Am Phi 1

MEC 557

The Finite Element Method for Solid [email protected]

[email protected]@lmt.ens cachan.fr

MEC 557, Ecole Polytechnique, September 16, 2011, Amphi 1

http://www.stru.polimi.it/home/frangi/MEC557.html

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Teachers:

Attilio Frangi: Department of Structural Engineering, Politecnico di [email protected]

Christian Rey: LMT, ENS [email protected]

• Amphis in english – italian english - Slides always available on web-site• PC in french • Book in french Chapters Amphis (book accurately reflects contents of Amphis)• Book in french. Chapters Amphis (book accurately reflects contents of Amphis)• New chapter related to Amphi 4 in english• Written exam: text in English. Use language you prefer, even mixed…

WHY??Course part of a Master in EnglishFinite Element Method (FEM) literature is in english.Finite Element Method (FEM) literature is in english. Research and work environments often adopt english as common language

MEC 557, Ecole Polytechnique, September 16, 2011, Amphi 1 2


Finite element procedures are an important and often indispensable part ofFinite element procedures are an important and often indispensable part ofengineering and design.

Finite Element computer programs are now widely used in almost all branches ofFinite Element computer programs are now widely used in almost all branches ofengineering for the analysis of structures, solids and fluids

FEM is by far the most important and employed general purpose numericalFEM is by far the most important and employed general purpose numericaltool in solid mechanics

• Replace and/or enrich expensive experimental campaigns in order to understandp / p p p gunderlying physics (after calibration of models)

• Optimization of existing procedures and products• Design of new products

Wealth of commercial codes:Abaqus, Ansys (international)

g p

q , y ( )Castem, Code Aster (typically french)and many, many more…..


Planning

Basic notions in linear elasticity – MEC 431 and beyondAmphi 1 Appro imate sol tion te hniq es in solid me hani sAmphi 1: Approximate solution techniques in solid mechanicsAmphi 2: The concept of Isoparametric Finite ElementsAmphi 3: The Finite Element Method

PC 1-2-3-4


Planning

Element EngineeringAmphi 4 Patholo ies and res of isoparametri finite elementsAmphi 4: Pathologies and cures of isoparametric finite elements

incompressibility.. rubber (tyres), stokes flow (MEMS!!), plasticity..


Planning

Non linear quasi-static problemsAmphi 5 Introd tion Appli ation to eometri al non linearitiesAmphi 5: Introduction. Application to geometrical non-linearities

buckling of structures


Planning

ElastoplasticityAmphi 6 “Lo al” iss es

Strong links withRupture et Plasticite’, J.J. Marigo(but self sufficient..)

Amphi 6: “Local” issuesAmphi 7: “Global” issues

plastic deformation in an exaust-pipe


Planning

Time dependent problemsAmphi 8 Thermal diff sion and thermoelasti itAmphi 8: Thermal diffusion and thermoelasticityAmphi 9: Dynamics

Impact of a tyre on a surface. Rolling of a tyreon an inclined surface

Temperature chart in an engine

on an inclined surface


Lesson 1: Approximate solution techniques in solid mechanicspp q

1. Governing equations in strong form

2. Weak formulation of the problem

3. Variational formulation of the problem

4. Approximate minimisation: Galerkin approach

Basically a smooth revision of some concepts already presented in MEC431 !!!(and an introduction to some english terminology)








Governing equations in strong formg q g

List of assumptions for Amphis 1-4

Small perturbations (HPP)Small perturbations (HPP)Actual configuration = Initial configurationLinearised strain tensorCauchy stress tensor

body forces

y

Quasi static evolutionNeglect inertia terms

body forcesLinear elastic, homogenoeus and isotropic material

stiffness tensor


Revision: properties of the constitutive law

elastic energy dens. x 2

elastic energy dens. x 2


Isotropic linear elastic material notation with lists!(freq. for implementations)

A1111 A1122 A1133 A1112 A1123 A1113A2222 A2233 A2212 A2223 A2213

A A A A

(freq. for implementations)

A3333 A3312 A3323 A3313A1212 A1223 A1213

SYM A2323 A2313A

222A13132

in case of isotropy

2( )2( )

2( )


2( )

Approximation by finite differences


Approximation by finite differences

see Amphis 8-9




2. Weak formulation of the equilibrium conditions

3. Variational formulation of the equilibrium conditions



Admissible spaces

Space of regular displacements (associated to a bounded energy)

Space of fields compatible with boundary datap p ykinematically admissible displacements:

statically admissible stresses

Space of displacements compatible with zero boundary displacements

Formulation of the equilibrium problem in linear elasticity (strong form):strain operator!


“Weak” problem formulation

Weak form of local equilibrium equations:T

stems from an integration by parts procedure of:

corresponds essentially to a form of the Principle of Virtual Power (PPV)

Compatibility equation and constitutive law enforced pointwise


“Weak” problem formulation

Remarks:• no explicit reference to how kinematic data on the boundary are enforced• presence of unknown T on S (reaction forces due to imposed displacements)presence of unknown T on Su (reaction forces due to imposed displacements)

T ibl difi ti ( i t d t i t f th ti b )Two possible modifications (associated to variants of the equation above):

• eliminate T on Su• weakly enforce displacement boundary data uD on S• weakly enforce displacement boundary data uD on Su


First variant: eliminate unknown tractions

A th t th t f ll i diti b t ( ill l t h )Assume that the two following conditions can be met (we will se later how)

restrict w to be kinematically admissible with zero boundary data: choose w in

u satisfies a priori boundary conditions in a strong form: u = uD on Su

Hence the problem formulation becomes:


Second variant: weak enforcement of displacement BC

no restrictions on wkinematic boundary conditions enforced inkinematic boundary conditions enforced in a weak manner with a new equation

h th t C’ f d i ibl t ti i d fi d b d lit ith t t Cwhere the set C’ of admissible tractions is defined by duality with respect to C:


classical example of “mixed” formulation



2. Weak formulation

3. Variational formulation



Variational formulation

The solution of a linear elastic problem can be characterized in terms of specificfunctionals (typically energy functionals)

this approach has already been presented in MEC431 (e.g. potential energy functional)


Energy functionals

Total Potential energy functional

The displacement field solution of the problem minimises P:of the problem minimises P:

Total Complementary energy functionalTotal Complementary energy functional

The stress field solution of the problem minimises P:of the problem minimises P:

It is clearly sufficient to perform one of the minimisations,d th t t i


e.g. and then compute stresses via

Stationarity of potential energy functionalConsider the displacement u solution of the problem and compute the variation of P when u is perturbed by ηw with

At the minimum point the variation vanishes up to the first order in η along any direction p p η g y

In the specific case of a linear elastic problem this yields:

which corresponds to the weak form, first variant (without unknown tractions)

weak formulation: multiply strong form of local equations by a test function and integrate


weak formulation: multiply strong form of local equations by a test function and integratevariational formulation: corresponds to the requirement that the variation of a functional vanishes








Approximate minimisation: Galerkin approach

The sets and of admissible fields areare spaces of infinite dimensions

The exact minimisations:

are impossible in practice (due to the complicated geometryare impossible in practice (due to the complicated geometry of real problems)

An approximate minimum is sought: Galerkin approach


Galerkin approach for potential energy

Minimisation of P(v) where v has the form

,


Galerkin approach for potential energy

is a square NxN matrix, symmetric and positive definite (if SUITABLE bc on u are imposed):

has one and only one minumum defined by

leading to the optimal generalized displacement

by construction

leading to the optimal generalized displacement

by construction


Galerkin approach for the weak formulation

Weak formulation of the linear elastic problem (revision)

The choice of the unknown and of the virtual fields:

l d t th li t f ti b fleads to the same linear system of equations as before:


Galerkin approach: general properties (1/3)

Let us express the solution u as u = uN +Δu(Δu is the “error” with respect to the exact solution)

Virtual field

Weak continuum formulation (written for the exact solution u)

Weak discrete formulation (written for the approximate solution uN)

The error Δu is orthogonal to every virtual field belonging to the space where theThe error Δu is orthogonal to every virtual field belonging to the space where the solution is sought (in the sense of the “energy norm”)



Deformation energy of remember u - uN = Δu

with arbitrary kinematically admissible y y

Property of best approximation: uN is the best approximation of the exact solution uin the selected space of approximation, in the sense of the energy norm:p f pp , f gy



Deformation energy of exact solution

Assumption: . Hence and then:

Property 3: if approximates u from below in the energy norm sense:


Galerkin approach: “complementary” energy

greatest difficulty!!!!!

Th i t l ti (i t f t ) i d fi d bThe approximate solution (in terms of stresses) is defined by

In general this cannot be integrated to yield a compatible displacement


Discussion

Galerkin approach: “displacement” versionBasis functions have to be kinematically admissible and are easy to buildBasis functions have to be kinematically admissible and are easy to build(regularity and finite energy)

Galerkin approach: “stress” versionppBasis functions have to be statically admissible admissible ( )and are difficult to build


Basis functions defined over the whole domain

The convergence of is very fast (exponential w.r.t. the number oft ) if i thterms) if u is smooth

Not well adapted to complicated geometries and to problems with limitedregularityregularity

Very little used in solid mechanics but very promising e.g. in fluid mechanics (spectral methods)


fluid mechanics (spectral methods)

Example of the hollow sphere

Hollow sphere (inner surface r = R1, outer surface r = R2) subjected to internal pressure p and displacement

( ) d h fu(R2)= d on the outer surface


Basis functions with local support: introduction to Finite Elements

Difficulties induced by “global” approaches

Not well adapted to complicated geometriesEnforcement of boundary conditions is difficult

Basics of the displacement Finite Element Method (FEM): G l ki h i h b i f i h i “ ll”Galerkin approach with basis functions having “small” support




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