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Politecnico di Milano, February 3, 2017, Lesson 1
Non-Linear Finite Element Methods in Solid MechanicsAttilio Frangi, attilio.frangi@polimi.it
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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”
Politecnico di Milano, February 3, 2017, Lesson 1 3
Planning
Basic notions in linear elasticity
Politecnico di Milano, February 3, 2017, Lesson 1 4
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
Politecnico di Milano, February 3, 2017, Lesson 1 5
Planning
Element EngineeringLesson 4: Pathologies and cures of isoparametric finite elements
incompressibility.. rubber (tyres), beams, shells, plasticity..
Politecnico di Milano, February 3, 2017, Lesson 1 6
Planning
Non linear quasi-static problemsLesson 5: Introduction. Application to geometrical non-linearities
buckling of structures
Politecnico di Milano, February 3, 2017, Lesson 1 7
Planning
ElastoplasticityLesson 6: “Local” issuesLesson 7: “Global” issues
plastic deformation in an exaust-pipe
Politecnico di Milano, February 3, 2017, Lesson 1 8
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
Politecnico di Milano, February 3, 2017, Lesson 1
Field equations:
Boundary conditions:
If material isotropic:
Governing equations in strong form
Politecnico di Milano, February 3, 2017, Lesson 1 10
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!
Politecnico di Milano, February 3, 2017, Lesson 1
Compatibility equation and constitutive law enforced pointwise
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“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
Politecnico di Milano, February 3, 2017, Lesson 1
Hence the problem formulation becomes:
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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
Politecnico di Milano, February 3, 2017, Lesson 1 13
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:
Politecnico di Milano, February 3, 2017, Lesson 1 14
Galerkin approach for the weak formulation
leads to the linear system of equations:
N
N
Politecnico di Milano, February 3, 2017, Lesson 1 15
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”)
Politecnico di Milano, February 3, 2017, Lesson 1
Deformation energy of remember u - uN = Δu
with arbitrary kinematically admissible
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Galerkin approach: general properties (2/3)
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:
Politecnico di Milano, February 3, 2017, Lesson 1 17
Galerkin approach: general properties (3/3)
Assumption: . Hence and then:
Deformation energy of exact solution
Property 3: if approximates u from below in the energy norm sense:
Politecnico di Milano, February 3, 2017, Lesson 1 18
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
Politecnico di Milano, February 3, 2017, Lesson 1 19
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)
Politecnico di Milano, February 3, 2017, Lesson 1 20
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!
Politecnico di Milano, February 3, 2017, Lesson 1 21
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)
Politecnico di Milano, February 3, 2017, Lesson 1 22
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
Politecnico di Milano, February 3, 2017, Lesson 1
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
Politecnico di Milano, February 3, 2017, Lesson 1
N1
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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
Politecnico di Milano, February 3, 2017, Lesson 1
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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Politecnico di Milano, February 3, 2017, Lesson 1 26
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
Politecnico di Milano, February 3, 2017, Lesson 1 27
Examples of master and physical elements
Politecnico di Milano, February 3, 2017, Lesson 1 28
Examples of master and physical elements
Politecnico di Milano, February 3, 2017, Lesson 1 29
Examples of master and physical elements
Politecnico di Milano, February 3, 2017, Lesson 1
Shape functions of isoparametric elements: properties
convergence
conformity
Politecnico di Milano, February 3, 2017, Lesson 1 31
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
Politecnico di Milano, February 3, 2017, Lesson 1 32
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
Politecnico di Milano, February 3, 2017, Lesson 1 33
List of degrees of freedom: DOF
Politecnico di Milano, February 3, 2017, Lesson 1 34
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
Politecnico di Milano, February 3, 2017, Lesson 1 35
Specific version fo Galerkin approach with:
Field dof associates a numberto every nodal component of displacement
Structure of dof to be memorised!!Galerkin interpolation
Politecnico di Milano, February 3, 2017, Lesson 1
Global problem formulation
Politecnico di Milano, February 3, 2017, Lesson 1
Global problem formulation
element by element computation
Politecnico di Milano, February 3, 2017, Lesson 1 38
Engineering notation
Politecnico di Milano, February 3, 2017, Lesson 1 39
Politecnico di Milano, February 3, 2017, Lesson 1 40
Politecnico di Milano, February 3, 2017, Lesson 1 41
Politecnico di Milano, February 3, 2017, Lesson 1 42
Element stiffness matrix
At this level no distinction is made between given and unknown displacements (see later)
Politecnico di Milano, February 3, 2017, Lesson 1 43
Numerical integration with Gauss quadrature(a) 1D integrals
Politecnico di Milano, February 3, 2017, Lesson 1 44
Numerical integration with Gauss quadrature
(a) 1D integrals(b) 2D or 3D integrals over squares or cubesCartesian products of 1D formulas
Politecnico di Milano, February 3, 2017, Lesson 1 45
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
Politecnico di Milano, February 3, 2017, Lesson 1 46
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!
Politecnico di Milano, February 3, 2017, Lesson 1 47
Politecnico di Milano, February 3, 2017, Lesson 1
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