Safeguards
Classical
Trust Region
Line Search
Heuristic
Eclipse Appleyard (EA)
Modified Appleyard (MA)
Modified Trust-region (X. Wang)
6
8
𝒰 Δ𝑡
ℛ
v
v
v
vv
𝜆
Zero level curve: ℛ 𝒰 𝜆 ,Δ𝑡 𝜆 = 0
Parameterized curve: 𝑑ℛ
𝑑𝜆= 𝒥
d𝒰
𝑑𝜆+𝜕ℛ 𝒰𝑛+1,𝒰𝑛;Δ𝑡
𝜕Δ𝑡
dΔ𝑡
𝑑𝜆= 0
𝑡 =𝑑𝒰
𝑑𝜆= −𝒥−1
𝜕ℛ
𝜕Δ𝑡
𝑑Δ𝑡
𝑑𝜆
10
Drawbacks:
1. Convergence neighborhood
2. Too many residual evaluations
3. No optimal steplength, 𝛼
Δ𝑡 parameterization:
-Requires no additional equation
Parameterization
unknown
Residual Equation Parameterization
𝜆 ℛ 𝒰 𝜆 , Δ𝑡 𝜆 = 0 𝑑ℛ
𝑑𝜆=𝜕ℛ
𝜕𝒰
d𝒰
𝑑𝜆+𝜕ℛ
𝜕Δ𝑡
dΔ𝑡
𝑑𝜆= 0
Δ𝑡 ℛ 𝒰 Δ𝑡 , Δ𝑡 = 0 𝑑ℛ
dΔ𝑡=𝜕ℛ
𝜕𝒰
d𝒰
𝜕Δ𝑡+𝜕ℛ
𝜕Δ𝑡= 0
𝒰
Δ𝑡
ℛ
𝜆12
Taylor series expansion :
𝒰pred = 𝒰0 + Δ𝑡𝑑𝒰
𝑑Δ𝑡+Δ𝑡2
2!
𝑑2𝒰
𝑑Δ𝑡2+Δ𝑡3
3!
𝑑3𝒰
𝑑Δ𝑡3+Δ𝑡4
4!
𝑑4𝒰
𝑑Δ𝑡4…
Order of
approximation
Terms
(1) Zero Order 𝒰pred = 𝒰0
(2) First Order𝒰pred = 𝒰0 + Δ𝑡
𝑑𝒰
𝑑Δ𝑡
(3) Second Order𝒰pred = 𝒰0 + Δ𝑡
𝑑𝒰
𝑑Δ𝑡+Δ𝑡2
2!
𝑑2𝒰
𝑑Δ𝑡2
Δ𝑡
(1) (2) (3)
𝒰 1 𝒰 2 𝒰 3
Solution
path
𝒰𝑛+1
14
Parameterization
unknown
First order
approximationSecond order approximation
Δ𝑡 𝑑𝒰
𝑑Δ𝑡= −𝒥−1
𝜕ℛ
𝜕Δ𝑡
𝑑2𝒰
𝑑Δ𝑡2= −𝒥−1
𝜕𝒥
𝜕𝒰⨂𝑑𝒰
𝑑Δ𝑡+𝜕𝒥
𝜕Δ𝑡
𝑑𝒰
𝑑Δ𝑡+𝜕𝐺
𝜕𝒰
𝑑𝒰
𝑑Δ𝑡+𝜕𝐺
𝜕Δ𝑡
𝐺 =𝜕ℛ
𝜕Δ𝑡⨂ - Tensor-vector multiplication
15
16
Δ𝑡 = 0.8
(1) (2) (3)
𝒰 1 𝒰 2 𝒰 3
Solution
path
𝒰𝑛+1
Order of
approximationTerms
Predicted
valueError, 𝜺𝟎
(1) Zero Order Spred = 𝑆0 0.0 -0.6255
(2) First Order 𝑆pred = 𝑆0 + Δ𝑡𝑑𝑆
𝑑Δ𝑡0.80 0.1745
(3) Second Order 𝑆pred = 𝑆0 + Δ𝑡𝑑𝑆
𝑑Δ𝑡+Δ𝑡2
2!
𝑑2𝑆
𝑑Δ𝑡20.7488 0.1233
𝑞𝑖𝑛𝑗 𝐹(𝑆)
𝑆𝑖𝑛𝑖𝑡 = 0.0
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑆𝑛+1 = 0.6255
𝜀0
𝑈 𝑡 + Δ𝑡 = 𝑈 𝑡 + 𝛼𝑑𝑈 𝑡
𝑑Δ𝑡+𝛼2
2!𝑘 𝑡 𝑁 𝑡 + 𝑂 𝛼3
𝑈𝑝𝑟𝑒𝑑 = 𝑈 𝑡 + 𝛼𝑑𝑈 𝑡
𝑑Δ𝑡
ℰ0 = 𝑈𝑘 − 𝑈𝑝𝑟𝑒𝑑 ≈𝛼2
2!𝑘 𝑡 𝑁 𝑡
Take 𝛼2
2!𝑘 𝑡 𝑁 𝑡 ≤ 𝛼
𝑑𝑈 𝑡
𝑑Δ𝑡
21
𝛼
𝑈𝑘 ≈ 𝑈∞ 𝑈0
22
𝜑 𝜀 =𝜀 + 10 − 𝜀2
5 − 𝜀2
𝜑 𝜀 =𝜀2
3 − 2𝜀
𝜑 𝜀 = 𝜀𝑝
Assume Quadratic convergence
(Newton-Kantorovich):
SuperLinear error model:
29
𝜀 𝑁 = 𝜀01 + exp 𝐵𝑀
1 + exp(−𝐵(𝑁 −𝑀)), 𝑤ℎ𝑒𝑟𝑒 𝐵 𝑎𝑛𝑑 𝑀 𝑎𝑟𝑒 𝑓𝑖𝑡𝑡𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠
𝐵 𝑑𝑒𝑝𝑒𝑛𝑑𝑎𝑛𝑐𝑦 𝑀 𝑑𝑒𝑝𝑒𝑛𝑑𝑎𝑛𝑐𝑦
33
𝑡1
𝑡2
𝑡3Δ𝑡𝑡𝑎𝑟𝑔𝑒𝑡
Δt = 0
𝒰n+1𝑆0 = (𝒰n, 0)
𝑆2
𝑆3
Solution path
𝑆1Tiny initial
steplength
Steplength from
Adaptation
36
𝐻𝑒𝑡𝑒𝑟𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 𝑔𝑟𝑖𝑑 𝑤𝑖𝑡ℎ𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 60 × 220 × 50
6 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑊𝑒𝑙𝑙𝑠𝑃𝑤𝑓 = 1500 𝑝𝑠𝑖.
2 𝐼𝑛𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑊𝑒𝑙𝑙𝑠𝑃𝑖𝑛𝑗 = 5000 𝑝𝑠𝑖.
𝐼𝑛𝑐𝑙𝑢𝑑𝑒𝑑 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 𝑎𝑛𝑑𝐶𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑒𝑓𝑓𝑒𝑐𝑡𝑠
P1
P2
P3
P4
P5
P6
Inj1
Inj1
http://www.spe.org/
P1
P2
P3
P4
P5
P6
Inj1
Inj1
Modified Continuation-Newton:
Never cuts the timestep due to lack of convergence;
Devised a robust and automatic steplength selection strategy;
Showed that high-order CN methods are not competitive;
Future work:
Test on complex physics;
Implement in GENSOL.
38
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