2. NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS 1)Mechanics of Materials Approach (A) Complex Beam Theory (i) Straight Beam (ii) Curved Beam (iii) Composite Beam From:Daviddarling.info

3. NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS Mechanics of Material Approach (Cont)

4. NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS (2) Finite Difference Method

5. NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS (2) Finite Difference Method (Contd) Consider an ordinary differential equation One of the difference equation method is using: To approximate the differential equation. Solution is:

6. APPLICATION OF FINITE ELEMENT METHOD TO BIOMECHANICS

7. Introduction lRe-invented around 1963 lInitially applied to engineering structures Concrete dams Aircraft structures (Civil engineers) (Aeronautical engineers)

8. Introduction lFEM is based on Energy Method of Residuals

9. Introduction lEnergy method Total potential energy must be stationary δ (U + W) = δ ( П ) = 0

10. Introduction lResidual method Differential equation governing the problem is given by A ( ø ) = 0 Minimise R = A ( ø* ) - A ( ø ) ø is actual solution ø* is assumed solution

11. Introduction l Both methods give us a set of equations [ K ] { a } = { f } Stiffness Matrix Displacement Matrix Force Matrix

12. Introduction - FEM Procedure l Continuum is separated by imaginary lines or surfaces into a number of “finite elements” Finite Elements

13. Introduction - FEM Procedure l Elements are assumed to be interconnected at a discrete number of “nodal points” situated on their boundaries Finite Elements Nodal Points Displacements at these nodal points will be the basic unknown

14. Introduction - FEM Procedure l A set of functions is chosen to define uniquely the state of displacement within each finite element ( U ) in terms of nodal displacements ( a 1, a 2, a 3 ) U = Σ N i a i i= 1, 3 x y a1a1 a2a2 a3a3 Finite Element Nodal Point

15. Introduction - FEM Procedure l This displacement function is input into either “energy equations” or “residual equations” to give us element equilibrium equation l [ K ] { a } = { f } x y a1a1 a2a2 a3a3 Finite Element Nodal Point Element Displacement Matrix Element Force Matrix Element Stiffness Matrix

16. Introduction - FEM Procedure lElement equilibrium equations are assembled taking care of displacement compatibility at the connecting nodes to give a set of equations that represents equilibrium of the entire continuum Finite Elements Nodal Points

17. Introduction - FEM Procedure lSolution for displacements are obtained after substituting boundary conditions in the continuum equilibrium equations Finite Elements Nodal Points Support Points

18. Introduction l Finite element method used to solve: l Elastic continuum l Heat conduction l Electric & Magnetic potential l Non-linear (Material & Geometric) -plasticity, creep l Vibration l Transient problems l Flow of fluids l Combination of above problems l Fracture mechanics

19. Introduction l Finite elements: l Truss, Cable and Beam elements l Two & Three solid elements l Axi-symmetric elements l Plate & Shell elements l Spring, Damper & Mass elements l Fluid elements

20. Application to Spine Biomechanics

21. Finite Element Mesh of C4-C7 IntactWith Graft at C5-C6 Level C4 C5 C6 C7 C5-C6 Graft Facet Joints

22. von Mises Stress in C4-C5 Annulus (Flexion) Neutral Graft Kyphotic Graft 5 MPa 6 MPaAnterior

23. Finite Element Mesh of L1-S1

24. Vertical Displacement Distribution in L1-S1

25. Finite Element Mesh of L2-L5 With 25% Translational Spondylolisthesis

26. Vertical Displacement Distribution in L2-L5 Under Flexion Moment (25% translational spondylolisthesis)

27. Application to Knee Implant Biomechanics

28. Finite Element Mesh to Represent Tibial Insert & Femoral Component

29. Contact Compressive Stress

30. Motion of Femoral Implant with respect to UHMWPE Knee Insert

31. Application to Femoral Implant Biomechanics

32. Finite Element Mesh of an Intact Femur

33. Distribution of SIGMA-ZZ in an intact femur

34. Finite Element Mesh of a Femur with Implant

35. SIGMA-ZZ in a Femur With Implant

36. Implant fixed with cement layer in a femur

37. Von Misses stress in cement layer

38. SIGMA-ZZ in cortical bone in a femur with implant attached using cement

39. Advantage of using FEM lIrregular complex geometry can be modeled lEffect of large number of variables in a problem can be easily analysed lMultiple phase problems can be modeled lEffect of various surgical techniques can be compared using appropriate FE models lBoth static and time dependent problems can be modeled lSolution to certain problems that cannot be (or difficult) obtained otherwise can be solved by FEM