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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
Institute of Structural Engineering Method of Finite Elements II 2
Learnig goals
Understanding the general solution process for nonlinearproblems
Understanding limitations of the process
Gaining a basic understanding of path following solutionmethods
Institute of Structural Engineering Method of Finite Elements II 3
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
Institute of Structural Engineering Method of Finite Elements II 4
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
Institute of Structural Engineering Method of Finite Elements II 5
The Newton-Raphson method in 1-D
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
Institute of Structural Engineering Method of Finite Elements II 6
The Newton-Raphson method in 1-D
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
Institute of Structural Engineering Method of Finite Elements II 7
The Newton-Raphson method in 1-D
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
Institute of Structural Engineering Method of Finite Elements II 8
The Newton-Raphson method in 1-D
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
Institute of Structural Engineering Method of Finite Elements II 9
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.
Institute of Structural Engineering Method of Finite Elements II 10
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
Institute of Structural Engineering Method of Finite Elements II 11
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
Institute of Structural Engineering Method of Finite Elements II 12
The Newton-Raphson method
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.
Institute of Structural Engineering Method of Finite Elements II 13
The Newton-Raphson method
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
Institute of Structural Engineering Method of Finite Elements II 14
The Newton-Raphson method
The above procedure is called load control and can be illustrated asfollows:
Increment load
Apply load increment
Solve withNewton-Raphson
Repeat for nextincrement
Institute of Structural Engineering Method of Finite Elements II 15
The Newton-Raphson method
The above procedure is called load control and can be illustrated asfollows:
Increment load
Apply load increment
Solve withNewton-Raphson
Repeat for nextincrement
Institute of Structural Engineering Method of Finite Elements II 15
The Newton-Raphson method
The above procedure is called load control and can be illustrated asfollows:
Increment load
Apply load increment
Solve withNewton-Raphson
Repeat for nextincrement
Institute of Structural Engineering Method of Finite Elements II 15
The Newton-Raphson method
The above procedure is called load control and can be illustrated asfollows:
Increment load
Apply load increment
Solve withNewton-Raphson
Repeat for nextincrement
Institute of Structural Engineering Method of Finite Elements II 15
The Newton-Raphson method
The above procedure is called load control and can be illustrated asfollows:
Increment load
Apply load increment
Solve withNewton-Raphson
Repeat for nextincrement
Institute of Structural Engineering Method of Finite Elements II 15
The Newton-Raphson method
The above procedure is called load control and can be illustrated asfollows:
Increment load
Apply load increment
Solve withNewton-Raphson
Repeat for nextincrement
Institute of Structural Engineering Method of Finite Elements II 15
The Newton-Raphson method
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
Institute of Structural Engineering Method of Finite Elements II 16
Example
Solution path involving snap-through:
Institute of Structural Engineering Method of Finite Elements II 17
Example
Solution path involving snap-through:
Institute of Structural Engineering Method of Finite Elements II 17
Example
Solution path involving snap-through:
Institute of Structural Engineering Method of Finite Elements II 17
Example
Solution path involving snap-through:
Institute of Structural Engineering Method of Finite Elements II 17
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
Institute of Structural Engineering Method of Finite Elements II 18
Path-following 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
Institute of Structural Engineering Method of Finite Elements II 19
Path-following methods
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
Institute of Structural Engineering Method of Finite Elements II 20
Path-following methods
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
Institute of Structural Engineering Method of Finite Elements II 20
Path-following methods
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
Institute of Structural Engineering Method of Finite Elements II 21
Path-following methods
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
Institute of Structural Engineering Method of Finite Elements II 22
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)
Institute of Structural Engineering Method of Finite Elements II 23
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
Institute of Structural Engineering Method of Finite Elements II 24
Path-following methods: Displacement control
Displacement control in a pathinvolving snap-through
Displacement control in a pathinvolving snap-back
Institute of Structural Engineering Method of Finite Elements II 25
Path-following methods: Displacement control
Displacement control in a pathinvolving snap-through
Displacement control in a pathinvolving snap-back
Institute of Structural Engineering Method of Finite Elements II 25
Path-following methods: Displacement control
Displacement control in a pathinvolving snap-through
Displacement control in a pathinvolving snap-back
Institute of Structural Engineering Method of Finite Elements II 25
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
Institute of Structural Engineering Method of Finite Elements II 26
Path-following methods: Arc-length
We observe that for i = 0 the system becomes:
KT −fext(u0 − u0)T
g
(λ0 − λ0)
g
[ ∆u∆λ
]=[−Ri
−g i
]
Institute of Structural Engineering Method of Finite Elements II 27
Path-following methods: Arc-length
We observe that for i = 0 the system becomes:
[KT −fext0 0
] [∆u∆λ
]=[−Ri
−g i
]
Institute of Structural Engineering Method of Finite Elements II 27
Path-following methods: Arc-length
We observe that for i = 0 the system becomes:
[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.
Institute of Structural Engineering Method of Finite Elements II 27
Path-following methods: Arc-length
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‖
Institute of Structural Engineering Method of Finite Elements II 28
Path-following methods: Arc-length
The sign of the increment is determined by the current stiffnessparameter:
κ = fextT ∆u
∆uT ∆u
Institute of Structural Engineering Method of Finite Elements II 29
Path-following methods: Arc-length
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
Institute of Structural Engineering Method of Finite Elements II 30
Path-following methods: Arc-length
Illustration of arc-length methods:
Crisfield: Riks:
Institute of Structural Engineering Method of Finite Elements II 31
