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Method of Finite Elements II - Introduction
Prof. Dr. Eleni ChatziKonstantinos Tatsis, Dr. Konstantinos Agathos
Lecture 1 - 17 September, 2020
Chair of Structural Mechanics & Monitoring, ETH Zurich
September 17, 2020
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 1
Outline
1 Course information
2 Learning goals
3 Linear vs. Nonlinear operators - Strong vs. Weak form
4 A simple finite element example with Matlab implementation -FEM I review
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 2
Course Information
InstructorsProf. Dr. Eleni Chatzi, email: [email protected]. Konstantinos Agathos, email: [email protected] Tatsis, email: [email protected]
AssistantsKostas Vlachas, HIF B21, Sergio Nicoli, HIL E19.4
Course WebsiteLecture Notes and Demos will be posted in: http://www.chatzi.ibk.ethz.ch/education/method-of-finite-elements-ii.html
Help us Structure the Course!Participate in our online survey: http://goo.gl/forms/ws0ASBLXiY
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 3
Course Information
Computer LabsComputer Labs take place online, as announced in the class schedule.They require your interactive participation!
Performance Evaluation - Final Project (100%)You can work with a partner (max groups of 2 allowed).Topics to be announced on 15.10.2020
Option for PhD students: You are free to shape your own project!
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 4
Course Information
Suggested reading:Finite Element Procedures” by K.J. Bathe, Prentice Hall, 1996, ETHZurich Library Portal
“Nonlinear Finite Elements for Continua and Structures”, by T.Belytschko, W. K. Liu, and B. Moran, John Wiley and Sons, 2000
“The Finite Element Method: Linear Static and Dynamic FiniteElement Analysis”, by T. J. R. Hughes, Dover Publications, 2000
“Nonlinear finite element analysis of solids and structures”,Crisfield,M.A., Remmers, J.J. and Verhoosel, C.V., John Wiley & Sons, 2012
“Computational methods for plasticity: theory and applications”, DeSouza Neto, E.A., Peric, D. and Owen, D.R., 2011
Lecture Notes by Carlos A. Felippa Nonlinear Finite Element Methods(ASEN 6107):http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d/Home.html
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 5
Course Outline
FEM II - Lecture ScheduleIntroduction - From Linear to Nonlinear ConsiderationsBlock #1 - Geometrical NonlinearitiesGeometrical Nonlinearities I - Nonlinear stress and strain measuresGeometrical Nonlinearities II - FE formulationGeometrical Ninlinearities III - Nonlinear solution methods
Computer Lab I
Block #2 - Material NonlinearitiesMaterial Nonlinearities I - Constitutive ModelingMaterial Nonlinearities II - FE Implementation
Computer Lab II
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 6
Course Outline
FEM II - Lecture Schedule
Block #3 - Nonlinear DynamicsNonlinear Dynamics I - The Newmark methodNonlinear Dynamics II - Modelling of Hysteresis
Computer Lab III
Block #4 - FractureFracture I - General principlesFracture II - The eXtended Finite Element MethodFracture III - Elastic/Plastic fracture
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 7
Today’s Lecture
Learning goalsBy the end of today’s lecture, the goal is to:
Be able to distinguish between linear and nonlinear behavior
Understand what a weak formulation of a problem is and why itis useful
Review basic concepts from Linear Finite Elements that will beof use
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 8
Linear maps
How can we in general define linear?
To help us answer the question we will use the followingmathematical definition:
DefinitionGiven two vector spaces V and W, a linear map is a functionf : V→W, such that for every u, v ∈ V and a scalar c, thefollowing conditions apply:
f (u + v) = f (u) + f (v)f (cu) = cf (u)
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 9
Linear maps
But what is a vector space?
DefinitionA vector space is a collection of objects for which addition andmultiplication with a scalar are defined.
The above is quite general and applies in the same way for e.g.:
Vectors in Rn
Functions defined in Rn, for instance all polynomials of order pdefined in R
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 10
Linear maps
Examples of linear maps:
An m × n matrix:
- Transforms elements from the vector space of vectors in Rn toelements of the vector space of vectors in Rm
- Preserves addition and multiplication by a scalar
Differentiation:
- Transforms elements from the function space of differentiablefunctions in R
- Preserves addition and multiplication by a scalar
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 11
Strong form
Consider the simple problem of a bar with:
Constant cross section A
Length L
A fixed left end
A distributed axial load P
A load R at the right end
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 12
Strong form
The equilibrium equation canbe obtained by:
Considering the stressesapplied at a segment ofinfinitesimal length
Considering the appliedforce constant in thatinterval
σ (x) A =[σ (x) + dσ (x)
dx dx]
A + P (x) dx A
⇒ dσdx + P = 0
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 13
Strong form
Additionally we consider:
Linear1elastic materialbehavior:
σ = Eε
where ε are the strains
A linear1 strain measure:
ε =(du
dx
)where u are thedisplacements
By substituting in the equilibrium equation we obtain:
E d2udx2 + P = 0
1As defined previouslyChair of Struct. Mechanics & Monitoring Method of Finite Elements II 14
Strong form
Finally, the boundary conditions are also considered:
E d2udx2 + P = 0
u = 0 for x = 0
Aσ = AE(du
dx
)= R for x = L
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 15
Linear equations
We can easily verify that the equilibrium equation is linear!
In general, differential equations of the form:
a0 (x) u (x) + a1 (x) u′ (x) + a2 (x) u′′ (x) + · · ·+ an (x) u(n) (x) = 0
are linear.
However, equations with products of derivatives of the same ordifferent degree are not, for instance:
u (x) u′′ (x) = 0
is not linear.
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 16
Sources of nonlinearity
The linear equilibrium equation can be turned into a nonlinear one iffor instance:
An alternative materialbehavior is considered:
σ = E1ε+ E2ε2
corresponding equation:
E1d2udx2 +2E2
dudx
d2udx2 +P = 0
An alternative strainmeasure is considered:
ε = dudx + 1
2
(dudx
)2
corresponding equation:
E[
d2udx2 + du
dxd2udx2
]+ P = 0
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 17
Sources of nonlinearity
Such occurrences are common in practice, and in general thefollowing types of nonlinearity can be identified:
Material nonlinearity, i.e. nonlinear material laws
Geometrical nonlinearity, i.e. nonlinearity associated with thegeometry of the deformation
Boundary associated nonlinearity, e.g. contact
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 18
Strong form solution
The solution of the differential equation would satisfyequilibrium at every point
Thus, the equation is called the “strong form” of the problem
Approximate solutions could also be obtained by:
- Assuming a specific form for the solution, e.g. polynomial of acertain degree
- Enforcing the equation and boundary conditions at selectedpoints to obtain unknown coefficients
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 19
Weak form
An alternative to the above is posing the problem in “weak form”.Then:
The equations are not satisfied at every point
Instead they are satisfied in an “average” (weak) sense
In the following, alternative ways of deriving such a “weak form” willbe presented.
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 20
Weak form - Galerkin method
Starting with the equilibrium equation:
Eu′′ + P = 0
we employ an arbitrary function w (x), called the weight function,and we demand that:∫ L
0
∫A
w(Eu′′ + P
)dAdx =
∫ L
0w(EAu′′ + q
)dx = 0
regardless of the values of the parameters involved in the definitionof w . In the above AP = q
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 21
Weak form - Galerkin method
Then by integrating the first term by parts we obtain:
−∫ L
0EAw ′u′dx +
(wu′)∣∣L
0 +∫ L
0wqdx = 0
By further assuming that w (0) = 0,w (L) = 0 and rearranging weobtain the weak form as:∫ L
0EAw ′u′dx −
∫ L
0wqdx = 0
We observe that the highest derivative appearing in the equation is 1as opposed to 2 for the initial equation.
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 22
Weak form - Potential energy minimization
The same result can be obtained by following a different route. Westart with the total potential energy functional for the bar:
Π =∫ L
0
12EA
(u′)2 dx −
∫ L
0qudx
but what is a functional?
DefinitionFunctional is a mapping from a function space to the real numbers.
In other words a functional maps functions to numbers, i.e. it is afunction of functions!
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 23
Weak form - Potential energy minimization
At the point of equilibrium, the total potential energy should beminimized.
To minimize a functional we need to define variations:
DefinitionThe variation of a function u (x) is defined as an arbitrary andsufficiently smooth function η (x) that vanishes at the points whereboundary conditions are applied:
δu = η
For the derivatives of η the following should apply:
dnη
dxn = dnδudxn = δ
(dnudxn
)
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 24
Weak form - Potential energy minimization
Then for a functional we have:
DefinitionThe variation of a functional F of a function u and its derivatives(u′, u′′, . . . , un) is defined as:
δF = limε→0
F[u + εη, (u + εη)′ , (u + εη)′′ , . . . , (u + εη)n
]− F [u, u′, u′′, . . . , un]
ε
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 25
Weak form - Potential energy minimization
The variational operator has several common properties todifferentiation, for instance:
δ (F + Q) = δF + δQ, δ (FQ) = (δF ) Q + (δQ) F
Also for functionals including integrals:
δ
∫F (x) dx =
∫δF (x) dx
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 26
Weak form - Potential energy minimization
Similar to functions, the stationary conditions for functionals is thattheir variations should vanish.
For the total potential energy functional, this yields:
δΠ =∫ L
0EAu′δu′dx −
∫ L
0qδudx = 0
For comparison, the Galerkin method yields:
∫ L
0EAw ′u′dx −
∫ L
0wqdx = 0
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 27
Weak form
We can observe:
Minimizing potential energy is equivalent to the principle ofvirtual work if δu are considered as the virtual displacements
Both of the above formulations are equivalent to the Galerkinmethod if δu are considered as the weights
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 28
Weak form
For all the formulations presented we can also remark the following:
The highest derivative of the displacement in the weak form isof degree 1 as opposed to 2 for the strong form
The above also relaxes the smoothness requirements forapproximate solutions
Similar to the strong form, the weak form is linear with respectto the displacements
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 29
Strong vs. Weak form
To better understand the concepts lets consider an example:
We set:R = 0
P = P0 ⇒ q = q0 = AP0
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 30
Strong vs. Weak form
Using the strong form:
Eu′′ + P0 = 0u = 0 for x = 0
Aσ = AE(du
dx
)= 0 for x = L
we can easily solve the problem by assuming the displacements are aquadratic polynomial:
u (x) = a0 + a1x + a2x2
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 31
Strong vs. Weak form
Then plugging displacements into the equilibrium equation yields:
E (a0 + a1x + a2x2)′′ + P0 = 0 ⇒ E (2a2) + P0 = 0 ⇒ a2 = P02E
and the boundary conditions:
u (0) = 0 ⇒ a0 = 0AEu′ (L) = 0 ⇒ a1 + 2a2L = 0 ⇒ a1 = −2a2L
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 32
Strong vs. Weak form
Putting everything together yields the solution:
u (x) = −P0LE x + P0
2E x2 = P02E x (2L− x)
or in terms of q:
u (x) = q02EAx (2L− x)
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 33
Strong vs. Weak form
To solve the same problem using potential energy minimization westart with the same assumption for the displacements:
u (x) = a0 + a1x + a2x2
Then the variations of the displacements and their derivatives are:
δu = δa0 + δa1x + δa2x2, δu′ = δa1 + 2δa2x
Notice that it is of the same form as the assumed displacements.
Also δu should vanish at x = 0, thus:
δu = δa1x + δa2x2
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 34
Strong vs. Weak form
Plugging these expressions into the weak form we obtain:
δΠ =∫ L
0EAu′δu′dx −
∫ L
0qδudx = 0
⇒δa1
[EAL (a1 + a2L)− L2q
2
]+
+δa2
[EAL2
(a1 + 4
3a2L)− L3q
3
]= 0
Since δa1 and δa2 are arbitrary parameters, both expressions inbrackets should vanish.
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 35
Strong vs. Weak form
By solving the system resulting form the above equations we obtain:
a1 = LqEA , a2 = − q
2EAApplying the boundary condition at x = 0 yields:
a0 = 0
By putting everything together, the same solution as with the strongform is obtained:
u (x) = q02EAx (2L− x)
We observe that the boundary condition at the right end is satisfiedautomatically!
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 36
Strong vs. Weak form
The same result as with the previous methods can be obtained usingthe Galerkin method if:
A quadratic polynomial is assumed for the displacements
The same form is assumed for the weights
The weights are forced to vanish at the locations whereboundary conditions are applied
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 37
A finite element example
A bar FE formulation can be obtained in exactly the same way by:
Removing the Dirichlet boundary conditions
Assuming the displacements to be FE shape functions
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 38
A finite element example
Weak form:
δΠ =∫ L
0EAu′δu′dx −
∫ L
0qδudx −
2∑i=1
δuiRi = 0
Shape functions:
u = Nun =[
xL
L− xL
] [ u1u2
]where u1, u2 are nodal displacements
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 39
A finite element example
By substituting and rearranging we obtain:
δΠ =[δu1 δu2
] ∫ L
0
xL
L− xL
′
EA[
xL
L− xL
]′dx[
u1u2
]′
−[δu1 δu2
] ∫ L
0
xL
L− xL
qdx
−[δu1 δu2
] [ R10
]−[δu1 δu2
] [ 0R1
]= 0
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 40
A finite element example
By carrying out the integrations and further rearranging:
EAL
[1 −1−1 1
]︸ ︷︷ ︸
K
[u1u2
]︸ ︷︷ ︸
un
= q0L2
[11
]+[
R1R2
]︸ ︷︷ ︸
f
We obtain the bar stiffness matrix and load vector!
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 41
Implementation example
Problem data:L = 1E = 1A = 1
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 42
Implementation example
Input:
Nodal coordinates
Element connectivities
Material parameters
Constraints
Nodal loads
Output:
Nodal displacements
Chair of Struct. Mechanics & Monitoring Method of Finite Elements II 43