Prof. Dr. Eleni Chatzi, Dr. Konstantinos AgathosPath following methods Prof. Dr. Eleni Chatzi, Dr. Konstantinos Agathos Lecture 4 - 10 October, 2019 Institute of Structural Engineering,

Path following methods

Prof. Dr. Eleni Chatzi, Dr. Konstantinos Agathos

Lecture 4 - 10 October, 2019

Institute of Structural Engineering, ETH Zurich

October 11, 2019

Institute of Structural Engineering Method of Finite Elements II 1

Outline

1 Introduction

2 The Newton-Raphson method

3 The Newton-Raphson method in structural mechanics

4 Path-following Methods


Learnig goals

Understanding the general solution process for nonlinearproblems

Understanding limitations of the process

Gaining a basic understanding of path following solutionmethods


Significance of the lecture

Applications include any problem described by nonlinear PDEs, suchas:

Structural mechanics problems involving:

Geometrical nonlinearities

Material nonlinearities

Material failure

Fluid mechanics


The Newton-Raphson method in 1-D

ProblemGiven the function:

f (x) : R→ R

find:

x : f (x) = 0

where:f (x) is a general nonlinear function

The derivative of f (x) is available



Solution:

If an initial estimate x0 is available, the function can be locallyapproximated by a truncated Taylor series:

f (x) ≈ f (x0) + dfdx |x0 (x − x0) = f (x0) + f ′ (x0) ∆x0

Then this equation can be solved to obtain a better approximationto the solution:

∆x0 = −(f ′ (x0)

)−1 f (x0) → x1 = x0 + ∆x0



The process can be repeated iteratively until a sufficient accuracy isobtained:

∆xi = −(f ′ (xi )

)−1 f (xi ) → xi+1 = xi + ∆xi

The absolute value of f (xi+1) can be used to determine theachieved level of accuracy

Alternatively the difference between two consecutive solutions(|xi+1 − xi |) could be used



The whole solution can be written as an algorithm:

Data: f (x),f ′ (x), x0, tol , maxitResult: x

for i ≤ maxit do∆xi = −

(dfdx |xi

)−1f (xi )

xi+1 = xi + ∆xiif |f | ≤ tol then

returnend

endreturn

Algorithm 1: Newton-Raphson algorithm



But how do we know that the method will eventually converge tothe correct value?

It can be shown that: (x − xi+1) ≤ C (x − xi )2

Provided that:

The initial estimate x0 is close enough to the solution x

f is sufficiently smooth


The Newton-Raphson method in n-DThe method can be extended to vector valued functions:

f (x) : Rn → Rn

and systems of nonlinear equations:

f (x) = 0

By considering that:

f (x) ≈ f(xi)

+ ∂f∂x |xi ∆xi

∆xi = −(∂f∂x |xi

)−1f(xi)

xi+1 = xi + ∆xi

For the n-D version of the method, we will use superscripts to indicate theiteration i and we will drop the index for the increment to simplify notation.


The Newton-Raphson method in n-DOr component wise:

f1 (x)f2 (x)

...fn (x)

n×1

≈

f1(xi)

f2(xi)...

fn(xi)

n×1

+

∂f1(xi)∂x1

∂f1(xi)∂x2

· · · ∂f1(xi)∂xn

∂f2(xi)∂x1

∂f2(xi)∂x2

· · · ∂f2(xi)∂xn...

... . . . ...∂fn(xi)

∂x1

∂fn(xi)∂x2

· · · ∂fn(xi)∂xn

n×n

∆x1∆x2

...∆xn

n×1

↓

x i+1

1x i+1

2...

x i+1n

n×1

=

x i

1x i

2...

x in

n×1

−

∂f1(xi)∂x1

∂f1(xi)∂x2

· · · ∂f1(xi)∂xn

∂f2(xi)∂x1

∂f2(xi)∂x2

· · · ∂f2(xi)∂xn...

... . . . ...∂fn(xi)

∂x1

∂fn(xi)∂x2

· · · ∂fn(xi)∂xn

−1

n×n

f1(xi)

f2(xi)...

fn(xi)

n×1


The Newton-Raphson method

We recall the linearized equilibrium equation:

KT∆un = −Ri = fext − finti

Typically external loads are scaled by a factor λ:

KT∆un = −Ri = λfext − finti



For a given value of λ, the above corresponds to a Newton-Raphsoniteration where:

R = Ri + ∂Ri

∂un ∆un = finti − λfext + KT∆un = 0

The iteration is repeated until

|R| <= tol

where tol is the tolerance.



Typically:

Different (increasing) values are given to the load factor

Displacements are obtained using the Newton-Raphson method

The response of the structure is obtained for progressivelyincreasing loading

The load-displacement curve of the structure can be obtained



The above procedure is called load control and can be illustrated asfollows:

Increment load

Apply load increment

Solve withNewton-Raphson

Repeat for nextincrement




Increment load







Increment load







Increment load







Increment load







Increment load






In many cases the solution path is not monotonic

Phenomena such as snap-through and snap-back can be present

The Newton-Raphson method will either fail or not provide thefull path in those cases

Such phenomena are often of interest in practice e.g.when theovercritical behavior of a structure needs to be known


Example

Solution path involving snap-through:


Example



Example



Example



Path-following methods

In this category of methods:

It is attempted to follow the full solution path includingsnap-back and snap-through phenomena

The load factor is included as an additional unknown in thenonlinear system of equations

An additional equation (constraint) is added in the system

The choice of this constraint leads to different methods



After adding ∆λ as an unknown and the constraint as an equationthe new system of equations is obtained:[

R (u, λ)g (u, λ)

]=[

00

]where:

u is the vector of nodal unknownsλ is the load factorR is the residual of the Newton-Raphson methodg is the additional constraint



The above system can be solved by employing the Newton-Raphsonmethod. The linearization of the system yields: R

(ui, λi

)+∂R

(ui, λi

)∂u ∆u + ∂R

(ui , λi)∂λ

∆λ

g(ui , λi)+ ∂g

(ui , λi)∂u ∆u + ∂g

(ui , λi)∂λ

∆λ

=[

00

]

where:∆u is the nodal displacement increment∆λ is the load factor increment



The above system can be solved by employing the Newton-Raphsonmethod. The linearization of the system yields:[

KT −fexthT s

] [∆u∆λ

]=[−Ri

−g i

]

where:∆u is the nodal displacement increment∆λ is the load factor increment

h Is the gradient of g with respect to u: h = ∂g∂u

s is the derivative of g with respect to λ: s = ∂g∂λ



The coefficient matrix in this system is not symmetric.

To exploit the symmetry and sparsity of KT the followingpartitioning procedure is employed:

∆uI = KT−1fext, ∆uII = −KT

−1Ri

∆λ = −g i + hT ∆uII

s + hT ∆uI , ∆u = ∆λ∆uI + ∆uII



The choice of the additional constraint g is crucial

Different alternatives exist

In some cases problem specific constraints are needed

Some widely used alternatives are demonstrated in the following


Path-following methods: Load control

Setting g = λ− λ results in load control:

g = λ− λ ⇒ h = ∂g∂u = 0, s = ∂g

∂λ= 1

g i = λi − λ

[KT −fext0 1

] [∆u∆λ

]=[−Ri

λ− λi

]

∆λ = λ− λi , ∆u = KT−1(λfext − fint

i)


Path-following methods: Displacement control

Setting g = T · u− u results in displacement control:

g = T · u− u ⇒ h = ∂g∂u = TT , s = ∂g

∂λ= 0

g i = T · ui − u

[KT −fextT 0

] [∆u∆λ

]=[

−Ri

u − T · ui

]

In the above T =[

0 0 . . . 1 . . . 0 0]

where the entry 1corresponds to a selected dof



Displacement control in a pathinvolving snap-through

Displacement control in a pathinvolving snap-back










Path-following methods: Arc-length

Setting g =√

(u− u0)T · (u− u0) + (λ− λ0)2 −∆s results in arclength control (Crisfield 1981):

g i =√

(ui − u0)T · (ui − u0) + (λi − λ0)2 −∆s

h = ∂g∂u =

(ui − u0)

g , s = ∂g∂λ

=(λi − λ0)

g

KT −fext(ui − u0)T

g

(λi − λ0)

g

[ ∆u∆λ

]=[

Ri

−g i

]

In the above, superscript 0 refers to the beginning of the step



We observe that for i = 0 the system becomes:

KT −fext(u0 − u0)T

g

(λ0 − λ0)

g

[ ∆u∆λ

]=[−Ri

−g i

]




[KT −fext0 0

] [∆u∆λ

]=[−Ri

−g i

]




[KT −fext0 0

] [∆u∆λ

]=[−Ri

−g i

]

The coefficient matrix is singular!

To obtain the load factor at the first iteration a predictor step isintroduced.



For the predictor step the system is solved for the external loads:

∆up = KT−1fext

Then the increment of the load factor is computed as:

∆λp = ± ∆s‖∆up‖



The sign of the increment is determined by the current stiffnessparameter:

κ = fextT ∆u

∆uT ∆u



Setting g = (∆up)T (u− u1)+ ∆λp (λ− λ1) results in analternative method for arc length control (Riks 1972):

g i = (∆up)T(ui − u1

)+ ∆λp

(λi − λ1

)

h = ∂g∂u = ∆up, s = ∂g

∂λ= ∆λp

[KT −fext

(∆up)T ∆λp

] [∆u∆λ

]=[−Ri

−g i

]

In the above, superscript 1 refers to the predictor step



Illustration of arc-length methods:

Crisfield: Riks:


Documents

Prof. Dr. Eleni Chatzi, Dr. Konstantinos AgathosPath following methods Prof. Dr. Eleni Chatzi, Dr. Konstantinos Agathos Lecture 4 - 10 October, 2019 Institute of Structural Engineering,