non-linear finite element methods in solid...

Politecnico di Milano, February 3, 2017, Lesson 1

Non-Linear Finite Element Methods in Solid MechanicsAttilio Frangi, [email protected]

1

Politecnico di Milano, February 3, 2017, Lesson 1 2

Outline

Lesson 1-2: introduction, linear problems in staticsLesson 3: dynamicsLesson 4: locking problemsLesson 5: geometrical non-linearitiesLesson 6-7: small strain plasticity

Final exam: project?

Slides on web-site (search page on www.dica.polimi.it)

Send e-mail with photo and name

Strong links with the course “Non-Linear Solid Mechanics”


Planning

Basic notions in linear elasticity


Planning

Time dependent problemsLesson 3: Dynamics (and diffusion)

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

Temperature chart in an engine


Planning

Element EngineeringLesson 4: Pathologies and cures of isoparametric finite elements

incompressibility.. rubber (tyres), beams, shells, plasticity..


Planning

Non linear quasi-static problemsLesson 5: Introduction. Application to geometrical non-linearities

buckling of structures


Planning

ElastoplasticityLesson 6: “Local” issuesLesson 7: “Global” issues

plastic deformation in an exaust-pipe


Textbook

originated from a course on FEMtaught at the Ecole Polytechnique for 5 years

9 modules (1h30 theory + 2h hands on sessions)

codes free for download fromhttp://www.ateneonline.it

AIM: couple a rigorous theoretical treatment with an the introduction to FEM coding

Not only toy codes, but “pilot” codes prior to “serious” implementations

http://www.ateneonline.it/


Field equations:

Boundary conditions:

If material isotropic:

Governing equations in strong form


Admissible spaces

Space of regular displacements (associated to a bounded energy)

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

Space of displacements compatible with zero boundary displacements

Space of fields compatible with boundary data kinematically admissible displacements:

statically admissible stresses

strain operator!


Compatibility equation and constitutive law enforced pointwise

11

“Weak” problem formulation

Weak form of local equilibrium equations:

stems from an integration by parts procedure of:

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

T tractions


Hence the problem formulation becomes:

12

Variant: eliminate unknown tractions

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


Galerkin approach for the weak formulation

Weak formulation of the linear elastic problem

The choice of the unknown and of the virtual fields:

leads to the linear system of equations:


Galerkin approach for the weak formulation

leads to the linear system of equations:

N

N


Galerkin approach: general properties (1/3)

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

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 the solution is sought (in the sense of the “energy norm”)


Deformation energy of remember u - uN = Δu

with arbitrary kinematically admissible

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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:



Assumption: . Hence and then:

Deformation energy of exact solution

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


Introduction to FEM Galerkin approach

element

node

a partition of Ω into triangular elementssharing nodesThis introduces a discretized Ωh

Neighbouring elements always sharenodes

forbidden

typically the mesh iscreated with dedicated codes.In our simple 2D case GMSH

notion of conformitytwo elements can only:- either be well separated- or share one node- or share one edge


nodal values are imposed by boundary conditions

x1

x2

v

Let us consider one scalar field(e.g. one component of displ. or a temperature field)

We draw “nodal values” of the field…

Admissible field, step 2: linear interpolation

The blue line denotes the discretization of an Su region (imposed displacements)


the three nodal values completely define the displacement field within the element

Linear interpolation at the local levelLet us now focus on a specific element

• the assumed displacement field is continuous• its restriction to each triangle is linear and depends only on nodal values

x1

x2

v

x(k)

x(l)

x(m)

v(k)

v(l)

v(m)

This is only a particular way to express a linear field!


Shape functions (local shape functions)

x(k)

x(l)

x(m)Nk

Nl

Nm

x(k)

x(l)

x(m)

x(k)

x(l)

x(m)

1

1

are called shape functions and are:

sometimes are called “local”or “elemental” shape functionsas opposed to “global” shape functions(see later)


Galerkin interpolation

Analysis domain: h

: kinematically admissible in the sense of FEM approximation

Global approximation: specific form of the Galerkin approach with:

x1

x2

v

global shape functions


x(1)

S1

S2

S3

x(2)

x(3)

Isoparametric elements: linear triangle revisited

physical triangle

ISO-parametric(geometry and displacement)

Moreover area coordinates coincide with shape functions!!!

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x

local mapping from master triangle onto physical triangle using area coordinates.Any physical triangle can be mapped on the same master triangle.The mapping is such that the red master nodes are mapped on the blue physical nodes fixed by the user.Linear mapping -> 3 parameters -> 3 nodes

1

linear mapping

a1

a2

2

3

(1,0)

(0,1)

(0,0)

master triangle

conventionalordering of nodeson master element


N1

12

1

3

N2

1

2

1

3

1

2

1

3

N3 Shape functionsare now imagined defined in the parametric space on the master element: Nk(a)=ak

From the parametric space to the physical space


x(1)

x(2)

x(3)

Isoparametric elements: quadratic triangle

a quadratic mapping from master triangle into physical triangle using area coordinates allows to better approximate curved boundaries!

physical trianglequadratic mapping

x(4)

x(5)

x(6)

real problem boundary

the discretized and real domain still DO NOT coincide everywhere but node position is respected exactly!

The same shape functions are then employed to generate the approximation space for displacements

a1

a2

1

2

3

(1,0)

(0,1)

(0,0)

master triangle

4

6

5 (0.5,0.5)

(0.5,0)

(0,0.5)

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N3

1

N2

1

N4

N1

1

24

1

63

5

N6N5

24

1

63

5

24

1

63

5

24

1

63

5

24

1

63

5

24

1

63

5

a1

a2

1

2

3

(1,0)

(0,1)

(0,0) 6

(0.5,0)

45 (0.5,0.5)

(0,0.5)

Isoparametric elements: quadratic triangle

Quadratic shapefunctions which satisfy the property:

Every shape function isassociated to a node and vanishes in all the other nodes


Examples of master and physical elements


Shape functions of isoparametric elements: properties

convergence

conformity


Conformal meshes

Two neighbouring elements are required not to overlap, nor to make holes at common boundaries.

If the families of isoparametric elements described before are employed, this is guaranteed if two elements are:• either be well separated• or share one node • or share one edge, and in this case they have the same number of nodes on the common edge with the same position

This is a fundamental benefit of isoparametric elements

non conformal non conformal conformal


Example

Important distinction between local and global numbering of nodes. E.g. the 4th local node in element 1 is the 7th global node

For each element we select the physical (global) node that will correspond to local node number 1 (this choice is not unique since any corner node will do). The rest of the connectivity flows from this choice

a1

a2

1

2

3

4

6

5

a1

a2

1 2

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6

5

7

8

connec

conventional (local)ordering of nodeson T6 master element

conventional (local)ordering of nodeson Q8 master element

global numbering of physical nodes


List of degrees of freedom: DOF


Galerkin interpolation

: kinematically admissible in the sense of FEM approximation

Global approximation: specific form of the Galerkin approach with:

x1

x2

v

global shape functions


Specific version fo Galerkin approach with:

Field dof associates a numberto every nodal component of displacement

Structure of dof to be memorised!!Galerkin interpolation


Global problem formulation


Global problem formulation

element by element computation


Engineering notation


Element stiffness matrix

At this level no distinction is made between given and unknown displacements (see later)


Numerical integration with Gauss quadrature(a) 1D integrals


Numerical integration with Gauss quadrature

(a) 1D integrals(b) 2D or 3D integrals over squares or cubesCartesian products of 1D formulas


Numerical integration with Gauss quadrature

(a) 1D integrals(b) 2D or 3D integrals over squares or cubes(c) 2D or 3D integrals over triangles or tetrahedraSpecific formulas not built from 1D formulas

Example (triangle, Gauss-Hammer rules with G=3)

Integrates exactly every second order polynomial


Numerical integration: bilinear quadrilateral

Choice of the order of quadrature

The numerical integration is said to be complete if, assuming the Jacobianmatrix is constant (non distorted element) the stiffness matrix is integrated exactly..

In our case?

G=2 is enough!


Homeworks for next lesson

Revise chapters 1-2-3 of the book

In particular:• isoparametric elements• element integrals and numerical integration• assemblage procedure• solution• convergence

operative knowledge of code genlin !!

play with examples and exercises provided and create your owns

non-linear finite element methods in solid...

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