thus, discretizing equilibrium equations leads to a system of nonlinear equations:

where

These equations can be approximated numerically by a relaxation algorithm:

ω > 0 is a relaxation coefficient. The quality of the result may be controlled, for example, using the parameter

For any given point of departure U(N,0), the results stay close to those of Table 1.1. Several values of ω also lead to similar results, all very close to each other. Thus, despite the inexistence of solutions, we observe an excellent numerical convergence, as if the problem had a solution. So, at least, from a numerical standpoint, it is not the same problem.

we obtain

where ∂U(x) is the subgradient of U at x (Definition 9.20). Likewise,

where W is the internal potential energy of the system,

Thus, the equilibrium equation is

Let

We obtain

This equality proves that the equilibrium equation should be

and Theorem 10.4 proves that

This problem does not admit solutions. Nonetheless, the relaxed problem

1.1.1.2. Unilateral contact

(1.8)

The part of the string D(x) in contact is one of the unknown factors of the problem. In general, the unknown cannot be determined until the equilibrium configuration has been found (that is to say, the solution):

(1.9)

The presence of an obstacle introduces a new constraint in the sense of classical mechanics. A reaction, or constraint reaction, R, being one of the unknown factors of the problem, is associated with this constraint.

Figure 1.4.Unilateral contact between the string and the obstacle

1.1.1.2.1. Case without friction

As we have already observed, the presence of the obstacle introduces a force of reaction R on the external load:

(1.10)

(1.11)

where n is the perpendicular angle orientated toward the interior of the admissible region. Assuming that ∇ψ ≠ 0 on the obstacle, we obtain

(1.12)

The reaction is directed toward the interior of the admissible region and is zero when there is no contact:

(1.13)

In this case, R is a normal vector to C (Definition 8.14) and the constraint is ideal, that is to say, the work of the reaction force R is zero for every virtual movement compatible with the constraint – that is to say, for all dx belonging to the tangent cone TC(C, x). R is an element of the normal cone NC(C, x) – which is formed by forces carrying out non-positive work (that is to say, negative or zero) in every compatible virtual movement, that is to say, elements which have a non-positive scalar product with all the elements of TC(C, x). In addition, R is orthogonal to TC(C, x). An ideal constraint does not dissipate energy.

Figure 1.5.Reaction of the obstacle

In order to determine the equilibrium configuration of the string in the presence of an obstacle, we have to solve the following problem.

PROBLEM 1.2. We have K > 0, ℓ > 0, and the load (c0, F) given. Determine (x, T, R, D(x)) verifying equations (1.2)–(1.6) and (1.8)–(1.13).

The main difficulty in the study of problems of unilateral contact resides in the multiplicity of values of D(x) and R (therefore, of R). For example, if we consider the situation where the obstacle is a cylinder:

Figure 1.6.Multiple solutions

If the string is long enough, we can construct several solutions (see Figure 1.6) which respect to different values of T , D(x) and R.

The multiplicity of solutions and, particularly, that of the values of D(x) and R is intimately related to convexity: assuming that the external load c0 is derived from a convex, continuous potential U0, that is to say,

Moreover, assuming that the admissible region is convex, then, on the one hand, C is a closed convex of V . On the other hand, the reaction of the obstacle R is associated with the subgradient (see Definition 9.20) of the indicator function ΨC of C (see Definition 10.1). As we have already seen, R is an element of the normal cone to C in x ∈ NC(C, x) (see Definition 8.14) and, since C is convex (see Lemma 10.2),

Thus, the external load verifies (see Theorem 9.15)

The potential U is convex and weak lower semi-continuous (Definition 9.10 and Theorem 9.4). Yet, as we have already pointed out, Theorem 10.4 proves that

In this case, the relaxed problem

admits a solution. We associate with this solution a tension field ∈ ∂W**(), reaction force ∈ −∂ΨC(), a load 0 ∈ ∂U0() and an area of contact D(). Using the convexity of the problem, it is possible to prove that

that is to say, that the tension field associated with x is and the total external load associated with x is +0. Thus, we obtain a partial uniqueness result concerning the internal and external forces. In certain specific situations, for example, a flat obstacle and a gravity-induced load, these equalities entail the uniqueness of the contact area (see also [CUR 87, CUR 85]).

1.1.1.2.2. The problem of “small perturbations” or “small displacements”

(a) Approximation of the contact area. This area is approximated by the area in contact in the reference configuration, that is to say,

(1.14)

Coherently with this approximation, we obtain

(1.15)

(b) Approximation of the admissible set. Being xR(a) ∉ D(xR), then ψ(xR(a))>0. Therefore we obtain

Thus, for every infinitesimal u(a),

in such a way that

Thus, for every infinitesimal u(a),

It follows from this analysis that

(1.16)

(c) Approximation of the obstacle reaction. In coherence with the approximation of the admissible set,

(1.17)

(d) Approximation of the external load unconnected to the obstacle. In coherence with the hypothesis of an infinitesimal displacement,

where u → Dc0(xR)(u) is a linear map. Therefore,

(1.18)

Thus, the problem of “small perturbations” of the equilibrium of string is formulated in the following way.

PROBLEM 1.3. We have K > 0, ℓ > 0, xR, c0 and F given. Determine (x, T, R) verifying equations (1.2)–(1.6) and (1.10), (1.14)–(1.18).

This formulation eliminates the difficulties connected to the non-convexity of the admissible region and to the non-convexity of the external potential U0: C is approximated by a closed convex set and U0 is approximated by the work of a constant force (that is to say, a linear and continuous functional). In this case, the uniqueness of the global external load entails that of the reaction of the obstacle R. During the numerical use, the indicator function must be approximated by a continuous function, using one of the methods explained in section 10.3.1 (see also [CUR 92]).

1.1.1.2.3. Contact with dry friction

If friction intervenes in contact, the reaction of the obstacle is not orthogonal to its surface:

(1.19)

φ is the friction force, tangent to the obstacle, and therefore the value is limited:

(1.20)

The function A is defined by the friction coefficient μ > 0 and the normal reaction:

(1.21)

Figure 1.7.Multiplicity of solutions in a convex situation

1.1.1.2.4. The problem of “small perturbations” with friction

The difficulty highlighted above (multiplicity of friction forces) has popularized the use of the “small perturbations” approximation for the analysis of situations with friction. In this case, apart from the previous approximations, we must also use the following.

(e) Approximation of friction. In coherence with the approximations (1.14) and (1.16), the maximal value of the friction force is approximated by

(1.22)

where AR is themaximal value in the reference configuration xR and RR is the normal part of the obstacle reaction in the same configuration.

Moreover, friction acts in a way which prevents movement. It is therefore natural to assume that the opposition of friction to movement is maximal and that a displacement only takes place when friction is incapable of preventing it – which also corresponds to the practical observations, that is to say, a non-zero displacement must arrive even with friction at its maximal value. This idea leads to Coulomb’s law, defined by the equations

(1.23)

where uT is the tangent displacement (let us recall that, hypothetically, n ≈ nR):

(1.24)

In this case, we obtain

1.1.2. The “elastica” problem: buckling of an inextensible beam

Let us consider a curvilinear medium with a horizontal segment as natural configuration and submitted to horizontal forces F and −F, with F > 0 (Figure 1.8).

Figure 1.8.The problem of the “elastica”

Assuming the medium is inextensible and dealing with the problem as if two-dimensional, the unit target vector is

in such a way that

The displacement of the particle a is

In the absence of any loads other than forces applied to the extremities, the external potential is

and let – assuming that the medium behaves as a beam submitted to pure flexion – the energy of elastic deformation be

Thus, we can determine the stable equilibrium configurations using the principle of minimum energy:

More generally, we can determine the Gâteaux derivative of J at θ (Definition 9.18):

and solve the variational equation:

Thus, for

Let us note that these conditions imply that

We have

We obtain

It is interesting to note that the presence of a flat obstacle does not modify this result (Figure 1.9).

Figure 1.9.Buckling in the presence of a flat obstacle

In this case, the admissible configurations verify that

in such a way that the admissible values of θ are the elements of

Since θ1 ∈ C, the stable solution is still θ1, for λ > λ1.

1.2. Supple membranes

A membrane is a continuous medium, geometrically modelized by a surface and physically described by the fact that the internal forces are tangent to its configuration. It is the two-dimensional similar to the medium studied in section 1.1, that is to say, a solid whose geometry is described by a mean surface Σ and a thickness h > 0. This thickness is considered small in comparison to the dimensions of Σ (see Figure 1.10).

Describing the geometry of such a solid is equivalent to describing the geometry of the surface of Σ – which usually requires resorting to differential geometry. In this context, it is useful to distinguish carefully between contravariant coordinates and covariant coordinates: if {e1, e2, e3} is a basis – possibly non-orthonormal – of 3 and x ∈ 3, we can represent x in a unique way as a linear combination

Figure 1.10.Description of the membrane configuration

Thus, for an orthonormal basis {I, J, K}, both the covariant and contravariant coordinates coincide. However, this does not apply to the general description of membranes, which most often makes intrinsic bases to the surface Σ intervene.

1.2.1. Curvilinear coordinates and charts

In this case, polar coordinates, for example, can be used:

Figure 1.11.An example of a geometrical description of a membrane. On the left: a circular membrane and polar coordinates. On the right: a configuration where the membrane is folded

Figure 1.12.Description of a membrane using two charts: natural state and two folded configurations

1.2.2. Metric tensor

Let us note

{e1, e2, e3} is a local reference point on the surface Σ. By stating that

Thus,

Figure 1.13.Determining angular variations and dilatations

Concerning local lengthening, we have

We write (gαβ)1≤α,β≤2 the inverse if the metric tensor

These two matrices serve to transform contravariant indexes to covariant indexes and reciprocally,

If we define

then we have

If N is a linear map associating a vector tangent to Σ with another vector tangent to Σ, then N(eβ) can be written as eα (that is to say, its contravariant coordinates are and ). We then have

and the covariant components of N(eβ) are

When we wish to use the covariant components of da, we write

Thus,

1.2.3. Internal efforts and constitutive law

When the natural state Σ0 corresponds to a chart X, we can define

and

The lengthening tensor (similar to that of Green-Lagrange [BUD 63]) is

Internal efforts are described by a tension tensor N which follows Hook’s law:

with

Here, E is the Young’s modulus of the material forming the membrane and ν is its Poisson’s ratio. We have (see [KOI 63])

The energy of elastic deformation is (see [BER 76])

The membrane only exerts traction forces:

(1.25)

1.2.4. Exterior efforts

We consider mass forces deriving from a potential. The external force which the particle a undergoes is f(a) such that

where ρ is the natural surface mass of the material forming the membrane (its thickness multiplied by its mass volume). This force applied on the edge of the membrane is assumed to be of the same nature and represented by a line density fb, deriving from a potential Fb. When the particles on the edge on the membrane are given by

the external potential is

We suppose, in what follows, that U is a convex, regular functional.

1.2.5. Infinitesimal deformations

According to the hypothesis of “small deformations” (in other words, infinitesimal deformations), the effects of shearing can be neglected (see [KOI 63]). This simplification rests on the hypothesis of a negligible energy cost for an infinitesimal angular variation, and it leads to

and

where

As part of this approximation, the energy of elastic deformation becomes

and equation (1.25) becomes

(1.26)

1.2.6. Principle of minimum energy

where, in small perturbations,

We seek the equilibrium configurations in such a way that

(1.27)

As in the case of the ideally flexible string, J is not convex and we must consider the relaxed problem

(1.28)

This model can be used for the computational simulation of the behavior of supple sails (see [LEM 97, LEM 98, CUR 99, CON 06]). The formulation shown can also be compared to that of [MOR 87, MOR 89] for dynamics.

Figure 1.14.Computational solutions for different shapes of the obstacle (sphere, cube, lath)

NOTE 1.2. The local form of the equilibrium equations is

Here,

Chapter 2

Comparison of conventional and regularized bridge weigh-in-motion algorithms

When a mechanical system contains unilateral aspects, its movement can become complex and include discontinuities. For example, in the case of an ideally flexible string, we must take into account discontinuities in the velocity field and kinks in the curve representing the geometry of the string. In the case of a mechanical system whose movement is hindered by the presence of an obstacle, against which contact may or may not occur, the velocity field can present discontinuities connected to impacts against the obstacle. In the study of such systems, three distinct situations are considered and are related on the one hand to the time scale considered and, on the other hand, to the study of movement.

– The study of discontinuous propagation analyzes the fastest phenomena, such as wave propagation or contact discontinuities. While this is the main interest of the study, methods from the theory of hyperbolic systems are required.

– The study of temporal evolution analyzes movements using methods from the dynamical systems theory, as well as temporal integration techniques. Within this framework, the main interest is, in general, the analysis of transitory movements (in other words, non-stabilized) of the system. However, periodic movements can also be taken into account.

– The study of the vibrations of the system analyzes the types of particular movements, possessing properties of periodicity, resonance or related to the modal analysis of movement. Also belonging to this category is the study of periodic responses, of small oscillations and its consequences on stability. The methods used are those for the time-frequency analysis or for determining proper values. The spectral analysis and modal decomposition are the classical methods in this framework. Let us note that the presence of nonlinearities – such as unilateral aspects – present difficulties in the analysis.

2.1. Dynamics of ideally flexible strings

(2.1)

2.1.1. Propagation of discontinuities

Yet,

in such a way that

Thus,

and equation (2.1) is

and B(u) is a 9 × 1 matrix such that

Thus,

and

The proper values of a(u) are given below (Table 2.1).

Table 2.1.Proper values and proper vectors of the system

the discontinuities associated with λ2, λ3, λ7, λ8 and u is constant through the discontinuities associated with λ1, λ9 (see [JEF 76, LER 81]).

If c is not regular – for example where c contains a term representing the reaction of an obstacle (see section 1.1.1) – penalization procedures (sections 10.3.1.1 and 10.3.1.2) and regularization procedures (section 10.3.1) can be used. For the numerical solution, we can resort to procedures suggested in the literature (see [GIL 89a, GIL 89b, HAN 02, CUR 05]).

2.1.2. Evolution

In this case, given xm and vm, xm+1 and vm+1 can be determined in the following way. We have

(2.2)

The conditions to the limits are, for example,

We have

We can determine the equilibrium configuration and the tension field associated with a string with Hook constant submitted to a load (see section 1.1.1), that is to say, the solution to

which determines vm+1, if necessary (this calculation is not useful when we focus exclusively on the sequence of configurations). At each stage of time, we use the approximations and the theory introduced in section 1.1.1. The situation is similar when other temporal discretization schemes are considered.

2.1.3. Vibrations

2.1.3.1. Harmonic response

and equation (2.1) becomes

(2.3)

where

(2.4)

while the conditions to the limits become, for example,

(2.5)

The condition of unilateral behavior is

(2.6)

2.1.3.2. Small oscillations

Thus, equation (2.1) becomes

(2.7)

where

(2.8)

The boundary values become

(2.9)

The condition of unilateral behavior is approximated by

that is to say,

(2.10)

We must then solve the boundary-value problem formed by equations (2.7)–(2.9) and select the solutions verifying equations (2.10).

2.2. Contact dynamics

This section presents some simple models for the movement simulation of a solid in the presence of an obstacle. The models presented do not reproduce entirely the complexity of the problems of contact, but help will be provided to illustrate the various applications of the convex analysis (see, e.g. [MOR 68, MOR 76, MOR 78, MOR 86, JEA 92, MOR 94, MOR 98, JEA 99, MOR 00]).

2.2.1. Evolution of a material point

As we have already seen, when an obstacle hinders the movement of a system, the velocity field of the latter can have discontinuities related to the impacts against the obstacle. These difficulties can be illustrated considering the very simple situation where a material point of mass m > 0 is freely falling onto a horizontal obstacle (see Figure 2.1).

Figure 2.1.Free falling material point onto a horizontal obstacle

In this situation, the movement equations are

As we have already seen above, the reaction of the obstacle R and the position x verify

Assuming that h > 0, there is no contact between the mass and the obstacle for t < , so the solution for the interval is

where

Figure 2.2.Comparison between the exact solution and the regularized solution for the material point

Figure 2.3.Comparison between the exact solution and the regularized solution for a material point

we can choose

(2.11)

Figure 2.4.A simple situation to illustrate Coulomb’s law

This methodology can be extended to situations involving friction, especially those where Coulomb’s law intervenes for dry friction. In this case, regularization, for example, can be used:

(2.12)

A solution strategy without regularization can also be used: the additional equation −φ ∈ ∂j(T) can be introduced into the model and, in certain situations, can be solved without regularization. For example, let us consider the situation in Figure 2.4: the material point is found on a moving plane whose inclination is α(t) and the friction coefficient between the mass and the plane is μ > 0.

The map t → α(t) is an increasing function, so the inclination of the plane increases. Taking the value of d as the main variable, d represents the relative position of the material point in relation to the origin (see Figure 2.4); we have the motion equations

and the initial conditions

We can determine that vm+1 is contingent upon the parameter

In effect,

Figure 2.5 gives the results for two values of c, chosen in order to illustrate the behaviors analyzed above.

Figure 2.5.Numerical solutions for two values of c

2.2.2. Evolution of deformable and non-deformable solids

There are numerous practical situations whose modelization needs to take into account contact between solids – that is to say, continuous three-dimensional media (see, e.g. [SAM 85]) – or between solids and an obstacle. This is the case, for example, of problems with pressing, impact, mechanisms, systems of solids, or granular media. Thus, these situations are the topics discussed in many works. The reader may find in the literature, numerous applications, approaches and examples.

It is interesting to note that the evolution of a solid may be studied using a generalization of the procedure explained for the material point. For example, let us consider the case of a non-deformable solid. From the point of view of the Mechanics of Continuous media, it is a three-dimensional medium whose kinetics are limited to rigidifying movements, that is to say, translation and rotation compositions. Adopting Lagrange’s point of view, let us consider a reference configuration Ωt and the configuration Ωt of the solid at the instant of time t > 0. Thus, Ωt is a transformation of Ω by a rigidifying movement and the position of every particle x ∈ Ωt is given by

the equation that corresponds to the Eulerian description of movement. The movement is usually described using displacement from the center of gravity xg of the solid:

and the resulting moment Υ in relation to the center of gravity:

The movement is then described in the Euclidian space by the classic equation:

In the presence of an obstacle, the first difficulty is related to the definition of the set of admissible configurations: the contact between the solid and the obstacle occurs on the boundary Σt and we have

These general principles are immediately extended to the case of a deformable solid: the obstacle reaction introduces a surface load acting on the boundary ∂Ω of the solid. This load can be regularized in the same way as in the non-deformable case. For example, Figure 2.6 shows the numerical results for the passing of a small purely elastic solid through a drawplate. The calculation is quasi-static and the contact between the piece and the drawplate occurs with friction – an approach by regularization is used. The advancing of the piece is imposed on its right side at a constant velocity and the equilibrium equations at each instant have been solved by the Newton-Raphson method. We can see that the state of tension and final deformation of the solid is identical to that at the beginning. This is characteristic of elastic media.

Figure 2.6.Passing of a purely elastic piece through a drawing plate

2.2.3.