1. ISO Geometric Analysis
Emad Shakour

2. Outline Engineering Problems Solving ODE/PDE using FEA
Solving ODE/PDE using IGA

3. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−𝐻𝑒𝑎𝑡 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟−

4. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠−
𝑚 𝜕 2 𝑦(𝑡) 𝜕 𝑡 2 +𝑐 𝜕𝑦(𝑡) 𝜕𝑡 +k 𝜕𝑦(𝑡) 𝜕𝑡 =𝑚𝑔

5. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠−
𝑚 𝜕 2 𝑦(𝑡) 𝜕 𝑡 2 +𝑐 𝜕𝑦(𝑡) 𝜕𝑡 +k 𝜕𝑦(𝑡) 𝜕𝑡 =𝑚𝑔

6. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−𝐹𝑙𝑢𝑖𝑑 𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠−

7. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−𝑆𝑡𝑎𝑡𝑖𝑐 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠−

8. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−𝑆𝑡𝑎𝑡𝑖𝑐 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠− 𝑷
𝐹 𝑥 =0
𝜎 𝑖𝑗 𝑥,𝑦 = ?
+ 𝜎 𝑥𝑥 + 𝜕 𝜎 𝑥𝑥 𝜕𝑥 𝑑𝑥 𝑑𝑦− 𝜎 𝑥𝑥 𝑑𝑦 + 𝜎 𝑥𝑦 + 𝜕 𝜎 𝑥𝑦 𝜕𝑦 𝑑𝑦 𝑑𝑥− 𝜎 𝑥𝑦 𝑑𝑥=0
𝜎 𝑥𝑥 + 𝜕 𝜎 𝑥𝑥 𝜕𝑥 𝑑𝑥
𝜎 𝑥𝑥
𝜎 𝑦𝑦
𝜎 𝑥𝑦 +𝜕 𝜎 𝑥𝑦
𝜎 𝑦𝑦 + 𝜕 𝜎 𝑦𝑦 𝜕𝑦 𝑑𝑦
𝜎 𝑥𝑦 𝑑𝑥 𝑑𝑦
𝜕 𝜎 𝑥𝑥 𝜕𝑥 + 𝜕 𝜎 𝑥𝑦 𝜕𝑦 =0
𝜕 𝜎 𝑦𝑦 𝜕𝑦 + 𝜕 𝜎 𝑥𝑦 𝜕𝑥 =0

9. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−𝑆𝑡𝑎𝑡𝑖𝑐 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠−
𝛻𝜎+𝑏=0

10. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐹𝐸𝐴− 1𝐷 𝐿
𝑙 𝑒 3𝐷 2𝐷

11. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝑤𝑒𝑖𝑔ℎ𝑒𝑑 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠−
𝑂𝐷𝐸: ℒ 𝑢 𝑥 −𝑓 𝑥 =0
Ω: 𝑎≤𝑥≤𝑏
𝑢 𝑥 = 𝑖=1 𝑁 𝑎 𝑖 𝜑 𝑖
∗𝑠𝑡𝑟𝑜𝑛𝑔 𝑓𝑜𝑟𝑚
ℒ 𝑢 𝑥 −𝑓 𝑥 =𝑅 𝑥
Ω 𝑅 𝑥 𝑑𝑥 =0
Ω 𝑅 𝑥 𝑤 𝑥 𝑑𝑥=0
∗𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑚𝑒𝑡ℎ𝑜𝑑 𝑖𝑛 𝑜𝑟𝑑𝑒𝑟
𝑡𝑜 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑡ℎ𝑒 𝑟𝑖𝑠𝑖𝑑𝑢𝑎𝑙
∗𝑤𝑒𝑎𝑘 𝑓𝑜𝑟𝑚 𝑎 𝑏

12. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐺𝑎𝑙𝑒𝑟𝑘𝑖𝑛−
Ω 𝑅 𝑥 𝑤 𝑥 𝑑𝑥= Ω ℒ 𝑢 𝑥 𝑤 𝑥 −𝑓 𝑥 𝑤 𝑥 𝑑𝑥=0
∗𝑤𝑒𝑎𝑘 𝑓𝑜𝑟𝑚
∗𝐴𝑐𝑐𝑜𝑟𝑑𝑖𝑛𝑔 𝑡𝑜 𝐺𝑎𝑙𝑒𝑟𝑘𝑖𝑛
𝑤 𝑥 = 𝑗=1 𝑁 𝜑 𝑗
∗𝑏𝑒𝑡𝑡𝑒𝑟 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 𝑓𝑜𝑟 ℎ𝑖𝑔ℎ𝑒𝑟 N
𝑢 𝑥 = 𝑖=1 𝑁 𝑎 𝑖 𝜑 𝑖
Ω ℒ 𝑖=1 𝑁 𝑎 𝑖 𝜑 𝑖 𝑗=1 𝑁 𝜑 𝑗 −𝑓 𝑥 𝑗=1 𝑁 𝜑 𝑗 𝑑𝑥=0 𝑎 𝑏

13. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐺𝑎𝑙𝑒𝑟𝑘𝑖𝑛−
Ω ℒ 𝑖=1 𝑁 𝑎 𝑖 𝜑 𝑖 𝑗=1 𝑁 𝜑 𝑗 −𝑓 𝑥 𝜑 𝑗 𝑑𝑥=0
Ω 𝑖=1 𝑁 𝑗=1 𝑁 ℒ 𝜑 𝑖 𝜑 𝑗 𝑑𝑥 𝑎 𝑗 − Ω 𝑗=1 𝑁 𝑓 𝑥 𝜑 𝑗 𝑑𝑥 =0
𝐾 𝑖𝑗
𝑓 𝑗
𝑲 𝐚 =[𝐟] 𝑎 𝑏
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7 1

14. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐸𝑥𝑎𝑚𝑝𝑙𝑒−
𝐴 𝑢 ′′ 𝑥 −𝑓 𝑥 =0
Ω: 0≤𝑥≤𝐿
𝑢 0 = 𝑢 0
𝑢 𝐿 = 𝑢 𝐿
Ω 𝑖=1 𝑁 𝑗=1 𝑁 ℒ 𝜑 𝑖 𝜑 𝑗 𝑑𝑥 𝑎 𝑗 − Ω 𝑗=1 𝑁 𝑓 𝑥 𝜑 𝑗 𝑑𝑥 =0
𝑲 𝐚 =[𝐟]
Ω 𝐴 𝑖=1 𝑁 𝑗=1 𝑁 𝜑 𝑖 ′′ 𝜑 𝑗 𝑑𝑥 𝑎 𝑗 − Ω 𝑗=1 𝑁 𝑓 𝑥 𝜑 𝑗 𝑑𝑥 =0

15. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐸𝑥𝑎𝑚𝑝𝑙𝑒−
Ω 𝐴 𝑖=1 𝑁 𝑗=1 𝑁 𝜑 𝑖 ′′ 𝜑 𝑗 𝑑𝑥 𝑎 𝑗 − Ω 𝑗=1 𝑁 𝑓 𝑥 𝜑 𝑗 𝑑𝑥 =0
Ω 𝑣′𝑢𝑑Ω = 𝑣𝑢 Γ − Ω 𝑣𝑢′ 𝑑Ω
∗ 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 𝑏𝑦 𝑝𝑎𝑟𝑡𝑠 𝑖𝑛 𝑜𝑟𝑑𝑒𝑟 𝑡𝑜 𝑟𝑒𝑑𝑢𝑐𝑒 𝑡ℎ𝑒 𝑜𝑟𝑑𝑒𝑟
Ω 𝜑 𝑖 ′′ 𝜑 𝑗 𝑑𝑥 = 𝜑 𝑖 ′ 𝜑 𝑗 Γ − Ω 𝜑 𝑖 ′ 𝜑 𝑗 ′𝑑𝑥
𝑲 𝐚 =[𝐟]
𝑖=1 𝑁 𝑗=1 𝑁 Ω 𝜑 𝑖 ′ 𝜑 𝑗 ′𝑑𝑥 𝑎 𝑗 = 𝑗=1 𝑁 𝑓 𝑥 𝜑 𝑗 𝑑𝑥 =0
𝐾 𝑖𝑗
𝑓 𝑗

16. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐸𝑥𝑎𝑚𝑝𝑙𝑒−
𝑠𝑜𝑙𝑣𝑒:𝐴 𝑢 ′′ 𝑥 = sin 𝜋𝑥 𝐿 cos 𝜋𝑥 𝐿 Ω: 0≤𝑥≤𝐿
𝐴=1
𝑢 0 =0
𝑢 𝐿 =1
𝑢 𝑥 = 𝑖=1 𝑁 𝑎 𝑖 𝜑 𝑖 𝐿
𝜑 1
𝜑 2
𝜑 3 1

17. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐸𝑥𝑎𝑚𝑝𝑙𝑒−
𝑢 𝑥 = 𝑖=1 𝑁 𝑎 𝑖 𝜑 𝑖 𝐿
𝜑 1
𝜑 2
𝜑 3 1

18. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐸𝑥𝑎𝑚𝑝𝑙𝑒− 𝐿
𝜑 1
𝜑 2
𝜑 3 1
Ω 𝐴 𝑖=1 𝑁 𝑗=1 𝑁 𝜑 𝑖 ′ 𝜑 𝑗 ′ 𝑑𝑥 𝑎 𝑗 = Ω 𝑗=1 𝑁 𝑓 𝑥 𝜑 𝑗 𝑑𝑥 =0
𝐾 𝑖𝑗
𝑓 𝑗
𝑲 𝐚 =[𝐟]

19. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐸𝑥𝑎𝑚𝑝𝑙𝑒−
𝑠𝑜𝑙𝑣𝑒:𝐴 𝑢 ′′ 𝑥 = sin 𝜋𝑥 𝐿 cos 𝜋𝑥 𝐿 Ω: 0≤𝑥≤𝐿
𝐴=1 𝐿
𝜑 1
𝜑 2
𝜑 3 1
𝑲 𝐚 = [𝐟]
𝑢 0 =0
𝑢 𝑥 = 𝑖=1 𝑁 𝑢 𝑖 𝜑 𝑖
𝑢 𝐿 =1
exact solution
approximate solution

20. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝐸𝑥𝑎𝑚𝑝𝑙𝑒−
𝑠𝑜𝑙𝑣𝑒:𝐴 𝑢 ′′ 𝑥 = sin 𝜋𝑥 𝐿 cos 𝜋𝑥 𝐿 Ω: 0≤𝑥≤𝐿
𝑒 𝑥 2 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠
𝑒 𝑥 4 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠

21. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠2𝐷

22. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠
−𝑇𝑜𝑝𝑜𝑙𝑜𝑔𝑦 𝑜𝑝𝑡𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛−
𝑜𝑑𝑒𝑑 2013

23. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠3𝐷

24. 𝐹𝑖𝑛𝑖𝑡𝑒 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠2𝐷

25. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝑠𝑜𝑙𝑣𝑒:𝐴 𝑢 ′′ 𝑥 = sin 𝜋𝑥 𝐿 cos 𝜋𝑥 𝐿 Ω: 0≤𝑥≤𝐿
𝑢:For example: Displacement or temperature…
𝑥=0 1
𝑥=𝐿

26. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝑠𝑜𝑙𝑣𝑒:𝐴 𝑢 ′′ 𝑥 = sin 𝜋𝑥 𝐿 cos 𝜋𝑥 𝐿 Ω: 0≤𝑥≤𝐿 𝐿
𝜑 1
𝜑 2
𝜑 3
𝑥 2 1 𝐿
𝜑 1
𝜑 2
𝜑 3 1 𝐿
𝜑 1
𝜑 2
𝜑 3 1
𝐻𝑖𝑔ℎ𝑒𝑟 𝑑𝑒𝑔𝑟𝑒𝑒
𝑥=0 1
𝑥=𝐿 𝑎 𝑏
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7 1 𝑎 𝑏
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝜑 5
𝜑 6
𝜑 7 1

27. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
What if the structure itself is presented as a B-splines?
𝑡= 𝑡 0
knot
𝑡= 𝑡 𝑝+𝑛+1
𝑡= 𝑡 𝑛
knot
𝑡= 𝑡 𝑝+𝑛+1
𝑡 0 , 𝑡 1 , 𝑡 2
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝑡 3
𝑡 4 , 𝑡 5 , 𝑡 6
Order = 3

28. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝑠𝑜𝑙𝑣𝑒: 𝑢 ′′ 𝑥 =6𝑥 Ω: 0≤𝑥≤2
𝑢 0 =0 𝑢 2 =0
𝑡= 0,0.5
𝜑 1 = 2𝑡− 𝜑 2 =−2𝑡 3𝑡− 𝜑 3 =2 𝑡 2
𝜑 4 =0
𝑡=[ ]
Order = 3
𝑡= 𝑡 𝑛
knot
𝑃 1 = 0, 𝑃 3 = 0.5,1 𝑃 2 = 1.5, 𝑃 4 = 2,1
𝑡= 0.5,1
𝜑 1 =0
𝜑 2 = 2𝑡−2 𝑡−1 𝜑 3 =−6 𝑡 2 +8𝑡−2 𝜑 4 = 2𝑡−1 2
𝑡 0 , 𝑡 1 , 𝑡 2
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝑡 3
𝑡 4 , 𝑡 5 , 𝑡 6

29. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝑡= 𝑡 𝑛
knot
𝑡 0 , 𝑡 1 , 𝑡 2
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝑡 3
𝑡 4 , 𝑡 5 , 𝑡 6
𝑠𝑜𝑙𝑣𝑒: 𝑢 ′′ 𝑥 =6𝑥 Ω: 0≤𝑥≤2
𝑢 0 =0 𝑢 2 =0
𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛:
𝐾 𝑖𝑗 = 𝑥 1 𝑥 2 𝜑 𝑖 ′ 𝑥 𝜑 𝑗 ′ 𝑥 𝑑𝑥 𝑓 𝑗 = 𝑥 1 𝑥 2 𝑓 𝑥 𝜑 𝑗 𝑥 𝑑𝑥
𝑥= 𝑖=0 𝑁 𝑃 𝑥 𝜑 𝑖 𝑡 →
𝜕𝑥 𝜕𝑡 = 𝑖=0 𝑁 𝑃 𝑥 𝜕 𝜑 𝑖 𝜕𝑡 →
𝜕𝑡 𝜕𝑥 = 𝜕𝑥 𝜕𝑡 −1
𝜑 𝑖 ′ 𝑥 = 𝜕 𝜑 𝑖 𝜕𝑡 𝜕𝑡 𝜕𝑥
𝐾 𝑖𝑗 = 𝑥 1 𝑥 2 𝜕 𝜑 𝑖 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕 𝜑 𝑗 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝑑𝑥 𝑓 𝑗 = 𝑥 1 𝑥 2 𝑓 𝑥 𝜑 𝑗 𝑥 𝑑𝑥

30. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝑡= 𝑡 𝑛
knot
𝑡 0 , 𝑡 1 , 𝑡 2
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝑡 3
𝑡 4 , 𝑡 5 , 𝑡 6
𝑠𝑜𝑙𝑣𝑒: 𝑢 ′′ 𝑥 =6𝑥 Ω: 0≤𝑥≤2
𝑢 0 =0 𝑢 2 =0
𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛:
𝐾 𝑖𝑗 = 𝑥 1 ( 𝑡 1 ) 𝑥 2 ( 𝑡 2 ) 𝜕 𝜑 𝑖 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕 𝜑 𝑗 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝑑𝑥 𝑓 𝑗 = 𝑥 1 ( 𝑡 1 ) 𝑥 2 ( 𝑡 2 ) 𝑓 𝑥 𝜑 𝑗 𝑥 𝑑𝑥
𝐽= 𝜕𝑥 𝜕𝑡 = 𝑗=1 4 𝑃 𝑥 𝑗 𝜕 𝜑 𝑗 𝜕𝑡
change variables:
𝐾 𝑖𝑗 = 𝑡 1 𝑡 2 𝜕 𝜑 𝑖 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕 𝜑 𝑗 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝐽 𝑑𝑡 𝑓 𝑗 = 𝑡 1 𝑡 2 𝑓 𝑥 𝑡 𝜑 𝑗 𝑥 𝑡 𝐽 𝑑𝑡

31. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝑡= 𝑡 𝑛
knot
𝑡 0 , 𝑡 1 , 𝑡 2
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝑡 3
𝑡 4 , 𝑡 5 , 𝑡 6
𝑠𝑜𝑙𝑣𝑒: 𝑢 ′′ 𝑥 =6𝑥 Ω: 0≤𝑥≤2
𝑢 0 =0 𝑢 2 =0
𝑛𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛:
𝐾 𝑖𝑗 = 𝑡 1 𝑡 2 𝜕 𝜑 𝑖 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕 𝜑 𝑗 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝐽 𝑑𝑡 𝑓 𝑗 = 𝑡 1 𝑡 2 𝑓 𝑥 𝑡 𝜑 𝑗 𝑥 𝑡 𝐽 𝑑𝑡
𝑡=[ ]
𝑁𝐷𝑂𝐹 𝑓𝑜𝑟 𝑜𝑛𝑒 𝑝𝑎𝑡𝑐ℎ=𝑝+1=3
𝑡 0 , 𝑡 1 , 𝑡 2
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝑡 3
𝑡 4 , 𝑡 5 , 𝑡 6
𝑁𝐷𝑂𝐹 𝐺𝑙𝑜𝑏𝑎𝑙=𝑁𝑜𝐶𝑜𝑛𝑃𝑛𝑡𝑠=4

32. ∪ 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠 −Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝐾 𝑖𝑗 = 𝑡 1 𝑡 2 𝜕 𝜑 𝑖 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝜕 𝜑 𝑗 𝜕𝑡 𝜕𝑡 𝜕𝑥 𝐽 𝑑𝑡 𝑓 𝑗 = 𝑡 1 𝑡 2 𝑓 𝑥 𝑡 𝜑 𝑗 𝑥 𝑡 𝐽 𝑑𝑡
𝑡 0 , 𝑡 1 , 𝑡 2
𝜑 1
𝜑 2
𝜑 3
𝑡 3
𝑡=[0−0.5]
𝜑 4
𝑡 4 , 𝑡 5 , 𝑡 6
𝑡 3
𝜑 2
𝜑 3
𝑡=[0.5−1]
𝑡=[ ] ∪
𝐾 𝐺 =
𝐹 𝐺 = − 1 2 −3 −5 −3 1 2
𝐾 𝐺 = −1 − − − 1 3 − −1 0 − 1 3 −1 4 3
𝐹 𝐺 =
𝑘 1 𝑒 = −1 − −1 −
𝑓 1 𝑒 = − 1 2 − − 3 4
𝑘 1 𝑒 = −1 − −1 −
𝑓 1 𝑒 = − − −3 1 2
𝜑 1 = 2𝑡− 𝜑 2 =−2𝑡 3𝑡− 𝜑 3 =2 𝑡 2
𝜑 2 = 2𝑡−2 𝑡−1 𝜑 3 =−6 𝑡 2 +8𝑡−2 𝜑 4 = 2𝑡−1 2

33. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝐾 𝐺 𝑎 =[ 𝐹 𝐺 ]
𝑎 1 𝑎 2 𝑎 3 𝑎 4 = − 1 2 −3 −5 −3 1 2
− 𝐴𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙
− 𝑁𝑢𝑚𝑒𝑟𝑖𝑐𝑎𝑙
𝑢 𝑥 = 𝑖=1 𝑁 𝑎 𝑖 𝜑 𝑖
𝑎 = 𝐾 𝐺 −1 [ 𝐹 𝐺 ]
𝑡 0 , 𝑡 1 , 𝑡 2
𝜑 1
𝜑 2
𝜑 3
𝜑 4
𝑡 3
𝑡 4 , 𝑡 5 , 𝑡 6

34. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

35. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

36. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

37. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

38. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

39. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

40. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

41. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

42. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

43. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑚𝑒𝑛𝑡

44. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−

45. 𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠`
𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐸𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑃𝑟𝑜𝑏𝑙𝑒𝑚𝑠
−Iso Geometric 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠−
Thank you